Secondary organic aerosol (SOA) accounts for a large fraction of submicron
particles in the atmosphere. SOA can occur in amorphous solid or semi-solid
phase states depending on chemical composition, relative humidity (RH), and
temperature. The phase transition between amorphous solid and semi-solid
states occurs at the glass transition temperature (Tg). We have
recently developed a method to estimate Tg of pure compounds
containing carbon, hydrogen, and oxygen atoms (CHO compounds) with molar mass
less than 450 g mol-1 based on their molar mass and atomic O : C
ratio. In this study, we refine and extend this method for CH and CHO
compounds with molar mass up to ∼ 1100 g mol-1 using the number
of carbon, hydrogen, and oxygen atoms. We predict viscosity from the
Tg-scaled Arrhenius plot of fragility (viscosity vs.
Tg/T) as a function of the fragility parameter D. We compiled
D values of organic compounds from the literature and found that D
approaches a lower limit of ∼ 10 (±1.7) as the molar mass
increases. We estimated the viscosity of α-pinene and isoprene SOA as
a function of RH by accounting for the hygroscopic growth of SOA and applying
the Gordon–Taylor mixing rule, reproducing previously published experimental
measurements very well. Sensitivity studies were conducted to evaluate
impacts of Tg, D, the hygroscopicity parameter (κ), and
the Gordon–Taylor constant on viscosity predictions. The viscosity of
toluene SOA was predicted using the elemental composition obtained by
high-resolution mass spectrometry (HRMS), resulting in a good agreement with
the measured viscosity. We also estimated the viscosity of biomass burning
particles using the chemical composition measured by HRMS with two different
ionization techniques: electrospray ionization (ESI) and atmospheric pressure
photoionization (APPI). Due to differences in detected organic compounds and
signal intensity, predicted viscosities at low RH based on ESI and APPI
measurements differ by 2–5 orders of magnitude. Complementary measurements
of viscosity and chemical composition are desired to further constrain
RH-dependent viscosity in future studies.
Introduction
Secondary organic aerosol (SOA) accounts for a large fraction of submicron
particles in the atmosphere and plays an important role in climate, air
quality, and public health (Goldstein and Galbally, 2007; Jimenez et al.,
2009). Traditionally, SOA particles were assumed to be liquid with dynamic
viscosity η below 102 Pa s, but a number of recent studies have
shown that they can also adopt amorphous semi-solid (102≤η≤1012 Pa s) or glassy solid (η>1012 Pa s) states depending
on chemical composition and temperature (Zobrist et al., 2008; Koop et al.,
2011; Huang et al., 2018; Reid et al., 2018). The phase state is also
strongly affected by relative humidity, as water can act as a plasticizer to
lower viscosity (Mikhailov et al., 2009). Ambient and laboratory-generated
SOA particles have been observed to bounce off the smooth hard surface of an
inertial impactor at low RH, implying a non-liquid state (Virtanen et al.,
2010; Saukko et al., 2012; Bateman et al., 2015; Jain and Petrucci, 2015),
whereas predominantly biogenic SOA particles in the Amazon basin did not
bounce off the impactor surface at high RH, implying that they are primarily
liquid (Bateman et al., 2016). Upon dilution or heating, SOA particles were
observed to evaporate unexpectedly slowly (Cappa and Wilson, 2011; Vaden et
al., 2011), and recent modeling studies have evaluated the contributions of
low diffusivity and volatility to slow evaporation rates (Roldin et al.,
2014; Yli-Juuti et al., 2017). Measurements of the viscosity of SOA bulk material
derived from the oxidation of α-pinene (Renbaum-Wolff et al., 2013; Zhang
et al., 2015; Hosny et al., 2016), limonene (Hinks et al., 2016), isoprene
(Song et al., 2015), and toluene (Song et al., 2016a) have confirmed that SOA
particles adopt a wide range of viscosities.
Viscosity can be directly converted to bulk diffusivity in organic molecules
using the Stokes–Einstein equation (Einstein, 1905; Atkins, 1998; Seinfeld
and Pandis, 2006; Schmelzer and Gutzow, 2011). This equation has been shown to
work well for organic molecules diffusing through materials with η
below ∼ 103 Pa s (Price et al., 2016; Chenyakin et al., 2017).
Note that this relation is not accurate for predicting the bulk diffusivity
of water and small molecules and it may also underestimate the diffusivity of
organic molecules in a highly viscous matrix by a few orders of magnitudes
(Champion et al., 2000; Shiraiwa et al., 2011; Power et al., 2013; Marshall
et al., 2016; Bastelberger et al., 2017; Reid et al., 2018). The particle phase state, viscosity, and bulk diffusivity have been shown to affect the gas
uptake and chemical transformation of organic compounds due to the kinetic
limitations of bulk diffusion (Shiraiwa et al., 2011; Abbatt et al., 2012;
Kuwata and Martin, 2012; Zhou et al., 2013; Slade and Knopf, 2014; Arangio et
al., 2015; Davies and Wilson, 2015; Wang et al., 2015; Berkemeier et al.,
2016; Marshall et al., 2016; Liu et al., 2018; Pratap et al., 2018; Zhang et
al., 2018), which may facilitate the long-range transport of organic compounds
embedded in viscous or glassy particles (Shrivastava et al., 2017b; Mu et
al., 2018). Molecular motion can be hindered in a highly viscous matrix,
slowing down photochemical reactions in particles (Lignell et al., 2014;
Hinks et al., 2016). Water diffusion can still be fast even in an amorphous
solid matrix under room temperature, but it can be hindered significantly
under low temperatures (Mikhailov et al., 2009; Zobrist et al., 2011; Bones
et al., 2012; Berkemeier et al., 2014; Price et al., 2014), affecting
homogeneous vs. heterogeneous ice nucleation pathways (Murray et al., 2010;
Wagner et al., 2012; Wang et al., 2012a, b; Wilson et al.,
2012; Baustian et al., 2013; Schill and Tolbert, 2013; Berkemeier et al.,
2014; Schill et al., 2014; Lienhard et al., 2015; Ignatius et al., 2016;
Knopf et al., 2018). Despite the substantial implications of the SOA particle phase state, its effects on gas–particle interactions have not yet been
considered explicitly in current climate and air quality models
(Shrivastava et al., 2017a).
