Introduction
The purpose of this Technical Note is to suggest definitions of terms for use
in describing ice nucleation. The suggested list of terms evolved from one
originally proposed by the authors to one containing substantial inputs from
reviewers and other contributors. Three successive drafts were posted on the
discussion page linked to this paper (Vali et al., 2014). Comments by
reviewers and others are on the same discussion page, as are responses to
those comments. Careful examination of this
material makes it clear that there have been different interpretations of
some terms and those uses continue to appear in the current literature. The
proposed list has evolved, and the introduction of a few new terms has become
necessary in order to recognize different perspectives and to allow for the
relatively unambiguous presentation of current knowledge. Nonetheless, it is
certain that the proposed list of terms will have to be revised, with some
terms becoming obsolete or ambiguous, and with the introduction of additional
terms to describe new discoveries.
The motivation for revising and expanding on the “Nucleation terminology”
article by Vali (1985) is that the progress made in the
intervening 30 years has revealed unexpected complexities of heterogeneous
ice nucleation and that the terminology applied in discussing these phenomena
has not evolved in a consistent and unambiguous way. These problems can be
seen in recent literature with overlapping, unclear, and in some cases,
contradictory usage of terms describing heterogeneous ice nucleation. More
detailed terminology is potentially helpful in eliminating some of the
problems and represents a step toward facilitating further progress. The
meanings of scientific terms evolve with time. A concise summary of all the
various interpretations attached to given terms by a broad spectrum of
researchers at any given time is not possible. Even so, a set of definitions
can perhaps be agreed upon for present usage, acknowledging that future
definitions will diverge to various degrees. However, it can be hoped that
the stated definitions will serve, at a minimum, to allow for more concise
identification of possible deviations from them.
With the aforementioned ideas in mind, the first version of the terminology
was posted in Atmospheric Chemistry and Physics – Discussions in
August 2014. Reviews of and comments on this paper indicated support for the
need to clarify the usage of controversial terms and included many
suggestions for changes and improvements. Based on those inputs, a second
draft was posted in February 2015 and a further exchanges of views followed.
A third draft was posted on 5 May 2015. Along with the second and third
drafts, responses were listed on specific points made by the reviewers and in
comments. All of this material is contained on the interactive discussion
page (Vali et al., 2014) for this paper. The reviewers of the discussion
paper were two anonymous referees and Dr. T. Koop. The authors of the
interactive comments were C. A. Knight, R. Jaenicke, Z. Kanji on behalf of
the Lohmann Ice Nucleation Group at ETH Zurich; H. Wex on behalf of
S. Augustin-Bauditz, H. Bieligk, T. Clauss, S. Hartmann, K. Ignatius,
L. Schenk, F. Stratmann, J. Voigtländer from the Cloud-group at the
Institute for Tropospheric Research, TROPOS; D. Niedermeier on behalf of
D. Ciochetto, C. Gurganus, R. Shaw and Y. Wang at Michigan Technical
University; B. Murray and A. Bogdan.
Heterogeneous ice nucleation is the main focus of the terminology proposed here as it is the topic where recent developments revealed most need for
clarifications of concepts. Homogenous ice nucleation and terms common to
both types of nucleation are included only for the sake of completeness and
no significant changes from accepted practice are proposed.
Following the naming of entries, a brief definition is
given in italics. Additional details are provided in the paragraph(s) that follow
in normal font. Cross references to other entries are given by section
numbers.
Homogeneous ice nucleation
Ice nucleation without any foreign substance aiding the process
Homogeneous deposition nucleation and homogeneous ice nucleation from water vapor
Ice nucleation from supersaturated vapor, without any foreign substance aiding the process
Deposition nucleation is the formation of ice
directly from water vapor. Because of the very high supersaturation required
for the homogeneous deposition nucleation of ice, it is not observed in the
atmosphere or in other natural systems. However, there is evidence for
homogeneous ice nucleation from water vapor via processes that involve the
intermediate step of homogeneous condensation of liquid, or an amorphous
phase, at supersaturations below that required for deposition (Murray and
Jensen, 2010).
List of symbols (with CGS units indicated).
