Size dependence of phase transitions in aerosol nanoparticles

Phase transitions of nanoparticles are of fundamental importance in atmospheric sciences, but current understanding is insufficient to explain observations at the nano-scale. In particular, discrepancies exist between observations and model predictions of deliquescence and efflorescence transitions and the hygroscopic growth of salt nanoparticles. Here we show that these discrepancies can be resolved by consideration of particle size effects with consistent thermodynamic data. We present a new method for the determination of water and solute activities and interfacial energies in highly supersaturated aqueous solution droplets (Differential Köhler Analysis). Our analysis reveals that particle size can strongly alter the characteristic concentration of phase separation in mixed systems, resembling the influence of temperature. Owing to similar effects, atmospheric secondary organic aerosol particles at room temperature are expected to be always liquid at diameters below ~20 nm. We thus propose and demonstrate that particle size should be included as an additional dimension in the equilibrium phase diagram of aerosol nanoparticles.

Avogadro constant, V m is the molar volume. The metal and water data are taken from Turnbull 18 while the rest are taken from Buckle and Ubbelohde 19,20 . The metals include gold, silver, mercury, lead, etc. The chloride, fluoride, bromide and iodide correspond to LiCl, NaCl, KCl, RbCl, CsCl; LiF, NaF, CsF; LiBr, NaBr, Kbr, CsBr; and KI, CsI, respectively.  Tables   Supplementary Table 1 | Fit equations for the DKA-retrieved water activity a w and liquidvapour interfacial energy  lv (N m -1 ) of ammonium sulphate (AS) at 298K. Here, b is the molality (mol kg -1 ) and x s is the AS mass fraction.

Supplementary Note 1. Differential Köhler Analysis (DKA)
Shape factor correction for HTDMA measurement data. The results of humidified tandem differential mobility analyzer (HTDMA) were usually reported in the form of the growth factor g f as a function of water vapour saturation ratio s w at each dry equivalent spherical mobility diameter D s . However, g f and D s determined by DMA need to be corrected for the particle shape (non-spherical) and porosity. For ammonium sulphate (AS), we adopted the method of Biskos et al. 21 , taking a shape factor of 1.02 for the dry AS particle sizes and 1.0 for the wet particle sizes (no correction). For sodium chloride (NaCl) dry particles, we adopted a size-dependent shape factor varying from 1.24 to 1.07 as suggested by DeCarlo et al. 22 and Biskos et al. 23 for 8-nm to 60-nm NaCl nanoparticles. For 6-nm NaCl particles, the same correction method failed to reproduce the observed g f over the whole hygroscopic growth curve. Biskos et al. 23 suggested that it is due to nanosize effects on thermodynamic properties and/or uncertainties in thermodynamic properties/shape factors. We notice that the deviation still exists for the largest droplet (e.g., with g f = 1.6 measured at 6-nm and thus D sol = 61.6 = 9.6 nm), where the size is beyond the range of significant nanosize effects (on thermodynamic properties). It is likely that the shape factor is the main reason for the deviation.
In the present study, a shape factor of ~1.37 was applied to 6-nm NaCl particles. It was determined by numerically searching for a value that is able to match the observed hygroscopic growth curve with relative humidity (RH) higher than the deliquescence point, an equivalent spherical mobility diameter about 5.14 nm is obtained for the dry particle after this shape factor correction. Considering the measurement uncertainty in RH and g f , the derived shape factor for 6-nm NaCl particle may vary from 1.28 to 1.45. This is slightly larger than the numerically calculated dynamic shape factor of ideal cubic NaCl particles (~1.24) in the free-molecular flow regime 22,24 , which may due to the imperfect nucleation and coagulation 25 .
Size dependence of  lv . It is known that the liquid-vapour interfacial energy ( lv ) depends not only on the solute concentration (mass fraction x s ) but also on the particle size 26 , (1) where  lv,bulk denotes the bulk interfacial energy, and  denotes the size-dependent curvature adjustment factor. The size-dependence of  lv will introduce another term sol lv 2 dD d RT v w  into the Köhler equation. The following deduction will show the origin of this size dependent term.
The change of Gibbs free energy (G) for the creation of a spherical droplet is where μ l and μ v represent the chemical potentials of the liquid and its vapor, respectively,  lv is the liquid-vapour interfacial energy, and A w is the surface area of the droplet, V w is the droplet volume, and v w is the molar volume of the liquid.
The criterion of equilibrium for a system of prescribed temperature and pressure is that ΔG has reached its minimum, so where D is the droplet diameter, then The difference in the chemical potential is a function of vapor pressure p over the droplet and the equilibrium vapor pressure p*, (μ l -μ v )= -RT ln(p/p*), so supplementary equation (4) becomes where s w = p/p*, and  lv is a function of the droplet size D. By neglecting the size dependence of  lv , i.e., d lv /dD=0, supplementary equation (7) becomes the tradition Kelvin equation In our study, particle sizes are larger than 6 nm. The size dependent term where A=4v w /(RTg f ), a w is the water activity, s w1 and s w2 are water saturation ratios measured at the same g f but at D s1 and D s2 , respectively. In this study,  is calculated according to Bahadur and Russell 26 . However, it is worth noticing that the size dependence of interfacial energy will not play a significant role unless the solution droplets are smaller than 5 to 6 nm 26 .
Partial molar volume of water v w . Partial molar volume of water v w can be expressed as a function of solution density 27 , where M w is the molar mass of water,  sol is the solution density and x s is the solute mass fraction.
Conversion of growth factor g f to solute mass fraction x s . In this section we show that if the concentrations of droplet solutions are the same, their growth factors g f would also be identical.
The symbols are defined as follows: m s and m sol are the mass of solute and solution, V s and V sol are the volume of solute and solution,  s and  sol are the density of solute and solution and x s is the mass fraction of solute, respectively. By definition, we have for spherical particles/droplets of homogeneous composition Here,  s is a constant and  sol is a function of solution concentration x s . Therefore, for a specific solute/solutes and with the assumption of a size-independent solution density of spherical particles, the same x s in general means the same g f and vice versa. Note that, nano-size effects on the density itself can result in a change in the density when the size gets really small. Such effects are, however, trivial (<1%) above a threshold diameter ~1 nm to 5 nm 23,28 , and are therefore negligible for 6 nm to 60 nm particles investigated in the present study.
The smallest growth factor of 6-nm ammonium sulphate particle observed by the HTDMA experiments during dehydration prior to crystallization is about 1.04 (upon efflorescence of 6-nm ammonium sulphate particle) 21 , which corresponds to a molality b of ~380 mol kg -1 . However, the retrieval using DKA is limited by the highest overlapped concentrations, corresponding to b ~160 mol kg -1 (x s ~0.96).

