1. Phase diagram of water
This line, for the hypothetical vapor pressure em of the melt water in equilibrium with ice, is slightly sloped to the left
Need to blow up this range to see vapor pressure differences -- important

2. hexagonal ice cubic ice
Water does not behave as a typical low-molecular-weight compound
Virtually all ice in the biosphere is ice Ih, with the exception only of a small amount of ice Ic which is occasionally present in the upper atmosphere
hexagonal ice
cubic ice
Vitreous [amorphous] ice
(glass / viscous liquid)
below 138 K
Cubic ice (Ic) is metastable with respect to hexagonal ice (Ih)

3. Aside: Cubic ice
The formation of cubic ice under conditions relevant to Earth's atmosphere Benjamin J. Murray, Daniel A. Knopf & Allan K. Bertram, Nature 434, 2005 An important mechanism for ice cloud formation in the Earth's atmosphere is homogeneous nucleation of ice in aqueous droplets, and this process is generally assumed to produce hexagonal ice1, 2. However, there are some reports that the metastable crystalline phase of ice, cubic ice, may form in the Earth's atmosphere3, 4, 5. Here we present laboratory experiments demonstrating that cubic ice forms when micrometre-sized droplets of pure water and aqueous solutions freeze homogeneously at cooling rates approaching those found in the atmosphere. We find that the formation of cubic ice is dominant when droplets freeze at temperatures below 190 K, which is in the temperature range relevant for polar stratospheric clouds and clouds in the tropical tropopause region. These results, together with heat transfer calculations, suggest that cubic ice will form in the Earth’s atmosphere. If there were a significant fraction of cubic ice in some cold clouds this could increase their water vapour pressure, and modify their microphysics and ice particle size distributions5. Under specific conditions this may lead to enhanced dehydration of the tropopause region5.

4. What about for transition liquid  ice?
Phase diagram of water
Recall for vapor  liquid phase change (nucleation of liquid droplets from the vapor), we had to supersaturate significantly to support homogeneous nucleation; also had to supersaturate for heterogeneous nucleation onto CCN
What about for transition liquid  ice?
It also seems possible to have a transition vapor  ice: does this happen?

5. Pathways to ice formation (pure water only)
Homogeneous nucleation of ice by sublimation
In this case, small ice crystals form by chance aggregation of vapor molecules. Molecules aggregate to form an ice-like embryo, which under appropriate conditions, remains stable and continue to grow into a small ice crystal.
Homogeneous nucleation of ice by freezing
At relatively small supercoolings, drop is observed not to freeze. As drop continues to cool, small ice-like embryos begin to form in the droplet associated with the chance aggregation of liquid molecules. At sufficiently low temperatures the ice-like embryo stays intact and the droplet is observed to freeze. Freezing of pure water occurs at –40°C, although there is some uncertainty in this value
Direct pathway is less likely because it requires molecules to go from disordered state  highly-ordered state
“Ostwald’s rule of stages” suggests that it’s easier to form intermediate (liquid) and then to form the solid
consider the formation of ice from supercooled water:
Supercooling DTs ≡ T0 − T serves as the thermodynamic driver for the phase change because of its direct relationship to chemical potential:

6. Hexagonal form clearly indicated.
Pruppacher and Klett 1997
Hexagonal form clearly indicated.
Note that the oxygen atoms are alternatively raised and lowered in a puckered manner.
c-axis is perpendicular to hexagonal plane
(perpendicular to basal face)
a-axis is parallel to hexagonal plane
(perpendicular to prism face)
basal face
c-axis
prism face
a-axis

7. Ih has a regular hexagonal structure:
black sphere – oxygen
open circles - hydrogen
Structure of ice Ih
Fletcher 1962
Ih has a regular hexagonal structure:
Each oxygen atom is surrounded tetrahedrally by four other oxygen atoms
The tetrahedrons are joined together by hydrogen bonding to form a hexagonal structure
Each oxygen has 2 hydrogen atoms as its nearest neighbors (H2O)

8. Homogeneous nucleation of ice from supercooled water
ice embryo formation in liquid phase
structure of ice-like embryo is not spherical; rather a hexagonal structure is more appropriate r
ice-like embryo
radius of inscribed sphere
Volume of embryo
Area of embryo
α and β are geometric factors to account for departure of embryo from sphericity
energy change associated with embryo formation (in supercooled liquid)
Our earlier knowledge suggests we include a thermodynamic and surface energy term,
chemical potential of liquid
# molecules/unit volume
chemical potential of solid
surface tension (liquid-solid interface)

9. This equation is usually rewritten as,
pL = saturation vapor pressure over liquid at temperature T pS = saturation vapor pressure over ice at temperature T Bulk thermodynamic term can be written as,
Therefore,
This equation is usually rewritten as,
Hobbs, 1978
Physics of Ice
i = # of molecules in the ice embryo
A0 is proportional to surface free energy of ice-water interface
B0 is proportional to bulk free energy difference between supercooled water and ice

10. >
Can chance aggregation deliver an embryo i* - at what temperature?
Hobbs 1978
embryo stability
Free energy required for nucleation initially increases sharply with increasing i, but after the embryo has achieved certain size (i*), energy associated with embryo growth (formation) decreases.

