Toward closure between predicted and observed particle viscosity over a wide range temperature and relative humidity

Atmospheric aerosols can exist in amorphous semi-solid or glassy phase states whose viscosity varies with atmospheric temperature and relative humidity. The temperature and humidity dependence of viscosity has been hypothesized to be predictable from the combination of a water-organic binary mixing rule of the glass transition temperature, a glass transition temperature scaled viscosity fragility parameterization, and a water uptake parameterization. This work presents a closure study between predicted and observed viscosity for sucrose and citric acid. Viscosity and glass transition temperature as a function of 5 water content are compiled from literature data and used to constrain the fragility parameterization. New measurements characterizing viscosity of sub-100 nm particles using the dimer relaxation method are presented. These measurements extend the available data of temperature and humidity dependent viscosity to−28oC. Predicted relationships agree well with observations at room temperature and with measured isopleths of constant viscosity at ~10 Pa s and warmer than −28oC. Discrepancies at colder temperatures are observed for sucrose particle. Simulations with the kinetic multi-layer model of gas-particle interac10 tions suggest that the observed deviations at colder temperature for sucrose can be attributed to kinetic limitations associated with water uptake at the timescales of the dimer relaxation experiments. Using the available information, updated equilibrium phase-state diagrams (−80oC < T < 40oC and 0<RH < 100%) for sucrose and citric acid are constructed and associated equilibration timescales are identified.

In summary, there are four types of empirical relationships that are commonly used to characterize the amorphous state: glass transition temperature vs. water weight fraction, viscosity vs. temperature, viscosity vs. RH at constant temperature, and viscosity isopleths as function of temperature and RH. These relationships are related through a phase diagram model that involves a water/organic binary mixing rule of the glass transition temperature, a glass transition temperature scaled viscosity fragility parameterization, and a water uptake parameterization.
In this work we perform a closure study that tests for consistency between these four state-spaces and the phase diagram 65 model for sucrose and citric acid. Sucrose and citric acid are selected because extensive data are available for a wide range of water contents and temperatures. These data are compiled from the literature and are used to constrain the inputs of the in mixing rule and fragility relationships. The resulting model is compared with the data in all of the state spaces. Limited data are available to constrain viscosity isopleths at temperatures below ambient. New measurements are performed to measure viscosity isopleths at T > −30ºC. To this end the dimer, coagulation, isolation, and coalescence (DCIC) method (Rothfuss 70 and Petters, 2016) was modified to extend the temperature range where this method can be applied. The method identifies the temperature and RH when dimer particles relax into spheres. Equilibrium water uptake is assumed when retrieving viscosity from this method. Model simulations using the kinetic multi-layer model of gas-particle interactions in aerosol (KM-GAP) model  are used to identify conditions where the equilibration timescale is similar to, or exceeds the time available for dimer relaxation. The combined work yields updated equilibrium phase-state diagrams (−80ºC < T < 40ºC and 75 0 < RH < 100%) and equilibration timescales for sucrose and citric acid.

Viscosity Measurement
The DCIC technique has been described extensively in prior publications Marsh et al., 2018;Petters, 2016, 2017a;Rothfuss et al., 2019;Petters, 2018;Tandon et al., 2019;Petters et al., 2019). The basic concept is 80 briefly introduced and then modifications made to enable low temperature measurements will be described. Dimer particles were generated using the dual tandem differential mobility analyzer (DMA) technique. Two DMAs were used to create mobility selected size-selected particles of opposite charge. The streams from the two DMAs were merged and passed into a coalescence chamber where some of the particles coagulated. Dimer particles formed from +1 and −1 or +2 and −2 coagulation events are charge neutral. The aerosol was passed through an electrostatic precipitator. Neutral dimer particles were transmitted, thus 85 isolating dimer particles. Coagulated and coalesced particles are spherical while coagulated and not coalesced particles are rod shaped. The dimer particles were passed through temperature controlled volume (thermal conditioner) to alter the thermodynamic state for a short amount of time. Subsequently the size distribution was measured using a scanning mobility particle sizer. Rod shaped particles have a large apparent mobility diameter due to increased drag force. The altered thermodynamic state, defined by an increase in either temperature or relative humidity, may induce partial or complete coalescence, which 90 was observed as a change in the mode diameter of the dimer mobility size distribution. The observed mode of the mobility size distribution was converted to a particle shape parameter, the shape parameter was graphed against temperature, and the observed relationship was fit to a logistic curve where ξ is the shape parameter, erf c is the complementary error function, σ is the spread parameter, T is the temperature 95 and T 0 is the midpoint of coalescence relaxation. The shape parameter was converted to viscosity using the Frenkel sintering theory (Pokluda et al., 1997;Rothfuss and Petters, 2016). Example logistic curves including data for sucrose and citric acid aerosol are given in prior work Petters, 2016, 2017a;Marsh et al., 2018;Rothfuss et al., 2019;Tandon et al., 2019;Petters et al., 2019) and the supporting information.
The relationship between shape factor and viscosity scales with particle diameter, surface tension, and residence time in Figure 1 depicts the schematic of the experimental setup. To facilitate cold-temperature measurements several modifications were made to the instrument. The particle drying technique was improved. All DMA flows were switched to N 2 boiled off from dewar. To accommodate the larger thermal load required to control the increased thermal conditioner volume, the cooling system of SMPS was revised. Settings specific to this study are now discussed.

