ACPAtmospheric Chemistry and PhysicsACPAtmos. Chem. Phys.1680-7324Copernicus PublicationsGöttingen, Germany10.5194/acp-16-5299-2016The rate of equilibration of viscous aerosol particlesO'MearaSimonhttps://orcid.org/0000-0001-7197-0651ToppingDavid O.https://orcid.org/0000-0001-8247-9649McFiggansGordong.mcfiggans@manchester.ac.ukhttps://orcid.org/0000-0002-3423-7896Centre for Atmospheric Science, School of Earth, Atmospheric &
Environmental Sciences, University of Manchester, Manchester,
M13 9PL, UKNational Centre for Atmospheric Science (NCAS),
University of Manchester, Manchester, M13 9PL, UKGordon McFiggans (g.mcfiggans@manchester.ac.uk)28April20161685299531314December201518January20164April20165April2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://acp.copernicus.org/articles/16/5299/2016/acp-16-5299-2016.htmlThe full text article is available as a PDF file from https://acp.copernicus.org/articles/16/5299/2016/acp-16-5299-2016.pdf
The proximity of atmospheric aerosol particles to
equilibrium with their surrounding condensable vapours can substantially
impact their transformations, fate and impacts and is the subject of vibrant
research activity. In this study we first compare equilibration timescales
estimated by three different models for diffusion through aerosol particles
to assess any sensitivity to choice of model framework. Equilibration times
for diffusion coefficients with varying dependencies on composition are
compared for the first time. We show that even under large changes in the
saturation ratio of a semi-volatile component (es) of 1–90 %
predicted equilibration timescales are in agreement, including when
diffusion coefficients vary with composition. For condensing water and a
diffusion coefficient dependent on composition, a plasticising effect is
observed, leading to a decreased estimated equilibration time with
increasing final es. Above 60 % final es maximum equilibration
times of around 1 s are estimated for comparatively large particles (10 µm)
containing a relatively low diffusivity component (1 × 10-25 m2 s-1 in pure form). This, as well as other results here,
questions whether particle-phase diffusion through water-soluble particles
can limit hygroscopic growth in the ambient atmosphere. In the second part
of this study, we explore sensitivities associated with the use of particle
radius measurements to infer diffusion coefficient dependencies on
composition using a diffusion model. Given quantified similarities between
models used in this study, our results confirm considerations that must be
taken into account when designing such experiments. Although quantitative
agreement of equilibration timescales between models is found, further work
is necessary to determine their suitability for assessing atmospheric
impacts, such as their inclusion in polydisperse aerosol simulations.
Introduction
Recent attention on the phase state of atmospheric particles has motivated
questions about the means to model diffusion through them. It had been
conventionally assumed that particles possess a liquid phase state, such
that timescales of diffusion were much less than their atmospheric residence
times. However, several recent studies present evidence that particles can
exist in an amorphous solid state (Smith
et al., 2002; Murray and Bertram, 2008; Virtanen et al., 2010; Vaden et al.,
2011). Viscosities for amorphous solid particles will be higher than for
liquid ones, resulting in lower condensed phase diffusion coefficients and
potentially limiting the rate of gas-particle partitioning for condensing or
evaporating components. For such particles it is important to critically
assess models that attempt to predict or infer the effects of diffusion
limitations in order to report findings with confidence.
Fick's first and second laws of diffusion state that the rate of transport
of a given component through a given area is proportional to the
concentration gradient normal to the area. The Fickian diffusion coefficient
(Di) is the proportionality constant between the diffusive flux and the
concentration gradient (Eq. 1).
Recent attempts at modelling diffusion through particles have centred on
Fick's second law, which in spherical coordinates is
∂Ci(r,t)∂t=1r2∂∂rr2Di∂Ci(r,t)∂r,
where C is the concentration of species i, r is the radius from the
particle centre and t is time. Fick's second law is applied when the
concentration gradient, and therefore flux, changes with time and distance,
i.e. non-steady state. This study analyses and compares three approximations
of Eq. (1) used to model diffusion through particles:
The ETH model presented by Zobrist et
al. (2011), based on the Euler forward step method;
The “kinetic multi-layer model of gas-particle interactions in aerosol
and clouds” (KM-GAP) (Shiraiwa et al., 2012), based
on coupled differential equations;
The Fick's Second Law solved by partial differential equation model
(hereafter referred to as Fi-PaD), a formulation of which was used in Smith et al. (2003).
A description of each model is provided in the method below. For a system
with Di independent of composition, it has been reported that Fi-PaD
and KM-GAP give very similar results (Shiraiwa et al., 2010).
To our knowledge however, no detailed comparison of all three approaches,
including cases of Di dependent on composition, has yet been published.
Despite this, a recent study by Lienhard et al. (2015) linked the impact of
particulate viscosity on ice nucleation using a composition dependent
Di. A critical review of these models is intended to guide those with
an interest in simulating particle evolution inside instruments, chamber
experiments, and the ambient atmosphere. For non-equilibrium viscous
particles, diffusivity (along with other properties such as volatility)
determines the temporal evolution of particle composition and size- and
number-distributions (Zaveri et al., 2014).
These are key factors determining aerosol impact on climate and health,
therefore the choice of diffusion model could have far-reaching consequences
(Pöschl, 2005). In addition to differences in
modelled particle size and composition change, inappropriate choice of model
formulation and assumptions therein could lead to differences in inferred
properties, such as diffusion coefficients from single particle levitation
measurements (Lienhard et al., 2014; Zobrist et al., 2011).
