(5.111)
The solution to CO must be obtained iteratively. This is known as the volatility basis set (VBS) approach (Donahue et al., 2006).
Solid particles
Equilibrium between the gas phase and solid particles is in general poorly understood, with the important exception of H2SO4–HNO3–NH3 aerosol for which well-established bulk thermodynamics apply (Martin, 2000). For example, in the simple case of dry ammonium nitrate (NH4NO3) aerosol, equilibrium can be expressed by a temperature-dependent equilibrium constant for condensation:
KNH4NO3 = pNH3pHNO3
(5.112)
If NH3 and HNO3 concentrations are sufficiently low that the product pNH3 pHNO3 is less than KNH4NO3, then no NH4NO3 aerosol forms. If pNH3 pHNO3 is greater than KNH4NO3, then NH3 and HNO3 are in excess and condense to produce NH4NO3 aerosol. This condensation decreases pNH3 and pHNO3 until equilibrium (5.112) is met, at which point the aerosol is in equilibrium with the gas phase. In this manner, knowledge of KNH4NO3 constrains how much NH4NO3 aerosol will form given initial concentrations of NH3 and HNO3.
Different equilibrium formulations apply in the case of a gas adsorbing heterogeneously onto a solid aerosol surface, for example HNO3 adsorbing on dry dust. One can then describe the uptake in terms of a number of available condensation sites on the aerosol surface, with kinetic expressions to compute adsorption/desorption rates at these sites. A simple model for monolayer uptake is the Langmuir isotherm:
(5.113)
where θ is the fraction of occupied condensation sites, and the equilibrium constant K′ is the ratio of the adsorption and desorption rate constants.
5.5.2 Mass Transfer Limitations
Achievement of gas–particle thermodynamic equilibrium as described in Section 5.5.1 may be limited by mass transfer in the gas and particles phases. The uptake rate of a gas-phase species by an aerosol can be expressed in a general form by
(5.114)
where ng is the number density in the bulk gas phase [molecules per cm3 of air], equivalent to n(g) in Section 5.5.1, A is the aerosol surface area concentration [cm2 per cm3 of air], and kT is a mass transfer rate coefficient [cm s–1] that depends on the thermal velocity of molecules, the probability that collision will result in uptake, and any limitations from molecular diffusion. The rate of volatilization from the aerosol surface to the gas phase is proportional to the surface concentration ns in the aerosol phase
(5.115)
where K is the thermodynamic equilibrium constant between the gas and particle phases, with units for ns and K chosen so that Kns has units of [molecules per cm3 of air]. The mass transfer rate constant kT is the same for uptake and for volatilization so that steady state [dng/dt]in + [dng/dt]out = 0 yields ns = K ng. The net rate of transfer at the interface is
(5.116)
and vanishes to zero at equilibrium. The form of kT depends on the Knudsen number Kn = λ/a, where a is the aerosol particle radius and λ is the mean free path for molecules in the gas phase. The mean free path for air is λ = 0.068 μm at standard conditions of temperature and pressure (STP: 273 K, 1 atm) and varies inversely with pressure. In the limit Kn ≫ 1, the gas-phase concentration in the immediate vicinity of the particle is the same as in the bulk, since the particle does not interfere with the random motion of the gas molecules. This is called the free molecular regime. From the kinetic theory of gases, we then have
(5.117)
where v = (8kT/πm)1/2 is the mean thermal velocity of molecules, α is the mass accommodation coefficient representing the probability that a gas molecule impacting the surface is absorbed in the bulk, k is the Boltzmann constant, and m is the molecular mass. α generally increases with the solubility of the gas and decreases with increasing temperature. It can approach unity for a highly soluble gas at low temperature, but may be several orders of magnitude lower for a gas of low solubility.
