08 Fakultät Mathematik und Physik
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Item Open Access Liquid-crystalline blue phase III and structures of broken icosahedral symmetry(1993) Longa, Lech; Fink, Werner; Trebin, Hans-RainerThe structure of the liquid-crystalline blue phase III (BPIII) is still unknown and remains one of the mysteries of liquid-crystal physics. We take all icosahedral space-group symmetries of the reciprocal space for BPIII and study their thermodynamic stability within the frame of an extended de Gennes–Ginzburg–Landau free-energy expansion. The stability of the icosahedral structures is compared with that of the cholesteric phase and of the cubic blue phases. Strikingly, even though the extended model contains three extra parameters, we could not detect a region of parameter space where icosahedral structures are absolutely stable just below the isotropic phase.Item Open Access Theory of liquid crystalline phases in biaxial systems(1992) Longa, Lech; Trebin, Hans-RainerGeneral properties of SO(3) - symmetric free- energy expansion for biaxial systems are studied. In particular, all invariants in powers of a traceless and symmetric quadrupole tensor order parameter and a vector order parameter are identified and their relation to possible local structures are found. A new class of polar, chiral biaxial phases are predicted.Item Open Access Spontaneous polarization in chiral biaxial liquid crystals(1990) Longa, Lech; Trebin, Hans-RainerA phenomenological theory of polar structures in chiral biaxial liquid crystals is constructed exploiting the properties of a symmetric and traceless tensor order parameter field Q αβ(r) and of a polar field Pα(r). Full advantage is taken of the symmetry of the order parameters by systematic use of the method of integrity bases, which allows us to establish an expansion of the most general SO(3)-invariant free-energy density to arbitrary powers in the components Qαβ and Pα. A coordinate-independent parametrization of the invariants is introduced that yields a classification of local polar structures and some predictions about possible topologies of phase diagrams without the necessity of performing numerical calculations. As one prominent result, the theory predicts a polar, chiral biaxial state that exists due to a piezoelectric coupling of a chiral biaxial tensor field and the polarization field and which disappears if tensor is uniaxial. We then provide a general theory of flexopolarization in biaxial systems. A general biaxial system is described by 12 fundamental flexopolarization modes. Special cases, obtained by imposing symmetry restrictions to the tensor field Q, reduce the number of modes. Finally, the theory is applied to chiral phases. Simple polar chiral structures including cholesteric and smectic-C* liquid crystals are analyzed. In particular, it is shown that if the smectic-C* phase is stabilized due to the piezoelectric coupling between Q, P, and a density wave, then it must be described as a biaxial uniform spiral with at least two nonvanishing commensurate harmonics. The minimization of the quadratic part of the Landau–de Gennes energy supplemented by (flexo)polarization terms may give rise to incommensurate two- or three-dimensional polar structures that can be stabilized by cubic terms.Item Open Access Bond orientational order in the blue phases of chiral liquid crystals(1993) Longa, Lech; Trebin, Hans-RainerIt is proposed to describe blue phases by two order parameters: the standard alignment tensor field Q αβ(r) and a bond orientational tensor order parameter of octahedral point group symmetry scrO(432). The yet mysterious blue fog then emerges as a liquid of purely cubic bond orientational order. In the transition from the cubic blue phases to the blue fog the cubic space group symmetry is being reduced to its octahedral factor group. Because of the new order parameter the scrO 5(scrI432) structure, which in all previous calculations proved most stable, but never has been detected in experiment, is eliminated from the phase diagram.Item Open Access Integrity basis approach to the elastic free energy functional of liquid crystals. 