Constraints on the Role of Laplace Pressure in Multiphase Reactions and Viscosity of Organic Aerosols

Aerosol chemistry has broad relevance for climate and global public health. The role of interfacial phenomena in condensed‐phase aerosol reactions remains poorly understood. In this work, liquid drop formalisms are coupled with high‐pressure transition state theory to formulate an expression for predicting the size‐dependence of aerosol reaction rates and viscosity. Insights from high‐pressure synthesis studies suggest that accretion and cyclization reactions are accelerated in 3–10‐nm particles smaller than 10 nm. Reactions of peroxide, epoxide, furanoid, aldol, and carbonyl functional groups are accelerated by up to tenfold. Effective rate enhancements are ranked as: cycloadditions >> aldol reactions > epoxide reactions > Baeyer‐Villiger oxidation >> imidazole formation (which is inhibited). Some reactions are enabled by the elevated pressure in particles. Viscosity increases for organic liquids but decreases for viscous or solid particles. Results suggest that internal pressure is an important consideration in studies of the physics and chemical evolution of nanoparticles.

PETTERS 10.1029/2022GL098959 2 of 12 Though condensed-phase reactions have a substantial influence on atmospheric organic chemistry (Herrmann et al., 2015;Su et al., 2020), these reactions remain largely unexplored. The aerosol matrix is partly to blame. Aerosol phases can include, for example, liquids, phase-separated liquids (Rastak et al., 2017;Renbaum-Wolff et al., 2016), viscous semisolids ( S. S. Petters et al., 2019;Renbaum-Wolff et al., 2013;Virtanen et al., 2010), crystalline solids, and the surface phase (Enami et al., 2017;Eugene et al., 2018;Wei et al., 2018). Water is generally present in atmospheric particles and controls aerosol pH and heterogeneous reaction rates (Ervens et al., 2011;T. B. Nguyen et al., 2014;Pye et al., 2020). Aerosol viscosity is thought to play a critical role in determining nucleation and growth rates, but the processes involved are not yet fully understood . Careful application of experimental, observational, and modeling insights is needed to understand how nanoscale phase controls heterogeneous reactions.
The size-dependent aerosol surface-to-volume ratio and Laplace pressure become especially important at the nanoscale. M. D. Petters and Kasparoglu (2020) and Cheng et al. (2015) explore the size-dependence changes to aerosol viscosity induced by surface tension (Kaptay, 2012). Riva et al. (2021) explore the pressure dependence of imidazole formation in aerosols. The present work fills gaps in (1) the development of thermodynamic formalisms describing the pressure-induced diameter dependence of particle-phase (a) reaction rates, and (b) viscosity; and (2) evidence and application of interdisciplinary high-pressure sciences to atmospheric organic chemistry.

Theoretical Development
Surface tension raises the vapor pressure adjacent to a curved liquid surface relative to its Clausius-Clapeyron or Arrhenius temperature dependence; this effect is described by the Kelvin equation for pure liquids and by Köhler theory for mixtures, for example, (Bilde et al., 2015;Kreidenweis et al., 2005): where subscript denotes the ith component, is the vapor pressure outside the interface, is activity, is surface tension, is molar volume, is the universal gas constant, is temperature, and is diameter; this is the Köhler equation when = water. Laplace pressure, Δ , is the pressure inside a droplet resulting from intermolecular forces and the interfacial curvature: Δ = 2 ∕ + ∕ , where is the droplet radius. A volumeless surface interface is assumed and is defined by the radius-of-tension, , for which ∕ = 0 , yielding (Laaksonen et al., 1999;Pruppacher & Klett, 2010;Rowlinson & Widom, 1982): Laplace pressure keeps the condensate in mechanical equilibrium with the gas and is equivalent to 4 ∕ in Köhler theory. Figure 1 (panel a) estimates Laplace pressure in 3-500 nm particles using Equation 2 for surface tensions corresponding to highly concentrated aqueous salt slurries (0.080 N m −1 ; S. S. Petters & Petters, 2016;Su et al., 2020), water (0.072 N m −1 ), and pure organic liquid or concentrated aqueous surfactant (0.030 N m −1 ; Hritz et al., 2016;Korosi & Kovats, 1981;Petters &. Petters, 2016). For 3-10 nm particles, Δ is ∼100-10 MPa. For 50-100 nm particles, Δ is ∼6-1 MPa. Reduced atmospheric pressure aloft alters these values by a negligible fraction. Laplace pressure in a 3-nm particle is 400-1000× higher than that of the same material in bulk.
