Secondary organic aerosols (SOAs) are ubiquitous in the atmosphere, comprising a major fraction of atmospheric aerosols . They have impacts on air quality, public health, and climate, and their representation in large-scale models remains a major challenge in atmospheric chemistry . SOA is generated through oxidation of volatile organic compounds and condensation of semi- and low-volatile products into preexisting particles. This involves a series of transport processes including gas-phase diffusion, gas–surface transfer via reversible adsorption, surface–bulk exchange, and bulk diffusion within the particle . These processes are often represented as very fast relative to the timescales of other atmospheric processes associated with SOA, so that the gas and particle are traditionally regarded as being in instantaneous equilibrium.

There has been mounting evidence that SOA particles can exist in amorphous semisolid or glassy states depending on chemical composition, relative humidity, and temperature . Global and regional modeling studies have shown that organic aerosols exist in an amorphous solid or glassy state , facilitating long-range transport of organic compounds in the atmosphere . The occurrence of a highly viscous state can lead to kinetic limitations of bulk diffusion within the particle to retard heterogeneous reactions and evaporation kinetics . Uptake coefficients of isoprene epoxydiols (IEPOXs) are observed to be reduced by organic coatings, affecting particle morphology and chemical evolution of SOA . Modeling work also demonstrates the impact of diffusion limitations in highly viscous aerosol on partitioning of semi-volatile compounds, thus affecting the evolution of particle size distribution .

Equilibration timescales can be prolonged in highly viscous particles under low relative humidities and temperatures . In fact, a recent field observation in Beijing demonstrated that partitioning of organic species was retarded to limit SOA mass when ambient particles adopt a semisolid state . Chamber experiments have observed limited particle–particle mixing of SOA, while some experimental studies indicate little kinetic limitations and fast mixing between miscible organic species . These contrasting results leave questions open about the conditions under which conditions and properties kinetically limited or equilibrium gas–particle partitioning can be expected in the atmosphere .

Atmospheric aerosol particles are generally complex mixtures of organics and inorganics and water . Such mixtures may exhibit non-ideal behavior possibly with phase separation when the organic components have low O : C ratio and often exist in core–shell structures . Aerosols composed of two organic populations may also phase-separate into two phases (usually a highly oxidized hydrophilic phase and a low O : C hydrophobic phase) . Very recently, even showed that the particles consisting of primary organic compounds, SOA, and inorganic compounds can phase-separate into three phases .

Many studies still assume a homogeneous well-mixed particle phase as well as ideal mixing between the particle and condensing gas-phase species. Because a dynamic consideration of gas–particle equilibrium would be significantly more computationally expensive especially in larger-scale models, it is important to understand under what conditions this consideration is most critical . Based on current experimental techniques, it is challenging to determine whether any limitations to SOA partitioning or mixing are due to diffusion limitations, small mass accommodation coefficients, or non-ideality and low miscibility between the populations studied. In this study, we investigate the combined effects of particle phase state and non-ideal mixing by modeling condensation of semi-volatile species into core–shell particles to evaluate equilibration timescales of SOA partitioning.

The kinetic multilayer model of gas–particle interactions in aerosols and clouds (KM-GAP) has been described in detail previously . A brief description and the specifics of the implementation for this work are addressed here. We assume a particle of a core–shell morphology, mimicking the shell phase with predominantly organics and the core phase as aqueous inorganics. KM-GAP consists of a gas phase, a sorption layer, a near-surface bulk layer (nsb), and a number of bulk layers. As shown in Fig. , we represent the shell phase with a near-surface bulk layer and four bulk layers, while the core is represented with one layer. Sensitivity studies were conducted with more layers (10 and 20 layers) to ensure that the simulation results were not sensitive to the number of layers chosen, but it only affects the resolution of bulk concentration profiles. KM-GAP explicitly treats the processes of gas-phase diffusion, adsorption and desorption, surface-to-bulk transport, and bulk diffusion within the shell and transport between the shell and core phases. A monodisperse particle population with an initial particle diameter of 200 nm and total particle concentration of 104 cm-3 is assumed, which yields a total organic aerosol mass of ∼40 µg m-3. The core and shell are composed of different non-volatile species each with a molar mass of 200 g mol-1. Initially, the total amount of mass in the core and shell are the same so that there will be an equal amount in each phase at equilibrium when the activity coefficients of the condensing species is 1 in both phases.

A gas-phase species is present with the same molar mass and density as the particle phase species but with pure compound saturation mass concentration of C0=10 µg m-3. Volatility or effective saturation mass concentration (C*) is expressed as C*=γshellC0, which controls the partitioning by driving the surface to bulk transport in the model . The initial gas-phase concentration of the condensing species is set to be small, 10-7 µg m-3, so that condensation of trace amounts of condensing species would not alter particle diameter and properties including phase state and non-ideality. It also ensures that the approximate volume of each layer is maintained even in circumstances with slow diffusion. We mainly consider a closed system, in which condensation of species would lead to a decrease in its gas-phase mass concentration and an increase in its particle phase mass concentration, while we also conduct additional simulations with an open system with a fixed gas-phase concentration. The surface accommodation coefficient is set to be 1 based on previous measurements and molecular dynamics simulations .

