Predicting the influence of particle size on the glass transition temperature and viscosity of secondary organic material

Atmospheric aerosols can assume liquid, amorphous semi-solid or glassy, and crystalline phase states. Particle phase state plays a critical role in understanding and predicting aerosol impacts on human health, visibility, cloud formation, and climate. Melting point depression increases with decreasing particle diameter and is predicted by the Gibbs–Thompson relationship. This work reviews existing data on the melting point depression to constrain a simple parameterization of the process. The parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document}ξ describes the degree to which particle size lowers the melting point and is found to vary between 300 and 1800 K nm for a wide range of particle compositions. The parameterization is used together with existing frameworks for modeling the temperature and RH dependence of viscosity to predict the influence of particle size on the glass transition temperature and viscosity of secondary organic aerosol formed from the oxidation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}α-pinene. Literature data are broadly consistent with the predictions. The model predicts a sharp decrease in viscosity for particles less than 100 nm in diameter. It is computationally efficient and suitable for inclusion in models to evaluate the potential influence of the phase change on atmospheric processes. New experimental data of the size-dependence of particle viscosity for atmospheric aerosol mimics are needed to thoroughly validate the predictions.

where ξ = 6σ s/l v s,m /� m S is a parameter that subsumes the prefactor, the unknown physicochemical properties and depends on the particle composition. Figure 1 summarizes available literature data of melting point depression for various composition categories. The data are obtained from a range of experimental techniques and include some molecular dynamic simulations. The data are approximately consistent with the Gibbs-Thompson slope of − 1 in log-log space for the majority of the data, with obvious deviations for lead, tin, and aqueous NaCl. The deviations from Gibbs-Thompson for led and tin below 10 nm have been noted before 64,65 . Reasons for the deviation from Gibbs-Thompson include variation of surface tension and entropy of melting with size, as well as particle shape effects 65 . There is significant variability between different compounds. For example, a melting point depression of 30 K is required to liquify 10 nm polyethylene while 200 K is required for aqueous ammonium sulfate, lead or gold particles. There is no obvious clustering of broad compound classes, i.e. metals, inorganic salts, and organics. Nonetheless, the Gibbs-Thompson slope of − 1 in log-log space is a reasonable approximation and there appears to be a limited range of ξ values, despite a large variability of bulk melting point temperature (414-1337 K) for the graphed compounds. For the data shown, ξ varies between 300 and 1800 K nm. www.nature.com/scientificreports/ Glass-transition and temperature-dependence of viscosity. The glass transition temperature is related to the bulk melting temperature via T g = gT m , which is known as the Boyer-Kauzmann rule. The value of g is 0.7 ± 0.2 (supplemental information). It follows that the size dependence of the glass transition temperature is where T bulk g is the glass transition temperature for the bulk material. To the authors' knowledge, no data on the melting point depression of amorphous glassy material is available. Equation 3 is therefore an untested hypothesis that will need to be validated against experimental data in future studies.
At temperatures warmer than T g , but colder than T m , viscosity decreases over many orders of magnitude. Angell 71,72 showed that viscosity scaled by T g can be modeled using a simple parameterization where η is the viscosity and D A is the fragility parameter. Figure 2 illustrates the Angell representation of viscosity. At T g /T = 1 the particle is a glass and viscosity is taken to be 10 12 Pa s . At T > T g , viscosity decreases. At very high temperature the viscosity approaches that of a gas, ∼ 10 −5 Pa s . The Arrhenius representation of the temperature dependence of viscosity corresponds to linear relationship in log-linear coordinates. Compounds following the Arrhenius law are dubbed "strong" glasses 71 . Fragile glasses deviate from the Arrhenius law and are described by Eq. 4. Shiraiwa et al. 32 estimate 5 < D A < 20 for typical compounds found in SOA. Only very limited data on the temperature dependence of viscosity for SOA are available 40 . The glass-transition temperature in that work is from an extrapolation of temperature-dependent viscosity data. Extrapolated T g from that method agrees within ∼ ±10 K of T g obtained from scanning calorimetry for citric acid and sucrose 44,45 . The fragility parameter for α-pinene SOA is consistent with D A = 7 . Importantly, aqueous solutions of sucrose may be described using a single fragility parameter 73 . This provides some justification to apply D A = 7 to dry SOA and aqueous solutions of SOA that form due to hygroscopic growth at elevated relative humidity.

