Extrapolation Length Research Articles

Surface melting and effective anchoring in nematics

AbstractWe study the equilibrium configuration of a nematic liquid crystal bounded by a rough surface. If ξ and η are, respectively, the nematic coherence length and the characteristic wavelength of the roughness, we show that in both limiting cases ξ ≪ η and η ≪ ξ the wrinkling of the surface induces a nontrivial structure in the equilibrium solution which may be interpreted in terms of an effective weak anchoring potential. We also determine how the effective surface extrapolation length is related to the microscopic anchoring parameters. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Theoretical investigation of polarization scaling in ultrathin epitaxial PbZrxTi1−xO3 films

We present a theoretical analysis of the scaling of the polarization and the static dielectric susceptibility through a mean-polarization approach for ultrathin epitaxial PbZrxTi1−xO3 thin films. We use the traditional Euler-Lagrangian framework applied to a Landau-Ginzburg-Devonshire (LGD) nonlinear thermodynamic treatment. The novelty of our approach is that the model hinges on using experimentally measured correlation lengths and temperature scaling relationships to give the size-dependent expansion parameters of the nonlinear thermodynamic potential. These are then used in a Taylor series expansion of the polarization at the center of the film. We show that this method is able to correctly predict experimentally observed scaling without the need for the so-called extrapolation length (which is impossible to measure experimentally). Furthermore, as no implicit correlation between the correlation length (ξ) and the coefficient of the gradient term in the LGD potential (g11) is assumed, the model thus involves fully experimentally measurable parameters and their systematic temperature dependence rather than implicit assumptions. The model finds that the Curie temperature in ultrathin films is more sensitive to epitaxial strain as compared to the polarization and that the critical thickness is strongly dependent on the “temperature-epitaxial strain” parameter space. Interestingly, while it finds that at lower temperatures the depolarization field does play a strong role in the thickness dependence as well as spatial profile of the polarization, with increasing temperature, a significant weakening of the role of depolarization fields occurs. Consequently the interface-induced suppression is lower and, as a result, the polarization profile is more homogenous at higher temperatures. This indicates that systematic temperature dependent studies are fundamental to further understanding of size effects in ferroelectrics.

Exact Expressions for Some Dielectric Properties of Ferroelectric Thin Films Based on the Tilley–Zeks Model

The extrapolation length adopted in the Tilley–Zeks model of ferroelectric thin films plays a substantial role in governing the dielectric properties of thin films through the boundary conditions. The boundary conditions under an applied field are reexamined. The exact formulas for the transition temperature, the dielectric susceptibility, and the Curie constant are obtained for the Tilley–Zeks model of ferroelectric thin films, where various combinations of the extrapolation length are taken into account. The polarization profiles obtained under the applied field may be considered to approximately represent the soft mode.

Ultraweak azimuthal anchoring of a nematic liquid crystal on a planar orienting photopolymer

The search of weak anchoring is an important issue for a whole class of liquid crystal displays. In this paper we present an orienting layer showing unreached weak planar azimuthal anchoring for 4-n-pentyl-4′-cyanobiphenyl nematic liquid crystal (5CB). Azimuthal extrapolation lengths as large as 80μm are easily obtained. Our layers are made with the commercial photocurable polymer Norland optical adhesive 60. The anisotropy of the film is induced by the adsorption of oriented liquid crystal molecules under a 2T magnetic field applied parallel to the surfaces. We use the width of surface π-walls and a high-field electro-optical method to measure, respectively, the azimuthal and the zenithal anchorings. The azimuthal anchoring is extremely sensitive to the ultraviolet (UV) dose and it also depends on the magnetic field application duration. On the opposite, the zenithal anchoring is only slightly sensitive to the preparation parameters. All these results are discussed in terms of the adsorption/desorption mechanisms of the liquid crystal molecules on the polymer layer and of the flexibility of the polymer network.