The numerical methods employed by all three models involve discretisation in
time and space. However, subtle differences in how they define concentration
gradients may induce variations in estimated diffusion rate. Therefore, it
is expected that any differences in rate will increase with greater
heterogeneity in the concentration-radius profile, i.e. an increasingly
steep diffusion front. Such fronts have been observed when water and glassy
organics diffuse through one another (Nowakowski et al., 2015). It is
currently unclear which of the models investigated here, if any, is suitable
to such a situation, given the paucity of experimental data available.
Indeed, the Fickian framework may not be appropriate for some systems; in
polymer studies it is well known that non-Fickian diffusion occurs for many
examples of liquids diffusing through glassy polymers (Thomas and Windle,
1982; Kee et al., 2005), and in such systems a narrow diffusion front is
often observed (e.g. Thomas and Windle, 1982). Non-Fickian diffusion results
from structural changes following diffusion and the resultant composition
change. It arises when the rate of deformation is comparable to that of
diffusion (Crank, 1975). Alternative models have been proposed, such as the
free volume model (He et al., 2006; Price et al., 2014) and the
Maxwell-Stefan model (Krishna and Wesselingh, 1997).
The aim of this study, however, is to compare the estimated equilibration
timescales of the Fickian diffusion models that are used in atmospheric
aerosol science and, in turn, assess sensitivities of derived diffusion
coefficients in such particles. In most test cases below the diffusing
semi-volatile component has the self-diffusion coefficient of water at room
temperature, and the resulting diffusion timescales are most relevant to
water and water-soluble particles, however, the findings regarding
consistency between models are applicable to components with self-diffusion
coefficients across the investigated range
(2 × 10-9–1 × 10-25 m2 s-1).
(a) A schematic of a particle split into shells, as used in the
diffusion models, with shell boundaries represented by lines, and symbols
relating to those used in the model equations (αi, Xi and
es are the accommodation coefficient, particle-phase mole fraction and
vapour-phase saturation ratio of the semi-volatile, respectively). The
relative width of the surface shell is shown larger than that used in models
for clarity. (b, c) The concentration-radius profiles estimated using the
ETH model at various times during diffusion for a semi-volatile component
diffusing inwards with an initial mole fraction of 0.01 and equilibrium mole
fraction of 0.90 (instantaneous change in saturation ratio of 1–90 %
assuming ideality), where te is the e-folding time.
Note that the axes are relative, and normalised by the total radius (Rp)
of the particle on the abscissa and by the equilibrium concentration
([i]eq) on the ordinate. (b) is for the diffusion coefficient
independent of composition and (c) is for the diffusion coefficient with a
logarithmic dependence on mole fraction (Dnv0=1×10-25, Dsv0=2×10-9 m2 s-1).
MethodModel description
The ETH model, KM-GAP and Fi-PaD used the same representation of an aerosol
particle: it was assumed spherical, and split into concentric shells. A
comparatively thin surface shell was assumed to equilibrate instantly with
the gas-phase in all simulations (for the purpose of comparing particle-phase
diffusion models and gaining insight into the limitation imposed by
particle-phase diffusion on mass-transfer this assumption is reasonable). The
initial concentration profile in bulk shells (those below the surface) was
homogeneous, and in equilibrium with the initial gas phase concentration.
Figure 1a demonstrates how the particle is represented in a 2-D view. Figure 1b
illustrates the concentration-radius profiles of a semi-volatile component at
several time steps using the ETH model in the case of an instantaneous
increase in saturation ratio from 1 to 90 % when the diffusion
coefficient is independent of composition. In contrast, Fig. 1c shows the
same information but when the diffusion coefficient has a logarithmic
dependence on composition and the self-diffusion coefficients of the two
components are very different, that of the non-volatile (Dnv0)=1×10-21 m2 s-1 and for the semi-volatile
(Dsv0)=2×10-9 m2 s-1. The “diffusion front” is clear
in this example and arises from the very different diffusion coefficient
values in neighbouring shells that result from variations in shell
composition.
The e-folding time for the difference in concentration of the semi-volatile
component at the surface ([sv]eq) and of its average concentration
across the particle bulk ([sv]‾b) was used as a metric for
diffusion time by Zaveri et
al. (2014). It is readily transferable to other studies, and is the chosen
metric for diffusion timescale here. The ratio of the concentration
difference in the surface and bulk-average of the semi-volatile component at
any time (t) to that difference at t=0 is
Q=[sv]eq-[sv]‾b,t≥0[sv]eq-[sv]‾b,t=0.
The e-folding time was therefore the time taken for Q to increase/decrease
by a factor of e1. Figure 1b and c therefore demonstrate the
concentration-radius profiles at several steps between Q=1, which occurs
at t=0, and Q=e-1, which marks the e-folding time. Comparing
e-folding times between models strictly only tests model consistency at this
particular stage of diffusion and not before this. However, e-folding time
agreement would indicate agreement at previous times (and future ones),
because the underlying equations are identical. For reassurance on this,
concentration-radius profiles at times prior to the e-folding time were
compared. All models were run on a 12 core Intel Core i7 processor with a
speed of 3.2 GHz.