The mass transfer limitation takes on a different form in the limit Kn ≪ 1. Under those conditions, gas molecules in the immediate vicinity of the surface undergo a large number of collisions with the surface before being able to escape the influence of the surface and migrate to the bulk. Thus, the gas concentration at the interface is controlled by local equilibrium with the aerosol, and transport between the surface and the bulk gas phase is controlled by molecular diffusion. This is called the continuum regime. A steady-state concentration gradient is established between the particle surface and the bulk gas phase, in contrast to the free molecular regime where there is no such gradient (Figure 5.14). Assuming a spherical aerosol, the gas-phase diffusion equation in spherical coordinates is
(5.118)
where Dg is the molecular diffusion coefficient and r is the distance from the center of the particle. Solving for the flux F at the gas–particle interface, we obtain
(5.119)
where ng(∞) is the bulk gas-phase concentration far away from the surface (ng in (5.116)). Thus, in the continuum regime,
(5.120)
Mass transfer in the continuum regime is not dependent on the mass accommodation coefficient α; this is because gas molecules trapped in the immediate vicinity of the surface collide many times with the surface and thus eventually become incorporated in the bulk aerosol phase even if α is low. By contrast, mass transfer in the free molecular regime is not dependent on the particle radius a, because the particles are too small to affect the motion of molecules.
Figure 5.14 Gas-phase concentration gradient in the vicinity of an aerosol particle of radius a.
The free molecular regime generally applies to stratospheric aerosols where a ~ 0.1 μm and λ ~ 1 μm. The continuum regime generally applies to tropospheric clouds where a ~ 10 μm and λ ~ 0.1 μm. Tropospheric aerosols are often in the transition regime since a ~ λ ~ 0.1–1 μm. Exact solution of mass transfer for the transition regime is complicated. Schwartz (1986) showed that the solution can be approximated to within 10% by harmonic addition of the mass transfer rate coefficients for the free molecular and continuum regimes as two conductances operating in series:
(5.121)
Equation (5.121) can be applied in the general case to calculations of gas–aerosol mass transfer. Unless in the free molecular regime, one should integrate over the aerosol size distribution in order to resolve the dependence of kT on the particle radius a.
The volatilization component of the gas–aerosol transfer flux was expressed in (5.115) as a function of the surface concentration ns. This is not in general a known quantity and we would like to relate it to the bulk aerosol phase concentration, which is more easily measured or modeled. The mixing timescale for a particle of radius a is given by τmix = a2/π2Da, where Da is the molecular diffusion coefficient in the aerosol phase. For a liquid aqueous phase Da ~ 10–4 cm2 s–1 and thus for a particle with a ~ 1 μm we have τmix ~ 10–5 s. This is in general sufficiently short to ensure complete mixing of the aerosol phase so that the surface concentration equals the bulk concentration. There are a few exceptions where the diffusion equation needs to be solved in the aerosol phase (Jacob, 2000).
5.5.3 Reactive Uptake Probability
Detailed treatment of heterogeneous chemistry in atmospheric models requires solution of the chemical evolution equations in the aerosol phase coupled to the gas phase through mass transfer. A simplified treatment is possible when the heterogeneous chemistry of interest can be reduced to a first-order chemical loss in the aerosol phase for a species transferred from the gas phase. Since the mass transfer processes are themselves first-order, the combined process can be encapsulated in a first-order loss equation. For this purpose, we define the reactive uptake coefficient γ as the probability that a molecule impacting the particle surface will undergo irreversible reaction rather than volatilization back to the gas phase. The rate of loss for the species from the gas phase can then be represented by (5.121) but with γ replacing α in the formulation of kT:
(5.122)
The reactive uptake coefficient γ combines the processes of inter
facial equilibrium, aerosol-phase diffusion, and reaction (Figure 5.15). It is a particularly helpful formulation because results from laboratory experiments can often be reported as γ. One can also relate γ to the actual chemical rate coefficient for loss in the aerosol phase. If the loss is a surface reaction, then γ simply compounds the mass accommodation α by the probability that the molecule will react on the surface [rate coefficient kS in unit of s–1] versus desorb [rate coefficient kD in unit of s–1]. Thus,
(5.123)
If the loss is a first-order reaction taking place in the bulk of a liquid aerosol phase [effective rate coefficient kC in unit of s–1], then the effect of diffusion in the aerosol phase needs to be considered. Solution of the diffusion equation for a spherical particle with a zero-flux boundary condition at the particle center yields
(5.124)
where
(5.125)
and q = a(kC/Da)1/2 is a dimensionless number called the diffuso-reactive parameter (Schwartz and Freiberg, 1981). f(q) represents a sphericity correction for the limitation of uptake by diffusion in the aerosol phase and is a monotonously increasing function of q. Limits are f(q) → q/3 for q → 0 and f(q) → 1 for q → ∞.
