Devil's staircase-type faceting of a cubic lyotropic liquid crystal

Mesoscopic theory for systems with competing interactions near a confining wall

Mesoscopic theory for self-assembling systems near a planar confining surface is developed. Euler-Lagrange equations and the boundary conditions (BCs) for the local volume fraction and the correlation function are derived from the density functional theory expression for the grand thermodynamic potential. Various levels of approximation can be considered for the obtained equations. The lowest-order nontrivial approximation [generic model (GM)] resembles the Landau-Brazovskii-type theory for a semi-infinite system. Unlike in the original phenomenological theory, however, all coefficients in our equations and BCs are expressed in terms of the interaction potential and the thermodynamic state. Analytical solutions of the linearized equations in the GM are presented and discussed on a general level and for a particular example of the double-Yukawa potential. We show exponentially damped oscillations of the volume fraction and the correlation function in the direction perpendicular to the confining surface. The correlations show oscillatory decay in directions parallel to this surface too, with the decay length increasing significantly when the system boundary is approached. The framework of our theory allows for a systematic improvement of the accuracy of the results.

Soft polyhedral particles based on cubic liquid crystalline emulsion droplets

Soft polyhedral particles based on variations of the cubic symmetry group are produced from a precursor emulsion by extracting solvent to grow facets on the droplets. The droplets transform into liquid crystals with solid-like rheology and controlled size and shape. Small-angle X-ray scattering confirms a bicontinuous cubic liquid crystalline phase forms from aqueous glycerol monoolein and is responsible for the particle faceting observed. Different polyhedra are produced by varying face growth rates through control of precursor droplet size, system temperature, and solubilization and adsorption of guest molecules. More exotic faceted shapes can be formed by the soft particles by applying asymmetric solvent removal gradients and by deforming the precursor droplets into, for example, ellipsoids before crystallization. The method is a powerful means to produce soft polyhedra, using continuous microfluidic or other approaches, or to act as templates for hard polyhedral particle synthesis.

Control of Nanomaterial Self-Assembly in Ultrasonically Levitated Droplets

We demonstrate that acoustic trapping can be used to levitate and manipulate droplets of soft matter, in particular, lyotropic mesophases formed from self-assembly of different surfactants and lipids, which can be analyzed in a contact-less manner by X-ray scattering in a controlled gas-phase environment. On the macroscopic length scale, the dimensions and the orientation of the particle are shaped by the ultrasonic field, while on the microscopic length scale the nanostructure can be controlled by varying the humidity of the atmosphere around the droplet. We demonstrate levitation and in situ phase transitions of micellar, hexagonal, bicontinuous cubic, and lamellar phases. The technique opens up a wide range of new experimental approaches of fundamental importance for environmental, biological, and chemical research.

Glycerol prevents dehydration in lipid cubic phases

Lipid cubic phase samples dry out and undergo phase transitions when exposed to air. We demonstrate experimentally and theoretically that adding glycerol controllably lowers the humidity at which cubic phases form. These results broaden the potential applications of cubic phases and open up the potential of a new humidity-responsive nanomaterial.

Predicting the orientation of lipid cubic phase films

Lipid cubic phase films are of increasingly widespread importance, both in the analysis of the cubic phases themselves by techniques including microscopy and X-ray scattering, and in their applications, especially as electrode coatings for electrochemical sensors and for templates for the electrodeposition of nanostructured metal. In this work we demonstrate that the crystallographic orientation adopted by these films is governed by minimization of interfacial energy. This is shown by the agreement between experimental data obtained using grazing-incidence small-angle X-ray scattering (GI-SAXS), and the predicted lowest energy orientation determined using a theoretical approach we have recently developed. GI-SAXS data show a high degree of orientation for films of both the double diamond phase and the gyroid phase, with the [111] and [110] directions respectively perpendicular to the planar substrate. In each case, this matches the lowest energy facet calculated for that particular phase.

Devil's staircase of facets on the surface of 4He crystals

According to Landau, at T=0 the equilibrium crystal surface consists of an infinite number of facets lying in all directions with rational Miller indices-the so-called devil's staircase phenomenon. We have discovered 11 new types of facets on the surface of 4He crystals, in addition to the three observed before. Some of the new facets are of very high order, lying at angles as small as 4 degrees to the basal c facet, thus forming the predicted devil's staircase. The estimated step energies depend rather weakly on interplanar distance which we explain by the strong anisotropy of the steps.