The bulk diffusivity (Db,shell) is varied in the range of 10-5 to 10-20 cm2 s-1 covering the range of particles being in a liquid, semisolid to amorphous solid, or glassy phase state . Db is fixed at any given depth in the particle bulk in each simulation, assuming that condensation would not alter particle viscosity and diffusivity, as only trace amounts of species condense to preexisting particles in these simulations. The activity coefficient of the condensing species in the shell is varied in the range of 10-3 to 105, covering and expanding the range of activity coefficients measured or modeled in between atmospherically relevant compounds . While many atmospherically relevant mixtures will form near-ideal solutions, with activity coefficients between 0.1 and 10, such as partitioning of similarly oxidized SOA components into an organic-rich phase , extremes of miscibility can be present as well. For example, the thermodynamic model AIOMFAC (Aerosol Inorganic-Organic Mixtures Functional groups Activity Coefficients; ) predicts an activity coefficient of as high as 108 for α-pinene oxidation products in an aqueous electrolyte-rich phase and ∼104 for a strongly hydrophobic species such as hexane in a solution of hydrophilic glyoxal and water . On the other end, AIOMFAC predicts an activity coefficient below 0.1 for small dicarboxylic acids in water–glyoxal solutions at high water activities, and similarly low activity coefficients have been experimentally determined for oxalic acid in acid–water solutions . The activity coefficient of the condensing species in the core is fixed to be 1 for all simulations. The e-folding equilibration timescale, τi, is calculated the first time (t) the following inequality is satisfied: 1|Ci(t)-Ci,eq||Ci,0-Ci,eq|<1e, where Ci is the mass concentration of the condensing species in the phase i (shell, core, or total particle) at e-folding equilibration time (τ), initial time (0), and equilibrium (eq).

We first show the simulation results of a few representative cases to illustrate how partitioning and mass transport of the condensing species in the semisolid shell of a phase-separated particle are affected by non-ideality. Then we discuss the overall combined effects of non-ideality and particle phase state on equilibration timescales of SOA partitioning.

The model simulations and analysis undoubtedly show the complex effects and interplay of non-ideality, particle phase state, and properties of the condensing species on equilibration timescales of SOA partitioning. Experimental techniques to investigate equilibration timescales rarely probe the mixing state of the aerosol population and can often only measure the total particle composition without information on particle morphology and potential concentration gradients within the particle. In this study we have demonstrated how concentration gradients can exist within the particle long after the particle as a whole has achieved equilibrium with the gas phase, particularly under circumstances where the shell is able to hold large amounts of the condensing species with low volatility or a low activity coefficient in the shell. This may have impacts on heterogeneous chemistry as under these circumstances much more of the condensing species is available at or near surface bulk than would be predicted assuming equilibrium partitioning.

This work can provide useful insights into some previous experimental work. observed that when toluene and α-pinene SOA are condensed sequentially on seed particles, the evaporation behavior is as if only SOA from the second precursor were present, consistent with the core–shell structure in particles . This then leads to minimal evaporation when the toluene SOA coats the α-pinene SOA, as very little evaporation of the toluene SOA was observed when it was the only constituent in the particle. If the activity coefficient of a condensing species in the shell phase (γshell) is low, then this condition can lead to long equilibration timescales in the whole semisolid particle, as transport to the core is slow. As suggested by , the coating of the second precursor may be caused by non-ideal mixing effects leading to phase separation or just very slow diffusivity and neither can be entirely ruled out. estimated an upper bound on the diffusivity in toluene SOA of D=10-17 cm2 s-1 based on an evaporation timescale longer than 14 h. Assuming an activity coefficient of a condensing species in toluene SOA of 1, in the simulations done here, the total particle equilibrium would be achieved in about 3 h (Fig. a), but the core equilibrated much more slowly on the order of about 1 d (Fig. b). However, a similarly long equilibration timescale of the core can be achieved using an activity coefficient of 10 in the shell and a diffusivity of D=10-18 cm2 s-1 (Fig. b). Even with a diffusivity of D=10-17 cm2 s-1, equilibrium in the core would not be achieved in 14 h with an activity coefficient of 10 in the shell (Fig. b).