Uncertainty analyses with Monte-Carlo simulation.
To properly estimate the uncertainty of the proposed DKA method, we performed a Monte Carlo simulation by randomly varying all or a section of input parameters/data for the retrievals. Deviations for these input parameters/data were considered to be normally distributed around the original value with one relative standard deviation (std). A range of 3std around the original value contains 99% of the values possible for the input parameter/data point according to the respective uncertainty due to assumptions in the parameterization or arisen from the experiments, which represent a conservative estimate of the maximum uncertainties and ensure that the overall uncertainty estimation is not based on excessively large outliers. The uncertainties of the different input parameters and data are summarised here as: 2% for relative humidity (RH), 1% for dry diameter sizing (D s ) of AS, 5% for D s of NaCl due to shape factor uncertainties, 1% for wet diameter sizing (D sol ), 2.5% for growth factor (g f ) 21, 23 .
40,000 runs of retrieval calculations were tested to be sufficient for convergence of the mean value and of the standard deviation. The discrepancy between the expected mean value of those 40,000 runs and the calculated value without any uncertainty variation of the inputs is less than 0.5%. Consequently, the retrieval calculations of  lv and a w were made by randomly choosing 20,000 different sets of input parameters and data (such as diameter, growth factor, and relative humidity). Three times relative standard deviation (3std) of the results from those 40,000 runs are considered as uncertainties (at 99% confidence level) for the output of the DKA method (i.e., a w and  lv ).