A
Total surface area of ice nucleating particles (INPs) in a sample unit (cm2)
f
Fraction of samples frozen (Sect. 4.5)
J(T)
Nucleation rate (probability of freezing) per unit time as a function of temperature (s-1)
Js(T)
Nucleation rate coefficient; per unit time and per unit surface area of INPs (cm-2s-1)
Jm(T)
Nucleation rate coefficient; per unit time and per unit mass of INPs (g-1s-1)
Jv(T)
Homogeneous nucleation rate coefficient, per unit time and per unit sample volume (Sect. 3) (cm-3s-1)
Jsite(T)
Nucleation rate on a specific site (Sect. 4.7.2) (s-1)
k(T)
Differential nucleus spectrum; number of sites active within a 1 ∘C interval at T per unit sample volume (cm-3∘C-1)
K(T)
Cumulative spectrum, or integrated volume density of active sites : number of sites active above T per unit sample volume (cm-3)
ns(T)
Surface density of sites (number per unit surface area of INPs) active above T (cm-2)
NU
Number of samples units in which no nucleation event has taken place
NF
Number of samples frozen
N0
Total number of samples in an experiment
R
Freezing rate per unit time (s-1)
S
Supersaturation
t
Time (s)
T
Temperature (∘C)
Tc
Characteristic temperature for a nucleating site (∘C)
V
Volume of sample unit (cm3)
CNT
Classical nucleation theory
INP
Ice nucleating particle
Homogeneous freezing nucleation
Ice nucleation within a body of supercooled liquid without any foreign substance aiding the process
Homogeneous nucleation rate coefficient
The probability, or observed frequency, of ice nucleation events in unit volume of supercooled liquid or supersaturated vapor within a unit of time
Classical nucleation theory (CNT) relates the nucleation rate coefficient to
the properties of the liquid and the net rate at which molecules are added to
the ice embryos. Empirically, the nucleation rate is determined from the
frequency of events as a function of supersaturation or temperature:
Jv=-1V⋅1NU⋅dNUdt, using NU to
denote the number of sample units in which no nucleation event has taken
place by time t, and V to denote the total volume
observed. Here the
subscript “v” is added to the usually employed symbol J in order to focus
on the fact that the homogeneous nucleation rate coefficient refers to unit
volume of vapor or liquid and to distinguish it from similar expressions for
heterogeneous nucleation. It is recommended to use the symbols
Jv(S) and Jv(T). The quantity Jv has
dimensions of L-3t-1 (CGS units of cm-3s-1).
While the concept of freezing rate (Sect. 4.6) has not been applied so far in
the literature on homogeneous ice nucleation, it is a valid representation of
experimental results or of predictions for both homogeneous and heterogeneous
ice nucleation. For homogeneous nucleation, the freezing rate is directly
proportional to the nucleation rate coefficient and the volume of the sample
units: R=Jv⋅V=-1NU⋅dNUdt.
In practice, it is not always possible to ensure that all sample units are
free of ice nucleating particles (INPs) so that the apparent freezing rate observed may lead to an
incorrect value for Jv; in such a case the observed freezing rate has to be seen as the sum of
various contributions (e.g., Koop et al., 1997).
Heterogeneous ice nucleation
Ice nucleation aided by the presence of a foreign substance so that nucleation takes place at lesser supersaturation or supercooling than is required for homogeneous ice nucleation
INP, INM, INE, etc.
Ice nucleating particle (INP), molecule (INM), entity (INE), material, substance, object, item, unit, or other, that is assumed to be the agent responsible for observed heterogeneous nucleation
Because of the variety of substances and forms that can be responsible for
heterogeneous ice nucleation, it is impractical to have a single designation
that covers all possibilities while being sufficiently informative. It is
suggested that authors refer to the specific nucleating agent in each
particular case in the manner most appropriate for the system studied. The
form of the designation IN, plus a third letter, may be helpful enough for
effective communication. The term “nucleator” is also used as a general
reference to an object whose presence is responsible for observed ice
nucleation.
