9.2.1 Size Distribution of Atmospheric Particles

The dynamics of particles covers by definition the processes that affect directly the size distribution of a population of aerosols, i.e., a population of particles in suspension in a gas. Therefore, emission and deposition processes are not included in aerosol dynamics; they are addressed separately in Chapters 2 and 11, respectively.

The size distribution of a population of particles may be represented in several ways. For example, the concentration of particles may be represented in terms of their number, N_p, their surface, S_p, their volume, V_p, or their mass, M_p. These different variables are interrelated via the Stokes diameter and the particle density. For example, for a population of particles with a same size, d_p,St (monodispersed population), and density, ρ_p:

Mp=Vp ρp=Np πdp,St36ρp(9.2)

However, atmospheric particles cover a large range of sizes. Therefore, the particle concentration must be represented by a concentration distribution as a function of size. Such distributions are typically presented as a function of the particle volume or diameter, and the particles are assumed to be spherical. It is a reasonable assumption for aged particles, which have undergone condensation of semi-volatile chemical species. It is, however, a poor assumption for freshly emitted primary particles, which may have fractal shapes. For the sake of simplicity, one uses here d_p rather than d_p,St to represent the particle diameter. The two particle size distributions that are the most commonly used are the number distribution and the mass distribution. The mass distribution is equivalent to the volume distribution if the particle density is uniform for all particles (then, ρ_p is the factor converting volume to mass). Let n_p(v_p) be the distribution of the number concentration of particles as a function of their volume, v_p, and n_p,d(d_p) be the distribution of the number concentration of particles as a function of their diameter d_p. The main equations for those number concentration distributions are as follows:

Np=∫0∞np(vp) dvp=∫0∞np,d(dp) ddp;  np,d(dp)=π2 dp2np(vp) Vp=∫0∞np(vp) vp dvp=∫0∞np,d(dp)π6dp3ddp Mp=∫0∞np(vp) vp ρp(vp) dvp=∫0∞np,d(dp)π6dp3 ρp(dp) ddp (9.3)

Here, N_p, V_p, and M_p represent the total concentration in terms of number, volume, and mass, respectively, of the polydispersed particle population. Typically, one distinguishes three modes in the particle size distribution (Whitby, 1978):

1. 
   

   – A nucleation mode

   
   
2. 
   

   – An accumulation mode

   
   
3. 
   

   – A coarse mode

The nucleation mode corresponds to particles that have been formed from gaseous molecules and have later grown via the condensation of other gaseous molecules and coagulation with other nucleated particles. This mode is located within the ultrafine fraction of PM.

The accumulation mode results from the emissions of fine particles and from dynamic processes such as condensation and coagulation. It is called the accumulation mode because these dynamic processes lead to the accumulation of particles in that size range (see Sections 9.2.3 and 9.2.4).

The coarse mode consists mostly of particles emitted via mechanical processes (abrasion, wind erosion, etc.). Condensation and coagulation have little effect on these particles, because their number concentration is low (although their mass concentration can be significant) and particle dynamics favors fine particles, which are typically present in greater numbers and have greater available surface area for condensation and coagulation processes to occur.

Figure 9.1 shows idealized volume and number concentration distributions of a typical urban particle population. Lognormal distributions are used here for the volume and number concentrations (see Section 9.7 for a description of the lognormal representation of particle concentration distributions). It appears that the volume (and, therefore, the mass) of the particles is present mostly in the accumulation and coarse modes and that the number concentration is dominated by the ultrafine particles of the nucleation mode.

Figure 9.1. Schematic representation of the size distribution of the volume concentration (top) and number concentration (bottom) of atmospheric particles typical of a polluted urban area.

Source of the data: Whitby (1978).

One should note that this classification into three modes is too simplistic. For example, several accumulation modes can be found in the atmosphere, because an accumulation mode resulting from the condensation of gaseous pollutants will have a different size distribution than an accumulation mode resulting from the evaporation of cloud and fog droplets where aqueous-phase chemical reactions have taken place (Hering and Friedlander, 1982).

Three processes are considered to govern particle dynamics and affect the size distribution of a population of particles:

1. 
   

   – Nucleation

   
   
2. 
   

   – Condensation (and the reverse process, which is evaporation)

   
   
3. 
   

   – Coagulation

The mathematical representation of these three processes is called the general dynamic equation, GDE (Friedlander, 2000). The different terms of this equation are presented in detail below.

9.2.2 Nucleation

Nucleation is the formation of a new particle from gaseous molecules. This nucleation process may involve two or three different chemical species. Nucleation involving two species (for example, sulfuric acid and water) is referred to as binary nucleation. Nucleation involving three species (for example, sulfuric acid, ammonia, and water) is referred to as ternary nucleation. Nucleation may occur spontaneously in the gas phase; then, one refers to homogeneous nucleation. In some cases, nucleation may be favored by the interaction of gaseous chemical species with a surface; then, one refers to heterogeneous nucleation. Nucleation rates are difficult to estimate theoretically and uncertainties may be of several orders of magnitude. The most robust algorithms have been developed for the homogeneous binary nucleation of sulfuric acid and water. A detailed review of nucleation algorithms has been conducted by Zhang et al. (2010), who recommended the algorithm of Kuang et al. (2008) for the nucleation rate, J_n (particles cm⁻³ s⁻¹), of sulfuric acid particles:

Jn=1.6×10−14 (NH2SO4)2 (9.4)

where N_H2SO4 is the number of gaseous sulfuric acid molecules (or other molecules containing sulfate, such as ammonium bisulfate) (molec cm⁻³).

The nucleation process is actually in competition with condensation (described in Section 9.2.3), which transfers gaseous molecules toward existing particles. If the particle concentration is important, then condensation tends to dominate. Therefore, nucleation is mostly observed in the atmosphere when the concentration of gaseous molecules of a chemical species with a low saturation vapor pressure is high and when the concentration of existing particles is low or moderate. Thus, one observes nucleation of new particles near emission sources (e.g., near vehicle exhaust) or under atmospheric conditions where the formation of a chemical species with a low saturation vapor pressure is important. This is the case, for example, in forested areas where the atmosphere is pristine (i.e., with low particle concentrations) and where emissions of biogenic VOC are important. These biogenic VOC can be oxidized rapidly to form compounds with low volatility, which may then nucleate. In addition, episodes of sulfate particle nucleation have been observed under conditions where the oxidation of sulfur dioxide (SO₂) is rapid and important.

Nucleation increases the number of particles, as well as the total mass of particles. However, the particle mass created by nucleation is typically very low compared to the mass of existing fine and coarse particles.

9.2.3 Condensation and Evaporation

Condensation consists of the transfer of gaseous molecules toward an existing particle. This particle can be liquid or solid and the new condensed mass can also be liquid or solid. The reverse process is evaporation; it consists of the transfer of molecules from the particle toward the gas phase. Condensation (or evaporation) occurs when the gas phase and the condensed phase are not in thermodynamic equilibrium and transfer between these phases must occur to establish this thermodynamic equilibrium. The two phases may be out of equilibrium, for example, because of a change of concentration in the gas phase (formation of a semi-volatile or non-volatile species) or because of a change of temperature, pressure or relative humidity.

The evolution of the distribution of the particle number concentration as a function of volume, n_p(v_p), where v_p is the particle volume, is given by the following equation:

∂np(vp)∂t=−∂Ivnp(vp)∂vpIv=dvpdt(9.5)

where I_v is called the growth law. It represents the rate of growth by condensation of a particle of volume v_p. This growth law corresponds to the flux of molecules from the gas phase to the particle and depends on (1) the mass transfer flux by diffusion of the molecules from the gas phase toward the particle surface and (2) the particle surface area available for condensation of these gaseous molecules. For spherical coarse particles, continuum mechanics implies that the mass transfer flux is proportional to the molecular diffusion coefficient and inversely proportional to a characteristic distance, which is the particle radius. Thus, the growth law is given as the product of three terms, which are the particle surface area, the mass transfer coefficient (diffusion coefficient/particle radius), and the condensing species concentration difference between the bulk gas phase and the particle surface (expressed in terms of molecular volume per unit volume of air):

dvpdt=π dp2 2Dmdp vm(Cg−Cg,e)=2πdp Dm vm(Cg−Cg,e)(9.6)

where v_p (cm³) is the particle volume, d_p is the particle diameter (cm), D_m is the diffusion coefficient of the condensing molecules in the air (cm² s⁻¹), v_m is the molecular volume of the condensing molecule (cm³ molec⁻¹), C_g is the gas-phase concentration of the condensing molecules (expressed as number of molecules per unit volume of air; here, molec cm⁻³), and C_g,e is the gas-phase concentration of the condensing molecules in thermodynamic equilibrium at the particle surface (molec cm⁻³). The molecular volume is obtained by dividing the molecular mass of the condensing species by its density in the condensed phase. Since the particle surface area is proportional to the square of the particle diameter and the mass transfer flux is inversely proportional to the particle diameter, the growth law is proportional to the particle diameter.