Scientific RepoRtS
| (2020) 10:15170 | https://doi.org/10.1038/s41598-020-71490-0 www.nature.com/scientificreports/ where w s is the weight fraction of solute, a w is the water activity ( a w = 0.01RH exp(A/D) −1 , A = 8.69251 × 10 −6 σ s/a /T 78 , σ s/a is the surface tension, RH is the relative humidity in %), κ m is the mass based hygroscopity parameter 79 , T g is the glass transition temperature of the mixture, T g,w the glass transition temperature of water, T g,s is the glass transition temperature of the solute and k GT is the Gordon-Taylor constant.
In contrast to the formulation described in Rothfuss and Petters 44 , Eq. 6 accounts for the effect of curvature on water content. The Gordon-Taylor mixing rule has its origin in polymer science and for some polymer systems the Gordon-Taylor constant can obtained from the density and expansion coefficient. This however fails when applying the rule to non-polymers and systems containing water. In such systems k GT becomes an empirical coefficient associated with the compound. A number of expressions having similar mathematical form have been proposed 80 . Variations of the mixing rule include the Fox 81 equation, the Couchman and Karasz 82 equation, which is as the Gordon-Taylor rule but using mole fraction as composition variable and heat capacity ratios to express k, and the Kwei 83 equation, which is as the Gordon-Taylor rule but also includes higher order terms. Secondary organic aerosol is a mixture comprising 100 s of components 84,85 . One approach is to group the organic compounds together and then use a binary Gordon-Taylor mixing rule to treat mixtures of water and the lumped organic fraction 86 , which is denoted as the quasi-binary assumption.
Here the Gordon-Taylor mixing rule is applied to SOA using the quasi-binary assumption with the following inputs. The glass transition temperature for α-pinene-derived SOA is T bulk g,s = 271.7 ± 10 K 40 . The ±10 K estimate denotes the accuracy of a single measurement and does not account for composition differences that arise from differences in SOA generation method between studies. Furthermore, T g,w = 136K 31 , k GT = 2.5 ± 1.5 22,31,87 and κ m = 0.04 ± 0.035 . The range κ m = 0.04 ± 0.035 neglects the potential systematic variation of κ m with relative humidity. Further details are provided in the methods and supplemental information.
Viscosity model. Let the generic function definitions T size g (T bulk g , D) denote Eq. 3, T bulk g (w s (RH), T bulk g,s ) denote Eq. 5, w s (RH ) denote Eq. 6, and η A (T, T g ) denote Eq. 4. Viscosity as a function of T, RH, and D is computed via function composition.
where the composite function η(T, RH, D) also depends on the solute parameters T bulk g,s = 271.7 ± 10K , κ m = 0.04 ± 0.035 , k GT = 2.5 ± 1.5 , D A = 7 , g = 0.7 and ξ = 1050 ± 750 K nm ; the values are estimates for α-pinene SOA, except for the range in ξ , which is based on the data in Fig. 1. In the limit of infinitely large particles, this model is identical to the model of DeRieux et al. 87 . However, the solute parameters differ from DeRieux et al. 87 due to newly available experimental constraints on fragility and different assumptions made about hygroscopicity. A detailed representation of Eq. 7 is given in the supplemental information.
