Chord length distributions and small-angle scattering

Chord length distributions describe size, shape and spatial arrangement of geometrical objects (particles). The chord length distribution is in principle proportional to the second derivative of the correlation function of small-angle scattering. It is calculable from a relative measurement of the scattering intensity I(h). In structure research, the characterization of numerous particle systems can be achieved by comparing experimental chord distributions with theoretical ones, provided the latter are available with sufficiently high precision for a lot of fundamental, universal shapes. Both sides of this concept are exemplified: – the step from a relative measurement of the scattering intensity of an isotropic two-phase sample to the chord length distribution (errors in and in , limited h-interval, corresponding to the region (1-2) nm < r in real space, must be observed); as well as the geometric matter of calculation of chord distributions as fingerprints for basic geometric figures, including the non-convex case. Received 15 March 1999 and Received in final form 26 April 2000     

A biaxial nematic gel phase in aqueous vanadium pentoxide suspensions

Aqueous suspensions of V₂O₅ ribbons are one of the very few examples of mineral liquid crystals. In the concentrated regime, we show that these ribbons organize in a biaxial nematic gel phase. A Couette shear cell was used to produce a well oriented sample for in situ synchrotron X-ray scattering studies. We observed two perpendicular anisotropic sections of reciprocal space, which proves the biaxial symmetry of the nematic order. The thermodynamic and flow properties of the biaxial nematic are well described by hard-core theories. We suggest the use of a shear geometry to produce and investigate single domains of other biaxial nematics, reported but still questioned in the literature. Received 26 May 1999     

Atomic Rayleigh scattering cross-sections and the associated anomalous dispersion in the X-ray regions

Elastic scattering cross-sections for Pd, Ag, Cd, In, Sn, Sb, Pt, Au and Pb are measured at an angle of 90 in the X-ray region 5.41 keV. These energies fall between the high-energy side of the L- and M-shell absorption edges of the atoms considered. The present atomic region is significant for solid X-rays to assess the contribution of resonance and solid-state environmental effects. Also it is the anomalous scattering region for many of the atoms of the periodic table. Experimental results are compared with theoretical calculations based on form factor formalisms including the anomalous corrections and available recent S-matrix values. Based on the experimental evidence, the present results indicate the influence of solid-state environmental effects, the importance of anomalous corrections nearer to absorption edges, the correctness of revised high-energy limit values, the superiority of S-matrix predictions over form factor values on measured elastic scattering cross-sections in the X-ray regime and also show the resonance behavior around K, L and M absorption edges. Received: 27 January 1998 / Received in final form: 4 January 1999     

A small-angle X-ray scattering study of the isotropic and nematic phases of V2O5 suspensions

We report small-angle X-ray scattering experiments performed in both the isotropic and nematic phases of aqueous V₂O₅ suspensions. We show that the scattering in the isotropic phase can be well described in the whole accessible q-range by only considering the form factor of non-interacting ribbons. We investigate the influence of concentration and pH on the dimensions of V₂O₅ ribbons and show that these parameters do not have any significant effect, as long as the system stays well within the chemical stability domain of the ribbons. We then show that nematic single domains display an anisotropic small-angle scattering pattern, even at scattering vectors small compared to that at which a characteristic correlation peak is observed. This feature is expected for a nematic phase, but was rarely observed. We finally try to describe this scattering within the framework of theories developed for the structure factor of a nematic polymer, and we reach the conclusion that chain ends are certainly important to understand this pattern. Received 21 July 1999 and Received in final form 17 December 1999     

A comprehensive,time-resolved SANS investigation of temperature-change-induced sponge-to-lamellar and lamellar-to-sponge phase transformations in comparison with <Superscript>2</Superscript>H -NMR results

