Measurements and Model Predictions for the Surface Tension of Aerosol Droplets Containing Mixtures of Strong Nonionic Surfactants and Organic Solutes

Abstract

Ambient aerosols are complex mixtures containing thousands of unique organic molecules. However, due to the large surface-area-to-volume ratio of aerosol droplets, surface-active molecules can become depleted in the droplet bulk, resulting in a size-dependent surface tension. Size-dependent partitioning models have been extensively tested for droplets containing a single surfactant and cosolute. However, these simple systems do not mimic the complexity of the ambient aerosols. Here, we compare surface tension predictions from mixed-surfactant partitioning schemes to experimentally measured droplet surface tensions for nonionic surfactants octyl-β-D-thioglucopyranoside and Tween20 with soluble organic glutaric acid. We first compare model predictions from two macroscopic mixed-surfactant partitioning schemes, the multicomponent Eberhart model and the ideal mixture of homologues model. We then pair these models with size-dependent partitioning models, which take into account the droplet’s surface-area-to-volume ratio. Additionally, we tested the applicability of treating a complex mixture with a surrogate surfactant or an effective Langmuir isotherm for the prediction of ambient aerosol surface tension. Our results suggest that an effective Langmuir isotherm, determined from macroscopic measurements of aerosol extracts, paired with a simple kinetic partitioning model, may be sufficient to describe the surface tension of the accumulation mode aerosol.

Introduction

Suspensions of solid and liquid particles in the atmosphere, termed aerosols, are chemically complex, containing a myriad of unique organic molecules. − Some organic molecules are highly soluble (e.g., sugars and dicarboxylic acids) while others preferentially partition to the droplet–air interface and may form micelles or aggregates in solution due to their low solubilities. − The concentration of organics in atmospheric aerosol droplets can vary considerably with aerosol source, , but also with time as aerosol particles move through air of varying relative humidity and evaporate or condense water to maintain equilibrium. − The partitioning of surface-active molecules (i.e., surfactants) to aerosol droplet interfaces has become a topic of great interest, as it may impact the spreading and drying of droplets deposited on surfaces as well as the hygroscopic growth and eventual activation of aerosols into cloud droplets. , Furthermore, surfactants at droplet surfaces have been shown to affect aerosol chemistry, − suggesting that an understanding of surfactant partitioning in atmospheric aerosol is necessary to make accurate predictions of yields of chemical reactions at aerosol interfaces.

The high surface-area-to-volume (SA-V) ratio of aerosol droplets greatly affects the partitioning of surfactants. In an aerosol droplet, the finite reservoir of surfactant molecules and the larger SA-V ratio compared to a macroscopic solution can lead to bulk surfactant concentration depletion. When surfactant becomes depleted in the droplet bulk, the equilibrium surface coverage of the droplet will be lower than that of a macroscopic solution with the same total concentration of surfactant. , Thus, to predict aerosol surface tension, macroscopic surface tension measurements must be paired with a surfactant partitioning model. ,− In order to have confidence in the surface tension predictions from these models, they should first be validated against direct measurements of aerosol surface tension.

New techniques have recently been developed to measure the surface tension of high SA-V ratio aerosol droplets. For example, atomic force microscopy has been used to measure the surface tension of >0.25 μm radius droplets deposited onto hydrophobic surfaces. − Several techniques have also been developed to measure the surface tension of micron-sized droplets in the contactless environment of optical traps ,,− ,− and electrodynamic balances, − as well as oscillating droplet streams. − However, only a few of these methods have been used to observe the partitioning behavior of strong surfactants in high SA-V ratio aerosol droplets. − ,, To date, experimental measurements have generally focused on droplets containing a single strong surfactant with a soluble cosolute. These data sets have been used to test the accuracy of surfactant partitioning models. ,− ,,,,

Droplets containing a single surfactant do not begin to capture the complexity of ambient aerosol. Aerosol extracts from field campaigns commonly find complex organic mixtures with many surface-active molecules present. − Therefore, to ascertain the impact of surfactants on hygroscopic growth and cloud droplet activation and to make predictions of the surface tension or surface concentration in these systems, we must be able to accurately predict the surface tension of droplets containing mixtures of soluble and surface-active organics. To date, only predictions from the monolayer partitioning model, initially developed by Malila and Prisle, and later extended for the mixed surfactant case, have been tested against experimentally determined measurements of surface tension for droplets containing two strong surfactants. In this work, we compare approaches to predict the surface tension in micrometer-sized droplets containing mixtures of two strong surfactants and a soluble organic cosolute. First, we utilize the Eberhart and ideal mixture of homologues approaches to predict the macroscopic surface tension of surfactant mixtures. We then pair the isotherm with a monolayer or simple kinetic partitioning model ,, to predict the size- and concentration-dependent surface tension and compare these model predictions with measurements of aerosol surface tension. Additionally, we investigate the potential of using an effective Langmuir isotherm or a surrogate surfactant description of surface tension, paired with a partitioning model, to predict aerosol surface tension from macroscopic solution measurements.