For particles that have a diameter significantly less than the mean free path in the air (which is about 70 nm at an atmospheric pressure of 1 atm), particle dynamics can no longer be represented by continuum mechanics, but must be treated as a free molecular flow. As a result, the mass flux of molecules condensing on a particle is governed by the kinetic theory of gases and does not depend on the particle size. Therefore, the particle diameter only appears in the particle surface area available for condensation, which is proportional to the square of the particle diameter:

dvpdt=π4 dp2(8kB Tπ mg )12 vm(Cg−Cg,e)(9.7)

where m_g is the mass of the condensing gaseous molecule (g), k_B is the Boltzmann constant (1.381 × 10⁻²³ J K⁻¹ or, here, 1.381 × 10⁻¹⁶ erg K⁻¹) and T is the temperature (K). The term in parentheses to the power 1/2 is the mean thermal velocity of the condensing molecule, ct¯ .

The Fuchs-Sutugin equation (Fuchs and Sutugin, 1971) is commonly used to calculate the growth law over the full range of particle sizes (there are other similar formulas to represent the growth law):

dvpdt=2π dp Dm1+(1.33Kn+0.711+Kn)Knvm(Cg−Cg,eexp(4σpvmdp kB T))(9.8)

where Kn is the Knudsen number, which is equal to (2 λ_m,a/d_p), λ_m,a is the mean free path of a molecule in the air (cm), and σ_p is the surface tension of the particle (erg cm⁻²). The term in the denominator (which is a function of Kn) accounts for the transition between the molecular regime for condensation on ultrafine particles and the continuum regime for condensation on coarse particles. Condensation on fine particles (i.e., d_p between about 0.1 and 2.5 μm) occurs in this intermediate regime. For ultrafine particles, Kn becomes important and the growth law becomes proportional to the square of the particle diameter. For coarse particles, Kn tends toward 0 and the growth law becomes proportional to the particle diameter.

The exponential term represents the Kelvin effect, which results in the possible evaporation of molecules from some ultrafine particles because of their strong curvature. For a sphere, the concentration above its surface will be greater than the thermodynamic equilibrium concentration obtained over a flat surface. Conceptually, this phenomenon is due to the fact that the curvature of the spherical surface implies that a molecule in the particle will be farther statistically from molecules in the gas phase, thereby allowing more space for molecules in the gas phase near the particle surface. This phenomenon depends on the surface tension of the particle. It can be significant: for example, it leads to a 10 % increase in the saturation vapor pressure of water over a 10 nm water droplet. It implies that condensation may not take place on the smallest ultrafine particles and that instead these particles may undergo evaporation of some species present in solution.

Figure 9.2 shows the growth law for the condensation of sulfuric acid, H₂SO₄, immediately neutralized into ammonium sulfate, (NH₄)₂SO₄. One notes that the growth law increases with the particle diameter, but that the slope decreases as the diameter increases, since the growth law is proportional to d_p² for ultrafine particles and proportional to d_p for coarse particles. However, in air pollution, one is interested in the overall condensation rate, i.e., the growth of a particle population due to condensation of gaseous molecules on all those particles. Figure 9.2 also shows the overall condensation rate, i.e., the growth law for a single particle of a given size multiplied by the number of particles of that size, for a particle population typical of a polluted urban area (see Figure 9.1). This condensation rate is maximum here for particles with diameters ranging from about 0.03 to 0.3 μm, which corresponds mostly to the accumulation mode. We will see in Section 9.2.4 that coagulation leads to a similar result, which explains why atmospheric particles accumulate in this size range.

Figure 9.2. Condensation. Top figure: growth law due to gas/particle conversion by condensation of 1 ppb of sulfuric acid (immediately neutralized as ammonium sulfate) as a function of the particle diameter; bottom figure: distribution of the condensation rate of 1 ppb of sulfuric acid (immediately neutralized as ammonium sulfate) as a function of particle size for a particle population typical of a polluted urban area (corresponding here to that of Figure 9.1).

Figure 9.3 depicts the Kelvin effect for an organic compound, nonadecane (an alkane with 19 carbon atoms), which is present in diesel engine exhaust. The saturation vapor pressure of nonadecane above a flat surface is 6.1 × 10⁻⁹ atm, i.e., 6.1 ppb at P = 1 atm. One investigates here the case where its gas-phase concentration is twice its saturation vapor pressure, i.e., 12.2 ppb. One notes that the Kelvin effect increases the saturation vapor pressure (or concentration) significantly below 0.025 μm. Thus, evaporation of nonadecane occurs for particles with a diameter less than about 0.025 μm. These particles shrink, whereas particles with a diameter greater than about 0.025 μm grow by nonadecane condensation. This process may occur in the exhaust of a diesel vehicle, where alkanes are initially very concentrated in the gas phase and condense on soot particles until the exhaust dilution leads to lower gas-phase concentrations, which may then become less than the saturation vapor pressure for the ultrafine particles. The decrease of the diameter of these ultrafine particles will favor their transfer toward the accumulation mode via coagulation (see Section 9.2.4).

Figure 9.3. Kelvin effect. Comparison of the gas-phase concentration and saturation vapor concentration of nonadecane as a function of particle diameter. The gas-phase concentration was taken equal to twice the saturation vapor concentration over a flat surface.

Source: Adapted from Devilliers et al. (2013); reprinted with authorization, © 2012 Elsevier Ltd.

Condensation increases the particle mass, but does not change the number of particles. Conversely, evaporation decreases the particle mass, but does not change the number of particles.

9.2.4 Coagulation

Coagulation corresponds to the collision of two particles and the formation of a single particle from the two original particles. One typically assumes that the collision of two particles results in the formation of a single particle and that, therefore, every collision leads to coagulation. Although this is likely to be the case for liquid or liquid-coated particles, it may not always be the case for solid particles. For the sake of simplicity, one will assume here that every collision leads to coagulation. The coagulation rate is minimum for two particles of a same size and increases as the difference between the sizes of the two coagulating particles increases. Therefore, freshly nucleated ultrafine particles will coagulate more readily with a fine or coarse particle than with an ultrafine particle. However, the number concentration of particles must be taken into account as well and, if the number of ultrafine particles is very high compared to those of fine and coarse particles, then coagulation among ultrafine particles becomes important. The change as a function of time of the distribution of the particle number concentration as a function of the particle volume, n_p(v_p), consists of two terms: one represents the increase of the number of particles of volume v_p due to coagulation of particles of smaller volumes (i.e., a particle of volume v_p’ coagulating with a particle of volume (v_p – v_p’) to create a particle of volume v_p) and the other represents the decrease of the number of particles of volume v_p due to coagulation of those particles with other particles. This overall change is given by the following equation, where the terms β(v_p, v_p’) are the coagulation coefficients between a particle of volume v_p and a particle of volume v_p’:

dnp(vp)dt=12∫0vpβ(vp‘,vp−vp‘)np(vp‘)np(vp−vp‘)dvp‘−∫0∞β(vp,vp‘)np(vp)np(vp‘)dvp‘(9.9)

As it was done for the growth law representing the transfer of gaseous molecules toward a particle, the collision between two particles can be addressed by considering the extreme cases of the continuous regime for coarse particles and the kinetic theory of gases for ultrafine particles in the free molecular regime. Between these two extremes, an intermediate regime must be parameterized for the size range that corresponds approximately to fine particles. For coarse particles, the coefficient of coagulation between two particles, β(v_p,i,v_p,j) (expressed here in units of cm³ per particle per second), is given by the following equation:

where, for particle i, v_p,i is the particle volume (cm³), D_p,i is the brownian diffusion coefficient in the air (cm² s⁻¹), and d_p,i is the particle diameter (cm). The coagulation coefficient, β(v_p,i,v_p,j), is minimum when d_p,i = d_p,j and increases when the difference in size of the two particles increases.

For particles of diameter less than the mean free path in the air, the kinetic theory of gases applies:

β(vp,i,vp,j)=π4(c¯t,i2+c¯t,j2)1/2(dp,i+dp,j)2 (9.11)

In the intermediate regime (which concerns fine particles), the Fuchs equation is commonly used (Fuchs, 1964):

β(vp,i,vp,j)=2π (Dp,i+Dp,j)(dp,i+dp,j)[dp,i+dp,jdp,i+dp,j+2(gi2+gj2)1/2+8(Dp,i+Dp,j)(c¯t,i2+c¯t,j2)1/2(dp,i+dp,j)]Dp,i=kB T cc3πμv,a dp,i;cc=1+2 λm,adp,i(1.257+0.400 exp(−0.55 dp,iλm,a))c¯t,i=1.6(kBTmp,i)12;gi=0.47dp,i λm,i[(dp,i+λm,i)3−(dp,i2+λm,i2)32]−dp,i;λm,i=8 Dp,iπ c¯t,i(9.12)

where c_c is the Cunningham coefficient and, for particle i, v_p,i is the volume (cm³), D_p,i is the brownian diffusion coefficient in the air (cm² s⁻¹), d_p,i is the diameter (cm), c¯t,i is the mean thermal velocity (cm s⁻¹), m_p,i is the particle mass, and g_i is a coefficient (cm) that is a function of the particle diameter i and its mean free path in the air λ_m,i. This formula tends toward the solution of the continuous regime when one of the diameters becomes large and toward the kinetic theory of gases when both diameters become less than the mean free path. This brownian coagulation coefficient is shown in Figure 9.4 for particles of diameters 0.001, 0.01, 0.1, 1, 10, and 100 μm coagulating with particles with diameters ranging from 0.001 to 100 μm in diameter. For a particle of a given diameter, the coagulation coefficient is minimum for coagulation with a particle of the same diameter.