The partitioning of semi-volatile compounds into viscous particles may result in
kinetically limited growth in contrast to quasi-equilibrium growth
(Perraud et al., 2012; Booth et al., 2014; Zaveri et al., 2014), which
also affects the evolution of particle size distribution upon SOA growth
(Shiraiwa et al., 2013; Zaveri et al., 2018). Note that the equilibration
timescale of SOA partitioning is determined by bulk diffusivity or viscosity,
but is also affected by other factors such as volatility, accommodation
coefficient, particle size, and mass loadings (Shiraiwa and Seinfeld, 2012;
Mai et al., 2015; Liu et al., 2016). Chamber experiments probing the mixing
timescales of SOA particles derived by the oxidation of various precursors such
as isoprene, terpene, and toluene have observed strong kinetic limitations at
low RH, but not at moderate and high RH (Loza et al., 2013; Ye et al., 2016, 2018). Several studies have observed the kinetic limitations of the bulk
diffusion of organic molecules including polycyclic aromatic hydrocarbons
(Abramson et al., 2013; Zhou et al., 2013) and isoprene-derived epoxydiols
(Zhang et al., 2018) in SOA, while Gorkowski et al. (2017)
did not observe significant diffusion limitations for glycerol and squalene
in α-pinene SOA. Quasi-equilibrium versus kinetically limited or
non-equilibrium SOA growth remains an open issue and warrants further
investigations.
Group contribution methods have been used to predict the viscosities of pure
compounds when the functionality and molecular structure are known
(Sastri and Rao, 1992; Rothfuss and Petters, 2017a).
Song et al. (2016b) showed that estimations from group contribution
approaches combined with either non-ideal or ideal mixing reproduced the
RH-dependent trends particularly well for the alcohol, dicarboxylic, and tricarboxylic
acid systems with viscosities of up to 104 Pa s. In contrast, model
calculations overestimated the viscosity of more viscous compounds including
monosaccharides, disaccharides, and trisaccharides by many orders of magnitude
(Song et al., 2016b). A recent study compiled the viscosity of
organic compounds with atmospherically relevant functional groups,
investigating the influence of the number and location of functional groups
on viscosity (Rothfuss and Petters, 2017a). These studies provide important
insights into estimating the viscosity of individual organic compounds.
The particle phase state can be characterized by a glass transition temperature
(Tg), which is a characteristic temperature representing a
non-equilibrium phase transition from a glassy solid state to a semi-solid
state as the temperature increases (Koop et al., 2011). Recently, we have
developed a parameterization to estimate Tg of pure organic
compounds comprised of carbon, hydrogen, and oxygen (CHO compounds) with
molar mass less than 450 g mol-1 based on their molar mass and atomic
O : C ratio (Shiraiwa et al., 2017). It has been applied successfully in a
global chemistry–climate model to predict Tg and the phase state
of atmospheric SOA, which indicated that SOA particles are mostly liquid or
semi-solid in the planetary boundary layer, while they should be glassy in
the middle and upper troposphere (Shiraiwa et al., 2017). A recent study
provided a consistent result, suggesting that the mixing timescales of organic
molecules within SOA are often < 1 h in a global planetary boundary layer
(Maclean et al., 2017).
It has been shown that SOA particles contain oligomeric compounds with molar
masses higher than 450 g mol-1 (Gao et al., 2004; Tolocka et al.,
2004; Nizkorodov et al., 2011; Nozière et al., 2015), which makes the
previously developed parameterization incomplete. In this study, we extend
the parameterization of Tg to higher-molar-mass compounds and
apply it to high-resolution mass spectrometry data for toluene SOA and
biomass burning particles. The Arrhenius approach and the Gordon–Taylor
mixing rules were applied to estimate the viscosity of SOA bulk materials to
compare with the literature-reported viscosity measurements. This method will
be useful for estimations of viscosity of organic particles, for which
high-resolution mass spectra are available. It can also be applied in global
or regional models to evaluate the impacts of the particle phase state on the
role of SOA in climate and air quality.
Parameterization developmentGlass transition temperature
Characteristic relationships between molecular properties and the
glass transition temperature (Tg) of organic compounds.
(a)Tg of organic compounds as measured (circles) and estimated with
the Boyer–Kauzmann rule (squares) plotted against molar mass. The markers are
color coded by atomic O : C ratio. (b) Measured (circles) and estimated
(squares) Tg of organic compounds plotted against the O : C ratio.
The markers are color coded by molar mass. (c) Predicted Tg for
CHO compounds using a parameterization (Eq. 2) developed in this study
compared to measured (circles) and estimated Tg by the
Boyer–Kauzmann rule (squares). The solid line shows the 1:1 line and the dashed
and dotted lines show 68 % confidence and prediction bands, respectively.
Figure 1a shows the dependence of Tg on the molar mass (M) of
organic compounds. Solid markers represent measured Tg of 258 CHO
compounds (Koop et al., 2011; Dette et al., 2014; Rothfuss and Petters,
2017a), while open markers represent 654 CHO compounds in SOA
(Shiraiwa et al., 2014). Markers are color coded by atomic
O : C ratio. Their melting points (Tm) were estimated by the
Estimation Programs Interface (EPI) Suite software version 4.1 (US-EPA, 2015)
and their Tg values were estimated using the Boyer–Kauzmann rule:
Tg=g⋅Tm with g=0.7
(Koop et al., 2011; Shiraiwa et al., 2017). This rule can provide good
estimates of Tg, as has been validated in previous work (Koop et
al., 2011) and is also shown in Fig. A2a. A subset of data shown in Fig. 1 was
originally published in Shiraiwa et al. (2017) for compounds with M<450 g mol-1. This version of the figure has been updated to include
a number of experimentally measured Tg values of larger compounds
with M up to 1153 g mol-1, including aliphatic compounds containing
OH and/or COOH groups. Specifically, data for 76 aliphatic alcohols, 39
carbohydrates and their derivatives, 4 carboxylic acids, and 4 hydroxy acids,
as compiled by Rothfuss and Petters (2017a), have been added to Fig. 1. Eight
of these compounds are carbohydrates with M>450 g mol-1. These
updates are critical for reliable parameterization of Tg based on
M. When M increases above ∼ 500 g mol-1, the slope of
Tg decreases, making it challenging to extrapolate the low-M
data from the original Shiraiwa et al. (2017) study to higher M values.
When M increases to ∼ 1000 g mol-1, the corresponding
Tg appears to level at around 420 K.
Such dependence on M has been described for polymers with the Fox–Flory
equation: Tg(M)=Tg,∞-KmM (Fox Jr. and Flory, 1950), where Km is
a constant and Tg,∞ is the asymptotic value of
Tg specific to the polymer. We conducted a literature search and
found that most of the reported Tg,∞ values fell below
∼ 500 K (Brandrup et al., 1999; Fox Jr. and Flory, 1950; Onder et al., 1972; Montserrat and
Colomer, 1984; Papadopoulos et al., 2004; Matsushima
et al., 2017). The Fox–Flory equation works very well for high-molar-mass
compounds and is also generally applicable to smaller compounds (Koop et al.,
2011), as supported by an approximately linear dependence of Tg
on the inverse molar mass in Fig. A1a. Figure 1b plots the values of
Tg as a function of the atomic O : C ratio of organic
molecules. Figure 1a and b clearly demonstrate that Tg depends
primarily on the molar mass with a weak dependence on the atomic O : C
ratio.