Methods

Experimental Methods

Glutaric acid (Acros Organics, 99%; Sigma-Aldrich, 99%; or Thermo Scientific, 99%), octyl-β-D-1-thioglucopyranoside (OTG, Sigma-Aldrich, > 98%; or Thermo Scientific, > 98%) and Tween20 (VWR BDH chemicals or Sigma-Aldrich) were used without any purification. Solutions with 0.9 M glutaric acid and OTG/Tween20 mol ratios of 1:0, 2:1, 1:1, 1:2 and 0:1 were used in experiments. These ratios were selected to span the range of mixtures between solutions with only the strong or weak surfactant. Total surfactant concentrations span orders of magnitude to cover the range from zero to beyond the critical micelle concentration (CMC). Solutions were made with highly accurate and precise concentrations using an analytical balance and analytical glassware. To reach the low surfactant concentrations required for these measurements, concentrated stock solutions were diluted. All solutions were made with deionized water.

Macroscopic surface tensions were measured by using the Wilhelmy plate method (DCATS, DataPhysics). All measurements were taken at room temperature, about 298 K, in triplicate, and average values are reported. Droplet surface tension data were collected using the droplet coalescence technique, which has previously been described in detail. ,, Briefly, two droplets are initially confined in two optical traps. The droplets are then brought close enough together for them to coalesce. Upon coalescence, the time-dependent elastic backscattered light is collected, which, after performing a fast Fourier transform, allows the oscillation frequency of the composite droplet to be determined. The droplet’s size and refractive index, which are related to its composition, are determined from the cavity-enhanced Raman spectrum taken concurrently. The concentrations of glutaric acid in the composite droplets were found to be 0.9 M, which corresponded to a water activity of 0.99. These two scattering spectra provide the necessary information to retrieve the surface tension of the composite droplet

$$\sigma =\frac{r^3\rho {\omega }_l^2}{l(l-1)(l+2)}$$ 1

where r is the droplet radius, ρ is its density, ω _l is the oscillation frequency, and l is set to 2 for the second-order surface mode.

Describing Macroscopic Surface Tension of Surfactant Mixtures

The surface tension data for macroscopic solutions containing two surfactants and an organic cosolute were fit using three methods to describe the concentration-dependent surface tension. All of the model calculations were performed at 298 K.

Ideal Mixture of Homologues

Fainerman and Miller developed a simple method to predict the surface tension of an ideal mixture of surfactant homologues. They used a generalized Szyszkowski equation, which closely resembles the Langmuir isotherm, to describe the surface pressure (Π = σ₀ – σ) of a surfactant solution. Here, this general approach is followed. For nonionic surfactants, the Langmuir isotherm can be written as

$$\mathrm{\Pi }={\sigma }_0-\sigma =-\mathit{RT}{\mathrm{\Gamma }}_{\max }\ln (1-\theta )$$ 2

$$\theta =\frac{\mathrm{\Gamma }}{{\mathrm{\Gamma }}_{\max }}=\frac{c{K}_{\mathrm{eq}}}{1+c{K}_{\mathrm{eq}}}$$ 3

where θ is the fractional surface coverage, σ₀ is the solvent surface tension, σ is the surface tension of the solution, R is the gas constant, T is the temperature, Γ_max is the maximum surface excess (mol/m²), Γ is the surface coverage at surfactant concentration c, and K _eq is the equilibrium surfactant partitioning constant for the adsorption and desorption of surfactant from the interface (m³/mol). The generalized Szyszkowski equation uses θ = cK _eq in eq rather than the expression given by eq .

For a mixture of two nonionic surfactants, the surface pressure can be described by utilizing the fractional surface coverage of each surfactant. Here, subscript i refers to one of the two different surfactants in the mixture. Note that the ratio of c ₁ to c ₂ is constant in an isotherm.

$${\mathrm{\Pi }}_{\mathrm{mix}}=-\mathit{RT}{\mathrm{\Gamma }}_{\max }\ln (1-{\theta }_1-{\theta }_2)$$ 4

$$c_i{K}_{\mathrm{eq},i}=\frac{{\theta }_i}{1-{\theta }_1-{\theta }_2}$$ 5

$${\mathrm{\Pi }}_{\mathrm{mix}}=\mathit{RT}{\mathrm{\Gamma }}_{\max }\ln (1+\frac{{\theta }_1+{\theta }_2}{1-{\theta }_1-{\theta }_2})=\mathit{RT}{\mathrm{\Gamma }}_{\max }\ln (1+c_1{K}_{\mathrm{eq},1}+c_2{K}_{\mathrm{eq},2})$$ 6