Figure 9.4. Brownian coagulation. Top figure: brownian coagulation coefficient between particles of diameters d_p,i and d_p,j; bottom figure: distribution of the coagulation rate of particle j coagulating with particles i of greater size using a particle population typical of a polluted urban area corresponding to that of Figure 9.1.

This coagulation coefficient includes the assumption that all collisions lead to coagulation of the two particles to form a single particle. If that is not the case, one must introduce a correction term to account for the fact that the coagulation probability is less than 1. In addition, this equation corresponds to the brownian motion of particles. For ultrafine particles, the van der Waals forces are no longer negligible and increase the coagulation rate. For electrically charged particles, one must account for the Coulomb forces. Equations are available to take into account these processes if they are relevant (Friedlander, 2000).

The coagulation rate between particles depends not only on the value of the coagulation coefficient, but also on their number concentrations. Therefore, coagulation is more important for ultrafine particles since they are typically present in greater number than fine and coarse particles (see Figure 9.1). For fine particles, coagulation will be important only near sources, because in the ambient background, their number concentrations are typically too low to lead to significant coagulation rates and coagulation can then be neglected. Figure 9.4 shows the distribution of the coagulation rate of a particle of a given diameter (0.001, 0.01, 0.1, 1, and 10 μm) coagulating with particles of a greater diameter using a particle population typical of a polluted urban area. This coagulation rate is equivalent to a pseudo-first-order coagulation rate coefficient, which is defined as the product of the coagulation coefficient and number concentration of the other particles involved in the coagulation process. The particle number concentrations presented in Figure 9.1 were used here. One notes that those coagulation rates are greater for ultrafine particles and that those particles coagulate preferentially with fine particles of diameters ranging from about 0.05 to 0.5 μm. The coagulation rates become negligible for coagulation with particles of diameter greater than 2.5 μm because their number concentrations are very low. Consequently, coagulation increases particulate mass in the accumulation mode (fine particles) and has almost no effect on coarse particles. As for condensation, particles tend to accumulate by coagulation in the size range corresponding to the accumulation mode. We will see in Chapter 11 that atmospheric deposition is less efficient for particles in that size range. Therefore, fine particles, i.e., those in the accumulation mode, have an atmospheric lifetime that is longer than those of ultrafine and coarse particles.

Coagulation decreases the number of particles, but does not affect their total mass.

9.3.1 Saturation Vapor Pressure

The saturation vapor pressure, P_s,i, is the pressure that a gas cannot exceed. If the partial pressure, P_i, of a gaseous species exceeds its saturation vapor pressure, then nucleation and/or condensation occur to decrease the partial pressure and bring the species back to thermodynamic equilibrium. One must note, however, that condensation on liquid particles may occur in cases where the partial pressure is less than the saturation vapor pressure (see Sections 9.3.2 and 9.3.3).

9.3.2 Henry’s Law

Henry’s law applies to dilute aqueous solutions. It relates the concentration of a chemical species in the gas phase (represented by its partial pressure, P_i) to its activity in the aqueous phase as follows:

where C_i is the concentration in the particle in moles per liter (M), γ_i is the activity coefficient of the species in the aqueous phase (γ_i C_i is the activity of the species), and H_i is the Henry’s law constant (which depends on temperature). The activity coefficient reflects the fact that the solution is not ideal and that interactions among species in solution (molecules and ions) affect their ability to be present in solution. By definition, the activity coefficients are equal to 1 in an ideal solution. If interactions with other species present in solution are repulsive, the species will be less present in solution than if the solution were ideal (i.e., without any interactions with other species); on the other hand, if the interactions are attractions, then the species will be more present in solution than if the solution were ideal. An activity coefficient greater than 1 corresponds to the case where the interactions are repulsive and an activity coefficient less than 1 corresponds to the case where interactions correspond to attractions. A dilute solution tends toward ideality; then, the activity coefficient tends toward 1, and the activity tends toward the concentration (Henry’s law was originally formulated for an ideal solution). The partial pressure is generally expressed in atm; then, the Henry’s law constant is expressed in M atm⁻¹.

9.3.3 Raoult’s Law

Raoult’s law applies to concentrated solutions. It relates the concentration of a chemical species in the gas phase (represented by its partial pressure, P_i) to its activity in the liquid phase as follows:

where x_i is the molar fraction of the chemical species in solution. If the solution is ideal, then the molar fraction of the species in solution is equal to the ratio of its partial pressure and its saturation vapor pressure. For a pure solution of the species, its molar fraction is equal to 1 and one obtains the definition of the saturation vapor pressure as being the maximum value of the partial pressure.

9.4.1 Sulfate and Ammonium

Sulfuric acid has a very low saturation vapor pressure and is not likely to remain in the gas phase. Therefore, it undergoes nucleation (either binary with water or ternary with water and ammonia) or condensation on existing particles. In the aqueous phase, sulfuric acid dissociates to form bisulfate (HSO₄⁻) and sulfate (SO₄^2-) ions:

H2SO4 ↔ H++HSO4−(R9.1)

HSO4− ↔ H++SO42−(R9.2)

H₂SO₄ is a strong acid and the equilibria are strongly displaced toward the formation of the sulfate ions. H₂SO₄ and its ions are partially or totally neutralized by ammonia (a base) present in the atmosphere. Depending on the ammonia and sulfate concentrations present in the atmosphere, there may be formation of ammonium bisulfate (NH₄HSO₄), ammonium sulfate ((NH₄)₂SO₄) or letovicite:

Ammonium bisulfate is formed when there are as many moles of ammonia as there are moles of sulfate. Ammonium sulfate is formed when there are at least two moles of ammonia available for each mole of sulfate (sulfuric acid being a diacid, it takes two moles of ammonia to neutralize one mole of sulfuric acid totally). Letovicite corresponds to the intermediate case when there are 1.5 moles of ammonia available per mole of sulfate and its chemical formula is (NH₄)₃H(SO₄)₂.

9.4.2 Nitrate and Ammonium

Nitric acid has a rather high saturation vapor pressure and, therefore, it remains preferentially in the gas phase. However, it can react with ammonia to form a semi-volatile species: ammonium nitrate (NH₄NO₃):

This species is subject to the following thermodynamic equilibrium:

Keq=[HNO3(g)][NH3(g)][NH4NO3(p)](9.15)

where the notations (g) and (p) indicate a gas-phase and particulate-phase species, respectively. In the case of a solid phase, the ammonium nitrate concentration is taken to be 1 by definition and we have the following equilibrium relationship:

The first comparison of this theoretical equilibrium relationship with experimental data in the Los Angeles basin, California, showed satisfactory results (Stelson et al., 1979). In the case of an aqueous phase, ammonium nitrate dissociates into an ammonium cation and a nitrate anion:

NH4NO3 ↔ NH4++NO3−(R9.6)

Then, the activities of the nitrate and ammonium ions must be taken into account and the equilibrium relationship is as follows:

Keq=[HNO3(g)][NH3(g)]γNH4+ [NH4+(aq)] γNO3−[NO3−(aq)] (9.17)

where the notation (aq) indicates an aqueous phase (this notation is generally ignored for ions, because they are typically present in the aqueous phase in the lower atmosphere). If the product of the concentrations of nitric acid and ammonia is low, then there is no formation of solid particulate ammonium nitrate (at low humidity) and negligible formation of ammonium nitrate in solution (at high humidity). A small fraction of nitric acid may dissolve in aqueous particles (e.g., in sulfate particles), but the liquid water content of particles is very low (<1 mg m⁻³) and the fraction of nitrate formed via dissolution in particles is negligible.

The equilibrium constant of ammonium nitrate decreases when the temperature decreases and when relative humidity increases (e.g., Stelson and Seinfeld, 1982, Mozurkewich, 1993). Therefore, cold and humid atmospheric conditions (for example, in wintertime) are conducive to the formation of particles containing ammonium nitrate. However, solar radiation is limited during winter. Therefore, the formation of HNO₃, which requires OH radicals or ozone, both of which are produced by photolysis, is low then. Thus, it is typically during springtime or in fall that the concentrations of ammonium nitrate will be highest, because there is sufficient production of HNO₃ and a moderate temperature favors the formation of particulate ammonium nitrate.

Ammonia will preferentially neutralize sulfuric acid before reacting with nitric acid. Therefore, the formation of ammonium nitrate will occur only if (1) there is sufficient ammonia to neutralize sulfuric acid first and (2) the product of the nitric acid and remaining ammonia concentrations is sufficiently high for ammonium nitrate formation.

