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Surface Tensions of Inorganic Multicomponent Aqueous Electrolyte Solutions and Melts Cari S. Dutcher,*,† Anthony S. Wexler,† and Simon L. Clegg*,†,‡ Air Quality Research Center, UniVersity of California at DaVis, DaVis, CA, United States, and School of EnVironmental Sciences, UniVersity of East Anglia, Norwich, United Kingdom ReceiVed: June 6, 2010; ReVised Manuscript ReceiVed: October 4, 2010

A semiempirical model is presented that predicts surface tensions (σ) of aqueous electrolyte solutions and their mixtures, for concentrations ranging from infinitely dilute solution to molten salt. The model requires, at most, only two temperature-dependent terms to represent surface tensions of either pure aqueous solutions, or aqueous or molten mixtures, over the entire composition range. A relationship was found for the coefficients of the equation σ ) c1 + c2T (where T (K) is temperature) for molten salts in terms of ion valency and radius, melting temperature, and salt molar volume. Hypothetical liquid surface tensions can thus be estimated for electrolytes for which there are no data, or which do not exist in molten form. Surface tensions of molten (single) salts, when extrapolated to normal temperatures, were found to be consistent with data for aqueous solutions. This allowed surface tensions of very concentrated, supersaturated, aqueous solutions to be estimated. The model has been applied to the following single electrolytes over the entire concentration range, using data for aqueous solutions over the temperature range 233-523 K, and extrapolated surface tensions of molten salts and pure liquid electrolytes: HCl, HNO3, H2SO4, NaCl, NaNO3, Na2SO4, NaHSO4, Na2CO3, NaHCO3, NaOH, NH4Cl, NH4NO3, (NH4)2SO4, NH4HCO3, NH4OH, KCl, KNO3, K2SO4, K2CO3, KHCO3, KOH, CaCl2, Ca(NO3)2, MgCl2, Mg(NO3)2, and MgSO4. The average absolute percentage error between calculated and experimental surface tensions is 0.80% (for 2389 data points). The model extrapolates smoothly to temperatures as low as 150 K. Also, the model successfully predicts surface tensions of ternary aqueous mixtures; the effect of salt-salt interactions in these calculations was explored. 1. Introduction Surface tension, the contractile force that always exists between two phases in equilibrium,1 is an important property in fields ranging from atmospheric sciences to microfluidics to endodontics. In the atmospheric sciences, a knowledge of surface tension is important for accurate predictions of the equilibrium of water and volatile components between the gas phase and sub-micrometer sized aerosol droplets, which typically contain ammonium, sulfate, components of seasalt and windblown dust, and organic compounds. However, surface tension dataseven for inorganic electrolyte solutionssare limited and do not extend to the temperatures and concentrations typical of many atmospheric conditions, for which instances of supersaturation (with respect to solid salts) and supercooling (with respect to ice) are common. In this work we develop a model that is able to represent surface tensions of inorganic electrolyte solutions and mixtures over the entire concentration range, including molten electrolyte systems, and which extrapolates well over very wide ranges of temperature. In principle, the model can also be applied to solutions containing nondissociating solutes, including dissolved organic compounds. All surface tensions are those of the interface between air and bulk aqueous solutions or melts, at 1 atm total pressure. Available surface tension data for the atmospherically significant electrolytes HCl, HNO3, H2SO4, NaCl, NaNO3, Na2SO4, NaHSO4, Na2CO3, NaHCO3, NaOH, NH4Cl, NH4NO3, (NH4)2SO4, NH4HCO3, NH4OH, KCl, KNO3, * To whom correspondence should be addressed. E-mail: csdutcher@ ucdavis.edu (C.S.D.), [email protected] (S.L.C.). † University of California, Davis. ‡ University of East Anglia.

K2SO4, K2CO3, KHCO3, KOH, CaCl2, Ca(NO3)2, MgCl2, Mg(NO3)2, and MgSO4 are reviewed and fitted to the model. The effects of mixture terms in the modelsparameters for the interactions of pairs of solutessare explored for both aqueous electrolyte mixtures and molten salt systems. The treatment of surface tension is extended to highly supersaturated solutions based upon extrapolated surface tensions of high temperature melts, and to very low temperature by using an equation for the surface tension of supercooled water based upon both available data2-5 and a recent modeling study.6 2. Surface Tension Surface tension (σ) is due to both the dispersive forces (forces across the phase boundary) and specific forces (forces within one phase, such as hydrogen bonding and other attractive and repulsive forces between ions) that occur at a surface of finite extent between two bulk phases in equilibrium.1 The surface phase, which in the current study separates the vapor from the aqueous or melt phases, will contain a higher or lower local concentration of ions than the bulk phase, depending on whether the solute is “positively” or “negatively” adsorbed, respectively. Inorganic salts, such as CaCl2, are generally negatively adsorbed in aqueous solutions, meaning that fewer ions are found at the interface than in the bulk. Thus, for entropic reasons, the interface seeks to minimize its volume, and this results in an increase in surface tension. Acids, such as HCl, are usually positively absorbed, which leads to a decrease in aqueous phase surface tension. When thermodynamically favorable, hydration shells are formed around ions in the bulk aqueous phase. Ions in dilute solutions can modify the structure of water beyond their

10.1021/jp105191z  2010 American Chemical Society Published on Web 11/02/2010

Surface Tensions of Inorganic Solutions and Melts TABLE 1: Reported Values of dσ/dM solute

dσ/dM and sourcea

HCl HNO3 H2SO4 NaCl NaHCO3 NaHSO4 NaNO3 NaOH Na2CO3 Na2SO4 KCl KHCO3 KNO3 KOH K2CO3 K2SO4 NH4Cl (NH4)HCO3 NH4NO3 NH3 (NH4)2SO4 MgCl2 Mg(NO3)2 MgSO4 CaCl2 Ca(NO3)2

-0.27,16 -0.2517 -0.83,16 -0.4517 0.44,16 0.5917 2.08,16 1.76,17 1.46,18 1.51,19 1.62,14 1.7,22 1.7523 1.35,17 1.01,18 1.1320 2.0317 2.9,16 2.99,17 2.96,18 2.6622 1.85,16 1.68,17 1.54,19 1.59,22 1.6523 1.1020 1.98,16 1.7517 2.2821 1.59,16 1.78,17 1.3918 1.15,16 1.17,17 1.03,18 0.38414 1.96,18 4.06,16 2.98,16 2.44,16 4.02,16 2.47,16

