Torque Balance Equation Research Articles

Effect of mode locking on toroidal flow evolution

The response of the toroidal flow velocity to being abruptly locked in the vicinity of a magnetohydrodynamic (MHD) tearing-mode-induced magnetic island is examined analytically and numerically using a diffusive transport model in a cylindrical plasma. The resultant toroidal momentum confinement is shown to degrade significantly after mode locking, as is often observed on many tokamaks and reversed field pinches (RFPs). The degraded toroidal momentum confinement time in the core and edge regions of the plasma are evaluated and compared with experimental observations. The mode locking time-scale itself is also investigated by using a model toroidal torque balance equation. The decrease of mode frequency during mode locking seems to be governed mainly by the electromagnetic torque exerted on the resistive layer. For tokamak plasmas, the mode locking time-scale is found to be much shorter than the diffusion time-scale, which is in general agreement with experimental observations

Director deformation of a twisted chiral nematic liquid crystal cell with weak anchoring boundaries.

On the basis of a generalization of the Rapini-Papoular expression [J. Phys. (Paris) Colloq. 30, C4-54 (1969)] for the anchoring energy, the rigorous expressions for the threshold and saturation fields are derived analytically, in detail, for a field-controlled twisted chiral nematic slab with weak boundary coupling. The surface anchoring energy of the Rapini-Papoular type for a twisted nematic slab should be generally expressed as the interfacial energy per unit area for a two-dimensional deformation which is a nonlinear combination of the azimuthal and polar angles. Applying a variational calculation method for the two-dimensional problem to the total free energy, that is, the sum of the bulk energy and the surface energy, we derive general torque balance equations which describe the equilibrium deformation of the director.

General Threshold Properties of Liquid Crystal Slab With Weak Anchoring Boundaries

Abstract On the basis of the Rapini-Papoular anchoring energy unifying the azimuthal and polar angle anchorings, the general surface torque balance equations for field-controlled twisted chiral nematic liquid crystal cells with weak anchoring boundary coupling have been derived analytically, The relationship between the anchoring energy and the threshold field of Freedericksz transition has been discussed rigorously.

Rotational vector magnetization measurements on Ni 74Mn 24Pt 2 alloy

Rotational vector magnetization measurements were carried out on thin disk-like samples of Ni 74Mn 24Pt 2 at T = 5, 12 and 25 K, respectively. The samples were first cooled in a high field to low temperatures and then longitudinal ( M L) and transverse ( M T) components of magnetization as a function of the sample rotational angle (Θ) were measured simultaneously in various fields. We have found that a multi-domain structure still persists in a field-cooled state. During sample rotations, each domain restabilizes the balancing opposite torques exerted on it by the external field H and anisotropy field ( H A), accompanying anisotropy field rotations relative to H, elastically at low temperatures, but dissipatively at higher temperatures. The experimental results were analyzed with a simple torque balance equation assuming that the anisotropy of the sample has only a purely directional component.

Equilibrium structures of planar nematic and cholesteric films in electric fields

Abstract Liquid crystal layers subjected to an electric or a magnetic field can have several types of instabilities. This paper reviews recent theoretical studies concerning equilibrium structures of planar layers. Using the Oseen-Frank elasticity theory, the Freedericksz transition and the transition to static periodic domains of planar nematic and cholesteric films are reconsidered. A perturbation treatment of nonlinear torque balance equations for the director is suitable to derive amplitude equations for the film distortions under the action of a field and to predict the topological features of phase diagrams for equilibrium states. The competition between the Freedericksz effect and the formation of periodic distortions is studied varying material and geometrical parameters. Some results are useful to optimize cholesteric mixtures for application in display devices in such a manner, that the occurrence of periodic domains is avoided.

On the Chevron Structure in Compressible SmC*Phases

Some numerical solutions corresponding to the chevron layer structure in compressible SmC * phases are obtained in terms of the torque balance equations on the basis of a phenomenological elastic free energy including the layer compression energy as well as the c -director deformation energy. Similar to the previously derived analytical solution (M. Nakagawa: Mol. Cryst. & Liq. Cryst. 174 (1989) 65), a soliton-like solution is found to exist as an equilibrium state in SmC * phase. In addition the electric field effect on the layer structure and the c -director configuration in the chevrons is presented taking account of the polarization electric field induced by the polarization charge.