Figure 5.15 The reactive uptake probability γ convolves processes of gas–aerosol interfacial equilibrium, aerosol-phase diffusion, and reaction.
It is important to recognize the sphericity correction when applying γ values measured in the laboratory to atmospheric aerosols, because the geometry used in the laboratory is different from that in the atmosphere. The laboratory measurements are often for a bulk liquid phase with planar surface (a → ∞ ⇒ q → ∞), so that the effective γ for an aerosol will be lower than the laboratory-reported value. Physically, this is because the small size of the aerosol particles does not allow diffusion of the dissolved gas into an infinite bulk but instead forces re-volatilization to the gas phase. A proper treatment requires that one relate γ measured in the laboratory to kC, which is the fundamental variable, and then apply (5.124) (with integration over the aerosol size distribution) for the actual atmospheric aerosol conditions. This is generally ignored in atmospheric models under the assumption that uncertainties in kC trump other uncertainties, so that only order-of-magnitude estimates of γ are possible in any case.
5.6 Aerosol Microphysics
The size distribution of aerosol particles evolves continuously in the atmosphere as a result of microphysical processes including particle nucleation, gas condensation, coagulation, activation to cloud droplets, and sedimentation. These processes are computationally challenging to represent in models because of the wide ranges of particle sizes, compositions, and morphologies that need to be resolved (Chapter 3). Processes of nucleation and aerosol–cloud interactions are also highly nonlinear, so that averaging in models can cause large errors.
Because of these difficulties, a common practice in chemical transport models is to simulate only the total mass concentrations of the different aerosol components (sulfate, nitrate, organic carbon, black carbon, dust, sea salt, etc.), integrating over all sizes or across fixed size ranges with no transfer between ranges. As pointed out in Section 4.3, the continuity equations for the aerosol components are then of the same form as for gases. The models must still assume a size distribution for the different aerosol components in order to compute radiative effects, heterogeneous chemical rates, and deposition rates. A log-normal size distribution is often assumed for the dry number size distribution function nN (Chapter 3):
(5.126)
Here Dp is the dry particle diameter, Dm is the median dry diameter, and σg is the geometric standard deviation characterizing the variance in log(Dp/Dm). Different values of Dm and σg are usually adopted for different chemical components of the aerosol on the basis of observations. Aerosol surface area and mass size distributions are deduced from the moments of the number size distribution function as described in Chapter 3. The aerosol size distribution is expressed in terms of dry sizes because aerosol water is highly fluctuating as a function of the local relative humidity. The contribution from aerosol water is derived by applying component-specific, multiplicative hygroscopic growth factors to the size distributions as a function of local relative humidity. See Martin et al. (2003) for an example of such an approach.
Assumption of fixed aerosol size distributions can be a large source of model error, for example in describing precipitation scavenging which is a strong function of particle size. Accurate computation of the time-evolving size distribution is also critical for addressing aerosol radiative effects, aerosol–cloud interactions, and air quality impacts. This requires solving the continuity equation for aerosols (Section 4.3) with terms describing the different aerosol microphysical processes forcing changes in the local size distribution. Here we present standard rate expressions and model formulations for this purpose.
