Rich and poor facetings

Faceting of Blue Phases I and II grownfrom the Isotropic phase. Thesis of RemiBarbet-Massin. Collaboration with PatCladis.

Faceting of inclusions of a micellar phasein a cubic lyotropic phase has beenmentioned by Winsor.

Faceting of an air bubble in the Ia3d cubicphase. Observation made by Paul Sotta.

Devil’s staircase faceting of the “free”surface of the Ia3d cubic crystals. (MixtureC12EO2/H2O)

“Poor” faceting of Pn3m crystalssurrounded by the L1 micellar phase.(C12EO2/H2O mixture)

Faceting of L1 inclusions in the Pn3mcubic phase. (C12EO2/H2O mixture)

Ia3d Structure Pn3m Structure Setup 1 Setup 2

Faceting of Cubic Liquid Crystals

C12EO2

Thesis of Rémi Barbet-Massin. Collaboration with Pat CladisFaceting of the Blue Phase I

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Faceting of the Bleue Phase IIThesis of Rémi Barbet-Massin. Collaboration with Pat Cladis

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Ia3d Structure

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Elementary patch Two interwoven labyrinths

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Ia3d Structure (suite)

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Octahedral habit ofA L1-in-Pn3m inclusion

Octahedral habitIn a cube

Elementary patch Two interwoven labyrinths

Pn3m Structure

Experimental setup 1

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In the first method, the composition of the sample is controlled through the contact with ahumid atmosphere.The metallic body of the external cell contains two channels, one dry and the second filledpartially with water, in which circulate two regulated fluxes of nitrogen. The two fluxes aremixed and send to the sample located in the internal cell.

Experimental setup 2

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In the second method, a classical one, theaverage composition of the sample is fixedand only the temperature is controlled.

Here, the sample is held in a flat glasscapillary situated inside a cell made ofglass plates.

The cell is supported by two Peltierelements which temperatures are controlledindependently.

Close-up of the cell

Devil’s staircase faceting of Ia3d crystals

Theoretically, in certain conditions(low temperature, strong interactionsbetween stiff steps on a crystalsurface), facets with arbitrary Millerindices (hkl) could occur on a crystalsurface.

This so-called "devil’s staircase"-typefaceting, never observed in crystalswith classical positional order ofatoms or molecules, has been recentlydiscovered on the "free surface" ofIa3d crystals in the binary mixtureC12EO6/water.

The explanation of this astonishingphenomenon has been proposedrecently by Nozières, Pistolesi andBalibar (EPJB 24 (2001) 387).

112

111

202

211

Collaboration with D. Rohe, P. Sotta, M. Clerc-Imperor, L. Sittler et C. Even

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Constellation of facets

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The “free” surface of Ia3d crystals is almostspherical and composed of 48 identical patchesrelated by the symmetry operations of theO(432) point group characteristic of the cubicsystem.

112-004

444

220-444

Some facets of the elementary patch areindexed. The occurrence of facets obeys to theselection rules identical with the extinctionrules for Bragg reflections.

Vicinity of the (444) facet

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Between (112) and (004) facets

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Between (220) and (444) facets

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Relationship between habits of BPI and the cubic LLC Ia3d

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Poor faceting of the Pn3m/L1 interface(Collaboration with C. Even, N. Ginestet, M. Bouchih, S. Popa-Nita)

Following Nozières, Pistolesi and Balibar, thedevil’s staircase faceting of the Ia3d/vaporinterface is due to the conjunction of twofactors : 1°- large surface tension g and 2°-extraordinary height

of steps h in the LLC.

Qualitatively, this is expressed by the formulafor the roughening temperature of (hkl) facets :

†

kBTR ª2p

gh2T R

where TR is a numerical factor. Remarkably, thesoftness of the Ia3d phase is not a pertinentfactor.

In order to test this model, we looked for otherinterfaces with a different surface tension.

Facets, steps and kinks

Phase diagram of the mixture C12EO2/H2O

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This phase diagram contains several domains of coexistence between the cubic and isotropic phases.Only the Pn3m/L1 interface is faceted and this faceting is “poor”.

Note the presence of three peritectic points.

Pn3m crystals surrounded by the L1 phaseThe crystal habit of Pn3m crystals contains (111)-type large facets and (200)-type smaller facetscoexisting, through angular junctions, with an everywhere else rough surface.

Top view of a Pn3m crystal obtained from a L2inclusion by the peritectic reactionL1+L2ÆL1+Pn3m. The crystal is located atthe upper glass wall of the flat capillary. Perspective view

Metastability of the sponge L3 phase

Upon cooling below the PPL3 peritecticpoint, sponge phase layers appear at thePn3m/L1 interfaces

Upon heating above the PPL3 peritecticpoint, the large droplet is recrystalisedbut the small droplet persists in themetastable state - the sponge phase.

Two Pn3m crystalssurrounded by the L1 phase

In the smallerdroplet, the Pn3mphase disappears

completely

Growth of the Pn3m crystals

When the temperature increases, thelever rule shifts the ratio between thevolume of Pn3m crystals and the L1micellar phase. When the averageconcentration is less than the neutralone, the Pn3m crystals grow.

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L1 inclusions in the Pn3m phase

Images obtained by focusing at different levels in the sample

bulktop bottom

L1 inclusions in the bulk of the Pn3m phaseThanks to the "solid"-like elasticity of the Pn3m phase, the L1 inclusion created in the bulk do notclimb by bouyancy like the L1 inclusions in the sponge L3 phase. Similar to the Sotta's air bubbles,they stay imprisoned in the Pn3m phase.

Growth of a L1 inclusion in the bulk of thePn3m phase

Octahedral shapes of a L1 inclusion grown inthe bulk of the Pn3m phase.

L1 inclusions at the capillary wall

Here, two L1 inclusions have been grown in alarge “pancake”-like Pn3m crystal.

Close-up of a L1 faceted inclusion

Growth of a L1 inclusion at the glass wall

Complementary shapes of Pn3m crystals and L1 inclusions

Complementary shapes and contact angle

Role of the contact angle

Pn3m crystal in a temperature gradient

hot

cold

Shape change in a temperature gradient

Pn3m crystal in a vertical gradient

Pn3m crystal in a reversed gradient

L1 inclusion in a temperature gradient