Introduction
Aerosol nucleation, or new particle formation (NPF), is an important
phenomenon taking place throughout the Earth's atmosphere (Kulmala et al.,
2004). The key parameter of interest is the nucleation rate, which is defined
as the formation rate (cm-3 s-1) of new particles at the critical
size. The critical size is the smallest size at which the growth rate of a
particle is on average faster than its evaporation rate. This size depends
mainly on the concentrations and other properties of the nucleating vapors,
as well as on temperature. However, it is generally agreed that the critical
size is somewhere below 2 nm mobility diameter under atmospheric conditions
(Kulmala et al., 2013). In fact it can be as small as two molecules in the
case of barrierless, kinetically limited particle formation, where the dimer
is already stable against evaporation (McMurry, 1980; Kürten et al.,
2014).
Until recently the smallest mobility diameter that could be measured by a
condensation particle counter (CPC) was 2.5 to 3 nm – which is
substantially larger than the critical size. However, the detection limit of
newly developed CPCs is as small as 1.2 nm in particle mobility diameter
(Sgro and Fernández de la Mora, 2004; Iida et al., 2009; Vanhanen et al.,
2011; Kuang et al., 2012a; Wimmer et al., 2013). Nevertheless, despite this
progress the most widely used CPCs have detection thresholds at 2.5 nm or
above. Moreover, care is needed when interpreting data from the
newly developed CPCs since they can be sensitive to the chemical composition
of the particles (Kangasluoma et al., 2014). Furthermore, CPC cutoff curves
do not have the shape of a step function. Instead, detection of particles
below the cutoff size (usually defined as the size d50, where 50 %
of the particles are detected) is occurring to some extent and, if this
includes clusters below the critical size, the accuracy of the derived
nucleation rates can be strongly affected. For this reason, under certain
conditions, it can still be more reliable to use a conventional CPC with a
nominal cutoff around 3 nm for determining NPF rates. On the other hand, in
order to minimize the corrections, it is advantageous to measure the
formation rates as close as possible to the critical size.
Kerminen and Kulmala (2002) derived an analytical formula for correcting
experimental particle formation rates to determine nucleation rates at a
given critical size (abbreviated as the KK method in the following). This
method was developed for atmospheric nucleation measurements, and a similar
formula was also used by the McMurry group (Weber et al., 1997; McMurry et
al., 2005). Several publications followed Kerminen and Kulmala (2002) to
include additional effects, like a better description of the coagulation sink
from particle size distribution measurements (Lehtinen et al., 2007),
self-coagulation (Anttila et al., 2010), and a size-dependent growth rate
(Korhonen et al., 2014). In addition to atmospheric measurements, nucleation
studies in aerosol chambers or flow reactors have tremendously helped in the
understanding of aerosol nucleation. Such experiments require an accurate
method to derive the NPF rates. In this study the applicability of the
previous methods to chamber experiments such as CLOUD (Cosmics Leaving
OUtdoor Droplets) at CERN will be discussed (Kirkby et al., 2011; Almeida et
al., 2013; Riccobono et al., 2014). Furthermore, we present here a new method
that yields accurate results for any environment – be they chamber or
atmospheric data – provided the particle size distribution above a certain
threshold size is known, as well as the particle growth rate, and where all
loss processes are quantified as a function of size. The new method is
verified with the results from a numeric aerosol model.
Methods
Review of methods previously used for correcting the measured
particle formation rate
A lack of suitable instrumentation for the measurement of the particle number
density at diameters below ∼ 3 nm required the application of a
correction to derive the NPF rates close to the critical size (Weber et al.,
1997; Kerminen and Kulmala, 2002). The corrections were derived for
atmospheric particle measurements where the sink of the particles is usually
dominated by the coagulation with larger pre-existing particles. In order to
derive their analytical formulae, Kerminen and Kulmala (2002) as well as
Lehtinen et al. (2007) made the following assumptions:
the only important sink for new particles is their coagulation with larger pre-existing particles,
the new particles grow at a constant rate,
the population of pre-existing particles remains unchanged during the new particle growth.
Finding an analytical expression for the relationship between the nucleation
rate at a smaller size (dp1) and a larger size (dp2) requires taking
into account the size dependency of the coagulation coefficient. However,
the coagulation coefficient does not follow an expression, which can be
analytically integrated (Seinfeld and Pandis, 2006). Therefore, KK made the
assumption that the coagulation coefficient decreases with particle size
dp to the second power, i.e.,
Kdp,dj⋅dp2=Kdp1,dj⋅dp12,
where K is the coagulation coefficient and dj is the diameter of
pre-existing particles. This assumption leads to the following analytical
expression, which connects the particle formation rates J at different sizes:
Jdp1=Jdp2⋅expCSdp1GR⋅dp12⋅1dp1-1dp2,
where
CSdp1=∑jKdp1,dj⋅Nj
is the coagulation sink for the nucleated particles due to larger
pre-existing particles Nj and GR is the particle growth rate (typically
expressed in nm h-1).
