Information on the rate of diffusion of organic molecules
within secondary organic aerosol (SOA) is needed to accurately predict the
effects of SOA on climate and air quality. Diffusion can be important for
predicting the growth, evaporation, and reaction rates of SOA under certain
atmospheric conditions. Often, researchers have predicted diffusion rates of
organic molecules within SOA using measurements of viscosity and the
Stokes–Einstein relation (D∝1/η, where D is the diffusion
coefficient and η is viscosity). However, the accuracy of this
relation for predicting diffusion in SOA remains uncertain. Using
rectangular area fluorescence recovery after photobleaching (rFRAP), we
determined diffusion coefficients of fluorescent organic molecules over
8 orders in magnitude in proxies of SOA including citric acid, sorbitol,
and a sucrose–citric acid mixture. These results were combined with
literature data to evaluate the Stokes–Einstein relation for predicting
the diffusion of organic molecules in SOA. Although almost all the data agree
with the Stokes–Einstein relation within a factor of 10, a fractional
Stokes–Einstein relation (D∝1/ηξ) with ξ=0.93
is a better model for predicting the diffusion of organic molecules in the SOA
proxies studied. In addition, based on the output from a chemical transport
model, the Stokes–Einstein relation can overpredict mixing times of organic
molecules within SOA by as much as 1 order of magnitude at an altitude
of ∼3 km compared to the fractional Stokes–Einstein relation with ξ=0.93. These results also have implications for other areas such as in
food sciences and the preservation of biomolecules.
Introduction
Atmospheric aerosols, suspensions of micrometer and sub-micrometer particles
in the Earth's atmosphere, modify climate by interacting with incoming solar
radiation and by altering cloud formation and cloud properties
(Stocker et al., 2013). These aerosols also
negatively impact air quality and may facilitate the long-range transport of
pollutants
(Friedman
et al., 2014; Mu et al., 2018; Shrivastava et al., 2017a; Vaden et al.,
2011; Zelenyuk et al., 2012).
A large fraction of atmospheric aerosols are classified as secondary organic
aerosol (SOA). SOA is formed in the atmosphere when volatile organic
molecules, emitted from both anthropogenic and natural sources, are oxidized
and partition to the particle phase
(Ervens
et al., 2011; Hallquist et al., 2009). The exact chemical composition of SOA
remains uncertain; however, measurements have shown that SOA contains thousands
of different organic molecules, and the average oxygen-to-carbon (O:C) ratio
of organic molecules in SOA ranges from 0.3 to 1.0 or even higher
(Aiken
et al., 2008; Cappa and Wilson, 2012; Chen et al., 2009; DeCarlo et al.,
2008; Ditto et al., 2018; Hawkins et al., 2010; Heald et al., 2010; Jimenez
et al., 2009; Laskin et al., 2018; Ng et al., 2010; Nozière et al.,
2015; Takahama et al., 2011; Tsimpidi et al., 2018). SOA also contains a
range of organic functional groups including alcohols and carboxylic acids
(Claeys
et al., 2004, 2007; Edney et al., 2005; Fisseha et al., 2004; Glasius et
al., 2000; Liu et al., 2011; Surratt et al., 2006, 2010).
In order to accurately predict the impacts of SOA on climate, air quality,
and the long-range transport of pollutants, information on the rate of
diffusion of organic molecules within SOA is needed. For example,
predictions of SOA particle size, which has implications for climate and
visibility, vary significantly in simulations as the diffusion rate of
organic molecules is varied from 10-17 to 10-19 m2 s-1
(Zaveri et
al., 2014). Lifetimes of polycyclic aromatic hydrocarbons (PAHs) in an SOA particle
increase as the bulk diffusion coefficient of PAHs decreases from 10-16 m2 s-1 at a relative humidity of 50 % to 10-18 m2 s-1 under dry conditions (Zhou et
al., 2019). Shrivastava et al. (2017a) have shown that including shielding by a viscous organic aerosol
coating (equivalent to a bulk diffusion limitation) results in better model
predictions of observed concentrations of PAHs. Reactivity in SOA can also
depend on the diffusion rates of organic molecules
(Davies
and Wilson, 2015; Lakey et al., 2016; Li et al., 2015; Liu et al., 2018;
Shiraiwa et al., 2011; Zhang et al., 2018; Zhou et al., 2013). For the cases
discussed above, the diffusion of organic molecules within SOA becomes a
rate-limiting step only when diffusion rates are small.