Fick's first law, which assumes a constant concentration gradient and
therefore flux, with time, is the basis of the ETH model. However, following
the Euler forward step method, if time steps are sufficiently short this
model should be able to capture changes in concentration gradient, and
therefore flux, and therefore replicate the second law. The flux between
shells is thus found by
Jbk,bk-1,i=-AbkDbk,bk-1,i([i]bk-[i]bk-1)0.5(δbk+δbk-1),
where J is the flux (mol s-1) between bulk (b) shell numbers k
and k-1 (ascending from k=1 for the near-surface shell to k=n at the
centre). If k=1, then k-1 is the surface layer (s). A is the surface
area of the shell's outer surface. Dbk,bk-1,i is the diffusion
coefficient at the shell boundary and here is found using one of the
dependencies on composition given in Sect. 2.3. [i] is concentration of
component i at the shell centre and δ is shell width. Figure 1a
demonstrates how these terms relate to the physical representation of the
particle. From Eq. (3) the change in number of moles in a shell is found by:
ΔNbk,i=(Jbk,bk-1,i-Jbk+1,bk,i)Δt,
where Δt is the time interval (setting of Δt is described in
Sect. 2.2). The version of the ETH model we have written was tested against
the model output in Zobrist et
al. (2011), and found to replicate their results accurately (Fig. A1 in the Appendix for the replica plot).
In KM-GAP the number of moles of a component in a shell is found by
integrating the following coupled ordinary differential equations (ode) with
respect to time:
dNs,idt=(Jb1,s,i-Js,b1,i),dNb1,idt=(Js,b1,i-Jb1,s,i)+(Jb2,b1,i-Jb1,b2,i),dNbk,idt=(Jbk+1,bk,i-Jbk,bk+1,i)+(Jbk-1,bk,i-Jbk,bk-1,i),(k=2,…,n-1),dNbn,idt=(Jbn-1,bn,i-Jbn,bn-1,i).
The flux is found by
Jbk,bk±1,i=Kbk,bk±1,i[i]bkAbk,
where K is the transport rate coefficient (m s-1):
Kbk,bk±1,i=2Dbk,bk±1,i(δbk+δbk±1),
where δ is shell width. Dbk,bk±1,i is the diffusion
coefficient at the shell boundary (Sect. 2.3). Note that in Eqs. (9) and (10),
if k=1, then k-1 is the surface shell (s).
Equations (5)–(10) were solved in Matlab software using the ode23tb
numerical solver, which has an adaptive time step. It was found that the
solver became increasingly unstable as the gradient of Di with r
increased, thus error tolerances (given in the Appendix) were increased
appropriately. ode23tb uses a Runge–Kutta method of two stages: a
trapezoidal rule followed by a backward differentiation formula stage.
Fi-PaD treats Eq. (1) as an initial-boundary problem, with initial
conditions:
Cbk,i(r<Rp,0)=Ci,eq0,Cs,i(Rp,0)=Ci,eq,
where eq represents the equilibrium condition. Equation (11) states that
initial concentrations in the bulk shells are in equilibrium with the
original es value, whilst Eq. (12) states that the initial
concentration at the surface is in equilibrium with the new es. The
boundary conditions were
∂Ni(0,t)∂t=0,∂Ci(Rp,t)∂t=0,
where Rp is the particle radius. Equation (13) states that there is no
flux at the centre of the particle and Eq. (14) states that the
concentration of components at the surface is constant. For Fi-PaD the
numerical solver pdepe in Matlab software was used. The solver uses the
method of lines, which discretises the problem in space to gain a system of
ordinary differential equations that are then solved using the numerical
solver ode15s in Matlab. ode15s is similar to ode23tb in that both are
designed for stiff systems, however, ode15s has a high order of accuracy for
a given error tolerance. The default error tolerances for pdepe were found
to provide stable solutions across the range of parameter spaces used here;
the contrast to the variable error tolerances used in KM-GAP is attributed
to the difference in the accuracy of their ode solvers.
Particle representation
Particles were assumed to consist of two components: a non-volatile (nv)
and semi-volatile (sv), which were assigned the molar mass and density of
sucrose and water, respectively. In general, components with relatively high
molar masses are expected to have comparatively low diffusion coefficients
(Haynes, 2015). To test the effect of using a high molar mass component
against using sucrose on equilibration times, a molar mass (M) of 700 g mol-1
(M of sucrose = 342.296 g mol-1) and density (p) of 2.0 × 103 kg m-3
(p of sucrose = 1.5805 × 103 kg m-3) was
assigned to the non-volatile component and its self-diffusion coefficient
was set relatively low: 1.0 × 10-25 m2 s-1. When the
saturation ratio of the semi-volatile component (es) increased from
1–90 % the e-folding times for all three models increased between 13–16 %
from those using sucrose values (since the molar volume of the non-volatile
component increased a decreased semi-volatile component concentration was
required to attain equilibrium, leading to a decreased concentration
gradient for the same change in es). Since these changes to e-folding
times are similar across the models and are for a comparatively large change
in es, our conclusions are expected to be applicable to a broad range
of component M and p values.
All models assumed ideality for most simulations (see later) so that at
equilibrium the value of es equalled the mole fraction of the
semi-volatile component in the condensed phase. While estimates of
accommodation coefficients for semi-volatiles cover a wide range, for the
purposes of this study we have held it constant at unity (as has been found
reasonable for that of water vapour on liquid water in multiple studies,
e.g. Kolb et al., 2010). Assuming ideality, the volume of a component was equal to the
product of its number of moles (N) and molar volume. The volume of a shell
was therefore given by
Vbk=Nnv,bkMnvpnv+Nsv,bkMsvpsv,
where M is molar mass and p is density.