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Dimer particles are composed of a polyethylene monomer and either a sucrose or citric acid monomer particle. Polyethylene particles were generated using an evaporation-condensation system that is described in Tandon  using Silica gel diffusion dryers and passed through a 210 Po neutralizer to establish charge equilibrium. Additional drying was achieved by passing particles through a U-type glasses embedded in a chilled bath held at ≈ −30ºC. The output flow from the U-type glass was split and passed to the dual tandem DMA. DMA 1 was connected to a negative polarity power supply to select positively charged particles, and DMA 2 was connected to a positive power supply to select negatively charged particles.
The DMAs were operated at a 5 L min −1 : 0.75 L min −1 sheath-to-sample flow ratio. The flows from the two DMAs were 120 merged and then split between a CPC with a flow rate of 0.5 L min −1 and the coalescence chamber with a flow rate of 1 L min −1 . The coalescence chamber was shielded with lead to prevent spontaneous particle decharging from naturally occurring ionizing radiation  and was placed inside the freezer to maintain dimers at cold enough temperature to prevent coalescence before reaching the thermal conditioner. The freezer temperature could be varied between −8ºC and 18ºC.
Temperature control was achieved by using a circulating bath chiller (Neslab ULT 90) that transported heat exchange fluid 125 through an aspirated car radiator placed inside the freezer. Also placed inside the freezer were the electrostatic precipitator (operated at 1500V potential) and the scanning mobility particle sizer, operated at a sheath to-sample-flow ratio of 3:0.5 L min −1 (DMA 3). Operation of the SMPS at subzero temperature was previously described in Wright et al. (2016) andPetters et al. (2019). Placing the coalescence chamber and SMPS in the freezer enables measurement of uncoalesced dimer particles for systems that otherwise would coalesce well below ambient temperature.

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The RH in the instrument was measured using three resistance based RH (Rotronic HC2) sensors placed at various points inside the flow path in the freezer. From the flow exiting the thermal conditioner, 0.5 L min −1 was transmitted through an RH sensor and subsequently to a chilled mirror hygrometer outside the freezer which was measuring dew or frost point temperature (GE Optica 4-stage with ±0.2ºC accuracy and a lower limit of detection of −65ºC). Readings < 0ºC were interpreted as frost point temperature. The critical value for the experiment are the vapor pressure and temperature inside the thermal condi-135 tioner. Temperature of the thermal conditioner was monitored using thermistors mounted at the outside of the metal chamber.

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The phase diagram model is described in detail in previous publications (Rothfuss and Petters, 2017a;DeRieux et al., 2018) and only briefly summarized here. The relationship between solute weight fraction and relative humidity is computed using a hygroscopicity parameterization (Mikhailov et al., 2013) where w s is the solute weight fraction, 1 − w s is the water weight fraction, a w is the water activity and taken to be a w = 145 RH/100%, κ m is the mass-based hygroscopicity parameter. The effect of particle curvature on water content is not considered.
The hygroscopicity parameter depends on water activity and temperature. This dependency is computed from the water activity parameterizations reported in Zobrist et al. (2008) and Lienhard et al. (2012) as described and visualized in the supporting information.
The glass transition temperature of the mixture is computed using the Gordon-Taylor mixing rule (Gordon and Taylor, 1952) 150 where T g,w is the glass transition temperature of water, T g,s is the glass transition temperature of solute, k GT is the Gordon-Taylor constant and w s (RH) is computed using Eq. (2).
where η(T, RH) is the viscosity, D A is the fragility parameter, T is the temperature, T g is computed using Eq. (3). If T g /T 155 exceeds 1 the compound is a glass and T g /T is set to 1 to ensure numerical stability for all input conditions. Table 1 summarizes the input parameters for sucrose and citric acid used in this study.