In some cases, the diffusion rates of organic molecules in SOA have been
measured or inferred from experiments
(Abramson
et al., 2013; Liu et al., 2016; Perraud et al., 2012; Ullmann et al., 2019;
Ye et al., 2016). However, in most cases researchers have predicted
diffusion rates of organic molecules within SOA using measurements of
viscosities and the Stokes–Einstein relation
(Booth
et al., 2014; Hosny et al., 2013; Koop et al., 2011; Maclean et al., 2017;
Power et al., 2013; Renbaum-Wolff et al., 2013; Shiraiwa et al., 2011; Song
et al., 2015, 2016a). This is due to the development and application of
several techniques that can measure the viscosity of ambient aerosol or small
volumes in the laboratory
(Grayson
et al., 2015; Pajunoja et al., 2014; Renbaum-Wolff et al., 2013; Song et
al., 2016b; Virtanen et al., 2010). The Stokes–Einstein relation (Eq. 1)
states that diffusion is inversely related to viscosity:
D=kT6πηRH,
where D is the diffusion coefficient, k is the Boltzmann constant, T is the
temperature in Kelvin, RH is the hydrodynamic radius of the diffusing
species, and η is the viscosity of the matrix. Until now, only a few
studies have investigated the accuracy of the Stokes–Einstein relation for
predicting the diffusion coefficients of organic molecules in SOA, and almost
all of these studies relied on sucrose as a proxy for SOA particles
(Bastelberger
et al., 2017; Chenyakin et al., 2017; Price et al., 2016). Sucrose was used
as a proxy for SOA in these studies because (1) sucrose has an O:C ratio
similar to that of highly oxidized components of SOA, and (2) viscosity and
diffusion data for sucrose exist in the literature (mainly from the food
science literature, as well as from Power et
al., 2013, who reported viscosities far outside the range of what had
previously been reported). However, studies with other proxies of SOA are
required to determine if the Stokes–Einstein relation can accurately
represent the diffusion of organic molecules in SOA and to more accurately
predict the role of SOA in climate, air quality, and the transport of pollutants
(Reid et
al., 2018; Shrivastava et al., 2017b).
In the following, we expand on previous studies with sucrose matrices by
testing the Stokes–Einstein relation in the following proxies for SOA:
2-hydroxypropane-1,2,3-tricarboxylic acid (i.e., citric acid),
1,2,3,4,5,6-hexanol (i.e., sorbitol), and a mixture of citric acid and
sucrose. These proxies have functional groups that have been identified in
SOA and O:C ratios similar to those ratios found in the most highly
oxidized components of SOA in the atmosphere (1.16, 1.0, and 0.92 for citric
acid, sorbitol, and sucrose, respectively). To test the Stokes–Einstein
relation, we first determined the diffusion coefficients of fluorescent organic
molecules as a function of water activity (aw) in these SOA proxies
using rectangular area fluorescence recovery after photobleaching (rFRAP;
Deschout et al., 2010). Studies as a function of
aw are critical because as the relative humidity (RH) changes in the
atmosphere, aw (and hence water content) in SOA will change to maintain
equilibrium with the gas phase. The diffusing organic molecules studied in
this work were the fluorescent organic molecules rhodamine 6G and cresyl
violet (Fig. S1 in the Supplement). Details of the experiments are given in the Methods
section. The experimental diffusion coefficients are compared with
predictions using literature viscosities
(Rovelli et al., 2019; Song et al.,
2016b) and the Stokes–Einstein relation. The results from the current study
are then combined with literature diffusion
(Champion
et al., 1997; Chenyakin et al., 2017; Price et al., 2016; Rampp et al.,
2000; Ullmann et al., 2019) and viscosity
(Först
et al., 2002; Grayson et al., 2017; Green and Perry, 2007; Haynes, 2015;
Lide, 2001; Migliori et al., 2007; Power et al., 2013; Quintas et al., 2006;
Rovelli et al., 2019; Swindells et al., 1958; Telis et al., 2007; Ullmann et
al., 2019) data to assess the ability of the Stokes–Einstein relation to
predict the diffusion of organic molecules in atmospheric SOA. The ability of
the fractional Stokes–Einstein relation (see below) to predict diffusion is
also tested.
In addition to atmospheric applications, the results from this study have
implications for other areas in which the diffusion of organic molecules within
organic–water matrices is important, such as the cryopreservation of
proteins (Cicerone and
Douglas, 2012; Fox, 1995; Miller et al., 1998), the storage of food products
(Champion et al., 1997; van der Sman
and Meinders, 2013), and the viability of pharmaceutical formulations
(Shamblin
et al., 1999). The results also have implications for our understanding of
the properties of deeply supercooled and supersaturated glass-forming
solutions, which are important for a wide range of applications and
technologies
(Angell,
1995; Debenedetti and Stillinger, 2001; Ediger, 2000).