Nitrate may occasionally be found in coarse particles, because nitric acid can react with sea salt and alkaline soil particles. The thermodynamics is favorable to the formation of nitrate salts and the following reactions may occur:

Sea salt:HNO3(g) + NaCl(p) → NaNO3(p) + HCl(g)(R9.7)

Soil particles:2 HNO3(g)+CaCO3(p) → Ca(NO3)2(p)+CO2(g)+H2O(g)(R9.8)

Similar reactions are possible with gaseous sulfuric acid. However, given its very low volatility, sulfuric acid tends to be in the particulate phase directly via nucleation or condensation, rather than via chemical reactions. Note that sea salt contains a fraction of primary sulfate. Sea salt and soil particle emissions are addressed in Chapter 11.

9.4.3 Deliquescence

When humidity increases, a solid salt (ammonium sulfate, ammonium nitrate …) will become liquid at a given humidity. This humidity is called the humidity of deliquescence. It depends on temperature. At 25 °C for example, it is 40 % for ammonium bisulfate, 62 % for ammonium nitrate, and 80 % for ammonium sulfate.

When the relative humidity decreases, a salt present in solution will become solid at a given humidity. This humidity of crystallization is less than the humidity of deliquescence and the solution is metastable between these two values of the relative humidity, being in a state of supersaturation for this salt. This phenomenon is called hysteresis.

Thermodynamic data pertaining to the inorganic equilibria of atmospheric aerosols are available in several books on atmospheric chemistry (e.g., Finlayson-Pitts and Pitts, 2000; Jacobson, 2005a; Seinfeld and Pandis, 2016).

9.5.1 General Considerations

The organic fraction of particulate matter consists of primary particulate matter, which has been emitted directly into the atmosphere as particles (for example, the organic fraction of particles emitted from biomass burning and diesel vehicles without particle filters), and secondary particulate matter, which was formed in the atmosphere. One typically refers to primary organic aerosols (POA) and secondary organic aerosols (SOA) to differentiate these two categories of particulate organic species. Regarding SOA, the semi-volatile organic compounds (SVOC) that condense on existing particles (or in some cases lead to new ultrafine particles via nucleation) may have been emitted into the atmosphere directly and have subsequently condensed because of a decrease in temperature that favors their partitioning toward the particulate phase or they may have been formed in the atmosphere via the oxidation of volatile or semi-volatile organic compounds (VOC and SVOC). A SVOC is a compound that partitions between the gas and particulate phases, whereas a VOC is entirely present in the gas phase. The European Union (EU) defines a VOC as an organic compound that has a boiling point equal to or less than 250 °C at a standard atmospheric pressure (1 atm) (the boiling point is anti-correlated with the saturation vapor pressure; the saturation vapor pressure increases as the boiling point decreases; Perry’s Chemical Engineers’ Handbook, 2008). There is no official definition of SVOC, but one may consider that they have saturation vapor pressures between 10⁻¹⁴ and 10⁻⁴ atm (Weschler and Nazaroff, 2008).

A potentially important source of SOA is the semi-volatile fraction of organic emissions from combustion processes (vehicle engines, biomass burning, etc.). This fraction may condense on soot particles, for example, during the cooling of the exhaust plume. It may also be oxidized, leading then to SVOC that are heavier (in terms of molar mass) and, therefore, more likely to condense. This SVOC source is currently uncertain, because only the particulate fraction of SVOC is typically included in emission inventories. The gas-phase fraction must, therefore, be estimated, which is difficult given the small number of available experimental data sets.

9.5.2 Chemistry of SOA Formation

Since the atmosphere is an oxidizing medium, VOC and SVOC undergo oxidation reactions, which lead to the formation of new compounds, which are oxygenated (e.g., Ziemann and Atkinson, 2012; Ng et al., 2017). The main oxidants are the hydroxyl radical (OH), the nitrate radical (NO₃), and for alkenes ozone (O₃). The oxidation step that consists of the addition of a functional group to the initial organic molecule and results in the addition of oxygen to that organic molecule is called functionalization. Once the organic molecule has been oxidized, other reactions may occur leading to successive functionalization steps with addition not only of oxygen, but also nitrogen (organic nitrates) and sulfur (organic sulfates). It is also possible that an oxidation reaction leads to the fragmentation of the organic molecule and the formation of two (or more) molecules that are smaller (in terms of the number of carbon atoms) than the original molecule. Then, there is a loss of carbon atoms from the original molecule.

The addition of oxygen atoms to an organic molecule will increase its molar mass because the oxygen atom replaces a hydrogen atom (for example, in the case of aldehydes, ketones, and acids) or is added to the organic molecule leaving the number of hydrogen atoms unchanged (for example, in the case of alcohols and peroxides). In general, given a VOC or SVOC with a given number of carbon atoms, the augmentation of the molar mass via an oxidation reaction leads to a less volatile compound, i.e., a compound with a lower saturation vapor pressure. Figure 9.5 shows the saturation vapor pressures of some organic compounds. There is little difference in terms of volatility among an alkane, an alkene, and an aromatic compound with the same number of carbon atoms. On the other hand, for a given number of carbon atoms, an aldehyde is less volatile than an alkane and a carboxylic acid is less volatile than an aldehyde. The oxidation products are, therefore, more likely to be present in the particulate phase than the original compound. As the number of oxidation steps increases, the number of functionalizations can increase and the saturation vapor pressure will decrease, favoring the partitioning of the compound toward the particulate phase. Similarly, the formation of an organic nitrate or sulfate will lead to a decrease in volatility compared to that of the original compound.

Figure 9.5. Saturation vapor pressures of selected organic compounds (atm). The saturation vapor pressures are shown for n-alkanes (methane to octadecane), n-hexene, benzene and its mono-substituted alkyl derivatives (toluene, ethyl-, and propyl-benzene), aldehydes (formaldehyde to hexanal), and monocarboxylic acids (acetic acid to hexanoic acid).

Source of the data: Schwarzenbach et al. (2003).

On the other hand, a reaction leading to the fragmentation of an organic compound will lead to compounds having lower molar masses and, therefore, higher saturation vapor pressures. The products of such a reaction are, therefore, more likely to be present in the gas phase than in the particulate phase, as compared to the original compound. The probability of fragmentation is significantly greater for compounds that are more functionalized, thereby limiting the number of functionalizations that are possible.

Figure 9.6 summarizes schematically the main processes involved in SOA formation. After a period leading to the formation of SVOC by functionalization of organic molecules, fragmentation reactions begin to occur and lead to more volatile products. These processes occur over several hours in the atmosphere. SVOC partitioning between the gas phase and the particulate phase may occur in an organic liquid phase or an aqueous phase depending on the hydrophobic or hydrophilic characteristics of the SVOC and those of the existing atmospheric particles (Saxena et al., 1995). The chemical reactions and products corresponding to these processes are described in detail below.

Figure 9.6. Schematic representation of the evolution of SOA formation from a VOC.

9.5.3 Experimental Estimates of SOA Formation Yields

Smog Chamber Experiments

The oxidation of VOC (and SVOC) has been studied experimentally in smog chambers, which allow one to follow the evolution of the concentrations of the main chemical species (reactants and products) over several hours. These experiments can be conducted in the presence of natural sunlight, artificial lighting or in the dark. The oxidants used are those present in the atmosphere and include the hydroxyl radical (OH), ozone (O₃), and the nitrate radical (NO₃). Most of these experiments have been conducted with OH as the oxidant. One may distinguish two main categories of experiments conducted with OH: (1) those where the OH radicals are produced by reaction between NO_x and the VOC being studied or by photolysis of nitrous acid (HONO) and (2) those where the OH radicals are produced in the absence of NO_x. In the former case, one will typically be in a high-NO_x regime and nitrogen oxides will then be involved in some reactions of the VOC oxidation products. In addition, the presence of NO_x is generally sufficient to titrate the ozone formed by reactions between NO_x and VOC, which allows one to study solely the OH oxidation pathway (in the case of alkenes). However, in some experiments, the NO_x concentrations are kept sufficiently low that one may consider those experiments to be typical of a low-NO_x regime. In the latter case, one is in a no-NO_x regime, which is similar to a low-NO_x regime in the atmosphere. The OH radicals can be produced by photolysis of hydrogen peroxide (H₂O₂) or by ozone (O₃) photolysis in the presence of water vapor (H₂O). The first oxidation step by OH will be identical in both cases, but the products of the subsequent reactions could be very different depending on the NO_x regime. Typically, the formation of organic nitrates prevails in a high-NO_x regime and the formation of peroxides prevails in a low-NO_x regime. Since the saturation vapor pressure of the peroxides and nitrates may differ significantly, different SOA formation yields may be obtained depending on whether the experiment was conducted under high- or low-NO_x conditions. Clearly, these two types of experiments are relevant, since VOC oxidation may occur in the atmosphere under high-NO_x (e.g., urban or industrial areas) or low-NO_x (e.g., rural areas) conditions.

Experiments conducted in smog chambers have initially targeted anthropogenic compounds (mostly aromatic compounds), but have rapidly been extended to include biogenic compounds. Today, experiments that have been conducted in smog chambers include, for example, biogenic compounds such as hemiterpenes and their products, monoterpenes, and sesquiterpenes, and anthropogenic compounds such as aromatic compounds, alkanes, alkenes, and polycyclic aromatic hydrocarbons (PAH). The terms terpenes and terpenoids have been defined in Chapter 8 and the term terpenes is used here to cover both terms.

