GEOPHYSICAL RESEARCH LETTERS, VOL. 31, L19102, doi:10.1029/2004GL020483, 2004
A new look at homogeneous freezing of water
M. B. Baker
Departments of Atmospheric Sciences and Earth and Space Sciences, University of Washington, Seattle, Washington, USA
M. Baker
Department of Physics, University of Washington, Seattle, Washington, USA
Received 9 May 2004; revised 1 August 2004; accepted 9 August 2004; published 7 October 2004.
[1] Despite its atmospheric importance, homogeneous
freezing of aqueous drops is poorly understood. Here we
provide evidence that at atmospheric pressures the
conditions leading to the initiation of freezing in pure
water are those for which the liquid compressibility and the
corresponding density fluctuations reach maxima. This
liquid-only criterion for the onset of freezing contrasts
with the traditional view that freezing temperatures depend
sensitively on the parameters of ice and the ice-water
interface. We generalize the connection between
compressibility maxima and freezing to aqueous solutions.
Utilizing compressibility data, we predict solution freezing
temperatures from those of pure water under pressure.
These predictions are almost coincident with the
parameterizations of droplet freezing temperatures in
terms of water activity [Koop et al., 2000] used to predict
ice formation in cirrus clouds. Our work then provides a
physical basis for this link between water activity in
INDEX
solution and homogeneous freezing temperatures.
TERMS: 0305 Atmospheric Composition and Structure: Aerosols
and particles (0345, 4801); 0320 Atmospheric Composition and
Structure: Cloud physics and chemistry; 0399 Atmospheric
Composition and Structure: General or miscellaneous.
Citation: Baker, M. B., and M. Baker (2004), A new look at
homogeneous freezing of water, Geophys. Res. Lett., 31,
L19102, doi:10.1029/2004GL020483.
1. Introduction
[2] In a thorough reexamination of measured homogeneous freezing temperatures of small aqueous solution
drops, Koop et al. [2000] found that for a wide range of
solutes and solute concentrations at ambient pressure p0 =
1 bar, the freezing temperatures Tf depend only on the water
activity aw of the solution. (For a solution of Ns solute
molecules at temperature T and pressure p, aw = exp[(mw(Ns,
p, T) mw(0, p, T))/kT], where mw is the chemical potential
of the water.) This finding provides a very practical way to
predict droplet freezing temperatures in the atmosphere,
and, since water activity is an equilibrium property of the
liquid solution, it suggests that the temperature of onset of
homogeneous freezing of aqueous solutions is a property of
the liquid phase alone. This poses the question of what
determines Tf(aw), the solution freezing curve at ambient
pressure.
[3] Earlier experiments on alkali halide solutions [Kanno
and Angell, 1977] had shown that the addition of solutes
Copyright 2004 by the American Geophysical Union.
0094-8276/04/2004GL020483$05.00
and applied pressure have similar impacts on homogeneous
freezing temperatures of water. Thus, to understand the
solution freezing curve, we take a new look at homogeneous freezing of pure water. The dependence of solution
freezing temperatures on an equilibrium liquid property
suggests that the freezing curve Tf(p) of pure water under
applied pressure Dp = p p0 is also a property of the liquid
phase alone.
[4] An isolated drop of pure water can remain liquid
while cooled (‘supercooled’) below its melting temperature
Tm to its homogeneous freezing temperature Tf, where the
freezing rate increases sharply. For pressures p 2.1 kbar
the achievable supercooling Tm Tf is about 70C [Kanno
et al., 1975], comparable with that in other liquids [Angell,
1982], whereas for p 1 kbar, the achievable supercooling
in water is substantially smaller than that in most simple
liquids. These anomalously high freezing temperatures were
initially explained [Speedy and Angell, 1976; Kanno and
Angell, 1979] by proximity of the freezing curve Tf(p) to a
postulated curve limiting the region of mechanical stability
of water, in which the isothermal compressibility is positive.
This limit was introduced to explain observed rapid
increases, with decreasing temperature, in the magnitudes
of the thermodynamic response functions (the isothermal
compressibility k @lnV/@pjT, the isobaric expansivity
a @lnV/@Tjp and the isobaric specific heat Cp). These
response functions determine the magnitudes of the associated equilibrium (thermal) fluctuations in volume and
entropy about their equilibrium values, h(DV)2i = kTkhVi;
h(DS)2i = Cpk, and their correlation hDSDVi = kTahVi. Later
theoretical studies [Poole et al., 1992; Sciortino et al., 1997;
Mishima and Stanley, 1998] cast doubt on the stability limit
hypothesis, however, leaving open the question of what
determines the freezing curve Tf (p) in pure water.
