The term photonic crystals appears because of the analogy between electron waves in crystals and the light waves in artificial periodic dielectric structures. During the recent years the investigation of one-, two-and three-dimensional periodic structures has attracted a widespread attention of the world optics community because of great potentiality of such structures in advanced applied optical fields. The interest in periodic structures has been stimulated by the fast development of semiconductor technology that now allows the fabrication of artificial structures, whose period is comparable with the wavelength of light in the visible and infrared ranges.

http://ieeexplore.ieee.org/iel5/6354/16969/00781867.pdf

Photonic crystals

This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

Most associate liquid crystals with their everyday use in laptop computers, mobile phones, digital cameras, and other electronic devices. However, in contrast to their rodlike (calamitic) counterparts, first described in 1907 by Vorlander, disklike (discotic, columnar) liquid crystals, which were discovered in 1977 by Chandrasekhar et al., offer further applications as a result of their orientation in the columnar mesophase, making them ideal candidates for molecular wires in various optical and electronic devices such as photocopiers, laser printers, photovoltaic cells, light-emitting diodes, field-effect transistors, and holographic data storage. Beginning with an overview of the various mesophases and characterization methods, this Review will focus on the major classes of columnar mesogens rather than presenting a library of columnar liquid crystals. Emphasis will be given to efficient synthetic procedures, and relevant mesomorphic and physical properties. Finally, some applications and perspectives in materials science and molecular electronics will be discussed.

Discotic Liquid Crystals: From Tailor-Made Synthesis to Plastic Electronics

An overview on the recent developments in the field of liquid crystalline bent-core molecules (so-called banana liquid crystals) is given. After some basic issues, dealing with general aspects of the systematisation of the mesophases, development of polar order and chirality in this class of LC systems and explaining some general structure–property relationships, we focus on fascinating new developments in this field, such as modulated, undulated and columnar phases, so-called B7 phases, phase biaxiality, ferroelectric and antiferroelectric polar order in smectic and columnar phases, amplification and switching of chirality and the spontaneous formation of superstructural and supramolecular chirality.

/pdf/bent-core-liquid-crystals-polar-order-superstructural-17m3bst2sc.pdf

Bent-core liquid crystals: polar order, superstructural chirality and spontaneous desymmetrisation in soft matter systems

Structures and properties of liquid crystalline phases formed by bent-core molecules are reviewed. At least eight phases designated as B1–B8 have been found, being unambiguously distinguished from phases formed by usual calamitic molecules due to a number of remarkable peculiarities. In addition to B1–B8 phases, smectic A-like phases and biaxial nematic phases formed by bent-core molecules are also reviewed. The most attractive aspects of this new class of liquid crystals are in polarity and chirality, despite being formed from achiral molecules. The bent-core mesogens are the first ferroelectric and antiferroelectric liquid crystals realized without introducing chirality. Spontaneous chiral deracemization at microscopic and macroscopic levels occurs and is controllable. Moreover, achiral bent-core molecules enhance system chirality. The interplay between polarity and chirality provides chiral nonlinear optic effects. Further interesting phenomena related to polarity and chirality are also reviewed.

https://iopscience.iop.org/article/10.1143/JJAP.45.597/pdf

Bent-Core Liquid Crystals: Their Mysterious and Attractive World

Broadband dielectric spectroscopy (10(3) Hz-1.8 × 10(9) Hz) has been performed on the odd nonsymmetric liquid crystal dimers of the series α-(4-cyanobiphenyl-4'-oxy)-ω-(1-pyreniminebenzylidene-4'-oxy) alkanes (CBOnO.Py) with n ranging from 3 to 11, as a function of temperature. A previous thermal behavior study through heat capacity measurements has been made. Dielectric measurements enable us to obtain information about the molecular dynamics in the nematic mesophase as well as in the isotropic phase. Two orientations (parallel and perpendicular) of the molecular director with regard to the probe electric field have been investigated. In the nematic mesophase, the dielectric anisotropy is revealed to be positive for all studied compounds. Measurements of the parallel component of the dielectric permittivity are well-explained by means of the molecular theory of dielectric relaxation in nematic dimers (Stocchero, M.; Ferrarini, A.; Moro, G. J.; Dunmur, D. A.; Luckhurst, G. R. J. Chem. Phys., 2004, 121(16), 8079). The dimer is seen as a mixture of cis and trans conformers, and the model allows us to estimate their relative populations at each temperature. The main molecular motions are interpreted by the model as independent end-overend rotations of each terminal semirigid unit of the dimer.