Figure 5.16 is a schematic representation of the different processes included in models of the aerosol size distribution. It shows the multimodal distribution of the aerosol as previously described in Chapter 3 with a nucleation mode (particle diameter less than 0.01 μm), Aitken nuclei mode (0.01–0.1 μm), accumulation mode (0.1–1 μm), and coarse mode (larger than 1 μm). The nucleation and Aitken nuclei modes dominate the number density but represent a very small fraction of the mass density. They are formed by nucleation, grow by gas condensation, and are lost by coagulation. The accumulation mode is produced from the smaller modes by coagulation and gas-condensation. Growth of particles beyond 1 μm is slow because gas condensation adds little mass and Brownian diffusion decreases, slowing down coagulation. This results in “accumulation” in the 0.1–1 μm range. The accumulation mode generally dominates the total aerosol surface area and makes a major contribution to total aerosol mass. Accumulation mode particles are also efficient cloud condensation nuclei (CCN) and drive formation of cloud droplets under supersaturated conditions. Particles in the coarse mode tend to be directly emitted to the atmosphere as dust or sea salt, and are removed relatively rapidly by precipitation scavenging and sedimentation.
Figure 5.16 Schematic representation of the microphysical processes that determine the evolution of aerosol particles from their formation through nucleation to their activation and conversion into cloud droplets. Particle diameter is in units of μm. Major aerosol modes are highlighted, including (in order of increasing size) the nucleation, Aitken, accumulation, and coarse modes.
Reproduced from Heintzenberg et al. (2003).
In the following discussion, we express the particle number distribution nN as a function of the particle volume (V) and omit the subscript N in the notation. The size distribution is thus noted n(V). We saw in Section 4.3 how the continuity equation could be applied to model the evolution of the aerosol size distribution in response to microphysical processes. We also saw how the local evolution in response to these processes could be described by separating the contributing terms:
(5.127)
Here we give formulations for these individual terms and describe their practical computation in models. The formulations are taken from Seinfeld and Pandis (2006), to which the reader is referred for a detailed presentation of aerosol microphysical processes.
5.6.1 Formulation of Aerosol Processes
Nucleation
Formation of new particles in the atmosphere is driven by clustering of molecules from the gas phase. Cluster formation requires very large supersaturations and takes place in a highly localized manner (nucleation bursts) when such supersaturations are achieved (and then relaxed through the nucleation process). Achievement of large supersaturations requires gas mixtures with very low vapor pressure. Binary nucleation can take place in the atmosphere from H2SO4–H2O mixtures, as H2SO4 has very low pressures over H2SO4–H2O solutions at all relative humidities of atmospheric relevance. Ternary nucleation involves a third component gas, typically ammonia (H2SO4–NH3–H2O mixture), for which the H2SO4 vapor pressure is even lower. See Chapter 3 for H2SO4 production me
chanisms. The possible role of organic molecules in contributing to nucleation is a topic of current research.
The critical step in nucleation is the formation of thermodynamically stable clusters that then grow rapidly by subsequent gas condensation. Clustering of molecules must overcome a nucleation barrier that can be expressed thermodynamically by the surface tension of the growing clusters, or at a molecular level in terms of the internal energy of successive clusters. In the continuity equation for aerosols, nucleation behaves as a flux boundary condition populating the bottom of the size distribution (the nucleation mode). If we assume that the size of particles produced by the nucleation process corresponds to a volume V0, and if J0 represents the nucleation rate, we write
(5.128)
See Seinfeld and Pandis (2006) for formulations of nucleation rates. The nucleation rate is a very strong function of the partial pressures of the nucleating gases, varying over orders of magnitude in response to relatively small changes in partial pressure. This nonlinear behavior is difficult to capture in models.
Condensation/evaporation
Gas condensation on existing particles causes these particles to grow. In Section 4.3 we expressed the corresponding term in the continuity equation for aerosols as
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