2.36214 3.73,17 3.5918 1.9521 2.37,17 1.6721 3.6417 2.6417

a dσ/dM,16-18 dσ/dm,14,19-23 although at low concentrations dσ/dm ∼ dσ/dM. Values obtained at or near 25 °C.

hydration shells and can be classified as “structure making” or “structure breaking”.7 “Structure makers” prefer to make complete structures in the bulk rather than to reside in the interface where such structures cannot be fully formed. “Structure breakers” are attracted to the interface as a way of avoiding the formation of hydration shells.8,9 Recently, a bulk/surface partitioning model has been used to explain the effects of atmospherically relevant cations and anions on surface tension.10 Surface tensions of single inorganic electrolyte solutions are often found to be linear functions of concentration or molality, over moderate ranges, and linear with respect to temperature.11 These simple relationships are used in a wide variety of applications, including the atmospheric sciences.12-15 However, they are not valid over large ranges of temperature or concentration. For example, surface tensions of aqueous HNO3 at approximately 298.15 K are linear (in mole fraction, x) only to xHNO3 equal to about 0.2. Equations for surface tension as linear functions of molarity (M), fitted only to measurements for 273.15 K are shown in Figure 2 (data sets labeled 4,

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Figure 2. Surface tensions of water (σw) as a function of temperature, T. (a) Data including measurements for supercooled water: (O) Floriano and Angell,3 (3) Hacker,4 (0) International Critical Tables,2 and (×) Trinh and Ohsaka.5 (Solid line) eq 11, (dotted line) equation of Vargaftik et al.24 Only the data sets labeled 4, 6, and 7 from Floriano and Angell3 are shown, as they agree most closely with the equation of Vargaftik et al. for T > 273.15 K. (b) Model predictions for supercooled water: (]) molecular dynamic simulations of Lu and Wei,6 ([) data from Floriano and Angell,3 Hacker,4 and the International Critical Tables2 (shown in part a). (Solid line) eq 11.

6, and 7 by Floriano and Angell). The results of Trinh and Ohsaka deviate from established values even for temperatures above the freezing points and are probably in error.5 Overall, the data for surface tensions of supercooled water shown in Figure 2a suggest that extrapolating eq 10 (dashed line) below 273.15 K yields values of σw that are too low (e.g., the value from eq 10 is 1.2 mN m-1 less than the measurements at 250 K). The homogeneous nucleation temperature of pure water is about 232.7-234.7 K,120,121 which is only about 15 K below the lowest temperatures at which the surface tension has been measured (Figure 2a). In the past, it was thought that the surface tension and other properties of supercooled water could only be defined to about -45 °C, where a “bond-breaking” transition and homogeneous crystallization occurred.15,122 A power law equation, with a singularity at -45 °C, was proposed to describe the surface tension of slightly supercooled water for use in atmospheric models.123 Recently, Archer and Carter have discussed in detail the possible causes for the anomalous behavior of water at very low T and present strong reasons for explanations other than a singularity.124 Murphy and Koop have examined the thermodynamic properties of supercooled water and note that experiments support a continuity in the property values from supercooled liquid water to amorphous ice at much lower temperatures.125 They used this to constrain an equation for heat capacity that extends to 180 K. Furthermore, the density of supercooled water has recently been measured at temperatures to below 200 K,126 and surface tensions for deeply supercooled water have been studied to similar temperatures using models.6,127 Small aqueous solution droplets containing dissolved electrolytes can be supercooled even more than pure water. The

Dutcher et al. work of Koop et al.,128 using aqueous droplets of 12 different electrolytes suspended in an emulsion, demonstrates that supercooling to temperatures as low as 180 K is possible.128 Estimates of surface tensions to similarly low temperatures are therefore desirable for practical applications, but how might surface tensions of aqueous electrolyte solutions vary with temperature close to and below the homogeneous nucleation limit of 232.7-234.7 K? Archer and Carter have measured heat capacities of aqueous NaCl,124 and later NaNO3,129 to temperatures as low as 236 K and show that the rise in the heat capacity of very dilute solutions as temperature is lowered (similar behavior to that of water) becomes much less as the solutions become more concentrated. At molalities greater than about 3 mol kg-1 no increase is shown, and the heat capacity appears to decrease smoothly with T, showing no anomalous behavior. By analogy, it seems reasonable to assume that surface tensions of solutions of at least moderate concentration will continue to increase as temperature is reduced to supercooled conditions without showing anomalous behavior. Our model for surface tension, eq 7, which requires a value for the surface tension of pure water for all calculations except those involving molten salts, should therefore be consistent with this assumption. Lu and Wei have calculated surface tensions from 193 to 398 K using molecular dynamics simulations,6 and their results are compared in Figure 2b with experimental data for temperatures above 250 K. There is good agreement, with the calculated values suggesting a smooth increase in surface tension to temperatures below 200 K. Lu and Wei have fitted both power law and polynomial equations to measured and calculated surface tensions at low temperatures, and in the latter (polynomial) case the equation does not capture the calculated surface tension at close to 193 K. Without this value, the model simulations do not seem to support the power law fit, which predicts a singularity near 180 K. For practical applications involving electrolyte solutions at very low temperatures a simple extrapolation of the surface tension of pure water below below 250 K, consistent with the simulations of Lu and Wei,6 seems advisible. We therefore combined the calculated surface tensions of Lu and Wei, for T > 193 K (Figure 2b, open symbols) with measurements from the International Critical Tables,2 Floriano and Angell,3 and Hacker4 to 250 K (Figure 2b, closed symbols) and fitted them with the following equation:

σ(T < 273.15 K) ) 75.65085 - 0.13350(T - 273.15) + 0.0019462(T - 273.15)2 (11) where T (K) is temperature. The fit excluded data from Trinh and Ohsaka5 due to their systematic disagreement with the Vargaftik et al.24 equation for T > 273.15 K. Surface tensions calculated using eq 11 are shown as a solid line in Figure 2b for temperatures to 150 K. 4.2. Molten Salts. The surface tensions of molten salts, at high temperatures and with zero water content, do not have a direct application in atmospheric science. However, if they can be extrapolated to lower temperatures they can potentially provide limiting values of σ as xw f 0 and enable surface tensions of extremely concentrated solutions to be estimated. The review of Janz25 summarizes literature data for the surface tensions of a very large number of molten electrolytes and their mixtures, and lists coefficients of fitted equations for surface tension for each data set, together with the temperature range of the fit, a percentage accuracy, and brief comments as to reliability. Surface tension data for some molten hydroxides and

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TABLE 3: Thermodynamic Parameters for Molten Saltsa molten salt

c1 (mN m-1)

c2 (mN m-1 K-1)

Tm (K)

