Elastic Vortex Research Articles

Comparison of the history effects in magnetization in weakly pinned crystals of high Tc and low Tc superconductors

A comparison of the history effects in weakly pinned single crystals of a high Tc YBa2Cu3O7−δ (for H || c) and a low Tc Ca3Rh4Sn13, which show anomalous variations in critical current density Jc(H), is presented via the tracings of the minor magnetization hysteresis loops using a vibrating sample magnetometer. The sample histories focussed are: (i) the field cooled (FC), (ii) the zero field cooled (ZFC) and (iii) an isothermal reversal of field from the normal state. An understanding of the results in terms of the modulation in the plastic deformation of the elastic vortex solid and supercooling across the order–disorder transitions is sought.

Multiple magnetization peaks in weakly pinnedCa3Rh4Sn13andYBa2Cu3O7−δ

The second magnetization peak and the peak effect anomaly coexisting in a given isothermal magnetization hysteresis loop show striking similarities in Ca 3 Rh 4 Sn 13 , a low-T c superconductor and YBa 2 Cu 3 O 7-δ , a high-T c superconductor. The observed variation of the hysteresis width with field could imply a modulation in the degree of the plastic deformation of the elastic vortex solid. The characteristics of the high-T c cuprates, such as large Ginzburg number, short coherence length, decoupling of the Josephson coupled pancake vortices, etc., are unlikely to be the cause of the observed behavior.

Effect of electron irradiation on vortex dynamics inYBa2Cu3O7−δsingle crystals

We report on drastic change of vortex dynamics with increase of quenched disorder: for rather weak disorder we found a single vortex creep regime, which we attribute to a Bragg-glass phase, while for enhanced disorder we found an increase of both the depinning current and activation energy with magnetic field, which we attribute to entangled vortex phase. We also found that introduction of additional defects always increases the depinning current, but it increases activation energy only for elastic vortex creep, while it decreases activation energy for plastic vortex creep.

The axisymmetric contraction–expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop

The flow of a polystyrene Boger fluid through axisymmetric contraction–expansions having various contraction ratios (2≤ β≤8) and varying degrees of re-entrant corner curvatures are studied experimentally over a large range of Deborah numbers. The ideal elastic fluid is dilute, monodisperse and well characterized in both shear and transient uniaxial extension. A large enhanced pressure drop above that of a Newtonian fluid is observed independent of contraction ratio and re-entrant corner curvature. Streak images, laser Doppler velocimetry (LDV) and digital particle image velocimetry (DPIV) are used to investigate the flow kinematics upstream of the contraction plane. LDV is used to measure velocity fluctuation in the mean flow field and to characterize a global elastic flow instability which occurs at large Deborah numbers. For a contraction ratio of β=2, a steady elastic lip vortex is observed while for contraction ratios of 4≤ β≤8, no lip vortex is observed and a corner vortex is seen. Rounding the re-entrant corner leads to shifts in the onset of the flow transitions at larger Deborah numbers, but does not qualitatively change the overall structure of the flow field. We describe a simple rescaling of the deformation rate which incorporates the effects of lip curvature and allows measurements of vortex size, enhanced pressure drop and critical Deborah number for the onset of elastic instability to be collapsed onto master curves. Transient extensional rheology measurements are utilized to explain the significant differences in vortex growth pathways (i.e. elastic corner vortex versus lip vortex growth) observed between the polystyrene Boger fluids used in this research and polyisobutylene and polyacrylamide Boger fluids used in previous contraction flow experiments. We show that the role of contraction ratio on vortex growth dynamics can be rationalized by considering the dimensionless ratio of the elastic normal stress difference in steady shear flow to those in transient uniaxial extension. It appears that the differences in this normal stress ratio for different fluids at a given Deborah number arise from variations in solvent quality or excluded volume effects.

Dislocations and the critical endpoint of the melting line of vortex line lattices

We develop a theory for dislocation-mediated structural transitions in the vortex lattice which allows for a unified description of phase transitions between the three phases, the elastic vortex glass, the amorphous vortex glass, and the vortex liquid, in terms of a free energy functional for the dislocation density. The origin of a critical endpoint of the melting line at high magnetic fields, which has been recently observed experimentally, is explained.