Defect Dynamics of a Nematic Polymer in a Magnetic Field

ABSTRACTAnalysis of the nonlinear dynamical response of a rod-like nematic liquid crystalline polymer, between parallel untreate plates, in a magnetic field is presented. Numerical solutions to the torque balance equations showthat when a uniformly aligned polymer is acted by a magnetic field normal to the initial polymer orientation an array of inversion walls develops. Stability analysis shows that the inversion walls are unstable to localized perturbations. The nonlinear resolution ofthe instability leads to the nucleation of an array of disclination line pairs of strength ±1/2. A perturbation analysis is used to obtain the evolution of the polymer structure towards equilibrium

Plate‐driving forces over the Cenozoic Era

Plate reconstructions are used to estimate plate torques for the Cenozoic Era, assuming a dynamical balance between active torques (slab‐pull and ridge‐push) and plate drag, as the passive torque. In a published one‐dimensional model for the motion of an oceanic plate the plate drag balances slab‐pull and ridge‐push; expressing each of these forces in terms of age and sub‐duction rate and using values at 15 trenches, coefficients are found which give a least squares fit. Here we have retained a common age factor, which gives a greater weight to older, more rapidly moving plates, and the contribution due to ridge‐push is larger than that found in the previous study, but still much smaller than the slab‐pull force. This one‐dimensional force model is generalized to two‐dimensional plates moving over the Earth's surface, and we obtain a torque balance equation in which slab‐pull and ridge‐push torques are proportional to boundary chord vectors, with the weights depending on powers of subduction velocity at the middle of the chords. For each plate we find the unique angular velocity which satisfies the torque balance requirement, and this balance torque is compared with the plate drag torque obtained from the plate's geometry and (reconstructed) angular velocity. Comparison of the torques is made for two published models for present‐day plate motions, as well as six published reconstructions for time intervals spanning the Cenozoic. For the present the mismatch between the balanced torque and drag torque is about the same as the difference between the two alternative published models. Torques are fairly stable throughout the Cenozoic, with the misfit increasing systematically for earlier reconstructions.

Nonlocal radial dependence of laser-induced molecular reorientation in a nematic liquid crystal: theory and experiment

We present a detailed theoretical calculation, with experimental verification, of the nonlocal molecular reorientation of the nematic-liquid-crystal director axis induced by a cw Gaussian laser beam. The natures of the torque balance equations and the solutions are significantly different for normally and nonnormally incident laser beams. The nonlocal effects resulting from molecular correlation effects are particularly important for laser spot sizes that are different (smaller or larger) from the sample thickness. Experimental measurements for the transverse dependence of the molecules and the dependence of the Freedericksz threshold as a function of the laser beam sizes are in excellent agreement with theoretical results. We also comment on the effect of these nonlocal effects on transverse optical bistability.

Determination of the anisotropic potential at the nematic liquid crystal-to-wall interface

The integrated birefringence of a surfactant (C16H33NH2)-aligned methoxybenzylidene butylaniline cell as a function of applied magnetic field (up to 100 kG) has been measured. The Oseen and Frank elastic continuum theory and a generalized torque balance equation at the interface have been utilized for the calculation to fit the measured data. A very good fit to the data was obtained using an anisotropic liquid crystal-to-wall interfacial potential of the form C cos2 θ+C4 cos4 θ for the calculation. It was found that C4/C=−0.38.

Weak Boundary Storage Effect in Homogeneous Liquid Crystal Cells

Based on the Oseen and Frank elastic continuum theory of nematic liquid crystals and on a generalized torque balance equation at the liquid crystal-to-substrate interface, static deformations of a semi-infinite liquid crystal (LC) medium and a homogeneous LC cell have been investigated using the anisotropic LC-to-substrate anchoring energy C as a parameter. Both the elastic and dielectric anisotropies are considered in obtaining the solutions. Storage effects are predicted for sufficiently small anchoring anisotropy. Numerical ranges of C, within which the weak boundary storage effect may occur, have been calculated for twenty commercially-avaialble nematic LC mixtures in the homo-geneous-cell configuration. An equation for the storage time as a function of C has also been derived. The treatment has also been extended to include the most general from of a LC-to-substrate interfacial potential. Such a weak boundary storage effect could provide a means for realizing low power, non-refreshed, matrix-addressed flat panel displays.

On the determination of liquid crystal-to-wall anchoring anisotropy by the surface-plasmon polariton technique

Based on the Oseen and Frank elastic continuum theory of nematic liquid crystals, and on a generalized torque balance equation at the interface, the magnitude and the gradient of the tilt angle of liquid crystal molecules at the interface have been calculated in a homogeneous cell as a function of applied voltage and liquid crystal-to-wall anchoring anisotropy C. Consequently, the reflectivities associated with the excitation of the surface-plasmon polariton propagating along the liquid crystal-to-wall interface can be evaluated for comparison with experimental results to determine a value for C. Numerical computations of reflectivities have been shown to fit the experimental results of Sprokel, Santo, and Swalen [Mol. Cryst. Liq. Cryst. 68, 29 (1981)]. An average value of 0.11±0.05 ergs/cm2 for C has been deduced.

