Showing papers in "Philosophical Magazine in 2007"

Self-consistent modelling of the mechanical behaviour of viscoplastic polycrystals incorporating intragranular field fluctuations

Abstract: We present a detailed description of the numerical implementation, within the widely used viscoplastic self-consistent (VPSC) code, of a rigorous second-order (SO) homogenization procedure for non-linear polycrystals. The method is based on a linearization scheme, making explicit use of the covariance of the fluctuations of the local fields in a certain linear comparison material, whose properties are, in turn, determined by means of a suitably designed variational principle. We discuss the differences between this second-order approach and several first-order self-consistent (SC) formulations (secant, tangent and affine approximations) by comparing their predictions with exact full-field solutions. We do so for crystals with different symmetries, as a function of anisotropy, number of independent slip systems and degree of non-linearity. In this comparison, the second-order estimates show the best overall agreement with the full-field solutions. Finally, the different SC approaches are applied to simulat...

Asymmetric tilt grain boundary structure and energy in copper and aluminium

Abstract: Atomistic simulations were employed to investigate the structure and energy of asymmetric tilt grain boundaries in Cu and Al. In this work, we examine the Σ5 and Σ13 systems with a boundary plane rotated about the ⟨ 100 ⟩ misorientation axis, and the Σ9 and Σ11 systems rotated about the ⟨ 110 ⟩ misorientation axis. Asymmetric tilt grain boundary energies are calculated as a function of inclination angle and compared with an energy relationship based on faceting into the two symmetric tilt grain boundaries in each system. We find that asymmetric tilt boundaries with low index normals do not necessarily have lower energies than boundaries with similar inclination angles, contrary to previous studies. Further analysis of grain boundary structures provides insight into the asymmetric tilt grain boundary energy. The Σ5 and Σ13 systems in the ⟨ 100 ⟩ system agree with the aforementioned energy relationship; structures confirm that these asymmetric boundaries facet into the symmetric tilt boundaries. The Σ9 and ...

Structures and energies of Σ 3 asymmetric tilt grain boundaries in copper and aluminium

Abstract: The objective of this research is to use atomistic simulations to investigate the energy and structure of symmetric and asymmetric Σ3 ⟨110⟩ tilt grain boundaries. A nonlinear conjugate gradient algorithm was employed along with an embedded atom method potential for Cu and Al to generate the equilibrium 0 K grain boundary structures. A total of 25 ⟨110⟩ grain boundary structures were explored to identify the various equilibrium and metastable structures. Simulation results show that the Σ3 asymmetric tilt grain boundaries in the ⟨110⟩ system are composed of only structural units of the two Σ3 symmetric tilt grain boundaries. The energies for the Σ3 grain boundaries are similar to previous experimental and calculated grain boundary energies. A structural unit and faceting model for Σ3 asymmetric tilt grain boundaries fits all of the calculated asymmetric grain boundary structures. The significance of these results is that the structural unit and facet description of all Σ3 asymmetric tilt grain boundaries m...

Force chain buckling, unjamming transitions and shear banding in dense granular assemblies

Abstract: Force chain buckling, leading to unjamming and shear banding, is examined quantitatively via a discrete element analysis of a two-dimensional, densely-packed, cohesionless granular assembly subject to quasistatic, boundary-driven biaxial compression. A range of properties associated with the confined buckling of force chains has been established, including: degree of buckling, buckling modes, spatial and strain evolution distributions, and relative contributions to non-affine deformation, dilatation and decrease in macroscopic shear strength and potential energy. Consecutive cycles of unjamming–jamming events, akin to slip–stick events arising in other granular systems, characterize the strain-softening regime and the shear band evolution. Peaks in the dissipation rate, kinetic energy and local non-affine strain are strongly correlated: the largest peaks coincide with each unjamming event that is evident in the concurrent drops in the macroscopic shear stress and potential energy. Unjamming nucleates from...

