Long-range Orientational Order Research Articles

Orientational order in disordered superconductors.

Orientational order in weakly pinned Abrikosov flux lattices is studied, taking into account two heretofore neglected effects: dislocations and orientational couplings to the underlying lattice. Without orientational couplings, arbitrarily weak pinning destroys long-ranged orientational order for all spatial dimensions {ital d}{lt}4. Orientational couplings stabilize long-ranged orientational order. For fields along an axis of {ital fourfold} symmetry, {ital sixfold} (hexatic) orientational order is described by a random-field Ising model, and so does not occur in {ital d}=2 (thin films) but does in {ital d}=3 (bulk).

Two-dimensional mixed crystals.

We show that the long-range positional order in a two-dimensional crystal ${\mathit{A}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathit{B}}_{\mathit{x}}$ is destroyed by size-mismatch disorder. The size-mismatch disorder is characterized by an effective temperature ${\mathit{k}}_{\mathit{B}}$${\mathit{T}}_{\mathit{D}}$=Kx(1-x)(${\mathit{L}}_{\mathit{B}}^{0}$-${\mathit{L}}_{\mathit{A}}^{0}$${)}^{2}$ associated with the strain energy, where K is a force constant and ${\mathit{L}}_{\mathit{B}}^{0}$-${\mathit{L}}_{\mathit{A}}^{0}$ is the length mismatch. We study a model that has a fixed triangular-net topology. By comparison with computer simulations, we show that a linearized small displacement theory is adequate for small size mismatches. The long-range orientational order remains. The positional correlation function decays algebraically, which leads to power-law peaks that replace the Bragg peaks in the diffraction pattern. We argue that this model should provide a reasonable qualitative description of real two-dimensional mixed crystals, in the limit of a small size mismatch.

Atomic force microscopy of hydrated phosphatidylethanolamine bilayers

We present images of the polar or headgroup regions of bilayers of dimyristoyl-phosphatidylethanolamine (DMPE), deposited by Langmuir-Blodgett deposition onto mica substrates at high surface pressures and imaged under water at room temperature with the optical lever atomic force microscope. The lattice structure of DMPE is visualized with sufficient resolution that the location of individual headgroups can be determined. The forces are sufficiently small that the same area can be repeatedly imaged with a minimum of damage. The DMPE molecules in the bilayer appear to have relatively good long-range orientational order, but rather short-range and poor positional order. These results are in good agreement with x-ray measurements of unsupported lipid monolayers on the water surface, and with electron diffraction of adsorbed monolayers.

Path-integral Monte Carlo study of the hard-disk solid.

We present a study of the properties of the quantum two-dimensional hard-disk solid at finite temperatures using path-integral Monte Carlo simulations. We performed extensive simulations on systems with 64, 144, 256, 576, and 1024 disks. We used the image approximation for the high-temperature density matrix and ignored exchange effects. Despite the discontinuous nature of the hard-disk interaction, a long-wavelength harmonic theory provided a good model for the solid. This theory was used as a guide to study the long-range translational and orientational order of the solid at low but finite temperatures where the zero-point motion and finite-temperature excitations are both important. The properties we have computed include the particle displacements, structure factor, angular order, and density of topological defects. We compare these properties with those of the same system in the classical regime and at absolute zero, thus gaining some understanding for the crossover from classical to quantum behavior.

Correlation analysis of gap junction lattice images

Fourier averages of connexon images computed from low-irradiation electron micrographs of isolated negatively stained gap junction domains exhibited differences in stain distribution and connexon orientation. To analyze these polymorphic structures, correlation averaging methods were applied to images from negatively stained and frozen-hydrated specimens. For the negatively stained specimens, separate averages over two subsets of connexons with differing degrees of stain accumulation in the axial channel were obtained. Two populations of connexons with opposite skew orientations were distinguishable within a single junctional domain of a frozen-hydrated specimen. Correlation maps calculated using the left- and right-skewed references showed that the selected connexons tend to locally cluster. Using correlation methods to analyze packing disorder in a typical connexon lattice, we estimated the root-mean-square variation in the nearest neighbor pair separation to be approximately 11% of the lattice constant. Displacements of the connexons relative to each other increased with increasing pair separation in the lattice, rather like a liquid, although long-range orientation order was conserved as in a crystal. These results support the hypothesis that the hexagonal ordering of the connexons results from short-range repulsive forces.