Results and discussion Figure 3 shows measured and predicted viscosity of α-pinene SOA at room temperature as a function of relative humidity without taking the potential size dependence of T g into account. The slope of the η-RH relationship is determined by k GT and κ m . Data from Pajunaja et al. 90 and Renbaum-Wolff et al. 30 show viscosity > 10 9 Pa s at RH < 30% and fall outside the shaded area. Increasing the range in T bulk g,s to ±25 K , as was assumed in DeRieux et al. 87 , would include those points inside the shaded area. T g,s depends on molecular weight and functional group composition and it small differences in these parameters have a large effect on T g,s . Thus composition differences between the studies likely explain the higher viscosity in these experiments 40 . Overall the model www.nature.com/scientificreports/ and parameter ranges are in reasonable agreement with most of the published data. The remainder of this work will use the parameter ranges T bulk g,s = 271.7 ± 10 K , κ m = 0.04 ± 0.035 , k GT = 2.5 ± 1.5 , D A = 7 , g = 0.7 and ξ = 1050 ± 750 K nm to explore the dependence of viscosity on particle size. Figure 4 shows the measured and predicted viscosity of α-pinene SOA at room temperature as a function of particle diameter. Data are as the same as in Fig. 3 for the lowest RH reported but plotted at the measurement diameter. The Grayson et al. 91 are obtained using a "bulk" poke-flow method and are plotted at 10 µm diameter for reference. Note that the Virtanen et al. 61 data differ in several respects. The data are taken at RH = 31%, significantly higher than the other values (0-2%). The SOA is from oxidation of pine emitted VOCs and not pure α-pinene and the data are not direct measurements of viscosity. Virtanen et al. observed a decrease in rebound fraction in an impactor for D < 30 nm . Upon impaction, a fraction of the energy is dissipated due to particle deformation (i.e. viscosity) while the remainder is available for rebound. Rebound occurs when the kinetic energy exceeds the energy adhesion 61 . Virtanen et al. show that the predominant impaction energy of the smaller particles is larger than the impaction energy of the larger particles, and thus that the decreasing rebound  www.nature.com/scientificreports/ fraction indicates a decrease in viscosity 36,92 . Assigning a viscosity to the Virtanen data in retrospect is difficult. Bateman et al. 36 calibrated a different impactor at a different size with sucrose and report that the viscosity range for that transition is 1-100 Pa s. That value range is expected to depend on the impactor surface properties and flow dynamics. Slade et al. 41 used a low pressure impactor similar in design to that in Virtanen et al. and classify particles with a rebound fraction < 0.2 as "liquid" while assigning "liquid" a viscosity < 100 Pa s . Therefore, the Virtanen et al. 61 observations, which show bounce fraction < 0.2 at ∼ 25 nm , are graphed in Fig. 4 having viscosity 1-100 Pa s. Future measurements are needed to confirm that value. As expected, the predicted dependence of viscosity on particle size depends on ξ , with larger values of ξ corresponding to a larger decrease in viscosity with decreasing size. The data are broadly consistent with 1050 < ξ < 1800 K nm . The exact value remains unclear due to the need to juxtapose data from different sources. This results in an the imperfect match of relative humidity. There are also differences in chemical composition due to differences in mass loading, oxidant exposure, size-dependence of composition, aerosol preparation method and aerosol age, all of which may affect the glass transition temperature and viscosity. Finally, only a single data point at D < 100 nm is available. This point is also taken at elevated relative humidity. If humidity influences the observed viscosity, the viscosity under dry conditions would have been higher, which in turn would alias as ξ < 1050 . Due to the very limited available aerosol viscosity data a preliminary range 300 < ξ < 1800 K nm for organic aerosol is proposed, which is based on the survey presented in Fig. 1. Systematic studies of the effect of particle size on viscosity are needed, including the study of pure compounds and of complex mixtures such as SOA.
The model predictions show a decrease in viscosity to that of a thin liquid, defined as < 10 −3 Pa s at 1.6, 5.5, and 9.3 nm for ξ = 300 , 1050, and 1800 K nm, respectively. Changing relative humidity or temperature in the model changes the viscosity at large size, but it has a negligible effect on the slope of viscosity vs. size at small diameter and the size of the intersection with 10 −3 Pa s . The uncertainty in the g value from application of the Boyer-Kauzmann rule ( g = 0.7 ± 0.2 ) slightly alters the intersection with 10 −3 Pa s . Sensitivity analysis to these parameters is provided in the supplemental information. The predicted decreased below 10 −3 Pa s implies that the mixture is behaving more like a gas and is equivalent to volatilization due to decreasing particle curvature and increasing Kelvin effect.