Is this a remarkable pressure? Figure 1 (panels b-d) compares particle internal pressures to pressures encountered in high-pressure chemistry, analytical chemistry, oceanography, and geology. Though aerosol internal pressure is too low to inactivate pathogens (Figure 1, panel c), capillary forces in the microscale could be important to bioaerosol viability (McRae et al., 2021). Pressures in particles 3-10 nm in diameter are close to those encountered in medium-to high-pressure synthetic organic chemistry, liquid chromatography, the ocean depths, and subsurface drilling.
How, then, does pressure affect reaction rates? The pressure dependence of a reaction rate constant, ( )∕ , is derived from the free energy change during compression (Evans & Polanyi, 1936;Van Eldik & Kelm, 1980): PETTERS 10.1029/2022GL098959 3 of 12 where = ∕ 0 is the rate enhancement factor, is the rate constant at high pressure, 0 is the rate constant at the reference pressure, Δ is the experimental gauge pressure, and Δ ‡ is the change in volume of the reaction transition state. The quantity Δ ‡ is parameterizes the pressure dependence of reaction rates in high-pressure synthesis. Reactions are accelerated under pressure if they decrease in volume during the transition state, Δ ‡ < 0 ; reactions with a positive Δ ‡ proceed more slowly (Klärner & Wurche, 2000). Cyclic transition states in concerted rearrangements tend to have lower Δ ‡ than acyclic intermediates in stepwise reactions (Van Eldik & Hubbard, 1997). Bond-breaking and neutralization reactions tend to have positive Δ ‡ , and bond-forming and ionogenic reactions tend to have negative Δ ‡ ; this is due to relaxation or constriction of the solvent cage (Isaacs, 1997). For this reason, the volume Δ ‡ is often split into intrinsic and the solvent-dependent molar volumes: Δ ‡ = Δ ‡ + Δ ‡ . Experimentally determined Δ ‡ ranges are compiled below.
The Laplace and Evans-Polanyi expressions are here combined. Setting Δ equal to Δ yields an expression describing the reaction rate change due to the aerosol Laplace pressure: Equation 4 is not an extension of Köhler theory (Equation 1); rather, the remarkable similarity between Equations 1 and 4 is due to their similar thermodynamics. Equation 1 is based on chemical equilibrium across phase boundaries and Equation 3, from which Equation 4 is derived, is based on chemical equilibrium between reactants and the transition state.
Viscosity is altered under high hydrostatic pressure, changing diffusion and vibration timescales and contributing to changes in observed reaction rates. The relationship between Δ ( ) and Δ becomes nonlinear as pressure becomes high (Chen et al., 2017). A rate constant expression accounting for viscosity is derived by Sumi (1991):  . House air = laboratory compressed air. LPG = liquefied petroleum gas. HPLC = high-performance liquid chromatography. UHPLC = ultrahigh-performance liquid chromatography. HPP = high-pressure Pasteurization of foods (Demazeau & Rivalain, 2011;Grove et al., 2006;Kovač et al., 2010). (d) Pressures encountered in oceanography, geomicrobiology, and petroleum geology; the lithostatic gradient is 22.7 MPa km −1 and the hydrostatic gradient is 10.5 MPa km −1 (Tiab & Donaldson, 2016 where is a proportionality constant. The isothermal pressure dependence of viscosity is predicted using the isothermal compressibility ( , ) and isobaric thermal expansion coefficient ( ) (Schmelzer et al., 2005): where is pressure, is a state variable describing the molecular-level order of the mixture (Schmelzer et al., 2011), and ( ∕ ) is the slope of ( ) at constant pressure. This last term, the temperature-dependent viscosity ( ) , is predicted using the Vogel-Fulcher-Tammann (VFT) equation: log 10 = + ∕ ( − 0) ; , and 0 are fitted (Fulcher, 1925). Incorporation of the VFT slope into Equation 6 yields the pressure dependence of viscosity: Equation 7 indicates that in the absence of changes in the microstructural order (including phase change), viscosity always increases with pressure. Equations 1-4 are used for liquid mixtures and Equations 5-7 become important at higher viscosities.

