ACPAtmospheric Chemistry and PhysicsACPAtmos. Chem. Phys.1680-7324Copernicus PublicationsGöttingen, Germany10.5194/acp-17-3525-2017Refreeze experiments with water droplets containing different types of ice
nuclei interpreted by classical nucleation theoryKaufmannLukasMarcolliClaudiaclaudia.marcolli@env.ethz.chhttps://orcid.org/0000-0002-9125-8722LuoBeipingPeterThomasInstitute for Atmospheric and Climate Science, ETH, Zurich, SwitzerlandMarcolli Chemistry and Physics Consulting GmbH, Zurich, SwitzerlandClaudia Marcolli (claudia.marcolli@env.ethz.ch)14March2017175352535522November20168November201614February201728February2017This 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/17/3525/2017/acp-17-3525-2017.htmlThe full text article is available as a PDF file from https://acp.copernicus.org/articles/17/3525/2017/acp-17-3525-2017.pdf
Homogeneous nucleation of ice in supercooled water
droplets is a stochastic process. In its classical description, the growth
of the ice phase requires the emergence of a critical embryo from random
fluctuations of water molecules between the water bulk and ice-like
clusters, which is associated with overcoming an energy barrier. For
heterogeneous ice nucleation on ice-nucleating surfaces both stochastic and
deterministic descriptions are in use. Deterministic (singular) descriptions
are often favored because the temperature dependence of ice nucleation on a
substrate usually dominates the stochastic time dependence, and the ease of
representation facilitates the incorporation in climate models. Conversely,
classical nucleation theory (CNT) describes heterogeneous ice nucleation as
a stochastic process with a reduced energy barrier for the formation of a
critical embryo in the presence of an ice-nucleating surface. The energy
reduction is conveniently parameterized in terms of a contact angle
α between the ice phase immersed in liquid water and the
heterogeneous surface. This study investigates various ice-nucleating agents
in immersion mode by subjecting them to repeated freezing cycles to
elucidate and discriminate the time and temperature dependences of
heterogeneous ice nucleation. Freezing rates determined from such refreeze
experiments are presented for Hoggar Mountain dust, birch pollen washing
water, Arizona test dust (ATD), and also nonadecanol coatings. For
the analysis of the experimental data with CNT, we assumed the same active
site to be always responsible for freezing. Three different CNT-based
parameterizations were used to describe rate coefficients for heterogeneous
ice nucleation as a function of temperature, all leading to very similar
results: for Hoggar Mountain dust, ATD, and larger nonadecanol-coated water
droplets, the experimentally determined increase in freezing rate with
decreasing temperature is too shallow to be described properly by CNT using
the contact angle α as the only fit parameter. Conversely,
birch pollen washing water and small nonadecanol-coated water droplets show
temperature dependencies of freezing rates steeper than predicted by all
three CNT parameterizations. Good agreement of observations and calculations
can be obtained when a pre-factor β is introduced to the rate
coefficient as a second fit parameter. Thus, the following microphysical
picture emerges: heterogeneous freezing occurs at ice-nucleating sites that
need a minimum (critical) surface area to host embryos of critical size to
grow into a crystal. Fits based on CNT suggest that the critical active site
area is in the range of 10–50 nm2, with the exact value depending on
sample, temperature, and CNT-based parameterization. Two fitting parameters
are needed to characterize individual active sites. The contact angle
α lowers the energy barrier that has to be
overcome to form the critical embryo at the site compared to the homogeneous
case where the critical embryo develops in the volume of water. The
pre-factor β is needed to adjust the calculated slope of
freezing rate increase with temperature decrease. When this slope is steep,
this can be interpreted as a high frequency of nucleation attempts, so that
nucleation occurs immediately when the temperature is low enough for the
active site to accommodate a critical embryo. This is the case for active
sites of birch pollen washing water and for small droplets coated with
nonadecanol. If the pre-factor is low, the frequency of nucleation attempts
is low and the increase in freezing rate with decreasing temperature is
shallow. This is the case for Hoggar Mountain dust, the large droplets
coated with nonadecanol, and ATD. Various hypotheses why the value of the
pre-factor depends on the nature of the active sites are discussed.
Introduction
Freezing of liquid droplets and subsequent ice crystal growth affects
optical cloud properties and precipitation (IPCC, 2013). Field measurements
show that ice nucleation in relatively warm cumulus and stratiform clouds
may begin at temperatures much higher than those associated with homogeneous
ice nucleation in pure water droplets. The glaciation of these clouds is
ascribed to heterogeneous ice nucleation occurring on the foreign surfaces
of ice-nucleating particles (INPs) present in the cloud droplets. Ice nucleation
induced by particles located within the body of water or aqueous droplets is
termed immersion freezing and is probably the most important nucleation
process turning liquid droplets in relatively warm clouds into ice crystals
(Murray et al., 2012).
Ice-nucleating surfaces are supposed to exhibit features or structures which
promote ice nucleation. However, it is not clear whether these structures
are extended over the whole surface or localized at specific sites. The
concept of epitaxy considers an extended surface with a close lattice match
to ice as responsible for ice nucleation. Ice nucleation is assumed to occur
at a random location on this uniform surface with a nucleation rate that
scales linearly with surface area. However, there is increasing evidence
that preferred locations present on surfaces are responsible for ice
nucleation (e.g., Vali, 2014; Vali et al., 2015). Such sites are thought to
be special surface regions such as crystal defects (Vonnegut, 1947), pores,
cracks, or ledges (Knight, 1979; Sear, 2011; Fletcher, 1969), although direct
evidence of the morphology, structure, and chemistry of active nucleation
sites is lacking up to now. If an ice embryo requires a critical size to
grow into a crystal, the area of the nucleating site needs to be above this
critical size. A point defect in a crystal lattice might be too small (e.g.,
Shevkunov, 2008). If ice nucleation occurred only at a few specific
locations, these have to be highly effective and characterized by high
nucleation rate coefficients. While observations of deposition nucleation on
a crystal may provide evidence for preferred locations for ice nucleation, only
indirect evidence from refreeze experiments exists for immersion freezing
(Vali et al., 2015).
In deterministic models, active sites are supposed to induce ice nucleation
at a characteristic temperature (Vali et al., 2015; Vali and Stansbury,
1966; Vali, 1971). The nucleation rate is equal to zero at temperatures
higher than the characteristic temperature of the site and equal to infinity
beyond that. This implies that no time dependence is involved in nucleation.
In a stochastic description (e.g., Bigg, 1953; Vali and Stansbury, 1966),
time dependence is introduced by assigning to each nucleation site a
characteristic nucleation rate which is a function of temperature. Ice
nucleation as a stochastic process occurring at specific sites can be
described by classical nucleation theory (CNT) assuming that heterogeneous
nucleation takes place in active site areas, which are often taken as the
areas needed by a critical embryo to develop (Marcolli et al., 2007).
When a droplet containing an ice-nucleating particle with an active site is
subjected to freezing cycles, the deterministic assumption predicts freezing
at exactly the same temperature for every cycle, independent of cooling
rate, whereas the stochastic approach predicts freezing temperatures, which
depend on the applied cooling rate. The site can be characterized by a
nucleation rate, which is a function of temperature and expected to increase
with decreasing temperature. When the droplet contains particles with many
sites but all of equal quality, the nucleation rate and the rise of the rate
with decreasing temperature is higher compared with the case of droplets
containing just one nucleation site. In such an idealized case, nucleation
rates derived from multiple droplets are nevertheless characteristic of a
specific nucleation site. However, in polydisperse samples, the surface area
of INPs present in a droplet can vary from droplet to droplet even if the mass is the same because a few small particles have a larger total surface area than a single
large one (Alpert and Knopf, 2016), leading to an additional variability of
freezing rates when the ice nucleation site density scales with surface area
(Hartmann et al., 2016). In investigations with continuous-flow diffusion
chambers (Welti et al., 2012; Lüönd et al., 2010), many particles are
investigated individually and a less steep temperature dependence of
heterogeneous nucleation rates compared with the homogeneous case is
observed. However, there is strong evidence that the surfaces of most
ice-nucleating particles are not uniform with respect to their ability to
nucleate ice (e.g., Marcolli et al., 2007; Vali, 2014). Refreeze experiments
show that variations of the freezing temperatures between runs are much smaller than the range covered by freezing experiments
with many droplets, in accordance with the assumption that specific sites
are responsible for freezing (Vali, 2008, 2014; Wright and Petters, 2013;
Peckhaus et al., 2016). If an ice-nucleating sample consists of particles
containing sites with different ice nucleation efficiencies, a rate derived
from freezing events of many droplets prepared from the sample cannot be considered as characteristic
of a specific nucleation site type; rather, it characterizes the whole sample
containing a variety of sites.
Moreover, the derived nucleation rate is not purely
stochastic, but it has a deterministic component given by the spread of ice
nucleation efficiencies of the different sites. If the ice nucleation
ability of a whole sample is wanted, measurements of many droplets are
convenient to give a result that is representative of the whole sample. For
most natural samples, the sample heterogeneity indeed leads to a large
spread of nucleation efficiencies of sites and the temperature dependence is
likely to exceed the time dependence (Marcolli et al., 2007; Wright et al.,
2013; Wright and Petters, 2013). This was confirmed by a sensitivity study
performed by Ervens and Feingold (2012) and is in agreement with Welti et
al. (2012), who found the time dependence to be of minor importance for
immersion freezing experiments with kaolinite particles. Therefore, a
singular or deterministic approach to describe ambient ice-nucleating
particles in models may be appropriate and justified. On the other hand, to
advance the microphysical understanding of ice nucleation, the presence and
properties of ice-nucleating sites need to be investigated in refreeze
experiments, where the same droplet is subjected to several freezing cycles.
Refreeze experiments with a droplet containing many different nucleation sites
probe only the best one. If a droplet is divided into different parts, only
one part will contain the particle with the best site, in the other portions
less effective sites come into action and will induce freezing at a slightly
lower temperature. Therefore, in more dilute droplets, less efficient sites
are probed.
If the slope of the nucleation rate increase at single sites is compatible with
the one for homogeneous nucleation, a description with CNT is possible by
just adjusting one parameter, namely the contact angle. However, when the
slope predicted by the parameterization of homogeneous nucleation deviates
from the measured one in refreeze experiments a second fit parameter is
needed to describe the nucleation rate as a function of temperature.