MethodsPreparation of fluorescent organic–water films
The technique used here to determine diffusion coefficients required thin films containing the organic matrix (i.e., citric acid or sorbitol or a
mixture of citric acid and sucrose), water, and trace amounts of the
diffusing organic molecules (i.e., fluorescent organic molecules). Citric acid (≥99 % purity) and sorbitol (≥98 % purity)
were purchased from Sigma-Aldrich and used as received. Rhodamine 6G
chloride (≥99 % purity) and cresyl violet acetate (≥75 %
purity) were purchased from Acros Organics and Santa Cruz Biotechnology,
respectively, and used as received. Solutions containing the organic matrix,
water, and the diffusing molecules were prepared gravimetrically; 55 wt % citric acid solutions and 30 wt % sorbitol and
sucrose–citric acid solutions were used to prepare the citric acid,
sorbitol, and sucrose–citric acid thin films, respectively. A mass ratio of
60:40 sucrose to citric acid was used for the sucrose–citric acid matrix.
The concentrations of rhodamine 6G and cresyl violet in the solutions were
0.06 and 0.08 mM, respectively. After the solutions were prepared
gravimetrically, the solutions were passed through a 0.02 µm filter
(Whatman™) to eliminate impurities. Droplets of the solution
were placed on cleaned siliconized hydrophobic slides (Hampton Research), by
either nebulizing the bulk solution or using the tip of a sterilized needle
(BD PrecisionGlide Needle, Franklin Lakes, NJ, USA). The generated
droplets ranged in diameter from ∼100 to ∼1300µm. After
the droplets were located on the hydrophobic slides, the hydrophobic slides
were placed inside sealed glass containers with a controlled water activity
(aw). The aw was set by placing saturated inorganic salt solutions
with known aw values within the sealed glass containers. The aw
values used ranged from 0.14 to 0.86. When the aw values were higher
than 0.86, recovery times were too fast to measure with the rFRAP setup. When
the aw values were lower than 0.14 or 0.23, depending on the organic
solute, solution droplets often crystallized. The slides holding the
droplets were left inside the sealed glass containers for an extended period
of time to allow the droplets to equilibrate with the surrounding aw.
The method used to calculate equilibration times is explained in Sect. S1 in the Supplement,
and conditioning times for all samples are given in Tables S2–S5 in the Supplement.
Experimental times for conditioning were a minimum of 3 times longer
than the calculated equilibration times.
After the droplets on the slides reached equilibrium with the aw of the
airspace over the salt solution, the sealed glass containers holding the
slides and conditioned droplets were brought into a Glove Bag™
(Glas-Col). The aw within the Glove Bag was controlled using a
humidified flow of N2 gas and monitored using a handheld hygrometer.
The aw within the Glove Bag™ was set to the same aw
as used to condition the droplets to prevent the droplets from being
exposed to an unknown and uncontrolled aw. To form a thin film,
aluminum spacers were placed on the siliconized glass slide holding the
droplets, followed by another siliconized glass slide, which sandwiched the
droplets and the aluminum spacers. The thickness of the aluminum spacers
(30–50 µm) determined the thickness of the thin film. The two slides
were sealed together by vacuum grease spread around the perimeter of one
slide before sandwiching (see Fig. S2 for details).
The organic matrices were often supersaturated with respect to crystalline
citric acid or sorbitol. Nevertheless, crystallization was not observed in
most cases until aw values 0.14–0.23, depending on the organic
matrix, because the solutions were passed through a 0.02 µm filter
and the glass slides used to make the thin films were covered with a
hydrophobic coating. Filtration likely removed heterogeneous nuclei that
could initiate crystallization, and the hydrophobic coating reduced the
ability of these surfaces to promote heterogeneous nucleation
(Bodsworth
et al., 2010; Pant et al., 2006; Price et al., 2014; Wheeler and Bertram,
2012). In the cases in which crystallization was observed, determined using
optical microscopy, the films were not used in rFRAP experiments. An image
demonstrating the difference in appearance between crystallized and
noncrystallized droplets is given in Fig. S3. We did not condition
droplets without fluorescent organic molecules to determine the effect of
the tracer molecules on crystallization. However, previous studies have
shown that droplets with the compositions and range of aw values
studied here can exist in the metastable liquid state if heterogeneous
nucleation by surfaces is reduced. Furthermore, since the concentration of
the tracers in the droplets were so low, the tracers are not expected to
change the driving force for crystallization in the droplets.
Rectangular area fluorescence recovery after photobleaching (rFRAP)
technique and extraction of diffusion coefficients
Diffusion coefficients were determined using the rFRAP technique reported by
Deschout et al. (2010). The technique uses a
confocal laser scanning microscope to photobleach fluorescent molecules in a
specified volume of an organic thin film containing fluorescent molecules.
The photobleaching event initially reduces the fluorescence intensity within
the bleached volume. Afterward, the fluorescence intensity within the
photobleached volume recovers due to the diffusion of fluorescent molecules
from outside the bleached region. From the time-dependent recovery of the
fluorescence intensity, diffusion coefficients are determined. All diffusion
experiments here were performed at 295±1 K.
