theory and equations: supplementary material table 1...

Oceanic macromolecular surfactant estimates

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Theory and equations: supplementary material 1 2 Our demonstration of marine surfactant/organic variability requires the application of 3 macromolecular physical chemistry concepts at divergent levels. Some readers will 4 already be comfortable with the full range of logic, but others will be more familiar with 5 either its marine or atmospheric aspects. In order to unify the concepts, we present an 6 equation list here in appendix form. The exposition is structured as an ocean-up narrative, 7 describing organic behaviors for major water-air interface types and the nascent aerosol. 8 Chemical reference points and indicators cited in the text tables are completely defined. 9 In order of appearance the quantities under consideration are: 10 11 Table 1 (general surface activity): Half saturation concentration C1/2 (moles carbon per 12 liter) and the maximum excess or surface concentration Γmax (carbon atoms per square 13 angstrom with conversion to moles carbon per square meter as needed) 14 15 Table 2 (ocean surfactants): Normalized bulk concentration C/C1/2 (dimensionless) and 16 the closely related relative excess measure Γmax C/C1/2 (moles carbon per square meter) 17 18 Table 3 (ocean to atmosphere transition): two dimensional phase state analogs (solid 19 versus expanded films etc.) 20 21 Table 3 (atmospheric surfactants): Driving force for spray coverage 1 Molar/C1/2 22 (dimensionless), surface to aqueous carbon ratios (Γmax/1 Molar)(A/V)( also 23 dimensionless), the surface pressure Π which is also a tension reduction (mJ/m2) 24 25 Table 3 (atmospheric macromolecular chemistry): Parameterized hygroscopicity κ 26 (dimensionless), organic to carbon mass ratio (dimensionless) and individual compound 27 density (g/cc). 28 29 For practical purposes, we restrict our thinking to a thin but global band embracing the 30 oceanic upper mixed layer, the sea surface and the immediate vicinity of sea spray 31 droplets. Conservation of free energy is a useful starting point in all situations. But it 32 must be conceived in the multicomponent multiphase sense -between solvent water, high 33 molecular weight organics and air all taken in the context of bubbles, the global ocean 34 boundary and finally the nascent aerosol. We adapt derivations primarily from two 35 prominent text book series –Adamson (1960) to Adamson and Gast (1997) and Seinfeld 36 (1986) to Seinfeld and Pandis (2006) for surfactant and aerosol chemical issues 37 respectively. An historical view is attempted, with reference to seminal research in 38 several relevant fields. For example the classic physical chemical contributions of Gibbs 39 and Langmuir are acknowledged, but simultaneously we draw on early and rapidly 40 evolving environmental studies of the marine aerosol organics. 41 42 Pointers have been inserted at several appropriate locations in the main text, directing 43 readers to the equations and theory. A more complete development is available from the 44 COSIM group at Los Alamos (Climate Ocean Sea Ice Modeling) as a technical report in 45


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the LAUR-LAMS series. Global mapping exercises and initial extensions to full, 46 competitive Langmuir equilibria are currently under review in Burrows et al (2014). 47 48 Table 1 (surface activity): C1/2 and Γmax 49 50 To establish our first indicator quantities, we take the point of view of a marine 51 macromolecule or polymer, possibly surface active, generated by phytoplanktonic cell 52 disruption in upper layers of the global ocean then transported toward or into the 53 atmosphere (schematics in Blanchard 1963 and 1989, Russell et al 2010). Within the 54 water column, organic structures are of course subject to an intense reactive chemistry -55 enzymatic and photolytic degradation plus recombination (Benner, 2002; Hansell et al 56 2012). But in the upper few meters they may also be confronted with aerobic interfaces, 57 associated with bubble plumes injected from above (Hoffman and Duce, 1976; Cunliffe 58 et al 2011). The wave-generated bubble distribution peaks between 30 and 300 microns 59 (Woolf 1997). At such sizes curvature may be neglected (Clift et al 1978), which 60 significantly simplifies our task at the outset. 61 62 To construct a basic thermochemical model, imagine a generic system consisting of 63 biomacromolecules and sea salt as solutes, solvent water surrounding them, a subset of 64 organics residing at the interface and then an overlying gas phase. Given bulk liquid and 65 gaseous mixtures of i components, total moles and Gibbs free energy may be expressed 66 as simple sums. But the surfactant reservoir is also accounted explicitly, and 67 rearrangement emphasizes its centrality. 68 69 𝑛!! + 𝑛!

