Highly dendritic snowflakes are an aesthetic source
of wonder, but the greater challenge for physicists lies in understanding the "simpler" prismatic
and planar forms.
Vast regions of our planet are seasonally or perennially
covered with snow and ice; the ground becomes rock hard, lakes and oceans freeze, and snow falls
from the sky. If you live in New England or Hokkaido, look out the window in winter and you will see
those inescapable consequences of Earth's seasonal cycles. For nearly everyone, snow crystals
induce a visceral response, be it an annoyance during one's commute or the enthusiasm of an impending
ski trip. The distribution of ice throughout the universe and close to home has central implications
in astrophysics and geophysics—from the agglomeration of matter in stellar nebulae to the
state and fate of Earth's climate. Like all materials, ice exhibits basic phase-transition phenomena.
But its study does not require ultrahigh-vacuum or cryogenic apparatus, so it is an ideal test bed
for physicists.
Snow crystals are simply ice crystals
grown from the water vapor present in air, and the varied forms they take during their transit through
the clouds are the basis of science, art, and the culture of cold regions. In his 1611 book A New
Year's Gift of Hexagonal Snow, Johannes Kepler considered the origin of the hexagonal shapes
of snow crystals, like those shown at left in photographs taken by one of us (Furukawa). Kepler argued
that the hexagonal form had to do with the packing of spheres, an idea that led to his famous conjecture
on the densest possible filling of space by spheres. During the winter of 1635, René Descartes
made strikingly detailed observations of snow crystals. Three centuries later the most extensive
catalog of its time was made by the Vermont farmer Wilson Bentley, who published the book Snow
Crystals with William Humphreys in 1931. Bentley's dedication and passion were evident:
"Was ever a life history written in more dainty or fairy-like hieroglyphics? How charming the task
of trying to decipher them."
The snowflake has substantially
shaped the fabric of life in Hokkaido, the northern island of Japan. The imagination of Hokkaido
University nuclear physicist Ukichiro Nakaya was captivated by Snow Crystals, and in
1932 he began his own observational program. In the course of taking more than 3000 photographs
in a few years, Nakaya classified snow crystals into some 40 morphological categories. He surmised
that the shapes were a consequence of the temperature and supersaturation of the atmosphere through
which the crystals moved, and he quantified his intuition by creating nearly all of the natural
morphological categories in the laboratory. His results, which have since been reproduced by
his students and by many others around the world, are elegantly summarized in the so-called Nakaya
diagram that appears on the next page. In the diagram, one can read the meteorological information
written on a snow crystal. That is, one can infer much of the temperature and humidity history of
the snow crystal by observing its morphology on the ground. Hence, Nakaya was often quoted as referring
to the snow crystal as "a letter from the sky."
Much of the focus on snowflakes
deals with the highly dendritic forms, but physicists generally understand how such shapes arise
out of diffusive instabilities associated with the vapor, as mediated by the underlying anisotropy
that shapes the inner hexagonal form. Nakaya clearly appreciated that the complex spacetime history
of an object in a turbulent, nonequilibrium environment cannot be captured in its entirety. Nonetheless,
the Nakaya diagram embodies the basic physics associated with the principal dynamic shape transitions,
even if it doesn't include all snow crystal forms. One of its salient features is that, depending
on the temperature regime of growth, the forms at small supersaturation are either hexagonal plates
or prisms, and the transitions from one regime to the other are abrupt. For some 50 years, many researchers
have grappled with the challenge of explaining that alternating plate–prism growth sequence.
A crystal's facets reflect
the symmetry of the underlying crystal structure. And its surface energy depends on the sum of the
energies of all the bonds broken per unit area during the creation of its surface by cleavage. Strictly
speaking, facets, beautifully reflected in the crystals at the bottom of the Nakaya diagram, are
orientations that are smooth on all scales down to the molecular scale. An ideal, dislocation-free
crystal is fully faceted at absolute zero and becomes rougher as its temperature increases. Some
facets persist right up to the bulk melting point Tm, and some become rough
at lower temperatures. The surface roughness, its equilibrium and nonequilibrium manifestations,
and liquidity itself all conspire to influence the transitions on the Nakaya diagram. Here we point
out the essential ingredients of the conspiracy.