Reference to an INP (or an equivalent) does not, in general, specify the
composition of the particle, but describes the unit that carries the
nucleating substrate. A number of different terms have been used in the
literature for this. For decades, the terms “ice nucleus” and “ice
nuclei” were used almost exclusively with reference to atmospheric aerosol
that could initiate ice, that is, individual particles, each of which
resulted in the formation of one ice crystal. While it was recognized that
only a specific location on the particle surface is actually where ice begins
to form, the entire particle was referred to as the ice nucleus. This led to
confusion. The concept of a “site” appeared in the literature to narrow the
identification of an ice nucleus. With the advent of ice nucleation studies
on systems other than clouds, also including biological substances
(bacteria, fungi, etc.), usage has become more confusing as focus has
expanded to nucleation by entities other than aerosol particles. In all, the
term “ice nucleus” has become both overused and vague. For atmospheric
applications, or more generally, when dealing with many separate entities, it
is more appropriate to use ice nucleating particle (INP) or the other forms
listed above to refer to individual units, and to
use the plural INPs or other forms to refer to a collection of them. Since ice
nucleation is more complicated than condensation nucleation, due to the
different modes it can follow, using “ice nuclei” in the general sense
similarly to “condensation nuclei” is overly ambiguous and can be misleading.
Site
Preferred location for ice nucleation on an INP, or equivalent
Direct experimental evidence for deposition nucleation (e.g.,
Mason, 1957; Anderson and Hallett, 1976) points to the role of specific
locations on surfaces which promote nucleation with greater effectiveness
than other locations. Similar evidence is available with freezing nucleation
in terms of repeated freezing of samples at nearly the same temperatures, but
this evidence is less direct than for deposition where the locations can be
visually identified. Sites are thought to arise due to non-uniform surface
properties of INPs that result in different binding energies to water
molecules and incipient ice structures. Sites are considered to be important for
both deposition and freezing nucleation. Observed nucleation on, or within, a
sample is understood to be due to the most effective site found in it. Sites of
various effectiveness are assumed to occur on the surfaces of most materials.
In principle, sites have identifiable properties distinct from the assumed
spontaneous formation of embryos at some unpredictable location on a surface.
Site density
The number of sites causing nucleation per unit surface area of the INP, or equivalent as functions of temperature or supersaturation; the quantitative measure of the abundance of sites of different ice nucleating effectiveness
A number of different methods have been used in the literature
to quantitate the frequency of occurrence of different temperatures or
supersaturations at which ice nucleation has been observed and/or modeled.
Most of these descriptions are direct representations of measurements. Time
is considered an implicit factor specific to each experiment, i.e., the singular
approximation (Sect. 4.7.1) is applied.
The density of sites is the number of sites per unit surface area of
INPs that have caused nucleation by the time some supercooling temperature or
supersaturation is reached. Connolly et al. (2009) and Niemand et al. (2012)
used “integrated site density”, and Hoose and Möhler (2012) used
INAS (ice nucleation active site density) to refer to this quantity. The
quantity is designated as ns(T), or ns(S), with dimension
of L-2 (CGS units of cm-2).
Interpreting the results of freezing experiments with subdivided sample units
(e.g., particles randomly distributed into liquid volumes), the number
concentrations of sites are defined (Vali, 1971) as the differential (k) and cumulative (K) nucleus spectra: k(T)=1V⋅(N0-NF(T))⋅dNF(T)dT and K(T)=-1V⋅ln(1-NF(T)N0). Site density with reference
to surface area and the cumulative nucleus spectrum, for freezing, are
related as K(T)=ns(T)⋅A.
The foregoing descriptions assume that the nucleation rate is equal to zero
at temperatures higher (supersaturations lower) than the characteristic
temperature (supersaturation) of the site and equal to infinity beyond that.
Thus, these definitions rely on the singular description (Sect. 4.7.1) with
Tc for each site replaced by the observed freezing temperature
T.
Marcolli et al. (2007) used contact angle as a surrogate to express site
effectiveness. That idea was further developed by Welti et al. (2012) in the
α-pdf model. Niedermeier et al. (2011, 2014) constructed the Soccer
Ball Model to describe the distribution of sites of different effectiveness.