Time-resolved small-angle neutron scattering (TR-SANS) was employed to observe temperature-induced phase transitions from the sponge (L ₃ to the lamellar ( L _α phase, and vice versa, in the water-oil (n -decane)-non-ionic surfactant ( C₁₂E₅ system using both bulk and film contrast. Samples of different bilayer volume fractions φ and solvent viscosities η were investigated applying various amplitudes of temperature jump ΔT . The findings of a previous ²H -NMR study could be confirmed, where the lamellar phase formation was determined to occur through a nucleation and growth process, while it was concluded that the L ₃ -phase develops in a mechanistically different and more rapid manner involving uncorrelated passage formation. Likewise, the kinetic trends of the nucleation and growth transition (decreased transition time with increase of φ and ΔT were witnessed once again. Additionally, NMR and SANS data that demonstrate a strong dependency of that process on solvent viscosity η are presented. Contrariwise, it is made evident via both SANS and NMR results that the L _α -to-L ₃ transition time is independent (within experimental sensitivity) of the varied parameters (φ , ΔT , η . Unusual scattering evolution in one experiment, originating from a highly ordered lamellar phase, intriguingly hints that a major rate determining factor is the disruption of long-range order. Furthermore, the bulk contrast investigations give insight into structure peak shifts/development during the transitions, while the film contrast experiments prove the bilayer thickness to be constant throughout the phase transitions and show that there is no evidence for a change in the short-range order of the bilayer structure. The latter was considered possible, due to the different topology of the L ₃ and L _α phases. Lastly, an unexpected yet consistent appearance of anisotropic scattering is detected in the L ₃ -to- L _α transitions.     

Polymorphism and chevron layer structure of an antiferroelectric liquid crystal studied by X-ray diffraction

We report X-ray diffraction experiments performed on an antiferroelectric compound exhibiting a very rich polymorphism (). The structural study of the unknown phases only allows us to exclude some phenomenological models. The use of oriented planar samples prepared between solid glass plates generate by cooling from the phase a chevron structure of tilted layers already well characterized for the phase. The extensive analysis of the evolution of the chevron structure through the numerous smectic-smectic phase transitions provides some original information in three distinct areas: fundamental data on the important physical parameters in the chevron structure formation, detection of the smectic-smectic phases transition by small change of the chevron structure, and information on the local molecular order induced by the alignment layer (interaction with a rubbed polymer). Received: 13 November 1996 / Received in final form: 19 January 1997 / Accepted: 30 January 1998     

Lamellar-to-nematic phase transition in a lipid-surfactant mixture

A lyotropic system, consisting of a lecithin (DMPC) and a non-ionic surfactant (C₁₂E₅) in water was studied. The system exhibits a lamellar-to-nematic phase transition. The nematic phase appears as the temperature is decreased and only exists in a very limited temperature and concentration range, for specific lipid-to-surfactant ratios. While a lamellar phase is found at higher temperatures in both mixed and pure C₁₂E₅ systems, the transition to the nematic phase at lower temperatures coincides with a micellar phase in the pure C₁₂E₅ system. The transition appears to be driven by the strong temperature dependence of the surfactant film spontaneous curvature. The structural properties of the lamellar phase close to the lamellar-to-nematic boundary have been studied by polarised light microscopy and small-angle neutron and X-ray scattering experiments. The signature of a helical defect with Burgers vector of magnitude 2 is apparent in our data, close to the lamellar-to-nematic phase transition. The proliferation of screw dislocations in the lamellar phase might be a plausible mechanism for driving this transition. Received 6 July 1999 and Received in final form 17 April 2000     

Ordering phenomena in C<Subscript>60</Subscript>-tetraphenylphosphonium bromide

The fulleride salt C₆₀-tetraphenylphosphonium bromide is investigated as a function of temperature by single crystal X-ray diffuse scattering and diffraction. At room temperature, the C₆₀ orientational disorder is found to be more complex than previously expected. Moreover, a structural phase transition, due to the C₆₀ orientational ordering, is evidenced around 120 K. Its relation with the stabilization of a static Jahn-Teller effect is discussed. Received 3 November 1999     

Structural evidence of charge renormalization in semi-dilute solutions of highly charged polyelectrolytes

We show experimentally that Manning counterion condensation also leads to a renormalization of the charge density at high concentrations of highly charged, flexible, hydrophilic polyelectrolytes. Investigations by small angle neutron and X-ray scattering of semi-dilute solutions of poly(acrylamide-co-sodium-2-acrylamido-2-methylpropane sulfonate) at different charge densities above the condensation threshold, show that the scattering function is invariant with the charge density. Received 16 June 1998     

Induced ferrielectricity in antiferroelectric liquid crystals

The behaviour of the antiferroelectric SmC_A liquid crystal phase under applied electric field is discussed theoretically. The phase diagram involving the SmA, SmC_A and SmC _A ^* phases is worked out and shown to exhibit a Lifshitz critical point. The deformation of the bilayer structures induced by the field transforms the SmC_A phases into a ferrielectric phase whose specific configuration is described. Received: 23 October 1997 / Revised: 8 April 1998 / Accepted: 14 July 1998     