! + 𝑛!!"#$%&' = 𝑛!; 𝐺! + 𝐺! + 𝐺!"#$%&' = 𝐺 (1) 70

𝑛!!"#$%&' = 𝑛! − 𝑛!! + 𝑛!

! ; 𝐺!"#$%&' = 𝐺 − 𝐺! + 𝐺! (2) 71 72 A general incremental change in free energy is just 73 74 𝑑𝐺 = −𝑆𝑑𝑇 + 𝑉𝑑𝑝 + 𝜎𝑑𝐴 + 𝜇!! 𝑑𝑛! (3) 75 76 where σ is the surface tension in units, for example, of N/m or J/m2 and other symbols 77 have their usual meanings. This is simply a textbook dG given addition of the surface 78 energy contribution. Under local conditions in the marine aqueous system, temperature 79 may be considered constant. Furthermore the volume of the dividing plane is zero by 80 definition (Adamson and Gast 1997, Tuckermann 2007). Since all chemical potentials are 81 in balance at equilibrium, the expression for surface free energy change simplifies and is 82 readily integrated. 83 84 𝜇!!"#$%&' = 𝜇!! = 𝜇!

! (4) 85 𝑑𝐺!"#$%&' = 𝜎𝑑𝐴 + 𝜇!! 𝑑𝑛!

!"#$%&' (5) 86 𝐺!"#$%&' = 𝜎𝐴 + 𝜇!! 𝑛!

!"#$%&' (6) 87 88


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We seek to understand tension as a function of composition. Application of the product 89 rule yields the expansion in 7. Gibbs Duhem-type arguments and matching with 5 then 90 provide the key relationship. 91 92 𝑑𝐺!"#$!"# = 𝜎𝑑𝐴 + 𝐴𝑑𝜎 + 𝑛!

!"#$%&'! 𝑑µμ! + 𝜇!! 𝑑𝑛!

!"#$%&' (7) 93 𝐴𝑑𝜎 = − 𝑛!

!"#$%&'! 𝑑µμ! (8) 94

95 At this point we focus for purposes of clarity on a single surface active organic. The 96 summation is obviated and it is natural to convert moles into a two dimensional 97 concentration, Γ = n/A. Typically this quantity is referred to as an “excess” of the 98 compound, because it is not present in the aqueous phase. For any dilute solute, chemical 99 potential is related to the logarithm of concentration C. Various forms of the Gibbs 100 surface equation therefore obtain: 101 102 𝑑𝜎 = −Γ𝑑𝜇;𝑑𝜇 = 𝑅𝑇𝑑𝑙𝑛𝐶; 𝑑𝜎 𝑑𝐶 = − Γ𝑅𝑇 𝐶 (9) 103 104 Now suppose that in a weak solution linearity is in force for the above relationship. 105 Macromolecules necessarily reduce the interfacial tension. Surface pressure Π is 106 typically defined as the difference in σ manifested by pure and water-solute situations. 107 108 𝜎 = 𝜎∗ +−𝑐𝑜𝑛𝑠𝑡.𝐶; Π = 𝑐𝑜𝑛𝑠𝑡.C; 𝑑𝜎 𝑑𝐶 = −𝑐𝑜𝑛𝑠𝑡. ; Π = Γ𝑅𝑇 (10) 109 110 The final expression 10 is sometimes referred to as the “two dimensional ideal gas law” 111 since back-substitution of the excess yields ΠA = nRT. In principle, some combination of 112 9 with the simple “law” might be used to model the behavior of hypothetical surfactants 113 in ocean/aerosol systems. But in fact the idealized relationships often break down under 114 ambient conditions. They refer to point solutes/adsorbers lacking structure and molecular 115 interaction. Real systems exhibit repulsions whether in the bulk or a concentrating film. 116 Polymers experience denaturing and wrapping in solution (Lipitov and Sergeeva 1984, 117 Birdi, 1989). Hydrophobic groups may tilt and rotate on the surface. It has long been 118 recognized that protein and lipid series begin to deviate from ideality as early as 10-5 and 119 10-3 J/m2. (Schofield and Rideal 1925, Guastalla 1939, Ter Minassian-Saraga 1956). And 120 in the present work we do not go so far as to consider two dimensional mixtures. 121 122 The Gibbs equation couples neatly to results from Langmuir adsorption kinetics to yield a 123 fit to the tension curve in the region of mounting surfactant influence. Rates of site 124 occupation and subsequent desorption are given by ka C, where the proportionality 125 constant has units of per (molar time), and kd in the reverse direction which is first order. 126 Fractional coverage is then a function of the ratio between the two, which in turn may be 127 thought of as an adsorptive equilibrium constant. The interested reader can readily 128 demonstrate that its reciprocal is simply the concentration at half coverage C1/2. 129 130 𝐾 = 𝑘! 𝑘! ; 𝜃 = 𝐾𝐶 1+ 𝐾𝐶 ; 𝐶! ! = 𝐾!! (11) 131 132