The surface pressure for each surfactant in the complex mixture (Π _i ) can be determined from an isotherm fit of measurements for solutions containing only a single surfactant. The partial molar surface area (1/Γ_max) is assumed to be the same for each surfactant in the mixture and is set to a value in the range typical for strong surfactants. Additionally, Fainerman and Miller report that the model results are insensitive to changes in partial molar surface area. This leaves only one fitting parameter (K _eq,i ) for each binary surfactant isotherm.

$${\mathrm{\Pi }}_i=\mathit{RT}{\mathrm{\Gamma }}_{\max }\ln (1+c_i{K}_{\mathrm{eq},i})$$ 7

Using eq , the surface tension of a mixture of two surfactants can then be determined utilizing only parameters determined from Langmuir fits of single surfactant solution data. Here, Π̅_mix is the reduced surface pressure of the surfactant mixture (eq ), and Π̅ _i can be determined in the same way.

$$\exp {\mathrm{\Pi ̅}}_{\mathrm{mix}}=\exp {\mathrm{\Pi ̅}}_1+\exp {\mathrm{\Pi ̅}}_2-1$$ 8

$${\mathrm{\Pi ̅}}_{\mathrm{mix}}=\frac{{\mathrm{\Pi }}_{\mathrm{mix}}}{\mathit{RT}{\mathrm{\Gamma }}_{\max }}$$ 9

Finally, an equilibrium partitioning constant for the mixed surfactant system (K _eq,mix) can be retrieved from Π̅_mix using equations , , and .

Initially, surface tension isotherms for binary or ternary surfactant solutions are fit with eq to determine fitting parameters K _eq,1 and K _eq,2. σ₀ is set equal to the surface tension of the solvent or cosolute solution when no surfactant is present (72.8 or 63.0 mN/m) in the calculations of Π. Binary isotherms describe one aqueous surfactant, and ternary isotherms describe one surfactant in solution with 0.9 M glutaric acid. Γ_max is set to 4.5 × 10^–6 m²/mol for binary isotherm fitting and 3.5 × 10^–6 m²/mol for ternary isotherm fitting. These values are within the range of typical values for strong amphiphilic surfactants and resulted in the lowest combined error over the two surfactants in both cases. As reported by Fainermann and Miller, predictions for the surface tension of a mixture of two surfactants are found to be insensitive to Γ_max. Isotherm fits are shown in Figure S1. Table shows the single surfactant isotherm fit parameter, and Table S1 shows the retrieved K _eq,mix for each surfactant mixture. Note that the large difference in best-fit Langmuir parameters for aqueous OTG and when it is in a mixture with 0.9 M glutaric acid is accompanied by a difference in CMC. The addition of glutaric acid appears to increase the solubility of the OTG.

1. Binary and Ternary Szyszkowski–Langmuir Equation Fits .

	two-parameter fit		single-parameter fit
	Γmax(mol/m2)	K eq(m3/mol)	Γmax(mol/m2)	K eq(m3/mol)
OTG (binary)	4.78 × 10–6	0.012	4.5 × 10–6	0.169
OTG (ternary)	3.90 × 10–6	0.85	3.5 × 10–6	0.67
tween20 (binary)	3.51 × 10–6	8.68 × 10–3	4.5 × 10–6	1.70 × 10–3
tween20 (ternary)	3.03 × 10–6	2.02 × 10–2	3.5 × 10–6	2.85 × 10–2

^aTwo-parameter fits are over Γ _max and K _eq, while single-parameter fits hold the Γ _max constant.

Ideal Multicomponent Eberhart Model

The multicomponent Eberhart model has been previously used to predict the surface tension of macroscopic solutions containing solute mixtures. The ideal multicomponent Eberhart model assumes that (1) the surface tension of a mixture can be determined from a surface mole fraction weighted linear combination of pure component surface tensions, (2) all solution components adsorb and desorb from the interface at equilibrium and are described by a distribution constant K _i = a _i /a _i , where a _i is the activity of substance i at the surface and in the bulk solution, and (3) ideal mixing occurs both at the surface and in the solution bulk. , Here, the ideal multicomponent Eberhart model is used, where n = 4 components: water, glutaric acid, Tween20, and OTG in a mixture, while n = 3 components: water, glutaric acid, Tween20, or OTG in a single surfactant mixture.

$$\sigma =\sum _{i=1}^n{\sigma }_i{x}_i+\sum _{i=1}^n\left(\frac{x_i}{\sum _{j=1}^n{x}_j/S_{\mathit{j}i}}\sum _{j=1}^n\frac{x_j}{S_{\mathit{j}i}}({\sigma }_j-{\sigma }_i)\right)$$ 10