Hartmann et al. (2013) modeled the distribution of sites among sample units.
In these cases, the site density is represented using distributions of
parameters in the CNT formulations of the nucleation rate coefficient thereby
linking these models to the stochastic description (Sect. 4.8.1 and 4.8.2),
while allowing the characterization of sites of different effectiveness to be
included.
Site frequency distributions should always include some indication of the
timescale of the experiment being interpreted. This allows various
experiments to be compared more effectively.
Modes of heterogeneous ice nucleation
Distinctions in the mode of nucleation made on the basis of the process envisaged to lead to nucleation
Definitions of nucleation modes were given by Vali (1985) with
a focus on atmospheric processes. Several of the definitions given below
broaden and alter those definitions.
Deposition nucleation
Ice nucleation from supersaturated vapor on an INP or equivalent without prior formation of liquid
It is difficult to ascertain whether or not ice nucleates from
(supersaturated) vapor without any liquid forming. Similar to the homogeneous
case, deposition nucleation may have a transitory stage in which liquid is
present but does not develop to a macroscopic, observable quantity. It has
also been theorized that condensation in voids and cavities followed by
freezing can account for many observations that appear to be deposition
(Marcolli, 2014), but this process is better viewed as freezing followed by
depositional growth. Observations of what is believed to be deposition
nucleation need to focus critically on identifying the details of the
process.
Freezing nucleation
Ice nucleation within a body of supercooled liquid ascribed to the presence of an INP, or equivalent
Further specifications of modes are as follows:
Immersion freezing refers to ice nucleation initiated by
an INP, or equivalent, located within the body of liquid.
Contact freezing is initiated by an INP, or
equivalent, at the air–water interface as the INP comes into contact with the
liquid, or forms at an air–liquid–particle triple interface.
This process is defined as separate from immersion freezing because of
empirical evidence that some INPs are more effective in this mode than when
immersed in liquid (Shaw et al., 2005). There is as of yet no
definite method for how to distinguish this mode from immersion
freezing. Some laboratory evidence points to a difference depending on
whether the particle is inside of or outside of the liquid; this is described
as inside-out versus outside-in nucleation. In the atmosphere, pre-activated
particles may cause freezing when coming into contact with supercooled liquid
droplets.
Condensation freezing is defined as taking place when freezing is
initiated concurrently with the initial formation of liquid on a cloud
condensation nucleus (CCN) at temperatures below the melting point of ice.
This was envisaged as a possible sequence in clouds but evidence for its
existence is minimal. Whether condensation freezing on a microscopic scale,
if it occurs, is truly different from deposition nucleation, or distinct from
immersion freezing, is not fully established. Hence, the use of this
term requires added circumspection.
Other modes of freezing nucleation
reported in the literature are electro-freezing, evaporation freezing,
mechanical shock freezing and collision freezing. Evidence available at this
time does not permit general definitions to be established for these
processes.
Fraction frozen
The ratio of the cumulative number of sample units frozen at
T to the original number N0: f=NFN0, with NF given as either a function of time or of temperature
The frozen fraction represents the results of experiments
with sample units drawn from the same original volume. It can be used when
the sample units are gradually cooled or when held at a fixed temperature.
Similar quantities can be readily defined for nucleation modes other than
freezing.
Freezing rate
Expresses the results obtained from an experiment in which the freezing of a number of sample units is observed
The freezing rate is expressed as a function of the number of sample units
frozen at time t: R=1N0-NF(t)⋅dNF(t)dt. The freezing rate for a
given T is given in units of inverse time, e.g., s-1. The
freezing rate function is a direct description of empirical observations with
distributed samples and can be used both for experiments in which the samples
are steadily cooled and others in which the temperature is held constant. The
freezing rate is related to the time derivative of the frozen fraction
(Sect. 4.5): R=11-f(t)⋅df(t)dt.
The freezing rate for any particular sample is dependent on the volumes of
the sample units and on the INP content (site density or nucleus spectrum)
of the liquid. It is also influenced by dissolved substances. For
polydisperse sample volumes, the freezing rate should be specified separately
for each volume range.