Discussion
Surface tension, the defining parameter in Equations 1-5, is uncertain for aerosols. Prior work on aerosol surface tension has largely been undertaken in the context of nucleation (Equation 1). Interfacial tension determines the energetic barrier to creating additional surface area and is therefore important not only in theoretical descriptions of new particle formation (Laaksonen et al., 1999), but also in CCN activation (S. S. Petters & Petters, 2016;Wex, Stratmann, Topping, & McFiggans, 2008) and the properties of cloud droplets (Dufour & Defay, 1963;Pruppacher & Klett, 2010). CCN activation theory and measurements have been used to understand surface tension at the nano-and microscale (Bzdek et al., 2020;Facchini et al., 1999;Forestieri et al., 2018;Nozière et al., 2014;Ovadnevaite et al., 2017;S. S. Petters & Petters, 2016;Prisle et al., 2008;Ruehl et al., 2016;Wex, Stratmann, Topping, & McFiggans, 2008). The growing consensus suggests that the surface tension is not inherently size-dependent, but that the activity term ( ) must be accurately known to estimate the surface tension of nanoparticles smaller than ∼100 nm from CCN activation data (Malila & Prisle, 2018;McGraw & Wang, 2021;M. D. Petters & Kreidenweis, 2013;Prisle et al., 2008). The implication of these studies is that nanoparticle surface tension depends in part on the phase state of the particle bulk.
Direct measurement of particle surface tension at the nanoscale has not been possible. However, simulations and microscale measurements can shed light on the surface tension at the nanoscale. Molecular dynamics simulations of liquid water in 1-2-nm nanopores indicate that the Young-Laplace equations are accurate at that scale, though tension was reduced (0.072-0.062 N m −1 ) (Liu & Cao, 2016). Molecular dynamics simulations of atmospherically relevant organic surfactants in 5-nm droplets are in good agreement with bulk measurements (Hede et al., 2011). Bzdek et al. (2020) used aerosol optical tweezers to confirm experimentally that the surface tension of 10-μm droplets containing strong surfactants is close to that of bulk measurements. Although direct evidence is limited, it is expected that the Laplace pressure predicted by Equations 1 and 2 is valid in particles larger than ∼1 nm in diameter.  (Kristensen et al., 2021;Nakao et al., 2013) and a molar mass of ∼250 g mol −1 (Hodzic et al., 2010), every 10 cm 3 mol −1 of Δ ‡ corresponds to 5.6%-6.6% of the molar volume. Yamashita et al. (2014) report high-pressure yields for lactone polymerization; the apparent rate constant after assuming a first-order reaction PETTERS 10.1029/2022GL098959 5 of 12 is used here with Equation 3 to produce an effective Δ ‡ of −7 cm 3 mol −1 . For ring-opening of propylene oxide, isobutylene oxide, and ethylene oxide, Koskikallio and Whalley (1959) report a Δ ‡ range of −9.2 to −6 cm 3 mol −1 . The Δ ‡ range for epoxide ring-opening of Isaacs is −20 to −15 cm 3 mol −1 , which is extended to −6 cm 3 mol −1 here to include the Yamashita and Koskikallio and Whalley results. The neutral aqueous Michael reaction of methyl vinyl ketone with nitromethane was conducted by Jenner (1999), who used the effective pseudo-second-order rate expression with Equation 3 to obtain the effective Δ ‡ range of −32 to −35 cm 3 mol −1 Jenner (2001) report a Δ ‡ range of −2 to −8 cm 3 mol −1 for a series of Bayer-Villiger reactions of peroxy acids with ketones, concluding that the reaction is concerted. The imidazole reaction of glyoxal with ammonium sulfate at different solution ionic strengths is described by Riva et al. (2021). Equation 3 is used here with an assumed pseudo-first-order rate of polymerization, quantified using light absorption, to calculate the effective Δ ‡ range of 2-27 cm 3 mol −1 . Due to the multi-step nature of many of the reactions presented (e.g., imidazole formation, Michael addition, ring-opening polymerization), Δ ‡ is an observed quantity variously referred to as effective, pseudo, or semi-empirical. Figure 