[5] In view of the above observations we seek a unified
explanation of both freezing curves (Tf (p) and Tf (aw)) in
terms of liquid only phenomena with no reference to the
parameters of the ice/water interface or of bulk ice. We ask
what is special about supercooled water itself in the vicinity
of these freezing curves.
2. Homogeneous Freezing of Pressurized
Pure Water
[6] In order to gain insight into the special features of
supercooled water and solutions near the freezing curve we
first consider a simple microphysical model representation
of pure water [Truskett et al., 1999] (hereafter, the TDST
model). The model includes hard-sphere and background
attractive interactions, as well as an orientation-dependent
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BAKER AND BAKER: HOMOGENEOUS FREEZING
Figure 1. Response functions of pure water calculated
from TDST model [Truskett et al., 1999]. (a) Isothermal
compressibility k [105/bar], (b) isobaric specific heat Cp
[kJ/mol-K], and (c) isobaric expansivity a[103/K].
attractive interaction representing hydrogen bonds. It permits only one bond per molecule but, in analogy with real
water, formation of a H-bond between two molecules is
possible only for a narrow range of intermolecular distances
and relative orientations of the bonding pair. ‘Strong’
H-bonds (strength 23 kJ/mole) form in low density
environments. The H-bond strength is decreased with
increasing local density. The ‘weak’ bonds represent
many-body effects on hydrogen bonding. In the model,
water is liquid above and below the homogeneous freezing
temperature and the thermodynamic functions can be calculated analytically. There are three adjustable model parameters describing the H-bond geometry; we use the set
identified by TDST that provides the best fit to thermodynamic data at T > 235K. Despite its simplicity, this analytic
model provides a useful first examination of the thermodynamic properties of water at lower temperatures as well.
[ 7 ] We have calculated the water density and the
response functions k, a, and Cp as functions of temperature
and pressure over the range 180 K T 280K, p0 = 1 bar p 2 kbar. In this range, according to the model, approximately 20– 50% of the water molecules are involved in
H-bonds, this fraction increasing as the temperature and
pressure decrease. The results, displayed in Figure 1, show
that the calculated k, a and Cp all exhibit extrema (maxima
or minima in both temperature and in pressure) along
curves from T = 235 K, p = p0 to lower temperatures and
higher pressures. At temperatures and/or pressures higher
than those at the extrema (e.g., higher than those at the
compressibility maxima in Figure 1a), most of the H-bond
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energy is contributed by the weak bonds. The strong
H-bonds begin to compete with the weak bonds as the
temperature and/or pressure is lowered toward the loci of
extrema. The loci of extrema thus reflect changes in the
character of the H-bonding.
[8] Figure 2 shows that the calculated positions of the
extrema in k lie very close to the observed freezing curve for
pressures p0 p 1.1 kbar; the extrema of a and Cp lie a few
degrees lower. At ambient pressure the compressibility
maximum occurs at T = 235K, and at p = 1 kbar the maximum
occurs at T = 218K. Both values are equal, within 1C, to the
measured freezing temperatures at those pressures.
[9] The line of compressibility maxima reproduces that
found in molecular dynamics simulations [Sciortino et al.,
1997] of supercooled water, where it marks a change in
local structure of the liquid. The near coincidence of
compressibility maxima with the observed freezing curve
for moderate pressures p 1 kbar in both the analytic
model and in the simulations suggests that water at these
pressures is a substance for which homogeneous freezing
temperatures can be approximately determined without
calculating the freezing rate; the criterion for the onset of
freezing is that the liquid compressibility reach a maximum.
[10] This criterion complements determinations of freezing temperatures based on the freezing rate. According to
classical nucleation theory [Pruppacher and Klett, 1997],
the rate is proportional to the probability of forming an ice
embryo of a temperature dependent critical size. This
probability depends sensitively on the properties of bulk
ice and the ice-water interface. The liquid-only criterion for
the initiation of freezing could then represent a reduction in
the barrier to nucleation near the compressibility maximum,
where there is a change in the H-bond structure of the
metastable liquid. Here we present a simple heuristic picture
showing how this reduction might come about.