Supplementary Note 2. Determination of a s ,  ls and  ls
The solute activity a s on a mole fraction basis is calculated by the Gibbs-Duhem equation 9,29,30 at constant temperature and pressure, where n s and n w represent the mole fraction of solute and water, respectively (12) Integrating supplementary equation (12)  shows negligible impact on the simulation, i.e., <0.5% differences in the predicted μ del and <0.2% differences in the DRH. It is hence neglected in our simulation.
The interfacial energy of solute embryo  sl is determined by the classical nucleation theory as Cohen et al. 29 and Gao et al. 34,35 .
where k j and  are geometrical constants dependent on the morphology of the particular crystal, ΔG v is the excess free energy of solute per unit volume in the crystalline phase over that in solution, k b is the Boltzmann constant, V sol is the volume of the droplet, K is the pre-exponential factor, and t i is the induction time interval (taken as 2 second for the HTDMA measurements 21 ).
All parameters except a w and a s in supplementary equation (15) were taken the same as in Cohen et al. 29 . It is still on debate if a closure should be expected between  sl and  sl 25 . We found that  sl >  sl for the investigated AS and NaCl salts particles which seems to hold for other compounds (e.g. BaSO 4 ) as well 36 . The fact that  sl >  sl appears to be in line with the general size dependence of interfacial energies 26 : decreasing interfacial energy is expected for embryos of sizes ~1 nm 29 while no significant difference is expected for particles larger than 6 nm.

Supplementary Note 3. Thermodynamic models and parameterizations
Overview of model selection. In the following, we listed the reasons why the particular models were chosen for model prediction and comparison in our study: (1) To represent the current study We use newly determined a w and  lv by the DKA method for AS. These data agree very well with literature observation data and also largely extend the concentration beyond the observation and existing parameterization methods as shown in Fig. 2.
For NaCl, we use a w from the modified TM model (more details as given below) and  lv from the PK model 5 . This is because the DKA method suffers from uncertainties in the shape factor correction for NaCl particles, and the modified TM and PK agrees well with the DKA derived a w and  lv for NaCl, as shown in Supplementary Fig. 1. The good agreement between modeled and observed deliquescence/efflorescence (Fig. 1) also confirms the reliability of modified TM and PK in the investigated concentration range. (2) To represent Biskos et al.
We choose the original model used by Biskos et al. 21,23 , i.e., a combination of TM and PK to represent their study. The molar volume of pure water was used instead of the partial molar volume as in Biskos et al. 21,23 .
(3) AIM The AIM growth curves in Fig. 1 are based on a w and σ lv from AIM (Aerosol Inorganics Model). There are two reasons for choosing AIM. First, AIM is one of the most commonly used and powerful models for studying the thermodynamic of inorganic aerosols. Second, AIM has the capacity to model highly supersaturated solution 37 , providing a good reference under conditions where no measurement data are available. Such comparison could help to illustrate the performance of our method and its potential in improving the AIM model for highly supersaturated solution.
(4) Further comparison of different parameterization methods Moreover, the TM and PK models are based on the best fit to the literature measurement data of a w and  lv ( Fig. 2 and Supplementary Fig. 1). As shown in Fig. 2a and Supplementary Fig.   1a, the agreement with measured a w is better for the TM model than for the KD model 8

and AIM.
The KD model is only used as a complete and proper reference to previous work.
The SP model 9 and the PK model were proposed to describe the concentration dependence of  lv . For NaCl, their difference is not significant within the concentration range of our simulation (Supplementary Fig. 1b). In Fig. 1, we showed only one simulation for clarity.