For homogeneous nucleation, the freezing rate is usually called nucleation
rate. In the stochastic description of heterogeneous nucleation, the terms
freezing rate and nucleation rate are interchangeable, but using freezing
rate makes it clearer that one refers to observed frequencies of events and
not to the more abstract rate coefficient or site nucleation rate (Sect. 4.7.2).
Some overlap exists in the use of the term freezing rate between discussing
nucleation of sample units and talking about growth of ice. The distinction
has to be clarified explicitly if not evident from the context.
Site-specific descriptions/models
Singular description (time independent)
Description/model of observed nucleation events for a population of sample units containing INPs (or equivalents) and assuming that the preferred sites have a spectrum of different nucleating abilities; also referred to as the deterministic description of ice nucleation; no time dependence is taken into account
This description is based on evidence that points to sites having
well-defined, albeit not perfectly stable, potentials for promoting
nucleation. Each site is then characterized by the temperature, or
supersaturation, at which it is observed to nucleate ice for a given mode.
For freezing, a characteristic temperature Tc is used to specify
the effectiveness of the site or INP. The time history of the sample is not
taken into account. In that sense, the singular description is often called
deterministic. The singular description is expressed quantitatively by site
density or by nucleus spectra (Sect. 4.3).
Site nucleation rate
Expresses the probability per unit time that nucleation takes place on a given site of an INP (or other) involved
Site nucleation rate, Jsite, is a function of temperature (for
freezing) and other factors reflecting the nature of the nucleating site. The
term is adopted from the description of homogeneous nucleation according to
which nucleation rate refers to an observed volume or ensemble of drops.
Applied to a site, the term has a narrower focus but the same meaning, i.e.,
the probability of nucleation within a time interval.
In cases where data fit the stochastic model (Sect. 4.8.1 and 4.8.2), i.e.,
all INP surface areas (or masses) appear to be entirely equivalent in their
ability to nucleate ice, the nucleation rate coefficient Js(T)
replaces the site nucleation frequency Jsite as the relevant
quantity.
Site nucleation rate has the dimension of inverse time (CGS units of
s-1).
Time-dependent site-specific descriptions/models
Descriptions that encompass definitions of site density distributions (Sect. 4.3) and account for the time dependence of freezing nucleation
Each site is assumed to be defined by its characteristic temperature and by
the site nucleation rate associated with it (Vali and
Stansbury, 1966, VS66; Vali, 2014). The abundance of sites of different
characteristic temperatures is specified by some characterization of the site
density (Sect. 4.3) such as the nucleus spectra. The site nucleation rate is
defined in Sect. 4.7.2.
The singular description (Sect. 4.7.1) is an approximate solution in which
the site nucleation rate is assumed to be a step function from 0 to ∞
at Tc.
The site nucleation rate function may depend on the value of Tc
and is thus designated as Jsite,Tc(T). The
characteristic temperature Tc for each site (Sect. 4.3) is
defined by the temperature at which the nucleation rate Jsite
has an arbitrarily chosen value C:
Jsite,Tc(Tc)=C. A convenient choice is C=1 s-1. Empirical evidence points to Jsite being a
steep function of temperature (similarly to homogeneous nucleation).
Empirical determination of Jsite has only been approached
indirectly (e.g., Vali, 2008; Wright and Petters, 2013) since multiple
examples of sites with the same Tc are not identifiable a priori.
Theoretical guidance is limited by the lack of detailed knowledge about the
nature of ice nucleating sites.
Stochastic description/model
Description/model of the frequency of nucleation events in a population of sites, or of sample units, which have equal probability for nucleation within a period of time
Stochastic description
This description assumes that there are large numbers of sites of equal
effectiveness on the surfaces of INPs, and interprets observations in terms
of a nucleation rate coefficient, i.e., the freezing rate per unit surface
area or per unit mass. Thus, this description employs Js(T) with
units of cm-2s-1, and Jm(T) with units of
g-1s-1. Empirical values of Js(T) are obtained
from R(T) via Js(T)=R(T)A. If the number of sites of
the same effectiveness per unit surface area is nS(T) then
Js(T)=ns(T)⋅Jsite(T), where
Jsite(T) is the site nucleation rate (probability of
nucleation) for given parameters. From these values of Js(T), the
site density ns can be derived if J(T) is taken from theory
(e.g., CNT) or is independently determined.