Appearance of ferroelectricity in thin films of incipient ferroelectric

AbstractThe consideration of a size‐induced ferroelectric–paraelectric phase transition for conventional and incipient ferroelectric thin films with perovskite structure was carried out in the phenomenological theory framework of Ginsburg–Landau–Devonshire. The more general form of surface free energy expansion that includes intrinsic surface stress tensor, the surface piezoelectric effect and electrostriction as well as quadratic and quartic powers of surface polarization has been considered. The analytical expressions for the thickness dependence of the transition temperature was derived both for the conventional and incipient ferroelectrics. It was shown that although there is no ferroelectricity in the bulk incipient ferroelectrics it appears in thin film for the negative extrapolation length that is realized, e.g., at positive surface stress coefficients and negative or zero misfit strain. In our consideration we came to the conclusion about thickness‐induced ferroelectricity in incipient ferroelectrics KTaO3 at room temperature for films that were sufficiently thin. The similar surface effects can strongly influence the phase transitions in the thin films of another incipient ferroelectric, SrTiO3. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Transport- and diffusion-based optical tomography in small domains: a comparative study

We compare reconstructions based on the radiative transport and diffusion equations in optical tomography for media of small sizes. While it is well known that the diffusion approximation is less accurate to describe light propagation in such media, it has not yet been shown how this inaccuracy affects the images obtained with model-based iterative image reconstructions schemes. Using synthetic nondifferential data we calculate the error in the reconstructed images of optical properties as functions of source modulation frequency, noise level in measurement, and diffusion extrapolation length. We observe that the differences between diffusion and transport reconstructions are large when high modulation frequencies and noise-free data are used in the reconstructions. When the noise in data reaches a certain level, approximately 12% in our simulations, the differences between diffusion- and transport-based reconstructions become almost indistinguishable. Given that state-of-the-art optical imaging systems operate at much lower noise levels, the benefits of transport-based reconstructions of small imaging domains can be realized with most of the currently available systems. However, transport-based reconstructions are considerably slower than diffusion-based reconstructions.

The diverse world of liquid crystals

Orientationally ordered soft matter is exceptionally responsive to a variety of excitations. That’s the basis for its great range of applications.

Hysteresis loops of first-order ferroelectric bilayers or superlattices and their size effect

Based on Landau–Ginzburg phenomenological theory, a first-order ferroelectric bilayer or superlattice with an antiferroelectric interface coupling has been studied by taking into account the spatial variation of polarization within each constituent film and the surface effect. The hysteresis loops are obtained and the size effect of a bilayer is discussed. The loop patterns vary between typically antiferroelectric and typically ferroelectric depending on the thickness ratio, the coupling constant, the thickness and the extrapolation length. The antiferroelectric coupling has a great effect on the phase transition of a bilayer. It is shown that the size-driven phase transition cannot be observed in a ferroelectric bilayer in the case of strong antiferroelectric coupling.

Size-induced appearance of ferroelectricity in thin antiferroelectric films

In the paper we consider size effects on phase transitions and polar properties of thin antiferroelectric films. We extend the phenomenological approach proposed by Kittel for thin films allowing for gradient (correlation) energy and depolarization field energy. Surface piezoelectric effect as well as misfit strain appear due to lattice constants mismatch between the film and its substrate. Direct variational method is used to derive the free energy with renormalized coefficients depending on the film thickness. Obtained free energy expression allows the calculation of phase diagrams and all electro-physical properties by a conventional minimization procedure. Approximate analytical expressions for the paraelectric–antiferroelectric–ferroelectric transition temperature dependences on film thickness, polarization gradient coefficient, and extrapolation lengths were obtained. The thickness dependence of the electric field critical value that causes antiferroelectric–ferroelectric phase transition was calculated. Under favorable conditions the antiferroelectric phase at first transforms into ferroelectric one and then into paraelectric phase with the decrease of the film thickness. Proposed theoretical consideration explains the experimental results obtained in antiferroelectric PbZrO 3 thin films.

