Elasticity, fluctuations and vortex pinning in ferromagnetic superconductors: A "columnar elastic glass"

We study the elasticity, fluctuations and pinning of a putative spontaneous vortex solid in ferromagnetic superconductors. Using a rigorous thermodynamic argument, we show that in the idealized case of vanishing crystalline pinning anisotropy the long-wavelength tilt modulus of such a vortex solid vanishes identically, as guaranteed by the underlying rotational invariance. The vanishing of the tilt modulus means that, to lowest order, the associated tension elasticity is replaced by the softer, curvature elasticity. The effect of this is to make the spontaneous vortex solid qualitatively more susceptible to the disordering effects of thermal fluctuations and random pinning. We study these effects, taking into account the nonlinear elasticity, that, in three dimensions, is important at sufficiently long length scales, and showing that a ``columnar elastic glass'' phase of vortices results. This phase is controlled by a previously unstudied zero-temperature fixed point and it is characterized by elastic moduli that have universal strong wave-vector dependence out to arbitrarily long length scales, leading to non-Hookean elasticity. We argue that, although translationally disordered for weak disorder, the columnar elastic glass is stable against the proliferation of dislocations and is therefore a topologically ordered {\em elastic} glass. As a result, the phenomenology of the spontaneous vortex state of isotropic magnetic superconductors differs qualitatively from a conventional, external-field-induced mixed state. For example, for weak external fields $H$, the magnetic induction scales {\em universally} like $B(H)\sim B(0)+ c H^{\alpha}$, with $\alpha\approx 0.72$.Comment: Minor editorial changes, version to be published in PRB, 39 pages, 7 figure

Dislocation based modelling of engineering alloys

Plastic deformation in crystalline materials arises due to the nucleation and motion of line defects known as dislocations. Simulating large numbers of dislocations is a useful tool for understanding the plastic behaviour of metals and alloys. Discrete dislocation dynamics (DDD) is a technique used to approximate the behaviour of dislocation networks in an infinite body. By combining DDD with Finite Element Modelling (FEM), it becomes possible to simulate dislocation behaviour in a finite domain allowing micromechanical tests to be performed virtually. With sufficient improvements and refinements, using this technique can predict how the microstructure will affect the plastic behaviour. I have been working on developing such improvements, including accurately and rapidly calculating the displacements caused by the growth of dislocation loops on a convex surface in order to satisfy FEM boundary conditions

Theory and applications of free-electron vortex states

Both classical and quantum waves can form vortices: with helical phase fronts and azimuthal current densities. These features determine the intrinsic orbital angular momentum carried by localized vortex states. In the past 25 years, optical vortex beams have become an inherent part of modern optics, with many remarkable achievements and applications. In the past decade, it has been realized and demonstrated that such vortex beams or wavepackets can also appear in free electron waves, in particular, in electron microscopy. Interest in free-electron vortex states quickly spread over different areas of physics: from basic aspects of quantum mechanics, via applications for fine probing of matter (including individual atoms), to high-energy particle collision and radiation processes. Here we provide a comprehensive review of theoretical and experimental studies in this emerging field of research. We describe the main properties of electron vortex states, experimental achievements and possible applications within transmission electron microscopy, as well as the possible role of vortex electrons in relativistic and high-energy processes. We aim to provide a balanced description including a pedagogical introduction, solid theoretical basis, and a wide range of practical details. Special attention is paid to translate theoretical insights into suggestions for future experiments, in electron microscopy and beyond, in any situation where free electrons occur.Comment: 87 pages, 34 figure

Mechanics of Materials

The rapid advances of modern fabrications technologies require a thorough understanding of physical and mechanical properties of materials as influenced by their atomic composition, processing history and structure at the micro- and nanometer length scales. Carbon nanotubes, nanometer sized crystals, thin films and coatings, MEMS, smart materials and bio-inspired multifunctional materials are current examples employing technologies and processes that heavily depend on material properties at very small length scales. Today’s leading materials for a range of applications are hierarchical, having characteristics of structure at multiple length scales to satisfy a complex set of performance requirements and constraints. Composite materials and advanced alloy systems for transportation and infrastructure increasingly must rely on theoretical understanding at each of a range of length scales from the atomic scale upward to improve existing materials and to develop new materials to meet critical societal needs. Modern day efforts in mechanics of materials exploit recent advances in mechanics of materials that draws upon concurrent use of solid state physics, mathematics and information technology, continuum and discrete (statistical) mechanics and materials chemistry. Advanced materials derive their outstanding properties, durability and multifunctionality from heterogeneity of their underlying microstructure. There is a richness of outstanding problem sets at the intersection of theoretical and applied mathematics and materials mechanics. This state of affairs motivates the central goals of this workshop, namely to explore new and emerging mathematical approaches to multiscale modelling of evolving microstructures and to identify new and emerging mathematical approaches to interfaces in materials