A parameterization for Tg calculation based on the molar mass and
atomic O : C ratio was developed in our recent work, which is applicable to
CH and CHO compounds with M<450 g mol-1 (Shiraiwa et al., 2017):
Tg=A+BM+CM2+D(O:C)+EM(O:C),
where A=-21.57 (±13.47) [K], B=1.51 (±0.14)
[K mol g-1], C=-1.7×10-3 (±3.0×10-4)
[K mol2 g-2], D=131.4 (±16.01) [K], and E=-0.25 (±0.085) [K mol g-1]. These values were obtained by
fitting the measured Tg of 179 CH and CHO compounds with M<450 g mol-1 with multi-linear least squares analysis. Note that the
application of Eq. (1) may provide unreasonable Tg values for
compounds with M>500 g mol-1 because it does not account for the
strong curvature in the Tg vs. M dependence shown in Fig. 1a.
In this study we have developed an improved parameterization to predict
Tg of CH and CHO compounds using the number of carbon
(nC), hydrogen (nH), and oxygen (nO) that can
also be applied to higher-molar-mass compounds. Motivated by a good
correlation between Tg and volatility (Fig. 1a in Shiraiwa et
al., 2017), we use an equation with a similar formulation to the equation
used to predict the saturation mass concentration or volatility (Donahue et
al., 2011; Li et al., 2016):
Tg=(nC0+ln(nC))bC+ln(nH)bH+ln(nC)ln(nH)bCH+ln(nO)bO+ln(nC)ln(nO)bCO,
where nC0 is the reference carbon number, bC,
bH and bO denote the contribution of each atom to
Tg, and bCH and bCO are coefficients that
reflect contributions from carbon–hydrogen and carbon–oxygen bonds,
respectively. These values were obtained by fitting the measured
Tg of 42 CH compounds and 258 CHO compounds with multi-linear
least squares analysis with 68 % prediction and confidence intervals. The
best-fit parameters are summarized in Table 1.
Composition classes and the nC0 and b values (K) for glass
transition temperature parameterizations obtained by least-squares
optimization using the measurements compiled in Koop et al. (2011), Dette et
al. (2014), and Rothfuss and Petters (2017a).
Note that the evaluation dataset used to derive Eq. (2) contains CH compounds
with M<260 g mol-1 (see Fig. A2b for comparison of measured and
predicted Tg). Thus, the application of Eq. (2) to higher-molar-mass compounds may require further refinement of the method when measured
Tg for higher-molar-mass CH compounds becomes available.
Figure 1c shows that the Tg values predicted using Eq. (2) are in
good agreement with the Tg values measured in experiments (see
also Fig. A1b) or estimated by the Boyer–Kauzmann rule as indicated by the
high correlation coefficient of 0.95. Tg of individual compounds
can be predicted within ±21 K as indicated by the prediction band
(dotted lines in Fig. 1c); however, this uncertainty may be much smaller for
multicomponent SOA mixtures under ideal mixing conditions as indicated in the
confidence band (dashed lines, almost overlapping with the 1:1 line).
These results are noteworthy given that the parameterization (Eq. 2) does not
consider either explicit molecular structures or functional groups. Previous
studies have shown that Tg can be especially sensitive to the
number of OH groups, which interact strongly through hydrogen bonding. For
example, Nakanishi and Nozaki (2011) found a direct relationship between
Tg and the number of hydroxyl groups in a molecule for sugar
alcohols; Tg increases as the number of OH groups increases. They
reported that the correlation between Tg and the number of OH
groups was much stronger than the correlation between Tg and the
number of carbons in a molecule. Such a trend is implicitly included in
Eqs. (1) and (2), which contain the O : C ratio and number of oxygen atoms
as parameters, respectively. Recently, Rothfuss and Petters (2017a) showed an
approximately linear relationship between the number of OH groups and
Tg for compounds with up to eight OH groups.
Grayson et al. (2017) showed that the addition of
hydroxyl functional groups increases viscosity, a conclusion supported by
both the experimental data and quantitative structure–property relationship
model. The correlation between Tg and the number of carbon atoms
is consistent with the free volume theory, in which molecular motion is
restricted by the difference between the space required for a molecule to
vibrate versus the space in which the molecule resides (i.e., the free
volume; White and Lipson, 2016). The correlation between Tg and
the number of OH groups is more consistent with the topological constraint
theory, in which the primary influence is the three-dimensional structure of the
molecule as determined by molecular bonds and hydrogen-bonding networks
(Nakanishi and Nozaki, 2011; van der Sman, 2013). Future experiments
targeting more comprehensive Tg data, especially for high-molar-mass compounds, would lead to further refinements of our Tg
parameterizations.
Comparing Eqs. (1) and (2), the two parameterizations give a similar
performance for compounds with M<450 g mol-1 as shown in Fig. A2c.
The statistical measures of correlation coefficient (R), mean bias (MB),
and root mean square error (RMSE) are 0.93, -6.45, and 25.64 K,
respectively, for the performance of Eq. (1), while for Eq. (2), they are
0.95, 3.15, and 21.11 K, respectively. It should be noted again that Eq. (1)
cannot be used to predict Tg for compounds with M>450 g mol-1. For example, Tg of stachyose (M=667 g mol-1) predicted by Eq. (1) is 198 K, while that by Eq. (2) is
394 K, which agrees much better with the measured mean Tg of
396 K (Rothfuss and Petters, 2017a). Equation (2) is more flexible than
Eq. (1) and can be potentially expanded to include compounds containing
hetero-atoms (e.g., nitrogen or sulfur) once substantial sets of
experimental values of Tg for such compounds become available.
Regarding the applications to air quality and climate models, Eq. (1) can be
applied in the volatility basis set (VBS; Donahue et al., 2006, 2011) and
the molecular corridor approach (Shiraiwa et al., 2014; Li et al., 2016) to
predict the Tg of SOA particles (Shiraiwa et al., 2017), while
the new parameterization may be suitable for coupling with the statistical
oxidation model, which characterizes the SOA evolution as a function of
nC and nO (Cappa and Wilson, 2012; Jathar et al., 2015).