Conceptually, it has been suggested to describe heterogeneous ice nucleation
in terms of a static factor, which is specific to the interaction between
the nucleating surface and the ice embryo, and a dynamic factor, which
accounts for the random timing of the formation of a stable (supercritical)
embryo (Vali, 2014). In the present study, we perform refreeze experiments
similar to those of Vali (2008) and Wright and Petters (2013) in order to
characterize and compare the properties of single nucleation sites. We
fitted the freezing rates from the refreeze experiments using three
different CNT-based parameterizations from Pruppacher and Klett (1997),
Zobrist et al. (2007), and Ickes et al. (2015) under the assumption that ice
nucleation occurs at a single site of critical size, namely the most
effective one in the sample.
The following samples have been investigated: Hoggar Mountain dust, Arizona
test dust (ATD), and birch pollen washing water. Hoggar Mountain dust
collected from the Sahara (Pinti et al., 2012) was chosen to represent
natural mineral dusts. It is a mixture of minerals originating from a source
region of dust aerosols with high shares of clay minerals. A number of field
studies have demonstrated the dominant role of mineral dusts to nucleate ice
in mixed phase clouds (Sassen et al., 2003; Ansmann et al., 2008; Pratt et
al., 2009; Choi et al., 2010; Creamean et al., 2013) and possibly also in
cirrus clouds (Cziczo et al., 2013). ATD is a commercial
dust sample that has been used by many groups as a proxy of natural
atmospheric mineral dust (Murray et al., 2012). It is a mixture of minerals
with a considerable share of microcline, a K feldspar with a high ice
nucleation ability as demonstrated in laboratory experiments (Atkinson et
al., 2013; Zolles et al., 2015; Harrison et al., 2016). Pollen is among the
primary biological aerosol particles that nucleate ice (Hader et al., 2014).
Its importance for precipitation on the regional scale has been suggested in
a number of studies (Pöschl et al., 2010; Prenni et al., 2013; Huffman
et al., 2013; Hader et al., 2014). Birch pollen is one of the most
efficient pollen species at nucleating ice as high as 264 K (Diehl et al.,
2001, 2002; von Blohn et al., 2005; Pummer et al., 2012, 2013; Augustin et
al., 2013). Pummer et al. (2012) have shown that macromolecules on or within
pollen grains are responsible for the ice nucleation activity. These
macromolecules can be extracted by suspending the pollen grains in water and
may be dispersed in the atmosphere during wetting and drying cycles (Pummer
et al., 2012; Hader et al., 2014) and also be transported to high altitudes.
Birch pollen washing water containing macromolecules with 100–300 kDa
show similar freezing temperatures to the whole pollen grains (Pummer et
al., 2012). Zobrist et al. (2007) performed refreeze experiments with water
droplets coated by a nonadecanol monolayer, which arranges itself in a 2-D
crystalline lattice on the water surface. The structural match of this 2-D
crystal with the ice lattice has been considered as a key reason for the good
ice nucleation ability of long-chain alcohol monolayers (Popovitz-Biro et
al., 1994; Majewski et al., 1995; Knopf and Forrester, 2011). The refreeze
experiments by Zobrist et al. (2007) are reevaluated here assuming that
instead of the whole monolayer only an active site in it is responsible for
ice nucleation.
The refreeze experiments are analyzed to tackle the following questions. (i) Is there experimental evidence that freezing starts from a nucleation site
rather than occurring at a random location on an extended ice-nucleating
surface? (ii) Is freezing always initiated by the same nucleation site for
each run of a refreeze experiment? (iii) Are nucleation sites stable over
the course of a refreeze experiment? (iv) Is one fit parameter enough to
describe the properties of an active site or are two fit parameters needed?
(v) What is the critical size of an ice-nucleating site? The
results are set in relation to the microphysical properties of the samples.
Classical nucleation theory
CNT formulates the Gibbs free energy to nucleate a solid phase from the
liquid as the sum of a volume term accounting for the energy released when a
molecule is incorporated from the liquid into the solid phase and a surface
term accounting for the energy needed to establish the interface between the
solid and the liquid phases. The critical size of the embryo is reached when
the probability of growth becomes equal to the probability of decay (Vali et
al., 2015). Nucleation is described as an activated process with an
Arrhenius-type equation, which yields nucleation rates as a function of the
activation energy needed to form the critical embryo (e.g., Fletcher, 1958;
Thomson et al., 2015):
ω(T)=Aexp-ΔG(T)kT,
with the pre-exponential factor A and the activation energy
ΔG(T); k is the
Boltzmann constant, and T is the absolute temperature. For first-order
reactions A has units of s-1 and is considered as attempt
frequency of a reaction. When the pre-factor A is low, the number of
nucleation attempts is low and an activated process may not be immediate
even if kT is large enough to overcome the energy barrier.
In the framework of CNT, the freezing due to homogeneous nucleation is
described by a nucleation rate coefficient given as (Pruppacher and Klett,
1997)
jhom(T)=ZkThexp-ΔFdiff(T)kTnvexp-ΔG(T)kT,
where h is the Planck constant, ΔFdiff(T) is the diffusion activation energy,
nv is the number density of water molecules, and Z is
the Zeldovich factor described by Zeldovich (1942), which is usually set to
1. ΔG(T) is the Gibbs free energy
described as
ΔG(T)=16πσ(T)3V(T)23kTlnS(T)2,
where σ(T) is the interfacial energy between ice
and the surrounding medium, V(T) is the volume of a water molecule
in ice, and S(T) is the ice saturation ratio. The critical
embryo radius can be calculated as
rc=2σ(T)V(T)kTln(S(T)).
The critical embryo radius calculated with CNT can be validated by testing
its consistency with the melting and freezing point depression of ice
observed in pores of mesoporous silica (Marcolli, 2014). When pores are too
narrow to incorporate an ice crystal of critical size, ice will not
nucleate. Subcritical ice clusters may be produced at a high rate but are
prevented from reaching a critical size by the confinement in the pores. Marcolli (2014) showed that the CNT-based parameterization by Zobrist et al. (2007)
is able to describe the observed melting point depression in pores as a
function of temperature and should therefore be well suited to estimate
critical embryo sizes. The critical embryo volume for homogeneous ice
nucleation predicted by this parameterization is 109 nm3 at 254 K and
increases to 1441 nm3 at 265 K. We therefore consider the energy
barrier and critical embryo size predicted by CNT as a quantity with a
physical basis.
CNT assumes that heterogeneous freezing occurs on ice-nucleating surfaces
that are able to reduce the interfacial energy between the ice embryo and
the surroundings. If the critical embryo forms on such a surface, the energy
needed to establish the interface is reduced. This leads to a decrease in
the energy barrier to form a critical ice embryo. This reduction is
described by the contact angle α between the ice
phase immersed in liquid water and the heterogeneous surface. The
heterogeneous nucleation rate coefficient describing nucleation in contact
with an ice-nucleating surface is given as
jhet(T)=ZkThexp-ΔFdiff(T)kTnsexp-ΔG(T)fhet(α)kT,
where ns is the number density of water molecules at the ice
embryo–water interface. fhet(α)
describes the change in the Gibbs free energy dependent on the contact angle
α due to the influence of ice-nucleating substrates and is
described as (Seinfeld and Pandis, 2006)
fhet(α)=142+cosα1-cosα2.
The volume of the critical embryo reduces to 50 % when the contact angle
is 90∘ and to 33 % for a contact angle of 45∘.
Several parameterizations of jhom(T) have been
proposed in the literature. Three different parameterizations are considered
in this study to evaluate the measured data, namely the parameterization
provided by Zobrist et al. (2007), hereafter referred to as Z07, the
parameterization given by Pruppacher and Klett (1997), P&K97, and the
parameterization from Ickes et al. (2015), Ick15. Differences between these
parameterizations are discussed by Ickes et al. (2015) and concern mainly
the treatment of ΔFdiff(T), σ(T), and ns. P&K97 fitted
ΔFdiff(T) from laboratory data and
estimated the interfacial energy. They assumed ns=5.85×1018 m-2. Z07 parameterized ΔFdiff(T) with measurements
from Smith and Kay (1999). The interfacial energy was used as a fit parameter to bring CNT
into accordance with homogeneous freezing experiments. They used ns=1019 m-2. Ickes et al. (2015) took ΔFdiff(T) from Zobrist et al. (2007) and the
interfacial energy from Reinhardt and Doye (2013). They also used
ns=1019 m-2.
If heterogeneous ice nucleation is described by a CNT-based formulation with
a reduced energy barrier for critical embryo formation given by
ΔG(T)fhet(α), the increase in the heterogeneous ice nucleation coefficient
jhet(T) with decreasing temperature is tied to the
corresponding homogeneous expression (jhom(T)) with
no possibility for an independent variation of the slope. In this study, we
therefore introduce an additional dimensionless pre-factor β
as a fit parameter to bring the temperature dependence of CNT-based nucleation
rates into agreement with the measured freezing rate increase:
jhet(T)=βZkThexp-ΔFdiff(T)kTnsexp-ΔG(T)fhet(α)kT.
We assume the pre-factor β to be independent of temperature
in the fitted temperature range. A pre-factor β < 1 is needed in the case of a shallow slope of the freezing rate
increase with decreasing temperature and implies a lower number of
nucleation attempts for heterogeneous ice nucleation than predicted from
jhom(T), so that even when the area of the site is
large enough to accommodate an ice embryo of critical size, nucleation is
not immediate. When β > 1, the number of
nucleation attempts is increased compared with the prediction based on
jhom(T) and nucleation is supposed to occur virtually
as soon as the temperature is low enough to accommodate a critical ice
embryo at the site.
We fit the measured data in two ways: in version V1, we use α as the only fit parameter and β is set to unity
(β=1); and in version V2, we use both α
and β as fit parameters.
Fits were performed directly to the nucleation rate ω(T) assuming that nucleation occurs at active sites of critical
size:
ω(T)=Acrit,hetjhet(T)=Acrit,hetβZkThexp-ΔFdiff(T)kTnsexp-ΔG(T)fhet(α)kT.Acrit,het=πrc2sin2α
describes the contact area of an ice embryo with the contact angle
α evaluated at the mean freezing temperature of a refreeze
experiment, and rc is the radius of a critical embryo for
homogeneous freezing given in Eq. (4). Critical site areas needed to
accommodate an ice embryo of critical size, Acrit,het, are
obtained as a result of the CNT fits to the experimentally determined
freezing rates using Eq. (8).