2 (panel b) summarizes expected changes in reaction rates in particles of 3 and 10 nm diameters based on Equation 4. Color regions trace the uncertainty for different types of reactions from panel (a). For cycloadditions (blue region), the potential impact of particle internal pressure is the highest and results in the maximum rate change factor of 10 in 3-nm particles with high salt content ( ∼ 0.08 N m −1 ). The imidazole reaction (purple region) results in a rate change factor of as low as 0.3 in 3-nm particles with high salt content, for example, a 70% reduction in rate. For diameters larger than 3 nm, reactions with smaller absolute Δ ‡ values, or surface tensions lower than 0.08 N m −1 , rate change factors trend away from these extremes toward unity. For example, the magenta bar in Figure 2 (panel b) traces the uncertainty in ∕ 0 for condensed-phase radical propagation reactions ( ∕ 0 ∼ 1.1-2.4). Radical propagation in atmospheric chemistry is most commonly understood as the key step in oxidation (Ehn et al., 2014;Matsunaga & Ziemann, 2010;Ziemann & Atkinson, 2012). Similar propagation reactions of peroxy radicals also occur in condensed-phase organic matrices (Ingold, 1961) and in fog and cloud droplets (Lim et al., 2010). Aldol and epoxide reactions are not shaded due to their overlap with other  Figure 3 shows the trend in aerosol viscosity with increasing internal pressure based physicochemical properties. Panel (a) compares compressibility, , , with organic surface tension, following Richard and Rogers (1971). Data are obtained for organic liquids with the formula C x H y O z at 298 K from (Marcus, 1998;Yaws, 2003;taken from;Caleman et al., 2012). Panels (b) and (c) extend the linear surface tension dependence to and ∕ . Panel (d) shows the range in pressure-dependent viscosity predicted by Equation 7. Ranges are calculated from predicted , ∕ , values (panel c). Calculations are made for 2.5%, 5%, and 10% sucrose in water (regions i, j, and k, respectively) following the detailed model of Kasparaglu et al. (2021). This model is based on liquids far from phase transition, and ∕ is always positive. Analogous to water melting under pressure, atmospheric aerosols are predicted to become less viscous at ultrafine sizes (Panel E) (Cheng et al., 2015;M. Petters & Kasparoglu, 2020). Experimental ∕ measurements for viscous atmospherically-relevant aerosols demonstrate a wider range of values indicative of more complex mixtures (Table S1 in Supporting Information S1) (Marsh et al., 2018;S. S. Petters et al., 2019;Rothfuss & Petters, 2017).

Implications for Multiphase Atmospheric Chemistry
This work combines the Laplace and Evans-Polyani expressions for application to atmospheric chemistry. It is predicted that all particle-phase reactions are affected by particle internal pressure as a function of particle diameter ( < ∼10 nm), surface tension, effective Δ ‡ , and pressure-induced changes in aerosol phase state. The following explores the effects of Laplace pressure on known reaction systems. The estimates presented here can then serve to start the discussion of reactions of similar highly functionalized molecules in nucleating or evaporating nanoparticles.
Highly oxygenated molecules (HOMs) are a class of low molecular weight compounds that participate in both nucleation and subsequent particle growth (Ehn et al., 2014). HOMs (e.g., multifunctional peroxides) can serve as oxidants in condensed-phase Baeyer-Villiger reactions, in which hydroperoxides or peroxyacids oxidize aldehydes or ketones. Bayer-Villiger reactions have a slightly negative Δ ‡ and have been observed in chamber experiments (Claflin et al., 2018) and in the field (Pospisilova et al., 2020). Because HOMs are an important early participant in new particle growth, they are likely present in 3-10-nm particles (Bianchi et al., 2019). Thus, pressure likely provide moderate enhancement of peroxide depletion rates and higher carboxylic acids yields, similar to the observations of Pospisilova et al.