Overall dielectric study on the odd members of a highly nonsymmetric pyrene-based series of liquid crystal dimers.

The piezoelectric polymeric sensors and actuators are widely valued because of their low density, availability to obtain complex shapes, and easy processing among other characteristics. However, there are not many useful piezoelectric polymers. Aromatic polyimides are high-performance polymeric materials characterized by thermal stability, chemical resistance, and excellent mechanical properties. Besides, some of these polymers have been reported as candidates to present piezoelectric properties. Our research focuses in the electric characterization of three piezoelectric amorphous aromatic polyimides, containing CN groups in different number and positions. The piezoelectricity in amorphous polymers is mainly due to the orientation polarization of the molecular dipoles, which is induced by the application of an external electrical field to a temperature over their glass transition temperature (Tg). Polyimides are measured by thermally stimulated depolarization currents and dielectric spectroscopy and the analysis of the dielectric data has been performed using the electric modulus formalism to separate conductive and dipolar processes. These measurements have evidenced that the frozen-in polarization is mainly due to the dipole orientation of the dipolar CN groups, allowing us to understand in depth the mechanism of polarization that contributes to the piezoelectric properties. This will facilitate the obtention of materials with the best possible piezoelectric properties, comparable to those currently available but with higher mechanical and thermal performance. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2012

Electric modulus and polarization studies on piezoelectric polyimides

Synthesis and study of .beta.-chlorhydrine and .alpha.-chloroacid ferroelectric liquid-crystal derivatives

The complex dielectric function of the orientationally disordered molecular mixed crystal [(CH3)3CCH2OH]0.7[(CH3)2C(CH2OH)2]0.3 (NPA0.7NPG0.3) covering 11 decades in frequency (10-2-109?Hz) is measured over a broad and continuous temperature range (163-333?K) which is close to the orientational glass transition (160.9?K). For the first time, the dynamics of the orientational disordered (OD), the supercooled orientational disordered and the orientational disordered glass is studied without a temperature breakdown as usually occurs due to the catastrophe of the crystallization. The orientational glass transition in the OD phase exhibits the essential features of the structural glass transition, a non-Arrhenius behaviour for the temperature variation of the relaxation time and a broadening of the width of the relaxation with decreasing temperature. The temperature dependence of the relaxation time is analysed by the empirical Vogel-Fulcher-Tammann law as well as the more recent mode-coupling theory. The suggested increase of the ratio between the crossover temperature Tc and glass transition temperature Tg with decrease of fragility of the system is in accordance with the present observation for this mixed crystal (Tc/Tg?1.45).

Relaxation dynamics in orientationally disordered molecular mixed crystal [(CH3)3CCH2OH]0.7[(CH3)2C(CH2OH)2]0.3

The dielectric relaxation spectra of NPA (CH2 OHC(CH3 )3 ) and TCE (CH2 OHCCl3 ) have been determined in the orientationally disordered state (ODIC) and for TCE also in the liquid state. The studied temperature ranges have been 313-233 K and 293-255 K for NPA and TCE respectively. The frequency dependence of the complex dielectric permittivity is well described in both cases by the Cole-Davidson function. From the fits to this equation, dielectric strengths, relaxation times and spectral shape parameters as functions of temperature have been obtained. The relaxation times are thermally activated following an Arrhenius law for NPA and a Vogel-Fulcher-Tammann law for TCE. Likewise, for NPA a new intermediate ODIC state has been stated by means of dielectric and x-ray diffraction measurements.

M. A. Perez-Jubindo

Papers

Overall dielectric study on the odd members of a highly nonsymmetric pyrene-based series of liquid crystal dimers.

Electric modulus and polarization studies on piezoelectric polyimides

Synthesis and study of .beta.-chlorhydrine and .alpha.-chloroacid ferroelectric liquid-crystal derivatives

Relaxation dynamics in orientationally disordered molecular mixed crystal [(CH3)3CCH2OH]0.7[(CH3)2C(CH2OH)2]0.3

Dielectric spectroscopy on two orientationally disordered crystals: NPA (CH2OHC(CH3)3) and TCE (CH2OHCCl3)