These parameterizations (Eqs. 1, 2) calculate Tg based on the
elemental composition of organic compounds. SOA particles contain a number of
organic compounds and a variable amount of liquid water, which has low
Tg (136 K) and can act as a plasticizer (Mikhailov et al., 2009;
Koop et al., 2011). Under humid conditions, SOA particles take up water by
hygroscopic growth in response to RH, lowering Tg and the viscosity
of SOA particles. Estimations of Tg for SOA–water mixtures were
discussed by Shiraiwa et al. (2017), who applied the Gordon–Taylor equation
validated for a wide range of mixtures of organics, polymers, and water
(Roos, 1993; Hancock and Zografi, 1994; Zobrist et al., 2008; Dette et
al., 2014; Dette and Koop, 2015). Briefly, Tg of mixtures of SOA
compounds under dry conditions (Tg,org) were calculated assuming
the Gordon–Taylor constant (kGT) of 1 (Dette et al., 2014):
Tg,org=∑iwiTg,i, where wi is
the mass fraction of organic compound i, which can be derived using mass
concentrations of SOA products. The Gordon–Taylor equation can also be
applied to calculate Tg of organic–water mixtures considering the
mass fraction of organics (worg) in SOA particles (Koop et al.,
2011):
Tg(worg)=(1-worg)Tg,w+1kGTworgTg,org(1-worg)+1kGTworg;worg can be calculated using the mass concentrations of water
(mH2O) and SOA (mSOA) as worg=mSOA/(mSOA+mH2O), and mH2O can be estimated using the
effective hygroscopicity parameter (κ) (Petters and Kreidenweis,
2007):
mH2O=κρwmSOAρSOA1aw-1.
The density of water (ρw) is 1 g cm-3, the density of
SOA particles (ρSOA) is assumed to be 1.2 g cm-3 (Kuwata
et al., 2012), mSOA is the total mass concentrations of SOA, and
aw is the water activity calculated as aw=RH/100. Pajunoja et al. (2015) found that water uptake in
subsaturated conditions is inhibited until RH is high enough for the dissolution
of water in SOA particles with relatively low O : C ratios. As the oxidation of
SOA increases, the solubility of water increases and dissolution occurs at lower
RH values. In both cases, the use of subsaturated hygroscopicity measurements
was supported.
Viscosity
The temperature dependence of viscosity (η) can be predicted using the
modified Vogel–Tammann–Fulcher (VTF) equation (Angell, 1991):
η=η∞eT0DT-T0,
where η∞ is viscosity at infinite temperature; T0
is the Vogel temperature; and T is the ambient temperature. The fragility
parameter, D, characterizes how rapidly the dynamics of a material slow
down as T approaches Tg, reflecting to what degree the
temperature dependence of the viscosity deviates from Arrhenius behavior.
When T is close to Tg (Tg/T≈1), smaller D
values indicate that viscosity is sensitive to temperature change (fragile
behavior), while larger D values indicate that viscosity is less sensitive
to temperature change (strong or Arrhenius behavior).
Assuming η∞=10-5 Pa s (Angell, 1991),
logη=-5+0.434T0DT-T0.
When T=Tg, η=1012 Pa s, which leads to
(Angell, 1991, 2002)
T0=39.17TgD+39.17.
As can be seen in Eq. (7), both Tg and D are required to
calculate η from Eq. (6) at a given temperature.
The Angell plot of viscosity (η) vs. Tg/T. The
lines represent different fragility parameter (D) values in the range of 5–100,
with D=10 (the solid line) used as a base case for this study. A
large fragility parameter value is associated with a strong glass former,
while fragile materials are associated with lower values. The black dashed
line at a viscosity of 102 Pa s indicates the approximate threshold
between liquid and semi-solid states.
Figure 2 shows the Tg-scaled Arrhenius plot of fragility
(viscosity versus Tg/T) referred to as an Angell plot (Angell,
1995). D values of organic compounds are typically in the range of
∼ 5–30 (Angell, 1997). To estimate D values that could be applied to
SOA compounds, we compiled measured fragility values. Fragility was often
measured in the form of the fragility steepness index (m), which represents
the slope of the Arrhenius plot at the point where T=Tg
(Boehmer et al., 1993). Compounds with lower m exhibit higher D values,
indicating stronger glass formers. The measured m values of 95 organic compounds
are included in the Supplement; m can be converted to D using the
following equation (see the full derivation of this equation in Appendix A):
D=665.89m-17.
Figure 3 shows the measured D as a function of (a) molar mass and
(b) the atomic O : C ratio of organic molecules. The molar mass exerts a
stronger effect on fragility, while there is little dependence of D on the
O : C ratio. As molar mass increases, D approaches a lower limit of 10.3
(±1.7), consistent with the value of 10 used in our recent study
(Shiraiwa et al., 2017). To evaluate the impact of the variations in D on
viscosity prediction, sensitivity calculations were conducted as described in
Sect. 3.
Fragility parameter of organic compounds (D) plotted against (a) molar
mass and (b) atomic O : C ratio. Error bars are standard deviations.
The solid red lines represent the fitted curves with fitted equations for (a)D=602.6/M+10.3 and (b)D=14.4–2.3 (O : C). Dashed
red lines indicate the 95 % confidence band.
Besides the VTF equation, another commonly used equation for describing the
temperature dependence of viscosity is the Williams–Landel–Ferry (WLF)
equation: logη(T)η(Tg)=-C1(T-Tg)C2+(T-Tg), where
empirical parameters C1 and C2 are adopted as 17.44 and 51.6 K,
respectively (Williams et al., 1955; Schill and Tolbert, 2013; Wang et al.,
2015). The two equations are mathematically equivalent; both are defined with
respect to a reference temperature, and their parameters are related through
C1=DT02.303(Tg-T0) and
C2=Tg-T0. For the WLF equation, Tg is the
reference temperature and there is a linear dependence assumed between
temperature and free volume (O'Connell and McKenna, 1999; Huang and McKenna,
2001; Metatla and Soldera, 2007). For the VTF equation, the reference is the
Vogel temperature (T0), a hypothetical temperature at which all
non-vibrational motion ceases and viscosity becomes infinite, and the
theoretical foundation of the VTF equation includes both thermodynamic and
kinetic considerations (O'Connell and McKenna, 1999; Huang and McKenna, 2001;
Metatla and Soldera, 2007). Recently, Rothfuss and Petters (2017b) applied a
similar approach to model viscosity for sucrose particles by applying the VTF
and Gordon–Taylor approaches. The calculations of the viscosity of
multi-component SOA mixtures in this study are based mainly on the VTF
equation, and the difference between calculated results from the two equations
will be briefly discussed in the following section.