R+ (10-10m)

Vom (cm3 mol-1)

KBr LiBr NaBr KCl KCl LiCl NaCl NaCl Cs2CO3 K2CO3 Li2CO3 Na2CO3 KF LiF NaF KI LiI NaI KNO2 NaNO2 CsNO3 KNO3 LiNO3 NaNO3 NH4NO3 NaOH Cs2SO4 K2SO4 Li2SO4 Na2SO4 CaBr2 SrBr2 CaCl2 CaCl2 MgCl2 SrCl2 CaI2 SrI2 Ba(NO3)2

158.69b 150.56 164.93 175.57 177.61 189.28 191.16 193.48 213.5 243.71 284.6 268.5 240.01 346.5 289.6 136.1 125.68 139.83 164.24 143.8 142.3 154.72 144.9 155.5 148.4 177.1c 180.94 245.2 301 269 165.6 190 195.67 189 65.343 230.7 103.4 145.3 147.8

-0.0681b -0.04988 -0.06276 -0.07321 -0.07519 -0.06973 -0.07188 -0.0747 -0.0731 -0.06368 -0.0406 -0.0502 -0.08478 -0.0988 -0.082 -0.06003 -0.04302 -0.0573 -0.08 -0.041 -0.074 -0.07171 -0.055 -0.0613 -0.105 -0.0402c -0.05509 -0.0765 -0.0672 -0.066 -0.0459 -0.0439 -0.04541 -0.03952 -0.003073 -0.0541 -0.0173 -0.0383 -0.015

1007.15 823.15 1020.15 1044.15 1044.15 883.15 1073.85 1073.85 1066.15 1172.15 1005.15 1129.15 1131.15 1121.35 1269.15 954.15 742.15 934.15 711.15 557.15 682.15 607.15 526.15 579.65 442.85 596.15 1278.15 1342.15 1133.15 1157.15 1015.15 930.15 1048.15 1048.15 987.15 1147.15 1056.15 811.15 863.15

1.38 0.76 1.02 1.38 1.38 0.76 1.02 1.02 1.67 1.38 0.76 1.02 1.38 0.76 1.02 1.38 0.76 1.02 1.38 1.02 1.67 1.38 0.76 1.02 1.43 1.02 1.67 1.38 0.76 1.02 0.99 1.12 0.99 0.99 0.72 1.12 0.99 1.12 1.35

43.43 25.07 32.15 37.50 37.50 20.74 26.93 26.93 76.84 60.35 35.02 41.73 23.43 9.94 15.10 53.21 32.97 40.84 44.44 31.80 53.25 48.03 28.97 37.59 46.54 18.78 85.35 65.51 49.75 52.61 59.14 58.69 51.62 51.62 40.95 51.91 74.21 75.04 80.66

a Values of the c1 and c2 parameters, including duplicate entries, were obtained from Janz;25 melting temperature (Tm), originally listed in °C, from the CRC Handbook;105 ion radii (R+) from o Shannon;132 and molar volumes of the solid salts (Vm ) were calculated from the densities listed in the CRC Handbook at 25 °C.105 b From Sato et al.133 c From Patrov and Yurkinskii.79

Figure 4. Predicted surface tensions of molten salts at the melting temperature, Tm, compared to values from the fitted equation of Janz25 (using coefficients listed in Table 3). Melting temperatures were taken from the CRC Handbook.105 Values of σ(Tm, pred.) were obtained using c1 and c2 calculated from eqs 13-15, and then inserted into eq 12 for the surface tension.

TABLE 4: Predicted Surface Tension Parameters for Molten Electrolytes and NH3a electrolyte

c1 (mN m-1)

c2 (mN m-1 K-1)

Tm (K)b

HCl HNO3c H2SO4d NaHCO3 NaHSO4 KHCO3 KHSO4 KOH NH4Cl NH4HCO3 NH4HSO4 NH3g (NH4)2CO3 (NH4)2SO4 Mg(NO3)2 Mg(OH)2 MgCO3 MgSO4 Ca(NO3)2 CaCO3 CaSO4

58.20 86.95 54.49 137.6 113.4 150.4 124.9 231.5 193.0 145.5 119.3 88.71 228.5 228.5 120.2 264.3 440.8 284.8 151.4 472.6 374.2

-0.091 -0.155 -0.0216 -0.141 -0.0489 -0.131 -0.0738 -0.108 -0.0979 -0.122 -0.0802 -0.228 -0.265 -0.146 -0.0988 -0.141 -0.114 -0.0615 -0.0496 -0.143 -0.0675

158.98 231.55 283.46 323.15e 588.15 373.15e 473.15 679.15 611.15f 380.15e 420.15 195.45 331.15e 523.15e 362.15 623.15 1263.15 1413.15 834.15 1098.15e 1733.15

a Equation 14. b Melting temperature (Tm), originally listed in °C, from the CRC Handbook.105 c Calculated from data for 99.8 wt% HNO3 solutions.2,96,101 d Determined in the Results section by a best fit to the available data. e Decomposes. f Sublimation point. g Calculated from data for the pure liquid.134,135

acids of interest can also be found in the literature (e.g., NaOH and liquid NH3),79 or can be extrapolated from data for extremely concentrated solutions (e.g., HNO3). The surface tensions of the majority of molten electrolytes, hydroxides, and acids are well described as linear functions of temperature T (K) with the equation:

σs(T) ) c1 + c2T

Figure 3. Coefficient c1, defined in eq 12, for the salts listed in Table 3. (a) Salts with valency ν+ g ν- plotted against ν+R+/Vm, with data for (b) ν+ ) 1, ν- ) 1; and (O) ν+ ) 2, ν- ) 1 salts. (Solid line) eq 13. (b) Salts with valency ν+ ) 1, ν- ) 2, as a function of R+. (Solid line) eq 14.

(12)

However, there are no data, or fitted coefficients, for surface tensions of the molten or liquid forms of a total of 12 of the compounds of interest in this study. For a number of compounds the lack of data for very concentrated and/or molten electrolytes is related their physical properties (e.g., example solid NH4Cl decomposes at 338 °C). To estimate surface tensions of all molten electrolytes of interest, and so enable us to extend calculations of σ over the entire concentration range, empirical

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Dutcher et al.