Vortex-glass phases in type-II superconductors

A review is given on the theory of vortex-glass phases in impure type-II superconductors in an external field. We begin with a brief discussion of the effects of thermal fluctuations on the spontaneously broken U(1) and translation symmetries, on the global phase diagram and on the critical behaviour. Introducing disorder we restrict ourselves to the experimentally most relevant case of weak uncorrelated randomness which is known to destroy the long-ranged translational order of the Abrikosov lattice in three dimensions. Elucidating possible residual glassy ordered phases, we distinguish betwee positional and phase-coherent vortex glasses. The discussion of elastic vortex glasses, in two and three dimensions occupy the main part of our review. In particular, in three dimensions there exists an elastic vortex-glass phase which still shows quasi-long-range translational order: the `Bragg glass'. It is shown that this phase is stable with respect to the formation of dislocations for intermediate fields. Preliminary results suggest that the Bragg-glass phase may not show phase-coherent vortex-glass order. The latter is expected to occur in systems with weak disorder only in higher dimensions. We further demonstrate that the linear resistivity vanishes in the vortex-glass phase. The vortex-glass transition is studied in detail for a superconducting film in a parallel field. Finally, we review some recent developments concerning driven vortex-line lattices moving in a random environment.

Entry flow and constitutive modelling of fluid S1

Experimental observations of the flow through an axisymmetric 4:1 contraction for the shear-thinning test fluid S1 are presented. In order to provide the basis for comparison with numerical simulations of the flow, the linear viscoelastic spectrum and the nonlinear parameters for three constitutive models are also obtained, based on the rheological characterisation of the fluid by Ooi and Sridhar [J. Non-Newtonian Fluid Mech. 52 (1994) 121–135]. The Phen-Thien-Tanner model provides the best fit to the data, but a good fit can also be obtained for the Giesekus model if the fit to the extensional stress growth coefficient is restricted to low strains. The results of flow visualisation experiments are presented for the flow of S1 through a 4:1 axisymmetric contraction with either an abrupt entry or a rounded re-entrant corner. The dimensionless vortex size increased with Weissenberg number for each geometry, with a larger vortex observed for the abrupt entry geometry for identical flow conditions. No elastic lip vortex or flow instability was observed for We ≤ 5.0.

Field Induced Vanishing of the Vortex Glass Temperature inTl2Ba2CaCu2O8Thin Films

The current density ${j}_{s}$ and relaxation rate $Q$ have been measured on a ring shaped ${\mathrm{Tl}}_{2}{\mathrm{Ba}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8}$ film between 4.2 K and the irreversibility temperature in magnetic fields up to 7 T. The temperature and field dependent exponent $\ensuremath{\mu}({T,B}_{e})$ used to parametrize the activation energy $U$ in the form $U({j}_{s}{,T,B}_{e})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({U}_{c}/\ensuremath{\mu})[({j}_{c}{/j}_{s}{)}^{\ensuremath{\mu}}\ensuremath{-}1]$ is determined for many different fields. In the ${B}_{e}\ensuremath{-}T$ phase diagram we associate the $\ensuremath{\mu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ line with the glass temperature ${T}_{g}$ and find that ${T}_{g}$ essentially drops to 0 K at ${B}_{e}\ensuremath{\approx}0.7\mathrm{T}$, which indicates a dimensional crossover at this field. The $\ensuremath{\mu}>0$ region $({B}_{e}<0.7\phantom{\rule{0ex}{0ex}}\mathrm{T},T\ensuremath{\le}55\mathrm{K})$ corresponds to 3D elastic vortex motin while the $\ensuremath{\mu}<0$ region corresponds to 2D dislocation mediated flux creep.

Flow regimes in model viscoelastic fluids in a circular couette system with independently rotating cylinders

Flow visualization of two highly elastic, nonshear-thinning polyisobutylene/polybutene fluids in the gap between concentric cylinders was performed over a range of shear rates and choices of relative cylinder rotations. The observed secondary flows are discussed in terms of destabilizing elastic and centrifugal forces. In the more viscous, more elastic fluid, instabilities are found to be independent of the choice of rotating cylinder and due entirely to elasticity. At the lowest shear rates examined, the first detectable secondary flows are steady counter-rotating vortices forming after a shearing time more than five orders of magnitude greater than the characteristic relaxation time of the fluid. At somewhat higher shear rates, a much more rapidly appearing oscillatory flow is observed to evolve into the steady vortex structure. In the less elastic fluid, the structure first detectable at the lowest shear rates is again steady vortices regardless of the choice of driving cylinder. At all shear rates examined, only elastic stationary vortices are observed in the absence of centrifugal destabilization (outer cylinder rotating). Secondary flows are significantly stronger in the presence of the centrifugal destabilization due to a rotating inner cylinder. Interaction of elasticity and centrifugal forces is found to generate a number of axially translating vortex structures, many of which are described here for the first time. At a shear rate more than five times the critical, another family of instability is observed which closely resembles a purely elastic instability observed by Baumert and Muller (1995). These experimental results are expected to provide a challenging test of numerical simulations of these viscoelastic flows.