In eq , x _i is the component mole fraction, σ _i is the pure component surface tension, and S _ji are separation factors, which describe the surface enrichment of component i in component j and correspond to the ratios of distribution constants (S _ji = K _i /K _j ). Thus, S _ii = 1, and S _ji = 1/S _ij . Solution component 1 is defined as water, and parameters S _1j can be found by fitting macroscopic surface tension data to the binary Eberhart model $(n=2,\sigma =\frac{{\sigma }_1(1-x_i)+{\sigma }_i{S}_{1i}{x}_i}{(1-x_i)+S_{1i}{x}_i})$ in the full surfactant concentration region, including concentrations beyond the CMC. Here, separation factors of solutes in other solutes S ₂₃, S ₂₄, and S ₃₄ are all set to 1 × 10^–15. Pure component surface tensions for surfactants are taken as the surface tension at the plateau (i.e., a full monolayer of surfactant at the interface). Model parameters for water and glutaric acid are taken from Kleinheins et al., and OTG and Tween20 surface tension data from Bain et al. are fit to find additional S _1i parameters. Figure S2 shows the binary Eberhart model fits for the aqueous OTG and Tween20. Table contains the binary Eberhart model fit parameters.

2. Physical Properties for the Pure Components in the Mixture and Binary Eberhart Model Fitting Parameters .

i	substance	ρ i (kg/m3)	M i (kg/mol)	v i N A (cm3/mol)	σ i (mN/m)	S 1i
1	water	997	0.018	18.054	72.0	1
2	glutaric acid	1219	0.132	108.29	53.1	67.3
3	OTG	998	0.308	309.05	30.0	5.3 × 104
4	tween20	1100	1.228	1115.9	37.5	9.6 × 106

^a ρ _i is the density, M _i is the molecular mass, ν _i is the molecular volume, and σ _i is the surface tension of pure component i.

Effective Langmuir Fit

The Langmuir isotherm, eqs and , has been extensively used in the surfactant design literature to describe the surface tension of binary surfactant solutions. It has also been used to describe the surface tension of surfactant solutions containing an additional cosolute by replacing the surface tension of the solvent, σ₀, with the surface tension of the cosolute solution without any surfactant present. ,, Here, the applicability of fitting one effective Langmuir isotherm for aqueous mixtures containing two strong surfactants and a soluble organic cosolute is tested. For the effective Langmuir isotherm fit, c in eq is the total surfactant concentration (i.e., the sum of the OTG and Tween20 concentrations), and σ₀ is set to the surface tension of the glutaric acid solution without any strong surfactant present, 63.0 mN/m, as predicted by Gaman et al. and confirmed with a laboratory measurement of 0.9 M glutaric acid without any surfactant present. The mixed surfactant data are then fit with eqs and using a 1-parameter fit (holding Γ_max,eff at 4 × 10^–6 mol/m²) to find an effective Langmuir surfactant partitioning constant, K _eq,eff, for the surfactant mixture. Changing the initial guesses for Γ_max,eff and K _eq,eff does not impact the retrieved best-fit parameters. Fits are shown in Figure S3, and fitting parameters are provided in Table S2.

Size-Dependent Surfactant Partitioning

The size-dependent surface tension of micron-sized droplets was predicted using either a simple kinetic partitioning model (ideal mixture of homologues and effective Langmuir fit) or a monolayer partitioning model (multicomponent Eberhart model). Surface tension measurements span droplet radii of 6–9 μm. All size-dependent surface tension model predictions are made for the 7.5 μm radius, the middle of the experimental size range.

Simple Kinetic Partitioning Model

The simple kinetic partitioning model used here was first developed by Alvarez et al. and has been shown to accurately predict the surface tension of micrometer-sized droplets containing a single surfactant and cosolute. , The simple kinetic model is used to calculate the effective bulk surfactant concentration (c _eff), accounting for a droplet’s surface area (A) and volume (V). Parameter f represents the ratio of the minimum bulk concentration required to reach full surface coverage to the equilibrium partitioning constant, and parameter ζ represents the maximum fractional loss of surfactant to the interface once equilibrium is reached. The simple kinetic model requires the Langmuir isotherm fitting parameters, obtained by fitting macroscopic solution data, as input.

$$\frac{c_{\mathrm{e}\mathrm{f}\mathrm{f}}}{c}=\frac{1}{2}\left(1-\zeta -\frac{\zeta }{f}\right)+\sqrt{{\left(1-\zeta -\frac{\zeta }{f}\right)}^2+4\frac{\zeta }{f}}$$ 11

$$\zeta =\frac{{\mathrm{\Gamma }}_{\max }A}{\mathit{cV}}$$ 12

$$f=\frac{K_{\mathrm{eq}}{\mathrm{\Gamma }}_{\max }A}{V}$$ 13

Here, Γ_max and K _eq are determined by either the ideal mixture of homologues or the effective Langmuir method (Table S1 or S2), and c is the total surfactant concentration for the mixed surfactant systems. The two-dimensional (2D) area that a surfactant molecule takes up at the interface is determined from Γ_max (A _surfactant = 1/Γ_max × N _A , where N _A is Avogadro’s number). For systems containing a single surfactant with 0.9 M glutaric acid, Γ_max and K _eq are taken from the 2-parameter, ternary, Langmuir fit of the macroscopic solution data (Table ).