Copper and copper oxide nanoparticles in a cellulose support studied using anomalous small-angle X-ray scattering

Microcrystalline cellulose is a porous natural material which can be used both as a support for nanoparticles and as a reducer of metal ions. Cellulose supported nanoparticles can act as catalysts in many reactions. Cu, CuO, and Cu₂O particles were prepared in microcrystalline cellulose by adding a solution of copper salt to the insoluble cellulose matrix and by reducing the copper ions with several reducers. The porous nanocomposites were studied using anomalous small angle X-ray scattering (ASAXS), X-ray absorption spectroscopy, and X-ray diffraction. Reduction of Cu²⁺ with cellulose in ammonium hydrate medium yielded crystalline CuO nanoparticles and the crystallite size was about 6–20 nm irrespective of the copper concentration. The size distribution of the CuO particles was determined with ASAXS measurements and coincided with the crystallite sizes. Using sodium borohydrate or hydrazine sulfate as a reducer both metallic Cu and Cu₂O nanoparticles were obtained and the crystallite size and the oxidation state depended on the amount of reducer.     

Structural characterization of Rh/pumice SMAD catalysts

The structure of Rh/pumice catalysts prepared by the SMAD (Solvated Metal Atoms Dispersion) technique at different metal loadings has been investigated by EXAFS (Extended X-ray Absorption Fine Structure Spectroscopy), XPS (X-ray Photoelectron Spectroscopy), SAXS (Small-Angle X-ray Scattering), WAXS (Wide-Angle X-ray Scattering) and TEM (Transmission Electron Microscopy). According to EXAFS and XPS, a fraction of the Rh atoms is oxidised, but a noticeable part is also present as Rh ⁰. The Rh oxidation is attributed to the interaction of the Rh atoms with the hydroxyls of the support; after the formation of the oxide, the nucleation of metallic rhodium becomes possible. The WAXS data do not show evidence of rhodium fcc crystallites; the metal-bearing particles are probably amorphous and/or very small, as results from the SAXS and TEM data analysis. The disagreement between the latter two techniques, resulting in a small-angle determination of the average size of the particles that is about half that of TEM in the catalyst with the higher Rh loading, is acknowledged and discussed. Preliminary catalytic tests are described, demonstrating the suitability of using a low surface area support for the preparation of SMAD catalysts. Received 2 February 1999     

Depletion effects in smectic phases of hard-rod-hard-sphere mixtures

It is known that when hard spheres are added to a pure system of hard rods the stability of the smectic phase may be greatly enhanced, and that this effect can be rationalised in terms of depletion forces. In the present paper we first study the effect of orientational order on depletion forces in this particular binary system, comparing our results with those obtained adopting the usual approximation of considering the rods parallel and their orientations frozen. We consider mixtures with rods of different aspect ratios and spheres of different diameters, and we treat them within Onsager theory. Our results indicate that depletion effects, and consequently smectic stability, decrease significantly as a result of orientational disorder in the smectic phase when compared with corresponding data based on the frozen-orientation approximation. These results are discussed in terms of the τ parameter, which has been proposed as a convenient measure of depletion strength. We present closed expressions for τ, and show that it is intimately connected with the depletion potential. We then analyse the effect of particle geometry by comparing results pertaining to systems of parallel rods of different shapes (spherocylinders, cylinders and parallelepipeds). We finally provide results based on the Zwanzig approximation of a fundamental-measure density-functional theory applied to mixtures of parallelepipeds and cubes of different sizes. In this case, we show that the τ parameter exhibits a linear asymptotic behaviour in the limit of large values of the hard-rod aspect ratio, in conformity with Onsager theory, as well as in the limit of large values of the ratio of rod breadth to cube side length, d, in contrast to Onsager approximation, which predicts τ ∼ d ³. Based on both this result and the Percus-Yevick approximation for the direct correlation function for a hard-sphere binary mixture in the same limit of infinite asymmetry, we speculate that, for spherocylinders and spheres, the τ parameter should be of order unity as d tends to infinity.     

