Nucleation of supercooled and superheated water in nature
Teemu Hölttä1, Timo Vesala2, Eero Nikinmaa1, Ari Laaksonen3, Markku Kulmala2
1
Department of Forest Sciences, University of Helsinki, Finland
2
Department of Physics, University of Helsinki, Finland
3
Finnish Meteorological Institute, Finland
Keywords: Heterogeneous nucleation, Cavitation, Boiling, Freezing, Surface energy, Xylem,
Methane
INTRODUCTION
A metastable state is a condition of a phase in which a disturbance will cause the appearance of a
new phase and the system will fall to a lower energy level. A liquid is supercooled when its
temperature is below normal freezing point (0 oC at atmospheric pressure for bulk pure water)
and superheated when its pressure is lower than its saturation vapour pressure. In these
circumstances the solid or vapour phases, respectively, would yield a lower energy state, but
before a phase change can occur, an energy barrier due to surface energy between the two phases
has to be overcome by nucleation. Nucleation can either be homogeneous where the energy
barriers to nucleation is overcome by random thermodynamical fluctuations in the bulk liquid
itself, or heterogeneous where impurities suspended in the liquid or irregularities in the walls in
contact with liquid can catalyze a phase change by lowering the surface energy barrier to
nucleation. According to the classical nucleation theory water can be superheated to above 3000 C
at atmospheric pressure before homogeneous boiling occurs, supercooled to nearly -40 oC at
atmospheric pressure before homogeneous freezing occurs, or its pressure can be decreased down
to approximately -150 MPa at ambient temperatures before homogeneous cavitation occurs.
Experimental evidence also supports the existence of such highly metastable states. However,
degrees of such high metastability are rarely reached in nature or in industrial applications, so
heterogeneous nucleation is typically the main driver for phase change. Here we describe the
nucleation process of supercooled and superheated liquid water, and take as examples the
cavitation of superheated and freezing of supercooled water in plants, and in methane bubble
formation in supersaturated solutions in the soil.
Nearly all terrestrial plants rely on metastable water for their water and nutrient supply. Large
trees can evaporate many hundreds of litres of water to the atmosphere in a day through small
pores in their leaves, the stomata. Evaporation of such large quantities of water is inevetiable as
CO2 assimilation for photosynthesis occurs through these same pores. Water lost by evaporation
in the leaves is replaced constantly by water being pulled from the soil in continuous water
columns in the xylem, the water transport tissue of plants, which consists mainly of dead hollow
cells specialised in transporting water. Water pressure in the xylem can decrease down –10 MPa
(Nobel 1991) so cavitation of the water columns is a common and frequent occurrence for plant.
Excessive cavitation during drought is one of the most important causes of declined plant
productivity worldwide. In general, the capability of the xylem to maintain water in the liquid
state is thought to be one of the main constraints on tree function and growth (Tyree and Sperry
1988, Koch et al. 2004).
Also the ability of plants to maintain liquid water below the equilibrium freezing point is
essential for plant survival. Water in the xylem below the equilibrium freezing point is especially
unique in the sense that it is metastable to two directions: it is simultaneously supercooled and
superheated, and hence called subtriple (Debenedetti 1996). Plants avoid freezing damage by two
strategies: freezing avoidance and freezing tolerance. Some winter hardy plants can avoid
freezing down to the homogeneous nucleation limit if impurities which can act as nuclei for
freezing are not present. Another strategy is to direct freezing to extracellular spaces. As the
equilibrium water vapour pressure of ice is lower than that of liquid below the equilibrium
freezing temperature, water is drawn from living cells to the extracellular spaces. The living cells
are dehydrated and thus become concentrated with solutes which lower the equilibrium freezing
temperature. But on the other hand, the dehydration causes pressure to decrease, and the
probability for cavition and other damage due to drying increses.
