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The step energy of a crystalline material.is the energy required to create the edge of a one-molecule-high 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 fruit-fly 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 phase-transition 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 nucleation-limited-growth model
To measure the ice step energies, we must first verify that the growth of ice facets is well represented by a nucleation-limited-growth 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 nucleation-limited-growth 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 nucleation-limited-growth. 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. Nucleation-limited-growth 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 semi-log plot means that purely nucleation-limited 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 nucleation-limited 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, 1-8 (2013).

[2] Libbrecht, KG, A Critical Look at Ice Crystal Growth Data, arXiv:cond-mat/0411662 (2004).