Figure 5. Predicted surface tensions σ(T, pred.) of the salts listed in Table 3 at two temperatures compared with values from the fitted equation of Janz.25 Values of σ(T, pred.) were obtained using eq 12, and values of c1 and c2 from eqs 13-15. Symbols: (O) at 1000 K, (b) at 150 K (extrapolated).

relationships for c1 and c2 in eq 12 as a function of ion and electrolyte properties are needed. Smirnov and Stepanov found a relationship between the surface tensions of molten alkali halide salts and their molar volumes.130 Kojima et al. found a similar relationship for molten alkali carbonates that involved molar volume and cation radius (R+).131 From these relationships, it appears that the surface tension is related to the distance between the cation and anion and the resultant Coulombic bond strength. However, the relationships are unique for specific anions and specific types of cation. Here, we searched for a more general relationship between surface tensions of molten salts and their physical properties related to bond strength. We found empirical relationships between c1 and c2 and the following ion and electrolyte properties: cation and anion valencies (ν+ and ν-), melting temperature (Tm), the molar o ), and cation radius (R+). We were volume of the solid salt (Vm interested in reducing the temperature dependence by comparing the molar volumes of a salt at a fixed temperature. The natural

choice was 298.15 K, for which data are readily available. However, the electrolytes are typically solid, not liquid, at this temperature, hence the use of Vom of the solid. Additionally, anion radius could have been explored instead of the cation radius, but radii of some of the anions of interest (e.g., OH-, NO3-, HCO3-, CO32-, HSO4-, SO42-) are not as well established as the elemental cation radii. The 38 inorganic electrolytes that were used for this purpose are listed in Table 3, together with their physical properties. First, a relationship was found between the intercept of eq o . 12 (c1, the hypothetical surface tension at 0 K) and R+ and Vm For the salts in Table 3 with ν- ) 1 and ν+ ) 1 or 2, c1 could be represented to within 10.8% or, excluding NaOH and CaI2, within 6.2% by: o c1 ) 3984.4(ν+R+ /Vm ) + 31.176

(13)

o is in where c1 is in mN m-1, R+ is in units of 10-8 cm, and Vm 3 -1 o cm mol . The slope of this equation, ν+R+/Vm, has units of charge/ area, appropriate for describing the area force of surface tension. Measured and fitted values of c1 are shown in Figure 3a. We assume that eq 13 generally applies to salts with ν+ g ν-. Salts in Table 3, for which ν- ) 2 and ν+ ) 1, do not follow eq 13. Instead, a simpler relationship between c1 and R+ was found to apply, similar to that determined by Kojima et al. for molten carbonate salts.131 The data in Table 3 for 1:2 salts were fitted to obtain:

c1 ) -100.27R+ + 371.89

(14)

The results of the fit are shown in Figure 3b. Predicted values of c1 agree with those based on measurements (Table 3) to within 4.1%. An expression for the temperature coefficient c2 was found for all electrolyte valencies by relating the surface tension at the melting temperature to c1 and Tm to obtain:

TABLE 5: Model Parameters for Pure Aqueous Solutions solute

aws (mN m-1)

bws (mN m-1 K-1)

asw (mN m-1)

bsw (mN m-1 K-1)

% erra

Nb

dσ/dmc (mN kg m-1 mol-1)

HCl HNO3 H2SO4 NaCl NaHCO3 NaHSO4 NaNO3 NaOH Na2CO3 Na2SO4 KCl KHCO3 KNO3 KOH K2CO3 K2SO4 NH4Cl NH4HCO3 NH4NO3 NH3 (NH4)2SO4 MgCl2 Mg(NO3)2 MgSO4 CaCl2 Ca(NO3)2

-199.88 -53.483 167.80 232.54 46.411 143.16 19.359 69.691 55.988 126.00 -117.33 70.133 0 0 0 84.065 -176.83 36.479 -24.245 -173.24 0 1069.9 -202.35 119.02 -19.766 101.05

0.746 0 0 -0.245 0 0 0 0 0 0 0.489 0 0.265 0.154 0.480 0 0.617 0 0.140 0.672 0.366 -2.86 1.24 0 0.575 0

19.424 42.195 -59.111 -142.42 0 0 0 0 0 -194.45 0 0 -74.056 82.957 0 0 0 0 0 0 0 0 0 -184.69 0 0

0 0 0.0800 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.289 0 -0.114 0 0 0

0.20 0.86 1.2 0.72 0.30 0 0.77 1.3 0.29 0.54 0.43 0.32 0.54 1.0 1.5 0.35 0.50 0.32 0.66 1.8 0.78 1.1 0.69 0.73 0.73 0.59

68 150 321 194 7 1 177 25 82 152 166 5 86 82 75 27 154 5 91 117 60 56 49 29 200 10

-0.06 -0.78 0.78 1.62 1.20 2.99 1.19 2.33 2.64 1.92 1.50 1.83 1.17 2.60 4.05 2.98 1.19 1.20 0.95 -1.13 2.38 3.76 2.70 2.31 3.94 2.65

a

Average absolute percentage error between measured and fitted surface tension. b The number of data points used in each fit. c Linear fit coefficients, dσ/dm, evaluated at m ) 0 and T ) 298.15 K can be used for quick estimates of the surface tension at low concentrations.

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Figure 6. Comparison between the predicted and measured surface tensions (σ) for 26 single electrolyte aqueous solutions (a total of 2389 data points). There is agreement to within an average (0.80% of the measured surface tension.

c2 )

c1(1 - 0.6685) - 13.54 95.97 - Tm

(15)

where Tm is in Kelvin. Values of c2 from this equation, when combined with those for c1 from eqs 13-14, yield σ at the melting temperature Tm that agree with values using c1 and c2 coefficients from Janz to within 15% (Figure 4). Table 4 lists

the complete set of surface tension parameters c1 and c2 predicted by eqs 13-15 for electrolytes which either do not exist in the molten state or for which there are no data. 4.2.1. Extrapolation to Lower Temperatures. The linear relationship between surface tension and temperature in eq 12 appears to be valid for most molten salts over a temperature range of hundreds of degrees above the melting point.25 Sada et al.136 found for several molten salt hydrates that this also applies to at least 5-10 °C below the melting point, without any discontinuity or change of slope. In the absence of data for surface tensions of molten electrolytes at very high degrees of supercooling (i.e., close to room temperature) we have simply assumed that eq 12, and its associated coefficients in Tables 3 and 4, applies at all temperatures. We have tested whether our fits of c1 and c1 are likely to introduce any bias into estimates of surface tensions of hypothetical molten electrolytes at very low T by comparing predicted values of σ at 1000 and 150 K obtained in two ways: first, using c1 and c1 obtained from Janz25 and, second, using c1 and c2 from eqs 13-15 above. The results are shown in Figure 5. The mean absolute difference between the two estimates is 12.8 mN m-1 at 1000 K and 17.6 mN m-1 at 150 K, and there are no systematic deviations from the 1:1 line. The largest differences at 150 K are for the compound