Monolayer Model

The monolayer partitioning model used here was first introduced by Malila and Prisle to predict surfactant partitioning in high SA-V ratio droplets, and later adapted by Kleinheins et al. to use the Eberhart model surface tension description. Here, the quaternary system multicomponent Eberhart-monolayer model example code for a mixed surfactant case is utilized. , Monolayer models divide the droplet into bulk and surface monolayer (surf) compartments, with a surface thickness $\delta ={\left(\frac{6}{\pi }\sum _i{v}_i{x}_i^{\mathrm{surf}}\right)}^{1/3}$ . The compositions of the bulk and surface compartments are determined through an iterative process. The total moles of each droplet component, i, (e.g., water, soluble organic, surfactant) must satisfy a conservation of mass constraint, n _i = n _i + n _i . The adapted monolayer model uses a volume additivity for all phases, V = ∑ _i n _i v _i , where v _i is the molecular volume of component i, which is determined from its molecular mass and density v _i = M _i /(ρ _i N _A ). N _A is Avogadro’s number. The surface tension of a bulk solution described by the multicomponent Eberhart model is then related to the surface tension of a droplet with the same surface composition using ,

$$\sigma =\frac{\sum _i{\sigma }_i{v}_i{x}_i^{\mathrm{surf}}}{\sum _i{v}_i{x}_i^{\mathrm{surf}}}$$ 14

Results and Discussion

The partitioning of surfactants in picoliter volume droplets containing mixtures of soluble and surface-active organics is investigated by comparing single droplet surface tension measurements to predictions from two surfactant partitioning schemes. Glutaric acid, a soluble dicarboxylic acid that has been identified in ambient aerosol, , as well as OTG (CMC = 9 mM and σ_CMC = 30.0 mN/m), and Tween20 (CMC = 0.06 mM and σ_CMC = 37.5 mN/m) were selected for this study. According to the surfactant strength categorization proposed by Kleinheins et al., glutaric acid is a weak surfactant, and OTG and Tween20 are strong surfactants. OTG and Tween20 have surface-active properties and oxygen to carbon (O/C) ratios in the range of surface-active material collected in aerosol samples.

Macroscopic Solution Predictions

The Eberhart and ideal mixture of homologues models are capable of making predictions of the surface tension of complex mixtures using the properties of the pure components of a mixture (Eberhart model) or Langmuir parameters determined for systems containing single surfactants (ideal mixture of homologues). First, these methods are tested for macroscopic solutions containing two surfactants. The effective Langmuir method is not considered here, as it is an empirical fit of surface tension data and does not make predictions of macroscopic surface tensions. Figure A shows the model predictions compared to surface tension measurements for ternary solutions containing a 1:1 molar ratio of OTG/Tween20, and panel B shows predictions for quaternary solutions containing the same surfactant mixture with the addition of 0.9 M glutaric acid. In panel A, binary surfactant–water Langmuir isotherm parameters (Table ) are used as input in the ideal mixture of homologues prediction, and in panel B, ternary 0.9 M glutaric acid–surfactant–water isotherm parameters are used as model input. Both models have been previously found to accurately predict the surface tensions of solutions containing surfactant mixtures. ,

1.

Macroscopic surface tension measurements for (A) aqueous 1:1 OTG/Tween20 mol mixtures and (B) 0.9 M glutaric acid cosolute and 1:1 OTG/Tween20 mol mixtures. Predictions from the Eberhart model and the ideal mixture of homologues are overlaid. Vertical error bars are shown where the standard deviation of three replicate measurements is larger than the data marker. Total surfactant concentration uncertainties are smaller than those of the data markers. The horizontal gray line shows the average measurement for water (A) or 0.9 M glutaric acid (B) without any surfactant present.

For both macroscopic mixtures, the Eberhart and ideal mixture of homologues models agree with the experimental data as well as one another in the region where surface tension decreases as total surfactant concentration is increased. The ideal mixture of the homologues method is only valid in this region and continues to predict lower surface tensions when the experimental surface tensions begin to plateau. If the minimum surface tension of the mixture is known from a measurement, it can be set as a boundary condition in the surface tension plateau region. Conversely, the Eberhart model makes predictions of the surface tension in the plateau region. This prediction is based on the pure component surface tensions of each component and their mole fraction weighting in the surface layer. In panel B, both the Eberhart and ideal mixtures of homologues methods overpredict the measured surface tensions. However, both models return the surface tension of 0.9 M glutaric acid (63 mN/m) when the surfactant concentration reaches zero. This suggests that there are nonideal interactions between the glutaric acid cosolute and the surfactants, as both models agree with the experimental data in panel A when the surfactants are present without glutaric acid.