It is challenging to summarize all the results obtained in a large number of smog chamber experiments, because those studies have been conducted under conditions that are sometimes very different (for example, in terms of chemical regime, lighting, duration, temperature, humidity, presence of seed particles, wall effects …). Some smog chambers are located indoor and use artificial lighting, which must then approximate sunlight; others are located outdoor, thereby using natural sunlight, but being affected by the possible presence of clouds. The size of the chamber impacts directly the duration of the experiments that can be conducted, because deposition to the walls is less in a large chamber (the surface/volume ratio is inversely proportional to the characteristic dimension of the chamber). The maximum duration of the experiments conducted in a large smog chamber is typically of several hours (but less than a day). In addition, the measurement of the SOA yield (the ratio of the particulate mass formed and that of the VOC reacted, expressed as a fraction or percentage) is subject to interpretation. First, this yield depends on the duration of the experiment. The longer the experiment, the more the yield tends to increase, because few experiments last long enough for fragmentation processes to prevail over functionalization reactions. Second, as the particulate mass increases, it favors the gas/particulate equilibrium toward the particulate phase, because a larger particulate volume becomes available for absorption of the semi-volatile compounds. Therefore, an experiment conducted with a large initial concentration of a precursor (for example, 1 ppm instead of 10 ppb) will lead to a greater yield, because the particulate mass will be more important (for example, ~1 mg m⁻³ instead of ~10 μg m⁻³) and will favor the absorption of semi-volatile compounds. If the results are not very sensitive to this initial concentration, it implies that the oxidation products have low volatility and are, therefore, mostly in the particulate phase (i.e., their particulate concentrations do not depend much on the particulate mass available for their absorption).

The temperature used for the experiments is important because, according to the van’t Hoff equation (see Chapter 10), the thermodynamic equilibrium favors the semi-volatile compounds toward the gas phase as the temperature increases. In addition, a low relative humidity does not allow hydrophilic compounds to condense on an aqueous phase. Such dissolution in aqueous particles may occur for some SVOC in the atmosphere (Saxena et al., 1995), in which case the smog chamber experiments conducted at low relative humidity may not be representative of atmospheric conditions.

Concentrations of the organic particulate mass present in the atmosphere are generally lower than those used in smog chamber experiments (higher concentrations make chemical measurements easier). Therefore, the maximum yields obtained in smog chamber experiments will typically overestimate the actual yields observed in the atmosphere. Normalizing those yields with respect to a particulate mass concentration of 10 μg m⁻³, for example, should provide more realistic SOA yields. One must also take into account the fact that gases and particles will deposit on the chamber walls. Therefore, some assumptions must be made regarding the absorption (or not) of semi-volatile compounds on the particles deposited on the walls; depending on those assumptions, yields may differ by a factor of two. Finally, measurement methods for SOA have uncertainties, which can affect the estimation of the SOA yield.

A summary of selected smog chamber experiments is presented in Figure 9.7 as an illustration of typical SOA yields for a range of VOC.

Figure 9.7. Examples of SOA yields obtained by OH oxidation under high-NO_x and low-NO_x regimes for several VOC categories. Other experiments have been conducted that can lead to significantly different yields for some of these compounds. Therefore, these yields must be seen in terms of comparison between high-NO_x and low-NO_x regimes or among VOC of a same category. Figures (a) to (d) from top to bottom:

1. 
   

   (a) SOA mass yields for aromatic compounds for an organic particulate mass of 10 μg m⁻³ (Ng et al., 2007a).

   
   
2. 
   

   (b) Maximum SOA mass yields for alkanes with 12 carbon atoms, which include a linear alkane (dodecane), a branched alkane (methylundecane), a cyclic alkane (cyclododecane), and a branched cyclic alkane (hexylcycloalkane) (Loza et al., 2014). These yields were obtained with high organic particulate mass concentrations (>100 μg m⁻³ in most cases) and, therefore, may not be representative of atmospheric yields.

   
   
3. 
   

   (c) SOA molar yields for terminal and internal alkenes (Matsunaga et al., 2009) calculated from measurements of molecular compounds in a high-NO_x regime. It was assumed that the dihydroxycarbonyls are semi-volatile and partition between the gas and particulate phases.

   
   
4. 
   

   (d) Maximum SOA mass yields for biogenic compounds (Kroll et al. (2006) for isoprene; Jaoui et al. (2012) for 2-methyl-3-buten-2-ol (MBO); Eddingsaas et al. (2012) for α-pinene; Lee et al. (2006) for β-pinene; Ng et al. (2007b) for longifolene).

9.6.1 Primary PM

Primary particles can be captured before being emitted in the atmosphere (see Chapter 2). In the case of diesel vehicle particulate emissions, the recent European regulatory emission standards require that particle filters be installed on all new on-road vehicles. There are also incentives provided by the French government for the replacement of old wood stoves (which are large sources of primary PM) by modern and more efficient stoves. The development of efficient control strategies for primary PM requires the identification of the major sources. A case study of a wintertime PM pollution in the Paris region is presented in Section 9.6.4 to illustrate this point.

9.6.2 Secondary Inorganic PM

The reduction of PM containing sulfate, nitrate, and ammonium can be challenging if one does not account for the relationships among these different species. The reduction of sulfate concentrations is obtained by decreasing the emissions of its precursor, SO₂. The strategy to reduce SO₂ emissions from coal-fired power plants in the U.S. has shown that this approach was efficient, in particular once SO₂ ambient concentrations became less than those of hydrogen peroxide (see Chapters 10, 13, and 15). However, the reduction of sulfate leads to a transfer of secondary particulate ammonium toward the gas phase as ammonia. Then, this gaseous ammonia becomes available for reaction with nitric acid to form ammonium nitrate. Therefore, a decrease in sulfate concentrations may lead to an increase in nitrate concentrations, thereby limiting the benefits of SO₂ emission controls. These antagonistic effects must be taken into account when selecting the emissions to be controlled, in order to avoid setting up an emission control strategy that could be inefficient.

The reduction of nitrate concentrations can occur either via the reduction of nitric acid or via the reduction of ammonia. Therefore, one must determine which of these two precursors will lead to the best result in terms of reducing the ammonium nitrate concentration. The amount of ammonia available for reaction with nitric acid, [NH₃]_a, is defined as follows:

where [NH₃]_t represents the total ammonia concentration, i.e., the concentration before any reaction or partitioning between the gas and particulate phases. This equation, expressed in moles or in ppb, reflects the fact that each mole of sulfuric acid requires two moles of ammonia to be neutralized and that, therefore, these moles of ammonia will not be available for reaction with nitric acid.

Next, one defines the excess ammonia concentration, [NH₃]_e, which represents the available ammonia concentration that is in excess of the total concentration of nitric acid:

One can show that if [NH₃]_e < 0, then, it is more efficient to reduce ammonia emissions rather than nitric acid emissions (ammonia-poor regime). On the other hand, if [NH₃]_e > 0, then, it is more efficient to reduce the nitric acid concentrations rather than those of ammonia (ammonia-rich regime).

Another way to express this relationship is to use the ratio of the gaseous precursors of ammonium nitrate, i.e., the available ammonia and total nitric acid (Ansari and Pandis, 1998):

  GR=[NH3]a[HNO3]t(9.20)

Thus, GR < 1 for the ammonia-poor regime and GR > 1 for the ammonia-rich regime.

The reduction of nitric acid concentrations is not straightforward. Nitric acid is formed by the oxidation of NO₂. A decrease in NO_x emissions may lead to an increase in oxidant concentrations in a high-NO_x regime (see Chapter 8); therefore, nitric acid formation, which is proportional to the concentrations of NO₂ and the oxidant (OH or NO₃), may not necessarily decrease and, in some cases, could increase.

9.6.3 Secondary Organic PM

A decrease in the anthropogenic precursors of particulate organic species can be an efficient emission control strategy for reducing PM_2.5 concentrations. However, secondary organic PM contains an important biogenic fraction. VOC emissions from vegetation cannot be reduced (one could consider the use of tree species that emit lower amounts of VOC that are precursors of SOA, but it is more likely that economic considerations rather than environmental considerations govern the selection of the tree species used in commercial forests). Nevertheless, it is possible to affect the PM mass available for SVOC absorption. SOA formation consists mostly of the partitioning between the gas phase and the particulate phase (organic and, to a lower extent, aqueous). Therefore, the SVOC fraction that is present in the particulate phase depends on the volume of the organic particulate phase available to absorb SVOC: The greater this particulate organic phase, the greater the SVOC absorption. Conversely, a decrease of the volume of this organic particulate phase leads, for a given total concentration (i.e., gas- + particulate-phase) of a SVOC, to a decrease of its particulate fraction.