Statistical description of the ice nucleation process
The statistical evaluation of bulk measurements follows the procedure
described in Koop et al. (1997). Here we summarize some of the key aspects
of this probability-based description. Ice nucleation is considered to be a
stochastic process. This can then be described in terms of a binomial
distribution, which provides the probability
Pk(m)=mkpk1-pm-k
to observe k nucleation events with the probability p for
m attempts to build a critical nucleus (or embryo). The variance
v can be calculated by the formula
v=mp(1-p).
Since each water molecule in a bulk sample can become the center of a
critical nucleus, m can be considered as the number of water
molecules in the bulk sample, yielding an m value for our bulk
measurements with droplet volumes of about 2 mm3 of ca. 1019. Due
to this large value, Stirling's approximation
k!≈2αkkke-k if m-k≫1 and p≪1
can be applied, and we obtain the Poisson distribution
Pk(m)≈(mp)kk!e-mp.
The nucleation rate for a single molecule can be written as p/t and
for the whole sample the nucleation rate becomes ω≡mp/t (in s-1). The Poisson distribution, given in
this formulation as
Pk(t)=(ωt)kk!e-ωt,
is a function of time and describes the probability to observe exactly k incidences of nucleation in the time
interval [0,t]. The probability of zero (k=0) incidences of nucleation, i.e., no freezing at
all, is
P0(t)=e-ωt.
Since there is only one incidence of nucleation needed to freeze a bulk
sample, freezing iterations have to be performed to obtain a statistically
relevant result.
We now consider a small temperature interval (a few tenths of a degree
Kelvin), and within this interval, p is assumed to be constant. In
the refreeze experiments, the sample passes this interval ntot
times (with constant cooling rate). When the number of passes with no
nucleation is defined as nliq(t), the probability of no nucleation becomes
P0(t)=e-ωt≈nliq(t)ntot.
If we assume that nnuc samples nucleate after times
tnuc,i (with i=0, 1, ..., nnuc) and nliq samples stay liquid over
times tliq,i (with i=0, 1, ..., nliq), the total time ttot is defined by
ttot=∑i=0nliqtliq,i+∑i=0nnuctnuc,i.
By means of the relation
nnuc=∑k=0∞kPkttot=∑k=1∞kωttotkk!e-ωttot=ωttot∑k=1∞ωttotk-1k-1!e-ωttot=ωttot∑k′=0∞ωttotk′k′!e-ωttot=ωttot∑k′=1∞Pk′ttot=ωttot,
the nucleation rate ω is obtained as
ω=nnucttot.
Here, nnuc is the
total number of freezing events observed within the considered
time ttot. The nucleation rate
coefficient
jhom=ωVsample or jhet=ωSIN
is calculated by dividing the nucleation rate by the sample volume
Vsample for homogeneous ice nucleation or by the
ice-nucleating surface SIN for heterogeneous ice nucleation.
The freezing rate ω can be calculated using Eq. (19), and the
uncertainty of freezing rates was calculated following Poisson statistics on
the 95 % level. The 95 % confidence level x for these
measurements assuming Poisson distribution can be calculated as described by
Koop et al. (1997). The lower confidence limit, ωlow,
is defined such that less than nnuc nucleation events would
occur with a probability x if ωlow were the
true nucleation rate:
x=e-ωlowttot∑k=0nnuc-1ωlowttotkk!.
Correspondingly, for the upper confidence limit, ωup:
x=1-e-ωupttot∑k=0nnucωupttotkk!,
where ωlow (ωup)Rfr,low is the lower (upper) confidence
limit for the nucleation Rfr,up rate, nnuc is
the number of observed freezing events within the considered
time ttot, and k is the
number of nucleation events
within ttot (Koop et al., 1997).
Experimental setup and proceduresTreatment of samples
Coarse particles were removed from the Hoggar Mountain dust sample by
sieving with a 32 µm grid prior to use. No pretreatment was applied to
the ATD sample. The concentration of Hoggar Mountain dust aqueous
suspensions was 0.5 or 5 wt %. The concentration of ATD was 5 wt %.
The birch pollen washing water was provided by Bernhard Pummer and is from
the same birch pollen batch as described in Pummer et al. (2012). The
concentration of the birch pollen suspension was 50 mg mL-1. Filtration of
this suspension was reported to give a 2.4 wt % birch pollen washing water
aqueous solution (Pummer et al., 2012). The birch pollen washing water was
further diluted with water (molecular biology reagent water from
Sigma-Aldrich) to obtain mass concentrations with respect to birch pollen
grains between 0.001 and 50 mg mL-1.
Differential scanning calorimetry
Experiments were conducted with a differential scanning calorimeter (DSC)
Q10 from TA Instruments. Refreeze experiments were carried out by placing
about 1.8–2 mg (volumes of 1.8–2.0 mm3) of the respective
suspension into an aluminum pan, covering the drop with mineral oil to avoid
evaporation and condensation and closing the pan hermetically. Water
molecular biology reagent from Sigma-Aldrich was used to prepare the
suspensions since it proved to have a lower average freezing temperature
compared with our Milli-Q water. The same sample was subjected to repeated
freezing runs with 10 and 1 K min-1 cooling rates. For Hoggar Mountain
dust measurements at constant temperature were also performed by cooling the
sample to the target temperature and keeping it there for 1 h. For
every refreeze experiment we took a fresh sample from our stock solution.
For the emulsion measurements 5 wt % lanolin was mixed with 95 wt %
mineral oil. Then, 80 vol % of this mixture and 20 vol % of aqueous suspension
were vigorously stirred to obtain an emulsion as described by Pinti et al. (2012). This suspension was subjected to repeated freezing cycles. The first
and third freezing cycles were conducted with a cooling rate of 10 K min-1 to
check the stability of the sample. The second cycle was performed with 1 K min-1 and was used for evaluation.
Refreeze experiments were carried out with bulk samples (single drops covered with mineral oil) which exhibit an
abrupt heat release when they freeze leading in the DSC thermograms to a
clear onset of the freezing peak which was taken as the nucleation
temperature. The evaluation was done using the implemented software TA
Universal Analysis of the instrument. By means of a thermocouple, the DSC is able to detect and
control tiny temperature differences between an
empty reference pan and the sample pan, which contains the sample of
interest. Due to the latent heat release during a freezing event, the
resulting temperature difference leads to a heat flux and to a signal in the
counteracting electric current applied to the thermocouple. The precision of
the DSC temperature measurement is nominally 0.01 K. Depending on the
cooling rate, heat transfer limitations result in a temperature gradient
within the droplet and the temperature measured at the bottom of the DSC pan
does not correspond exactly with the temperature inside the droplet. Thus,
for such measurements additional uncertainties have to be considered. To
estimate these uncertainties it was assumed that the cooling or heating of
the droplet is fully controlled by the contact to the bottom of the pan and
that the surrounding air has a negligible influence. The temperature
gradient inside the droplet can then be estimated by the thermal
conductivity of water. For bulk measurements with 1.8 mm3 droplets and
the assumption that the droplet is a semisphere the temperature gradient is
about 0.7 K for a 10 K min-1 cooling rate and 0.07 K for a 1 K min-1 cooling
rate.
EvaluationNucleation rates
To calculate freezing rates from the refreeze experiments, the measured
freezing temperatures were divided into bins of equal interval width. The
interval width was optimized for each dataset subject to the freezing range,
the number of freezing events, and the resolution of the DSC, which depends
on the cooling rate. The Hoggar Mountain dust measurements utilize 0.2 K
bins for a 1 K min-1 cooling rate and 0.3 K bins for a 10 K min-1 cooling rate, such
that data points were distributed between at least five temperature bins.
For birch pollen washing water, freezing temperatures for a sample were
spread over a smaller range than for Hoggar Mountain dust, resulting in bin widths between 0.08 and 0.2 K. For the evaluation of the water
droplets covered with a nonadecanol monolayer, the same bin sizes as in
Zobrist et al. (2007) were used. Bins were between 0.5 and 1.5 K in width.
At least four temperature bins were populated by freezing events, allowing us to
estimate the temperature dependence of the nucleation rate coefficient. We
assume that these observed freezing rates are equivalent to nucleation
rates. To calculate nucleation rates ω (s-1), the
procedure described by Koop et al. (1997) was applied as summarized in Sect. 3.
Nucleation rate coefficientsHoggar Mountain dust and ATD
To calculate nucleation rate coefficients jhet
(cm-2 s-1) from the nucleation rates, two opposing assumptions were
applied for Hoggar Mountain dust and ATD, yielding two possible extremes:
In the conventional manner, a lower limit for jhet was
obtained by assuming that the whole sample surface is active at nucleating
ice. For Hoggar Mountain dust and ATD, the available surface area per sample
was calculated based on the mass present in the sample and the BET
(Brunauer–Emmett–Teller) surface area, namely 46.3 m2 g-1 for Hoggar
Mountain dust (Pinti et al., 2012) and 85 m2 g-1 for the ATD sample
(Bedjanian et al., 2013).
An upper limit for jhet was obtained by assuming only one
single site per sample and assuming a critical site size
Acrit,het calculated at the mean freezing temperature of the
experiment with Eq. (8).
Birch pollen washing water
We obtained birch pollen washing water from Pummer et al. (2012), which was
prepared by filtration of suspensions of birch pollen grains. The washing
water contains macromolecules with an upper limit of 300 kDa mass
corresponding to diameters of 10 nm (assuming a spherical shape and a density
of 1.5 g cm-3). For the birch pollen washing water, three very different
assumptions were used to calculate the active surface of the macromolecules:
The lower limit of the nucleation rate coefficient was obtained by assuming
that the surfaces of all macromolecules present in the birch pollen washing
water contribute to freezing. Based on the information given by Pummer et
al. (2012), we calculated the number of macromolecules present in the
suspensions assuming that a 50 mg mL-1 birch pollen suspension yields a 2.4 wt % birch pollen washing water solution consisting of macromolecules of
300 kDa. Assuming a spherical shape, the surface of a single macromolecule was
calculated and multiplied by the number of macromolecules present in the
solution. This yields a value of 1014 macromolecules present in a 50 mg mL-1 pollen bulk droplet. From this, the total surface Atot
of all macromolecules was estimated to be about 300 cm2, i.e., some 14 orders of magnitude larger than Acrit,het.
The upper limit of the nucleation rate coefficient was obtained by assuming
one active site per sample. The area of the active site was taken as
Acrit,het.