Direct observation of a "devil's staircase" in wave-particle interaction

We report the experimental observation of a "devil's staircase" in a time-dependent system considered as a paradigm for the transition to large-scale chaos in the universality class of Hamiltonian systems. A test electron beam is used to observe its non-self-consistent interaction with externally excited wave(s) in a traveling wave tube (TWT). A trochoidal energy analyzer records the beam energy distribution at the output of the interaction line. An arbitrary waveform generator is used to launch a prescribed spectrum of waves along the slow wave structure (a 4 m long helix) of the TWT. The resonant velocity domain associated to a single wave is observed, as well as the transition to large-scale chaos when the resonant domains of two waves and their secondary resonances overlap. This transition exhibits a "devil's staircase" behavior for increasing excitation amplitude, due to the nonlinear forcing by the second wave on the pendulum-like motion of a charged particle in one electrostatic wave.

Steps and dislocations in cubic lyotropic crystals

It has been shown recently that lyotropic systems are convenient for studies of faceting, growth or anisotropic surface melting of crystals. All these phenomena imply the active contribution of surface steps and bulk dislocations. We show here that steps can be observed in situ and in real time by means of a new method combining hygroscopy with phase contrast. First results raise interesting issues about the consequences of bicontinuous topology on the structure and dynamical behaviour of steps and dislocations.

Exploring the facets of "soft crystals" using an Atomic Force Microscope

We obtained monocrystalline droplets in a thermotropic cubic phase, of approximate size 100 microm, deposited on a flat surface. The facets of these soft crystals are explored using both an optical microscope and an AFM. The height of individual steps on the principal facets and the lateral distance between steps in vicinal facets are measured using AFM in imaging (tapping) mode. Moreover, the elastic modulus is measured locally, using the AFM tip (in contact mode) as a local rheological probe.

Anisotropic surface melting in lyotropic cubic crystals: part 2: facet-by-facet melting at Ia3d/vapor interfaces

From experiments with metal crystals, in the vicinity of their crystal/liquid/vapor triple points, it is known that melting of crystals starts on their surfaces and is anisotropic. Recently, we have shown that anisotropic surface melting occurs also in lyotropic systems. In our previous paper (Eur. Phys. J. E 19, 223 (2006)), we have focused on the case of poor faceting at the Pn3m/L1 interface in C12EO2/water binary mixtures. There anisotropic melting occurs in the vicinity of a Pn3m/L3/L1 triple point. In the present paper, we focus on the opposite case of a rich devil's-staircase-type faceting at Ia3d/vapor interfaces in monoolein/water and phytantriol/water mixtures. We show that anisotropic surface melting takes place in these systems in a narrow humidity range close to the Ia3d-L2 transition. As whole (hkl) sets of facets disappear one after another when the transition is approached, surface melting occurs in a facet-by-facet type.

Faceting and stability of smectic A droplets on a solid substrate

It is shown that a smectic A droplet deposited on a solid substrate treated for strong homeotropic anchoring is faceted at the top in spite of the fact that there are no steps at the free surface, but instead edge dislocations in the bulk. The radius of the facet and the full profile of the curved part of the droplet are determined as a function of the temperature in the vicinity of a nematic-smectic A phase transition. It is shown that the observed profiles do not correspond to the actual equilibrium shape, but to metastable configurations close to their point of marginal stability. In addition, we predict that the profiles must be different for a given temperature depending on whether the droplet has been heated or cooled down to reach this temperature. Finally, we discuss the problem of the formation of giant dislocations in big droplets (Grandjean terraces).