Dislocation structures. Part I. Grain orientation dependence

Abstract: To clarify the effect of grain orientation on the evolution of dislocation structures in metals of medium-to-high stacking fault energy, detailed TEM characterization of structures was carried out for more than 350 individual grains in Al and Cu deformed in tension or by cold rolling up to moderate strain levels (ϵvM ≤ 0.8). Efforts were made to obtain a precise description of the three-dimensional arrangement of the dislocation structures and to determine the crystallographic plane of extended dislocation boundaries (geometrically necessary boundaries). A universal pattern of structural evolution characterized by a formation of three types of structure was found in both metals, irrespective of material parameters (stacking fault energy, grain size and impurity) and deformation conditions (deformation mode, strain and strain rate). The key parameter controlling the formation of the different structural types was found to be grain orientation with respect to the deformation axis (axes) and a clear relation...

Strain-rate dependence of the brittle-to-ductile transition temperature in tungsten

Abstract: We have investigated the strain-rate dependence of the brittle-to-ductile transition (BDT) temperature in pre-cracked tungsten single-crystals and polycrystals. There is an unambiguous Arrhenius relationship over four decades of strain rate, giving an activation energy for the process controlling the BDT of 1.05 eV. This is equal to the activation energy for double-kink formation on screw dislocations, suggesting that their motion controls the brittle–ductile transition.

Estimating the strength of single-ended dislocation sources in micron-sized single crystals

Abstract: Three-dimensional (3D) discrete dislocation dynamics simulations were used to calculate the effects of anisotropy of dislocation line tension (increasing Poisson's ratio, ν) on the strength of single-ended dislocation sources in micron-sized volumes with free surfaces and to compare them with the strength of double-ended sources of equal length. Their plastic response was directly modelled within a 1 µm3 volume composed of a single crystal fcc metal. In general, double-ended sources are stronger than single-ended sources of an equal length and exhibit no significant effects from truncating the long-range elastic fields at this scale. The double-ended source strength increases with ν, exhibiting an increase of about 50% at ν = 0.38 (value for Ni) as compared to the value at ν = 0. Independent of dislocation line direction, for ν greater than 0.20, the strengths of single-ended sources depend upon the sense of the stress applied. The value for α in the expression for strength, τ = α(L)µb/L is shown to vary ...

A three-dimensional continuum theory of dislocation systems: kinematics and mean-field formulation

Abstract: We propose a dislocation density measure which is able to account for the evolution of systems of three-dimensional curved dislocations. The definition and evolution equation of this measure arise as direct generalizations of the definition and kinematic evolution equation of the classical dislocation density tensor. The evolution of this measure allows us to determine the plastic distortion rate in a natural fashion and therefore yields a kinematically closed dislocation-based theory of plasticity. A self-consistent theory is built upon the measure which accounts for both the long-range interactions of dislocations and their short-range self-interaction which is incorporated via a line-tension approximation. A two-dimensional kinematic example illustrates the definitions and their relations to the classical theory.

Composition dependent crystallography of α''-martensite in Ti-Nb-based β-titanium alloy

Abstract: Composition dependence of crystallographic features of α″-martensite in Ti–Nb-based alloy was systematically examined in Ti–Xmol%Nb–3mol%Al alloy, where X is 10–30. One of the lattice deformation s...

Radiation defects in graphite

Abstract: This article discusses the nature of radiation defects in graphite, reviewing past and recent developments in understanding their structure, interactions and effect on physical properties. The principal focus is on behaviour at the atomic and microstructural level, with an interest both in understanding graphite moderator damage in nuclear reactors and building a foundation for the range of emerging technological applications of defect-engineered graphitic materials. It is hoped that this article will both clarify the picture that has emerged over the last 50 years and provide a useful background to ongoing efforts.

Brittle and ductile fracture of semiconductor nanowires – molecular dynamics simulations

Abstract: Fracture of silicon and germanium nanowires in tension at room temperature is studied by molecular dynamics simulations using several interatomic potential models. While some potentials predict brittle fracture initiated by crack nucleation from the surface, most potentials predict ductile fracture initiated by dislocation nucleation and slip. A simple parameter based on the ratio between the ideal tensile strength and the ideal shear strength is found to correlate very well with the observed brittle versus ductile behaviours for all the potentials used in this study. This parameter is then computed by ab initio methods, which predict brittle fracture at room temperature. A brittle-to-ductile transition (BDT) is observed in MD simulations at higher temperature. The BDT mechanism in semiconductor nanowires is different from that in the bulk, due to the lack of a pre-existing macrocrack that is always assumed in bulk BDT models.