The presence of a homogeneous freezing peak in emulsion measurements of
birch pollen washing water reveals that not all macromolecules are active at
nucleating ice (Fig. 1). Accounting only for the active macromolecules and assuming that all
active ones induce freezing at the same temperature leads to an
intermediate value for the nucleation rate coefficient. Knowing the droplet
size distribution of the emulsions and the size of particles in the pollen
washing water, the theoretical homogeneous and heterogeneous freezing peak
area can be estimated. The probability Pj for a macromolecule
to be in a droplet j with a volume Vj is
Pj=Vj/Vtot, where
Vtot is the total volume of all droplets in the emulsion.
Assuming n macromolecules in the emulsion, which are all
distributed among the water droplets, the probability of no macromolecule
in a droplet j with a volume Vj is
(1-Vj/Vtot)n. The contribution of droplet
j to the total heterogeneous and homogeneous peak area
Atot is proportional to Vj/Vtot.
The percentage of homogeneous freezing, phom, can then be
written as
phom=∑j=1kVjVtot×1-VjVtotn,
where k is the number of droplets. The percentage of heterogeneous
freezing phet is given by
phet=1-phom.
DSC thermograms of emulsion experiments for 1 K min-1 cooling
rate. (a) Hoggar Mountain dust emulsion sample with homogeneous
freezing peak at ∼ 234 K and heterogeneous freezing peaking at ∼ 244 K. The spikes around 253 K are due to latent heat release by a few
very large water droplets containing Hoggar Mountain dust. (b) ATD
emulsion sample. The broad peak at ∼ 248 K stems from heterogeneous
freezing induced by ATD particles present in many droplets, the narrow peak
at ∼ 235 K from homogeneous freezing of water droplets.
(c) Birch pollen washing water emulsion sample with the
heterogeneous freezing peak at ∼ 250 K.
Comparing the theoretical and measured values gives an estimate for the
fraction of birch pollen macromolecules that are active. This fraction is
about 10-7. Estimates based on bulk measurements with different
dilutions (see Fig. 2) lead to a similar result. For these bulk measurements
the washing water was diluted to 5×10-6 mg mL-1 until freezing
occurred at temperatures at which also pure water may start to freeze
(indicated by the horizontal line in Fig. 2). Assuming that at these
concentrations, hardly any ice-nucleating macromolecules are left in a bulk
sample, an ice nucleation active fraction of about 10-7 macromolecules was
obtained again. Therefore, the number of active macromolecules can be
estimated by dividing the total number of macromolecules in a bulk sample
(1014) by 107, yielding a value of 107 ice nucleation active
macromolecules present in a 50 mg mL-1 pollen suspension.
Dependence of freezing temperatures of bulk birch pollen washing
water drops (1.8–2 mg) on concentration. Symbols: mean freezing
temperature of five freezing runs performed at a cooling rate of
10 K min-1. Uncertainty ranges are given on the 68 % confidence
level. The concentration scale refers to the concentration of the birch
pollen in suspension. Horizontal solid line: uppermost limit below which pure
water drops may start to freeze.
Nonadecanol-coated droplets
For nonadecanol-coated droplets, the assumptions that the whole nonadecanol
monolayer was ice nucleation active and that there was only one active site
in the monolayer were used to convert from nucleation rate to nucleation
rate coefficient.
Spearman's rank correlation coefficient
The Spearman's rank correlation coefficient assesses how well the
relationship between two variables can be described using a monotonic
function. To evaluate whether a trend is present in the refreeze experiments
during repeated freezing cycles, the Spearman's rank correlation coefficient
was calculated using time and freezing temperatures as variables. The values
are presented in Tables 1 and 2. Numbers around zero indicate hardly any
monotonic trend. Numbers close to 1 or -1 indicate a strong monotonic trend.
Spearman's rank correlation coefficients for Hoggar Mountain dust
samples (H1–H12) and an ATD sample (A5). Experiments with cooling rates 1
and 10 K min-1 are flagged by (1) and (10), respectively. The
uncertainties represent a 68 % confidence interval.
Refreeze experiments with the Hoggar Mountain dust samples H1–H12.
Freezing onsets as a function of freezing run no. Squares: freezing runs
with 10 K min-1 cooling rate. Circles: runs with 1 K min-1.
Filled symbols: 5 wt % samples. Open symbols: 0.5 wt % samples.
Gray lines: bin intervals for runs with 1 K min-1. Black lines: bin
intervals for 10 K min-1. Bin intervals are shown only for evaluated
samples H1–H9 (see text). Error bars given for the first data points are
representative of all following data points acquired with the same cooling
rate. They represent the instrumental temperature uncertainty as explained in
Sect. 4.2. For 1 K min-1 runs the error bars are smaller than the
symbol.
ResultsHoggar Mountain dust
Figure 3 shows refreeze experiments performed with Hoggar Mountain dust
samples (H1–H12). With each sample between 21 and 97 freezing cycles were
performed. The sequences of freezing temperatures for several samples reveal
clear signs of non-stochastic behavior, such as trends or jumps. Therefore,
following Zobrist et al. (2007), samples were tested by means of a linear
fit for the presence of a trend. When the 95 % confidence level of the
slope included zero, the samples were considered to be constant in their
freezing behavior over the conducted freezing cycles, a necessary (but not
sufficient) condition for a stochastic behavior. Samples H1–H9 performed
with 1 K min-1 cooling rate satisfy the condition of the absence of an overall
trend (see Fig. 3), although not all freezing series appear to be purely
stochastic. Furthermore, for runs of samples H6 and H9 performed at 10 K min-1
only a 99.7 % confidence level of the slope included zero.
Samples H10–H12 are examples of refreeze experiments which show
non-stochastic features. This may be a decrease in freezing temperatures by
almost 3 K over about eight freezing runs (H10) or an abrupt jump to lower
freezing temperature by almost 2 K from one freezing run to the next (H11),
or a slow increase in freezing temperature by 4.5 K over 35 freezing cycles
(H12). Such features are clearly non-stochastic and must have been due to
modifications (deteriorations or improvements) of the site, which might be
due to coagulation, settling, or breakup of aggregates (Emersic et al.,
2015).
We calculated Spearman's rank correlation coefficients to check the data
concerning a monotonic trend. Table 1 displays the results for Hoggar
Mountain dust. The indicated uncertainties represent a 68 % confidence
interval. Spearman's rank correlation coefficients for Hoggar Mountain dust
samples are within ±0.1 for H3, H4, and H7, indicating hardly any
monotonic trend. For H1, H2, H5, H6, H8, H9, and H12 performed at 10 K min-1,
correlation coefficients are within ±0.5, indicating at most a weak
monotonic trend. Samples H10 and H11 with correlation coefficients close to
1 show a very strong monotonic trend. Even in the absence of a monotonic
trend, refreeze data series do not need to be stochastic. Sample H5 has a
weak monotonic trend with a Spearman coefficient of 0.21 ± 0.09, but
the four runs with the highest freezing temperatures all occurred in a row. The
probability of this happening is low (2.04×10-4).
Nevertheless, we cannot exclude that this improbable sequence occurred by
chance. We therefore use the samples H1–H9 for further evaluation.
Panel (a) of Fig. 4 shows the freezing rates for refreeze experiments H1–H9. While most of the samples freeze at temperatures between 258 and 263 K,
samples H4 and H6 freeze at significantly higher temperatures of 263–265 K. There are also differences in the slopes of nucleation rates. The
steepest increase is observed for the samples H4 and H6, which freeze at the
highest temperatures. Freezing rates for H8 and H9 align well for a wide
range of cooling rates (10 K min-1, 1 K min-1, and at constant temperature).
Hoggar Mountain dust samples prepared from 5 wt % (H3–H6, H8,
H9; filled symbols) and 0.5 wt % suspensions (H1, H2, H7, open symbols).
(a) Freezing rates. (b) Nucleation rate coefficients
evaluated with respect to the total dust surface present in a sample as
determined by BET measurements. (c) Nucleation rate coefficients
with respect to the surface of one single site. Squares: 10 K min-1
cooling rate. Circles: 1 K min-1 cooling rate. Triangles: constant
temperatures.
CNT-based fits for Hoggar Mountain dust samples H1–H9 and ATD
sample A5 using the parameterizations Ick15, P&K97, and Z07 for
version V1: fitting only contact angle α (∘) and setting the
pre-factor β=1. Acrit,het is the calculated critical
active site area in square nanometers. All uncertainties correspond to 68 %
confidence intervals.
CNT-based fits for Hoggar Mountain dust samples H1–H9 and ATD
sample A5 using the parameterizations Ick15, P&K97, and Z07 for
version V2: simultaneous fit of contact angle α (∘) and
pre-factor β (dimensionless). Acrit,het is the calculated
critical active site area in square nanometers.
Next, we calculate nucleation rate coefficients (per square centimeters per second) from
the measured freezing rates (per second). Values on the order of 10-6–1 cm-2 s-1 (in the temperature range between 258 and 265 K)
are obtained when assuming the total surface to be ice-nucleating (panel b) and on the order of 109–1013 cm-2 s-1 when only one
active site per sample is assumed to be responsible for ice nucleation
(panel c).
Figure 5 presents fitting results for the refreeze experiment H9 using the
three CNT-based parameterizations Ick15, P&K97, and Z07. Analogous figures
for the other samples are displayed in the Appendix Fig. A1. Fit parameters
for all evaluated refreeze experiments H1–H9 are listed in Table 3, using
only the contact angle as a fit parameter and setting the pre-factor β to
1 (Version V1), and in Table 4 with pre-factor β and the contact angle
simultaneously fitted (version V2). The tables also list the values of the
critical active area Acrit,het, which is not a fit parameter but
a result of the calculation. For all parameterizations, the Hoggar Mountain
dust samples show slopes much shallower than predicted by CNT when only the
contact angle is used as a fit parameter (V1). Very good fits can be obtained when the pre-factor β is used as a second fit parameter (V2). For all
Hoggar Mountain dust samples, the fitted β values are < 1.
For a given parameterization, contact angle and pre-factor values show
significant differences between samples. The samples H1 and H5 have the same contact angles (α(H1) = 36.5 ± 0.7∘ and α(H5) = 36.0 ± 1.7∘) and pre-factors (β(H1) = (3.77–11.96) ×10-6 and β(H5) = (1.36–13.42) ×10-6 for the Ick15 parameterization) within statistical variability. The same is the case for
samples H4 and H6 and samples H8 and H9. All other samples can be
discriminated from one another in terms of contact angles and pre-factors. We
therefore conclude that Hoggar Mountain dust samples taken from the same
suspension show distinctly different behaviors in terms of freezing
temperatures and freezing rate increase with decreasing temperature. This
supports the assumption that ice nucleation occurs within these samples at
nucleation sites that differ distinctly from each other.