Anisotropic surface melting in lyotropic cubic crystals. Part 1: Pn3m/L1 interface, poor faceting

From experiments with ice or metal crystals, in the vicinity of their crystal/liquid/vapor triple points, it is known that melting of crystals starts on their surfaces and is anisotropic. It is shown here by direct observations under an optical microscope that this anisotropic surface melting phenomenon occurs also in lyotropic systems. In the case of C12EO2/water mixture, it takes place in the vicinity of the peritectic Pn3m/L3/L1 triple point. Above the peritectic triple point, where the Pn3m and L1 phases coexist in the bulk, the surface of a Pn3m-in-L1 crystal is composed of (111)-type facets surrounded by rough surfaces. The angular junction suggests that rough surfaces are wet by a L3-like layer while facets stay "dry". This is analogous to the pre-melting at rough surfaces in solid crystals. Upon cooling below the peritectic triple point, where L3 and L1 phases coexist in the bulk, a thick layer of the L3 phase grows from the pre-melted, rough Pn3m/L1 interface. Simultaneously, facets stay dry and their radius decreases. In this tri-phasic configuration, stable in a narrow temperature range, the L3/L1 and L3/Pn3m interfaces have shapes of constant mean curvature surfaces having common borders: edges of facets.

Ratchet effect in faceting: a growth of perfect lyotropic crystals by temperature cycles

The paper deals with a new phenomenon, named ratchet effect, envisioned theoretically as a likely consequence of metastability of crystal facets and expected to occur upon a temperature cycling. In experiments, Pn3m lyotropic crystals surrounded by the isotropic L1 phase in the mixture C(12)EO(2)/water are used. At equilibrium, the Pn3m/L1 interface contains small (111)-type facets in coexistence with rough surfaces. In agreement with theoretical expectations, it is shown that upon a saw-tooth-shaped temperature cycling, facets are growing until the rough surfaces are completely eliminated. A model of the ratchet effect is proposed.

Concave and convex shapes of the Pn3m/L1 interface

Shapes of the interface between the L1 and cubic Pn3 m phases in the mixture C(12)EO(2)/water are studied. The concave and convex variants of the interface are realised using Pn3 m crystals surrounded by the L1 phase and L1 inclusions on surfaces and in the bulk of the Pn3 m phase. It is shown that both variants of the Pn3 m/L1 interface contain the (111)-type facets in coexistence with everywhere else rough surfaces. The matching between facets and curved parts of the interface is angular. In the vicinity of the upper limit of the L1 + Pn3 m coexistence domain, additional (200)-type facets appear on the interface. The influence of the contact angle at glass walls on shapes of crystals and of inclusions is discussed.

Devil's staircase in kinetically limited growth

The devil's staircase is a term used to describe surface or an equilibrium phase diagram in which various ordered facets or phases are infinitely closely packed as a function of some model parameter. A classic example is a one-dimensional Ising model [P. Bak and R. Bruinsma, Phys. Rev. Lett. 49, 249 (1982)] wherein long-range and short-range forces compete, and the periodicity of the gaps between minority species covers all rational values. In many physical cases, crystal growth proceeds by adding surface layers that have the lowest energy, but are then frozen in place. The emerging layered structure is not the thermodynamic ground state, but is uniquely defined by the growth kinetics. It is shown that for such a system, the grown structure tends to the equilibrium ground state via a devil's staircase traversing an infinity of intermediate phases. It would be extremely difficult to deduce the simple growth law based on measurement made on such a grown structure.

Faceting on He crystals

It has been predicted by Landau that, ideally at low temperatures, crystals should show many different types of facets, i.e., flat smooth faces on their surface, but this so-called "devil's staircase" phenomenon has been difficult to observe experimentally. In this paper we describe our recent experiments, in which altogether 11 different types of facets have been identified on growing (3)He crystals at the temperature of 0.55 mK by using a unique low-temperature Fabry-Pérot interferometer. Previously only 3 types of facets had been seen in this system. We have also measured the growth velocities of different facets, and our interpretation of the obtained results yields the conclusion that (3)He has much stronger coupling of the liquid-solid interface to the crystal lattice than has been expected. After an introduction we present a short theoretical background about the equilibrium crystal shape and the roughening transitions, which is followed by the description of our experimental results and discussion.

Observation of higher order facets on 3He crystals

Faceting has been observed on 3He crystals investigated with a low-temperature Fabry-Pérot interferometer. Nine types of facets were clearly identified during growth of a bcc- 3He single crystal at a temperature of 0.55 mK, while previously only three types of facets have been seen. Because of the weak coupling between the liquid-solid interface and the solid lattice in 3He the facets are apparently too small to be observed in equilibrium. The number of facets observed in our experimental conditions is consistent with the theory of dynamic roughening.