However, depending on the ambient particle size spectrum, the power
dependency from Eq. (1) can be weaker; for example, Fig. 1 shows the calculated
coagulation coefficient between nanometer-sized particles and particles of
100 nm in diameter (solid black line, upper panel). The power dependency
follows a value of -1.5 rather than -2 (see Eq. 1), and for smaller
particles the magnitude of the slope becomes even smaller (colored lines in
Fig. 1, upper panel). The indicated slopes are reported for the size range
between dp1 = 1.7 nm and dp2 = 3.2 nm (mobility
diameters) because these are used in the CLOUD experiment (Kirkby et al.,
2011; Almeida et al., 2013; Riccobono et al., 2014) and also in the later
sections of this text. Note that the mobility diameter can be obtained by adding a
constant value of 0.3 nm to the geometric diameter (Ku and Fernandez de la
Mora, 2009).
Coagulation coefficient, K, as a function of particle size, dp
(upper panel). Coagulation coefficients are calculated between two particles,
where one particle has a constant size (indicated in the legend of the
figure) and the second particle diameter varies between 1 and 10 nm. The
wall loss rate for the CLOUD chamber as a function of particle size is shown by
the dashed curve (lower panel), whereas the dilution rate is indicated by the
dash-dotted line. Slopes of the curves are indicated for the range between
dp1 (1.4 nm, i.e., 1.7 nm in mobility diameter) and dp2 (2.9 nm,
i.e., 3.2 nm in mobility diameter).
Realizing that the power dependency from Eq. (1) depends on the conditions
during a nucleation event, Lehtinen et al. (2007), in a follow-up publication, dealt with introducing the real power dependency derived from atmospheric
size distribution measurements. This led to the following formulation for the
size correction:
Jdp1=Jdp2⋅expγ⋅dp1⋅CSdp1GR,
with
γ=1s+1⋅dp2dp1s+1-1,
where the parameter s is the slope of the coagulation coefficient with
particle size.
Furthermore, recent findings from atmospheric growth rate measurements
indicate that the GR can be a function of particle size (Kuang et al., 2012b;
Kulmala et al., 2013). Therefore, Korhonen et al. (2014) extended the
analytical solution from Eqs. (4) and (5) and included the effect of a
size-dependent GR, which can either vary linearly with particle size or
according to a power-law dependency. Another effect that can become important
when the population of particles between dp1 and dp2 becomes large
is self-coagulation. This effect was considered recently by Anttila et
al. (2010). While we will also deal with the effects of a size-dependent GR
(Sect. 3.1) and self-coagulation (Sect. 3.3), we will first focus on the
question of how far atmospheric nucleation and nucleation within a chamber
experiment are comparable in terms of their loss processes in the next
section.
Relevant losses in chamber experiments
The dominant particle loss mechanism for seedless chamber nucleation
experiments is generally due to collisions with the walls of the vessel and
possibly also due to dilution of the chamber gas. Large (3 m) chambers such
as CLOUD have wall loss rates (around 0.001 s-1 at 1 nm) similar to
the loss rates onto pre-existing aerosols in a pristine atmospheric
environment. We will address here to what extent these two environments are
equivalent.
The wall loss rate in chamber experiments can be expressed by (Crump and
Seinfeld, 1981; Metzger et al., 2010)
kwdp=C⋅Ddp,
where D(dp) is the diffusivity of a particle with size dp and C
is an empirical factor that depends on the chamber dimensions and turbulent
mixing. The diffusivity of a particle can be calculated from the
Stokes–Einstein relationship according to (Hinds, 1999)
Ddp=kB⋅T⋅CC3⋅π⋅η⋅dp,
which depends on the Boltzmann constant kB, the temperature T,
gas viscosity η, and the Cunningham correction factor CC. The
latter is a function of the gas mean free path and the particle diameter. At
small particle sizes the Cunningham correction factor is approximately
proportional to dp-1, and so the wall loss rate can be approximated
by
kwdp=C′dp,
where C′ is an empirical constant determined from a least-squares fit by
taking into account measured wall loss rates of sulfuric acid monomers and
particles in different size bins. Figure 1 shows the wall loss rate for the
CLOUD chamber as a function of dp (dashed curve, lower panel), where the
value of C′ is approximately 0.001 nm s-1. The wall loss rate
decreases by ∼ dp-1, which is much weaker than the originally
assumed power dependency of ∼ dp-2 for loss to atmospheric
particles (Eq. 1).
In addition to wall loss, another mechanism which affects the particle number
density in a chamber experiment is dilution of the chamber gas. Instruments
can take considerable amounts of the chamber gas, and this gas needs to be
replenished in order to maintain a constant pressure. The CLOUD chamber has a
volume of 26.1 m3, while the instruments typically use
150 L min-1. This leads
to a dilution rate of
kdil = 9.6 × 10-5 s-1, which is independent
of particle size (see dash-dotted black line in the lower panel of Fig. 1).