CNT-based fits of freezing rates for the Hoggar Mountain dust sample
H9 with the parameterizations by Ick15, P&K97, and Z07. Version V1: CNT
fits performed with contact angle α as the only fit parameter.
Version V2: modified CNT fits performed with α and β as fit
parameters. Note that Z07 (V2) mostly overlaps Ick15 (V2).
ATD
Figure 6 presents the refreeze experiments performed with ATD for 5 wt %
suspensions with freezing temperatures in the range of T=264–268 K. The freezing temperature of the first run was always distinctly lower
than that of the subsequent ones. This memory effect ranged from 1 to 4 K.
Refreeze experiments with the ATD
samples A1–A5. Freezing onsets as a function of freezing run no. Squares:
freezing runs with 10 K min-1 cooling rate. Circles: with
1 K min-1 cooling rate. All samples had a suspension concentration of
5 wt % and show a memory effect. Error bars given for the first data
points are representative of all following data points acquired with the
same cooling rate. They represent the instrumental temperature uncertainty as
explained in Sect. 4.2. For 1 K min-1 runs the error bars are smaller
than the symbols.
The sample A5 was evaluated for freezing rates. To evaluate A5 with respect
to its stochastic behavior, the first freezing point was omitted. Spearman's
rank correlation coefficients for the A5 sample is close to zero. Therefore,
this sample can be assumed to be without a monotonic trend, after removing
the initial memory effect by omitting the first point.
Figure 7 shows the CNT-based fits to the freezing rates for sample A5
assuming that freezing occurred at a single nucleation site. The increase in
freezing rate with decreasing temperature is much shallower than can be
fitted with the CNT-based parameterizations if the contact angle is the
only fit parameter (V1). If contact angle α and pre-factor
β are used simultaneously as fit parameters (V2), very good
fits are obtained for all CNT-based parameterizations. The last lines in
Tables 3 and 4 show the fit parameters and the critical heterogeneous
surface Acrit,het for the different CNT-based
parameterizations. Similar to Hoggar dust, the pre-factor β
needs to be very small to reach agreement with the shallow increase in
freezing rate with decreasing temperature.
ATD sample A5 (suspension with 5 wt % dust). CNT-based fits of
freezing rates for the parameterizations Ick15, P&K97, and Z07. V1: fits
with contact angle α as the only fit parameter. V2: fits with α and β as fit parameters. Fitting curves belonging to the same
version are partly overlapping.
Refreeze experiments with birch pollen washing water samples P1–P9.
Freezing onsets as a function of freezing run no. Filled symbols:
50 mg mL-1 samples. Half-filled symbols: 0.1 mg mL-1 samples.
Open symbols: 0.001 mg mL-1 samples. Squares: freezing runs with
10 K min-1 cooling rate. Circles: runs with 1 K min-1. Thin
gray lines: bin intervals for runs with 1 K min-1. Thick black lines:
bin intervals for 10 K min-1. Error bars given for the first data
points of 10 K min-1 runs are representative of all following data
points acquired with the same cooling rate. They represent the instrumental
temperature uncertainty as explained in Sect. 4.2. For 1 K min-1 runs
the error bars are smaller than the symbols.
Birch pollen washing water
Figure 8 shows refreeze experiments with birch pollen washing water drops performed with a cooling rate of 1 and 10 K min-1. In all experiments,
freezing occurred in the temperature range from 254 to 261 K, but individual
samples froze over a much narrower temperature range of typically < 1 K. The samples were again tested with a linear fit for trends. For samples
P1 and P8 the first few runs showed lower freezing temperatures than all
subsequent ones, which might be due to a memory effect. These first runs
were therefore excluded from the test. Samples P1–P7 satisfy the 95 %
confidence level condition but not so P8 and P9. For samples P6 and P7
refreeze experiments were also performed at a cooling rate of
10 K min-1, giving distinctly lower freezing temperatures. There is no overlap in
freezing temperatures for these two cooling rates.
Birch pollen washing water samples P1–P9. Freezing
rate (a) and freezing rate coefficients (b–d).
(b) Lower limit of nucleation rate coefficients jhet
considering the whole surface of all macromolecules as ice nucleation active.
(c) Nucleation rate coefficient jhet considering a
fraction of 10-7 of the macromolecules to be ice nucleation active.
(d) Upper limit of nucleation rate coefficients jhet
with respect to the surface of one single active site. Filled symbols:
50 mg mL-1 samples. Half-filled symbols: 0.1 mg mL-1 samples.
Open symbols: 0.001 mg mL-1 samples. Squares: 10 K min-1
cooling rate. Circles: 1 K min-1 cooling rate.
Spearman's rank correlation coefficients were calculated to check the data
concerning a monotonic trend. The results for the birch pollen washing water
are shown in Table 2. The uncertainties given in the table represent a
68 % confidence interval. Spearman's rank correlation coefficients for
birch pollen washing water samples P1–P5, P6 (10 K min-1), and P7 (10 K min-1)
are ±0.2, indicative of a very weak monotonic trend. Correlation
coefficients for P6 (1 K min-1), P7 (1 K min-1), P8, and P9 are significant
different from zero.
Panel (a) of Fig. 9 shows freezing rates for the samples P1–P9. Sample P3
(50 mg mL-1 pollen) freezes at distinctly higher temperature than all other
samples. There is no significant difference in freezing temperatures for 50 and 0.1 mg mL-1 samples. However, the more
dilute P6 sample (0.001 mg mL-1) freezes at an almost 2 K lower temperature than all other ones. Figure 2
shows the dependence of freezing temperatures on suspension concentration.
It can be seen that the average freezing temperature of birch pollen washing
water first decreases gradually from 257 to 253 K for a dilution from 50 to 5×10-6 mg mL-1 and upon further dilution abruptly
drops into the range where pure water bulk samples may also freeze as
indicated by the black line at T=252.5 K in Fig. 2.
Freezing rates evaluated for birch pollen washing water samples
P6 (a) and P7 (b). Dependence of freezing rates on the
choice of bin size for samples exposed to 10 K min-1 cooling rate
(squares) and 1 K min-1 cooling rate (circles). Bin widths were varied
between 0.15 and 1 K (color-coded). Horizontal error bars: temperature
uncertainty within the droplet due to the precision of DSC temperature
measurement. Vertical error bars: uncertainty due to the Poisson
distribution.
The slope of freezing rate with temperature is similar for all refreeze
experiments irrespective of solution concentration or cooling rate, with the
exception of P8, which shows a distinctly stronger freezing rate increase
with decreasing temperature. However, experiments performed with cooling
rates of 10 and 1 K min-1 do not fall on one line but occur with
similar freezing rates at a temperature just ∼ 1 K lower for 10 K min-1
compared with 1 K min-1. This behavior of the birch pollen samples is in clear
contrast to the behavior of Hoggar Mountain dust samples, which showed a
good alignment of freezing rates acquired with different cooling rates (see
Fig. 4). To check whether the misalignment of the 10 and 1 K min-1
freezing rates of the birch pollen samples is influenced by the very narrow
bin intervals (0.15 K), we varied the bin widths for the 10 K min-1
experiments. The results in Fig. 10 show that freezing rates are independent
of the choice of bin widths (ΔT=0.15–1 K). An
alternative explanation might be an induction time required for the ice
embryo to grow large enough to be detected in the DSC instrument or due to
heat transfer limitations in the pan as discussed in Sect. 4.2.
CNT-based fits for the birch pollen washing water samples P1–P7
using the parameterizations Ick15, P&K97, and Z07 for version V1: fitting
only contact angle α (∘) and setting the pre-factor β=1. Acrit,het is the critical active site area in square nanometers. For
samples P6 and P7, refreeze runs carried out with 10 K min-1 (P6(10),
P7(10)) and 1 K min-1 (P6(1), P7(1)) are evaluated separately.
Similar to the derivation of nucleation rate coefficients for the Hoggar
Mountain dust samples, we also applied different assumptions to the pollen washing water to convert freezing rates to freezing rate
coefficients as described in detail in Sect. 5.2.2, yielding very different
values for jhet, as is shown in panels (b–d) of Fig. 9.
Panel (b) shows freezing rate coefficients in the range 10-4–103 cm-2 s-1 (for temperature between 254 and 261 K), when
assuming that the whole birch pollen washing water consists of
macromolecules and that the whole surface of all macromolecules is
ice-nucleating. With this assumption, the sample P6, when cooled with 1 K min-1, has higher nucleation rate coefficients than the other samples because it has the lowest concentration and thus the lowest active area.
Conversely, assuming only one active site per sample (Fig. 9d), nucleation
rate coefficients on the order of 109–1013 cm-2 s-1
are obtained for the temperature range between 254 and 261 K. In Fig. 9c we
assume that a small fraction of the birch pollen washing water contains
active macromolecules. In Sect. 5.2.2 we estimated this fraction to be
10-7. With this assumption, the resulting nucleation rate coefficients
are in the range 101–1010 cm-2s-1 for temperatures
between 254 and 261 K.
Birch pollen washing water sample P7 (50 mg mL-1). CNT-based
fits of freezing rates with the parameterizations Ick15, P&K97, and Z07.
Left: fits for a cooling rate of 10 K min-1. Right: fits for a cooling
rate of 1 K min-1. Version V1: fits performed with the contact angle
α as the only fit parameter. Version V2: fits performed with α and β as fit parameters. Fitting curves belonging to the same
version are partly overlapping. Values of fit parameters are given in
Tables 5 and 6.