Furanoic compounds are an important group of five-membered aromatic heterocyclic gases emitted during biomass burning (Hartikainen et al., 2018). New particle formation is observed in biomass burning plumes during photochemical oxidation (Hennigan et al., 2012). The uptake of furans by aerosols and hydrometeors can result in condensed-phase reactions adding to particulate mass (Liang et al., 2018;Tomaz et al., 2018). Diels-Alder cycloadditions of furans with alkenes have enhanced yields under high pressure, due in part to the prevention of thermally-accelerated reverse reactions (Dauben & Krabbenhoft, 1976;Isaacs, 1997;Oppolzer, 1991). Furanoic compounds present during the growth of these new particles can partition into the condensed phase and experience elevated pressures. Note that in-flame nucleation and soot formation results in particles greater than ∼10 nm (Johansson et al., 2018); nanosphere soot and tarballs have diameters of ∼50 and ∼250 nm, respectively (Adachi et al., 2019). Though compaction of soot occurs by capillary forces (Bhandari et al., 2019;D. Hu et al., 2021;Leung et al., 2017;Ma et al., 2013), the pressure in these accumulation-mode particles is also outside the range for which condensed-phase reaction rates are altered ( Figure 1, panel a). Given that cycloadditions including the Diels-Alder reaction are the most dramatically accelerated by increased pressure, oligomerization of furanoid compounds during the nucleation and growth of new particles in sunlit biomass burning plumes is likely affected by particle internal pressure.
Carbonyls such as pyruvic acid and glyoxal are highly water-soluble and can dimerize (Loeffler et al., 2006;Perkins et al., 2016) or undergo multi-step oligomerization in the presence of ammonium (Maxut et al., 2015). Riva et al. (2021) show that high hydrostatic pressure reduces the oligomerization rate of glyoxal in the presence of ammonium sulfate. Pyruvic acid undergoes imidazole formation and this reaction rate is likely reduced in the nucleation-mode aerosol. Hazen et al. (2002) show that pyruvic acid decarboxylates under pressure to form acetic acid and carbon dioxide and undergoes aldol condensation to form dimers and trimers; a large fraction of these decarboxylate to form methyl succinate, and some undergo Diels-Alder cyclization. Pyruvic acid dimers are important in atmospheric chemistry because they are surface-active (B. P. Gordon et al., 2019;Hazen et al., 2002) and cyclize in droplets (Perkins et al., 2016;. Aldol reactions are a hypothesized route for reactive uptake of gases or modification of aerosol optical properties when catalyzed by inorganic salts (Nozière et al., 2010), and have been suggested for highly acidic particles and in evaporating droplets (De Haan et al., 2009;Herrmann et al., 2015;Jang et al., 2002).
Characterization of molecules and reaction mechanisms in aerosols is challenging and often informed by studies of bulk samples at standard temperature and pressure (Glasius & Goldstein, 2016). Higher pressures may enable the formation of different reaction products. For example, high pressure can prevent backwards reactions or enable otherwise-unfavorable ring-opening reactions. This work adjusts the existing assumptions of aerosol phase state, opening the door for future and retrospective studies seeking to match observations to reaction models.
Aerosol viscosity is predicted to change due to Laplace pressure. Organic solvents become more viscous under pressure (Cook et al., 1994;Herbst et al., 1993;Isaacs, 1997). In contrast, supercooled water becomes less viscous as the hydrogen bond network collapses (∼300 MPa) (Singh et al., 2017). Further, melting points of various organic and inorganic substances are depressed in sub-100-nm particles (M. D. Petters & Kasparoglu, 2020), and secondary organic aerosols below 20 nm are predicted to liquefy (Cheng et al., 2015).

Conclusions
This work combines thermodynamics from aerosol physics and organic physical chemistry to formulate expressions for the diameter-dependent reaction rates and viscosity in particles. Application of this equation is guided by a survey of synthetic organic chemistry and suggests that heterogeneous reaction rates are accelerated by up to tenfold in 3-10-nm particles. Elevated pressure is explored as a primary contributor to size-dependent reaction rates and mechanisms of HOMs and other peroxides, furans and furanoids, epoxides and other lactones, and carbonyls. Rate enhancements are ranked as follows: cycloadditions >> aldol reactions > epoxide ring-opening reactions > Baeyer-Villiger oxidation by peroxides >> imidazole oligomerization of carbonyls. High-pressure changes to aerosol viscosity depend on aerosol composition and temperature. Careful work is needed to separate PETTERS 10.1029/2022GL098959 8 of 12 the effects of particle internal pressure, viscosity, curvature, pH, phase separation, surface tension, and other factors on reactions within particles.