British Association For Crystal Growth Annual Conference 2017
Ice Crystal Growth from Liquid Water
Charles A. Knight
National Center for Atmospheric Research, U. S. A.
[email protected]
Much of crystal growth theory is based upon the initial idealization and analysis of Burton, Cabrera
and Frank [1]: atoms or molecules attaching either at kinks on steps on a smooth interface, or anywhere on
a rough one, resulting in either faceted or smoothly rounded growth interfaces (Fig. 1). The conceptual
illustration involves stacking space-filling blocks representing individual atoms or molecules, though
molecular crystals in general do not grow by adding unit cells. Ice (1h, 4 molecules per unit cell, an open
structure) growing from liquid water has facets on basal orientations but other interfaces are rounded.
Jackson [2,3] proposed an equilibrium interface model to explain this, but his treatment evidently is not
based upon the structural differences between the ice interfaces. Here we idealize growth on the three, lowindex interfaces, basal {0001} and primary and secondary prism {101� 0} and {112� 0} in terms of sequential
bonding of water molecules into the ice structure. At all three orientations, half of the ice surface molecules
already have all four bonds in the ice structure, and half have three, with one bond “unsatisfied”. (There is
convincing evidence that dislocation step sources are not involved in ice growth from the melt. Hillig [4]
measured a supercooling threshold for growth on {0001} at about -0.03oC.)
With this simple growth model but using the actual ice structure and idealizing the interface as
abrupt, the advance of steps on the basal and primary prism faces is geometrically similar. Most molecules
added at a step (except when a new layer is very small) acquire two bonds, one to a molecule in the layer
and the other to one on the original plane. Two bonds fix the molecule’s location and orientation in the ice
structure. However, in nucleating a new layer the first molecule can have only one bond in the ice structure,
leaving it free to rotate. In layer initiation on the basal plane, a second molecule for the new layer cannot
bond both to the first one and to the basal surface (because the initially unsatisfied bonds are directed
normal to the interface, and the bonding is tetrahedral). Thus a two-molecule “island” must be a twomolecule tail, attached to the crystal by a single bond at one end. This tail is not constrained into the ice
structure, but is mostly surrounded by liquid. (A third molecule may then bond to the end of the tail and to
the original surface if it is lucky, but it has about a 50:50 chance of doing it with the second molecule in
position to start a stacking fault.) In contrast, the prism planes are structured so that all of its unsatisfied
bonds are arranged in pairs, such that two-molecule islands are “bridges” completely constrained in the ice
structure.
This may be (probably is?) the reason why the growth at basal surfaces is faceted, but there are no
growth facets at prism surfaces. (Growth at a secondary prism interface in this idealized model would be
one-dimensional; a single-bonded addition starting linear growth parallel to the c-axis along a half-hexagonal
trough in the interface, with each trough needing one single-bonded addition in order to start the process.) In
crystal growth theory, an energy barrier to nucleating a new layer of molecules causes growth facets. In the
theory, the energy for island formation as a function of island radius starts from zero at zero size and rises to
a maximum value, which is the energy barrier to nucleation. The critical size of an island increases without
limit as the supercooling approaches zero. This growth model obviously does not work for ice on the basal
plane, since the “island” does not even have the ice structure until it comprises three or four molecules.
Jackson’s approach to explaining the growth habits was to estimate the roughness of equilibrium
interfaces. Following that strategy in principle, for ice in water one should take into account not only the ice
crystal structure but also the high degree of association within liquid water itself, which has many, transient
hydrogen bonds [e.g.5,6]. The interfacial “unsatisfied” bonds must then be in reality “satisfied” much of the
time but also transiently. In an instantaneous view an interface might have a half of these bonds on the
basal face satisfied, and many of the two-molecule “bridges” on prism orientations also would exist
transiently at equilibrium. This may give the “roughness” on the prism orientations a two-dimensional
character, since these bridges are easily connected together by single-molecule additions. These transient
structures are automatically accounted for in values of surface energy, defined as an equilibrium quantity.
This view of the interfaces makes the phenomena of ice growth from liquid water seem more understandable
in principle: the faceting only on the basal orientation, the very low threshold for layer nucleation there, and
the roughness of prism faces coupled with the fact that dendritic branching in growth parallel to the basal
plane appears to be accurately oriented parallel to ice’s a-axes [7] instead of being not oriented
British Association For Crystal Growth Annual Conference 2017
crystallographically at all, which might have been expected of dendritic growth at a more three-dimensionally
rough interface.
Fig. 1: Ice single crystal growing around a cold finger (6 mm diameter, maintained at -0.1oC at about 0.2
mm/hour), from purified liquid water. Background illumination diffuse light with black stripes to reveal interface
features. The c-axis is E-W in the photo, but a few degrees out of the plane of the photo to show the basal
facets.
References:
[1] Burton, W. K., N. Cabrera, and F. C. Frank, Phil. Trans. Roy. Soc. A 1951, 243, 299.
[2] Jackson, K. A, in Liquid Metals and Solidification, American Society for Metals, Cleveland, 174, 1958.
[3] Jackson, K. A, Kinetic Processes, Wiley-VCH, FRG, 2010.
[4] Hillig, W. B., in Growth and Perfection of Crystals, Wiley, New York, 350, 1958.
[5] Eisenberg, D. and W. Kauzmann, The Structure and Properties of Water, Oxford U. P, New York and London, 1969.
[6] Zielkiewicz, J., J. Chem. Physics 2005, 123, 104501.
[7] Glen, J. W. and M. F. Perutz, J. of Glaciology 1954, 2, 397.