Figure 7. Measured and fitted surface tensions of single electrolyte aqueous solutions, converted to the dimensionless quantity σ′/σ° - (T′/T° 300)/2 described in the text (including adjustment to nearest “round” temperatures T′, which are labeled on the graphs). The solid lines are calculations from eq 7 using parameters in Table 2. Data correspond to the nearest isotherm, except where indicated by an arrow, and are plotted against electrolyte mole fraction, x. Sources of data are as follows: NH4NO3: (b) 2, (O) 103, (1) 40, (4) 16, (9) 18, (0) 17; NaNO3: (b) 2, (O) 103, (1) 40, (4) 20, (9) 18, (0) 104, ([) 49, (]) 70, (2) 100; HNO3: (b) 2, (O) 96,101, (1) 101, (4) 16, (9) 52, (0) 2; KNO3: (b) 102, (O) 104, (1) 40, (4) 84, (9) 20; Ca(NO3)2: (b) 16, (O) 59; Mg(NO3)2: (b) 16, (O) 75, (1) 59, (4) 64.

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NaOH, with c1 and c2 estimated from Patrov and Yurkinskii,79 not Janz,25 and CaI2, which is not atmospherically relevant. 4.3. Pure Aqueous Solutions. Surface tensions of aqueous single electrolytes and NH3, at all temperatures, were fitted to eq 5. This equation contains the quantities σw(T) (given by eqs 10 and 11), the surface tension of the liquid or hypothetical molten solute σs(T) (given by eqs 13-15 and parameters listed in Tables 3 and 4), and the two pairs of water-solute interaction parameters aws and bws, and asw and bsw. All the data for each of the solutes were fitted simultaneously by minimizing the sum of the squares of the residuals (Marquardt-Levenberg algorithm). Due to the limited temperature or concentration ranges of some data sets, and to the unique surface tension profiles of particular electrolytes, not all parameters aws, bws, asw and bsw are significant. Every possible permutation of fit parameters was explored for each electrolyte to obtain the lowest value of the overall standard error of estimate using only parameters with a dependency of less than 0.075. Dependency is defined as 1 (variance of the parameter, other parameters constant)/(variance of the parameter, other parameters changing). Parameters with dependencies near unity are strongly dependent on one another, whereas parameters with dependencies near zero are uncoupled. It is important to note that even in the absence of, for example, bws and bsw, surface tensions can still be extrapolated based on

Dutcher et al. the known temperature dependence of surface tension of water and the molten salt. Likewise, even in the absence of asw and bsw, calculations of surface tension over a range of concentration are still possible, based on the σs value. Values of the fitted parameters, and the quality of the fit, are listed in Table 5. Measured surface tensions are usually reported to no more than 3 or 4 significant figures, and the model parameters are given to a precision that generally exceeds this in order to avoid inadvertent errors due to rounding. The average error in the fitted surface tensions is 0.80% for the approximately 2300 fitted measurements in the data sets for the 26 solutes. The average errors for the individual solutes listed in Table 5 are comparable to those obtained by Li and Lu.27 Both models use the same number of fitted parameters (2) to describe surface tensions at a single temperature, but our equation extends to the pure liquid electrolyte, is fit to a significantly larger number of data points, and does not require values of water or solute activities. Measured and fitted surface tensions are compared in Figure 6 (all data), and in Figures 7-10 for each individual solute. Surface tensions of aqueous solutions of nitrate salts are shown in Figure 7; chloride salts in Figure 8; carbonates and sulfates in Figure 9; and hydroxides, bicarbonates, and H2SO4 in Figure 10. To clearly display the scatter of the data and agreement with the model, while limiting the number of plotted isotherms, data points were adjusted to the nearest average temperature,

Figure 8. Measured and fitted surface tensions of single electrolyte aqueous solutions, plotted in the same manner as Figure 7. Sources of data are as follows: MgCl2: (b) 50, 102, (O) 102, (1) 16, (4) 75, (9) 59, (0) 17; NaCl: (b) 78, (O) 49, (1) 39, (4) 16, (9) 59, (0) 103, ([) 103, (]) 103, (2) 54, (3) 100; KCl: (b) 2, (O) 59, (1) 50, 102, (4) 102, (9) 47, 102, (0) 39, ([) 60, (]) 78, (2) 98, (3) 16, (`) 100; CaCl2: (b) 2, (O) 68, (2) 16, (4) 104, (9) 101, (0) 55, ([) 100; HCl: (b) 101, (O) 2, (1) 69, (4) 56, (9) 16, (0) 99, ([) 52; NH4Cl: (b) 2, (O) 103, (1) 103, (4) 39, (9) 16, (0) 59, ([) 98, (]) 72, (2) 17, (3) 18, (`) 100.

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Figure 9. Measured and fitted surface tensions of single electrolyte aqueous solutions, plotted in the same manner as Figure 7. Sources of data are as follows: Na2SO4: (b) 2, (O) 49, 103, (1) 82, 103, (4) 103, (9) 19, (0) 16, ([) 59, (]) 18, (2) 93; MgSO4: (b) 16, (O) 59, (1) 17, (4) 100, (9) 93, (0) 102; (NH4)2SO4: (b) 80, 103, (O) 51, 103, (1) 59, (4) 104, (9) 58; K2SO4: (b) 2, (O) 102, (1) 69, (4) 100, (9) 93; Na2CO3: (b) 2, (O) 19, (1) 48, (4) 77, (9) 59, (0) 104; K2CO3: (b) 70, (O) 102, (1) 104, (4) 84, (9) 77, (0) 100.