The rFRAP experiments were performed on a Zeiss Axio Observer LSM 510MP
laser scanning microscope with a 10X, 0.3 NA objective, and a pinhole setting
between 80 and 120 µm. Photobleaching and the subsequent acquisition of
recovery images were done using a 543 nm helium–neon (HeNe) laser. The
bleach parameters (e.g., laser intensity, iterations, laser speed) were
varied for each experiment so that the fraction of fluorescent molecules
being photobleached in the bleach region was about 30 %. A photobleaching
of about 30 % was suggested by Deschout et al. (2010), who report that diffusion coefficients determined using the rFRAP
technique are independent of the extent of photobleaching up to a bleach
depth of 50 %. The energy absorbed by the thin film during photobleaching
is not expected to affect experimental diffusion coefficients. Although
local heating may occur during photobleaching, the thermal diffusivity in
the samples is orders of magnitude greater than the molecular diffusivity,
and the heat resulting from photobleaching will dissipate to the
surroundings on a timescale much faster than the diffusion of molecules will
occur (Chenyakin et
al., 2017). Measurements as a function of photobleaching size and power are
consistent with this expectation
(Chenyakin
et al., 2017; Ullmann et al., 2019).
Bleached areas ranged from 20 to 400 µm2. The
geometry of the photobleached region was a square with sides of length
lx and ly ranging from 4.5 to 20 µm. Smaller bleach areas
were used in experiments in which diffusion was slower in order to shorten
recovery times.
Chenyakin et al. (2017) showed that experimental diffusion coefficients varied by less than
the experimental uncertainty when the bleach area was varied from 1 to 2500 µm2 in sucrose–water films. Similarly,
Deschout et al. (2010) demonstrated that diffusion
coefficients varied by less than the experimental uncertainty when the
bleach area was varied from approximately 4 to 144 µm2 in sucrose–water films. The images collected during an rFRAP
experiment represent fluorescence intensities as a function of x and y
coordinates and are taken at regular time intervals after photobleaching.
An example of images recorded during an rFRAP experiment is shown in Fig. S4. Every image taken following the photobleaching event is normalized
relative to an image taken before photobleaching. To reduce noise, all
images are downsized by averaging from a resolution of 512×512 to
128×128 pixels.
The mathematical description of the fluorescence intensity as a function of
position (x and y) and time (t) after photobleaching a rectangular area in a
thin film was given by Deschout et al. (2010):
F(x,y,t)F0(x,y)=1-K04⋅erfx+lx2r2+4Dt-erfx-lx2r2+4Dt⋅erfy+ly2r2+4Dt-erfy-ly2r2+4Dt,
where F(x,y,t) is the fluorescence intensity at position x and y after a time t,
F0(x,y) corresponds to the initial intensity at position x and y before
photobleaching, K0 is related to the initial fraction of photobleached
molecules in the bleach region, and lx and ly correspond to the size
(length) of the bleach region in the x and y directions. The parameter r
represents the resolution of the microscope, t is the time after
photobleaching, and D is the diffusion coefficient.
The entire images (128×128 pixels following downsizing) collected during an
rFRAP experiment were fit to Eq. (2) using a MATLAB script (The Mathworks,
Natick, MA, USA), with the terms K0 and r2+4Dt left as free
parameters. An additional normalization factor was also left as a free
parameter and returned a value close to 1, since images recorded after
photobleaching were normalized to the pre-bleach image before fitting. To
determine the bleach width (lx, ly), Eq. (2) was fit to the first
five images recorded after photobleaching a film with the bleach width
(lx,ly) left as a free parameter. The bleach width returned by the
fit to the first five frames was then used as input in Eq. (2) to analyze
the full set of images.
From the fitting procedure, a value for r2+4Dt was determined for each
image and plotted as a function of time after photobleaching. A
straight line was then fit to the r2+4Dt vs. t plot, and from the slope of
the line D was calculated. An example is shown in Fig. S5. As the intensity
of the fluorescence in the bleached region recovers, the noise in the data
becomes large relative to the difference in fluorescence intensity between
the bleached and non-bleached regions (i.e., signal). To ensure that we only use
data with a reasonable signal-to-noise ratio, images were not used if this signal
was less than 3 times the standard deviation of the noise.
Figure S6 shows a cross section of the fluorescence intensity along the x
direction from the data in Fig. S4. Figure S6 is given only to visualize the
fit of the equation to the data, and the cross-sectional fit was not used to
determine diffusion coefficients. As mentioned above, the entire images
(128×128 pixels following downsizing) were used to determine diffusion
coefficients. To generate the cross-sectional view, at each position x, the
measured fluorescence intensity is averaged over the width of the
photobleached region in the y direction (black squares). Also included in
Fig. S6 are cross-sectional views of the calculated fluorescence intensity
along the x direction generated from the fitting procedure (solid red lines).