Very strong azimuthal anchoring of nematic liquid crystals on uv-aligned polyimide layers

The azimuthal anchoring energy of the nematic liquid crystal 4- n -pentyl- 4' -cyanobiphenyl (5CB) on a uv-aligned polyimide substrate with in-plane order parameter S'=0.2 is measured. The measurements are performed at temperature T=24 degrees C using simultaneously a high accuracy reflectometric method and a high accuracy transmitted light method. With both the methods, we observe an apparent surface director rotation opposite to the orienting torque that would correspond to a negative extrapolation length. It is shown that this unusual behavior is due to the relatively high birefringence of the uv-aligned polyimide layers. Taking into account for this birefringence, we find a small but positive extrapolation length. The experimental results are interpreted in terms of a simple mesoscopic model where the nematic molecules are assumed to be rigidly attached on the polymer surface and the measured extrapolation length is entirely due to the order parameter variation in a thin interfacial layer where the nematic order parameter passes from the surface value to the bulk value within a few nematic correlation lengths. Assuming the surface order parameter is S(0)=0.37 , the correlation length of the nematic liquid crystal is estimated to be xi'(c)=2.4+/-1 nm . The corresponding thermodynamic extrapolation length is de=2.8+/-1.2 nm that corresponds to a very strong azimuthal anchoring.

Vortex arrays in superconducting cylinders

We solve numerically the nonlinear TDGL equations to study the vortex dynamics in a mesoscopic type II cylinder superconductor containing one rectangular hole in the presence of an external field applied perpendicular to the surfaces. We calculate the spatial distribution of the superconducting electron density and the phase of the superconducting order parameter using the numerical method based on the technique of gauge invariant variables. It is assumed that the inner hole edge is in contact with a material whose properties are accounted in the de Gennes boundary conditions with a parameter b (de Gennes extrapolation length), while the outer edge is in contact with vacuum. The dynamics of different vortex states and the magnetization curves are studied as a function of the external magnetic field.

Simulation of characteristics of phase transitions in ferroelectric thin films

Characteristics of phase transition in ferroelectric thin films are studied using the time dependent Ginzburg–Landau equation. Properties such as the spontaneous polarization, Curie temperature, critical thickness and susceptibility are investigated as a function of the extrapolation length, film size and the misfit strain. For free-standing PbTiO 3 thin film, the supercooling and superheating temperatures can be separately determined from the spontaneous polarization changes, under temperature heating-up and cooling-down conditions, respectively. When the film grown on compressive substrate, clamped effect on film may cause film made of first-order ferroelectric material to undergo a second-order transition. In addition, we have also comprehensively analyzed the effects of film size, the extrapolation length and substrate on characteristics of phase transitions. For free standing PbTiO 3 film, the different of superheating and supercooling temperature is insensitive to the film thickness and extrapolation length. The decrease of extrapolation length would not decrease the Curie temperature, but also decrease the critical thickness of the ferroelectricity. When film is thinner, the effect of the extrapolation length is larger on the change of the Curie temperature.

Technique for azimuthal anchoring measurement of nematic liquid crystals using magnetic field induced deformation

The authors devised a technique for measuring the azimuthal anchoring coefficient of nematic liquid crystals. The coefficient was estimated calculating the varying ratio of magnetic coherence length to extrapolation length using light transmission to magnetic field. The paralleled control of the directors on the entrance surface, polarizer axis and magnetic field enables the transmission to sensitively reflect the anchoring. The extrapolation length was 0.48μm for polyimide rubbed surface, 0.56μm for rubbed poly-vinyl alcohol and 5.6μm for photo-aligned polyimide. This technique has advantages such as insensitiveness to sample adjustment, covering wide range of anchoring and independence on the cell gap, electrodes and impurity.

The effects of different expansions of the exit distribution on the extrapolation length for linearly anisotropic scattering

Abstract HN and singular eigenfunction methods are used to determine the neutron distribution everywhere in a source-free half space with zero incident flux for a linearly anisotropic scattering kernel. The singular eigenfunction expansion of the method of elementary solutions is used. The orthogonality relations of the discrete and continuous eigenfunctions for linearly anisotropic scattering provides the determination of the expansion coefficients. Different expansions of the exit distribution are used: the expansion in powers of μ, the expansion in terms of Legendre polynomials and the expansion in powers of 1/(1 + μ). The results are compared to each other. In the second part of our work, the transport equation and the infinite medium Green function are used. The numerical results of the extrapolation length obtained for the different expansions is discussed.