Data Summary
Experimental data were fitted with Eq.
(1) to determine T 0 and σ. Temperature and RH corresponding to shape factors ξ = 1.5, ξ = 2.5, and ξ = 3.5 and the associated viscosities are summarized in Table 2. To compute fragility, the temperature 160 Table 2. Data summary for each experiment. The viscosity is evaluated at ξ = 1.5, 2.5 and 3.5. The corresponding viscosity is calculated based on the surface tension (Table 1)  was normalized by the average T g calculated from Eq. (3). Detailed data for each experiment listed in Table 2 are provided in the supporting online information and data repository. Discussion of these data is provided in the Results section.

KM-GAP Simulations
The time to achieve gas-particle equilibrium is computed using the simulations of the kinetic multi-layer model of gas-particle interactions in aerosols and clouds (KM-GAP) (Li and Shiraiwa, 2019;. KM-GAP consists of multiple 165 model layers: gas phase, near-surface gas phase, sorption layer, quasi-static surface layer, and a number of bulk layers. The model includes gas-phase diffusion, adsorption and desorption at the particle surface-bulk exchange, and diffusion through bulk layers. The diffusion coefficient of water (D H20 ) is treated as composition dependent. Parameterizations of D H20 for the sucrose-water system applies the method in Zobrist et al. (2011). The estimation of D H20 in citric acid particles uses a semi-empirical method in Berkemeier et al. (2014) that utilizes the VFT equation and assumes the citric acid aqueous solutions 170 are chemically similar to the sucrose-water system. A total of 2500 simulations were performed for a single compound. The model was initialized with T , RH on an equidistant grid with −90 < T < 90ºC and 0 < RH < 100% and 50 points in each dimension. The time to reach equilibrium, τ , is defined when the variability in composition throughout the drop is < ±1%.
This equilibrium timescale depends on the volatility, bulk-phase diffusivity, surface accommodation coefficient, and particle size (Shiraiwa and Seinfeld, 2012). Contour curves denoting constant time to equilibrium are derived from the 2D scalar field.   (2008) where the data are presented as viscosity and temperature value with respect to the weight percent. The data were normalized by T g calculated from the Gordon-Taylor mixing rule Eq.
(3) and the parameters in Table 1. Then, the viscosity and T g /T points are plotted where the temperature range is between −5ºC to 60ºC for 50 wt % of solutions, −10ºC to 94ºC for 60 wt % of solutions, and 5ºC to 80ºC for 86 wt % of solutions. The data points denoted as ref 3 are calculated as follows: The viscosity 185 and T values are read from Table S1 in Rothfuss and Petters (2017a) for 100 % dry data points (where RH is assumed as 0.8 %), then by using the hygroscopicity parameter, κ m which is calculated through the polynomial coefficients from Table   S1.1 in Marsh et al. (2018), the weight percent of the solution is calculated. Then, it is converted into T g by Eq.
(3) by using the corresponding constants for sucrose taken from Table 1. Thus, the viscosity and T g /T points are plotted against viscosity where the temperature range is between 78.5ºC to 86.8ºC.   Table S1 in Rothfuss and Petters (2017b) and in Marsh et al. (2018), respectively, note that their error range corresponds to the values 200 given in Table S1 in Rothfuss and Petters (2017b).   (Apelblat, 2014;Först et al., 2002;Haynes and Lide, 2011;Laguerie et al., 1976;Omran et al., 2019;Quintas et al., 2006;Simion et al., 2014;Swindells et al., 1958;Telis et al., 2007).  Table 131 in Swindells et al. (1958). RH(aw) data points are read directly from the Table 1 in Mazurkiewicz et al. (2001) where the viscosity is measured by viscometer, and from Table SI Table 2 in Luyet and Rasmussen (1968), from Table 2 in Saleki-Gerhardt and Zografi (1994); read from Table 2 in Elamin et al. (1995). The citric acid data for Figure 4b are tabulated in Marsh et al. (2018); read from Table   225 1 in Lu and Zografi (1997); from Table 2 in Lienhard et al. (2012); from Table1 in Summers and Enever (1980); the literature range for T g of citric acid is taken from Rothfuss and Petters (2017b).