To generate the line, Eq. (2) was first fit to the images. The resulting fit
was then averaged over the width of the photobleached region in the y
direction. The good agreement between the measured cross section and the
predicted cross section illustrates that Eq. (2) describes the rFRAP data
well.
Equation (2) assumes that there is no net diffusion in the axial direction
(i.e., z direction). Deschout et al. (2010) have
shown that Eq. (2) gives accurate diffusion coefficients when the numerical
aperture of the microscope is low (≤0.45) and the thickness of the
fluorescent films is small (≤120µm), which is consistent with
the numerical aperture of 0.30 and film thickness of 30–50 µm used
here.
Results and discussionDiffusion coefficients of organic molecules in citric acid, sorbitol,
and sucrose–citric acid matrices
The experimental diffusion coefficients of organic molecules in matrices of
citric acid, sorbitol, and sucrose–citric acid as a function of water
activity (aw) are shown in Fig. 1 (and listed in Tables S2–S5).
The experimental diffusion coefficients depend strongly on aw for all
three proxies of SOA. As aw increases from 0.23 (0.14 in one case) to
0.86, diffusion coefficients increase by between 5 and 8 orders of
magnitude. This dependence on aw arises from the plasticizing influence
of water on these matrices; as aw increases (and hence the water
content increases) the viscosity decreases
(Koop et al., 2011). In addition, the
experimental diffusion coefficients varied significantly from matrix to
matrix at the same aw (Fig. 1). As an example, at aw=0.23
the diffusion coefficient of rhodamine 6G is about 4 orders of magnitude
larger in citric acid compared to the sucrose–citric acid mixture.
Experimental diffusion coefficients of fluorescent organic
molecules in various organic matrices as a function of water activity
(aw). The x error bars represent the uncertainty in the measured aw
(±0.025) and y error bars correspond to 2 times the standard
deviation in the diffusion measurements. Each data point is the average of a
minimum of four measurements. Indicated in the legend are the fluorescent
organic molecules studied and the corresponding matrices.
We also considered the relationship between log(D)-log(kT/6πRH) and log (η), a comparison that allows for the identification of
deviations from the Stokes–Einstein relation (Fig. 2). By plotting log(D)-log(kT/6πRH), we account for differences in the hydrodynamic radii
of diffusing species and small differences in temperature (within a range of
6 K). The viscosity corresponding to each diffusion coefficient was
determined from relationships between aw and viscosity developed from
literature data (Figs. S7–S9). The solid line in Fig. 2 corresponds to the
relationship between log(D)-log(kT/6πRH) and log (η) if
the Stokes–Einstein relation (Eq. 1) is obeyed. Figure 2 shows that the
diffusion coefficients of the fluorescent organic molecules depend strongly
on viscosity, with the diffusion coefficients varying by approximately
8 orders of magnitude as viscosity varied by 8 orders of magnitude. If the
uncertainties of the measurements are considered, all the data points except
three (89 % of the data) are consistent with predictions from the
Stokes–Einstein relation (meaning that the error bars on the measurements
overlap the solid line in Fig. 2) over 8 orders of magnitude of change
in diffusion coefficients. This finding is remarkable considering the
assumptions inherent in the Stokes–Einstein relation (e.g., the diffusing
species is a hard sphere that experiences the fluid as a homogeneous
continuum and no slip at the boundary of the diffusing species).
Comparison with relevant literature data
Previous studies have used sucrose to evaluate the ability of the
Stokes–Einstein relation to predict the diffusion coefficients of organic
molecules in SOA
(Bastelberger
et al., 2017; Chenyakin et al., 2017; Price et al., 2016). In addition, a
recent study (Ullmann et al.,
2019) used SOA generated in the laboratory from the oxidation of limonene,
subsequently exposed to NH3(g) (i.e., brown limonene SOA), to
evaluate the Stokes–Einstein relation. Although studies with SOA generated
in the laboratory are especially interesting, that previous study was
limited to relatively low viscosities (≤102 Pa s), whereby a
breakdown of the Stokes–Einstein relation is less expected. In Fig. 3a, we
have combined the results from the current study (i.e., the results from Fig. 2) with previous studies of diffusion and viscosity in sucrose and brown
limonene SOA
(Champion
et al., 1997; Chenyakin et al., 2017; Price et al., 2016; Rampp et al.,
2000; Ullmann et al., 2019). To be consistent with the current study, we
have not included data in Fig. 3a if the diffusion coefficients and
viscosities were measured at, or calculated using, temperatures outside the
range of 292–298 K and if the radius of the diffusing molecule was
smaller than the radius of the organic molecules in the fluid matrix. Previous work
has shown that the Stokes–Einstein relation is not applicable when the
radius of the diffusing molecule is less than the radius of the matrix
molecules, and those cases are beyond the scope of this work
(Bastelberger
et al., 2017; Davies and Wilson, 2016; Marshall et al., 2016; Power et al.,
2013; Price et al., 2016; Shiraiwa et al., 2011). Additional details for the
data shown in Fig. 3a are included in Sect. S2 and Table S6.