Aerosol Inorganics model (AIM
( 1 (18) The polynomial coefficients A q for AS at 298 K are A 1 = -2.17510 -3 , A 2 = 3.11310 -5 , A 3 = -2.33610 -6 , and A 4 = 1.41210 -8 . The respective coefficients for NaCl are A 1 = -6.36610 -3 ,   (19) in which k s = 2.1710 -3 N m -1 L mol -1 for AS and k s = 1.6210 -3 N m -1 L mol -1 for NaCl. c s is the molarity of the solution, which is defined as the amount of substance divided by the volume of the solution in units of mol L -1 .  w is the surface density of pure water. In the present study,  w is treated as temperature-dependent and is described by 9 :
Since we are dealing with highly supersaturated AS solution, we adopted a parameterization suggested by Clegg and Wexler 39 in which the extended solution density was constrained by the density of molten salt. The density of molten AS is estimated to be ~1.6110 3 kg m -3 at 298.15K. It shows general good agreement with non-constrained density parameterization 38,39 when x s is below ~0.45, but when x s approaches 1.0, the deviations increases to up to 3-4% 39 . We also found that the smallest growth factor of 6-nm AS particles observed during dehydration prior to crystallization (g f ~1.041, corresponding to a AS:H 2 O molar ratio of 7:1) 21 cannot be explained by the non-constrained AS solution density. The minimum  sol required to explain such small g f is ~1.5710 3 kg m -3 , falling into the range of the  sol constraint by the molten salt properties.
Disjoining pressure. Among earlier efforts to explain the deliquescence/efflorescence, Djikaev et al. 40 and Shchekin et al. 41 introduced a "disjoining pressure" in their models. The effect of "disjoining pressure" makes the existence of partially dissolved solute possible, which will result in a continuous deliquescence as RH increases. To explain the continuous deliquescence is one of the major goals of introducing the disjoining pressure in Djikaev et al. 40 . In other words, a prompt deliquescence would suggest that the solute is fully dissolved and the effect of disjoining pressure is negligible 40 . This is exactly the case for the HTDMA data that we were using. Therefore, the disjoining pressure was not considered in our model simulation. The negligibility of the disjoining pressure was further confirmed by the good agreement between the observation and our model prediction.  Cappa and Wilson 42 indicate that the secondary organic aerosol (SOA) particles with median volume weighted diameter of ~100 nm formed through -pinene ozonolysis might be in a solid amorphous state rather than liquid. On the other hand, bouncing experiments clearly show that particle bounce decreases with decreasing particle size in sub 30 nm size range, suggesting a different phase state of larger (> 30 nm, solid-like) and smaller (17-30 nm, liquid-like) particles 43,44 . In order to explore the size effect on the biogenic SOA particles (such as pinederived SOA) and to estimate the relevant critical diameter range that is able to depress its melting temperature down to the ambient conditions (~298 K) according to the T bulk -D s,c relationship ( Fig. 5), we need to know the general melting temperature range of such biogenic SOA particles. The relatively low-volatility oxidation products of pinenes include multifunctionalized acids, such as pionic acid, pinic acid, 1,2,3-propane-tricarboxylic acid, 1,2,4butane-tricarboxylic acid, 3-methyl-1,2,3-butane-tricarboxylic acid, etc. Koop et al. 45 summarized the melting temperature of some of these oxidation products in the range of ~378-439 K. They also estimated that the glass transition temperature (T g ) of pinene-derived SOA in the range of ~240-300 K, and the ratio between glass transition and melting temperature is about 0.7. So, the melting temperature of pinene-derived SOA can be calculated to be about 340-430 K, accordingly. Combining these two ranges, we consider the melting temperature range of pinenederived SOA to be in the range of 340 K to 440 K, as a conservative estimation.

Supplementary Discussion. Heuristic viewpoint concerning the phase transition
In this section, we will discuss the size dependence of phase transition concerning the following aspects: (1) the relationship between different compounds, (2) the relationship between different phase transition processes (melting, glass transition, etc.), and (3) an outlook for future work in these directions.

(1) Relationship between different compounds
The similarity in the T bulk -D s,c relations suggests the following supplementary equation (25), a close relationship between  sl (the interfacial energy at the solid-liquid interface) and ΔH (enthalpy of phase transition)  constant (25) where V m is the molar volume of solid. The deduction of supplementary equation (25) According to supplementary equation (28), the validation of supplementary equation (29) would require  constant, suggesting a close relationship between the interfacial energy and the enthalpy of phase transition. This relationship, supplementary equation (25), reminds us the Turnbull empirical equation 18 (30) which differs slightly from our equation by a cube root of V m . For comparison, we reanalyse the data of Turnbull 18 along with corresponding data for molten salts (Supplementary Fig. 7a). We find these data also fit well to our equation (Supplementary Fig. 7b). Note that Supplementary   Fig. 7 contains a variety of metals (17 metals such as mercury, gold, silver, lead, etc) and molten salts, and our results contain organics and different water-salt mixed systems. These facts give us confidence that we might be able to generalize supplementary equation (25) for a large number of compounds in both melting and dissolution processes.
(2) Relationship between different phase transition processes Beside dissolution and melting processes, size dependence has been found for other phase transition processes, such as glass transition, nucleation and spinodal decomposition 46 (Supplementary Fig. 4 and references therein). There are few studies providing size dependent data from different phase transition processes for the same compound. The freezing and melting temperatures of water in Supplementary Fig. 4 are the only data we found in literature. The measured T m (melting temperature) and simulated T f (freezing temperature) show very similar size dependence. The reason for the similarity is still not clear, but it has been found that the characteristic temperatures for different phase transition processes often show simple relations.
For example, T g 0.7*T m [45] and T f 0.82*T m [18] , despite of different theories involved (thermodynamics equilibrium for melting and nucleation theory for freezing). If such relations hold at each size range, i.e.,