T′, so that σ′(T′) ) σmeasured(T) + (σmodel(T′) - σmodel(T)) where σ′(T′) (mN m-1) is the adjusted measured surface tension. To efficiently display multiple isotherms on single graphs, the surface tensions are plotted as the dimensionless quantity σ′/σ° - (T′/T° - 300)/2, where σ° and T° are equal to 1 mN m-1 and 1 K, respectively. Data from references listed in Table 2 but not shown in Figures 7-10 were excluded from the fits due to significant deviations from other measurements at similar temperatures and concentrations. In some cases, not enough data were available to confidently eliminate some measurements, leading to higher scatter (e.g., MgSO4 and NaHCO3). Aqueous H2SO4 has one of the most complex surface tension profiles, with a non-monotonic dependence on concentration and temperature (Figure 10). The effect of the sulfate-bisulfate equilibrium was explored to determine if a better expression for H2SO4 solutions could be obtained, especially for dilute solutions at low temperatures where a slight minimum in surface tension is seen. In these cases the H2SO4 solution was described by eq 8 with the two solutes, 2H+ + SO42-, and H+ + HSO4-. The mole fraction of H+ + HSO4- was determined from the dissociation constants given by Knopf et al.137 However, despite the addition of 8 nonzero fit parameters, this approach was found to improve the fit only from 1.2 to 1.0% error and did not capture the minimum of the surface tension curve at low concentrations. The best fit for the complex surface tension

profile was found by allowing the parameters c1 and c2 for the pure liquid to be optimized as a part of the overall fit. Figure 10 shows the results with c1 ) 54.49 mN m-1 and c2 ) -0.0216 mN m-1 K-1. Figure 11 displays contour plots of the predicted surface tensions of the atmospherically relevant electrolytes (NH4)NO3, H2SO4, (NH4)2SO4, and NaCl extrapolated to supercooled and supersaturated conditions. Solid/liquid equilibrium is indicated on the plots, as described in the figure captions. In all cases the surface tensions extrapolate satisfactorily to extreme concentration and to very low temperatures. 4.4. Mixtures. In this section we first consider the application of the model to liquid mixtures not containing water (molten salts), and then to aqueous mixtures. 4.4.1. Molten Electrolytes. We have found that surface tensions of binary molten mixtures are well described by eq 6. Data for the coefficients c1 and c2 in eq 12 can be found in the study of Janz for many molten salt mixtures at various concentrations.25 Table 6 lists parameters of the Fij function derived from some of these values. Once the interaction parameters for salts s1 and s2, s2 and s3, and s1 and s3 are known (in addition to the surface tensions of the pure salts), surface tensions of the ternary system containing all three salts can be predicted quite accurately. For example, Figure 12a shows measured and fitted surface tensions of five binary chloride salt

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Figure 10. Measured and fitted surface tensions of single electrolyte aqueous solutions, plotted in the same manner as Figure 7. Sources of data are as follows: NaOH: (b) 48, (O) 49, 108, (1) 66; KHCO3: (O) 104; NH4HCO3: (b) 77; KOH: (b) 45, (O) 48, (1) 16, (4) 104, (9) 86; NaHCO3: (b) 77, (O) 2, 103; NH3: (b) 106, (O) 93, (1) 63, (4) 85, (9) 58, (0) 92, ([) 74, (]) 44, (2) 105, (3) 89, (`) 62; H2SO4: (b) 73 (O) 36 (1) 36, 71, 101, (4) 88, 101, (9) 101, (0) 87, ([) 36,53, (]) 94, (2) 16, (3) 17, (`) 57.

mixtures at 1073 K. The relevant single salt parameters c1 and c2 are listed in Table 3 (where two sets of parameters are listed, those that appear first were used), and the salt-salt interaction parameters as1, s2 and bs1,s2 are listed in Table 6. Predicted surface tensions for the ternary systems MgCl2-NaCl-CaCl2 and MgCl2-CaCl2-KCl are compared with values from the equations of Janz25 in Figure 12b. There is satisfactory agreement, with a maximum average absolute percent error of 3.3% found for the xMgCl2 ) 0.4, xNaCl ) 0.4, xCaCl2 ) 0.2 mixture. The equation used for the ternary system MgCl2-NaCl-CaCl2 is: ln(σ(xMgCl2, xNaCl, xCaCl2, T)) ) xMgCl2 ln(σMgCl2(T) + xNaClFMgCl2,NaCl(T) + xCaCl2FMgCl2,CaCl2(T)) + xNaCl ln(σNaCl(T) + xMgCl2FNaCl,MgCl2(T) + xCaCl2FNaCl,CaCl2(T)) + xCaCl2 ln(σCaCl2(T) + xMgCl2FCaCl2,MgCl2(T) + xNaClFCaCl2,NaCl(T))

(16) Equation 16 above, which contains six binary mixture terms, predicts surface tensions of the ternary mixture to within 1.8%. This is an improvement over predictions of the mixture model of Kojima et al.,37 which is also based on the surface tensions of the pure and binary salts and agrees with the data to within 6%. In our model, if the binary mixture terms are set to zero, which might be necessary for many systems where data are

lacking, then the maximum errors rise to about 6.6%. Using the binary data, both our model and that of Kojima et al. are able to predict surface tensions of the molten carbonate system Li2CO3-Na2CO3-K2CO3 with an averaged absolute percent error of 0.54%. If the mixture terms in our model are set to zero, the level of agreement falls to 5.6%. Likewise, if only the pure salt and molar volume data are used in the model of Kojima et al., the average error is 4.9%. 4.4.2. Aqueous Solutions. Equation 7 can be applied to aqueous mixtures in a similar way to that described above for molten salts, except that water is treated as one of the mixture components. Surface tensions of water and the pure liquid eletrolytes (Tables 3 and 4) are used, together with single solution terms from Table 5 and mixture parameters obtained from data for molten mixtures (Table 6) where available. For example, surface tensions of Na2SO4-NaClH2O solutions at 298.15 K, up to 1.0 mol kg-1 total molality,19 are described to within 0.07% using the parameters from Tables 5 and 6. The absolute difference between the predicted and observed surface tension is no greater than 0.09 mN m-1. The mixture coefficients in Table 6 were determined by fits to the expressions for the surface tensions of molten mixtures given by Janz.25 If data for mixtures of molten salts are not available, then the Fij terms associated with the salt-salt interactions can be

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Figure 11. Calculated surface tensions (contour lines, labeled in mN m-1) of four atmospherically important electrolytes, as functions of temperature (T) and mole fraction (x) or weight percent (wt%) in solution, and including highly supercooled and supersaturated solutions. (a) NH4NO3, (b) H2SO4, (c) (NH4)2SO4, and (d) NaCl. Data for freezing points and solid-liquid equilibrium are superimposed and were taken from either Timmermans106 (aqueous NH4NO3, H2SO4, and (NH4)2SO4) or were calculated using the extended UNIQUAC model142 and Archer143 (aqueous NaCl).