Plot of log(D)-log(kT/6πRH) as a function of log
(η) for the diffusion coefficients shown in Fig. 1. Viscosities (η)
were determined from relationships between viscosity and aw (Figs. S7–S9). T corresponds to the experimental temperature and RH corresponds
to the radius of each diffusing species (see Table S6). The x error bars
were calculated using the uncertainty in aw at which the samples were
conditioned (±0.025) and uncertainties in the viscosity–aw
parameterizations. The y error bars represent 2 times the standard
deviation of the experimental diffusion coefficients. The black line
represents the relationship between log(D)-log(kT/6πRH) and
log (η) predicted by the Stokes–Einstein relation (slope =-1).
Shown at the bottom of the figure are various substances and their
approximate room temperature viscosities to provide context, as in
Koop et al. (2011). The image of tar
pitch is part of an image from the pitch drop experiment (image courtesy of
Wikimedia Commons, GNU Free Documentation License, University of Queensland,
John Mainstone).
Based on Fig. 3a the diffusion coefficients of the organic molecules in
sucrose matrices and matrices consisting of SOA generated in the laboratory
depend strongly on viscosity, similar to the results shown in Fig. 2. In
addition, almost all the data agree with the Stokes–Einstein relation (solid
line in Fig. 3a) within a factor of 10. This finding is in stark contrast
to the diffusion of water in organic–water mixtures, wherein much larger
deviations between experimental and predicted diffusion coefficients were
observed over the same viscosity range
(Davies and Wilson, 2016;
Marshall et al., 2016; Price et al., 2016).
(a) Plot of log(D)-log(kT/6πRH) as a function of log
(η) for experimental diffusion coefficients reported in this work and
literature data. Indicated in the legend are the diffusing organic molecules
studied and the corresponding matrices. T corresponds to the experimental
temperature of each diffusion coefficient and RH corresponds to the
radius of each diffusing species (Sect. S2 and Table S6). The symbols
represent experimental data points. The solid line represents the
relationship between log(D)-log(kT/6πRH) and log (η)
predicted by the Stokes–Einstein relation, while the dashed line represents
the relationship between log(D)-log(kT/6πRH) and log (η) predicted by a fractional Stokes–Einstein relation with a slope of -0.93
and crossover viscosity of 10-3 Pa s. Panels (b) and (c) are plots of
the differences (i.e., residuals) between experimental and predicted values
of log(D)-log(kT/6πRH) using the Stokes–Einstein relation and
the fractional Stokes–Einstein relation, respectively. The sum of squared
residuals for the Stokes–Einstein relation is 19.7 and the sum of squared
residuals for the fractional Stokes–Einstein relation is 10.8.
In Fig. 3b, we show the differences between the experimental values and the
solid line in Fig. 3a as a function of viscosity. If the Stokes–Einstein
relation describes the data well, these differences (i.e., residuals) should
be scattered symmetrically about zero, while the magnitude of the residuals
should be less than or equal to the uncertainty in the measurements.
However, the residuals are skewed to be positive, especially as viscosity
increases, with experimental diffusion faster than expected based on the
Stokes–Einstein relation. Figure 3b suggests that the Stokes–Einstein
relation may not be the optimal model for predicting diffusion coefficients
in SOA, particularly at high viscosities.
Fractional Stokes–Einstein relation
When deviation from the Stokes–Einstein relation has been observed in the
past, a fractional Stokes–Einstein relation (D∝1/ηξ, where ξ is an empirical fit parameter) has often been used to
quantify the relationship between diffusion and viscosity. For example,
Price et al. (2016) showed that a fractional
Stokes–Einstein relation can accurately represent the diffusion of sucrose
in a sucrose matrix over a wide range of viscosities (from roughly 100–106 Pa s) with ξ=0.90. Building on that work, the data in
Fig. 3a were fit to the following fractional Stokes–Einstein relation:
D=Dcηcηξ,
where ξ is an empirical fit parameter, ηc is the crossover
viscosity, and Dc is the crossover diffusion coefficient. The crossover
viscosity is the viscosity at which the Stokes–Einstein relation and the
fractional Stokes–Einstein relation predict the same diffusion coefficient.
Based on the data in Fig. 3 we have chosen ηc=10-3 Pa s.