Experimental work to probe timescales of particle–particle mixing has also been conducted by mixing two populations of particles. showed that below 20 % RH, toluene SOA does not mix appreciably but mixes readily with a deuterated toluene SOA population at higher RH. They also showed that even at low RH, α-pinene SOA mixes with D-toluene SOA within 1 h. extended this to include different SOA populations from isoprene, limonene, and β-caryophyllene. While the semi-volatile organic compound (SVOC) components of isoprene and α-pinene SOA mix rapidly in the presence of another SOA population, they showed that in some cases the properties of the other SOA population can inhibit this rapid mixing as in the case of toluene SOA mixing with limonene or β-caryophyllene SOA. observed mixing between erythritol-coated black carbon and sugar-coated ammonium sulfate. When using a small (and presumably less viscous) sugar under high RH and temperature conditions, the erythritol achieves a steady state in the sugar-coated ammonium sulfate population in minutes. When these conditions are changed to increase the viscosity of the particles (lower RH, lower temperature, and larger sugars), equilibrium is prolonged to a few hours. Here we discuss in detail a few cases where the results of this work may help explain some findings of . However, in this work we have only simulated one particle population, and thus our interpretation is limited. Future work will focus on simulating two particle populations to represent these results.

observed that when mixing toluene SOA and limonene SOA, the limonene SOA initially took up a substantial amount of the toluene SOA, leading to the limonene SOA containing 15 % toluene SOA almost immediately, but that was followed by much slower uptake for the next 3 h . This could be evidence of the overshooting behavior as seen in this work, where the equilibrium (or steady-state) concentration is achieved rapidly as the shell layers are able to accommodate a large amount of the condensing material due to the low volatility of the condensing species and/or non-ideal mixing with a low activity coefficient. If diffusivity within the particle is slow, the large concentration gradient in the particle will dissipate slowly leading to slow continuous uptake of the gas phase until the full equilibrium is achieved.

In addition, performed experiments to show that even at high RH, mass exchange between SOA populations derived from toluene and β-caryophyllene SOA remained very low, despite both SOA populations containing significant fractions of semi-volatile species. In addition, β-caryophyllene SOA has been shown to have low viscosity at high RH, without inhibiting the timescale for mixing . Therefore, this slow mixing could be explained by low miscibility between the two SOA populations. In this case, equilibrium may be achieved rapidly, but very little mass actually ends up mixing with the other population. As shown in Fig. c, if the activity coefficient in the shell is high, the shell will achieve equilibrium rapidly even if the shell is viscous. Therefore, there was likely an activity barrier to the populations mixing at high RH as observed in their experiments.

We have conducted simulations to address the complex effect of non-ideal mixing and particle phase state on equilibration timescales of semi-volatile species into a core–shell particle. This effect is investigated by varying the bulk diffusivity in the particle shell to cover different particle phase states from liquid and semisolid to amorphous solid and varying the activity coefficient of the condensing species in the shell. We found that the particle can achieve equilibrium with the gas phase long before the particle itself has reached full equilibrium without concentration gradients, especially when the gas-phase species is preferentially miscible with the shell. If the condensing species is immiscible with the shell phase, the shell rapidly reaches quasi-equilibrium or steady state with the gas phase, while the full equilibrium in the core, the shell as a whole, and the whole particle with the gas phase is achieved much more slowly. If the condensing species is favorably miscible with the shell, the total particle equilibrium will reflect the equilibrium between the shell and gas phase, but the dissolution of concentration gradients in the shell and the equilibration of the core will take much longer when the shell is highly viscous with low bulk diffusivity. We have also shown that comparing these simulations to experimental work may help to reconcile apparent discrepancies between experimental observations of fast particle–particle mixing and predictions of long mixing timescales in viscous particles based on viscosity measurements.

Here we have shown that non-ideality, viscosity, and volatility all impact equilibration timescales and that combining these effects can lead to similar equilibration timescales due to their complex interplay. For example, we would expect that a semisolid particle with very high viscosity would equilibrate with a gas phase slowly. In some circumstances, however, equilibrium can also be achieved rapidly for a condensing species with low C0 or an immiscible species with high γ. Experimental work to probe the equilibrium timescales of semi-volatile species has focused on mixing two particle populations. Thus, to compare these results to experimental results, we must consider both the parameters affecting transport through and evaporation from one population as well as condensation and subsequent transport through a second population of aerosol, which is planned for future work.

Large-scale models often assume both instantaneous equilibrium between a surrounding gas phase and aerosol particles, which are ideally well-mixed liquids independent of the physical properties of those systems. We show that these assumptions are justified in cases where the particle phase is liquid and non-ideal mixing has slight impacts on equilibration timescale between the gas phase and the particle. If the particle is viscous, condensation of miscible species in the shell can prevent mass transport to the core, prolonging equilibration timescales of full internal mixing and equilibrium of the particle with the gas phase. Hence, poor representation of particle phase state and internal mixing state can lead to overestimation of particle growth as well as heterogeneous and multiphase chemistry.

We have only considered monodispersed particles and a low gas-phase concentration of the condensing species so that the size of the particle remains constant throughout any simulations. Future work should consider the combined effect of phase state and non-ideality on poly-dispersed particles on the evolution of particle size distribution in atmospherically relevant concentrations and conditions.