Partial order in phospholipid monolayers

Phospholipid monolayers at the air/water interface are studied by fluorescence microscopy and by X-ray scattering techniques. Due to long-range electrostatic forces superlattices of domains of ordered lipid are observed. For lateral densities between an isotropic fluid (LE) and a nearly incompressible (S) phase, an additinal phase is detected and characterized. This phase (LC) exhibits long-range bond orientational and short-range positional order as expected for a hexatic phase. Yet it has lower than sixfold symmetry due to uniform tilt of the aliphatic tails with tilt angle decreasing on compression. Reflectivity data are in agreement with the view that the transition to the S phase involves head group ordering. At lower temperature, the LE phase is suppressed but the LC phase exhibits the same features as at room temperature with respect to lattice spacings, positional coherence length and tilt angle. These features appear to be rather general for many phospholipids and for single chain surfactants like fatty acids.

Landau-model for the λ-transition in NaNo3

Abstract A Landau free energy model with two biquadratically coupled order parameters adequately describes the non-classical (β = 0.22) temperature evolution of the orientational long-range order in NaNO3, and explains the cross-over to quasi-tricritical (β = 0.25) behaviour below 460K and the observed tail of excess order above the critical temperature of the λ-transition.

(KI)1-x(NH4I)x: A dipolar glass.

Solid solutions of (KI${)}_{1\mathrm{\ensuremath{-}}\mathit{x}}$(${\mathrm{NH}}_{4}$I${)}_{\mathit{x}}$ with concentrations x=0.14, 0.43, and 1 were investigated using dielectric spectroscopy. The high-temperature rocksalt structure of these compounds exhibits a dipole moment of 1.4 D. For x=0.14 and 0.43 the NaCl structure is stable down to the lowest temperatures and the reorienting dipoles undergo a cooperative freezing transition devoid of long-range orientational order.

Dynamics of orientational glass formation in Rb(CN)xBr1-x mixed crystals.

Measurements of the ultrasonic attenuation \ensuremath{\alpha} associated with the ${\mathit{c}}_{44}$ shear mode in Rb(CN${)}_{\mathit{x}}$${\mathrm{Br}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$ single crystals are presented. The cooperative quadrupolar clamped (constant-strain) relaxation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{s}}$ is determined as a function of temperature and frequency. In a sample with x>${\mathit{x}}_{\mathit{c}}$\ensuremath{\approxeq}0.55, long-range orientational ordering occurs and the free (constant-stress) time \ensuremath{\tau} diverges with the dynamical critical exponent z\ensuremath{\nu}\ensuremath{\approxeq}3.7. In samples with x${\mathit{x}}_{\mathit{c}}$, no phase transition occurs and an orientational glass is obtained at low temperatures. The dramatic increase observed in ${\mathrm{\ensuremath{\tau}}}_{\mathit{s}}$ on cooling toward the glass transition temperature is discussed in terms of several models.

Elastic stability for a discotic liquid crystal in the hexatic N+6 phase

The elastic stability of a discotic liquid crystal in the theoretically predicted hexatic N+6 phase against fluctuations of orientational order is tested. For this purpose, the author derives the elastic stability conditions for Frank constants of a discotic liquid crystal, and finds that the coupling constant gamma 3, which couples the director distortions to local rotations of the hexagonal two-dimensional lattice, is forced to be weak with respect to the other Frank constants, in order to preserve the long-range sixfold orientational order. He then shows that critical enhancements of Frank elastic constants, in the hexatic N+6 phase near the supposed continuous transition to the hexagonal discotic phase, fulfil the elastic stability conditions previously derived. Critical fluctuations of orientational order, therefore, are not able either to decorrelate the hexatic phase or to drive the transition to first order. Such a fluctuation-induced first-order transition, on the contrary, is known to play a role in smectic liquid crystals. We conclude that the N+6 phase remains orientationally correlated in the critical region, which proves the self-consistency of the model assuming such an intermediate phase, as yet experimentally undiscovered, between nematic and hexagonal discotic phases.