A major proportion of earth’s methane emissions to the atmosphere are released from northern
peatlands where methane is formed under anaerobic conditions in the soil. Methane is released to
the atmosphere by diffusion through the soil and aeronchyma tissue of plants, but at least the
former mechanism is probably rather minor in importance as diffusion in the wet peat soil is so
slow that methane is oxidised on its way up. Another important, and less understood, mechanism
for methane release to the atmosphere is the nucleation and the subsequent growth and movement
of methane bubbles to the surface. Once the soil water reaches a high enough level of
supersaturation with regards to methane and other dissolved gasses such as carbon dioxide and
air, nucleation of bubbles occurs, and high and short-lived burst of methane emissions are
observed for example in relation to decreasing ambient air pressure (Tokida et al. 2007).
MATERIALS AND METHODS
Homogeneous nucleation
Change in the Gibbs free energy ( G) of a spherical nucleus in a phase change from liquid to
either solid or gas is (Pruppacher and Klett 1997)
G
4 3
r Gv
3
4 r2
(1)
where r is radius of the nucleus of the new phase, Gv is difference in the volume free energy
between the two phases, and is surface energy between the two phases. The first term on the
right side is the bulk energy term and the second is the surface energy term.
Bubble formation
For a phase change from liquid to gas, the bulk term is (Brennen 1996)
N
Gv
P
PL - Pg
PL
Pv (T )
C i H i (T )
(2)
i
where P is the difference in pressure, i.e., tension, between the liquid and gas nuclei, PL is liquid
phase pressure, Pg is gas phase pressure, which is a sum of the saturation vapor pressure Pv of
water and the partial pressure of all of the gasses dissolved in the liquid, which can be expressed
as a function of their concentrations in the liquid phases Ci and their Henry’s laws coefficients.
Both the saturation vapor pressure of water the Henry’s laws constants are a function of
temperature. Here it is assumed that the nucleus is long lived enough for the dissolved gases to
diffuse into the gas nuclei at their equilibrium pressures. It is also assumed that the different
gasses do not interact with each other.
Differentiating eq. (1) with respect to r gives the critical radius of the nuclei r*
rbubble
*
2
(3)
P
The critical radius denotes an unstable equilibrium state where bubbles smaller than this critical
size will dissolve due to surface tension and larger bubble will grow.
Freezing
For freezing (Pruppacher and Klett 1997)
L T
TE
Gv
(4)
where L is latent heat of fusion, TE is equilibrium freezing temperature,
temperature (T-TE). The critical size nucleus is then
r freeze
*
T is supercooling
2 TE
L T
(5)
Nucleation rate and nucleation probability
Homogeneous nucleation rate J is the rate at which nuclei larger than r * are formed spontaneously
by random thermodynamic fluctuations in the bulk of the liquid and it is written (Pruppacher and
Klett 1997)
J hom
J 0 exp
G (r * )
kT
(6)
where k is the Bolzmann constant and J0 is a kinetic prefactor. The kinetic prefactor is
proportional to molecular density of water (Debenedetti 1996).
Heterogeneous nucleation
For heterogeneous nucleation the size of the critical nucleus r * is the same as for homogeneous
nucleation, but the energy barrier for nucleation is lowered by a factor of (1-f), which is
dependent on geometry and surface energy between all the phases involved (Debenedetti 1996).
J het
J hom
a
exp
N 1/ 3
G (r * )
1
kT
f
(7)
a is surface area available for heterogeneous nucleation per unit bulk volume of liquid.
The factor f can be estimated from the interfacial surface energies between solid surface and the
phases concerned (contact angle), plus geometrical considerations. For heterogeneous nucleation
on a smooth surface f is (Debenedetti 1996)
f smooth
where
1 cos
2
2 cos
4
(8)
is contact angle between the solid surface and the nucleating phase as defined in fig. 1.
Fig 1. Contact angle is a function of the interfacial energies between all of the phases concerned.
Calculations of f which are used in the following calculations for a spherical seed particle
(concave surface), and convex surface are from Winkler et al. (2008) and Pruppacher and Klett
(1997).
