
The step energy
of a crystalline material.is the energy required to create the edge of
a onemoleculehigh terrace on a faceted surface. This energy is
proportional to the length of the edge, so the step energy is measured
in
Joules/meter. Like the bulk energy (aka
latent heat, measured in Joules/meter\(^3\)) and the surface energy (aka
surface tension, measured in Joules/meter\(^2\)), the step energy is a
fundamental equilibrium property of ice. Understanding step energies is
key to understanding the overall structure and dynamics of the ice
surface. This page describes our measurements of ice step energies, and
what we know so far.
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Equilibrium energetics
Once the lattice structure of
a crystalline material is known (see ***The Ice Crystal), one can begin to
relate bulk material properties directly to underlying molecular
interactions. For example, the energetics of chemical bonding between
water molecules allows an ab
initio
calculation of the ice unit cell (the lowest energy lattice
configuration), as well as bulk properties such as latent heats and
specific heats. With ever more sophisticated computational methods, it
becomes possible to directly calculate material properties with ever
greater precision, thus increasing our basic understanding of these
properties, and increasing our ability to predict the properties of
more complex and yet undiscovered materials. I view ice as something of
a case study in materials science, the fruitfly of crystalline
materials, serving as a fertile testing ground for developing more
advanced computational tools for studying material properties. 

Relating bulk, surface, and step
energies in ice
Bulk energy
Also
known as the latent heat of deposition (sometimes latent heat of
vaporization), the bulk energy of the ice/vapor transition is the
amount of energy needed to break all the molecular bonds within a ice
crystal, transforming it to a gas of water vapor molecules. Near the
triple point, the bulk energy is \(L_0\approx 2.6\times 10^9\ J/m^3\).
At other temperatures, \(L\) can be calculated from \(L_0\) along with
the specific heat of ice (giving the energy needed to change the ice
temperature) along with ice density changes with temperature. Of
course, \(L_0\) describes the phasetransition energy at just one point
in the ice phase diagram, which includes many other forms of ice and
many additional phase boundaries. But our focus here is simply on ice
Ih and its growth from water vapor.
Surface energy
This is
the energy required to break and ice crystal into two pieces, thus
breaking the molecular bonds that formerly held the two crystal pieces
together. (Again, our focus here is on the ice/vapor interface.) The
number of bonds broken is proportional to the new surface area created,
so this can also be thought of as the amount of energy needed to create
an ice surface. The ice/vapor surface energy has not been accurately
measured, but is about \(\gamma\approx 0.11\) J/m\(^2\).
Step energy
Usually
defined only on a faceted surface, the step energy is the work needed
to break a molecular terrace (see 2D Nucleation) into two pieces.
Again, this means breaking the molecular bonds that held the former
terrace together, which is proportional to the length of the new
terrace edge created. Thus the step energy is measured in Joules/meter.
The only known way to measure the ice step energies (with any kind of
accuracy) is through measurements of the growth rates of faceted
surfaces, as described in 2D Nucleation.


Verifying
the nucleationlimitedgrowth model
To measure the ice step energies, we must first verify that the growth
of ice facets is well represented by a nucleationlimitedgrowth model.
The graph at right shows some example measurements of the growth
velocity of the basal surface of a single ice crystal as a function of
\(\sigma_{surf}\) (the supersaturation near the surface) at 12 C. Data
points in the main graph show measurements of a single ice crystal
as \(\sigma_{surf}\) was slowly
increased. The line through the points shows a
nucleationlimitedgrowth model \(v = \alpha v_{kin}
\sigma_{surf}\) with \(\alpha = exp(\sigma_0/\sigma_{surf})\) and
\(\sigma_0 = 2.3\)
percent. The surrounding dotted lines show the same
model with \(\sigma_0 = 2.3\pm 0.2\) percent. The dashed line shows an
alternative spiral
dislocation growth model with \(v\sim \sigma_{surf}^2\), which gives
a poor fit to the data. The inset graph shows an unweighted histogram
of measured \(\sigma_0\) values for 23 measured crystals. A weighted
fit to these data gives an estimated mean \(\sigma_0 = 1.95\pm 0.15\)
percent. See ***Ice Growth Measurements for a description of the
experimental apparatus used to take these measurements, and [1] for
more details.
The
upshot, after measuring many ice crystals over a range of temperatures
from 40 C to 2 C, is that the growth measurements clearly show the
\(exp(\sigma_0/\sigma_{surf})\) behavior expected from
nucleationlimitedgrowth. This is not a surprising result; the
existence of faceted surfaces itself suggests that the growth is
limited by 2D nucleation, as this is model has been verified in many
crystal systems. Nucleationlimitedgrowth is not the only theoretical
possibility, but it certainly the most likely explanation for the
formation of faceted surfaces, including those in ice.