Figure 11 presents curves fitted to the refreeze experiment P7 for cooling
rates of 10 and 1 K min-1 for the three CNT-based parameterizations
Ick15, P&K97, and Z07. For both cooling rates, P7 shows a slightly steeper
slope than could be fitted when only the contact angle was used as a fit
parameter (V1). Analogous figures for the other samples are given in the
Appendix Fig. A2. Fit parameters for all evaluated refreeze experiments P1–P9 are summarized in Tables 5 and 6. When only the contact angle is used
as a fit parameter, the fitted contact angles for most experiments are
significantly different from each other (Table 5), but the steep increase in
freezing rates with decreasing temperature could not be realized for all
samples (see Fig. A2). When the contact angle and the pre-factor β are
used as fit parameters (V2), good agreement is obtained. For most birch
pollen washing water samples the fitted β values are > 1, implying a steeper increase in freezing rate with decreasing temperatures
than predicted by the CNT-based parameterizations. However, the
β values are not well constrained by the fit as can
be seen from the large uncertainties associated with them (Table 6). Worth
mentioning are sample P4 with the lowest pre-factor β=0.006–0.0855 and sample P8 with a huge pre-factor of (6.2–513) ×1041 (Ick15 parameterization). Fits of version V2 to samples P1, P2, and P5 yield contact angles
that are identical within the observed variability, while the other samples
can be differentiated from one another based on their α and
β values. This suggests that for some of the birch
pollen washing water samples, ice nucleation always occurs at the same site,
i.e., on the same macromolecule. However, for the samples P4, P5, P7, and P9
with a concentration of 50 mg mL-1, it is likely that nucleation alternated
between macromolecules from run to run. The freezing rates of the samples
measured with cooling rates of 10 and 1 K min-1 (P6 and P7) do not
coincide, but those measured with 1 K min-1 freeze at a ∼ 1 K higher
temperature. Nevertheless, fitting the freezing rates with CNT gives the same contact angles within
the observed variability for the two cooling rates
(see Table 6), in agreement with ice nucleation occurring at the same site
for both cooling rates.
CNT-based fits for birch pollen washing water samples P1–P7 using
the parameterizations Ick15, P&K97, and Z07 for version V2: simultaneous
fit of contact angle α (∘) and pre-factor β
(dimensionless). Acrit,het is the critical active site area in square nanometers.
Nonadecanol samples N2 (large droplet, a) and N6 (small
droplet, b). CNT-based fits of freezing rates measured by Zobrist et
al. (2007) with the parameterizations Ick15, P&K97, and Z07. V1: fits with
contact angle α as the only fit parameter. V2: fits with α
and β as fit parameters. V1 parameterizations overlap completely; V2 parameterizations overlap partly.
CNT-based fits for nonadecanol droplets N1–N6 with radii r=31–1100 µm measured by Zobrist et al. (2007) using the CNT-based
parameterizations Ick15, P&K97, and Z07 for version V1: fitting only
contact angle α (∘) and setting the pre-factor β=1.
Acrit,het is the calculated active site area in square nanometers.
CNT fits for nonadecanol samples N1–N6 with radii r=31–1100 µm measured by Zobrist et al. (2007) using the
parameterizations Ick15, P&K97, and Z07 for version V2: simultaneous fit
of contact angle α (∘) and pre-factor β
(dimensionless). Acrit,het is the calculated active site area in square nanometers.
Zobrist et al. (2007) performed refreeze experiments with water droplets
coated by a nonadecanol monolayer for droplets with radii between 31 and 1100 µm. They calculated nucleation rate coefficients from the
freezing rates assuming that the whole surface of the nonadecanol monolayer
is nucleating ice and tried to describe the nucleation rate coefficients as
a function of temperature with CNT using the contact angle as a fit parameter.
They could reconcile their measurements with CNT only by assuming a
temperature-dependent contact angle. We reevaluate their freezing rate data
for the nonadecanol samples N1–N6, assuming that single sites were the
location of freezing instead of the whole surface. Each sample is therefore
fitted separately with Eq. (8). Figure 12 shows the refreeze experiments for
droplet N2 with a radius r=1100µm in panel (a) and for
droplet N6 with r=31µm in panel (b). Analogous figures
for experiments N1 (r=1100µm), N3 (r=370µm), N4 (r=320µm), and N5 (r=48µm)
are shown in Fig. A3 in the Appendix. Fit parameters for nonadecanol
droplets N1–N6 are given in Tables 7 (V1) and 8 (V2). The freezing
temperature of nonadecanol-coated water droplets decreases significantly
with decreasing surface area of the droplets. The droplets (N1 and N2) with
r=1100µm freeze at Tfr=260–265 K,
the droplets with r=370 or 320 µm between
Tfr=256–262 K, and the ones with r=48 or 31 µm at Tfr=248–252 K. Fits with β=1 (V1) show much too steep slopes compared with the measurements for the
samples N1–N4. The samples N5 and N6 show a steeper slope, reasonably represented by V1. When the pre-factor β is fitted
as well, the fits of the droplets N1–N4 improve; however, the freezing
rates at the highest temperatures are still not reproduced well. Only a few
runs populate the bins at higher temperatures, and their freezing rates are
associated with large uncertainty ranges. Therefore, they were given less
weight for the fits shown in Figs. 12 and A3. However, when the fitted
curves were forced to pass through the lowest and highest data points by
increasing their weighting (not shown), the fit quality decreased for the
points measured in between since the resulting curves were too bowed. An
improved fit could also not be obtained when the whole surface was
considered to be ice-nucleating. Table 8 for V2 shows that the contact angle
and the pre-factor β increase with decreasing droplet size.
For the smallest droplets (N5, N6), the pre-factor β is on the order of unity (0.001–1000) and the contact angle is above 50∘ for all parameterizations. For the largest droplets (N1, N2) the pre-factor
is around 10-7–10-8 and the contact angle is below
32∘ for all parameterizations. Droplets of similar sizes (N1/N2;
N3/N4; N5/N6) have similar contact angles and pre-factors.
Emulsion measurements
In Fig. 1 typical thermograms of emulsion measurements with Hoggar Mountain
dust (panel a), ATD (panel b), and birch pollen washing water (panel c) are
shown. For ATD, Marcolli et al. (2007) showed that the observed range of
heterogeneous freezing temperatures cannot be described by assuming the same
contact angle for all ATD particles. Rather, the ice-nucleating sites of ATD
particles are required to be of different qualities. Note, that the refreeze
experiments were performed with single droplets weighing 1.8–2 mg which
contain a high number of particles. The best nucleation sites probed in the
refreeze experiments with bulk samples are active from 260 to 268 K, i.e., at
distinctly higher temperatures than the average sites probed in the emulsion
experiments, which nucleate ice below 252 K. In contrast to the bulk
measurements, no memory effect was observed for ATD emulsions. Hoggar
Mountain dust is a mixture of various minerals which nucleate ice at
quite different temperatures (Pinti et al., 2012; Kaufmann et al., 2016), giving rise to the broad freezing signal starting below 257 K with the
freezing of single large emulsion droplets as shown in panel (a). Again, there is no overlap in freezing temperatures between emulsion measurements
and the refreeze experiments performed with large single droplets which
froze from 258 to 265 K. With an onset of 255 K, the heterogeneous freezing
peak of the emulsion made from the birch pollen washing water exhibits a
clear overlap with the freezing temperatures observed for bulk measurements, which indicates that the ice nucleation active macromolecules present in the
birch pollen washing water contain nucleation sites of quite uniform quality.
DiscussionNucleation on active sites
In the following, we investigate the refreeze experiments for evidence
against or in favor of ice nucleation at active sites. Sudden jumps of
freezing temperature during refreeze experiments are evidence that specific
singular features in the samples are the nucleating entity, which might be
fragile and can vanish or emerge during the course of a refreeze experiment.
Hoggar Mountain dust
Refreeze experiments with Hoggar Mountain dust showing sequences with trends
or even jumps are strong evidence that freezing occurs at particular sites
in these samples. For the H11 sample shown in Fig. 3, the freezing
temperature first shows a decrease when after about 40 runs it suddenly
drops and remains quite constant for the rest of the experiment at a value
∼ 3 K lower than before. Such a drop points to freezing
occurring at a single site, which suddenly becomes inactive possibly due to
blocking by an impurity, and from then on freezing occurred at the next best
site. Furthermore, freezing temperatures before the drop give the impression
that freezing at this site was not fully stochastic. The samples H10 and H12
show less abrupt transitions, which might be related to a site that remained
dominant but underwent modifications during the course of the experiment.
The samples H1, H2, H5, H6, H8, and H9 show a weak monotonic trend, which
could be due to slight modifications or some kind of aging of the
ice-nucleating site during the course of the experiment. Nevertheless,
nucleation sequences with such trends fulfill the criteria for evaluation
with CNT as long as nucleation supposedly occurred always at the same site,
even if this site is not completely stable in time.
Hoggar Mountain dust consists of a mixture of minerals with high shares of
the clay minerals smectite and montmorillonite, illite, and kaolinite and minor
contributions of quartz and the feldspars sanidine and plagioclase (Kaufmann
et al., 2016). However, nucleation at the best sites present in bulk samples
(Pinti et al., 2012) does not need to be closely related to the prevailing
minerals in the sample. It is therefore not clear whether a specific mineral
component or rather a non-mineralogical component present in the collected
dust is responsible for ice nucleation. This further supports the
interpretation that freezing occurs at distinct sites that are different for
different samples. The evaluation with CNT of the refreeze experiments with
Hoggar Mountain dust shows that individual samples taken from the same stock
solution can be discriminated based on their contact angles and pre-factors.
This together with the heterogeneity of the sample and the jumps and trends
observed for the time sequences of some samples supports the notion that
ice nucleation occurs at specific sites on the sample surface. However, it
is not clear whether these active sites originate from a specific mineral
component or even biogenic components in the dust sample (Conen et al.,
2011; Tobo et al., 2014; O'Sullivan et al., 2014). Moreover, the activity of
sites could be influenced by coagulation or the breakup of aggregates
(Emersic et al., 2015).
ATD
Only a few refreeze experiments were performed with ATD. For this limited
dataset, we did not observe non-stochastic behavior such as trends or
unexpected jumps, but all samples showed a pronounced memory effect. Wright
and Petters (2013) performed refreeze experiments with ATD and observed
jumps similar to the ones that we observed for the Hoggar Mountain dust
sample, but they did not mention a memory effect. ATD is a complex mixture of
minerals with a considerable share of microcline (20–30 %) (Atkinson
et al., 2013), which is a K feldspar with exceptionally high heterogeneous
ice nucleation temperatures. Microcline samples showed high freezing
temperatures from T=264–272 K in bulk freezing experiments
(Kaufmann et al., 2016) similar to the ones performed in this study.
Therefore, microcline is most probably the mineral component responsible for
freezing of bulk samples. The experiments by Wright and Petters (2013) were
performed with smaller droplets (15–120 µm diameter) containing
only few particles. Freezing occurred at T=236–253 K.