Lagrangian Aspects of the Kirchhoff Elastic Rod

The relation between Euler–s planar elastic curves and vortex filaments evolving by the localized induction equation (LIE) of hydrodynamics was discovered by Hasimoto in 1971. Basic facts about (an integrable case of) Kirchhoff elastic rods are described here, which amplify the connection between the variational problem for rods and the soliton equation LIE. In particular, it is shown that the centerline of the Kirchhoff rod is an equilibrium for a linear combination of the first three conserved Hamiltonians in the LIE hierarchy.

The flux-line lattice in high-T c superconductors

The flux-line lattice in type-II superconductors has unusual nonlocal elastic properties which make it soft for short wavelengths of distortion. This softening is particularly pronounced in the highly anisotropic high-T c superconductors (HTSC) where it leads to large thermal fluctuations and to thermally activated depinning of the Abrikosov vortices. Numerous transitions are predicted for these layered HTSC when the temperatureT, magnetic inductionB, or current densityJ are changed. In particular, the flux lines are now chains of two-dimensional (2D) “pancake vortices” which may “evaporate” by thermal fluctuations or may depin individually. At sufficiently highT, ohmic resistivityρ(T, B) is observed down toJ → 0. This indicates that the flux lines are in a “liquid state” with no shear stiffness and with small depinning energy or that the 2D vortices can move independently. At lowerT, ρ(T, B, J) is nonlinear since the pinning energy of an elastic vortex lattice or “vortex glass” increases with decreasingJ as predicted by theories of collective pinning and by “vortex glass” scaling.

Thermally activated coherent vortex motion in YBa2Cu3O7- delta thin film microbridges.

Microbridges with dimensions smaller than the London penetration depth \ensuremath{\lambda} have been prepared in epitaxial ${\mathrm{YBaCu}}_{2}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ thin films by means of electron beam lithography. Typical peaks and kinks are observed in their differential resistance versus current characteristics which can be attributed to coherent flux motion. A vortex activation energy proportional to the inverse current 1/I is obtained. Both the current and the temperature dependence of the activation energy are consistent with the collective flux creep model if the dimensionality of the elastic vortex lattice equals 2.

Nonlinear dynamics of viscoelastic flow in axisymmetric abrupt contractions

The steady-state and time-dependent flow transitions observed in a well-characterized viscoelastic fluid flowing through an abrupt axisymmetric contraction are characterized in terms of the Deborah number and contraction ratio by laser-Doppler velocimetry and flow visualization measurements. A sequence of flow transitions are identified that lead to time-periodic, quasi-periodic and aperiodic dynamics near the lip of the contraction and to the formation of an elastic vortex at the lip entrance. This lip vortex increases in intensity and expands outwards into the upstream tube as the Deborah number is increased, until a further flow instability leads to unsteady oscillations of the large elastic vortex. The values of the critical Deborah number for the onset of each of these transitions depends on the contraction ratio β, defined as the ratio of the radii of the large and small tubes. Time-dependent, three-dimensional flow near the contraction lip is observed only for contraction ratios 2 [les ] β [les ] 5, and the flow remains steady for higher contraction ratios. Rounding the corner of the 4:1 abrupt contraction leads to increased values of Deborah number for the onset of these flow transitions, but does not change the general structure of the transitions.

Collective flux pinning in a defective flux vortex lattice

The Larkin and Ovchinnikov (LO) theory for the summation of local pinning forces on flux vortices in type II superconductors describes the collective interaction between a perfectly elastic vortex lattice and a subthreshold pinning potential. Using a similar minimization procedure, it is shown that the elastic solution derived by LO does not always represent the lowest energy state of the vortex lattice. It is argued that as the magnetic field approaches Bc2, the vortex lattice inevitably becomes defective and that the defect structure determines the total pinning force, the static and dynamic response of the vortex lattice, and history effects when they occur. Calculations are presented for two and three dimensions, in two dimensions, the equilibrium defect structure predicts a peak effect in good agreement with observed critical currents. It is suggested that the situation in three dimensions is always nonequilibrium.

The pinning of flux lines in superconductors by randomly-spaced pinning centres

Abstract A simple theory is developed to explain the pinning of a vortex array by a large number of widely spaced pinning centres. By making certain simplifying assumptions, which do not alter the qualitative conclusions, it is possible to obtain analytic expressions for most of the parameters that have been measured on such a system. These include the critical current density, the voltage-current (V-I) characteristics, noise voltages and flux creep. It is shown that there is no regime in which the pinning is proportional to the square of the basic interaction, and also that it is impossible to reconcile the theory with all the experimental results. It is concluded that no theory based on non-interacting pinning centres in an elastic vortex lattice is tenable.

Elastic Filaments and Vortex Waves

The analogy between a vortex line and a stretched string is examined by considering the equation of motion of an elastic filament surrounded with circulating irrotational fluid. A generalization to filaments with discontinuities in density provides a model of an impurity trapped on a vortex line. The corresponding long-wavelength reflection probability reproduces that found previously.