Comparison of predicted viscosity with measurementsSOA formed from <inline-formula><mml:math id="M254" display="inline"><mml:mi mathvariant="italic">α</mml:mi></mml:math></inline-formula>-pinene and isoprene
The purpose of this section is to demonstrate that the viscosity of SOA material
can be predicted over a broad range of RH values from four parameters:
Tg of dry SOA (Tg,org), fragility (D),
hygroscopicity (κ), and the Gordon–Taylor constant for mixing SOA and
water (kGT). The viscosity of α-pinene SOA has been measured
or estimated as a function of RH by several groups using multiple
experimental techniques as shown in Fig. 4a (Abramson et al., 2013;
Renbaum-Wolff et al., 2013; Kidd et al., 2014; Pajunoja et al., 2014; Bateman
et al., 2015; Zhang et al., 2015; Grayson et al., 2016). The wide range of
experimentally measured viscosities reported for α-pinene SOA,
particularly from 30–60 % RH, is most likely a consequence of the different
experimental approaches, mass loadings, and O : C ratios for each
experiment. For instance, Grayson et al. (2016) used mass loadings of 121 to
14 000 µg m-3 and observed that viscosity decreased as mass
loading increased. Higher mass loadings would lead to greater partitioning of
semi-volatile and lower-molar-mass compounds into the particle phase, which
would lead to the decrease in Tg and the viscosity of the resulting
SOA mixture, as very recently demonstrated experimentally by Jain et
al. (2018). Grayson et al. (2016) concluded that their results should be
considered a lower limit for the viscosity of α-pinene SOA in the
atmosphere. It should also be noted that the viscosity measurements from
Renbaum-Wolff et al. (2013) were for the water-soluble portion of the SOA.
These datasets suggest that the viscosity of α-pinene SOA approaches very
high values (∼> 108 Pa s) below 20–30 % RH and decreases
with an increase in RH, reaching a value of ∼ 10 Pa s at 80 % RH. As
can be seen in Fig. 4b, PAM-generated isoprene SOA is less viscous with η<106 Pa s even under dry conditions, undergoing a phase transition
from a semi-solid phase to a liquid phase at ∼ 55 % RH (Bateman et
al., 2015; Song et al., 2015).
Comparison of the measured and predicted viscosity of (a)α-pinene SOA and (b) isoprene SOA at 295 K as a function of RH. The solid
lines represent base simulations with the VTF equation, while the dotted line
represents viscosity predicted using the WLF equation. Parameters are the glass
transition temperature of dry SOA (Tg,org), fragility (D),
hygroscopicity (κ), and the Gordon–Taylor constant (kGT): (a) 278.5 K,
0.1, 10, and 2.5; (b) 255 K, 0.1, 10, and 2.5. The shaded regions were
determined by varying these parameters. (a) Upper (lower) limit:
Tg,org=300 K (278.5 K), κ=0.1 (0.1), D=20 (10),
kGT=2.5 (2.0); (b) upper (lower limit): Tg,org=255 K (255 K), κ=0.10 (0.15), D=15 (8), kGT=2.5 (4.0). Panel (a): Renbaum-Wolff et al. (2013) data represent
viscosity for the water-soluble portion of SOA; Grayson et al. (2016) data in
panel (a) represent two different mass loadings (121 µg m-3;
520 µg m-3). Panel (b): the gray box represents
estimated viscosity for isoprene SOA based on the bounce measurements of Bateman
et al. (2015).
The solid lines with the shaded areas in Fig. 4 are viscosity values
predicted using Tg,org, D, κ, and kGT.
Tg,org values were adopted by Berkemeier et al. (2014), who
estimated Tg,org with the Boyer–Kauzmann rule using the melting
point of representative SOA oxidation products. Note that Eqs. (1) and (2)
were not used to estimate Tg,org, which should be done in future
studies by obtaining their elemental composition using high-resolution mass
spectrometry. For α-pinene, Tg,org was assumed to be
278 K, corresponding to an O : C ratio of 0.5 (Berkemeier et al., 2014),
which is a typical O : C ratio of α-pinene SOA (Aiken et al., 2008;
Chen et al., 2011; Putman et al., 2012).
The Tg,org selected for isoprene SOA was 255 K, corresponding to
the O : C ratio of 0.6. Although no measurements of the O : C ratio for
the experimental isoprene SOA data were reported, Song et al. (2015)
estimated O : C of 0.64–1.1 based on literature values. As O : C ratios
are useful in estimating Tg,org, we encourage the measurement of
the O : C ratio of SOA when conducting viscosity measurements. In contrast
to α-pinene SOA, there are limited viscosity measurements for
isoprene SOA. While the predicted viscosity is consistent with the
experimental data, comparison of our model predictions to additional
measurements is strongly recommended. Song et al. (2015) prepared isoprene
SOA in a potential aerosol mass (PAM) reactor, while the data produced by Bateman et
al. (2015) were for isoprene SOA generated in a smog chamber. It has been
suggested that under ambient conditions, the majority of isoprene-derived SOA
can be derived through heterogeneous interactions with acidic sulfate
particles forming oligomers (Surratt et al., 2010; Lin et al., 2013; Gaston
et al., 2014), which may increase viscosity compared to the model SOA
generated in PAM or a chamber. Further studies are warranted to compare
laboratory-generated and ambient isoprene SOA and to investigate the effect
of the acidic seed on the viscosity.
For both α-pinene and isoprene SOA, D was set to 10 based on the
analysis presented in Fig. 3a. κ was set to 0.1 based on field and
laboratory measurements (Gunthe et al., 2009; Lambe et al., 2011b; Pajunoja
et al., 2014; Petters et al., 2017) and kGT was assumed to be 2.5
(Zobrist et al., 2008; Koop et al., 2011). Using these parameters, the
predicted viscosities match the magnitude and the RH-dependence of the
measured viscosity of α-pinene and isoprene SOA. Figure 4 also shows
predicted viscosities (dotted lines) using the WLF equation, which shows
similar values as the VTF equation, but slightly underestimates the viscosity
of α-pinene SOA at low RH and overestimates the viscosity of isoprene
SOA at high RH.
Sensitivity studies were conducted to examine the effects of
Tg,org, D, κ, and kGT on the calculated
viscosity. In these studies, Tg,org values of α-pinene and
isoprene SOA were varied within 229–328 and 255–316 K, respectively,
representing Tg,org of different oxidation states (Berkemeier et
al., 2014). D was varied between 5 and 30, which is the range
characteristic for organic compounds (see Fig. 3a). κ values of 0.05–0.15
were used for α-pinene and isoprene SOA (Lambe et al., 2011b;
Pajunoja et al., 2015). For the Gordon–Taylor constant, values of
2.5 ± 1.5 were considered (Zobrist et al., 2008; Koop et al., 2011;
Dette et al., 2014; Dette and Koop, 2015).
The effect of varying each parameter on the calculated viscosity of α-pinene SOA is illustrated in Fig. 5. Variations of ±50 K in
Tg,org result in 3–6 order of magnitude differences in
calculated values at dry conditions, indicating that Tg,org is a
critical parameter for viscosity estimations. Decreasing D from 10 to 5 led
to a decrease in calculated values by more than 1 order of magnitude. The
calculated results were within the upper limit of measurements when
increasing D from 10 to 20, and the predicted values were only slightly
enhanced when further increasing D from 20 to 30. Calculated values with
variations in κ from 0.05 to 0.15 and kGT from 1.0 to 4.0
were all within the measured ranges.