The model formulated in Eq. 7 only relies on simple algebraic functions. It is computationally efficient and thus suitable for inclusion in atmospheric models. Input parameters such as T g/s and κ m can be predicted from various proxies for organic aerosol composition 32,87,[93][94][95] . However, prediction of the fragility parameter ( D A ), Gordon-Taylor constant ( k GT ) and ξ from composition data are not yet available. Nevertheless, reasonably constrained ranges for these parameters exist and have been identified here or in previous studies. The parameter ξ = 6σ s/l v s,m /� m S can be predicted from first principles provided σ s/l , v s,m , and m S data are available. For example, crystalline pyrene has σ s/l = 22.8 × 10 −3 J m −2 (ref. 96 ), v s,m = 1.59 × 10 −4 m 3 mol −1 (based on molecular weight and density), and m S = 40.97 J mol −1 K −1 (ref. 97 ). Therefore ξ = 6σ s/l v s,m /� m S = 531 K nm is predicted. A fit to the pyrene data from MD simulations in Fig. 1 yields 560 K nm. Systematic molecular dynamics simulations 66,98 systematic evaluation of available surface tension and entropy of melting data may provide additional constraints on ξ and its dependence on composition.

Conclusions
Data of the melting point depression as a function of particle size have been reviewed. A parameter ξ is abstracted from the Gibbs-Thompson relationship, where ξ characterizes the size dependence of the melting point for a fixed particle composition. The parameterization is applied to predict changes in glass transition temperature and viscosity as a function of size. Only limited aerosol data are available to validate the model. From these data a preliminary range 300 < ξ < 1800 K nm for organic aerosol is proposed. New experimental data of the sizedependence of particle viscosity for atmospheric aerosol mimics are needed to thoroughly validate the predictions made herein. The model framework, together with the identified parameter ranges, can be used to evaluate the potential influence of melting point depression on atmospheric processes.

Methods
Melting-point depression. Melting point depression data collated in Fig. 1 were digitized from data displayed in figures in the literature. Details about the digitization method are given in the supplemental information. The digitized data plotted are summarized in a comma delineated text file that contains the melting point depression ( T ), particle diameter (D), the source citation, the source figure that was digitized, and the method that was used to generate the data. Tin data are taken from Fig. 2 in Koppes et al. 99 including data from Wronski 68 and Lai et al. 100 . Gold data correspond to spherical gold particles summarized in Fig. 2 by Lu et al. 101 with data from Castro et al. 69 and Dick et al. 102 . Lead data are taken from Fig. 2 in David et al. 67 . Pyrene and coronene data are taken from Fig. 5 in Chen et al. 98 ; naphthalene data are taken from Fig. 4 in Jackson and McKenna 103 . Polyethylene data is from Fig. 1 in Jiang et al. 66 including data for linear alkanes 104 , cyclic alkanes 105 , and chainextended polyethylene 70 . Data for aqueous salt solutions is taken from Fig. 5 in Cheng et al. 62 with data derived from Biskos et al. 52,55 . Hygroscopicity data. Volume-based hygroscopicity parameter 106 data for α-pinene SOA were digitized from data displayed in figures and tables in the literature 38,107,108 . The digitized data plotted are summarized in a comma delineated text file that contains relative humidity, κ v value, and the source citation. Volume-based hygroscopicity parameter 106 data are converted to mass-based hygroscopicity 79  www.nature.com/scientificreports/ where κ m is the mass-based hygroscopicity parameter 79 , ρ w and ρ s are the density of water and solute, respectively. The density of α-pinene SOA in these studies is unknown. Kuwata et al. 109 report 1.23 < ρ s < 1.46 g cm −3 for α -pinene SOA. The uncertainty due to unknown density in the conversion from κ v to κ m is unimportant relative to the reported range in values. The data are graphed in the supplemental information.
Viscosity data. Viscosity data collated in Figs. 2, 3 and 4 were digitized from data displayed in figures and tables in the literature 27,30,40,[88][89][90][91] . The digitized data plotted are summarized in comma delineated text files. Viscosity data points in Fig. 2 were generated as follows. The logistic fit to the particle shape parameter measurement for α-pinene (supplemental information ref. 40 ) was solved for the interval [x 0 − σ , x 0 + σ ] , where x 0 corresponds to the mean and σ to the slope of the transition, and x 0 = 20.2 • C and σ = 4.93 for α-pinene. The interval was discretized into 2.5 • C steps. The interval only determines the density of points plotted in Fig. 2. The shape parameter was converted to viscosity using sintering theory 25,110 and the temperature was normalized by the T g extrapolated from the same measurements.

Data availability
All data and scripts used to create the figures in this manuscript are freely available through GitHub and Dock-erHub. Data and scriptsare archived in an online data repository at https ://doi.org/10.5281/zenod o.38242 14.). Further details are in the supplemental information.