[11] As the temperature decreases toward Tf (p) water
density fluctuations rise, so that it is increasingly likely to
find regions in which the density of the metastable water
approaches that of ice. At p = p0, T = 235K the volume of a
critical nucleus Vc 4 1027[m3] [Pruppacher and
Klett, 1997]. Using the value k (1 2) 104 [bar1]
(obtained from the TDST model or the simulations) gives
(hDVi)0.5/Vc 0.03, comparable to the relative difference
in ice and water molecular volumes, (vi vw)/vw 0.05.
Figure 2. Measured freezing temperatures and loci of
extrema in the calculated response functions for pure
water. Dots: measured homogeneous freezing temperatures
[Kanno et al., 1975]. Positions of extrema in k shown by
thick solid line: extrema in Cp, thin line: and extrema in a,
dashed line.
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BAKER AND BAKER: HOMOGENEOUS FREEZING
Figure 3. Measured compressibilities and freezing temperatures. (a) Measured isothermal compressibilities at
temperature T = 298K [Rogers and Pitzer, 1982]. Large
light blue dots: k[105 bar1] for pure water under applied
pressure Dp. Small red dots: ksoln[105 bar1] for NaCl
solutions of concentrations up to 5 M, plotted against 10 ,
where (aw, T) (1/vw(T))kTln(aw) is approximate
osmotic pressure. (See text.) (b) Freezing temperatures of
solutions at ambient pressure vs water activity aw. Dots: data
from a wide range of solutes and concentrations including
strongly dissociating acids and salts and weakly dissociating
organics [Koop et al., 2000]. Dashed red line: prediction
Tf (aw) = Tf (p), for Dp = 10 (aw, Tf). Thin black line
(almost coincident with red line): curve used as basis for
parameterization of freezing temperatures in cloud models
[Haag et al., 2003; Kärcher and Lohmann, 2002].
Thus there are regions of size comparable to Vc in which the
metastable water has ice-like density. When embryos form
in these regions, the decreased density difference between
embryos and surrounding liquid facilitates embryo growth
to volume V Vc and freezing is initiated. (In solutions the
onset of freezing is most probable in those parts of the fluid
that are relatively little affected by the solute particles [Omta
et al., 2003], when density fluctuations produce a sufficiently large volume of low density, solute free water.)
3. Homogeneous Freezing of Aqueous Solutions
[12] We now apply these ideas to the initiation of freezing
in aqueous solutions, guided by the fact that solutes and
applied pressure can have similar impacts on H-bond networks [Leberman and Soper, 1995] and on compressibility
[Tammann, 1923] in water.
[13] At fixed temperature the compressibility of pure
water, k, increases as pressure is reduced toward the
freezing curve (Figure 1a). Analogously, at fixed temperature the solution compressibility ksoln should increase as
solute concentration is lowered, reaching a maximum where
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there is a change in the character of the H-bonding. The
value of the concentration at which this maximum occurs
depends on the response of the water to the particular solute.
We assume that the solution freezes at that temperature that
maximizes ksoln.
[14] To predict the freezing temperatures of solutions, we
compare the dependence of ksoln on solute content to the
dependence of k on pressure. We express the solute content
in terms of a pressure scale , the approximate osmotic
pressure. For a solution of activity aw at temperature T and
ambient pressure, (aw, T) (1/vw(T))kT ln (aw), where
vw(T) is the molecular volume of water. In Figure 3a we
show measured room temperature compressibilities of NaCl
solutions and of pure water under applied pressure up to
1 kbar at T = 25C [Rogers and Pitzer, 1982]. (A 2 M
solution corresponds to 100 bar.) Figure 3a shows that
in the parameter ranges where the data overlap dksoln/d 10dk/dDp. That is, adding solute to pure water to increase
the osmotic pressure from 0 to produces the same change
in the water compressibility as does the application of
pressure Dp 10 (aw, T).
[15] We assume that the approximate relationship between dksoln/d and dk/dDp is maintained as the samples
are cooled, so that if k(Dp, T) is maximized at T = Tf, then
ksoln( 0.1 Dp, T) also reaches a maximum at T = Tf,
where both samples will freeze. Hence
Tf ðaw Þ Tf Dp ¼ 10 aw ; Tf
:
ð1Þ
[16] Thus we can determine the freezing temperature Tf at
ambient pressure of an NaCl solution of water activity aw
(having osmotic pressure (aw, Tf)) by setting it equal to the
measured freezing temperature of pure water under applied
pressure Dp = 10 (aw, Tf). To the extent that the response
of ksoln to an increase of is similar for other solutions, we
can use equation (1) to predict their freezing temperatures as
well. These predictions are presented in Figure 3b and
compared with the measured solution droplet freezing
temperatures discussed by [Koop et al., 2000]. The resulting
freezing curve is almost indistinguishable from the empirical fit of these droplet freezing temperatures to measured
water activities presented in that paper. The link between
compressibility maxima and the onset of freezing appears to
hold for all these solution droplets as well as for pure water
under pressure, so that we have a unified explanation of
both freezing curves (Tf ( p) and Tf (aw)) in terms of bulk
liquid properties.