If coagulation with larger pre-existing aerosols is neglected, which is
well justified in a seedless chamber experiment, the two main loss mechanisms
– wall loss and dilution – can be used to derive an analytical solution for the
NPF rate at a small size. This is achieved by replacing the coagulation loss
term in Eq. (4) from Lehtinen et al. (2007) with kw(dp) and
kdil:
dJdpddp=-loss rateGR⋅Jdp=-1GR⋅C′dp+kdil⋅Jdp.
In this case, integration yields
Jdp1=Jdp2⋅exp1GR⋅C′⋅lndp2dp1+kdil⋅dp2-dp1.
The identical result would follow from Eqs. (4) and (5) by taking the limit
for s→-1 to take into account wall loss, and by taking s=0 for loss
due to dilution.
In conclusion, the KK method and also the follow-up versions should only be
applied to chamber nucleation experiments after applying the necessary
adjustments. Equation (10) provides a useful analytical formula for conditions in
which coagulation can be neglected. The data from Fig. 1 provide a guide as to
the relative importance of the different loss mechanisms for the CLOUD
chamber. The wall loss rate for the relevant sizes between 1.4 and 2.9 nm is
on the order of 10-3 s-1. Depending on particle size the
coagulation coefficient is in the range 10-9 to
10-8 cm-3 s-1, which indicates that particle number
densities between 105 and 106 cm-3 are required in order to
reach similar effects for coagulation and wall loss. At this point it is also
worth mentioning that all the expressions derived so far are based on the
assumption that nucleation and particle growth is driven by the condensation
of monomers (Lehtinen et al., 2007) and that cluster–cluster collisions are
unimportant. The effect of cluster–cluster collisions will be discussed in
Sect. 2.4.
The important conclusion that follows from the comparison of Eqs. (2), (4),
and (10) is that experiments and atmospheric environments with similar sink
rates cannot be directly compared before corrections are applied, because not
only the magnitude of the sink is important but also the dependency of the
loss rate as a function of particle size. Despite the practicability of
Eq. (10), a new method is required which additionally takes into account
coagulation as well as self-coagulation.
New method to derive the nucleation rate from the experimental
formation rate
We will assume that the size distribution above a certain threshold size
(dp2) is known, and furthermore that the size between two adjacent bins
differs by one molecule only. For the following discussion it is useful to
add m to all bin indices, although the original size distribution contains
n size bins ranging from 1 to n. In this case the size dp2
corresponds to the bin with the index m+1 (Fig. 2). The formation rate of
particles at and above dp2 can then be calculated from
J≥m+1=dN≥m+1dt+∑i=m+1n+mkw,i⋅Ni+kdil⋅N≥m+1+∑i=m+1n+m∑j=in+mδi,j⋅Ki,j⋅Nj⋅Ni,
where double counting of collisions between particles in the same size bin is
avoided by the factor (Seinfeld and Pandis, 2006)
The original size distribution above the cutoff size dp2 (size
bin m+1) is shown in light grey. The loss rate of particles and the rate of
change of the particle concentration in this size range must be compensated
for by the formation rate due to smaller particles growing into the measured size
range. This knowledge can be incrementally extended to bins at smaller sizes
in a stepwise process, finally reaching the smaller size, dp1 (size bin
x).
δi,j=0.5ifi=j1ifi≠j.
The first term on the right-hand side takes into account non-steady-state conditions
by means of the time derivative of the total particle number density (sum of the
particle concentrations from bin m+1 to n+m). The remaining three terms on
the right-hand side describe the loss processes of neutral particles in a chamber
experiment: wall loss, loss due to dilution of the chamber gas (independent
of particle size), and coagulation loss between particles of all size bins.
Note that the index i runs from m+1 to n+m and the index j from i to
n+m. In this way, the collisions between the bins i and j are not counted
twice. Since we are looking at formation rates larger than a certain size,
collision products will remain in the size range under consideration and
therefore loss due to coagulation between bins i and j only has to be taken into
account once. The formation rate at dp2 can also be calculated from
J≥m+1=J≥m+2+dNm+1dt+kw,m+1+kdil+∑j=m+1n+mδm+1,j⋅Km+1,j⋅Nj⋅Nm+1.
This equation allows for calculation of the formation rate at a smaller size from
the NPF rate at the next larger size. Here the time derivative of Nm+1
refers to the concentration of particles in the size bin m+1, whereas in
Eq. (11) it refers to all particles at and above index m+1.
In order to calculate the formation rate dp1 let us now introduce x+1
new size bins, which extends the size distribution towards the smaller sizes
(Fig. 2). Kerminen et al. (2004) also introduced extra size bins in order to
increase the accuracy of their analytical formula, which connects the
nucleation rate and a formation rate at a larger size similar to Eq. (2).
Extending the size distribution towards smaller sizes requires calculation of
the number concentration Nm in the first new bin. However, Eq. (13) does
not allow this directly; therefore additional information is required. This
information can be taken from the growth rate of the particles. The formation
rate and the growth rate (GR) are connected by the following equation
(Heisler and Friedlander, 1977; Lehtinen et al., 2007):
Jm+1=GRm⋅Nmdp,m+1-dp,m.