An illustrative graph The
graph at right shows some additional crystal growth data, this time
plotting measurements of \(\alpha_{basal}\) as a function of the
inverse surface supersaturation \(1/\sigma_{surf}\). Displaying
\(\alpha\) as a function of \(1/\sigma_{surf}\) on a semilog plot
means that purely nucleationlimited growth data will appear as
straight lines on the plot (because 2D Nucleation
theory gives \( \alpha = A\exp (\sigma_0/\sigma_{surf})\) with \(A
\approx \) constant). So here again we see that growth of faceted basal
surfaces is well described by a nucleationlimited model. The different
slopes of the four lines shown indicate that \(\sigma_0\) changes quite
a bit with temperature.
The \(A\) factor can be seen immediately
in this plot by extrapolating the data to \(1/\sigma_{surf} \rightarrow
0\). Thus we see that the measurements are all consistent with
\(A\approx 1\), independent of temperature. This means that \(\alpha
\rightarrow 1\) when \(\sigma_{surf} \gg \sigma_0 \), and this in turn
means that the surface grows as fast as a rough surface (\(\alpha =1\))
once the rate of nucleation of new terraces is high. Considering the
underlying physics of 2D Nucleation, this all makes pretty good sense. 

More measurements We
next measure many crystals over a range of temperatures, looking at the
growth of both the basal and prism facets. Extract \(\sigma_0\) for
each crystal by fitting to the nucleation model above, and combine all
the data to yield \(\sigma_0\) as a function of temperature for
each facet. The details are given in [1], and the results are shown in
the graph at right.
We spent a long time acquiring these data,
examining a host of tricky experimental problems and subtle systematic
errors along the way. We discovered along the way that all previous ice
growth experiments were not sufficiently accurate to extract useful
\(\sigma_0\) measurements [2]. 
 Step energies The
last step is to convert \(\sigma_0(T)\) to the step energies for
the two facets as a function of temperature, using classical nucleation
theory (see 2D Nucleation). This gives the results on the right.
As
described at the top of this page, the step energies \(\beta(T)\) are
fundamental, equilibrium properties of the ice surface. Through
nucleation theory, \(\beta(T)\) can be used to describe the growth
rates of ice crystal facets as a function of temperature and
\(\sigma_{surf}\).
The bad news is that the step energies are
not the whole story. For example, to explain the growth of thin ice
plates at 15C, we need to include an ***Edge Sharpening Instability.
And to explain the growth of slender columnar crystals at 5C, we need
to include **Terrace Erosion on the prism facets. Both these phenomena
occur when ice crystals grow in air at one atmospheric pressure, and
not in the absence of air. And both result from the fact that classical
nucleation theory is not sufficient to adequately describe ice growth
under all circumstances.


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References
[1] Libbrecht, KG and Rickerby, ME, Measurements of surface attachment kinetics for faceted ice crystal growth, J. Cryst. Growth 377, 18 (2013).
[2] Libbrecht, KG, A Critical Look at Ice Crystal Growth Data, arXiv:condmat/0411662 (2004). 