To accurately simulate the size change in particles resulting from
condensation (growth) or evaporation (shrinkage), at the end of each time
interval, shell volumes were recalculated using the new values of
Ni,bk. In KM-GAP and Fi-PaD, a maximum change to the particle radius of
0.1 % was allowed per time step; if the radius change exceeded 0.1 % the
interval was iteratively shortened until the change was acceptable.
Decreasing this maximum acceptable change did not change e-folding times
significantly (< 2 % for both KM-GAP and Fi-PaD when a maximum
radius change of 0.01 % was used instead), thus it was considered
sufficient to account for volume change. For the ETH model, it has been
recommended that to ensure model stability, the number of moles inside any
shell should not change by more than 2 % over a single time interval
(Zobrist et al., 2011). The same condition was
used here because values below 2 % did not change predicted e-folding times
significantly (< 1 % change when maximum change in number of moles
was 0.01 % instead).
Bulk shells (those below the surface) were initially set to have equal
widths. The surface shell represents the sorption layer, where transfer
between the condensed and gas-phase occurs. Since the surface shell is
contained within the initial particle diameter, the width should be
sufficiently thin to not significantly affect the e-folding time, i.e. one
must not decrease the width of bulk shells such that diffusion is
accelerated. A factor of 1 × 10-3 of the particle radius was chosen to
calculate the surface shell width because using lower factors resulted in no
significant change to estimated e-folding time.
During condensation the surface shell expands; however, since this shell
simulates the boundary between the shell and the gas-phase it should remain
comparatively thin. Therefore, if the surface shell grew to double its
initial width, it was reduced back to its initial width by transferring the
excess volume to the near-surface shell, or, if this near-surface shell had
a width greater than the total radius divided by the number of shells, the
transferred material was used to make a new near-surface shell. The
concentration of components in the transferred material was the same as in
the surface shell (i.e. at equilibrium with the gas phase). This approach
had potential to introduce numerical diffusion by decreasing the distance
for diffusion in the case of introducing a new shell and decreasing the
concentration gradient in the case of transfer.
To gain an indication of whether numerical diffusion influenced one model
more than another, e-folding times were found with this approach (transfer
on) and without it (transfer off). For the latter case the surface shell was
allowed to grow without adjustment, leading to an unrealistically wide shell
and comparatively longer equilibration times, but eliminating the
possibility of numerical diffusion. If numerical diffusion affected one
model more than another we would expect the difference in e-folding times
between the transfer on and transfer off cases to vary between them.
However, there was no substantial difference between models: for a change in
es of 1–90 % all models had an increase of 20–30 % in e-folding
times from the transfer on to the transfer off case; and for a change in
es of 60–80 % the increase was between 6 and 10 %. These differences
are negligible in comparison to the several orders of magnitude change in
e-folding times seen across the range of non-volatile component diffusivity
used below.
During evaporation the width of the surface shell decreased and the mass of
non-volatile component in the surface shell tended toward zero. If the
surface shell decreased below a factor of 1 × 10-1 of its initial width
it was returned to its initial width by transferring a sufficient volume
from the near-surface shell. The concentration of components in the
transferred material was equal to that in the surface shell, thus the
concentration in the surface shell was maintained and any excess
semi-volatile component was presumed to evaporate. Similarly, if the
near-surface shell shrank to below a factor of 1 × 10-1 of the initial
width of the surface shell, then the two shells were coalesced into a new
surface shell at equilibrium concentration. It was found that decreasing the
width at which transfer and coalescence were invoked led to a decrease and
convergence of predicted e-folding times, indicating decreasing numerical
diffusion (which could occur due to steepening of the concentration gradient
through either coalescence or transfer). A decrease of no more than 1 %
was seen across models and changes in es when using lower factors than
1 × 10-1 of the initial width of the surface shell, thus this factor was
concluded to be sufficiently low to effectively prevent numerical diffusion.
D Dependence
At any point in the particle the diffusion coefficient of both components
was the same, i.e., we assumed symmetrical diffusion coefficients, which is
valid for an ideal binary mixture (Wesselingh and Bollen,
1997) . We compared models using three functions of Di:
Di independent of the semi-volatile mole fraction (xsv) and
therefore fixed throughout the simulation;
Di with a logarithmic dependence on semi-volatile mole fraction,
which has been observed for ideal systems by Vignes (1966):
Di(xsv)=Dsv0xsvDnv0(1-xsv),
where Dsv0 is the self-diffusion coefficient of the semi-volatile
component and Dnv0 is the self-diffusion coefficient of the
non-volatile component;
Di with a sigmoidal dependence on xsv, which was observed for
the citric acid-water system by Lienhard et al. (2014):
Di(xsv)=Dsv0xsv∝Dnv0(1-xsv∝),
where ∝ is a correction parameter given by
ln(∝)=(1-xsv)2[C+3D-4D(1-xsv)].
Where the values of C and D were chosen as -3.105 and 3.300
respectively. These provided a relatively steep “cliff-edge” sigmoidal
dependence and therefore a substantial variation from the logarithmic
dependence, enabling a test of consistency between models across a wide
range of dependencies. Examples of these dependencies are shown in Fig. 2.
Example dependencies of Di on the mole fraction of the
semi-volatile component. For the constant case both components have a value
of 2 × 10-9 m2 s-1 and for the other cases the self-diffusion
coefficient of the semi-volatile component is set to
2 × 10-9 m2 s-1 and that of the non-volatile is
1 × 10-25 m2 s-1.
