Faceting of Soft Crystals

I. Introduction :

Thanks to their unusual structures and properties, the lyotropic liquid crystals bring interesting contributions to the problem of faceting of crystals. The figure below summarizes a few circumstances in which faceting has been observed.


(J. de Physique II 1 (1991) 763)










It has been also discovered that faceted shapes of crystals and inclusions are affected by temperature gradients.


II."Devil's Staircase" faceting of Ia3d crystals
in the mixture C12EO2/water

Theoretically, in certain conditions (low temperature, strong interactions between stiff steps on a crystal surface), facets with arbitrary Miller indices (hkl) could occur on a crystal surface. This so-called "devil’s staircase"-type faceting, never observed in crystals with classical positional order of atoms or molecules, has been recently discovered on the "free surface" of Ia3d crystals in the binary mixture C12EO2/water.
more details in :
P. Pieranski, P. Sotta, D. Rohe, M. Imperor-Clerc, Phys. Rev. Lett. 84 (2000) 2409)
P. Pieranski, L. Siitler, P. Sotta, M. Imperor-Clerc, EPJE 5 (2001) 317
APS “Focus”



(a)-------------------------------------------------------------(b)

On the photograph (a), one sees a constellation of facets at the interface between an Ia3d crystal and a gas atmosphere made of nitrogen and water vapours. This is only a small portion of the crystal surface. About 60 types of facets with different Miller indices (without permutations or sign changes) have been identified by interferometry (b).



The almost spherical crystal habit of a Ia3d crystal is composed of 48 identical triangular patches related by the symmetry elements of the point group O(432). Some of facets from the constellation covering one elementary patch are indexed. The occurence of facets is obeying to the selection rules identical to the extinction rules for the Bragg reflections. The explanation of this astonishing phenomenon has been proposed recently by Nozières, Pistolesi and Balibar (EPJB 24 (2001) 387)


III."Poor" faceting of the Pn3m/L1 interface
in the mixture C12EO2/water

The “devil’s staircase” faceting is characteristic of the Ia3d/vapor interface but is not a general rule for other interfaces such as, for example, the interface between the cubic Pn3m phase and the micellar L1 phase in the mixture C12EO2/water. Due to a small surface tension at the L1/Pn3m interface, the faceting is “poor” ; only (111)- or (200)-type facets occur here.

------
a) L1 in Pn3m _____________________________ b) Pn3m in L1

These are photographs of a L1 inclusion in the Pn3m cubic phase (left) and of a Pn3m crystal surrounded by the micellar L1 phase (right). The scheme in the center explains that the inclusion (a) and the crystal (b) are truncated due to the contact with the glass wall of a flat capillary containing the sample.
Inclusions of the L1 micellar phase can also be grown in the bulk of the Pn3m phase. In such a case they have octahedral shapes.

--

The photograph on the left illustrates the growth of a L1 inclusion in the bulk of the Pn3m phase. The drawing b, c and d on the right explicite positions of facets on the surface of the inclusion. In order to obtain perfect octahedral shapes, the sample is first cooled below the peritectic point of the sponge phase ; the layer of the sponge phase that forms at the L1/Pn3m interface supresses facets and the L1 inclusion gets a perfect spherical shape (a).


IV. Effects of temperature gradients


Crystal habits of lyotropic cubic phases are very sensitive to temperature gradients. Below, we show three photographs of L1 inclusions in the Pn3m phase of the C12EO2/water mixture.


The photograph in the center shows three inclusions at a uniform temperature. The two other photographs prove that the temperature gradient breaks the threefold symmetry of the inclusion’s habit.