Dislocation loops and bond-orientational order in the Abrikosov flux-line lattice.

An Abrikosov flux-line lattice with an equilibrium concentration of unbound dislocation loops is considered as a way to describe the entangled flux liquid that arises in high-${T}_{c}$ superconductors. The long-wavelength properties of this dislocation loop gas are discussed using continuum elastic theory. We show explicitly that edge dislocations drive the long-wavelength shear modulus to zero, i.e., melt the lattice, analogous to what happens in two dimensions. Dislocations do not, however, destroy the sixfold long-range orientational order of the crystal in the xy plane. The flux-line lattice with dislocations is therefore a hexatic liquid crystal of lines rather than an isotropic liquid. The expected signature of an entangled hexatic liquid from neutron diffraction is discussed. Long-range orientational order relaxes in the z direction with a correlation length due to flux-line entanglement mediated by screw dislocations.

Lattice-gas model of particles with orientational and positional degrees of freedom: Mean-field treatment.

We consider a lattice-gas model of particles with internal orientational degrees of freedom. In addition to antiferromagnetic nearest-neighbor (NN) and next-nearest-neighbor (NNN) positional interactions we also consider NN and NNN interactions arising from the internal state of the particles. The system then shows positional and orientational ordering modes with associated phase transitions at ${T}_{p}$ and ${T}_{o}$ temperatures at which long-range positional and orientational ordering are, respectively, lost. We use mean-field techniques to obtain a general approach to the study of these systems. By considering particular forms of the orientational interaction function we study coupling effects between both phase transitions arising from the interplay between orientational and positional degrees of freedom. In mean-field approximation coupling effects appear only for the phase transition taking place at lower temperatures. The strength of the coupling depends on the value of the long-range order parameter that remains finite at that temperature.

Atomistic growth of two-dimensional quasicrystals.

An atomistic growth model based on local growth rules and bond lengths defined by fivefold symmetry is constructed to obtain both perfect and defective structures that exhibit long-range orientational order. The interplay of randomness and energy in the calculation of growth priority is investigated and a proof is given for the condition of perfect structures. Variation of the energy parameters representing the depth of the potential well results in different phases characterized by the dominance of different local clusters. The perfect structure is a special phase resembling a multiple twinning structure made from stretched hexagons and stripes. It can be described as the projection from higher dimension with randomness in the form of random stacking of layers. Defects are illustrated and discussed for various initial clusters to show that the general low-temperature phase is a defective one. All defects found can be traced back from the occurrence of two elementary defects, whose existence arises from the geometric definition of the model. The existence of a new kind of defect not describable by Burgers vectors is illustrated by the growth from the interior surface of an annulus. The ground-state property is investigated and the perfect structure is shown to have a lower energy than structures with only rhombic defects. We conjecture that the perfect structure is indeed the ground state for a wide range of energy.

An investigation of the spot profiles in transmission electron diffraction from Langmuir-Blodgett films of aliphatic chain compounds

The profiles of the transmission diffraction spots from a Langmuir-Blodgett film of aliphatic chain compound are derived from three plausible models of molecular organisation : a polycrystalline phase with long-range orientational order; a paracrystal possessing a density of point defects which do not interrupt the lattice rows ; and a Nelson and Halperin hexatic smectic phase in the «Debye-Huckel» limit of weakly-interacting dislocations. The three resulting predictions are distinctly different. Experimental results are presented for the room-temperature diffraction patterns from Langmuir-Blodgett films of a lipid, DMPE, and the cadmium soap of a fatty acid, 22-tricosenoic. Only the Nelson and Halperin theory gives a satisfactory fit, although there are deviations attributable to paracrystal-type defects or to non-ergodic behaviour of the layer. Dislocation densities of between 10-4 and 10 -5 per molecule are deduced.

Super-roughening: A new phase transition on the surfaces of crystals with quenched bulk disorder.