Magnetization Process of Nearly 2-Dimensional Cu(HCOO)2·4H2O and Cu(HCOO)2·2(NH2)2CO·2H2O. I Mechanism of the Field-Induced Transition

Magnetization process is studied theoretically on the basis of the 4-sublattice hamiltonian obtained by adding a weak interlayer exchange to the 2-sublattice hamiltonian introduced by Seehra, and Castner. Equations of torque balance derived by applying the molecular field approximation at T =0 are truncated by neglecting smaller terms. The resulting simplified equations are solved analytically or numerically and the magnetization curve is obtained. The result indicates that the unique mangetization process of these salts is attributable to the field-induced transition from a 4- to a 2-sublattice state which is related closely to the 2-dimensionality of the crystal. The jumping field for H //b is given approximately by twice of the interlayer exchange field divided by the a''b-element of the effective g-tensor.

ELECTROHYDRODYNAMIC INSTABILITY IN PLANAR, POSITIVE DIELECTRIC ANISOTROPY NEMATIC LAYERS AT D.C. EXCITATION

A generalisation of de Gennes model for the electrohydrodynamic instability at strong unipolar injection is developed. Including in the torque balance equation the elastic torque gives the possibility for description of positive materials as well. Two domains of dielectric anisotropy values (small and big) are found with qualitatively different behaviour of the instability. The case of no injection is considered as well. In conclusion some other mechanisms, active at d.c. excitation are briefly discussed, especially the flexo-electric effect-gradient and linear one. 1 . Theory. The considerations given below originate from a model developed by de Gennes in 1970 [I]. Let a nematic, placed in a planar capacitor with interelectrode distance d and oriented parallel to the electrodes, is subjected to a strong unipolar injection by a d.c. voltage V, applied to the electrodes. The condition for strong injection means that the density of injected charges go near to the injecting electrode (the cathode usually) is big enough to compensate a great part of the applied electric field in that region. Numerically it means that The development of an instability will be studied on the basis of the charge balance equation. The time evolution of a charge density fluctuation is described by where an assumption is made for the convective term at strong injection that In the second term 7 is the dielectric relaxation time : o is the mean value of the conductivity and the coefficient 2 arises because the conductivity itself is proportional to 69 [I]. The third term is due to the Carr-Helfrich effect. The transverse current is determined by the angle of deviation cp of the director from the Ox axis. For small angles this connection is In order to express the fluid velocity v, via 69 the Stokes equation is used (with neglecting of the inertial effects) : q : mean viscosity of the nematic. In the case of spatial variations of v, in the following form (determined by the presence of electrodes) : 2 nx v, = vt cos ( , ) cos (7) , (6) Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979359 ELECTROHYDRODYNAMIC INSTABILITY IN POSITIVE NEMATICS C3-311 it follows from (5) that In the original version of the model the deviation angle cp was calculated from a torque balance equation containing only dielectric (re) and hydrodynamic (T,) torque with neglecting of the elastic one. The complete equation should read Here rk,,, the y-component of the elastic torque, in isotropic elasticity approximation (K,, = K,, = K) is The last result can be obtained if the same form as for v, is adopted for the spatial variation of 9 : (P = cp, cOS (T) cos (T) . (9) With de Gennes' results for both other torques the torque balance equation reads A differentiation of this equation with respect to x together with de Gennes' estimation : and the result from (7)

Dielectric alignment in an electrically conducting nematic liquid crystal

For small deformations the local macroscopic electric field strength in a deformed uniaxial crystal is derived. For relative high frequencies (≫ σ/ε0ε) the electric field strength is determined by the anisotropic dielectric properties and for relative low frequencies (≪ σ/ε0ε) by the anisotropic electric conductivity. The local electric field strength is frequency independent if the dielectric and the electric conductivity tensor have the same anisotropy. The nonhydrodynamic deformation of a nematic liquid crystal under the influence of an electric field is derived from the torque balance equation in the small-angle approximation. This dielectric alignment is frequency dependent corresponding to the frequency dependence of the local electric field strength.

Transient times and multiplex behavior of nematic liquid crystals in the electric field

The change of the optical axis in a nematic liquid crystal (NLC) is connected with a macroscopic material flow (backflow), therefore two basic equations (force balance and torque balance equation) have to be discussed to get the transient behavior of the NLC in an electric field. For a homogeneous and a homeotropically aligned NLC the transient times are inversely proportional to the square of the sample thickness and inversely proportional to the normalized voltage (U/U 0)2−1.U 0 is the threshold voltage. The agreement between the experimental and the theoretical results is quite good. From the calculated formula the possible duty-cycle for multiplexing a NLC can be estimated.