Liquid and vapor interfaces
It is commonly thought that ice skating
is possible because of pressure melting; the pressure applied by the skate melts the ice below it,
and the water film allows the blade to glide. That fact alone, however, cannot explain the slipperiness
of ice; the answer involves not only frictional melting but also surface melting, which describes
the natural state of the free surface of ice as being covered by a thin liquid film of water.
Detailed experiments
on the surfaces of semiconductors, metals, rare-gas solids, and molecular solids, including
ice, show that the process of surface melting begins at a temperature below Tm.
It is due to the weaker binding of the surface atoms or molecules, which are thus more susceptible
to thermal disorder than the bulk interior. Further temperature increases lead to the breakdown
of the surface layers and an increase in the defect density; hence the atomic and molecular mobility
increases. That mobility eventually becomes the same as in the bulk liquid phase, and finally the
liquid film thickness becomes arbitrarily large at Tm; surface melting is
completed, and bulk coexistence is reached.
Liquidlike layers persist
on ice at temperatures as low as a few tens of degrees below 0 °C when, as under atmospheric conditions,
appreciable surface impurities are present. The different facets of an ice crystal in the Nakaya
diagram have microscopic films, but the film thickness depends on the facet. As the temperature
is lowered, the films thin. The surface energy of a "dry" facet depends on the distribution of ledges,
corners, edges, and point defects such as single molecules and vacancies in the surface. The cost
of all of those contributions determines whether the surface is faceted or rough: The transition
between the two situations is called the roughening transition.
On an infinite two-dimensional
surface, the energy required to form a step vanishes at a roughening temperature Tr ≤ Tm,
and thermal fluctuations liberate the surface from the ordering influence of the underlying lattice.
Although finite crystals are different in detail, it is qualitatively accurate to say that if T > Tr,
the crystal's equilibrium shape will be rough.
When a crystal grows, it
receives additional molecules from the vapor phase. If a molecule lands on a region that is molecularly
rough, it will find a ready place for incorporation into the solid. In contrast, a molecule landing
on a facet has many options: It can diffuse along the facet until it finds a ledge, corner, or edge
to attach to; it can meet up with a number of other diffusing molecules and create an expanding island
on the surface; or it can detach and return to the vapor. Moreover, if the growth is so rapid that new
material can't settle before being buried by more molecules, a surface will kinetically roughen.
Thus the growth process depends sensitively on the nature of the surface, which sets the stage for
anisotropic growth. Much of the discussion of the Nakaya diagram does and must deal with the confluence
of surface growth processes and their dependence on vapor density, surface temperature, and the
size of the crystal.
Here's the rub. Independent
experiments by Michael Elbaum and Minoru Maruyama established that roughening depends on whether
ice is in contact with water or water vapor. Therefore, as the temperature rises, surface melting
begins to become important—in natural snow the precise temperature depends sensitively
on the impurity level. A thin film develops between ice and vapor, the nature of the roughening transition
changes, and that change depends on which facet one scrutinizes. Is the change dramatic? Evidently
so, but the key issue is to extract the underlying physics that weds surface melting to roughening
in systems driven away from equilibrium. If you succeed in doing it, there will be a diagram named
after you.
Yoshinori
Furukawa (frkw@lowtem.hokudai.ac.jp) is a professor of physics at Hokkaido University's
Institute of Low Temperature Science in Sapporo, Japan. John Wettlaufer (john.wettlaufer@yale.edu)
is a professor of geophysics and physics at Yale University in New Haven, Connecticut.
Additional resources
J. G. Dash, A. W. Rempel, J. S. Wettlaufer, "The Physics of Premelted Ice and Its Geophysical Consequences," Rev. Mod. Phys.78, 695 (2006)[SPIN].
Y. Furukawa, "Fascination of Snow Crystals: How Are Their Beautiful Patterns Created?" available at [LINK].
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