For the latter two cases Di(xsv) was calculated within the numerical
solvers of KM-GAP and Fi-PaD, whilst for the ETH model it was calculated at
the start of each time step. xsv at a shell boundary was found using
the arithmetic mean concentration of the semi-volatile component across the
bounding shells.
In the first part of the study we compare the equilibration timescales
estimated by models when the diffusion coefficient is constant and when it
follows the logarithmic and sigmoidal dependencies on composition given
above. Self-diffusion coefficients of the non-volatile and semi-volatile
components range between that of water at room temperature,
2.0 × 10-9 m2 s-1 (Starr et al., 1999), and a
comparatively low value of 1.0 × 10-25 m2 s-1, which according
to the Stokes–Einstein relationship between diffusivity and viscosity, is
representative of a glassy material (Debenedetti and Stillinger, 2001).
e-folding times are found for several changes in the vapour-phase saturation
ratio of a semi-volatile component, and across a range of particle sizes and
differences in the self-diffusion coefficient of components.
Finally, we present an example of the differences in modelled particle size
change with time when different dependencies of Di on composition are
assumed, thereby providing guidance on the most effective experimental
procedure for inference of diffusion coefficient dependencies on
composition. For actual inferences one would preferably have good knowledge
of the system's deviation from ideality. In an attempt to replicate a real
system, we therefore use the estimation for water activity and density as a
function of sucrose weight fraction presented in Zobrist et al. (2011). The
initial and surface shell water activity were set equal to the initial and
current gas-phase saturation ratio of water (the saturation ratio changed
with time), respectively, with the accommodation coefficient of water
assumed to be one.
Results
Numerical convergence of e-folding times was observed with increasing spatial
resolution for all three models due to improved resolution of concentration
gradients and therefore changes in Di (when dependent on composition)
and flux with space. e-folding times showed an exponential relationship with
shell number (e.g. Fig. A2), thus the criteria for shell
number was that at which the e-folding time was within 10 % of the
asymptote. Generally as the gradient of Di with particle radius
increased, the shell number increased to maintain convergence (Table A2).
However, increasing the shell number increases the
possibility of accumulating significant round-off error, in addition to
requiring greater computer time. The round-off error at the chosen
resolution was investigated by halving the number of significant numbers
assigned to variable values. The difference in predicted e-folding times
between the two precisions was found to be negligible, with a maximum of
2 %, indicating that round-off error was not a substantial source of
inaccuracy.
Zobrist et al. (2011) reported requiring up to several thousand shells in the
ETH model to resolve concentration gradients. However, we found that using of
the order of hundreds gave convergence for the cases with steepest
concentration gradients (Fig. A2). The difference in required
shell resolution between the studies could be due to differences in Di
dependence on composition. Using the Matlab software it was found that
computational time for the case of diffusion coefficient independent of
composition was quickest, gradually increasing as the steepness of the
diffusion coefficient dependence on composition increased, largely due to the
greater spatial resolution. For Di independent of composition the ETH
model took of the order 1 s to reach the e-folding state while KM-GAP and
Fi-PaD were of the order 102 s. For a steep diffusion coefficient
dependence, the chosen example was the logarithmic dependence, with
Dnv0=1 × 10-25 and
Dsv0=2 × 10-9 m2 s-1 and
es instantaneously increased from 1 to 90 %: to
reach e-folding states the ETH model took of the order 102 s while
both KM-GAP and Fi-PaD took of the order 104 s.
In the first model comparison, e-folding times were found when Di was
independent of xsv. For a complete analysis of model output, initial
particle diameters (Dp,t=0) were varied between
1 × 10-5 and 1 × 10-8 m, which covers most of the size range
observed in the ambient atmosphere (Seinfeld and Pandis,
2006) and Di ranged between 2.0 × 10-9 and
1.0 × 10-25 m2 s-1. e-folding times were found across this
parameter space for a change in es of 1–90 and 90–1 % for all
three models. This relatively large change in es was chosen to create a
large concentration gradient, as this would most likely induce disagreement
between models. However, all models agreed very well across the whole range
of particle size and Di (Fig. A3).
e-folding time contour plots for different changes to the
saturation ratio of the semi-volatile component (Δes)
and different diffusion coefficient dependencies:
(a)Δes= 1–90 %
logarithmic dependence, (b)Δes= 90–1 %, logarithmic dependence,
(c)Δes= 1–90 %, sigmoidal
dependence, and (d)Δes= 90–1 %
sigmoidal dependence. Dnv0 is
the diffusion coefficient at a semi-volatile mole fraction of 0, while
Dsv0 (diffusion coefficient at a
semi-volatile mole fraction of 1) was fixed at 2.0 × 10-9 m2 s-1.
In the next case Di varied logarithmically with mole fraction of the
semi-volatile, between a maximum of 2.0 × 10-9 m2 s-1 at
xsv=1 and a minimum given by Dnv0 (i.e. Di at
xsv=0). Dnv0 ranged between 2.0 × 10-9 and
1.0 × 10-25 m2 s-1. Contour plots of e-folding times as a
function of Dnv0 and Dp,t=0 and for a
1–90 and a 90–1 % change in es are shown in Fig. 3a and b,
respectively.