Sensitivity calculations for the viscosity of α-pinene SOA at
295 K as a function of RH by varying (a) the glass transition temperature of dry
SOA (Tg,org), (b) fragility (D), (c) hygroscopicity (κ), and (d) the Gordon–Taylor constant (kGT).
Sensitivity calculations for the viscosity of isoprene SOA at 295 K as
a function of RH by varying (a) the glass transition temperature of dry SOA
(Tg,org), (b) fragility (D), (c) hygroscopicity (κ),
and (d) the Gordon–Taylor constant (kGT). Data points are measured
viscosity by Song et al. (2015) and the gray box represents estimated
viscosity based on the bounce measurements of Bateman et al. (2015).
For isoprene SOA, an increase in Tg,org to 287 K, which
represents a higher oxidation state (Berkemeier et al., 2014), led
calculated values to be several orders of magnitude higher than the upper
limit of measurements (Fig. 6a). When Tg,org reaches 316 K,
isoprene SOA can occur as a solid for RH lower than ∼ 40 %. Compared
to α-pinene SOA, a variation in D has a larger effect on the
calculated viscosity (Fig. 6b). For a range of 5–30 for D, calculations
with the D value of 10 agreed well with the measurements, while other D
values resulted in calculated viscosity outside of the measured ranges.
Figure 6c and d show that by decreasing κ and kGT below the
reference values, the predictions overestimate the measured η by 1 or
2 orders of magnitude. The latter is most evident at RH > 60 %, at
which
the calculated values were higher than the upper limit of measurements.
Modeling results with κ and kGT increasing to 0.15 and
4.0, respectively, were within the lower limit of measurements.
The above comparison between the measured and predicted viscosity
demonstrates that the method described in this study can reproduce reasonably
well the measured RH-dependent viscosity of SOA formed from α-pinene
and isoprene. The sensitivity calculations showed that Tg,org
contributed the most to the uncertainty in the viscosity estimates. Previous
studies have shown that the experimental conditions such as particle mass
concentrations (Grayson et al., 2016; Jain et al., 2018) and RH upon SOA
formation (Kidd et al., 2014; Hinks et al., 2018) can impact the chemical
composition of SOA and hence the phase state and viscosity. Further efforts
to constrain the uncertainties are needed both in experiments and
parameterizations.
SOA formed from toluene
In this and the following sections, we examine the feasibility of calculating
the value of Tg,org from mass spectrometry data on SOA. Hinks et
al. (2018) measured the elemental composition of toluene SOA using nanospray
desorption electrospray ionization high-resolution mass spectrometry
(nano-DESI-HRMS; Roach et al., 2010a, b). Toluene SOA was formed by OH
photooxidation in an aerosol smog chamber at < 2 % RH (mass loading =
23 µg m-3) and 75 % RH (mass loading =
8 µg m-3) to investigate the effect of RH on the chemical
composition of toluene SOA formed under low-NOx conditions. Measurements
revealed a significant reduction in the fraction of oligomers present in
toluene SOA generated under high RH conditions compared to SOA generated
under low RH conditions (Hinks et al., 2018). The detected molar mass of
individual oxidation products spanned a range of 102–570 g mol-1 at
high RH, which increased up to 726 g mol-1 at low RH.
(a) Molecular corridor of molar mass plotted against the volatility of
toluene SOA formed under dry conditions (Hinks et al., 2018) color coded by
the glass transition temperature (Tg) estimated using Eq. (2). The
upper dashed line indicates the low O : C bound of the molecular corridor
(linear alkanes CnH2n+2 with O : C = 0), and the lower
dotted line indicates the high O : C bound (sugar alcohols
CnH2n+2On with O : C = 1). (b) Comparison of
the measured (markers) and modeled (lines) viscosity of toluene SOA at 295 K as a
function of RH. Viscosities were calculated using a fragility (D) of 13,
hygroscopicity (κ) of 0.25, and a Gordon–Taylor constant
(kGT) of 3.0 with different glass transition temperatures of dry
SOA (Tg,org) as estimated using Eqs. (1) or (2) under low and high
RH conditions. The shaded regions were calculated by varying those
parameters: Tg,org=313 K (295 K), κ=0.20 (0.25), D=13 (10), kGT=2.5 (3.5) for the upper (lower)
limit. Mass loadings were 23 µg m-3 for low RH and
8 µg m-3 for high RH (Hinks et al., 2018).
Mass spectra of biomass burning organic particles collected from
test burns of (a) subalpine fir and (b) lodgepole pine as measured by high-resolution mass spectrometry with two ionization techniques: electron spray
ionization (ESI, red) and atmospheric pressure photoionization (APPI; blue).
The numbers of elemental formulas identified by ESI (red), APPI (blue), and both
modes (black) are also specified. Van Krevelen plots of the compounds
identified by ESI (red) and APPI (blue) mode in BBOA from the burning of (c) subalpine fir and (d) lodgepole pine.
Figure 7a shows the interdependence of glass transition temperature,
volatility, and the molar mass of the detected toluene SOA compounds. Glass
transition temperatures were calculated using Eq. (2). The saturation mass
concentrations or volatilities of detected compounds were estimated from the
elemental composition by using the parameterization of Li et al. (2016). The
analysis is based on the molecular corridor approach – a two-dimensional
framework of volatility and molar mass of SOA components constrained by
boundary lines of low and high atomic O : C ratios corresponding to
n-alkanes (CnH2n+2, O : C = 0) and sugar alcohols
(CnH2n+2On, O : C = 1), respectively (Shiraiwa et
al., 2014; Li et al., 2016). The toluene SOA constituents are well
constrained by the molecular corridor and Tg values are higher for
compounds with a higher molar mass and lower volatility.
Equation (1) was used to calculate Tg for individual compounds
with M<450 g mol-1, while excluding compounds with a molar mass
higher than 450 g mol-1. This approach was deemed reasonable as such
high-molar-mass compounds account for < 10 % of all toluene SOA products
formed at low RH and for < 2 % formed at high RH. Equation (2) was used
to calculate Tg for all the detected compounds. Tg of
dry toluene SOA (Tg,org) was then computed using the
Gordon–Taylor approach with kGT=1 (Sect. 2.1). The relative
mass concentrations of individual components were assumed to be proportional
to their relative abundance in the nano-DESI-HRMS spectrum. This assumption
has a number of caveats (Bateman et al., 2012; Nguyen et al., 2013), and as
we will see below, it results in deviations between the predicted and
measured viscosity. Table 2 summarizes the results of such calculations,
showing that the Tg,org by Eq. (1) – excluding high-molar-mass
compounds – is about 10 K lower compared to Tg,org by
Eq. (2). Tg,org at low RH is predicted to be higher than
Tg,org at high RH, which results from a lower abundance of high-molar-mass compounds observed at high RH. This trend is consistent with Kidd
et al. (2014), who showed that SOA material formed under dry conditions is
more viscous than that formed under wet conditions.