Droplet Predictions

Next, size-dependent surfactant partitioning in droplets containing surfactant mixtures is investigated. Droplets confined in the optical trap require a cosolute to lower the water activity and keep the droplet stable. All droplet measurements are for mixtures of 0.9 M glutaric acid with varying concentrations and ratios of OTG and Tween20. Figure shows the surface tension of 6–9 μm radius droplets containing OTG, Tween20, or a 1:1 mol ratio of OTG and Tween20 previously reported by Bain et al. Surface tension predictions for 7.5 μm radius droplets using the Eberhart-monolayer (panel A), ideal mixture of homologues-simple kinetic (panel B), and effective Langmuir-simple kinetic models (panel B) are overlaid.

2.

Surface tension measurements for 6–9 μm radius droplets containing 0.9 M glutaric acid with OTG (circles), Tween20 (triangles) and 1:1 molar ratio OTG/Tween20 (squares) from Bain et al. with partitioning model predictions for 7.5 μm radius droplets from (A) the Eberhart-monolayer model and (B) the simple kinetic model predictions. Ternary OTG and Tween20 model predictions in panel B, shown as solid lines, use the ternary Langmuir parameters in Table . The 1:1 mol mixture prediction from the ideal mixture of homologues model is shown in the solid line, and the effective Langmuir method prediction is shown with a dotted line. The effective Langmuir method is only applied to the surfactant mixture. In panel B, the Tween20 and 1:1 OTG/Tween20 model predictions are nearly identical, obscuring the Tween20 model line.

In Figure A, the Eberhart-monolayer model more accurately predicts the surface tension of droplets containing glutaric acid and OTG than the simple kinetic model. However, the Eberhart-monolayer model underpredicts the surface tension of the droplets containing Tween20 and a 1:1 mol ratio OTG/Tween20 throughout the region of surface tension reduction with increased surfactant concentration. In panel B, the ideal mixture of homologues-simple kinetic model overpredicts the surface tension of droplets containing glutaric acid and OTG and overpredicts the surface tension of droplets containing Tween20, but agrees with the experimental 1:1 OTG/Tween20 measurements when the total concentration of surfactant is less than about 1.5 mM. The droplet surface tension predictions are nearly identical from the simple kinetic model paired with the ideal mixture of homologues and the 1-parameter effective Langmuir method, where Γ_max,eff is set to 4 × 10^–6 mol/m² (Table S2). Since the Eberhart model and the ideal mixture of homologues surfactant mixture predictions for macroscopic solutions are nearly identical (Figure B), the differences here can be attributed to the bulk concentration depletion predicted by the paired surfactant partitioning model due to the different methods used to determine the molecular size of the surfactants at the interface.

Figure S4 compares 1-parameter (where Γ_max,eff is held constant a 4 × 10^–6 mol/m²) and 2-parameter effective Langmuir droplet surface tension predictions using the simple kinetic model. Although the 2-parameter fit agrees better with the macroscopic surface tension measurements (Figure S3), these fitting parameters result in greatly underpredicting the surface tension compared to the droplet measurements. This is due to the smaller Γ_max,eff that results from the 2-parameter fit (Table S2), which corresponds to a large area taken up by each surfactant molecule at the interface. Γ_max of strong surfactants is generally found to be in the range of 3–4 × 10^–6 mol/m². , The Γ_max,eff of 1.6 × 10^–6 mol/m² determined in the 2-parameter fit corresponds to a molecular area at the interface of 104 Ų, which is approximately twice as large as when Γ_max is 3–4 × 10^–6 mol/m² (42–55 Ų). It is not expected that mixing two nonionic surfactants would lead to each surfactant taking up twice as much room at the interface. A smaller Γ_max results in the simple kinetic partitioning model partitioning fewer molecules to the interface to reach a specific surface coverage and surface tension. Additionally, using the two-parameter fits, K _eq,eff in Table S2 are more than an order of magnitude larger than those determined using the ideal mixture of homologues method. When using the 1-parameter fit holding Γ_max,eff = 4 × 10^–6 mol/m², K _eq,eff are the same order of magnitude as K _eq,mix. Figure S5 shows surface tension predictions for a 7.5 μm droplet at a range of Γ_max,eff.

Figure S6 shows size-dependent surface tension predictions for droplets containing a 1:1 mol ratio of OTG/Tween20 for a radius range of 5–10 μm. The Eberhart-monolayer model predicts a strong size dependence. However, even the model result for a 5 μm radius droplet significantly underpredicts the droplet surface tension data. This indicates that the offsets between measurements and predictions observed in Figure A are due to the predicted partitioning strength or mole fraction of each component in the surface layer rather than droplet size. Figure S7 shows the predicted surface layer composition as a function of the total surfactant concentration. The Eberhart-monolayer model predicts that Tween20, the stronger surfactant, dominates the surface as the total surfactant concentration increases, while the surface mole fraction of OTG is always less than 0.05.