For both changes in es there is good agreement of e-folding times
between all models, with a maximum variation of 10 %, which is well
within the uncertainty caused by varying degrees of numerical convergence
and potential numerical diffusion. Diffusion times are much shorter than in
the constant Di case due to the high diffusivity of the semi-volatile
component. Figure 3a shows that even when starting with a glassy particle, if
the saturation ratio of a plasticising semi-volatile component increases
sufficiently, the e-folding state can be reached in less than 1 s. For the
decreasing es used in Fig. 3b a low diffusivity outer casing will form
on the particle, impeding diffusion and evaporation. However, Fig. 3b shows
that if a particle initially of water-like diffusivity is quickly dried, the
e-folding state is reached within 10 s, even when the non-volatile component
has a relatively low diffusivity.
e-folding times for 1–90 and 90–1 % changes in es were also
found using the sigmoidal dependence of Di on xsv; the results are
given in Fig. 3c and d, respectively. In the 90–1 % case an unpractical
computer time (> 12 h) was required to attain numerical
convergence at low values of Dnv0, therefore the minimum
Dnv0 is 1 × 10-20 m2 s-1. For this relatively large
change in es the sigmoidal dependence induces a steeper diffusion front
than the logarithmic dependence. Despite this, the models show good agreement
here also. In the 1–90 % case, a maximum variation in e-folding times
of 5 % is seen while for 90–1 % this value is 30 %. This latter
variation is between KM-GAP and the other two models and is greater than
expected from different degrees of numerical convergence. However, given the
gradual divergence of the e-folding isolines in Fig. 3d, we do not
attribute the discrepancy to model framework differences, but to an
insufficient shell resolution in KM-GAP. Diffusion is quicker using the
sigmoidal dependence than the logarithmic dependence, particularly for the
90–1 % scenario. This is explained by the higher Di values at
xsv > 0.5 (Fig. 2).
e-folding time contour plots for different changes to the
saturation ratio of the semi-volatile component (Δes)
and different diffusion coefficient dependencies:
(a)Δes= 10–20 %
logarithmic dependence, (b)Δes= 20–10 %, logarithmic dependence,
(c)Δes= 10–20 %, sigmoidal
dependence, and (d)Δes= 20–10 % sigmoidal dependence. Dnv0
is the diffusion coefficient at a semi-volatile mole fraction of 0, while
Dsv0 (diffusion coefficient at a
semi-volatile mole fraction of 1) was fixed at 2.0 × 10-9 m2 s-1.
es changes more realistic of the atmosphere were also tested. Results
for 60–80 and 80–60 % (Fig. A4) are similar to those for 1–90
and 90–1 % for their respective Di dependency; there is good model
agreement, and across the Dp,t=0 and Dnv0
range and for both dependencies, e-folding time is less than 1 s. Results
for 10–20 and 20–10 %, given in Fig. 4, also show agreement between
models. For both dependencies diffusion is much slower than in the 1–90
and 60–80 % simulations, approaching 1 ky at low Dnv0 and
high Dp,t=0. This shows that at low saturation ratios
of semi-volatile component, gas-particle partitioning can be limited by
condensed-phase diffusion in viscous particles.
e-folding times between models were also found to be in good agreement for
these changes in es when Dnv0 was fixed at
1.0 × 10-25 m2 s-1 and Dsv0 was varied between
1.0 × 10-25 and 2 × 10-9 m2 s-1.
As discussed, the agreement between models in
estimating e-folding times indicates that the estimated profiles of
concentration with particle radius prior to the e-folding state are
consistent between models because the underlying equations are the same. By
comparing concentration-radius profiles at various stages of diffusion we
indeed found good model agreement across all cases. In Fig. 5 we show the
example of the logarithmic dependence of Di on xsv, an
instantaneous change in saturation ratio of 1–90 % and with
Dnv0=1×10-25 and
Dsv0= 2×10-9 m2 s-1. At several
times preceding and including e-folding time the concentration-radius
profiles are in good agreement.
The concentration ([sv])-radius (r) profiles of the
semi-volatile component at times (t) preceding and including the e-folding
time (te) as estimated by the three models given in the legend, for the
case of logarithmic dependence of Di on
xsv, an instantaneous increase in
es from 1 to 90 % and with
Dnv0= 1 × 10-25 and
Dsv0= 2 × 10-9 m2 s-1. Note
that for clarity the radius axis begins at 2 × 10-6 m and not 0 m.
In the final part of this study the estimated temporal profile of particle
radius was compared between the sigmoidal and logarithmic Di
dependencies. We have used the water activity and density dependence on
sucrose weight fraction as described in Zobrist et al. (2011) for the
sucrose-water system in an attempt to replicate a non-ideal system. The ETH
model was employed, though the results above indicate that KM-GAP and Fi-PaD
would produce identical profiles. For the inference of Di dependency
from radius measurements the signal to noise ratio is minimised by inducing
a large change in radius relative to the measurement accuracy over a
time-span that is large compared to the measurement frequency.
The radius (Rp) change with time for a single
particle subject to the changes in es shown by the
orange curve and right vertical axis. Here
Dsv0= 2 × 10-9
and Dnv0= 1 × 10-25 m2 s-1
using the Di dependencies given in the
legend.
(a) Radius (Rp) change with time for a single
particle experiencing the changes in saturation ratio of the semi-volatile
component (es) shown by the orange curve (allied with
the right vertical axis), for Dsv0= 2 × 10-9
and Dnv0= 1 × 10-25 m2 s-1
and using the Di dependencies given in the legend.