This relationship was used to describe the flux of particles due to
collisions with monomers. In such a case particles can grow only from one
size bin to the next larger bin without “jumping” into an even larger bin
due to cluster–cluster collisions. However, for the moment we will assume
that Eq. (14) is also valid for the case where cluster–cluster collisions are
relevant if appropriate definitions for the growth rate and NPF rate are
being used, and we will justify this assumption later in Sect. 2.4. Using of the
formation rate and growth rate relationship, the particle number concentration
can be calculated for the first new size bin (Nm) from the following
relationship:
Nm=dp,m+1-dp,mGRm⋅(J≥m+2+dNm+1dt+(kw,m+1+kdil+∑j=m+1n+mδm+1,j⋅Km+1,j⋅Nj)⋅Nm+1).
In the limiting case where particle formation and growth is dominated by the
addition of monomers, this method is accurate at steady state provided that
knowledge about the growth rate is available initially.
When applying the method, the particle growth rate GRm is required for
calculating the first unknown concentration. Strictly, the growth rate is not
known at the index m (because the known size distribution starts at index
m+1 by definition; see Fig. 2) but can only be calculated at the next
larger index using Eq. (14) by adjusting all indices to the next larger bin.
According to Eq. (15), the GR would need to be updated in every
reconstruction step. Nevertheless, we have found from numerical simulation
(see later sections) that the method is numerically more stable if a constant
GR at index m+1 is used for all iterative steps. However, if accurate
knowledge about a size-dependent GR is available, it can be easily implemented
in the method.
In Eq. (15) all quantities are known except the value of Nm (if GRm
is approximated by GRm+1). Once Nm is found, the formation rate
Jm can be calculated and the process repeated with the next
smaller size bin (index m-1). In this way the complete particle spectrum
above dp1 (containing now n+x+1 size bins) can be recreated until the
final formation rate Jdp1 (at index x) is calculated. The
underlying assumption is that growth above this size is purely kinetic (no
evaporation), which is likely a good assumption for most chemical systems and
the atmospheric data (e.g., Chen et al., 2012). A similar approximation was made by Nieminen et al. (2010) when deriving an analytical formula for
calculating growth rates where the vapor pressure of the condensing species
has been set to zero. However, in future studies, one could examine the effect
of evaporation at sizes larger than the critical diameter on the method and
attempt to implement it in a similar fashion as Olenius et al. (2014) in
their study about the effect of monomer collisions on the growth rates.
In order to test the relative importance of self-coagulation on the magnitude
of the formation rate correction it is also possible to take into account
only particles at and above m+1 in all reconstruction steps in the last
term on the right-hand side of Eq. (15). In Sect. 3.3 we will discuss under which
circumstances this can be done without sacrificing too much accuracy.
Relationship between particle formation rate and growth rate
including cluster–cluster collisions
In a recent publication, Olenius et al. (2014) investigated the
relationship between J and GR as well as different methods for deriving the
GR due to monomer collisions. The method introduced here should also be
applied to conditions where new particle formation is proceeding at the
kinetic limit, i.e., where all cluster evaporation rates are zero. Under such
conditions the cluster concentrations are quite high in comparison to the
monomer concentration, e.g., the dimer concentration can be ∼ 20 %
of the monomer concentration (McMurry, 1980; Chen et al., 2012; Kürten et
al., 2014). In this case, the particle formation as well as the particle
growth cannot be described by monomer collisions only and cluster–cluster
collisions have to be taken into account. Therefore, Eq. (14) might not be
valid anymore. In the following we will investigate whether the relationship from
Eq. (14) can still be used. Following a similar approach as Olenius et
al. (2014) but taking into account cluster collisions and neglecting the
effect of evaporation, the particle growth rate for particles in the size bin
m can be defined as
GRm=ddp,mdt=∑i=1mδi,j⋅dp,m3+dp,i31/3-dp,m⋅Ki,m⋅Ni.
Equation (16) indicates that the clusters in the size bin m can grow by
collisions with all smaller clusters. We will assume that a particle in size
bin m contains m monomers with a mass u and density ρ:
dp,m=6⋅m⋅uπ⋅ρ1/3=dp,mono⋅m1/3,
where dp,mono is the diameter of the monomer. Multiplication of
the growth rate from Eq. (16) by Nm/(dp,m+1-dp,m) and using
the Taylor expansion
m+i1/3-m1/3=m1/3⋅1+im1/3-1≈i3⋅m2/3
leads to the following expression:
GRm⋅Nmdp,m+1-dp,m≈∑i=1mδi,j⋅i⋅Ki,m⋅Ni⋅Nm.
On the other hand, the particle formation rate J≥m+1 can be defined as
J≥m+1=0.5⋅∑i+j≥m+1δi,j⋅Ki,j⋅Ni⋅Nj,
where i and j have to be smaller than m+1. From this definition it
follows that Eq. (19) cannot be cast into the form of Eq. (20) because the
equation involving the growth rate only considers collisions where one
collision partner always belongs to the size bin m. Instead, Eq. (20)
involves collisions where none of the collision partners is fixed to one size
bin in the summation. Therefore, we were not able to find an analytical
solution in terms of bringing Eqs. (19) and (20) into agreement. However, we
can argue qualitatively that the two equations are yielding approximately the
same results for certain conditions.