The ideal mixture of homologues-simple kinetic surface tension predictions also show a strong size dependence, but the 5 and 10 μm radius model predictions envelop the experimental surface tension data. In Figure B, the Tween20 and 1:1 mol ratio predictions are nearly identical. Unlike the monolayer model, which defines a surface layer comprised of all solution components, the surface layer in the simple kinetic model is only composed of surfactants partitioned at the interface. Using the Langmuir approach, only the fraction of surface sites occupied by each surfactant can be determined. Figure S8 shows that Tween20 is predicted to occupy greater than 2 orders of magnitude more of the surface than OTG.

Figure shows the surface tension as a function of total surfactant concentration in the 1–2 mM range for droplets containing OTG, Tween20, and 2:1, 1:1, and 1:2 mol ratios of OTG/Tween20. This total surfactant concentration range was selected, as it is in the range where droplets containing Tween20 (the stronger surfactant) approach their minimum surface tension, and the surface tensions of droplets containing OTG or the 1:1 mol mixture of the two surfactants are clearly offset to higher values. Droplet measurements are grouped into 0.2 mM concentration bins, with an average and standard deviation of measurements in each bin shown by the data markers. The experimental data show the expected trend of decreasing surface tension with increasing Tween20 mol ratio.

3.

Surface tension measurements of 6–9 μm radius droplets containing 0.9 M glutaric acid with OTG, Tween20, and 2:1, 1:1, and 1:2 molar ratio OTG/Tween20. Data points represent an average surface tension in 0.2 M concentration bins, and error bars show the standard deviation of measurements in a bin. Partitioning model predictions for 7.5 μm radius droplets from (A) Eberhart–monolayer model, (B) simple kinetic-ideal mixture homologues model, and (C) effective Langmuir isotherm where Γ_max,eff = 4 × 10^–6 mol/m² have been overlaid.

In Figure A, the surface tension predictions from the Eberhart-monolayer model are shown over the experimental data. The model predictions for 2:1, 1:1, and 1:2 OTG/Tween20 mol ratios are nearly identical with the predictions for droplets containing only Tween20. The surface mole fraction of Tween20 is predicted to dominate over that of the OTG, regardless of the total mole ratio. In Figure B, the surface tension predictions from the ideal mixture of homologues-simple kinetic model are shown over the experimental data. Again, the expected general trend of reduced surface tension with increasing mole fractions of Tween20 is predicted for the mixtures. However, the surface activity of Tween20 dominates the surface tension of mixed surfactant droplets, with 2:1, 1:1, and 1:2 mol ratios of OTG/Tween20 being closely grouped near the Tween20 prediction. In panel C, predictions from the effective Langmuir-simple kinetic model are shown for a 1-parameter fit where Γ_max,eff = 4 × 10^–6 mol/m². These predictions have the same shape as the predictions in panel B, but the surface tensions of the mixtures are shifted to higher values for total surfactant concentrations less than 1.6 mM. All of the models tested predict a smaller difference between the surface tensions of the different mole mixtures than is observed in the droplet measurements. However, the 1-parameter effective Langmuir-simple kinetic method predicts surface tensions in the region between the OTG and Tween20 predictions, while the other two methods predict all mixtures to have nearly identical surface tensions to Tween20. The predicted trend in surface tension with surfactant mole ratio is nonlinear for all partitioning schemes. Figure S9 shows the surface tension predictions as a function of total Tween20 concentration over a wider concentration range. The trend observed is similar to that of an isotherm.

Approaches to Predict the Surface Tension of Ambient Aerosol

In the previous section, we investigated the ability of three surfactant partitioning schemes to predict the surface tension of micrometer-sized droplets containing mixtures of strong, nonionic surfactants and an organic cosolute. In the micrometer size range, the Eberhart-monolayer and the ideal mixture of homologues-simple kinetic models predict that different amounts of surfactant are required to lower a droplet’s surface tension. Additionally, an effective Langmuir isotherm and an ideal mixture of homologues, representing the surfactant mixture, predict that the surface tension of a mixture is dominated by the stronger surfactant even when there is twice as much of the weaker surfactant in the droplet.

Ambient aerosol droplets can contain thousands of different organic molecules. Identifying each component, its mole fraction, and how it influences surface tension (required for the Eberhart-monolayer and the ideal mixture of homologues, simple kinetic models) is not a realistic approach to predicting the surface tension of ambient aerosol. It has previously been suggested that since bulk concentration depletion results in strong surfactants of similar sizes requiring similar total concentrations to reduce the surface tension in accumulation mode aerosol, a single strong surfactant could be used to approximate all of the surfactant in a monolayer or simple kinetic partitioning model. , Alternatively, the surface tension of aerosol extracts can be measured in the lab, ,,, providing macroscopic surface tension data of the complex organic mixture that could be used to inform a partitioning model. In Figure , these approaches are tested by comparing surface tension predictions for a droplet with a 100 nm radius containing OTG, Tween20 and a 1:1 mol mixture of OTG/Tween20 using the Eberhart-monolayer, ideal mixture of homologues-simple kinetics, and effective Langmuir-simple kinetic partitioning approaches.