(b–e) Time intervals of (a) over
select changes in es (as shown by their orange curve
and right vertical axes).
Taking the case of water as the semi-volatile component, from Fig. 3 it is
clear that for certain values of Dnv0 and certain changes in
es attaining a large ratio of equilibrium time to measurement frequency
may be difficult, even if the change in radius is large. Indeed, the
radius-time profiles in Figs. 6 and 7 for instantaneous changes in es
and a Dnv0= 1 × 10-25 m2 s-1 confirm that for changes
with a high final es, significant radius change is estimated to occur
over less than 1 s, while the measurement frequency reported in the studies
of Zobrist et al. (2011) and Lienhard et al. (2014) is approximately 15 s.
Nevertheless, for the es change of 1–90 % in Fig. 5, there is a
notable difference in the radius profiles between the dependencies. Despite
having lower Di at low xsv, the radius change from the sigmoidal
dependence is more rapid than the logarithmic, indicating that the Di at
higher xsv has a dominating effect on the profile. The inference of
Di dependency using such a large change in es is therefore poorly
constrained for lower xsv. For better constraint smaller changes in
es are required, such as those used in Lienhard et al. (2014). An
example of the radius profiles following incremental changes in es,
Dnv0= 1 × 10-25,
Dsv0= 2 × 10-9 m2 s-1
and using both dependencies is shown in Fig. 7. This plot
demonstrates the need for consideration of the time a given es is
maintained in measurement experiments, since the difference in the
equilibrium timescales between the es increments covers several orders
of magnitude. Indeed, over low changes in es such as between 1–10 %,
equilibration time may be too long to be practical for gaining a useful
measurement of radius change. It is worthwhile to note that the rate of
change of es over an increment is preferably much greater than the rate
of equilibration, as this provides the greatest potential for a clear
signature of the Di dependence and therefore greatest constraint on
inference.
Discussion and conclusion
The results above show that despite variations in their numerical methods,
all three Fickian-based diffusion models tested here: the ETH model, KM-GAP
and Fi-PaD give good agreement of estimated e-folding timescales over a wide
range of changes to the saturation ratio of the semi-volatile component and
over a wide range of differences in the self-diffusion coefficient of the
semi-volatile and non-volatile components. Furthermore, there is good
agreement between models when different dependencies of diffusion
coefficient on composition are used. This result has not been reported
before to our knowledge and verifies consistency between existing Fickian
diffusion models. The maximum disagreement in e-folding times for results
gained with satisfactory shell resolution is 10 %, which is within the
uncertainty generated by varying degrees of numerical convergence and
potential numerical diffusion. The consistency in modelled
concentration-radius profiles at times preceding and including the
e-folding state (Fig. 5) shows that if used for a polydisperse aerosol
population, the models would give agreement in changes to the size
distribution. In addition, if the diffusing component were reactive the rate
of particle-phase reaction would depend on its concentration; therefore
model agreement in concentration-radius profiles would give consistent
reaction rates across the particle (which in turn could affect diffusion
rate).
Using the three diffusion models as described above and with the spatial
resolutions presented in the Appendix, the ETH model takes approximately 2
orders of magnitude less computer time than Fi-PaD or KM-GAP for a given
diffusion scenario. With the models giving consistent estimates of
diffusion, the ETH model therefore appears to be favourable.
The e-folding times given in Fig. 3 for changes in es of 1–90 and
90–1 %, and in Fig. A4 for changes of 60–80 and 80–60 %, show that
for a semi-volatile component with water-like (at room temperature)
diffusivity, given a sufficiently high starting/finishing es,
attainment of the e-folding state is effectively instant compared to
residence times in the atmosphere and chamber experiments. This is due to
the plasticising effect of water (and applies to any semi-volatile component
with a sufficiently high self-diffusion coefficient). At lower values of
es diffusion time can be much longer (Fig. 4), consistent with
measurement studies (e.g. Zobrist et al., 2011; Lienhard et al., 2014).
The question therefore arises that for a given Dsv0, at what es
can equilibration be assumed instant? Figure 7 indicates that for water
condensing at room temperature equilibration time is less than 1 s when the
final es is greater than 50 % for the sigmoidal dependence used here
and when it is greater than 60 % for the logarithmic dependence. These
results indicate no limitation on mass transfer of water from particle-phase
diffusion at high relative humidity and at ambient temperature, and
therefore no impediment to the formation of cloud droplets. Experimental
results from Lienhard et al. (2014) and Zobrist et al. (2011) indicate that
this is also true down to ∼250 K.
For a hygroscopicity tandem differential mobility analyser (HTDMA), which
has a typical residence time of 20–25 s (Zardini et al., 2008) and
es increase of ∼90 % from an initial value <10 %,
Figs. 3 and 5 show that equilibration is attained even when sampling relatively large
(∼1×10-5 m) particles containing components of relatively low
diffusivity (self-diffusion coefficients ∼1×10-25 m2 s-1).
Note, however, that we have not considered extreme dependencies of Di
on composition. If, for example, a very high mole fraction of water were
required before the “cliff-edge” in the sigmoidal dependence (Fig. 2) was
reached, longer diffusion times than those shown here would be expected.
Regarding the inference of diffusion coefficient dependence on composition
from particle radius measurements, we have shown that incremental changes in
es provide the best constraint, and note that changes should occur over
a short time compared to the equilibration time. The consistency between the
diffusion models shown here indicates that the choice of model does not
affect the accuracy of the inferred dependence (as long as sufficient
spatial and temporal resolution is used).