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An assymptote for monolayer coverage Γmax may also be assumed, although even this is 133 not a given as polymers reconfigure, stack and react (Ter Minassian-Saraga 1956, 134 McGregor and Barnes 1978, Graham and Phillips 1979a through c). A particularly 135 compact mathematical form is then attained. Substitution and integration yield what is 136 often called the Langmuir-Szyszkowski equation (LS), after major contributors working 137 near the beginning of the last century. 138 139 Γ = Γ!"#𝜃 = Γ!"#𝐾𝐶 1+ 𝐾𝐶 (12) 140 𝑑𝜎 = − Γ𝑅𝑇𝑑𝐶 𝐶 ; 𝑑𝜎 = −Γ!"#𝐾𝑅𝑇 𝑑𝐶/ 1+ 𝐾𝐶 (13) 141 Π = Γ!"#𝑅𝑇𝑙𝑛 1+ 𝐾𝐶 (14) 142 143 Equilibrium constants C1/2 in text Table 1 were obtained by fitting laboratory surface 144 tension or pressure data to the Gibbs equation in forms 13 and 14, with one significant 145 exception. Stearic acid is sufficiently insoluble in seawater that a kinetic dissolution 146 approach had to be applied (Ter Minassian-Seraga 1956, Brzozowska et al 2012) . The 147 limiting 2D concentrations Γmax are measurable directly, by weighing out a particular 148 sample then generating the spread film on a flat water surface (Graham and Phillips 149 1979b). Values are often reported in mg/m2 but we convert to atomic or mole units for 150 convenience, first in visualizing the microscopic system then conducting comparisons 151 with the bulk. 152 153 Note that in a true Langmuir-kinetic situation, temporal dependencies are strongly 154 implied (Graham and Phillips 1979a, Lipitov and Sergeeva 1984, Adamson and Gast 155 1997). The rate constants ka will enfold differing polymeric diffusion rates, and 156 reciprocal kd are a set of desorption time scales depending on binding energies at diverse 157 sites. The potential for disequilibrium is real but is bypassed intentionally in our study. 158 We plan to address this point as it becomes possible and necessary to conduct simulations 159 at the molecular scale. 160 161 Table 2 (ocean surfactants): C/C1/2 and Γmax C/C1/2 162 163 If the distinctions “i” were maintained and propagated beyond 8 in order to represent 164 multiple adsorbing agents, our manipulations would reflect competition for the available 165 bubble surface area (Burrows et al 2014). Multicomponent coverage takes the chemically 166 resolved form 167 168 𝜃! = 𝐾!𝐶! 1+ 𝐾!! 𝐶! ; 𝜃!"!#$ = 𝜃!! (15) 169 170 The derivation once again involves the kinetics of approach to (and desorption from) a 171 water-air interface. It is given in many physical chemistry source books (e.g. Adamson, 172 1960, Laider 1965). In the context of the simultaneous equilibria of 15, an individual 173 ratio KjCj = C/C1/2 for the jth species provides a quick assessment of relative coverage. 174 This is our strategy as a preliminary analysis method, and the same arguments apply to 175 equation 12. Therefore the quantity ΓmaxC/C1/2 can be thought of as a mass weighted 176 coverage analog. The usual caveats with regard to nonlineary and nonideality must 177 naturally be kept in mind. The surfactant systems are in reality structurally complex and 178