Glass transition temperatures calculated using Eqs. (1) and (2) for
toluene SOA produced at low relative humidity (< 2 %) and high
relative humidity (75 %) conditions.
* Compounds with M>450 g mol-1 were excluded from the
analysis.
Figure 7b shows the predicted viscosity of toluene SOA as a function of RH
compared to the measured viscosity of toluene SOA formed in an oxidation
flow reactor at 13 % RH (Song et al., 2016a). Indirect viscosity
measurements are also included in shaded boxes for toluene-derived SOA
(Bateman et al., 2015; Li et al., 2015). Lines with shaded areas are
calculated viscosities using Tg,org as described above. κ
was assumed to be 0.25 based on laboratory measurements (Lambe et al., 2011a;
Hildebrandt Ruiz et al., 2015). To achieve good fit, D was set to 13 and
kGT was assumed to be 3.0 (Dette et al., 2014). Estimations with
Eq. (1) match the measured viscosity values very well over the entire RH
range. Predictions with Eq. (2) overestimated the measurements by 1 or 2
orders of magnitude at moderate RH between 30 and 50 %, while they agreed
with the measurements derived at RH ≥ 60 % and at dry conditions.
Sensitivity of the predicted viscosity for biomass burning
particles of (a) subalpine fir and (b) lodgepole pine trees as measured by
high-resolution mass spectrometry to the ionization technique: electrospray
ionization (ESI, red) and atmospheric pressure photoionization (APPI; blue).
Tg,org values are specified in the figure legend and the other
parameters used are fixed to κ=0.1, D=10, and kGT=2.5.
Different results are obtained for the same sample because ESI and APPI probe
a different subset of compounds (Fig. 8).
There are several possible reasons for the difference between the
measurements and predictions. First, the relative abundance of high-molar-mass compounds observed in HRMS measurements may be overestimated, as
high-molar-mass compounds tend to have higher (yet generally unknown) ionization
efficiencies compared to lower-molar-mass compounds. Second, the
nano-DESI-HRMS analysis of toluene SOA was limited to an m/z range of
100–1000 (Hinks et al., 2018). It is possible that some SOA products with
lower molar mass were present in particles but not detected, which would lead
to an overestimation of Tg. Third, the chemical composition of
toluene SOA is likely different between Hinks et al. (2018) and Song et
al. (2016a) because of the differences in the experimental conditions.
Specifically, toluene SOA was formed in a Teflon chamber in Hinks et al.,
while Song et al. used an oxidation flow reactor to generate toluene SOA. The
O : C ratios are 0.71 at low RH and 0.63 at high RH based on nano-DESI-HRMS
measurements in Hinks et al. (2018), while the O : C ratio was 1.06 in Song
et al. (2016a) based on the aerosol mass spectrometry (AMS) measurements.
In addition, different mass loadings may have affected viscosity. Song et
al. (2016a) measured viscosity at two different mass loadings (60–100 and
600–1000 µg m-3) and compared their results to the toluene SOA
data in Bateman et al. (2015; 30–50 µg m-3) and Li et
al. (2015; 44–125 µg m-3), observing little impact of mass
loadings on viscosity. We carried out a sensitivity study of mass loadings on
viscosity using a set of compounds detected by HRMS. The saturation mass
concentration was predicted for each component using the molecular corridor
approach (Li et al., 2016). Assuming that the mass signal intensity is
proportional to the total mass concentration of the compound in the mixture
and applying the absorptive partitioning theory (Pankow, 1994),
the particle-phase concentrations of each compound were predicted to estimate
Tg at different organic aerosol mass loading values
(1–1000 µg m-3). The glass transition temperature of the SOA
mixture decreases as mass loading increases. Viscosity decreases up to 2
orders of magnitude at low RH, while at high RH there is little difference
as shown in Fig. A3. Simultaneous measurements of viscosity and chemical
composition with different mass loadings should be performed in future
studies.
Biomass burning particles
To further explore the applicability of our viscosity prediction method using
elemental composition as measured by HRMS, we performed similar calculations
for biomass burning organic particles emitted from test facility burns of
subalpine fir and lodgepole pine trees, which were conducted as a part of the FIREX 2016
campaign (Selimovic et al., 2018). These samples were analyzed by HRMS using
two different ionization sources: electrospray ionization (ESI) and
atmospheric pressure photoionization (APPI). The mass spectra shown in Fig. 8a
and b indicate that a substantial number of compounds were detected by both
methods (109 and 170 compounds for subalpine fur and lodgepole pine,
respectively). However, pronounced differences are also observed between the
ESI and APPI spectra both in terms of the identity and signal intensities of
the detected compounds.
Glass transition temperatures for the assigned CH and CHO compounds were
computed using Eq. (2). Nitrogen- and sulfur-containing compounds (CHON and
CHOS) are not yet covered by Eq. (2) and were therefore excluded from the
analysis. CHON and CHOS compounds comprised less than 10 % of the detected
ion intensity and < 15 % of the assigned compounds. Note that we do not
intend to provide accurate estimates of the viscosity of ambient biomass burning
particles (as inorganic components are also not included in this analysis),
but we investigate how the use of different ionization methods would lead to
variations in our viscosity predictions. Tg values of organic mixtures
(Tg,org) were then calculated using the Gordon–Taylor approach
with kGT= 1, assuming that the relative concentration of each
compound is proportional to its MS signal intensity. The calculated
Tg,org values for the mixtures are specified in the legend of
Fig. 9. For both types of mixtures, the calculated Tg,org for the
APPI MS data is lower than the value calculated based on the ESI MS data with
a difference of 32 K for subalpine fir and 11 K for the lodgepole pine.
Figure 9 shows the predicted viscosity as a function of RH, assuming D=10, κ=0.10, and kGT=2.5. The difference in
Tg,org derived from ESI and APPI results in a variation of
predicted viscosity at low RH by up to 5 and 2 orders of magnitude for
subalpine fir and lodgepole pine, respectively.