In a follow-up study we intend to investigate the implementation of
composition-dependent Di in the Model for Simulating Aerosol
Interactions and Chemistry (MOSAIC) (Zaveri
et al., 2008). MOSAIC is used for chamber and ambient studies and can
therefore include, among other factors, multiple components, chemistry and
volatility. Furthermore, it can model polydisperse aerosol, providing
insight into how composition-dependent diffusion coefficients affect the
evolution of size distributions.
As mentioned, Fickian-diffusion is, strictly speaking, limited to
ideal-systems. Thus, for cases where dissolution occurs, for example, the
employed or derived diffusion coefficients are actually effective values of
Di. As we mention briefly in the introduction, numerous alternative
theories to Fickian diffusion exist. Although an analysis of such frameworks
is beyond the scope of this study, a similar critical analysis may be useful
in the future when data from more complex multicomponent systems exist.
The water mole fraction as a function of time and distance through a
single particle using the ETH model and the same inputs as for Fig. 3 of
Zobrist et al. (2011).
To validate our version of the ETH model Fig. 3d of Zobrist et al. (2011)
was reproduced using our version of the model and the relative humidity
measurements presented in their Fig. 3a. Note that in reproducing this figure
the dependence of diffusion coefficient on water activity given in Zobrist et
al. (2011) was used. Furthermore, water activity and density estimated as a
function of sucrose weight fraction, as described in Zobrist et al. (2011),
was used. Our reproduction is given in Fig. A1.
Figure A2 shows the convergence of e-folding times with increasing shell
number for the ETH model. Results are for self-diffusion coefficients of the
semi-volatile and non-volatile components of 2.0 × 10-9
and 1.0 × 10-25 m2 s-1, respectively, with a logarithmic
dependence of Di on composition and change to the vapour-phase
saturation ratio of the semi-volatile of 1–90 (Fig. A2a) and 90–1 %
(Fig. A2b). These cases were chosen because they are expected to have the
strongest concentration and diffusion coefficient gradients through the
particle (compared to other cases in this study) and should therefore
require greatest spatial resolution. The exponential fits in Fig. A2 were
obtained using Igor Pro software.
The e-folding time convergence with increasing shell number for
the ETH model: change in the semi-volatile component saturation ratio in
(a) was 1–90 % and in (b) was 90–1 %. The
self-diffusion coefficient of the semi-volatile was
2.0 × 10-9 m2 s-1 and that of the non-volatile was
1.0 × 10-25 m2 s-1. Orange curves are the
exponential best fits.
e-folding times (isolines) for the three models given in the
legend. Where Dcon0 is the constant diffusion coefficient used
throughout the simulation. Dp,t=0 is the initial particle
diameter. In (a) the saturation ratio of the semi-volatile is
increased from 1 to 90 % instantaneously, whilst in (b) it is
decreased from 90 to 1 % instantaneously.
e-folding time contour plots for different instantaneous changes
in the saturation ratio of the semi-volatile component (Δes)
and different diffusion coefficient dependencies: (a)Δes= 60–80 % and logarithmic dependence, (b)Δes= 80–60 % and logarithmic dependence, (c)Δes= 60–80 % and sigmoidal dependence, and (d)Δes= 80–60 % and sigmoidal dependence. Models are given in the
legend.
Absolute tolerances used in KM-GAP whilst the self-diffusion
coefficient of the semi-volatile component was held constant at
2.0 × 10-9 m2 s-1 and the saturation ratio of the
semi-volatile component was increased and decreased from 1–90 and
90–1 %. The absolute tolerance required for stability depended on the
self-diffusion coefficient of the non-volatile component (Dnv0)
and the initial particle diameter (Dp,t=0).
The absolute tolerances that were required to attain stability in the KM-GAP
model are given in Table A1. The tolerance was dependent on the
self-diffusion coefficient of the non-volatile and the initial particle
diameter. The relative tolerance was kept fixed at 1.0 × 10-12. These
tolerances were used for changes to the semi-volatile saturation ratio of
1–90 and 90–1 %. Since these represent the largest changes in
saturation ratio used in this study, the tolerances in Table A1 are
conservative values for all other cases presented in the study.
The number of shells used for each model is given in Table A2. The optimum
number of shells required for acceptable numerical convergence was found to
be dependent on the change in semi-volatile saturation ratio and the
difference in the self-diffusion coefficients of the components.
Results for model estimates of e-folding times when the diffusion
coefficient was kept constant are given in Fig. A3. Good agreement can be
seen between all three models across the values of constant diffusion
coefficient and initial particle diameter. In Fig. A4 are the e-folding
times for changes of 60–80 and 80–60 % in the saturation ratio of the
semi-volatile component using the logarithmic and sigmoidal dependencies of
diffusion coefficient on semi-volatile component mole fraction.
Number of shells used in each model for each change in the
vapour-phase saturation ratio of the semi-volatile component (Δes)
and for different values of non-volatile component self-diffusion coefficient
(Dnv0).
Simon O'Meara wrote the model codes, ran the simulations, created plots and tables and
wrote the manuscript. David O. Topping and Gordon McFiggans provided
substantial input to the method, model development and manuscript.
Acknowledgements
The Natural Environment Research Council has funded this work through the
PhD studentship of Simon O'Meara, grant number NE/K500859/1, and through grant
number NE/J02175X/1. Edited by:
M. Petters
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