TABLE 6: Model Parameters for Molten Salt Mixtures salt i NaCl KCl K2CO3 MgCl2 CaCl2 CaCl2 CaCl2

salt j

aij bij aji bji (mN m-1) (mN m-1 K-1) (mN m-1) (mN m-1 K-1)

Na2SO4 1.54 MgCl2 -144 Na2CO3 -5.79 NaCl -19.0 NaCl 14.5 KCl -11.2 MgCl2 -2.13

0.00389 0.176 -0.00958 0.0145 -0.0185 -0.00136 0.000103

1.28 139 -5.38 -7.11 20.2 -11.6 -6.54

0.00173 -0.158 -0.00695 0.00495 -0.0256 -0.00760 0.00165

T (K)a 980-1473 1010-1160 1000-1279 980-1220 1075-1193 1050-1193 1010-1193

a The maximum reported temperature range over which the equations of Janz25 are valid. (The equations were used to generate surface tensions fitted by the model.)

set to zero. For instance, surface tensions of the KNO3-NH4ClH2O, NaNO3-(NH4)2SO4-H2O, and NH4Cl-(NH4)2SO4-H2O systems reported by Abramzon and Gaukhberg138 can be predicted within 0.18, 0.49, and 0.28%, respectively, using only the parameters for single electrolyte solutions from Table 5. Likewise, surface tensions of H2SO4-NH3-(NH4)2SO4-H2O determined by Hyvarinen et al.58 can be predicted to within 2.9% using only the parameters in Table 5. The results for these mixtures are summarized in Table 7. The effect of solute-solute interaction terms, such as those obtained from the binary molten salt data or from measurements for concentrated aqueous mixtures, is shown in Figure 13 for HNO3-KNO3-H2O solutions at 293.15 K. Using parameters from Table 5 only, measured surface tensions for this system (Abramzon and Gaukhberg138) are predicted to within 2.7% over a broad range of electrolyte concentrations (the solid lines in Figure 13). Fitting constant values of FHNO3,KNO3 and FKNO3,HNO3 to the data yields calculated values of surface tension that agree

Figure 12. Surface tensions (σ) of seven molten chloride salt mixtures at 1073 K from the fitted equations of Janz25 compared to predictions of eq 7 with coefficients from Table 6. (a) Values for binary mixtures, plotted against either MgCl2 or CaCl2 mole fraction. Symbols (values from Janz): (b) MgCl2-CaCl2 (plotted against xCaCl2), (3) NaCl-CaCl2, (9) CaCl2-KCl, (]) MgCl2-NaCl, and (2) MgCl2-KCl. Lines: eq 7. (b) Surface tensions from eq 7 (σ(pred.)) for ternary mixtures compared with values from Janz (σ(obs.)). Symbols: (b) MgCl2NaCl-CaCl2, and (3) MgCl2-CaCl2-KCl.

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TABLE 7: Application of the Model to Aqueous and Molten Mixtures mixture MgCl2-NaCl-CaCl2 MgCl2-CaCl2-KCl Li2CO3-Na2CO3-K2CO3 Na2SO4-NaCl-H2O KNO3-NH4Cl-H2O NaNO3-(NH4)2SO4-H2O NH4Cl-(NH4)2SO4-H2O HNO3-KNO3-H2O KNO3-(NH4)NO3-H2O H2SO4-(NH4)2SO4-H2O, or NH3-(NH4)2SO4-H2O H2SO4-HNO3-H2O

T (K)a 1084-1216 (1073) 1081-1221 (1073) 822-1211 (1073) 298 291.15 291.15 291.15 293.15-353.15 291.15-303.15 298.15 253-293

available Fs1,s2b

note

25

FMgCl2,NaCl, FMgCl2,CaCl2, FNaCl,CaCl2, FNaCl,MgCl2, FCaCl2,MgCl2, FCaCl2,NaCl

e

1.6

6

25

FKCl,MgCl2, FCaCl2,MgCl2, FCaCl2,KCl, FMgCl2,KCl, FMgCl2,CaCl2, FKCl,CaCl2

e

2.1

6

37

FNa2CO3,Li2CO3, FK2CO3,Na2CO3, FLi2CO3,K2CO3 FLi2CO3,Na2CO3, FNa2CO3,K2CO3, FK2CO3,Li2CO3 FNa2SO4,NaCl, FNaCl,Na2SO4

f

0.54

7

e

0.07 0.18 0.49 0.28 1.5 1.5 2.9

12 10 9 10 244 143 139

3.3

60

source

19 138 138 138 138 138 58 73

g g g

FHNO3,KNO3, FKNO3,HNO3 FKNO3,NH4NO3, FNH4NO3,KNO3

h

FH2SO4,HNO3, FHNO3,H2SO4

j

a

i g

% errc

Nd

b

The temperature range is followed by the temperature used in this study (1073 K). This column lists Fi,j, Fj,i (where i and j are electrolytes or NH3) for which values were used in the calculations. c Average absolute percentage error in the calculated surface tension. d The number of data points for each mixture. e Calculated from the model parameters given in Table 6. f Calculated from the model parameters given in Table 6, except salts involving Li. Here, FNa2CO3,Li2CO3 ) -3.232, FLi2CO3,K2CO3 ) -30.41, FLi2CO3,Na2CO3 ) -2.863, and FK2CO3,Li2CO3 ) -44.01 at 1073 K were determined from the ternary data. g All values set to zero. h Values aHNO3,KNO3 ) 131, bHNO3,KNO3 ) -0.213, aKNO3,HNO3 ) -61.2, bKNO3,HNO3 ) -0.156 were determined from the ternary data. i Values aKNO3,NH4NO3 ) -149, bKNO3,NH4NO3 ) 0.0588, aNH4NO3,KNO3 ) 31.1, bNH4NO3,KNO3 ) -0.377 were determined from the ternary data. j Values aH2SO4,HNO3 ) -52.7, bH2SO4,HNO3 ) 0.0178, aHNO3,H2SO4 ) -87.0, bHNO3,H2SO4 ) 0.154 were determined from the ternary data.

Figure 13. Surface tensions (σ) of aqueous HNO3-KNO3 mixtures at 293.15 K, as a function of KNO3 mole fraction, xKNO3. Symbols: data of Abramzon and Gaukhberg138 for the following mole fractions of HNO3 in the solvent: (•) 0.256, (O) 0.300, (1) 0.347, (4) 0.400, (9) 0.462, (0) 0.618, ([) 0.720, and (]) 0.845. Solid line, calculated using eq 7 with parameters from Table 5 for water-electrolyte interactions only; dashed line, calculated using eq 7 and including the ternary interaction terms FHNO3,KNO3 ) 68.39 and FKNO3,HNO3 ) -107.02.

to within 1.5%, as shown by the dashed lines in Figure 13. The equation for the surface tension of the mixture is:

(17)