Accordingly, the strategies implemented to control primary PM emissions (for example, reduction of soot particles from diesel engines and wood burning) and secondary inorganic PM (i.e., the liquid phase content of those deliquescent particles) are beneficial and contribute to a decrease in particulate organic species concentrations, including those corresponding to SVOC that have unchanged total concentrations. In addition, a decrease in oxidant concentrations (such as ozone and the OH and NO₃ radicals produced by ozone reactions) will lead to a slower oxidation of VOC into SVOC and, therefore, a reduction in SVOC concentrations and a decrease of their contribution to organic PM.

9.6.4 Winter Air Pollution Episodes in Paris

We describe here the characteristics of the chemical composition and origin of PM leading to typical winter air pollution episodes in the Paris region, using a case study from December 2016, which provides a comprehensive data set. This episode lasted more than a week and was due to particularly strong anticyclonic conditions, i.e., the presence of a high-pressure system over the Paris region. For example, on December 5 and 6, the wind speed at 10 m was less than 2 m s⁻¹ and the planetary boundary layer (PBL) height was about 100 m at night and 150 m during the day according to measurements available at Sirta (Site instrumental de recherche par télédétection atmosphérique), an experimental meteorological station located in Saclay, about 15 km southwest of Paris. Therefore, primary PM emissions were not dispersed much due to the low wind speeds and the limited volume of the PBL in which they were diluted, thereby leading to high concentrations. Figure 9.10 depicts the temporal evolution of the fine PM concentration (here PM₁, i.e., particles with an aerodynamic diameter ≤1 μm) and their chemical composition measured at Sirta. One notes the importance of carbonaceous PM, including black carbon and organic compounds. Although a fraction of organic carbon may be secondary, the low photochemical activity during winter suggests that primary organic carbon dominates the organic PM fraction. It is possible, however, that SVOC emissions may have led to condensation of a fraction of SVOC on primary particles because of the very low ambient temperature (near 0 °C). The main sources of primary carbonaceous PM are diesel vehicles without particle filters and residential wood burning.

Figure 9.10. Temporal evolution of the chemical composition of fine PM (PM₁) during the air pollution episode of December 2016 in the Paris region. BC: black carbon measured with a light absorption method, OM: organic matter, SO₄²⁻: sulfate, NO₃⁻: nitrate, NH₄⁺: ammonium.

Data sources: SIRTA / IPSL – LSCE / INERIS.

Inorganic species are less important, but are nevertheless present in notable amounts. Although ammonium sulfate contributes little to fine PM mass, ammonium nitrate contributes about 1/3 of fine PM mass on December 5 and 6. This ammonium nitrate fraction results from the oxidation of NO_x into HNO₃ and the subsequent reaction of HNO₃ with NH₃, which is favored here by the very low temperature. The inorganic fraction is less than 50 % of fine PM mass during winter episodes; however, it becomes dominant during spring episodes as discussed in Section 9.6.5.

Numerical modeling of this air pollution episode with the Polyphemus/Polair3D chemical-transport model confirmed that local emissions were the main source of air pollutants (>90 %) and that the imported fraction was a minor contribution.

9.6.5 Spring Air Pollution Episodes in Paris

The meteorology of a spring air pollution episode in the Paris region is described in Chapter 3. We are interested here in investigating which emission control strategies can be developed to reduce the concentrations of fine PM during such episodes. Figure 9.11 depicts the chemical composition of fine PM measured at Sirta, southwest of Paris, during the first half of March 2014 (see also Fritz et al., 2015). PM₁₀ hourly concentrations exceeded the daily regulatory value of 50 μg m⁻³, since the hourly PM_2.5 concentrations exceeded that concentration several times from March 11 to 15 (there is no daily regulatory concentration for PM_2.5 in Europe, see Chapter 15). One notes that the secondary fraction of fine PM dominates and that the contributions of black carbon and primary organic matter are low, particularly during the peak events. Therefore, this PM episode is mostly secondary and dominated by ammonium nitrate and SOA. This episode is interesting to investigate because it involves relationships between secondary PM and its gaseous precursors, as well as relationships between oxidants and precursors. Table 9.1 summarizes simulation results obtained for March 13, 2014 with the Polyphemus/Polair3D numerical chemical-transport model. These simulations used the EMEP emission inventory, which has a relatively coarse spatial resolution. Nevertheless, these simulations are useful to illustrate the main aspects of the relationships between pollutant sources and the ambient levels of air pollution.

Figure 9.11. Temporal evolution of the chemical composition of fine PM (PM₁) during the air pollution episode of March 2014 in the Paris region. BC: black carbon measured with a light absorption method, OM: organic matter, SO₄²⁻: sulfate, NO₃⁻: nitrate, NH₄⁺: ammonium. Some periods have missing data: part of March 11 for inorganic and organic compounds and some periods from March 13 to 15 for black carbon.

Data sources: SIRTA / IPSL – LSCE / INERIS.

The first emission control scenario targeted primary PM emissions within the Paris region with a 43 % reduction in those emissions. The decrease of the PM₁₀ concentration is only 11 %, which is consistent with the primary fraction of PM shown in Figure 9.11. It appears clearly that it is necessary to target emission controls of gaseous precursors of the secondary fraction of PM.

The second scenario targeted the reduction of gaseous precursors corresponding to the main chemical species present in the particles during the air pollution episode (reductions of the Paris region emissions are indicated in parentheses), i.e., NO_x (−33 %) for nitrate, NH₃ (−76 %) for ammonium, SO₂ (−7 %) for sulfate, and VOC (−23 %) for the organic fraction. The effect of these reductions is negligible, which seems surprising at first. However, a reduction in NO_x emissions leads to an increase in the oxidant levels in a high-NO_x regime (see Chapter 8) and, therefore, the daytime reaction leading to nitric acid formation shows little change in its kinetics ([NO₂] decreases, but [OH] increases). The important reduction in NH₃ emissions shows little effect (−6 % only for ammonium), because (1) NH₃ is not the limiting species for ammonium nitrate formation in the Paris region and (2) NH₃ has an atmospheric lifetime on the order of a week, which implies that the NH₃ concentrations observed in the Paris region are due in great part to long-range atmospheric transport originating from other regions and/or countries. The emissions of SO₂, the precursor of sulfate, are low in the Paris region and sulfate is transported over long distances from other regions. The VOC emission reductions only target anthropogenic VOC (which decrease by 7 %), but has only a small effect on biogenic SOA, which are major contributors to organic PM in spring.

The third scenario takes into account the high-NO_x regime of the Paris region and, therefore, targets a stronger reduction of VOC emissions (−85 %) in order to decrease the oxidant levels, and a more limited reduction (−15 %) of NO_x emissions. This strategy is efficient (−19 % for [PM₁₀]), but still fails to decrease PM₁₀ concentrations down to their daily regulatory value. The reason is that a significant fraction of the atmospheric PM concentrations in the Paris region originates from other regions via long-range atmospheric transport.

Therefore, a fourth scenario targeted a larger domain for reductions of PM and gaseous precursor emissions. A reduction by one third of the PM₁₀ concentration is then simulated. Although this decrease is still not sufficient to reach the regulatory value, it suggests that a strategy that targets a reduction of some selected gaseous precursors combined with a decrease (or at a minimum a status quo) of the oxidant concentrations over a domain that is much larger than the Paris region is needed to reduce PM₁₀ levels significantly during spring air pollution episodes.

9.7.1 Particle Size Distribution

The particle size distribution can be represented by a continuous distribution with a range of atmospheric particle sizes ranging from a few nanometers to several micrometers. However, such a representation is not applicable to air quality modeling because of the very large computational times that would be required to calculate its evolution due to nucleation, coagulation, condensation, and evaporation. Therefore, simpler representations are used. They use the logarithm (typically with base 10, i.e., decimal logarithm) of the particle diameter in order to cover the full spectrum of particle sizes in an optimal manner. There are two main approaches used in air quality simulation models: the modal approach and the sectional approach.

The Modal Approach

This approach is based on the assumption that atmospheric particles can be represented by distinct modes, which include mainly the nucleation mode (ultrafine particles), the accumulation mode (mostly fine particles), and the coarse mode (mostly primary particles produced by mechanical processes) (Whitby, 1978). The observation of atmospheric particle concentrations suggests the presence of such modes, although in some cases the use of only three modes is too simplistic, because there may be more than three modes present in the atmosphere (see Section 9.2.1). Measurements suggest that the number or volume distribution of particles in a mode can be approximately represented by a lognormal distribution of the particle concentration as a function of the logarithm of the particle diameter. Therefore, the distribution of the particle concentrations can be represented by three lognormal functions, each representing a mode and being characterized by a median diameter (or mean diameter since these two diameters are equivalent for a lognormal function), a standard deviation, and a total (or maximum) concentration of the mode. The standard mathematical representation of a lognormal distribution typically uses the natural logarithm. However, in practice, the decimal logarithm is used for the particle diameters. Therefore, the decimal logarithm is used here, except for the relationships between properties of the number and volume concentration distributions, for which the natural logarithm is more appropriate.