2.5.4 Figure S1 (Supporting Information) Figure S1a: The data is read from Table 2 in Elamin et al. (1995), from Table 2 Table 1 in Summers and Enever (1980); Fig. 5 in Murray (2008); read from Table 2 and digitized from Fig. 1 in Lienhard et al. (2012); read from Table 1 in Lu and Zografi (1997). 3 Results Figure 2 shows the Angell representation of viscosity scaled by T g for sucrose and citric acid. The sucrose data show two apparent regimes, one for aqueous solutions and one for dry sucrose. Dry sucrose thermally decomposes at 456 K (Šimkovic 235 et al., 2003) or T g /T = 0.74, thus explaining the limited range of available measurements. The stratification along a single fragility parameter model for aqueous sucrose solutions over a wide range of solute wt % is remarkable. Angell (2002) noted the systematic difference in fragility between dry sucrose and aqueous solutions based on the two data points at~10 2 to~10 3 Pa s. Since these measurements approach the decomposition temperature, Angell (2002) considered the possibility that these measurements are not representative of sucrose. The more recent measurements by Rothfuss and Petters (2017a) using the 240 DCIC method suggest that dry sucrose has indeed a lower fragility than aqueous sucrose solutions. The data for citric acid aqueous solutions are more scattered. The observed data from this study using sub-100 nm particles are within the range of the published bulk data. Dry citric acid data are not available in the literature or from our DCIC measurements. The dimer relaxation occurs at T ≈ 10ºC in the DCIC system. Drying to RH = 0% at that temperature is currently not possible and low fractions of hygroscopically bound water are expected to be present. Despite the scatter, no systematic variation of fragility 245 with solute wt % is observed.  presented elsewhere (Song et al., 2016;Gervasi et al., 2020). In general, the data investigating the same viscosity range agree within 0.5 orders of magnitude. Few measurements at η > 10 2 P a s are available. Song et al. (2016) and Power et al. (2013) report viscosity for sucrose in this range. Song et al. (2016), Marsh et al. (2018) and this study show citric acid measurements 250 at η > 10 6 P a s and approaching dry conditions. These measurements agree within ±24% when viscosity is expressed as log10 of its value. Predicted viscosity computed via Eq. (4) evaluated at T = 20ºC and fragility parameters in Figure 2 are in excellent agreement with the data at RH > 40%. At RH < 40% the model slightly overestimates the data. The reasons for the small discrepancy are not entirely clear. Possible explanations are a bias in the holographic optical tweezer data, variations in fragility that are not captured with a single parameter representation of fragility, and uncertainty in the aerosol 255 water content associated with the particle. For the latter, the variability in κ m with RH may be insufficiently described by the water activity parameterization. Accurate estimates of water content at low RH, especially below 10%, remain highly uncertain due to experimental challenges (Kreidenweis et al., 2008). Whichever the case, the difference between measurements and model does not appear to be substantially larger than the scatter in the data. Figure 4 shows the variation of viscosity with temperature and RH. The glass transition data coincide with 10 12 Pa s line 260 by the definitions of the T g vs. wt % fits and the wt % vs. water activity relationships used to relate RH and mass fraction.