We present and study a model for surface fluctuations and equilibrium crystal shapes in solids with quenched bulk translational disorder but infinitely long-ranged orientational order. Strictly speaking, such surfaces have no sharp surface phase transition. However, for reasonable values of the bulk correlation length ${\ensuremath{\xi}}_{B}$ (${\ensuremath{\xi}}_{B}$\ensuremath{\gtrsim}30 A\r{} should be sufficient), an experimentally sharp ``super-roughening'' transition occurs at a temperature ${T}_{\mathrm{SR}}$. This transition separates a high-temperature ``rough'' phase of the surface from a low-temperature ``super-rough'' phase that, counterintuitively, is even rougher. Specifically, the root-mean-square equilibrium vertical fluctuation in the position of the interface 〈${h}^{2}$〉\ifmmode\bar\else\textasciimacron\fi{} $^{1/2}$ diverge like \ensuremath{\surd}lnL as the length L of the surface \ensuremath{\rightarrow}\ensuremath{\infty} for Tg${T}_{\mathrm{SR}}$ (just as in ordered solids for T greater than the roughening temperature ${T}_{R}$), while 〈${h}^{2}$〉\ifmmode\bar\else\textasciimacron\fi{} $^{1/2}\mathrm{ln}\mathit{L}$ for T${T}_{\mathrm{SR}}$.

Hexatic vortex glass in disordered superconductors.

It is shown that interaction of the flux-line lattice with randomly arranged pinning centers should destroy the long-range positional order in the lattice, but not the long-range orientational order. A new phase: hexatic vortex glass, is suggested for the mixed state of disordered, type-II superconductors. Relevance to amorphous and high-{ital T}{sub {ital c}} superconductors is discussed.

Crystallographic texture of melt-spun AlMn based and AlLiCu icosahedral quasicrystals

The atomic structure of the icosahedral phase is characterized both by long-range icosahedral orientational order and by a quasiperiodic translational one. Because the long-range orientational order generates anisotropy, it is natural that the icosahedral phase forms a crystallographic texture during solidification through preferred growth and/or nucleation. No texture was, however, observed in melt-spun quasicrystalline Al{sub 74}Si{sub 6}Mn{sub 20} by x-ray photography and diffractometry. The present work reports the observation of textures by pole figure measurements for Al-Mn based and Al-Li-Cu quasicrystals produced by melt-spinning.

Defect-mediated melting of pentagonal quasicrystals

The theories of dislocation- and disclination-mediated melting in two-dimensional systems, due to Kosterlitz, Thouless, Nelson, Halperin and Young (1979), are applied to pentagonal quasicrystals. As in triangular lattices, the transition from solid to liquid occurs in two stages. First, dissociation of neutral pairs of dislocations causes a transition out of the low-temperature solid phase, which is characterised by algebraically decaying quasiperiodic translational order and long-range fivefold orientational order, into a 'pentahedratic' phase with exponentially decaying translational order and algebraically decaying orientational order. A second transition, caused by dissociation of pairs of disclinations, leads to an isotropic fluid whereby orientational order also decays exponentially. The authors present the relevant recursion relations and critical exponents.

Second-harmonic generation and absorption studies of polymer–dye films oriented by corona-onset poling at elevated temperatures

Second-harmonic generation and spectroscopic absorption measurements are used to study the nonlinear-optical thin-film properties of azo dye guest molecules oriented in a polymer host by corona-onset poling at elevated temperatures (COPET). Parallel-wire electrode and needle electrode configurations are studied. The orientational order of the nonlinear molecules, the internal electric field, and the stability of the second-harmonic properties are measured. The nonlinear properties stabilize after 10 days and remain relatively constant over a period of at least 8 months, and the films tolerate power densities of at least 60MW/cm2. Compared with other poling methods, COPET produces a more efficient long-range and long-term orientational order in the guest–host system studied, resulting in large, stable second-harmonic properties.

Order-Disorder Transition in Capillary Ripples

A well-defined order-disorder transition occurs in the capillary waves on a fluid layer driven by vertical oscillation. The transition is characterized by a sharp decline in both the translational correlation length and the long-range orientational order of the pattern, and an onset in the characteristic frequency ${f}^{*}$ of chaotic fluctuations varying approximately as ${(A\ensuremath{-}{A}_{d})}^{\frac{1}{2}}$, for driving amplitudes above a threshold ${A}_{d}$. The transition is geometry dependent even when the container size is 50-100 times the wavelength.