The accurate definition of J≥m+1 (Eq. 20) is visualized schematically
in Fig. 3a, whereas Eq. (19) is indicated in Fig. 3b. For the monomer there
is only one possibility for contributing to J≥m+1 in both cases.
However, the dimer can contribute to J≥m+1 due to collisions with
particles in bin m and bin m-1. The approximation (Eq. 19) only takes into
account collisions between dimers and particles in bin m. However, the
dimer collision is taken twice and the second collision can therefore
compensate for the collision between dimers and particles in bin m-1 from
Eq. (20). For the trimer the situation is similar; in the accurate case the
trimer has three possibilities (i.e., three different collisions) to
contribute to J≥m+1. The approximation (Eq. 19) is taking into
account only one collision, but it is multiplied by a factor of 3. This
mechanism is the same for the collisions involving larger clusters.
Therefore, we can conclude that
GRm⋅Nmdp,m+1-dp,m≈J≥m+1
applies also for conditions where cluster–cluster collisions become important
(note the “≥” sign on the right-hand side). The requirement is that the cluster
concentrations do not change strongly in the region around bin m and that
the contribution of clusters to new particle formation and growth becomes
negligible at some index smaller than m. Under what circumstances the
relationship from Eq. (21) is valid needs to be studied in more detail in the
future. However, the benefit of the method from Sect. 2.3 is that even if the
relationship from Eq. (21) introduces inaccuracies, these are very likely
small because its effects should cancel out. GRm+1 for Eq. (15) is
calculated from the relationship in Eq. (21), and the same relationship is
used to calculate Nm in Eq. (15). Therefore, we expect the error due to
this approximation to be small, and the numerical simulations shown in the
following sections support this assumption.
(a) Particle formation rate J≥m+1 due to collisions
of monomers and clusters. (b) Approximation of the particle
formation rate including the growth rate definition according to Eq. (19).
See text for details.
Kinetic model for testing the universal method
A numerical model was developed recently for the CLOUD chamber to
simulate the formation and growth of uncharged sulfuric acid–dimethylamine
particles (Kürten et al., 2014). The model assumes that particles grow
from monomers by condensation and coagulation. Due to the arguments presented
by Kürten et al. (2014), it has been concluded that
H2SO4 ⋅ ((CH3)2NH clusters (abbreviated as
SA ⋅ DMA) constitute the basic “monomer” for the formation of
particles in a system of sulfuric acid (SA) and dimethylamine (DMA). Assuming
unit sticking efficiency and zero evaporation rate, good agreement is found
between the model and the experimentally measured neutral clusters.
The kinetic model is based on McMurry (1980). The time-dependent balance
equation for the monomer concentration N1 is
dN1dt=P1-k1,w+kdil+∑j=1NK1,j⋅Nj⋅N1
and, for all larger clusters (k≥2),
dNkdt=12⋅∑i+j=kKi,j⋅Ni⋅Nj-kw,k+kdil+∑j=1NKk,j⋅Nj⋅Nk.
Here, P1 is the production rate of the monomers, kw the wall
loss rate, kdil the dilution rate, and K the coagulation
coefficient.
The original model calculated concentrations of clusters ranging from dimer
up to clusters of several thousand molecules. Each size bin was represented
by a single cluster with a fixed number of molecules (or SA ⋅ DMA
clusters, which are each treated as one molecule). The maximum particle size
that can be reached with reasonable computation time is a few nanometers, which is
too small for the current study. Therefore we incremented the size by one
molecule for the first 100 bins (linear bins) and by a constant geometrical
factor for the next 100 bins (geometric bins; see, for example, Landgrebe and
Pratsinis, 1990, or Lovejoy et al., 2004). With this method, a size of
∼ 30 nm can be reached using a geometrical factor of 1.023, which is
suitable for the present study. The sizes of dp1 and dp2 do,
however, fall into the size range of the linear bins.
In addition to the kinetic modeling, we have also introduced evaporation rates
for the dimer and the trimer (evaporation rates not included in Eqs. 22 and
23 for simplicity). These simulations are used to investigate situations
where nucleation and particle growth is dominated by the addition of
monomers, because if the evaporation rates for the smallest clusters are sufficiently
high, their concentrations become very small and will therefore not contribute
significantly to NPF and growth. Although not directly relevant for the
sulfuric acid–dimethylamine system, we have calculated the dimer and trimer
evaporation rates at 223.15, 248.15, and 278.15 K at 38 % RH from the
data presented by Hanson and Lovejoy (2006). Their thermodynamic data were
derived for the binary system of sulfuric acid and water. However, the
calculated formation rates are not meant to be representative of binary
nucleation; rather, they only serve to demonstrate the effect of going from purely
kinetic nucleation to nucleation with a relatively large barrier (278 K
data). Kinetic nucleation will include collisions with monomers and also show
a significant effect from clusters, whereas the new particle formation at
278 K will be dominated by monomer collisions. The other two temperatures
show the transition from purely kinetic nucleation to nucleation dominated by
monomer additions.