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dynamic, entailing polymer conformation shifts, molecular interactions, chemical 179 reactions and more so that Gibbs-Langmuir is really only an approximation. 180 181 Table 3 (transition to atmosphere): The surfactant phase state 182 183 As the bubble field advances through the water column carrying long chain carbon 184 upward, a global ocean-atmosphere boundary is approached which is physically quite 185 heterogeneous (Liss 1975, Hardy 1982, Cunliffe et al 2011). Photochemistry, gels, foams, 186 organisms and even entire ecosystems are supported there. Organic pollutants accumulate 187 because they too tend to be surface active. Particles are continually deposited from the 188 atmosphere and contribute to the mix. But a true aqueous salt system is interspersed, and 189 this particular portion of the interface is entirely amenable to the current logic. We find 190 ourselves firmly in the realm of Σini

surface in equations 4 through 6. In the real ocean 191 vertical microchemical gradients must exist just below the atmosphere, and such factors 192 can ultimately be parameterized. For present purposes however, we make the zeroth order 193 assumption that two dimensional ocean-atmosphere chemistry resembles that of the 194 approaching bubbles. 195 196 As the excess Γ is increased during a given laboratory experiment, planar versions of 197 familiar “phase transitions” are documented. Order may accumulate as the adsorbing 198 entities align head to tail or effectively crystallize within a monolayer. Analogs of three 199 dimensional liquid and solid states appear as kinks in surface pressure versus area 200 experiments (Christodoulou and Rosano 1968, Liss 1975, Frew 1997, Donaldson and 201 Vaida 2006; Brozozowska et al 2012). Early work in the marine realm invoked the Van 202 der Waals equation of state as a framework for comprehending such phase phenomenon 203 in coastal waters (Barger and Means 1985). This is one of the ways in which oceanic 204 microlayer surfactants were first shown to be imperfect. In fact their behavior is often 205 nonsolid and expanded but not chaotic enough to be referred to as “gaseous” –the level of 206 disorder is intermediate on the Gibbs plane. Hence the classifications adopted in Table 3. 207 208 The film state is partially responsible for uncertainties in species-dependent sea-air gas 209 transfer. Multiple mechanisms are at play. Macromolecules introduce purely mechanical 210 barriers to the mobility of small molecules (Clift et al 1976, Leifer and Patro 2002). They 211 also alter viscoelastic properties of the laminar layer across which molecular diffusion 212 must take place (Frew et al 1990). In overlap with (and/or feedback onto) the 213 fundamental arguments here, surfactants distributed along the integrated bubble field 214 modulate dissolution and therefore the Henry’s Law augmentation to gas transport (Frew 215 1997). At high wind speed, this is the strongest atmospheric reintroduction channel for 216 insoluble compounds. The relative piston velocity effects of a lipid or marine biopolymer 217 class may well depend on solid versus expanded forms. The sequence of Table 3 entries 218 labeled “2D Phase State” indicates that geographic dependence is expected. All major 219 and trace dissolved gases of the sea are susceptible including those critical to the climate 220 system –carbon dioxide, nitrous oxide, methane, dimethyl sulfide and others (Frew 1997, 221 Leifer and Patro 2002, Tsai and Liu 2003). 222 223