For example, the representation of the particle number concentration distribution as a function of the diameter, η_p,a(d_p), is expressed as follows for one of the modes (the accumulation mode is used here; it is indicated by the subscript a):

ηp,a(dp)=Na(2π)1/2log(σa) exp(−(log(dp)−log(dp,a,n))22(log(σa))2)(9.21)

where d_p,a,n is the median (also mean) diameter of the mode, σ_a is the standard deviation and N_a is the total number concentration of particles in that mode. If the distribution of the particle number concentration is lognormal, then the distributions of the surface and volume concentrations (and, assuming uniform particle density, the distribution of the mass concentration) are also lognormal. For example, the particle volume concentration is given as follows:

υp,a(dp)=Va(2π)1/2log(σa) exp(−(log(dp)−log(dp,a,v))22(log(σa))2)(9.22)

where υ_p,a(d_p) is the log-normal distribution of the volume concentration as function of the particle diameter, d_p, d_p,a,v is the median (also mean) diameter of the mode, σ_a is the standard deviation, and V_a is the total volume concentration of particles in that mode. The standard deviation is identical in all distributions (number, surface, and volume). However, the median diameter differs depending on the variable considered in the distribution. The relationship between the median diameters of the number and volume concentration distributions is as follows:

ln(d_p,a,υ) = ln(d_p,a,n) + 3(ln(σ_a))²

ln(dp,a,v)=ln(dp,a,n)+3(ln(σa))2(9.23)

The relationship between the total number concentration and the total volume concentration is as follows:

Va=π6 Na exp(3ln(dp,a,n)+4.5(ln(σa))2)(9.24)

The advantage of the modal representation is that it is consistent with the conceptual model of atmospheric aerosols (three modes representing three distinct groups of particles) and it is mathematically simple (lognormal functions allow one to convert from number to surface or volume concentrations analytically). However, it is essential to use variable median diameters and standard deviations when solving the general dynamic equation.

There are a few disadvantages. There is no analytical solution to the general dynamic equation, because the concentration distribution is no longer lognormal following coagulation and condensation/evaporation processes. Therefore, some approximations are necessary to constrain the solution and maintain lognormal distributions for all modes. Such approximations are typically appropriate for the accumulation and coarse modes, but they can lead to erroneous results when simulating ultrafine particles. Indeed, (1) the effect of coagulation on the nucleation mode is poorly represented with lognormal distributions and (2) the Kelvin effect cannot be simulated with a single nucleation mode. Taking into account a larger number of modes could help minimize such problems, but the model formulation then becomes more difficult to develop because of significant overlaps between modes.

Despite these shortcomings (which affect mostly ultrafine particles), the modal approach is used in a large number of air quality models, including that of the U.S. Environmental Protection Agency (EPA) (CMAQ model), and many global models, due to its efficient computational times.

The Sectional Approach

The sectional approach consists of discretizing the distribution (number, volume, mass concentration …) as a function of the particle diameter or its logarithm (Gelbard and Seinfeld, 1980). Therefore, the distribution corresponds to a histogram of the concentrations by particle size sections. The particle number and volume concentrations for particle size section i are expressed as follows:

Ni=∫dp,i−1dp,inp,d(dp) ddp(9.25)

Vi=∫dp,i−1dp,inp,d(dp) π6 dp3 ddp(9.26)

where n_p,d (d_p) is the distribution of the number concentration as a function of the diameter, d_p, and the subscript i corresponds to the section bounded by d_p,i-1 and d_p,i. Generally, the number and volume (or mass) distributions are represented as a function of the decimal logarithm of the particle diameter, log(d_p). If the sections are selected to be of equal range on a logarithmic scale, then (d_p,i / d_p,i-1) = constant. However, it is not necessary that all sections be of equal range.

The advantage of this approach is that it is mathematically simple to manage, since it consists of a discretization of the particle diameter spectrum into sections. Then, to solve the general dynamic equation, it suffices to discretize the equations for coagulation and condensation/evaporation (see Section 9.7.4). Nucleation is treated simply as the creation of new particles in the section with the smallest particle diameter.

The disadvantages of this approach are as follows. If a small number of sections is used, (1) the representation of the particle distribution is not very accurate (for example, nucleated particles may appear in a section that covers a range of particles that is too large) and (2) the numerical solution may be inaccurate (for example, due to numerical diffusion when solving for condensation). If a large number of sections is used, then the representation of the particle distribution should be sufficiently fine and the solution of the aerosol dynamics should be fairly accurate. However, the computational costs may become large. Therefore, one must optimize between accuracy of the solution and computational burden.

Another disadvantage of the sectional approach is that the solution is obtained for only one variable of the particle concentration (number or mass, in most cases). Then, the calculation of the other variables (for example, the number concentration from the mass concentration) leads to a change of that variable with time that is not consistent with the general dynamic equation (except if the number of sections is infinitely large). A solution to this problem involves solving the dynamic equation for the number concentration for the sections with small diameters (since the mass of those sections is negligible) and the dynamic equation for the volume (or mass) for the sections with large diameters (since the number concentration in those sections is negligible). The transition diameter between these two solutions is typically about 100 nm. Another approach is to let the diameter representative of a given section vary, which allows one to maintain the relationship between the number and mass concentrations of each section with this moving diameter and to jointly simulate the number and mass concentrations (see “Numerical modeling” in Section 9.7.4).

An advantage of the sectional approach is that it is possible to use a large number of sections, unlike the modal approach. Then, a good resolution of the chemical composition of particles as a function of their size may be obtained. In addition, it is possible to discretize also the chemical composition of the particles, which presents the advantage of simulating particles of a same size, but of different chemical compositions (i.e., an external aerosol mixture).

The sectional approach is currently used in most air quality models that are applied at regional and urban scales, as well as in some global models. The number of sections being used varies among models and also depending on the application. Typically, a number of sections on the order of six to eight provides a good compromise between computational time and accuracy.

Comparative evaluation studies for lognormal and sectional approaches have been performed, which illustrate the advantages and shortcomings of these two approaches (Zhang et al., 1999; Sartelet et al., 2006; Devilliers et al., 2013).

9.7.2 Inorganic Aerosols

The inorganic chemical composition of particles may vary when there is condensation of secondary chemical species (sulfate, nitrate, ammonium) or evaporation of semi-volatile species (for example, ammonium nitrate). A particle will tend toward equilibrium with the surrounding gas phase to minimize its energy. Thus, it suffices to calculate the Gibbs energy of that particle, which should be minimum at thermodynamic equilibrium. This calculation requires calculating the chemical potential of each species present in the particle and minimizing the Gibbs energy of the particulate system. When a large number of species must be taken into account, such a minimization can be difficult to perform (see “Numerical modeling” in Section 9.7.4). Therefore, the solution tends to be costly in terms of computational time.

Another approach consists of simplifying the chemical system by using conceptual models. For example, if the system is ammonia-poor, one assumes that there will not be any ammonium nitrate formation. One may also estimate, based on the relative molar amounts of ammonia and sulfate, whether there will be formation of ammonium bisulfate or ammonium sulfate. Thus, it is possible to eliminate some species from the system, thereby simplifying the system of equations to be solved. Next, one writes the system thermodynamics in terms of thermodynamic equilibria, which are then solved numerically as a system of algebraic equations. This approach is computationally more efficient than the former one. Clearly, it is not as accurate since some approximations are involved when simplifying the system. However, a comparison of these two distinct approaches has shown that the latter approach can be sufficiently accurate for most cases. Several models have been developed using this latter approach, such as MARS, EQUISOLV, and ISORROPIA. ISORROPIA (Fountoukis and Nenes, 2007) is currently the most widely used inorganic aerosol model in air quality simulation models.

9.7.3 Organic Aerosols

Organic aerosols contain a very large number of species, most of which have not yet been identified experimentally. Therefore, organic aerosol models must necessarily involve a larger number of assumptions than used in inorganic aerosol models.

The first studies of SOA formation in smog chambers have been analyzed using parameterizations of the SOA yield as a function of the particulate organic matter formed. If one assumes that there is partitioning of semi-volatile compounds between a gas phase and an organic particulate phase, then Raoult’s law can be applied to quantify this partitioning. When the particulate organic mass increases, the partitioning of the semi-volatile compounds is displaced toward the particulate phase, since there is more organic phase volume available for SVOC absorption. It is possible to parameterize this relationship using only two surrogate species in order to obtain quantitative relationships for the SOA yield as a function of the particulate organic mass. The yield increases with the particulate organic mass until it tends toward an asymptotic value, usually for high particulate mass concentrations. One should note that these high concentrations are typically not representative of atmospheric conditions and that yields can be significantly lower in the ambient atmosphere than those obtained from smog chamber experiments. Nevertheless, this type of parameterization, generally referred to as two-compound Odum parameterization, allows one to represent SOA formation from gaseous precursors taking into account kinetic aspects, which depend on the first oxidation step of the VOC studied, and thermodynamic aspects, which depend on the partitioning estimated from the statistical regression performed on the experimental data (Odum et al., 1996). The four parameters obtained from the statistical regression are the stoichiometric coefficients and the partitioning coefficients of the two SVOC. The SOA yield, Y, resulting from the oxidation of one (or several) VOC is defined as the ratio of the particulate organic mass formed, ΔM_o in g m⁻³, and the VOC mass that has reacted, Δ[VOC] in g m⁻³:

Y=ΔMoΔ[VOC](9.27)

One assumes that the particulate organic mass is composed of a limited number (typically two) of SVOC and that their partitioning between the gas phase and the particulate phase is governed by Raoult’s law, which is expressed as follows for SVOC_i (Pankow, 1994):

Kom,i=R TMMomγiPs,i(9.28)

where K_om,i is the gas/particle partitioning coefficient in m³ g⁻¹, R is the ideal gas law constant in m³ atm mol⁻¹ K⁻¹, T is the temperature in K, MM_om is the molar mass of the particulate organic phase in g mol⁻¹, γ_i is the activity coefficient of SVOC_i in the particulate phase, and P_s,i is the saturation vapor pressure of SVOC_i in atm.