Interfaces in crystalline materials

Interfaces such as grain boundaries in polycrystalline as well as heterointerfaces in multiphase solids are ubiquitous in materials science and engineering. Far from being featureless dividing surfaces between neighboring crystals, elucidating features of solid-solid interfaces is challenging and requires theoretical and numerical strategies to describe the physical and mechanical characteristics of these internal interfaces. The first part of this manuscript is concerned with interface-dominated microstructures emerging from polymorphic structural (diffusionless) phase transformations. Under high hydrostatic compression and shock-wave conditions, the pressure-driven phase transitions and the formation of internal diffuse interfaces in iron are captured by a thermodynamically consistent framework for combining nonlinear elastoplasticity and multivariant phase-field approach at large strains. The calculations investigate the crucial role played by the plastic deformation in the morphological and microstructure evolution processes under high hydrostatic compression and shock-wave conditions. The second section is intended to describe such imperfect interfaces at a finer scale, for which the semicoherent interfaces are described by misfit dislocation networks that produce a lattice-invariant deformation which disrupts the uniformity of the lattice correspondence across the interfaces and thereby reduces coherency. For the past ten years, the constant effort has been devoted to combining the closely related Frank-Bilby and O-lattice techniques with the Stroh sextic formalism for the anisotropic elasticity theory of interfacial dislocation patterns. The structures and energetics are quantified and used for rapid computational design of interfaces with tailored misfit dislocation patterns, including the interface sink strength for radiation-induced point defects and semicoherent interfaces.Comment: 138 pages, 70 figure

Optical manipulation and imaging of assemblies of topological defects and colloids in liquid crystals

Liquid Crystals (LCs) have proven to be important for electro-optic device applications such as displays, spatial light modulators, non-mechanical beam-steerers, etc. Owing to their unique mechanical, electrical, and optical properties, they are also being explored for wide array of advanced technological applications such as biosensors, tunable lenses, distributed feedback lasers, muscle-like actuators, etc. The thesis explores LC media from the standpoint of controlling their elastic and optical properties by generating and manipulating assemblies of defects and colloidal particles. To achieve the goal of optically manipulating these configurations comprising defects and particles at microscale with an unprecedented control, and then to visualize the resultant molecular director patterns, requires development of powerful optical system. The thesis discusses design and implementation of such an integrated system capable of 3D holographic optical manipulation and multi-modal 3D imaging (in nonlinear optical modes like multiphoton fluorescence, coherent anti-Stokes Raman scattering, etc.) and how they are used to extensively study a vast number of LC based systems. Understanding of LCs and topological defects go hand in hand. Appreciation of defects leads to their precise control, which in turn can lead to applications. The thesis describes discovery of optically generated stable, quasiparticle-like, localized defect structures in a LC cell, that we call "Torons". Torons enable twist of molecules in three dimensions and resemble both Skyrmion-like and Hopf fibration features. Under different conditions of generation, we optically realize an intriguing variety of novel solitonic defect structures comprising rather complicated configurations of point and line topological defects. Introducing colloidal particles to LC systems imparts to these hybrid material system a fascinating degree of richness of properties on account of colloidal assemblies supported by networks of LC defects as well as variety of localized defects supported by colloidal particles. To fully understand and exploit the resultant interactions involving colloids and defects in LC systems to achieve the full potential of their practical applications, it is required that these be explored on the level of individual particles and defects. We explore a multitude of interactions mediated by defects over different length scales and demonstrate for the first time, creation of several types of colloidal assemblies such as sparse colloidal structures and three-dimensional defect-bound colloidal structures

A field dislocation mechanics approach to emergent properties in two-phase nickel-based superalloys