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A given wind-generated bubble rises steadily back toward the atmosphere, and ultimately 224 it penetrates the overlying ocean-air boundary. There it acquires resident surface material, 225 in the sense of a target cross section. The process can be represented conceptually as the 226 multiplication of the several surfactant monolayers (Oppo et al 1999, Burrows et al 2014). 227 Domed structures form and stretch, leading to a familiar enrichment of marine 228 macromolecules, by many orders of magnitude relative to the total of dissolved organic 229 carbon in seawater (Blanchard 1963 and 1989, Hoffman and Duce 1976, Russell et al 230 2010, Burrows et al 2014). In what follows we sidestep issues such as foam delays, 231 drainage of the aqueous phase from the stressed film or estimation of its thickness 232 (Sellegri et al 2006, Modini et al 2013). This is accomplished by relying primarily on 233 empirical data. 234 235 Table 3 (atmospheric surfactants): 1 Molar/C1/2, (Γmax/1 Molar)(A/V), Πmax 236 237 Carbon enters the marine aerosol system in primary form via bubble breaking, with later 238 additions from secondary level processing through the gas phase. In the main text plus 239 supplemental equations 1 through 15, we attempt to capture the essence of surfactant 240 chemistry along the bubble driven channel. Shifting to the atmosphere, our focus 241 therefore falls naturally upon the primary organic source term. Direct injections become 242 nascent spray particles, and their chemistry remains distinct for an extended period 243 during boundary layer processing (O’Dowd and de Leeuw 2007, Westervelt et al 2012). 244 Meanwhile by many indications, the fractional mass contribution of biomolecules may be 245 comparable to that of sea salt (O’Dowd and de Leeuw 2007, Russell et al 2010, Lapina et 246 al 2011). 247 248 It is readily demonstrated under these circumstances that sea spray organics approach or 249 exceed unit carbon molarity overall. As such, we adopt this value in Table 3 and the main 250 text as a convenient reference concentration level. For example, one can apply a series of 251 traditional rules of thumb to nascent material. 252 253 𝑂:𝐶 ≈ 2;𝜌 ≈ 1.5𝑔 𝑐𝑐 ; 𝑟! ≈ 2𝑟!" ≈ 4𝑟!"# (16) 254 255 (Svenningson et al 2006, O’Dowd and de Leeuw 2007, Westervelt et al 2012). The first 256 entry on line 16 is an average organic to carbon mass relationship for partially oxidized 257 compounds (Turpin and Lim 2003, Russell 2003). Likewise the density provided is a 258 round figure often applied collectively to the remote biogenics (Meskhidze et al 2011). 259 Decreasing radii are defined working from the point of particle generation (ro) then 260 representing water vapor equilibrium at ambient relative humidity and finally the size of 261 the chemistry-determining dry particle core. For equivalent masses of dry organics and 262 sodium chloride diluted to RH 80, the value of C is 5 Molar. Our approximation leaves 263 ample room for reductions to the relative carbon mass or mole fraction. 264 265 Thus the simple half saturation reciprocal 1 (molar carbon)/C1/2 offers a convenient 266 reflection of surface coverage tendency for remote spray. Since KC >> unity in all the 267 Table 3 cases, marine surfactants are initially capable of occupying all positions at the 268 aerosol perimeter. Equations 11 and 15 are turned “inside out”, so to speak, since 269