Microcline will therefore just be one among the various mineral components
responsible for freezing in these experiments. The freezing can therefore
not be ascribed to microcline alone in these experiments, in contrast to the
experiments performed in this study. If the memory effect is due to the
microcline component, it may explain why Wright and Petters (2013) did not
observe it. Zolles et al. (2015) attribute the high ice nucleation activity
of K feldspars to an intrinsic property of the surface. They hypothesize
that the surface cations released into the surface bilayer may interact with
water to enhance or inhibit ice formation. Also, the ion charge density of
the cations of the mineral was suggested to influence ice nucleation. The
memory effect might therefore be related to surface characteristics
involving the cation distributions, which might change once the surface has
been covered with ice. Indeed, the memory effect in our ATD samples is
typically confined to the very first run. The limited number of refreeze
experiments with ATD performed for this study does not allow for a detailed
characterization of the ice nucleation activity of microcline. A dedicated
study with refreeze experiments performed on pure microcline samples might
help to elucidate whether this mineral possesses surfaces with small patches
of high ice nucleation probability or larger surface areas with lower but
uniform ice nucleation probability.
Birch pollen washing water
The molecular identity of the macromolecules in birch pollen washing water
is still unknown. Pummer et al. (2012) suspected them to be polysaccharides
or glycoproteins based on their resistivity against denaturation by 6 M
guanidinium chloride and heating to 400 K. The ice nucleation activity
differs slightly between birch pollen washing water from different
geographical regions. Augustin et al. (2013) found that Swedish birch pollen
washing water shows a second plateau in the temperature range between 249
and 256 K, which is absent in Czech birch pollen washing water. In the
present study, we investigate Czech birch pollen washing water.
There is clear evidence from the emulsion measurements that only a small
fraction of the birch pollen macromolecules are ice nucleation active. We
estimate this fraction to be on the order of 10-7 (see Sect. 5.2.2)
based on emulsion freezing experiments and the dilution series shown in Fig. 2. A 50 mg mL-1 sample weighing 2 mg should therefore contain on the order of
107 active macromolecules while this number reduces to about 1 for 2 mg
of a 10-5 mg mL-1 sample. These numbers are consistent with the freezing
experiments with water droplets activated from a birch pollen washing water
aerosol performed by Augustin et al. (2013). They observed a frozen fraction
of 0.03 for 800 nm particles at 254 K, which translates into an
ice-nucleating fraction of macromolecules of 4×10-8 assuming
that the whole sample consists of macromolecules with 300 kDa. While our 50 mg mL-1 samples contain a high number of ice-nucleating macromolecules, not
all of them induce freezing at the same temperature. The emulsion
measurement (Fig. 1c) shows a heterogeneous freezing peak with onset at
about 255 K that stretches to below 245 K and then fades away.
Heterogeneous freezing occurring in this temperature range is in agreement
with Augustin et al. (2013). They observed the highest freezing temperatures
at 254 K with frozen fractions of 0.007 and 0.02 for 500 and 800 nm
particles, respectively. The frozen fraction increased, when temperature was
lowered, reaching a plateau with no further increase at 245 K. Augustin et
al. (2013) further reported results from Pummer et al. (2012), who
investigated droplets in the size range from 10 to 200 µm diameter
and observed an increase in frozen fraction from 2.5×10-3 at
257 K to full activation at 253 K for 50 mg mL-1 samples. We can therefore
assume that only few macromolecules are active at the highest temperature.
This conclusion is supported by the fits of freezing rates obtained from the
different CNT-based parameterizations, which yield significantly different
contact angles α and pre-factors β
between some samples. This indicates that for some samples ice nucleation might
have occurred always on the same macromolecule during the course of a
refreeze experiment.
Nonadecanol-coated droplets
The refreeze experiments with water droplets coated with a nonadecanol
monolayer show a clear decrease in freezing temperature with decreasing
surface area of the droplets. The 1100 µm radius droplets freeze
between 260 and 265 K, the droplets with radii of 370 and 320 µm between 256 and 262 K and the ones with radii of 48 and 31 µm between 248
and 252 K. Zobrist et al. (2007) evaluated these results
within the framework of CNT assuming that the whole surface of the
nonadecanol monolayer is ice nucleation active. They obtained best agreement
assuming a temperature dependence of the effective contact angle described
by the linear function α(T)= 571.50–2.015 ×(T/K), yielding contact angles from
37.5∘ at T=265 K to 71.8∘ at T=248 K. They explained this temperature dependence by assuming a reduced
compatibility of the alcohol monolayer with the ice embryo as the
temperature decreases due to the decreasing mobility of the alcohol
molecules on the water surface, which inhibits the rearrangement of the alcohol
molecules at the water surface. Vali (2014), on the other hand, speculated
that the monolayers formed by long-chain alcohols are not simple, smooth
surfaces but may have discontinuities of various kinds such that ice
nucleation occurs at specific nucleation sites and not on the whole
monolayer surface. In this study, we reevaluated the freezing rates
determined by Zobrist et al. (2007) assuming that freezing occurred at sites
of critical size. Fitting the freezing rates separately for the individual
refreeze experiments using the contact angle α and
pre-factor β as fit parameters, yielded pre-factors β around 1 and contact angles above
50∘ for the smallest
droplets, irrespective of the choice of CNT parameterization. For the
largest droplets the pre-factor is on the order of 10-8 and the contact
angle is below 32∘. Droplets of a similar size gave contact angles
that are identical within the observed variability. This
indistinguishability supports the notion that long-chain alcohol monolayers
provide an extended surface with a relatively uniform ability to nucleate
ice. However, to substantiate this conjecture, more refreeze experiments with droplets of the same size would be needed.
Critical site area
In the framework of CNT, freezing only occurs, when the embryos developing
at a site can reach the critical size to grow into a crystal. Because the
critical embryo size increases with increasing temperature, the
critical size of a nucleating site also increases with temperature. In this
study, critical site areas needed to accommodate an ice embryo of critical
size, Acrit,het, are obtained as a result of the fits to the
experimentally determined freezing rates using Eq. (8). All three CNT-based
parameterizations yield critical areas in the same size range. This is an
indication that the determined values are well constrained and might indeed
have a physical basis. Critical site areas, calculated with the three
CNT-based parameterizations, are Acrit,het=16–39 nm2 for Hoggar Mountain dust with freezing temperatures
Tfr=258–265 K. For the ATD sample with
Tfr=267–268 K the critical site area ranges from
Acrit,het=39–52 nm2 for the different CNT-based
parameterizations. Birch pollen washing water samples freeze in the range
Tfr=254–261 K with Acrit,het=20–50 nm2. Finally, Acrit,het for the nonadecanol samples
decrease from 16.1 to 27.2 nm2 for the r=1100µm
droplets with Tfr=260–265 K, to 13.2–21.6 nm2
for the r=370/320 µm droplets with Tfr=256–262 K, and finally to Acrit,het=10.4–16.1 nm2 for the r=48/31 µm droplets with
Tfr=248–252 K. These critical site areas show a
temperature dependence and are larger at higher temperatures. They are in
the same size range as the ice nucleation active area of proteins expressed
by the bacteria Pseudomonas syringae (P. syringae) and
Erwinia herbicola, which are active at 263–265 K and have a mass
of 150 kDa (Yankofsky et al., 1981; Govindarajan and Lindow, 1988; Budke and
Koop, 2015; Pandey et al., 2016). Kajava and Lindow (1993) determined the
area of the minimum ice-nucleating site of P. Syringae as 25 nm × 2.5 nm = 62.5 nm2, corresponding to the area on the
protein that shows a lattice match with ice. Critical nucleus surface areas,
Acrit,het, estimated in this study are in general agreement
with this number.
Fit parameters α and β
In this study, we fitted the observed freezing rates of refreeze experiments
using three different CNT-based parameterizations (Ick15, P&K97, and Z07)
together with the assumption that freezing occurs at single sites of
critical size at the mean freezing temperature of the refreeze experiment.
The different parameterizations gave slightly different values of contact
angles, pre-factors, and Acrit,het but were very similar in
their ability to fit the data. When the contact angle was used as the only fit
parameter (V1), the parameterizations underrated or overrated the increase in freezing rate with decreasing temperature depending on the sample. If the
contact angle α and the pre-factor β were
used as fit parameters (V2), good fits could be obtained for most refreeze
experiments. This shows that ice-nucleating sites need to be characterized
by two parameters. While the α parameter describes
the reduction in the energy barrier in the presence of an ice-nucleating
surface, the interpretation of the pre-factor β is
less obvious. There are different explanations conceivable for the need of a
pre-factor β as an additional fit parameter:
If some sites were not constant in quality from one freezing cycle to the
next, ice nucleation at such sites would not be fully stochastic. In this
case, it would not be correct to describe the freezing temperature sequences
with a constant contact angle α. When variability of
α is mistaken as random fluctuations of freezing
temperatures, a low value of pre-factor β would be fitted.
The presence of a monotonic trend or an improbable sequence of freezing
temperatures are indications that nucleation indeed was not fully
stochastic. However, there is no criterion available to discriminate
stochastic variations of freezing temperature from variations due to the variability of α.
A high number of sites active at the same temperature instead of only one or
a few would result in a pre-factor β > 1
because each site would contribute to the total frequency of nucleation
attempts.
For homogeneous ice nucleation the kinetic pre-factor is considered to
account for the rate at which water molecules are transferred into an ice
germ (e.g., Ickes et al., 2015). If the presence of a surface changed this
rate because it, e.g., influences the orientation of water molecules, the
additional fit parameter β could account for this. A
pre-factor β < 1 would describe an
unfavorable orientation of water molecules for the transfer into the growing
ice embryo leading to a reduced number of successful nucleation attempts. A
pre-factor β > 1, on the other hand,
would mean a favorable orientation of water molecules for incorporation into
the ice embryo leading to an accelerated nucleation process.
If different orientations of water molecules on a surface were energetically
similar but only one of them were suited to develop into an ice embryo,
nucleation could only occur at times when this favorable arrangement is
realized. This would correspond to a reduction in the number of nucleation
attempts compared to a case when one preferred orientation of water
molecules exists on a surface that promotes ice nucleation. In such a case,
β would be < 1.
Kinks, cracks, or screw dislocations next to a site could orient water
molecules favorably to develop critical ice embryos at a site. This would
increase β compared with the case of a site on a flat
surface.
In the case of explanations (i) and (ii), the pre-factor β is
just a correction factor lacking a fundamental physical meaning but it accounts for inadequacies of the conjectures for the fit, namely for the
assumptions of an ice nucleation active area of critical size (point i) and
for the assumption of constant α during the course of the
experiment (point ii). Explanations (iii)–(v) imply that the number of
nucleation attempts can be lower or higher than predicted by
jhom(T) and should be considered as a characteristic
of a nucleation site. Note that the values fitted for β range from
10-9 to 1043 for all refreeze experiments and show uncertainties
of a factor of 100 for individual fits to refreeze experiments. This
shows that the exact value of β is not well constrained.