The objective of this study is the development of a theoretical framework for treating the flow stress response of two-phase alloys as emergent behaviour arising from fundamental dislocation interactions. To this end a field dislocation mechanics (FDM) formulation has been developed to model heterogeneous slip within a computational domain representative of a two-phase nickel-based superalloy crystal at elevated temperature. A transport equation for the statistically stored dislocation (SSD) field is presented and implemented within a plane strain finite element scheme. Elastic interactions between dislocations and the microstructure are explicitly accounted for in this formulation. The theory has been supplemented with constitutive rules for dislocation glide and climb, as well as local cutting conditions for the γ’ particles by the dislocation field. Numerical simulations show that γ’ precipitates reduced the effective dislocation mobility by both acting as discrete slip barriers and providing a drag effect through line tension. The effect of varying microstructural parameters on the crystal deformation behaviour is investigated for simple shear loading boundary conditions. It is demonstrated that slip band propagation can be simulated by the proposed FDM approach. Emergent behaviour is predicted and includes: domain size yield dependence (Hall-Petch relationship), γ’ volume fraction yield dependence (along with more complex γ’ dispersion-related yield and post-yield flow stress phenomena), and hardening related to dislocation source distribution at the grain boundary. From these simulations, scaling laws are derived. Also, the emergence of internal back stresses associated with non-homogeneous plastic deformation is predicted. Prediction of these back stresses, due to sub-grain stress partitioning across elastic/plastic zones, is an important result which can provide useful information for the calibration of phenomenological macroscale models. Validation for the presented model is provided through comparison to experimental micro-shear tests that can be found in published literature

Localized transition states in many-particle systems

This thesis addresses the investigation of the transition from order to chaos in two different systems. In this context, both numerical simulations and theoretical considerations are applied. Popular examples of such transitions are, among others, the melting of a crystal or the transition from a laminar flow to turbulence. They have in common that the variation of an external parameter, for example temperature, results in an abrupt change in the properties of the system: The ordered, well-defined structure of a crystal transforms to the unordered, random configuration of a liquid. In the first part of this thesis, we investigate the influence of a linear, periodic shear on a system of mutually repelling particles. This can be considered as a model for a well known phenomenon, the mixing of a blob of dye in a liquid confined between two concentric cylinders. If the cylinders are rotated back and forth slowly enough so that the flow remains in the laminar regime, a demixing is possible and after one period we retrieve the original blob. Something similar occurs in the simple two-dimensional model system that we investigate: For small shear rates, we observe a self-organization of the particles, ending up in a lattice configuration. Above a critical shear rate, an abrupt change takes place, and the system is found - and remains - in an unordered, chaotic state. We are able to associate the transition with a loss of stability of the sheared lattice. In the chaotic regime, the spatially resolved correlations reveal more details and allow us to extract phase information. Additionally, they provide a possible explanation why diffusion parallel to the shear is enhanced beyond the known advection-diffusion coupling. In the further course of this work, we turn toward a transition known from everyday life, the melting of a solid. There, we again restrict ourselves to a two-dimensional model similar to the one in the first part. We are especially interested in the microscopic processes which eventually result in the melting of the crystal. Therefore, we perform molecular-dynamics (MD) simulations which confirm a two-step process of melting. Furthermore, the computed trajectories allow us deeper insights into the dynamics of the system. In the critical temperature range, we initially observe isolated localized processes where several particles exchange their positions. With the help of a projection on the energetically lowest configuration, these transitions can be identified as hopping events on the hexagonal lattice. As temperature is increased, more processes occur simultaneously, and eventually secondary, more complex transitions are stimulated which result in the melting of the solid. On this account, we investigate the melting transition in view of a rate activated process induced by localized reorganizations of a few particles. We identify possible transitions and the corresponding transition states which comprise up to 18 particles and only affect up to approximately 25 particles, implying that the states are well localized. We characterize the states, both by their arrangement in configuration space as well as by their thermodynamically relevant properties such as the energy barriers. We determine the dependence of the transition rates on the temperature, and compare them to the melting temperature of the system. Apparently, the rates are too low to explain melting on their own. Nevertheless, this is in accordance with previous observations in the simulations, where localized reorganizations lead to secondary transitions which break up the lattice structure and hence initiate the melting process. In the course of our studies, we also investigate the elastic properties of the system. The crystal, consisting of individual particles, can be described by the elastic constants of a continuous solid body. We show that the displacement field induced by the local disturbance of the transition state can be approximated by a superposition of several displacement fields of singular forces acting on an elastic medium. The screening of the potential not only gives rise to a rescaling of the energy of the system, but alters its elastic properties as well. This is partly reflected in the transition states. Though their basic configuration remains unaffected, energy barriers and displacement fields change considerably. We once again refer to the rate model in order to determine the transition rate at the critical temperature. A concluding comparison with results from MD-simulations and other predictions reveals that the model captures the dependence of the melting temperature on the screening parameter of the potential very well

Mechanics of Materials: Towards Predictive Methods for Kinetics in Plasticity, Fracture, and Damage

The workshop dealt with current advances of computational methods, mathematics and continuum mechanics directed at thermodynamically consistent forms of constitutive equations for complex evolutionary phenomena in modern materials such as plasticity, fracture and damage. The main aspects addressed in presentations and discussions were multiphysical description of new materials, (visco)plasticity, fracture, damage, structural mechanics, mechanics of materials and dislocation dynamics