1, Classification of basic elastic modes(1989) Longa, Lech; Trebin, Hans-RainerUsing the method integrity basis, the most general SO(3)-invariant free energy density up to all powers in xβ and up to second order in Q xβ,y is established. The method provides all analytically independent elastic modes for nematics and cholesterics in the form of 33 so-called, irreducible invariants. Interestingly, among the irreducible invariants there are only three chiral terms (i.e. linear in Q δ,β,y ). They give rise locally to three independent helix modes in chiral, biaxial liquid crystals. This conclusion generalizes results of Trebin and Govers and Vertogen and contradicts a statement of Pleiner and Brandt, according to which only one twist term is supposed to exist. The most general free energy expansion can be written as sum of 39 additive invariants, which are multiplied by arbitrary polynomials in TrQ 2 and TrQ 3.Item Open Access Structure of the elastic free energy for chiral nematic liquid crystals(1989) Longa, Lech; Trebin, Hans-RainerIn Landau–de Gennes theory, the free energy f of liquid crystals is expanded into powers of a symmetric, traceless tensor order parameter Q αβ and its derivatives Q αβ,γ. The expansion is subject to the condition that f is a scalar, i.e., invariant under all rotations of the group SO(3). Using the method of integrity basis, we have established the most general SO(3)-invariant free-energy density up to all powers in Q αβ and up to second order in Q αβ,γ. It turns out that this free-energy density is composed of 39 invariants, which are multiplied by arbitrary polynomials in TrQ 2 and TrQ 3. On the other hand, these 39 invariants can be expressed as polynomials of 33 so-called irreducible invariants. Interestingly, among the irreducible invariants there are only three chiral terms (i.e., linear in Q αβ,γ). They locally give rise to three independent helix modes in chiral, biaxial liquid crystals. This conclusion generalizes results of Trebin [J. Phys. (Paris) 42, 1573 (1981)] and Govers and Vertogen [Phys. Rev. A 31, 1957 (1985); 34, 2520 (1986)] and contradicts a statement of Pleiner and Brand [Phys. Rev. A 24, 2777 (1981); 34, 2528 (1986)], according to which only one twist term is supposed to exist.Item Open Access Biaxiality of chiral liquid crystals(1994) Longa, Lech; Fink, Werner; Trebin, Hans-RainerUsing the extended de Gennes–Ginzburg–Landau free energy expansion in terms of the anisotropic part Q αβ(x) of the dielectric tensor field, a connection between the phase biaxiality and the stability of various chiral liquid crystalline phases is studied. In particular, the cholesteric phase, the cubic blue phases, and the phases characterized by an icosahedral space group symmetry are analyzed in detail. Also, a general question concerning the applicability of the mean-field approximation in describing the chiral phases is addressed. By an extensive study of the model over a wide range of the parameters, a class of phenomena, not present in the original de Gennes–Ginzburg–Landau model, has been found. These include (a) reentrant phase transitions between the cholesteric and the cubic blue phases and (b) the existence of distinct phases of the same symmetry but of different biaxialities. The phase biaxiality serves here as an extra scalar order parameter. Furthermore, it has been shown that, due to the presence of competing bulk terms in the free energy, the stable phases may acquire a large degree of biaxiality, also in liquid crystalline materials composed of effectively uniaxial molecules. A study of icosahedral space group symmetries provides a partial answer to the question of whether or not an icosahedral quasicrystalline state can be stabilized in liquid crystals. Although, in general, the stability of icosahedral structures could be enhanced by the extra terms in the free energy, no absolutely stable icosahedral phase has been found.Item Open Access Phase diagrams of cholesteric liquid crystals obtained with a generalized Landau-de Gennes theory(1989) Longa, Lech; Monselesan, Didier; Trebin, Hans-RainerPhase diagrams of chiral nematic liquid crystals are studied within the framework of a generalized Landau-Ginzburg-de Gennes theory. Using the parametrization of Grebel, Hornreich, and Shtrikman for the tensor order parameter Q, all relevant elastic terms are included for the helicoidal phase and the blue phases of chiral nematic liquid crystals up to fourth order in Q and its gradient ∂Q. The influence of the additional elastic terms on the phase diagrams of the chiral nematic phases is then investigated. The theory correctly describes the variation of the pitch with temperature and the induced biaxiality of the cholesteric phase. The results resolve the discrepancies encountered by Hornreich and Shtrikman in the comparison of experiment and theory. New features in the topology of the phase diagrams of blue phases, like re-entrant phase transitions, are predicted.