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hydrophobic groups now populate the exterior of the microscopic object of interest. 270 Liquid dispersed from a rupturing film/dome into the atmosphere will be under severe 271 mechanical disturbance initially (Spiel 1997a and b, 1998). But fragments soon resolve 272 themselves into spheres and then both vapor and surfactant equilibria are reestablished. 273 Depletion of macromolecular carbon from the interior must of course be accounted 274 (Sorjamaa et al 2004, Petters and Kreidenweis 2013). In the language of our first few 275 supplementary equations ng = 0, n = nl + nsurface = constant. Complete atmospheric models 276 may ultimately deal with this form of mass conservation directly. But here we are content 277 to handle the issue in a zeroth order manner. 278 279 The upper limiting surface reservoir (moles) and round figure bulk reservoir (moles) 280 follow from multiplication through by droplet surface area and volume respectively. A bit 281 of algebra then yields -the ratio of two and three dimensional concentrations followed by 282 a surface to volume term, which brings us at last to spherical geometry. 283 284 𝑛!"#$%&' = Γ!"#𝐴; 𝑛! = 1 Molar V (17) 285 𝑛!"#$%&' 𝑛! = Γ!"#/1 𝑀𝑜𝑙𝑎𝑟 𝐴 𝑉 (18) 286 𝐴 = 4𝜋𝑟!;𝑉 = 4 3 𝜋𝑟!; 𝐴 𝑉 = 3 𝑟 (19) 287 288 The sea spray injection mode radius can be rounded to the half decade as ro = 0.3 microns 289 (equation 16, Gong 2003, O’Dowd and de Leeuw 2007), so that the quantity 3/r is about 290 107 per meter. One mole per liter of carbon is 1000 moles per cubic meter. Thus 291 nsurface/nlΓmax must be of order 104 m2/mole. The fractions in Table 3 arise from our 292 reliance on microscopically intuitive round values quoted for the excess. For example, 293 one atom per square angstrom is very close to 5/3 x 10-4 moles per square meter. But 294 since the final values for quantity 18 are all small, we can presume that areal saturation is 295 supportable. Barriers to mass transfer and alterations to heterogeneous atmospheric 296 photochemical reaction rates are implicit (Feingold and Chuang 2002, Donaldson and 297 Vaida 2006). Reductions to surface tension follow from the Gibbs equation (8 and 9). Per 298 the two dimensional gas law, they may be expressed as surface pressures (10). 299 300 To assess the effects of Πmax as cited in the main text and Table 3, it becomes necessary 301 to bring curvature into our development for the first time. The physical scale has been 302 shrinking steadily as our carbon moves upward. A swarm of submerged bubbles and the 303 planetary phase boundary may be considered flat collectively. But basic equations 1 304 through 3 must be revisited in light of global aerosolization. The spotlight now falls upon 305 water vapor equilibrium, which determines particle growth versus shrinkage in a rising 306 marine air mass (Pruppacher and Klett 1997, Seinfeld and Pandis 2006). Surfactant 307 terms are subsumed, in fact deleted, in order to focus on the solvent in its two major 308 physical states. The Gibbs phase plane is ignored, but constant temperature and pressure 309 can nonetheless be invoked to achieve standard free energy simplifications. We recognize 310 that the central reaction process has become a bulk or three dimensional phase transition, 311 from liquid to vapor so that -dnl = dng. As always equilibrium is defined by dG =0. The 312 chemical potentials are those of water itself. From 1 through 3 and the definition of µ, 313 314 𝑑𝐺 = 𝜎𝑑𝐴 + 𝜇!𝑑𝑛! + 𝜇!𝑑𝑛!; 𝜎𝑑𝐴/𝑑𝑛! = 𝜇! − 𝜇! (20) 315


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𝜇! − 𝜇! = 𝜇!"#$%#&%! − 𝜇!"#$%#&%! + 𝑅𝑇𝑙𝑛 𝑝 𝑎 (21) 316