The difference in the calculated Tg,org values is attributed to
the chemical profile of the species detected using different ionization
techniques as shown in the mass spectra in Fig. 8a and b. The van Krevelen diagrams
in Fig. 8c and d illustrate these compositional differences between chemical
species detected by ESI and APPI. ESI is more efficient at detection of polar
compounds (Kiontke et al., 2016), which typically have higher O : C ratios
and therefore would result in higher predicted values of glass transition
temperature (Koop et al., 2011; Saukko et al., 2012). APPI enables the
detecting nonpolar compounds with lower O : C ratios, in particular
polycyclic aromatic hydrocarbons (PAHs) that have low ionization
efficiencies when analyzed by ESI MS (Raffaelli and Saba, 2003; Itoh et al.,
2006). Due to the complementary nature of these ionization methods, it is
most likely that the actual glass transition temperature and viscosity of
each type of organic components in biomass burning aerosols are somewhere in
between the values inferred from ESI and APPI datasets: ESI MS may be viewed
as providing the upper limit of viscosity, while APPI MS gives the lower
limit. Our results indicate that the use of complementary ionization
techniques may help evaluate the associated uncertainty for the prediction of
viscosity values based on elemental composition as measured by HRMS.
Conclusions
We have developed a parameterization for the calculation of the glass transition
temperature of individual SOA compounds with molar mass up to
∼ 1100 g mol-1 using the number of carbon, oxygen, and hydrogen
atoms. The viscosity of SOA was estimated using the Tg-scaled
Arrhenius plot of viscosity versus Tg/T and the Gordon–Taylor
approach to account for mixtures of SOA and water. The fragility parameter
D was compiled for organic compounds and we found that D approaches a
lower limit of ∼ 10 (±1.7) as the molar mass increases. The
resulting viscosity estimations agree well with the measured viscosity of α-pinene and isoprene SOA, validating our method. Using HRMS data, the glass
transition temperatures of individual components and the viscosity of toluene SOA
were predicted, also resulting in a good agreement with measurements.
However, we note that the predicted viscosities were higher than the measured
values, suggesting that additional considerations may need to be taken into
account. For example, the ionization efficiency of both low- and high-molar-mass compounds may have a pronounced effect on the relative abundance of
different classes of compounds in HRMS data. The viscosity prediction method
was also applied to biomass burning particles whose elemental composition
was measured using HRMS with two different ionization techniques. Substantial
differences in viscosity estimations were obtained using ESI and APPI mass
spectra because these two ionization methods probe different subsets of
compounds.
Summary of the predicted range of viscosity for α-pinene SOA
(red), isoprene SOA (blue), toluene SOA (purple), and biomass burning
particles (green).
Figure 10 summarizes the predicted range of the viscosity of α-pinene
SOA, isoprene SOA (generated by PAM), toluene SOA, and biomass burning
particles. Isoprene SOA has lower viscosity, reflecting a lower glass
transition temperature due to the relatively low molar mass of isoprene oxidation
products. α-Pinene and toluene SOA have much higher viscosity with a
different shape of the RH dependence due to differences in glass transition
temperatures and hygroscopicity. Biomass burning particles have moderate
viscosity between the two extreme cases. Currently, both predictions and
measurements are subject to large uncertainties and variations. Complementary
measurements of viscosity and chemical composition employing different
ionization techniques are desired to further constrain RH-dependent viscosity
in future studies. Current Tg parameterizations do not consider
functionality or molecular structure explicitly and further measurements of
Tg and viscosity of SOA would allow us to refine the method
presented in this study. Nevertheless, the current results offer a promising
starting point and such simple parameterizations are practical for predicting
the viscosity of particles as measured by HRMS. The developed viscosity
prediction method should also be useful in recent efforts to simulate the
distribution of SOA phase state and related properties in regional or global
air quality models (e.g., Maclean et al., 2017; Shiraiwa et al., 2017).
Data availability
Data are available upon request to the corresponding author.
Conversion of fragility steepness index (<inline-formula><mml:math id="M457" display="inline"><mml:mrow><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> to fragility (<inline-formula><mml:math id="M458" display="inline"><mml:mrow><mml:mi>D</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>
The fragility steepness index (m) is defined as
m=limT→TgdlogηdTgTgTT.
Combining Eq. (A1) with Eq. (6) gives
m=limT→Tgdd(TgTgTT)-5+0.434T0DT-T0.
Considering that η=1012 Pa s at T=Tg (Angell,
1991) and by defining Δx=1-Tg/T, a combination with
Eq. (7) leads to
m=limΔx→01Δx12--5+0.43439.17TgD+39.17DTg1-Δx-39.17TgD+39.17=limΔx→01Δx17-0.43439.17TgD(1-Δx)DTg+39.17TgΔx=limΔx→0(665.89+17D)(D+39.17Δx)=665.89+17DD.
Note that Eq. (A3) is derived assuming that the high temperature limit of
viscosity η∞ is equal to 10-5 Pa s (Angell,
1991) in the VTF equation (Eq. 5). Similar equations for the relation between
m and D were given by previous studies using different η∞ and units (Angell et al., 1994; Angell, 2002; Bones et
al., 2012) and applying those gave very similar results in our study.
(a)Tg of organic compounds as measured (circles) and
estimated with the Boyer–Kauzmann rule (squares) plotted against the inverse
molar mass. The markers are color coded by atomic O : C ratio. (b) Predicted
Tg for CHO compounds using a parameterization (Eq. 2)
developed in this study compared to measured Tg (circles). The
solid line shows the 1:1 line and the dashed and dotted lines show 68 %
confidence and prediction bands, respectively.
(a) Comparison of measured and estimated Tg by the
Boyer–Kauzmann rule for 251 organic compounds (Koop et al., 2011; Dette et
al., 2014; Rothfuss and Petters, 2017a) with their measured Tm
available. The markers are color coded by molar mass. (b, c) Predicted
Tg using Eq. (2) compared with (b) measured Tg for CH
compounds and (c) predicted Tg using Eq. (1) for CHO compounds.
The solid line shows the 1:1 line. Solid circle markers represent organic
compounds as compiled in Koop et al. (2011) and open square markers represent
SOA oxidation products in Shiraiwa et al. (2014) in panel (c).
Effect of mass loading on predicted viscosity for toluene SOA.
Solid lines represent the predicted viscosity with Eq. (2) using the chemical
composition of toluene SOA formed at low RH. Viscosity was predicted with
different mass loadings ranging from 1–1000 µg m-3. Markers
and shaded boxes represent experimentally measured viscosity values. The Song et
al. (2016a) mass loadings were 60–100 and 600–1000 µg m-3.
The Bateman et al. (2015) and Li et al. (2015) mass loadings were
30–50 and 44–125 µg m-3, respectively.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
This work was funded by the National Science Foundation (AGS-1654104) and the
Department of Energy (DE-SC0018349). The Purdue group and Sergey A.
Nizkorodov acknowledge additional support by the US Department of Commerce
and the
National Oceanic and Atmospheric Administration through the Climate Program
Office AC4 program, awards NA16OAR4310101 and NA16OAR4310102. We thank
Ulrich Pöschl and Thomas Koop for stimulating
discussions. Edited by: Jason Surratt
Reviewed by: two anonymous referees
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