As mentioned in Sect. 4.7.2, the heterogeneous nucleation rate coefficient
can be applied to cases where all INP surface areas appear to nucleate ice
with equal effectiveness. In other cases the site nucleation rate
Jsite should be used.
Stochastic description for multi-component systems
The foregoing description (Sect. 4.8.1) is valid when all sites are
considered identical. This case is termed the single component model in
Broadley et al. (2012) and Herbert et al. (2014). Sites of different
effectiveness are considered in the multi-component stochastic models
(MCSMs). The essence of this approach is to allow for different site
characteristics by varying critical parameters (usually the contact angle) in
the CNT formulation of the nucleation rate coefficient and assuming some
frequency distribution for that parameter. Variations of this approach have
been presented by Marcolli et al. (2007), Niedermeier et al. (2011, 2014),
Welti et al. (2012) and Ekman (2015) among others. In these formulations,
sites are characterized by their frequency, say ns,i, and a
corresponding function Js,i, using the second subscript, i,
to indicate assignment to a specific value of the contact angle (or other
parameter). In effect, this is very similar to the time-dependent
site-specific description (Sect. 4.7.3) with a theory-based nucleation rate
coefficient instead of Jsite and an assumed, or fitted,
frequency distribution instead of an empirical one (Sect. 4.3).
Comparison of stochastic and site-specific descriptions
While the CNT-derived nucleation rate coefficient in the stochastic
description (Sect. 4.8.1) and the site nucleation rate in the site-specific
description (Sect. 4.7.3) arise from the same need to describe the
probability of nucleation, different underlying assumptions are incorporated
in these two descriptions. To apply the stochastic description to a given
data set, the function Js is assumed to have finite values over the range of observed
freezing temperatures in the sample. For the site-specific description, the
function Jsite is assumed to rise very rapidly over a narrow
range of temperatures and the spread in observed freezing temperatures is
ascribed to differences in the effectiveness of sites. The two descriptions
lead to divergent predictions about the time dependence of nucleation (Vali,
2014; Herbert et al., 2014). This time dependence can not be determined from
a single continuously cooled experiment with a sample, but requires more
elaborate tests.
The multi-component (and similar) descriptions (MCSM; Sect. 4.8.2) present
a view similar to the time-dependent site-specific description (VS66;
Sect. 4.7.3). The degree of similarity is determined by the range of
temperatures over which the nucleation rate coefficients Js,i(T) for
species i assume empirically relevant values in the MCSMs in comparison
with the range of <1 ∘C for the site nucleation rate
Jsite(T) in VS66.
Aqueous solutions
Dissolved substances in water change the equilibrium
phase boundaries and influence ice nucleation.
In many atmospheric and other natural systems, dissolved materials are
present in water and alter the conditions for ice nucleation. The magnitudes
of the changes in freezing rates depend on the type and concentration of the
solute. Water activity has been shown to provide a good representation of
these dependences for homogeneous freezing nucleation and several
heterogeneous systems (Koop et al., 2000; Knopf and Alpert, 2013).
Modifications of INP surfaces by some solutes may introduce additional
changes in heterogeneous freezing nucleation rates Js or
Jsite (e.g., Reischel and Vali, 1975; Wex et al., 2014).
Pre-activation and memory effects
The mode or efficacy of observed nucleation may be influenced or altered by
the previous temperature/humidity
history of the INP, or equivalent.
Experiments have shown that prior exposure to low temperature or high
humidity, or a combination of both, leads to enhanced activity in comparison
to what the INP or equivalent would exhibit otherwise. Such effects may
introduce ambiguity in the diagnosis of the mode of activity (Sect. 4.4) in
laboratory experiments or in atmospheric or other natural systems. Certain
INP characteristics (composition, configuration, surface properties) may
favor such effects. Explanations of the effects focus on the potential for
cracks, pores and other features on surfaces to retain ice even under
conditions where bulk ice would be unstable.