Defects in a TGBA phase: A theoretical approach

Following our experimental observations of disclination lines in freely suspended droplets and free-standing films (Yu.A. Nastishin et al., Eur. Phys. J. E 5 353 (2001)), topological defects of the twist grain boundary (TGBA) phase are considered according to two aspects: topological and energetical. There are two classes of line defects, disclinations (as in the cholesteric (N*) phase and the liquid vortices phase (NL*), relating to the directors tripod symmetries) and dispirations (relating to the translation-rotation symmetries); there are no topological point defects. Differences between N*, NL* and TGBA disclinations are physical, not topological. The absence of focal conic domains in the TGBA phase is an immediate consequence of the materialization of the helical axis (along the χ-director); the same feature, coupled to the trend to parallelism of the smectic layers, accounts for the predominance of λ-lines. Finally, the presence of defects akin to developable domains is explained in the frame of the leastcurvaturemodel, that requires the introduction of a third type of defects: the densitiesofedgedislocations of the smectic layers. Received 20 February 2002     

Macroscopic properties of smectic liquid crystals

We discuss the macroscopic behavior of smectic C_G liquid crystals. Smectic C_G is the most general tilted smectic phase that is fluid in the layers. It is characterized by global C₁ symmetry. Consequently, it is ferroelectric, pyroelectric and piezoelectric, opening up a number of possible applications for such a phase. As smectic C_G-phase has a macroscopic hand due to its structure, it is a natural candidate to explain the recent experimental observations of left and right-handed helices in a system composed of achiral molecules. We also discuss critically to what extent smectic C_G could be important for liquid crystalline phases formed by banana-shaped molecules. Phase transitions involving a smectic C_G phase and defects of its in-plane director are briefly discussed. Received: 25 March 1998 / Revised: 15 June 1998 / Accepted: 15 July 1998     

Optical reflectivity study of synclinic and anticlinic structures in thin freely suspended smectic films

Optical reflectivity studies have been conducted on freely suspended films with synclinic and anticlinic structures. For the first time quantitative data were obtained on orientational ordering in films with anticlinic structure. In the same wide temperature range of more than 30 K we observed both transverse and longitudinal ferroelectricity in the films of the same thickness. Equilibrium transition temperatures between structures with transverse and longitudinal polarization were above the bulk transition temperature to the SmC phase. The molecule tilt was determined in superthin films with synclinic and anticlinic ordering. Received 29 October 1999 and Received in final form 17 February 2000     

Spontaneous condensation in DNA-polystyrene- b-poly(l-lysine) polyelectrolyte block copolymer mixtures

We investigated the condensation of calf thymus DNA by amphiphilic polystyrene_m-b-poly(l-lysine)_n block copolymers ( PS_m-b- PLys_n, m, n = degree of polymerization), using small-angle X-ray scattering, polarized optical microscopy and laser scanning confocal microscopy. Microscopy studies showed that the DNA condenses in the form of fibrillar precipitates, with an irregular structure, due to electrostatic interactions between PLys and DNA. This is not modified by the presence of hydrophobic PS block. Scattering experiments show that the structure of the polyplexes corresponds to a local order of DNA rods which becomes more compact upon increasing n. It can be concluded that for DNA/ PS_m-b- PLys_n polyplexes, the balance between the PLys block length and the excess charge in the system plays an essential role in the formation of a liquid crystalline phase.     

Attractive forces between anisotropic inclusions in the membrane of a vesicle

The fluctuation-induced interaction between two rod-like, rigid inclusions in a fluid vesicle is studied by means of canonical ensemble Monte-Carlo simulations. The vesicle membrane is represented by a triangulated network of hard spheres. Five rigidly connected hard spheres form rod-like inclusions that can leap between sites of the triangular network. Their effective interaction potential is computed as a function of mutual distance and angle of the inclusions. On account of the hard-core potential among these, the nature of the potential is purely entropic. Special precaution is taken to reduce lattice artifacts and the influence of finite-size effects due to the spherical geometry. Our results show that the effective potential is attractive and short-range compared with the rod length L. Its well depth is of the order of , where is the bending modulus. Received 5 February 1999 and Received in final form 14 May 1999     

Competition between ferroelectric and flexoelectric polarization in freely suspended smectic films

We report a detailed ellipsometric study of freely suspended films of chiral liquid-crystal compounds possessing smectic-A and smectic-C phases. In the temperature region between the smectic-A - smectic-C bulk and surface transitions, a discontinuous reconstruction of the tilt profile across the film is observed in the presence of a constant d.c. electric field. Comparison of the measured ellipsometric quantities with values calculated from model tilt profiles reveals a competition between a structure possessing a homogeneous tilt direction and large ferroelectric polarization and a structure with opposite tilt direction in the two film halfs and large flexoelectric polarization. Received 21 October 1998