Nevertheless, the β value can be used as an indicator of a steeper
(β > 1) or a shallower (β < 1) increase in
nucleation rate coefficients with decreasing temperature than predicted by
CNT. In the following, we will relate the fit parameters α
and β to the specific properties of the investigated ice
nuclei.
Hoggar Mountain dust and ATD
For Hoggar Mountain dust and ATD, the pre-factor β is low
(10-2–10-9). There might be a low bias of β if the variability of α is taken as random fluctuations of
freezing temperatures (point i). Nevertheless, this is likely to be a minor
effect because there is no correlation evident between monotonic trends of
the time series and β values. A low value of the
pre-factor β indicates that the ice-nucleating
surface is not effective at growing ice embryos of critical size. Even if
the temperature has dropped low enough to overcome the energy barrier to
form a critical ice embryo at the nucleation sites of Hoggar Mountain dust
and ATD, embryos of a critical size might form only infrequently. Pedevilla et
al. (2016) investigated the most easily cleaved (001) surface of the microcline
with ab initio density functional calculations. They demonstrated that water
does not form ice-like overlayers in the contact layer; however, they
identified contact layer structures of water that induce ice-like ordering
in the second overlayer. If these structures are only very few among several
water structures and develop only infrequently, this might explain a low
frequency of freezing attempts, i.e., β < 1.
Birch pollen washing water
For the macromolecules present in the birch pollen washing water the
pre-factor β ranges from 1 to 10 000 for most samples at
Tfr=256–261 K and increases even to ∼ 1040 for P8, which freezes at Tfr=258.5–259 K. The
high values of β may indicate that many macromolecules
induce freezing at similar temperatures so that they alternate in inducing
ice nucleation from run to run and thus increase the effective surface on
which ice nucleation may take place (point ii). This might explain the high
value of the pre-factor for some samples but not for all. The sample P3 with
the highest freezing temperatures (260–261 K), which is probably due to
nucleation at a rare and especially effective nucleation site, also has
β > 1. This indicates that nucleation attempts
are very frequent and the sample freezes immediately once the temperature
has dropped low enough to overcome the energy barrier for critical embryo
formation.
Assuming sizes of the birch pollen macromolecules of 100–300 kDa as
inferred by Pummer et al. (2012), the surface area should range from 111 to 232 nm2 assuming a density of 1.5 g cm-3 for the
macromolecules. If we compare this area with the range of a calculated
critical site area of 20–50 nm2, a considerable part of the
macromolecules' surfaces should be involved in ice nucleation. Pummer et al. (2015), who consider the macromolecules to be polysaccharides, attribute the
ice nucleation ability to a hydration shell around the polysaccharides. This
hydration shell might form an ice template that does not randomly dissociate
like ice embryos in homogeneous ice nucleation. Such a stable shell might
indeed be a reason for the high β values.
Nonadecanol-coated droplets
For the larger droplets with radii r=1100µm and
r=370/320 µm covered with the nonadecanol monolayer, the
pre-factor β is small (10-6–10-8), but for the
small droplets with r=48/31 µm, it is quite large
(10-3–102). The measured slope of freezing rate increase with
decreasing temperature was even flatter than could be fitted with a
pre-factor β=1. It can be seen in Fig. 1 of
Zobrist et al. (2007) that all experiments have random outliers to higher
temperatures which populate the highest temperature bins. This would mean
that at high temperatures, the freezing is limited by the frequency of
nucleation attempts because the surface does not offer features that
facilitate the aggregation of water molecules into ice-like subcritical
clusters that eventually grow to critical size. Investigating
C31H63OH alcohol monolayers, which induce freezing at about 271 K,
by grazing incidence X-ray diffraction showed that the coherence length
between the monolayer and the ice lattice was only ∼ 2.5 nm
corresponding to about five lattice spacings and was rationalized by
assuming multiple ice nucleation sites separated on average by about 5–6 nm (Popovitz-Biro et al., 1994; Majewski et al., 1995). A close match
between the ice lattice and the monolayer only extends 3 nm in the a
and 5 nm in the b direction. These values yield critical site areas in
the same range as the ones calculated for the nonadecanol monolayers from
the CNT-based fits (see Table 8). The spacing of the 2-D lattice of the
nanodecanol monolayer might be temperature dependent such that the lattice
fit between the monolayer and ice deteriorates with decreasing temperature.
The memory effect observed for this sample is discussed as structural
rearrangement within the alcohol monolayer (Seeley and Seidler, 2001;
Zobrist et al., 2007). The interaction between the lattice of ice and the 2-D
crystalline monolayer might lead to a rearrangement of the long-chain
alcohols into a structure with improved lattice match and enhanced ice
nucleation efficiency. This supports the interpretation given in Zobrist et
al. (2007) that the formation of a critical embryo is favored by lower
temperatures and the molecular rearrangement is favored by higher
temperatures because the flexibility of the monolayer to adapt to the ice
structure decreases with decreasing temperature.
Summary and conclusions
This study presents freezing rates determined from refreeze experiments
using Hoggar Mountain dust, ATD, and birch pollen washing
water as heterogeneous ice nuclei. These samples were analyzed using three
parameterizations of CNT. Additionally, nonadecanol refreeze experiments
from Zobrist et al. (2007) were reevaluated. The presented analysis leads
to the following microphysical insights:
Presence of preferred nucleation sites. For Hoggar Mountain dust,
ATD and the pollen washing water, there were significant differences in
freezing temperatures between samples taken from the same stock solution.
Such differences are not compatible with the assumption that ice nucleation
occurs at a random location of a large uniform surface. The experimental
basis for the nonadecanol monolayers was too small to come to the same
conclusion. Six time sequences of refreeze experiments from droplets of
different sizes were analyzed. Droplets of the same radius were
indistinguishable from each other with respect to their freezing
temperatures. This is compatible with the assumption that freezing takes
place at a random location on a large surface.
Stability of sites and randomness of nucleation. While some of the
time sequences observed for Hoggar Mountain dust, ATD, and birch pollen
washing water were in accordance with stochastic freezing, others showed
jumps and trends in the sequence of freezing temperatures indicating that
some sites were not stable during the course of the experiment. This is in
accordance with Vali (2014) and Wright and Petters (2013), who also evidenced
limitations of the stability of sites.
Description with CNT. For the analysis of the experimental data
with CNT, it was assumed that the same site is always responsible for
freezing and that this site is stable and of the critical size. Three CNT-based
parameterizations were used to describe freezing rates as a function of
temperature. All of them led to similar results. For Hoggar Mountain dust,
ATD, and larger nonadecanol-coated water droplets the experimentally
determined increase in freezing rate with decreasing temperature is
shallower than can be described by CNT using the contact angle as the only fit
parameter. The opposite is true for birch pollen washing water and small
nonadecanol-coated water droplets: the observed increase in freezing rate is
steeper than can be fitted by CNT-based parameterization relying on the
contact angle as the only fit parameter. Good agreement of observations and
calculations for most experiments were obtained when a pre-factor
β was introduced as a second fit parameter.
Critical site size. The description of heterogeneous nucleation
with CNT implies that nucleation occurs at sites with a minimum (critical)
surface area so that embryos that develop on them can reach the critical
size to grow into ice crystals. CNT provides an estimate of the size that is
needed to accommodate the critical embryo. This size is in the range of 10–50 nm2 for the investigated ice nuclei. The required size decreases
with decreasing nucleation temperature. Sizes in this order of magnitude are
in agreement with the area of the minimum ice-nucleating site that was
determined for P. Syringae. We therefore suggest that
ice-nucleating surfaces have to be searched for features in this size range
to identify ice-nucleating sites.
Interpretation of fit parameters. The energy barrier of nucleation
is reduced when the ice embryo forms at an ice-nucleating surface. The
reduction in Gibbs energy is described by the contact angle α, which was used in this study as a first fit parameter. To adjust
the slope of freezing rate increase with decreasing temperature predicted by
the three CNT-based parameterizations to the measured one, a second fit
parameter in the form of a pre-factor β was needed.
If the assumption of the nucleating area of a critical size and constant
α is valid, the pre-factor β modifies the
frequency of nucleation attempts predicted by CNT. If β > 1, there are many nucleation attempts and nucleation occurs
immediately when the temperature is low enough so that the active site area
is large enough to accommodate a critical embryo. This is the case for the
birch pollen washing water and the small droplets coated with nanodecanol.
If β < 1, the number of nucleation attempts
is low and the increase in freezing rate with decreasing temperature is
shallow. This is the case for Hoggar Mountain dust, ATD, and the large
droplets coated with nonadecanol.
To get access to the data, please contact Claudia Marcolli (claudia.marcolli@env.ethz.ch).
The figures in this appendix present fitting results for the three CNT-based
parameterizations Ick15, P&K97, and Z07 to refreeze experiments with Hoggar
Mountain dust containing water droplets (Fig. A1), birch pollen washing
water droplets (Fig. A2), and nonadecanol-coated droplets (Fig. A3). The
fitting results for the samples presented in this appendix are in accordance
with the fitting results for samples presented in the main part of this
publication.
CNT-based fits of freezing rates for the Hoggar Mountain dust H1–H8
samples with the parameterizations Ick15, P&K97, and Z07. See Fig. 5 for
H9. V1: fits with the contact angle α as the only fit parameter. V2:
fits with α and β as fit parameters. Values of fit parameters
are given in Tables 3 and 4.
Birch pollen washing water samples P1–P9. CNT-based fits of
freezing rates with the parameterizations Ick15, P&K97, and Z07. V1: fits
with the contact angle α as the only fit parameter. V2: fits with
α and β as fit parameters. Values of fit parameters are given
in Tables 5 and 6.
Nonadecanol samples N1, N3, N4, and N5 measured by Zobrist et
al. (2007). CNT-based fits of freezing rates with the parameterizations
Ick15, P&K97, and Z07. V1: fits with the contact angle α as the
only fit parameter. V2: fits with α and β as fit parameters.
Values of fit parameters are given in Tables 7 and 8.
The authors declare that they have no conflict of interest.
Acknowledgements
This work was supported by the Swiss National Foundation, project
no. 200021_138039. We thank Ulrich Krieger and Anand Kumar for
fruitful discussions.
Edited by: D. Knopf
Reviewed by: two anonymous referees
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