Particle formation rates that have been calculated from the model serve as
the reference formation rates to which the reconstructed formation rates can
be compared to. We have implemented two separate procedures to calculate the
NPF rates, where the first one is following the approach based on Eq. (11) by
taking into account all loss processes, while the second one follows the
production of particles from two smaller clusters (Eq. 20). The two methods
yield exactly the same result, which is a good verification of the kinetic model in
this respect.
Discussion
Figure 4 shows the result of the kinetic model simulation for a monomer
(molecular weight of 143 g mol-1 and density of 1.47 g cm-3) production rate of
8.8 × 104 cm-3 s-1, after
1.5 × 104 s. Integration of Eqs. (22) and (23) yields the
displayed size distribution (grey bars). Although the particles continue to
grow, the populations at smaller sizes (below about 10 nm) are close to
steady state. Since the total particle number concentration is dominated by
these smaller particles, time dependency can be neglected in the following,
but will be revisited in Sect. 3.2. The size distribution (grey bars in
Fig. 4) is obtained after normalizing the concentrations by the number of
molecules per bin.
Modeled and reconstructed particle size distribution for kinetic
nucleation. The model uses different definitions for the first 100 size bins
(up to ∼ 3.1 nm) and the last 100 size bins (> ∼ 3.1 nm). In
the first 100 size bins, the number of molecules in the particles increases
by one between each bin, whereas in the next 100 bins the particle diameter
is increased by a constant factor between each bin. Normalizing the
concentration by the number of molecules per bin leads to the shown size
distribution (grey bars). The reconstructed size distribution using the new
method described here is shown by the solid red line, starting from the
particle distribution above 2.9 nm.
The new universal method to derive a particle formation rate at a smaller
size dp1 has been applied to the data shown in Fig. 4. A threshold size
dp2 = 3.2 nm (corresponding to 2.9 nm geometric diameter) has been
chosen. Starting with the size distribution for particles equal to or larger
than 2.9 nm, 71 new bins were introduced to reach the size dp1 at
1.7 nm (1.4 nm geometric diameter). The red line shows the recreated size
distribution obtained by this method. A constant growth rate of
3.81 nm h-1 was chosen, corresponding to the value given by a numeric
model calculation for a particle in the size bin m+1. As can be seen, the
reconstruction works well for the first few size bins and then starts to
deviate somewhat from the correct values. This occurs since the GR is not
exactly constant with size, and slightly increases when approaching dp1
(see Sect. 3.1).
Size-dependent growth rate
The growth rate which is used for the reconstruction is calculated from
GRm+1=J≥m+2⋅dp,m+2-dp,m+1Nm+1.
Our studies with the kinetic model indicate that GR is only weakly dependent
on particle size in the range between critical size and detection threshold.
In the example shown in Fig. 4 there is less than 20 % variation.
However, the model does not include the effects of evaporation or of a
spectrum of condensable vapors with different volatilities. Therefore care
has to be taken when applying size corrections to atmospheric particle
formation rates. The GR should ideally be measured over a wide range of
diameters (Kulmala et al., 2013). In this case analytical solutions for the
KK method can be found for certain size-dependent GRs (Korhonen et al.,
2014). These considerations underscore the importance of directly measuring
the particle GR in the sub-3 nm size range, as well as at larger sizes. With
this information the effect of particle evaporation can be separated from the
uncertainties due to size-dependent particle GR. In the absence of such
measurements, a detailed error analysis is required to bracket the range of
GR uncertainty and its impact on the derived nucleation rates.
A comparison between the accurate solution for the NPF rates and the ones
from the reconstruction method as a function of particle size is shown in
Fig. 5. The accurate solution from the kinetic model is shown by the
solid green line, while the results from the reconstruction method are indicated by
the red triangles. Due to the slight size dependency of the growth rate (it
increases slightly with decreasing size), the reconstructed NPF rates are
somewhat higher than the accurately calculated values. The maximum deviation
occurs at the smallest size and reaches ∼ 17 % in this example.
Given the fact that the formation rate J(dp1) is more than a factor of
12 higher than J(dp2), this is a rather small deviation.
Formation rates as a function of particle size for kinetic nucleation.
Formation rates simulated with the kinetic model are shown by the green line.
Reconstructed particle formation rates starting at dp2 = 2.9 nm and
ending at dp1 = 1.4 nm using a constant GR (taken at dp2) are
shown by the red triangles.
Time evolution in a simulated chamber nucleation
experiment
Using a kinetic model simulation, we show in Fig. 6 an example of the
time-dependent formation rates J for the particle sizes dp1 (1.4 nm
geometric diameter; solid green line) and dp2 (2.9 nm geometric
diameter; solid blue line). In addition, the rate of change of particle
concentration dN/dt (dashed lines) above the size thresholds
dp1 and dp2 are shown. The formation rates J are directly
obtained from the model using Eq. (11) and the size distribution.