317 We combine equations 20 and 21 then exponentiate. Standard states drop out in favor of 318 the water vapor pressure since ΔGo concerns products minus reactants and the activity of 319 pure water is one. Otherwise the solvent term reverts to its mole fraction since our 320 mixtures are dilute. So long as we do not enter the realm of molecular clusters, tension σ 321 may be considered fixed -only at the nanometer scale do surface Van der Waals 322 interactions begin to alter with curvature (Seinfeld and Pandis, 2006). Computing the 323 number of moles and drawing on the geometries of 19 together yield 324 325 𝑛! = 𝜌! 𝑀𝑊 𝑉 (22) 326 𝑑𝑛! = 𝜌! 𝑀𝑊 4𝜋𝑟!𝑑𝑟;𝑑𝐴 = 8𝜋𝑟𝑑𝑟; 𝑑𝐴 𝑑𝑛! = 2𝑀𝑊 𝜌! 𝑟 (23) 327 𝑝 𝑣𝑝 = 𝑆𝑅(𝐷) = 𝑎𝑒𝑥𝑝 4𝜎𝑀𝑊 𝑅𝑇𝜌!𝐷 ; 𝑆𝑅(𝐷) = 𝑎𝑒𝑥𝑝(𝐾𝑒𝑙𝑣𝑖𝑛 𝐷) (24) 328 𝐾𝑒𝑙𝑣𝑖𝑛 = 𝐷!"# = 4𝜎𝑀𝑊/𝑅𝑇𝜌! (25) 329 330 where we also take advantage of the opportunity to apply the conversion 2r = D and refer 331 to the equilibrium supersaturation ratio as SR. Both of these adjustments are made for 332 consistency with familiar aerosol science presentations (Petters and Kreidenweis 2007). 333 The Kelvin coefficient is often referred to in the aerosol literature as A (Seinfeld and 334 Pandis 2006, Petters and Kreidenweis 2007, Ghan et al 2011), which is in conflict with 335 our multidisciplinary need to represent total surface area using this symbol. Hence we 336 invoke a reference length instead. 337 338 Equations 24 and 25 highlight the crucial curvature effect on atmospheric droplet vapor 339 pressure. The pure water surface tension is 72 mJ/m2 and so a convenient round figure 340 Dref is 2 nm. If we now round/convert the spray mode injection size r80 = 0.1 to D = 0.2 341 microns (Gong 2003) and linearize the exponential, it is immediately clear that 342 equilibrium could be as high as 1 % supersaturation over nascent material (SR = 1.01). 343 Modulation by the activity term pulls this down by perhaps a factor of three due to the 344 presence of a chemically active core (Seinfeld and Pandis 2006). But with the tendencies 345 1 Molar/C1/2 enforcing surface pressures greater than 30 in some cases (Table 3), SR 346 could be virtually halved over local ecosystems. Pruppacher and Klett (1997) give 0.03 to 347 1 % as the range of maximum supersaturations encountered during updrafts, in marine 348 stratiform to cumulus cloud structures. By this reckoning the macromolecular 349 biogeography of Πmax is central to early cloud formation. 350 351 Table 3 (chemistry in the atmospheric bulk): κ, O:C and ρ 352 353 Activity coefficient reductions in the marine boundary layer are a direct manifestation of 354 Raoult’s Law. They become dominant in the aerosol fine mode, and in the remote 355 oceanic regime aqueous phase chemistry of the macromolecules is necessarily reflected 356 (Meskhidze et al 2011, Westervelt et al 2012). A popular parameterization of the effects 357 allows us to connect with well known physical properties of the organic material (Petters 358 and Kreidenweis 2007). The quantities involved are divergent among themselves and also 359 across the scenarios of Table 3, underscoring again the potential for ecogeographic 360 variation. 361