The crossover diffusion coefficient corresponds to the diffusion coefficient
at ηc (which can be calculated with the Stokes–Einstein
relation). The value of ξ is determined as the slope of the dashed line
in Fig. 3a. The best fit to the data (represented by the dashed line in Fig. 3a) resulted in a ξ value of 0.93. Each data point was weighted equally
when performing the fitting.
Mixing times of organic molecules within a 200 nm particle as a
function of viscosity using the Stokes–Einstein relation (black line) and a
fractional Stokes–Einstein relation (red line). The dashed lines indicate
that the relations were extrapolated to viscosities beyond the tested range
of viscosities (≥4×106 Pa s).
Mixing times (in hours) of organic molecules in 200 nm SOA
particles at (a) the surface, (b) 850 hPa or ∼1.4 km of altitude, and (c) 700 hPa or
∼3.2 km of altitude using diffusion coefficients calculated
with the Stokes–Einstein relation (solid black lines) and the fractional
Stokes–Einstein relation (dashed black lines). A 1 h mixing time, which
is often assumed in chemical transport models, is also indicated in each
figure with a horizontal dotted line.
In Fig. 3c, we plotted the difference between the experimental values shown
in Fig. 3a and the predicted values using the fractional Stokes–Einstein
relation (dashed line in Fig. 3a). These residuals are more symmetrically
scattered about zero compared to the residuals plotted in Fig. 3b. In
addition, the sum of squared residuals (r2) in Fig. 3c was less than the
sum of squared residuals in Fig. 3b (r2=10.8 compared to 19.7).
Beyond the sum of squared residuals test we have performed a reduced
chi-squared (χ2) test, which takes into account the extra fitting
variable present in the fractional Stokes–Einstein relation. Assuming a
variance of 0.25, the reduced χ2 value is 1.24 for the
Stokes–Einstein relation and 0.67 for the fractional Stokes–Einstein
relation. This information suggests that the fractional Stokes–Einstein
relation with an exponent value of ξ=0.93 may be the better model
for predicting the diffusion coefficients of organic molecules in SOA compared
to the traditional Stokes–Einstein relation. This is in close agreement with
the findings of Price et al. (2016), who showed that the
diffusion of sucrose in a sucrose–water matrix could be modeled using a
fractional Stokes–Einstein relation with ξ=0.90 over a large range
in viscosity. The new fractional Stokes–Einstein relation, which builds on
the work of Price et al. (2016), was derived using
diffusion data of several large organic molecules in several types of
organic–water matrices and thus demonstrates a broader utility of the
fractional Stokes–Einstein relation.
For the case of large diffusing molecules such as those included in this
work (i.e., the radius of the diffusing molecule is equal to or larger than
the radius of the organic molecules in the matrix), we do not observe a
strong dependence of ξ on the size or nature of the diffusing molecule.
For smaller molecules, ξ is expected to change significantly. For
example, Price et al. (2016) showed that ξ=0.57 for the diffusion of water in a sucrose–water matrix, and
Pollack (1981) showed that ξ=0.63 for the diffusion of
xenon in a sucrose–water matrix. The development of a relationship between
ξ and the size of small diffusing molecules is beyond the scope of this
work.
Implications for atmospheric mixing times
To investigate the atmospheric implications of these results, we considered
the mixing times of organic molecules within SOA in the atmosphere as a
function of viscosity using both the Stokes–Einstein relation (Eq. 1) and
the fractional Stokes–Einstein relation (Eq. 3) with ξ=0.93. Mixing
times were calculated with the following equation
(Seinfeld and Pandis, 2006; Shiraiwa et
al., 2011):
τmix=dp24π2D,
where τmix is the characteristic mixing time, dp is the SOA
particle diameter, and D is the diffusion coefficient. τmix
corresponds to the time at which the concentration of the diffusing
molecules at the center of the particle deviates by less than a factor of
1/e from the equilibrium concentration. We assumed a dp of 200 nm,
which is roughly the median diameter in the volume distribution of ambient
SOA
(Martin
et al., 2010; Pöschl et al., 2010; Riipinen et al., 2011). We assumed a
value of 0.38 nm for RH based on literature values for molecular weight
(175 g mol-1; Huff
Hartz et al., 2005) and the density (1.3 g cm-3;
Chen
and Hopke, 2009; Saathoff et al., 2009) of SOA molecules and assuming a
spherical symmetry of the diffusing species.
Figure 4 shows the calculated mixing times of 200 nm particles as a function
of the viscosity of the matrix. The mixing time of 1 h is highlighted,
since when calculating the growth and evaporation of SOA and the long-range
transport of pollutants using chemical transport models, a mixing time of
< 1 h for organic molecules within SOA is often assumed
(Hallquist et
al., 2009). At a viscosity of 5×106 Pa s, the mixing time is
> 1 h based on the Stokes–Einstein relation but remains
< 1 h based on the fractional Stokes–Einstein relation.