The concentration of SVOC_i formed is (in g m⁻³):

where α_i is the stoichiometric coefficient of the reaction leading to the oxidation of VOC into SVOC_i. Since SVOC_i partitions between the gas and particulate phases according to Raoult’s law, given a particulate organic mass M_o, the amount of SVOC_i can be written in terms of an amount present in the gas phase and an amount present in the particulate phase:

where [SVOC_i,g] is the concentration of SVOC_i in the gas phase in g m⁻³ and [SVOC_i,om] is the concentration of SVOC_i present in the particulate phase in g m⁻³. The partitioning between these two phases is quantified using a partitioning constant that is the ratio of the concentrations of SVOC_i in the particulate phase (i.e., expressed in g of SVOC_i per g of M_o) and the concentration of SVOC_i in the gas phase. Therefore, this partitioning constant, K_om,i, is expressed in m³ of air per g of particulate organic mass:

Kom,i=[SVOCi,om] Mo [SVOCi,g](9.31)

Thus:

[SVOCi]=αi Δ[VOC]=[SVOCi,om] +[SVOCi,om] Mo Kom,i(9.32)

[SVOCi,om]=αi Mo Kom,iΔ[VOC](1+MoKom,i)(9.33)

In the case where there are several SVOC_i, the particulate organic mass formed is as follows:

ΔMo=∑i[SVOCi,om]=Δ[VOC]∑iαi Mo Kom,i(1+Mo Kom,i)  (9.34)

The yield may then be expressed in terms of the stoichiometric coefficients of formation of the SVOC_i, α_i, their partitioning coefficients, K_om,i, and the particulate organic mass formed, M_o:

Y=ΔMoΔ[VOC]=∑i(αi Mo Kom,i(1+Mo Kom,i))(9.35)

If there is no particulate organic mass present at the start of the experiment, ΔM_o = M_o. Y and M_o are measured at different times of the experiment and the coefficients α_i and K_om,I are calculated by regression. Therefore, in the case of two SVOC_i, four coefficients must be estimated and, therefore, measurements must be made at four distinct times at the minimum (a greater number of measurement sets provides a regression that is statistically more robust).

One must note that most of those experiments have been conducted at fairly low relative humidity. Therefore, only absorption into an organic phase was taken into account and absorption into an aqueous phase was not considered. For some compounds such as those obtained from isoprene oxidation, aqueous absorption (dissolution) prevails over the hydrophobic absorption into an organic phase in the ambient atmosphere. Then, this type of parameterization would underestimate the yields in the ambient atmosphere under humid conditions (Couvidat and Seigneur, 2011).

Modeling of SOA formation from experimental data obtained in smog chambers can be done with other approaches than the two-compound Odum approach. Two main approaches are currently used: (1) the VBS approach (Donahue et al., 2006) and (2) the surrogate molecule approach (Pun et al., 2006). The VBS approach involves using a fixed number of organic compounds with predefined saturation vapor pressures (or concentrations): it is the volatility basis set (VBS). The regression is then performed with these compounds for a given experimental data set to obtain the stoichiometric coefficients (their partitioning coefficients are predefined based on their saturation vapor concentrations; see Equation 9.28). The surrogate molecule approach involves identifying the main compounds present in SOA and to use a limited number of these products to represent the whole SOA. The partitioning coefficients are estimated using theoretical or semi-empirical methods (functional group methods) and the stoichiometric coefficients are obtained from smog chamber data. One advantage of the surrogate molecule approach is that it can treat both hydrophobic SVOC partitioning into an organic phase and hydrophilic SVOC partitioning into an aqueous phase.

The addition of oxygen atoms tends to increase the water solubility of the compounds. Both methods have been modified to account for additional functionalization of the organic molecules following successive oxidation steps. The VBS method uses a second dimension (the first dimension being the saturation vapor concentrations), which corresponds to the O:C ratio (Donahue et al., 2011). The surrogate molecule approach adds additional oxidation steps and surrogate molecules that are more oxidized than those of the first oxidation step (Couvidat et al., 2012). Other methods have been developed to parameterize the evolution of SOA as a function of their oxidation state and volatility; they have been summarized by Seinfeld and Pandis (2016).

9.7.4 Numerical Modeling

Numerical modeling of atmospheric particles consists of two components: (1) the solution of the general dynamic equation and (2) the solution of the chemical composition of particles.

The general dynamic equation is an integro-differential equation. The term representing the condensation and evaporation processes is hyperbolic. Therefore, the numerical diffusion problem found for the advection term of the atmospheric diffusion equation (see Chapter 6) is also present here. Therefore, the numerical algorithms presented in Chapter 6 to address this problem can also be used to solve the condensation/evaporation term of the particle dynamic equation. The other two terms, i.e., nucleation and coagulation do not present any particular numerical difficulty once the size distribution has been defined. Zhang et al. (1999) conducted a comparative evaluation of several numerical algorithms for the various terms of the general dynamic equation. The moving diameter method of Jacobson and Turco (1995) seems to be the most robust (Zhang et al., 1999; Devilliers et al., 2013) and is, therefore, recommended for the simulation of the particle size distribution with a sectional representation.

The solution of the chemical composition of particles requires treating the thermodynamic equilibria among the various chemical species present in the particles (in both the liquid and solid phases) and those between the particle phase and the gas phase. In addition, it may be necessary to treat the mass transfer between the gas phase and the particles explicitly, i.e., by accounting for the kinetics of this mass transfer. For fine particles, this mass transfer is sufficiently fast that the assumption of equilibrium between the particles and the gas phase is appropriate. On the other hand, the mass transfer kinetics slows down for coarse particles (see Chapter 11) and mass transfer must be taken into account for coarse particles. It is possible to separate fine and coarse particles in terms of their numerical treatment in order to assume equilibrium for fine particles and explicitly treat the mass transfer kinetics for coarse particles (Capaldo et al., 2000). The numerical solution of the mass transfer equation (see Chapter 11) does not present any particular difficulty, but it may affect the numerical stability of the solution of the equations treating the chemical composition of particles.

The solution of the equations that govern the chemical composition of particles can be obtained according to the second law of thermodynamics by minimizing the Gibbs energy of the chemical system (assuming that pressure and temperature are locally constant). However, this system of equations may be large and difficult to minimize. In particular, it is essential to find the global minimum of the system and not a local minimum, which requires using appropriate numerical algorithms. Therefore, this approach may be computationally demanding and other approaches have been developed that are less accurate, but computationally more efficient. Such approaches, which were mentioned in Section 9.7.2, are based on a system of equations based on thermodynamic equilibrium constants to obtain the particulate-phase concentrations of a reduced set of chemical species. The electroneutrality of the particulate phase and mass conservation between the gas phase and the particulate phase must of course be verified. Zhang et al. (2000) compared different numerical models available to calculate the chemical composition of inorganic particles. Currently, the ISORROPIA model (Fountoukis and Nenes, 2007), which uses simplifying assumptions based on various chemical regimes, is widely used in air quality models. Mass transfer between the gas and particulate phases of several chemical species may lead to numerical oscillations arising from opposite modifications of some chemical concentrations affected by (1) mass transfer and (2) chemical equilibria. The PNG-EQUISOLV II model (Jacobson, 2005b) avoids those numerical oscillations, while maintaining long integration time steps. For the chemical equilibria calculation, PNG-EQUISOLV II is based on the solution of chemical equilibria (rather than a minimization of the Gibbs energy), but it uses fewer simplifying assumptions than ISORROPIA II.

Solving for the chemical composition of the organic fraction of particles follows the same approach as used for inorganic species. However, it may be more complicated if one takes into account the possible separation of several organic phases (for example, distinguishing between hydrophobic and hydrophilic organic species) and/or the viscosity of particles, which may affect the diffusion of chemical species within the particles. Couvidat and Sartelet (2015) have developed a model, which treats these processes; other similar models are referenced in their article.

The joint solution of the particle size distribution and chemical composition is a difficult problem, unless one assumes that all particles of a given size (or size range) have the same chemical composition (internal mixing assumption). If particles of a given size have different chemical compositions, particles are then considered to be externally mixed. Such external mixing has been observed in the atmosphere, in particular near emission sources where freshly emitted particles are present near aged particles (Hughes et al., 2000). A bi-dimensional discretization (for both particle size and chemical composition) can be used to solve this problem (Zhu et al., 2015).