4.

Surface tension predictions utilizing the Eberhart-monolayer and simple kinetic models for 100 nm radius droplets containing OTG, Tween20 or a 1:1 mol mixture of OTG/Tween20.

Figure shows that for a 100 nm radius droplet, the Eberhart-monolayer model predicts large differences in surface tensions for OTG, Tween20, and their 1:1 mol mixture, even though both surfactants are classified as strong surfactants. The difference in surface tension is >10 mN/m at some total surfactant concentrations. Conversely, the simple kinetic partitioning model predicts nearly identical surface tensions for 100 nm radius droplets containing OTG and Tween20, as well as for the ideal mixture of homologues and effective Langmuir treatments of the 1:1 mol mixture of OTG and Tween20. At total surfactant concentrations where the simple kinetic partitioning model starts to predict a reduction in surface tension from σ₀, the Eberhart-monolayer model predictions already approach the minimum surface tension. Similar to the results for micrometer-sized aerosol droplets in Figure , the difference in surfactant area or volume at the interface that the two models use to predict the point of full surface coverage leads to a lower total surfactant concentration predicted by the monolayer model than the simple kinetic model to reduce the surface tension.

The effective Langmuir isotherm approach described here performs as well as the two approaches for predicting the surface tension of mixtures of two strong surfactants and a soluble organic cosolute in picoliter volume droplets. Additionally, the predictions of surface tension for accumulation mode droplets by this approach are nearly identical to the ideal mixture of homologues’ surfactant mixing approach, suggesting this could provide a simple method to predict the surface tension of ambient aerosol from extracted filter samples. However, the effective Langmuir isotherm approach does not provide the same level of information about the surface composition of a droplet as the other approaches do.

While it has been suggested that a surfactant with similar surfactant properties can be used as a proxy when the SA-V ratio becomes large, like in the case of accumulation mode aerosol, , Figure shows that the Eberhart-monolayer model predicts quite different surface tensions for droplets containing one of OTG or Tween20, or their mixture, even though both of these surfactants are classified as strong surfactants. Conversely, the predictions from the ideal mixture of homologues and simple kinetic model show only small differences in surface tension for droplets with these surfactants and their mixture. Using the ideal mixture of homologues-simple kinetic framework, Langmuir parameters from either an OTG or Tween20 mixture with glutaric acid are sufficient to predict the surface tension of the strong surfactant mixture. For applications where the surface concentration of a surface-active species in a complex mixture is required, the Eberhart-monolayer model or the ideal mixture of homologues, a simple kinetic model, can provide additional information about the surface composition of the droplet. However, it is necessary to confirm the accuracy of the selected partitioning model for the system of interest. The additional surface composition information that these models provide requires knowledge of the droplet components and their physical properties. The large difference observed between the Eberhart-monolayer and the simple kinetic model surface tension predictions for 100 nm radius aerosol highlights the importance of validating model predictions against experimental data before applying a model to predict cloud droplet concentrations.

Accurate representation of aerosol surface tension is necessary to ascertain its impact on cloud droplet numbers and the aerosol indirect effect. , Since ambient aerosol contains complex mixtures of organic species, it is necessary to develop and test models to predict the size-dependent surface tension in mixed surfactant systems. At this time, direct surface tension measurements for droplets with known concentrations of strong surfactants are limited to radii >4 μm. Thus, surface tension predictions from surfactant partitioning models cannot be directly validated for the accumulation mode aerosol. However, comparing model surface tension predictions for micrometer-sized droplets provides a route for model validation before these predictions are extrapolated to ambient-sized droplets. For mixtures of two strong surfactants and an organic cosolute, the simple kinetic model was in closer agreement with the experimentally determined droplet surface tensions than the Eberhart-monolayer model predictions. We also find that an effective Langmuir isotherm paired with a simple kinetic model performs similarly to the ideal mixtures of homologues approach. This suggests its utility to predict the surface tension of ambient aerosol from surface tension measurements of field-collected aerosol extracts. Further validation of partitioning models for predictions of ambient aerosol surface tension should explore higher complexity mixtures and mixtures containing nonionic and ionic surfactants, as field studies generally identify large quantities of anionic and nonionic surfactants in aerosol samples. ,

Macroscopic and droplet surface tension data underlying the figures are available through the ScholarsArchive@OSU repository, https://doi.org/10.7267/h128np654.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.est.5c06630.

• Supporting figures showing Eberhart and Langmuir fits to macroscopic surface tension data; partitioning model droplet size dependence; and model predictions of droplet surface composition as a function of total surfactant concentration (PDF)

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K.L.R. and E.K.W. contributed equally to this work.

The authors declare no competing financial interest.