4. Discussion
[17] The parameterization of droplet freezing temperatures in terms of water activity [Koop et al., 2000] is used to
predict homogeneous nucleation of ice in the atmosphere
[e.g., Haag et al., 2003; Kärcher and Lohmann, 2002], a
process of importance to climate and radiative balance
[Baker, 1997; Intergovernmental Panel on Climate Change,
2001]. Our work suggests the physical basis for this water
activity criterion is that the value of the water activity at the
freezing temperature identifies the position of the compressibility maximum in supercooled solutions. The compressibility reflects features of the H-bond networks that control
the nucleation barrier.
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BAKER AND BAKER: HOMOGENEOUS FREEZING
[18] Figure 3b shows there is some spread in Tf (aw)
values for the data set used by [Koop et al., 2000].
Moreover, recent laboratory measurements [e.g., Cziczo et
al., 2004, and references therein] of Tf for aerosol droplets
show some larger variations, particularly at low activities
and for some solute species. While these variations may in
part be due to differences in laboratory technique and in
kinetic factors, they may also be useful as tests and
extension of our ideas. The scale factor 10 in equation (1)
was obtained from NaCl solution data at room temperature.
Small variations in this factor with temperature and/or
composition would, according to our ideas, correspond to
some variation in freezing temperatures for different solutes
at a given water activity. Extension of our analysis to a
range of solutions and temperatures would allow further
tests of the link between compressibility maxima and
freezing temperatures.
[19] The ideas presented here can also be tested by and
applied to other issues of current interest related to supercooled water and droplet freezing, as follows.
4.1. Phase Diagram of Water
[20] One of the obstacles to further theoretical exploration of the homogeneous freezing criterion is uncertainty in
the underlying phase diagram of low temperature liquid
water. Several molecular dynamics simulations yield first
order phase transitions between liquid phases of different
densities in supercooled water [e.g., Poole et al., 1992;
Tanaka, 1996; Sciortino et al., 1997; Brovchenko et al.,
2003]. Each liquid-liquid coexistence curve ends at a critical
point, where the compressibility and density fluctuations
become infinite and above which the two metastable phases
of liquid water become indistinguishable. The simulations
differ significantly in the predicted locations and numbers of
these liquid-liquid critical points. We suggest that an additional constraint be placed on input parameters for the
simulations; namely, that the calculated locus of maxima
in k be coincident with the observed homogeneous freezing
curve Tf (p).
4.2. Size Effects on Freezing
[21] Homogeneous freezing temperatures depend on
sample size, even if the sample surface has no impact on
individual ice nucleation events within it. Moreover, it has
recently been suggested [Tabazadeh et al., 2002] that
freezing of small aqueous droplets might initiate near the
droplet surface. If so, this could partially explain the spread
in measured Tf values for aerosol particles. According to our
picture, enhanced density fluctuations near the surface
might contribute to reduction of the nucleation barrier there
relative to that in the bulk. Examination of the spatial
distribution of density fluctuations inside small droplets
could then shed light on the possible role of the droplet
surface in the onset of freezing and hence on freezing rates
in atmospheric clouds.
[22]
Acknowledgments. We gratefully acknowledge useful discussions with V. Tsemekhman, who first noticed the interesting results of the
TDST model calculations, and with P. Debenedetti, J. Nelson, and S. Wood,
who made very helpful comments on the manuscript. This work was
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motivated by a stimulating visit to the Atmospheric Physics Group
at ETH, Zurich, and we thank the institute for that opportunity. We are
deeply grateful to Th. Peter for the many interesting scientific ideas and
opportunities he has provided us, and to T. Koop, for his help in
formulating this project and in supplying the data in Figure 3b. Supported
by NSF ATM-02-11247 (M. B. B.)
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M. Baker, Department of Physics, University of Washington, Seattle,
WA, USA.
M. B. Baker, Departments of Atmospheric Sciences and Earth and Space
Sciences, University of Washington, Seattle, WA 98195-1310, USA.
([email protected])
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