General flux to a trap in one and three dimensions

The problem of the flux to a spherical trap in one and three dimensions, for diffusingparticles undergoing discrete-time jumps with a given radial probability distribution, issolved in general, verifying the Smoluchowski-like solution in which the effective trap radiusis reduced by an amount proportional to the jump length. This reduction in the effectivetrap radius corresponds to the Milne extrapolation length.

Antiferroelectric thin films phase diagrams

In this article the size effects of phase transitions and polar properties of the thin antiferroelectric films are considered. We modified the phenomenological approach proposed by Kittel. The Euler--Lagrange equations were solved by direct variational method. The free energy with renormalized coefficients depending on the film thickness has been derived. The approximate analytical expression for the coefficients dependence on film thickness, temperature, polarization gradient coefficient and extrapolation lengths were obtained. It is shown how the anti-ferroelectric “double” hysteresis loop transforms into the ferroelectric “single” one under the film thickness decrease. Proposed theoretical consideration explains the experimental results obtained in antiferroelectric PbZrO3 thin films.

Ferroelectricity enhancement in ferroelectric nanotubes

In this article we study the size effects of the ferroelectric nanotube phase diagrams and polar properties allowing for effective surface tension and depolarization field influence. An approximate analytical expression was derived for the dependence of the paraelectric-ferroelectric transition temperature on the radii of nanotube, polarization gradient coefficient, extrapolation length, surface tension and electrostriction coefficient. It was shown that the transition temperature could be higher than the one of the bulk material for the negative electrostriction coefficient. Therefore, we predict conservation and enhancement of polarization in long ferroelectric nanotubes. The obtained results explain the observed conservation and enhancement of ferroelectricity in Pb(Zr,Ti)O3 and BaTiO3 nanotubes.

Monte Carlo simulations of intrinsic anchoring in nematic liquid crystals based on spatially anisotropic pair potential

The free surfaces of nematic liquid crystals are studied based upon the molecular pair potential model, which is spatially anisotropic and dependent on elastic constants of liquid crystals. The study is based on the simple cubic lattice model with the aid of Monte Carlo simulation. An elastic deformation is imposed, forming a hybrid cell-like nematic sample so that the anchoring at free nematic interfaces (intrinsic anchoring) is well studied. It is found that the preferred orientation at the free interface and the corresponding extrapolation length change with the modification of potential parameters, but are not dependent on temperature.

Boundary‐Roughness Effects in Nematic Liquid Crystals

We study the equilibrium configuration of a nematic liquid crystal bounded by a rough surface. The wrinkling of the surface induces a partial melting in the degree of orientation. This softened region penetrates the bulk up to a length scale which turns out to coincide with the characteristic wavelength of the corrugation. Within the boundary layer where the nematic degree of orientation decreases, the tilt angle steepens and gives rise to a nontrivial structure, which may be interpreted in terms of an effective weak anchoring potential. We determine how the effective surface extrapolation length is related to the microscopic anchoring parameters. We also analyze the crucial role played by the boundary conditions assumed on the degree of orientation. Quite different features emerge depending on whether they are Neumann‐ or Dirichlet‐like. These features may be useful to ascertain experimentally how the degree of orientation interacts with an external boundary.

Diffraction analysis of LC gratings based on linearly graded phase model for surface anchoring energies

We reported the diffraction analysis of the liquid crystal (LC) binary gratings and the surface anchoring energies for planar and homeotropic alignments. The planar and homeotropic anchoring energies were directly derived based on the linearly distorted director distribution near domain boundaries, in which the distorted lengths correspond to the extrapolation lengths into both planar and homeotropic regions. From the diffraction analysis for the LC binary gratings with various grating periods based on the linearly graded phase model, both distorted lengths into planar and homeotropic regions were simultaneously obtained. In this work, the planar and homeotropic anchoring energies were found to be about 1.4 × 10-4 J/m2 and 0.9 × 10-5 J/m2, respectively.