Nucleation probability
The probability that heterogeneous nucleation will occur in a volume V within a time t is
(Lazaridis 1992)
Pnucl
1 exp
J het Vat
(9)
Non-classical nucleation in the case of bubble formation
In practise, the gas nuclei above the critical size (eq. 3) are often not created by thermodynamic
fluctuations, but they already exist in the liquid suspended in impurities or solids confining the
liquid (Jones et al. 1991, Brennen 1996). Once the liquid pressure falls or temperature rises, the
gas nuclei are seeded into the liquid and induce cavitation or boiling. The general condition for
the “non-classical nucleation” (Jones et al. 1991) of such gas nuclei from pores is
P
2 cos
rbubble
*
(10)
The above condition will be more complex when the real geometry of the pore is accounted for.
RESULTS AND DISCUSSION
Bubble formation by nucleation (Fig 2a) is strongly aided by solid hydrophobic surfaces (the
adhesive forces between water molecules and the solid surface are weaker than cohesive forces
between water molecules), and freezing nucleation (Fig. 2b) by solid surfaces in which the ice
molecules experience a stronger force towards the solid surfaces in comparison to other ice
molecules. In addition, convex seed particles induce both bubble formation and freezing at lower
levels of metastability in comparison to smooth, and especially to concave surfaces found on
surface of spherical particles. For the methane bubble formation case, the tension between the gas
and liquid phase of the nuclei is caused mainly by the increase in the partial pressures of methane
and other gases in the bubble nuclei (the last term in Eq. 2), whereas nucleation by cavitation in
the xylem is primarily induced by the decrease in the liquid pressure (first term in Eq. 2). For the
case of boiling of relatively pure water, the tension would be induced by an increase in the
saturation vapor pressure (second term in Eq. 2) due to temperature increase.
Fig. 2. Contact angle between the nucleated phase and solid seed particle where a) freezing by
heterogeneous nucleation and b) cavitation by heterogeneous nucleation becomes probable (P=0.5) in one
second in a cell of 20µm in radius and 1 cm in length. For bubble formation (a) we used the kinetic
prefactor J0 defined in Blander and Katz (1975) and for freezing (b) the one given in Turnbull and Fisher
(1949). For both cases the surface area available for nucleation was the cell wall area.
In plants, true heterogeneous nucleation is likely to play a minor role in cavitation induction,
whereas the “non-classical” nucleation is likely to be more important based on both theoretical
considerations and experimental manipulations (e.g. Tyree & Sperry 1988, Cochard et al. 2009).
The xylem always contains air filled conduits, and in practise cavitation in thought to spread from
one xylem conduit to another when the condition in Eq. (10) is fulfilled in a process termed airseeding. Heterogeneous nucleation at the tensions where cavitation is typically experienced (less
than 5 MPa) would require extremely hydrophobic surfaces which are unlikely to be found in the
xylem. On the other hand, the pore sizes separating air and water filled conduits are known to be
a few nanometers in radius so air-seeding according to Eq. (10) will induce cavitation at tensions
typically experienced in plants. The nuclei inducing freezing in plants could vary much in their
composition as the heterogeneous freezing temperature in plants can vary anywhere from close to
00C to -400C and only little is known about them. However, for freezing occurring near zero and
inducing most freezing damage in agricultural plants, ice nucleation inductive bacteria are
believed to seed nucleation most commonly. The same bacterial species are also found in the
upper atmosphere where they nucleate cloud formation.
The exact mechanism and seed nuclei for methane bubble formation in soils remains unknown. It
has also been observed that the soil water need not to be even saturated with respect to methane
in order for bubble formation to occur (Baird et al. 2004). Nevertheless, the sum of the partial
pressures of all of gases in the bubble nuclei need to be in excess of the liquid pressure, or surface
tension will crush the gas nuclei. Even supersaturations as high as 10 (corresponding to a tension
of 1 MPa) need very hydrophopic surfaces with a contact angle approaching 1800 for nucleation
to occur within the surface geometries considered here (see fig. 2b). Therefore methane bubble
nucleation in the “non-classical” sense from pre-existing gas pockets in the peat soil crevices
should also be considered as a possible mechanism initiating methane bubbles. Preliminary
calculations demonstrate that the dynamics of subsequent buble growth and bubble rise in the
peat soil matrix are also likely to have high significance in determining the relative importance,
timing and environmental variability in wetland methane emissions by bubbles.
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