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362 The solvent water mole fraction may be modeled as a volume mixing ratio, with 363 appropriate solute weightings accounting for ion dissociation, complexation, 364 reconfiguration, gelling or other critical behaviors that may be observed empirically. 365 Factors involved in the chemical and mass effects are referred to as κ and ε respectively. 366 Since our goal is to extract a solvent vapor pressure, we denote water substance by the 367 special marker w. Individual dissolved species whether organic or otherwise become i 368 and the full complement of solutes is indicated by s. In this final section of the 369 supplement, superscripts are applied to collective components which exist as separate 370 phases only in the conceptual sense, or else under extreme conditions –solutes gather to 371 become a solid core on complete drying. Activity is constructed stepwise assuming that 372 contributing volumes can be summed. Insertion into 24 provides the desired expression. 373 Note geometric cancellations follow once again from 19 (with conversion to D). 374 375 𝑎 = 𝑉! 𝑉! + 𝜅!! 𝑉! = 𝑉! 𝑉! + 𝑉! 𝜅!! 𝜖! = 𝑉!/ 𝑉! + 𝑉!𝜅 (26) 376 𝜅 = 𝜅!𝜖!! ; 𝑉! = 𝑉!! ; 𝜖! = 𝑉! 𝑉! (27) 377 𝑉 = 𝑉! + 𝑉!; 𝐷! = 𝐷!! + 𝐷!! (28) 378 𝑆𝑅(𝐷) = 𝑉! 𝑉! + 𝑉!𝜅 𝑒𝑥𝑝 4𝜎𝑀𝑊 𝑅𝑇𝜌!𝐷 (30) 379 𝑆𝑅(𝐷) = 𝐷! − 𝐷!! 𝐷! − 𝐷!! 1− 𝜅 𝑒𝑥𝑝 𝐷!"! 𝐷 (31) 380 381 The result in 31 is convenient but overly simple, in fact constituting a parameterization of 382 a series of approximations. Nonetheless, it permits us to demonstrate several crucial 383 points with regard to the marine organics. Extensions have been proposed in order to 384 strictly conserve surfactant mass (Sorjamaa et al 2004, Petters and Kreidenweis 2013). 385 Based on conclusions from equation 18, however, this issue may be decoupled for 386 preliminary purposes. 387 388 An effective way to explore the function SR is by plotting versus D, for example fixing 389 Dref = 2 nm while adopting κ values at partial logarithmic intervals. Note that for zero 390 hygroscopicity, aerosol water behaves as though it were pure. Moreover, κi >1 are 391 permitted and in fact documented for certain inert salts. Chemical interactions can 392 produce bumps, kinks and irregularities in the D space (Seinfeld and Pandis 2006). But 393 for purposes here we will analyze the single peak in form 31. Its location can be 394 estimated by 1.) assuming total solute volume is negligible, 2.) application of a binomial 395 approximation to the activity term, 3.) log transformation overall, 4.) linearization, and 396 ultimately 5.) setting the derivative to zero. This procedure produces a relation between 397 the greatest or critical equilibrium supersaturation SRc plus hygroscopicity and the 398 chemical core size (Seinfeld and Pandis 2006, Petters and Kreidenweis 2007, Ghan et al 399 2011). 400 401 𝑙𝑛 𝑆𝑅! = 4𝐷!"#! 27𝜅𝐷!! ! !

; 𝑆! ≈ 1+ 4𝐷!"#! 27𝜅𝐷!! ! ! (32) 402

403 Equation 32 is acceptable as a learning device over much of the Table 3 range in κ 404 (compare with data in Petters and Kreidenweis 2007). Total hygroscopicity enters 405 directly into the quantity in brackets. Based on our text discussion of the inherent 406


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uncertainties, the central round figure κ = 0.1 can be adopted for an arbitrary baseline 407 organic. Next the averages from equation 16 are introduced conceptually. Working from 408 the rough Gong (2003) injection radius r80 = 0.1 (D80 = 0.2), a round value core is about 409 Ds = 0.1 microns in size. For the baseline Dref of 2 nm a central Sc 0.3 % follows, which 410 is not unreasonable. But the main issue is sensitivity in any case. The product κ(Ds)3 can 411 be adjusted upward and downward by a factor of 3 to address simultaneously the multiple 412 sources of uncertainty in κ, O:C and density (with the latter two playing into Vs). Sc 413 varies in turn from 0.2 to 0.5 %. Again a significant fraction of the marine updraft 414 spectrum is spanned (Pruppacher and Klett 1997). 415 416 Alert readers from the aerosol community may notice an implicit assumption hidden in 417 the above paragraph -that the organics sometimes possess their own externally mixed 418 state. Dilution with salt weakens the influence for all factors contained in κ(Ds)3. But 419 there will be compensation. For example, high ionic strength likely enhances lipid or 420 protein surface tendencies. Evidence for chemical segregation of the aerosol is strong in 421 any case (Russell et al 2010, Hawkins and Russell 2010, Wex et al 2010). Plus 422 processing toward uniform composition takes time (Westervelt et al 2012), leaving open 423 the possibility for ecosystem scale regimes of biogeochemical influence. Clearly back of 424 the envelope calculations are inadequate to the task at hand, and the problem must be 425 recast in the form of numerical modeling. But this is precisely the point of the entire 426 exercise, and so here we rest our case. 427 428 Supplementary References 429 430 Adamson A 1960 Physical Chemistry of Surfaces (Easton PA: Interscience Publishers) 431 Adamson A and Gast A 1997 Physical Chemistry of Surfaces (New York: Wiley 432

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theory and equations: supplementary material table 1...

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