11. Alternative expression (Lamb & Verlinde)
Role of supercooling is easier to see in this form

12. Deriving the homogeneous freezing rate
The ice nucleation rate Ji [m−3 s−1] is equal to the rate J1 that a critical embryo acquires molecules from the parent phase, times the number n(ri*) of such embryos:
The germ concentration is related to the concentration nL of liquid-phase molecules by the Boltzmann factor:
For the vaporliquid transition, we know the limiting rate of addition of molecules to the drop is the impingement of vapor molecules on the surface
For freezing, however, the supply of molecules is not a limiting factor; rather it is their ability to leave the liquid and enter the solid phase
Notice there is an energy barrier to be overcome
This is the same barrier responsible for restrictions on liquid-phase diffusion
Freq of molecular incorporation into ice, per sec per molecule

13. ~1

14. Exponential dependence arises from energy term
T dependence from this term

15. Evaluating “threshold” conditions
if we have a certain volume Vd of supercooled water, say that of a cloud droplet or a raindrop, at what temperature should we expect it to freeze?
First, note that only a single nucleation event suffices to cause the entire volume of the drop, regardless of size, to freeze.
The relevant variable is therefore the nucleation frequency ωi = Vd Ji (T ), the product of liquid volume and nucleation rate, not Ji (T ) alone.
A larger liquid volume, Vd , implies that a smaller volumetric rate, Ji (T ), is needed to get the same freezing effect; equivalently, the chance of finding one critical ice embryo in a large volume is greater than that in a small volume of supercooled water.
The choice of a significant nucleation rate, and an estimate of threshold conditions, thus depends on the volume of supercooled liquid, as well as the temperature.
A nucleation frequency of ωi = 1 s−1 is often chosen as a reasonable criterion for homogeneous freezing

16. Threshold conditions and rates
Once the threshold condition has been met the drops don’t all freeze at once. The process is visualized as stochastic (non-deterministic)
Consider N0 drops, each with identical volume Vd at the temperature T
All begin unfrozen at t=0; how many have frozen at time t?
The

17. Threshold temperature

18. Threshold temperature depends on drop size, cooling rate

19. Uncertainties in theory
Calculation of T at which J becomes appreciable is difficult.
known at 0°C
not known accurately for T < 0°C
Conclude
For droplets >1µm, probability of homogeneous nucleation should be appreciable when T is somewhere in range of °C.
* -40°C commonly accepted
Measurements of supercooled water at T < -40°C?
uncertainty in
freezing point depression associated with solution (need to consider)
Freezing of droplet occurs when critically-sized embryo is formed, and then is ‘nucleated’ by attachment of additional water molecules. This process occurs with high probability with significant supercoolings.

20. Heterogeneous freezing -40 C Height of critical free energy barrier
Fletcher 1962
-40 C
-40°C Homogeneous nucleation?
This temperature marks the “lower limit”, temperature wise, for supercooled water.

21. Homogeneous nucleation of ice by sublimation
Formation of ice by vapor molecules going directly to the ice phase—very similar to homogeneous nucleation of water from the vapor.
Consider hexagonal embryo
Can immediately write down
p = ambient vapor pressure
ps = equilibrium vapor pressure over solid
This type of nucleation not likely to produce ice particles in the Earth’s atmosphere
T < -60°C RHice ~ 150%
Mother-of-pearl clouds in stratosphere?
0.025Jm-2
homogeneous freezing favored over deposition
0.100Jm-2

22. Heterogeneous Nucleation of Ice
Suspect presence of foreign particle may aid nucleation of ice by allowing ice to form at higher temperatures compared to homogeneous nucleation of freezing –and at lower humidities compared to formation of ice by homogeneous nucleation from the vapor.
Nucleation on a planar substrate
Nucleation by deposition
vapor
solid
From force balance,
contact angle
If large; contact angle is large
large curvature

23. “Similar” substrates cause a strain in the ice.
Both ice and substrate have lattice structure so must examine microscopic properties of the nucleation process.
Ice
Fletcher 1962
Substrate
Dislocation occurs if ice retains its lattice structure right down to the surface of the substrate, or:
Ice lattice will deform elastically to coherently join the lattice of the substrate—this results in a strain in the ice lattice.
Dislocations normally occur when substrate has a considerably different lattice structure than ice.
“Similar” substrates cause a strain in the ice.
Dislocation – increases increases contact angle
Strain – raises
What is the best nucleating substrate?

24. In general a combination of strain and dislocation will exist.
small increase in µs
ice
substrate
Young 1983
increase in σcs
In general a combination of strain and dislocation will exist.
So µs increased due to strain and σcs increased due to dislocations

25. Why is σcs increased due to dislocations?= O H+
Why is σcs increased due to dislocations?
H2O – polar molecule
Interfacial energy σcs is minimized when H+ - O- dipole aligns with the local field at the surface of the substrate. _ +
arises due to ionic nature of nucleating material
Misalignment results in electrical ‘torque’