Furthermore, at high viscosities > 5×106 Pa s, the mixing
times predicted with the traditional Stokes–Einstein relation are at least a
factor of 5 greater than those predicted with the fractional Stokes–Einstein
relation.
Recently, Shiraiwa et al. (2017) estimated mixing
times of organic molecules in SOA particles in the global atmosphere using
the global chemistry climate model EMAC
(Jöckel et al., 2006) and the
organic module ORACLE
(Tsimpidi
et al., 2014). Glass transition temperatures of SOA compounds were predicted
based on molar mass and the O:C ratio of SOA components, followed by
predictions of viscosity. Diffusion coefficients and mixing times were
predicted using the Stokes–Einstein relation. To further explore the
implications of our results, we calculated mixing times of organic molecules
in SOA globally using the same approach as Shiraiwa
et al. (2017) and compared predictions using the Stokes–Einstein relation
and predictions using the fractional Stokes–Einstein relation with ξ=0.93. Shown in Fig. 5 are the results from these calculations. At all
latitudes at the surface, the mixing times are well below the 1 h often
assumed in chemical transport models regardless of whether the Stokes–Einstein
relation or the fractional Stokes–Einstein relation is used (Fig. 5a). On
the other hand, at an altitude of approximately 1.4 km, the latitudes at which
the mixing times exceed 1 h will depend on whether the Stokes–Einstein
relation or fractional Stokes–Einstein relation is used (Fig. 5b). At an
altitude of 3.2 km the mixing times are well above the 1 h cutoff
regardless of what relation is used, and the Stokes–Einstein relation can
overpredict mixing times of SOA particles by as much as 1 order of
magnitude compared to the fractional Stokes–Einstein relation (Fig. 5c). A
caveat is that the predictions at 3.2 km are based on viscosities higher
than the viscosities studied in the current work. Hence, at 3.2 km the
Stokes–Einstein and fractional Stokes–Einstein relations are being used
outside the viscosity range tested here. Although experimentally
challenging, additional studies are recommended to determine if the
fractional Stokes–Einstein relation with ξ=0.93 is able to
accurately predict the diffusion coefficients of organic molecules in proxies of
SOA at viscosities higher than investigated in the current study.
Summary and conclusions
We report experimental diffusion coefficients of fluorescent organic
molecules in a variety of SOA proxies. The reported diffusion coefficients
varied by about 8 orders of magnitude as the water activity in the SOA
proxies varied from 0.23 (0.14 in one case) to 0.86. By combining the new
diffusion coefficients with literature data, we have shown that, in almost
all cases, the Stokes–Einstein relation correctly predicts the diffusion
coefficients of organic molecules in SOA proxies within a factor of 10.
This finding is in stark contrast to the diffusion of water in SOA
proxies, whereby much larger deviations between experimental and predicted
diffusion coefficients have been observed over the same viscosity range.
Even though the Stokes–Einstein relation correctly predicts the diffusion of
organic molecules in the majority of cases within a factor of 10, both a
sum of squared residuals analysis and a reduced chi-squared test show that a fractional Stokes–Einstein
relation with an exponent of ξ=0.93 is a better model for
predicting diffusion coefficients in SOA proxies for the range of
viscosities included in this study. This is consistent with earlier work
that showed the fractional Stokes–Einstein relation is able to reproduce
experimental diffusion coefficients of sucrose in sucrose–water matrices.
The fractional Stokes–Einstein relation predicts faster diffusion
coefficients and therefore shorter mixing times of SOA particles in the
atmosphere. At an altitude of ∼3.2 km, the difference in mixing times
predicted by the two relations is as much as 1 order of magnitude.
Data availability
The underlying data and related material for this paper are located in the
Supplement.
The supplement related to this article is available online at: https://doi.org/10.5194/acp-19-10073-2019-supplement.
Author contributions
EE performed the diffusion experiments. AMM, YL, APT, VAK, JL, and MS
provided calculations of mixing times as a function of altitude and
latitude. GR and JPR provided viscosity data. SK provided assistance with
the diffusion experiments. EE and AKB conceived the study and wrote the
paper. All authors contributed to revising the paper. All
authors read and approved the final paper.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
Diffusion experiments were performed in the LASIR
facility at UBC, funded by the Canadian Foundation for Innovation.
Financial support
This research has been supported by the Natural Sciences and Engineering Research Council of Canada, a Deutsche Forschungsgemeinschaft individual grant program
(project reference TS 335/2-1), the Natural Environmental Research Council of the UK (grant nos. NE/N013700/1 and NE/M004600/1), the U.S. National Science Foundation (AGS-1654104), and the U.S. Department of Energy (DE-SC0018349).
Review statement
This paper was edited by Thorsten Bartels-Rausch and reviewed by three anonymous referees.
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