RH (%)
Deviations from the 10 12 Pa s line of three out of four citric acid data points reported by Summers and Enever (1980) mirror the deviations in Gordon-Taylor mixing plot (see Figure 1 in the supporting information). The viscosity isopleth corresponding to the measurements are shown as a shaded band due to the mismatch in viscosity from prior studies (Rothfuss and Petters, 2017a;   Rothfuss and Petters (2017a) and this study are from the DCIC method. Data lines correspond contours computed from the computed η(T, RH) field evaluated for 10 12 Pa s (blue), 5 × 10 6 to 4.8 × 10 7 Pa s for sucrose (gold shaded area) and 5 × 10 6 to 3.8 × 10 7 Pa s for citric acid (gold shaded area) and 10 4 to 10 −2 Pa s (black). Marsh et al., 2018) using shorter coalescence time than in this study. The uncertainty due to this mismatch is less than the scatter 265 in the data. The measured viscosity data are in reasonable agreement with the modeled isopleth, except for the coldest data point (T = −28.5ºC). Sucrose measurements at even colder temperature were attempted by unsuccessful; no dimer coalescence was observed. The citric acid isopleths show significant curvature when compared to sucrose, with an upward inflection near −20ºC.
This curvature is caused by a decrease in hygroscopicity, i.e. κ m in that temperature range. (The temperature dependence of κ m is shown in Figure 2 in the supporting information). Excellent agreement between measurements and the modeled isopleth 270 is observed between 8.7ºC and 28.5ºC. Measurements between −14.9ºC and 8.7ºC were attempted but unsuccessful due to lacking ability to precisely control the dewpoint temperature of the flows exiting the coalescence chamber. At T < −14.9ºC, the inferred viscosity from the DCIC method is in good agreement with the viscosity isoplethes predicted by the model (Eq. 4), but the data are more scattered. to 4.8 × 10 7 (sucrose, blue shaded area) and 5 × 10 6 to 3.8 × 10 7 (citric acid, red shaded area).
timescale approaches the coalescence timescale (60 s). Figure 6 shows the modeled equilibration timescale evaluate along the viscosity isopleths characterized by the DCIC experiments. The figure suggests that sucrose equilibration timescales approach and exceed the available coalescence time at T −30ºC. Thus the systematic divergence of the data and viscosity model for sucrose at the coldest measured temperatures shown in Figure 4 is explained by kinetic limitations of water uptake. Figure 6 also shows that no kinetic limitations are expected for citric acid at the temperatures probed in this work, which is consistent with the data. The strong curvature of the viscosity isopleth in the temperature/RH state space for citric acid is caused by the temperature and water content dependence of κ m . As noted in previous work (Rothfuss and Petters, 2017a), there is a need to accurately predict the mass fraction of water at low temperature and at low RH to fully characterize the phase diagram. Although this is not an issue here, there are only few studies that quantify equilibrium water content in that regime. For example, measurements of water uptake by secondary organic aerosol (Varutbangkul et al., 2006;Jurányi et al., 2009;Petters et al., 2009;Good et al., 295 2010; Massoli et al., 2010;Chu et al., 2014;Pajunoja et al., 2015) have focused on RH > 50% and room temperature. The lack of appropriate water activity data remains an impediment to fully characterize the phase diagram for other organic aerosols, including limiting full confidence in predictions of phase state in large scale atmospheric models.
The central tenet of this study is the hypothesis that a phase diagram model that involves a water/organic binary mixing rule of the glass transition temperature, a glass transition temperature scaled viscosity fragility parameterization, and a water uptake 300 parameterization connects the four common state spaces used to characterize the amorphous state: glass transition temperature vs. water weight fraction, viscosity vs. temperature, viscosity vs. RH at constant temperature, and viscosity isopleths as function of temperature and RH. This hypothesis is supported for sucrose and citric acid, based on the synthesis of data from a wide range of sources. A key ingredient of the phase diagram model is that the fragility of sucrose and citric acid only weakly depend on water content. This is an empirical and perhaps surprising observation noted previously (Angell, 2002;Longinotti 305 and Corti, 2008) that greatly simplifies the prediction of viscosity. Differences in fragility for dry and aqueous sucrose are apparent in Figure 3. However, the potential dependence of the fragility of organic compounds on water content and the potential dependence of fragility in systems with multiple dry components at different mass ratios will require rigorous testing before it can be confidently applied for other compound classes.
This work also reports new measurements of viscosity at sub-zero temperatures using the dimer coagulation isolation and 310 coalescence (DCIC) method. Several modifications were made to the system to enable measurements to~−30ºC. These include longer coalescence timescale and a more involved drying and temperature control system. Although the data shown here Temperature (  demonstrate that it is feasible, precise temperature and humidity control through the entire system remain challenging. A consequence of this limitation is that the data collected with the DCIC method are not uniformly spaced along the entire~10 7 Pa s viscosity isopleth shown in Figure 4. In addition to technical constraints related to cooling and drying the DMA and 315 coalescence system, water equilibration timescales present a limit to the lowest temperature that can be studied. Increasing the time in the coalescence chamber to >> 60 s is challenging due to the nature of the flow-through system. The exact lower temperature limit depends on the system studied. Lienhard et al. (2015) summarize the temperature dependence of the water diffusion coefficient in pure component model mixtures.
In their compilation of model systems sucrose has the most resistance to diffusion. Based on this we expect that the DCIC method can probably safely be used for T −30ºC.

Conclusions
Literature data of four common empirical state-spaces used to characterize the amorphous phase diagram were compiled including (1) glass transition temperature vs. water weight fraction, (2) viscosity vs. temperature, (3) viscosity vs. RH at constant temperature, and (4) viscosity isopleths as function of temperature and RH. The dimer coagulation, isolation, and coalescence method (DCIC) was modified to enable measurement of viscosity isopleths at sub-zero temperatures down to 325 −30ºC. Colder temperatures might be accessible using this method for substances that do not show kinetic limitations to water uptake. New measurements of the temperature and RH dependence for sucrose and citric acid were obtained using the DCIC method. The new measurements experimentally constrain the~10 7 Pa s viscosity isopleths. Closure between the four state spaces using a semi-empirical phase diagram model is achieved. A critical component of the phase diagram model is the