Interestingly, the formation rates overshoot before they reach an almost
constant value. This overshoot is explained by the absence of larger
particles at the beginning of the experiment. Therefore the loss rate is
smaller at the beginning, which allows for faster formation rates. Once the
larger particles start to form, the loss rate increases until eventually
there are only small changes in particle concentrations and formation rates.
This overshoot can be quite large and, in this example, reaches almost a
factor of 3 for the maximum J compared with its steady-state value.
Particle formation rates J (cm-3 s-1, solid lines) and
change in particle concentration dN/dt (cm-3 s-1,
dashed lines) shown for two different sizes, dp1 = 1.4 nm (green
lines) and dp2 = 2.9 nm (blue lines). The data are from a kinetic
model calculation. The reconstructed J(dp1) is shown by the solid red
line. Through use of a time correction, the reconstructed J(dp1) are shifted to
earlier times (dash-dotted red line).
Using the size distribution as a function of time for particle sizes equal to
or larger than dp2 (not shown), as well as the growth rates GRm+1
(not shown) and the time derivative of the total number concentration of
particles dNm+1/dt, the size-corrected formation rate
Jdp1 can be derived by the method described in Sect. 2.3 (solid red
line). The derived formation rate agrees closely with the accurate solution
from the kinetic model (solid green line) for conditions close to
steady state. However, when evaluating J at dp1 from the formation
rate at dp2 and time t, one needs to consider that the particles that
appear at dp2 were passing the size dp1 at an earlier time t′.
This time can be approximated by
t′=t-dp2-dp1GRm+1(t)
if the time dependency of the GR is considered. Displaying the reconstructed
formation rate J(dp1) against the corrected time axis yields the
dash-dotted red line, which shows a very similar time dependency to the
accurate J(dp1). The overestimation (difference between the red and
green lines in Fig. 6) is due to the size dependency of the growth rate (see
previous section). An accurate determination of J(dp1) can only be
obtained after the particles have formed at and above dp2.
Formation rates as a function of the sulfuric monomer
concentration
Kinetic limit
In the preceding section, the universal method has only been tested for one
sulfuric acid monomer concentration. Variation of the monomer production rate
P1 in Eq. (22) will result in different sulfuric acid concentrations.
The resulting size distributions (N), growth rates (GR), and rates of change
of particle concentration (dN/dt) as a function of particle
size can be used to test the reconstruction method. Figure 7 shows the
results for 8 × 105 to 2 × 107 cm-3
sulfuric acid concentration (lines denoting “kinetic limit”). The
accurate solution for dp2 is shown by the solid blue line, while
J(dp1) is shown by the solid green line. Using a constant GR,
corresponding to its value at dp2, the reconstruction method yields the
results shown by the solid red line. For the high nucleation rates (above
several hundred) the accuracy is quite good. For the lower formation rates,
the required corrections are quite large because the growth between 1.4 and
2.9 nm is slow and therefore losses are high. The effect of the
size-dependent growth rate therefore has a relatively large impact on the
reconstructed NPF rates. The curved shape of the formation rates displayed
against the sulfuric acid concentration on a log–log plot is due to the fact
that losses are much more relevant when particle growth is slow (see Ehrhart
and Curtius, 2013).
Formation rates as a function of the sulfuric acid monomer
concentration. The solid blue curves show the formation rates at dp2
calculated from the model. The simulated formation rates J(dp1) from the
model are indicated by the green lines. The reconstructed formation rates at
dp1 are shown by the red lines. Varying the constant GR by both a factor of
1.5 and 0.9 results in the error band shown in light red.
Neglecting self-coagulation yields the dashed black line. A complete set of
all curves is shown for four different scenarios (kinetic limit, “223 K”,
“248 K”, and “278 K”). See text for details.
In practice, GR will always be subject to measurement uncertainties. In order
to test the sensitivity of the method, the constant GR was multiplied by both a
factor of 1.5 and 0.9. The faster GR leads to an
underestimation in the reconstructed J(dp1), while the slower GR leads
to an overestimation. The variation in the GR is indicated by the light-red
bands in Fig. 7. It can be seen that the reconstructed J(dp1) is highly
sensitive to GR, especially when the particle growth is slow. In this
example,
a GR underestimation of only 10 % can lead to a substantial
overestimation of J(dp1) due to the exponential dependence on GR.
Therefore, accurate growth rate measurements are essential to reliably
reconstruct the particle formation rate at a smaller size.
In order to test the effect of self-coagulation, coagulation has only been
taken into account to occur with particles at and above dp2 (dashed
black lines). As long as the formation rate is close to
∼ 100 cm-3 s-1 at dp1, the effect of neglecting
self-coagulation is quite small. For larger formation rates the deviation
progressively increases because self-coagulational loss becomes competitive
and eventually dominant compared to wall loss, dilution, and coagulation with
larger particles. However, these numbers are relevant for the CLOUD chamber
experiment and are not necessarily applicable to other chambers with other
wall loss and dilution characteristics. Performing the corrections twice –
once by including self-coagulation and a second time by neglecting it – over
a range of formation rates can help to find the formation rate at which
self-coagulation becomes important in other chambers. The advantage of
neglecting coagulation is that the reconstruction is computationally much
less demanding. One major difference between most experiments and the example
calculations shown in Fig. 7 is that nucleation is generally not proceeding
at the kinetic limit, even though this is the case for nucleation of sulfuric
acid and dimethylamine (Kürten et al., 2014). In order to evaluate the
method for NPF which is not proceeding at the kinetic limit, we have also
simulated NPF with nonzero dimer and trimer evaporation rates.