Figure 14. Comparison between the modeled and observed surface tensions (σ) of 11 ternary aqueous electrolyte solutions from 253 to 1073 K. (a) Predicted surface tensions plotted against measured values (more than 600 data points). (b) The difference between the measured and predicted surface tensions of 8 ternary aqueous mixtures (a subset of those shown in part a) as a function of total electrolyte mole fraction, 1 - xw. Symbols: (b) H2SO4-HNO3-H2O,73 (O) HNO3-KNO3H2O,138 (1) H2SO4-NH3-(NH4)2SO4-H2O,58 (4) NH4Cl-(NH4)2SO4H2O,138 (9) KNO3-NH4Cl-H2O,138 (0) KNO3-NH4NO3-H2O,138 ([) NaNO3-(NH4)2SO4-H2O,138 and (]) Na2SO4-NaCl-H2O.19

Results for MgCl2-NaCl-CaCl2, MgCl2-CaCl2-KCl, Li2CO3-Na2CO3-K2CO3, Na2SO4-NaCl-H2O, KNO3NH4Cl-H2O, NaNO3-(NH4)2SO4-H2O, NH4Cl-(NH4)2SO4H2O, HNO3-KNO3-H2O, KNO3-(NH4)NO3-H2O, H2SO4NH3-(NH4)2SO4-H2O, and H2SO4-HNO3-H2O systems are compared in Figure 14 to measurements from Janz,25 Kojima et al.,37 Matubayasi et al.,19 Abramzon and Gaukhberg,138 Hyvarinen et al.,58 and Martin et al.73 and to values from the fitted equations of Janz.25 The solute-solute Fij terms are set

to zero for all aqueous systems except Na2SO4-NaCl-H2O, HNO3-KNO3-H2O, KNO3-(NH4)NO3-H2O, and H2SO4HNO3-H2O. Values for the salt pair (Na2SO4, NaCl) are given in Table 6, and for the other three systems are as follows at 298.15 K: FHNO3,KNO3 ) 67.33, FKNO3,HNO3 ) -107.8, FKNO3,NH4NO3 ) -131.5, FNH4NO3,KNO3 ) -81.18, FH2SO4,HNO3 ) -47.37, and FHNO3,H2SO4 ) -40.92. The results of the comparisons made in this section, and the data upon which they are based, are summarized in Table 7.

ln(σ(xH2O, xHNO3, xKNO3, T)) ) xH2O ln(σH2O(T) + xHNO3FH2O,HNO3(T) + xKNO3FH2O,KNO3(T)) + xHNO3 ln(σHNO3(T) + xH2OFHNO3,H2O(T) + xKNO3FHNO3,KNO3(T)) + xKNO3 ln(σKNO3(T) + xH2OFKNO3,H2O(T) + xHNO3FKNO3,HNO3(T))

Surface Tensions of Inorganic Solutions and Melts 5. Summary A model for calculating surface tensions of inorganic aqueous and molten salt solutions has been developed, based on a simple mixing scheme and contributions to the surface tension from water, the dissolved electrolyte or uncharged solute, and the effects of interactions between pairs of solution components. The presence in the model of solvent-solute and solute-solute interaction terms allows it to be applied both to aqueous solutions over the entire concentration range and to molten electrolyte systems. Surface tensions of aqueous mixtures of inorganic electrolytes are satisfactorily described using parameters fitted to data for pure (single solute) solutions. The performance of the present model compares favorably to other models for aqueous solutions, such as those of Li and Lu27 and Hu and Lee,28 and has fewer limitations. For molten salt mixtures it also yields excellent results when compared to the equations of Guggenheim38 and Kojima et al.37 Model coefficients for 26 single electrolyte solutions and 11 mixed electrolytes, and their variation with temperature, were determined from literature data. The average deviation between the fitted and experimental surface tensions was 0.80%. Unlike simple polynomial fits, the present model, based on a physicochemical rationale of the mixing properties of water and salt, predicts reasonable values of surface tension over the entire concentration range from infinitely dilute solution to the liquid melt. It also appears to extrapolate well to the high concentrations and very low temperatures (below 200 K) needed for atmospheric modeling. To facilitate these and other applications, the model has been incorporated into the Extended Aerosol Inorganics Model (E-AIM) and is made available on the web.139,140 Acknowledgment. The authors gratefully acknowledge the support of the National Oceanic and Atmospheric Administration (grant NA07OAR4310192), the Natural Environment Research Council of the U.K. (grant NE/E002641/1), and the Department of Energy (grant DE-FG02-08ER64530). References and Notes (1) Hiemenz, P. C. Principles of Colloid and Surface Chemistry; Marcel Dekker, Inc.: New York, 1977. (2) Washburn, E. International Critical Tables of Numerical Data, Physics, Chemistry and Technology; McGraw-Hill: New York, 1926-1930; Vol. 4. (3) Floriano, M. A.; Angell, C. A. J. Phys. Chem. 1990, 94, 4199– 4202. (4) Hacker, P. T. Experimental Values of the Surface Tension of Supercooled Water, Technical Note 2510; National Advisory Committee for Aeronautics, Lewis Flight Propulsion Laboratory, 1951. (5) Trinh, E. H.; Ohsaka, K. Int. J. Thermophys. 1995, 16, 545–555. (6) Lu, Y. J.; Wei, B. Appl. Phys. Lett. 2006, 89, 164106. (7) Marcus, Y. Chem. ReV. 2009, 109, 1346–1370. (8) Du, H.; Liu, J.; Ozdemir, O.; Nguyen, A. V.; Miller, J. D. J. Colloid Interface Sci. 2008, 318, 271–277. (9) Hey, M. J.; Shield, D. W.; Speight, J. M.; Will, M. C. J. Chem. Soc. Faraday Trans. 1981, 77, 123–128. (10) Pegram, L. M.; Record, M. T. Proc. Natl. Acad. Sci. U. S. A. 2006, 103, 14278–14281. (11) Horvath, A. L. Handbook of Aqueous Electrolyte Solutions Physical Properties, Estimation and Correlation Methods; Ellis Horwood Series in Physical Chemistry; Ellis Horwood Limited: New York, 1985. (12) Gibson, E. R.; Hudson, P. K.; Grassian, V. H. J. Phys. Chem. A 2006, 110, 11785–11799. (13) Lewis, E. R. J. Aerosol Sci. 2006, 37, 1605–1617. (14) Svenningsson, B.; Rissler, J.; Swietlicki, E.; Mircea, M.; Bilde, M.; Facchini, M. C.; Decesari, S.; Fuzzi, S.; Zhou, J.; Monster, J.; Rosenorn, T. Atmos. Chem. Phys. 2006, 6, 1937–1952. (15) Pruppacher, H. R.; Klett, J. D. Microphysics of Clouds and Precipitation, 2nd ed.; Kluwer Academic Publishers: Dordrecht; Boston, 1997.

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