223 K
The dimer evaporation rate has been set to 2.9 s-1 and the trimer
evaporation rate to 0.024 s-1 (corresponding to conditions in the
binary system at 223.15 K and 38 % RH; see Hanson and Lovejoy, 2006). At
these relatively low evaporation rates the effect of cluster–cluster
collisions is still pronounced, which can be seen for the high sulfuric acid
concentrations where a relatively large difference between J(dp2) and
J(dp1) can be seen. This difference is due to the strong effect of
self-coagulation, which leads to high loss rates. Although the GR is
increasing with higher sulfuric acid concentration, self-coagulation
increases as well because the cluster concentrations increase. Therefore, the
two opposing effects cancel out, which leads to a rather constant factor
between J(dp2) and J(dp1). The maximum deviation between the
reconstructed and the accurate J reaches a factor of 4 at the lowest
sulfuric acid concentration of 2 × 106 cm-3. As the
growth rate becomes higher and the corrections smaller with increasing
sulfuric acid concentration, the effect of the size-dependent GR becomes less
relevant and the accuracy increases.
248 K
Evaporation rates of respectively 181 and 3.1 s-1 were used for the dimer and the trimer (Hanson and Lovejoy, 2006). Because of these relatively
high evaporation rates, particle formation and growth is dominated by
collisions with monomers. The growth rates are quite high and
self-coagulation can be neglected for most conditions (monomer concentration
below ∼ 1 × 108 cm-3); therefore the correction
factors are lower than for the previous two conditions discussed. The maximum
error due to the size-dependent GR is a factor of 2 at
5 × 106 cm-3 of sulfuric acid.
278 K
When evaporation rates of respectively 10 060 and 360 s-1 are used for the dimer and the
trimer, conditions can be simulated where monomer
collisions are by far the dominant process for nucleation and growth due to
very low cluster concentrations. In this case, quite high sulfuric acid
monomer concentrations are required to yield substantial NPF. At these
conditions the GRs are very high (up to ∼ 100 nm h-1) and
self-coagulation is irrelevant. Therefore, the correction factor between
J(dp2) and J(dp1) approaches a value of 1. Only at the low sulfuric
acid monomer concentrations is a significant correction necessary.
Conclusions
The Kerminen and Kulmala (2002) method, and its refinements presented in
subsequent publications (Lehtinen et al., 2007; Anttila et al., 2010;
Korhonen et al., 2014), is widely used in atmospheric and chamber experiments
to derive nucleation rates from experimentally measured formation rates at
larger particle sizes. However, it was not designed to be applied to
chamber nucleation experiments where self-coagulation can be important.
We have therefore presented a new method that yields representative results
in any general environment, provided certain quantities are known. The new
method requires knowledge of the particle size spectrum above the detection
threshold, the particle growth rate, and all loss processes as a function of
particle size. With this information the size spectrum and the formation rate
can be reconstructed in a stepwise process to a smaller size, where the
nucleation rate is determined. The method can give accurate results and,
furthermore, takes into account self-coagulation among newly formed
particles, which can be an important effect, recognized previously by Anttila
et al. (2010). Additionally, if the size-dependent growth rate is available
from measurements, it can be readily incorporated during the reconstruction
of the size distribution.
The proposed new method allows extrapolation of the particle formation rate
measured at one threshold size (dp2) to a second, smaller size
(dp1). In this way, a precise quantitative comparison can be made
between formation rates measured simultaneously by several counters operating
in the 1 to 3 nm threshold range and, where differences emerge, a deeper
understanding of fundamental quantities such as cluster critical sizes,
growth rates, and evaporation rates can be obtained.
One general issue with all methods that extrapolate formation rates
towards smaller sizes arises from the uncertainty in the growth rate. In
most cases no measurement of the GR will be available down to the very small
size since the particle number concentrations are also not available
(otherwise no extrapolation of the formation rate would be necessary). A
small size dependency of the GR that is not taken into account can therefore
lead to large uncertainty. In addition, the critical size of the nucleating
particles is generally not known. Ideally, one would choose dp1 to
correspond to the critical size. However, since this is not possible, a
reasonable solution to this issue is to choose a size for dp1 which is
safely at or above the critical size in order to avoid extending the size distribution
into the subcritical size regime. For this reason the CLOUD experiment has
reported particle formation rates at a size of 1.7 nm in mobility diameter
rather than nucleation rates (Kirkby et al., 2011).
Further studies using the new method will focus on the effect of using
larger size bins and its application to experimental data measured with
condensation particle counters (CPCs) and scanning mobility particle sizer
(SMPS) systems.