Multiscale modelling of dislocation/grain boundary interactions. II. Screw dislocations impinging on tilt boundaries in Al

Abstract: The interaction of dislocations with grain boundaries (GBs) determines a number of important aspects of the mechanical performance of materials, including strengthening and fatigue resistance. Here, the coupled atomistic/discrete-dislocation (CADD) multiscale method, which couples a discrete dislocation continuum region to a fully atomistic region, is used to study screw-dislocations interacting with Σ3, Σ11, and Σ9 symmetric tilt boundaries in Al. The low-energy Σ3 and Σ11 boundaries absorb lattice dislocations and generate extrinsic grain boundary dislocations (GBDs). As multiple screw dislocations impinge on the GB, the GBDs form a pile-up along the GB and provide a back stress that requires increasing applied load to push the lattice dislocations into the GB. Dislocation transmission is never observed, even with large GBD pile-ups near the dislocation/GB intersection. Results are compared with experiments and previous, related simulations. The Σ9 grain boundary, composed from a more complex set of str...

The effect of Zn on precipitation in Al–Mg–Si alloys

Abstract: Effects of addition of Zn (up to 1 wt%) on microstructure, precipitate structure and intergranular corrosion (IGC) in an Al–Mg–Si alloys were investigated. During ageing at 185 °C, the alloys showed modest increases in hardness as function of Zn content, corresponding to increased number densities of needle-shaped precipitates in the Al–Mg–Si alloy system. No precipitates of the Al–Zn–Mg alloy system were found. Using high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM), the Zn atoms were incorporated in the precipitate structures at different atomic sites with various atomic column occupancies. Zn atoms segregated along grain boundaries, forming continuous film. It correlates to high IGC susceptibility when Zn concentration is ~1wt% and the materials in peak-aged condition.

Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices

Abstract: We report conjectures on the three-dimensional (3D) Ising model of simple orthorhombic lattices, together with details of calculations for a putative exact solution. Two conjectures, an additional rotation in the fourth curled-up dimension and weight factors on the eigenvectors, are proposed to serve as a boundary condition to deal with the topologic problem of the 3D Ising model. The partition function of the 3D simple orthorhombic Ising model is evaluated by spinor analysis, employing these conjectures. Based on the validity of the conjectures, the critical temperature of the simple orthorhombic Ising lattices could be determined by the relation of KK* = KK ' + KK '' + K ' K '' or sinh 2K - sinh 2(K ' + K '' + (K ' K ''/K)) = 1. For a simple cubic Ising lattice, the critical point is putatively determined to locate exactly at the golden ratio x(c) = e(-2kc) = ((root 5 - 1)/2), as derived from K* = 3K or sinh 2K = sinh 6K = 1. If the conjectures would be true, the specific heat of the simple orthorhombic Ising system would show a logarithmic singularity at the critical point of the phase transition. The spontaneous magnetization of the simple orthorhombic Ising ferromagnet is derived explicitly by the perturbation procedure, following the conjectures. The spin correlation functions are discussed on the terms of the Pfaffians, by defining the effective skew-symmetric matrix A(eff). The true range k(x) of the correlation and the susceptibility of the simple orthorhombic Ising system are determined by procedures similar to those used for the two-dimensional Ising system. The putative critical exponents derived explicitly for the simple orthorhombic Ising lattices are alpha = 0, beta = 3/8, gamma = 5/4, delta = 13/3, eta = 1/8 and nu = 2/3, showing the universality behaviour and satisfying the scaling laws. The cooperative phenomena near the critical point are studied and the results based on the conjectures are compared with those of approximation methods and experimental findings. The putative solutions have been judged by several criteria. The deviations of the approximation results and the experimental data from the solutions are interpreted. Based on the solution, it is found that the 3D-to-2D crossover phenomenon differs with the 2D-to-1D crossover phenomenon and there is a gradual crossover of the exponents from the 3D to the 2D values. Special attention is also paid to the extra energy caused by the introduction of the fourth curled-up dimension and the states at/near infinite temperature revealed by the weight factors of the eigenvectors. The physics beyond the conjectures and the existence of the extra dimension are discussed. The present work is not only significant for statistical and condensed matter physics, but also fill the gap between the quantum field theory, cosmology theory, high-energy particle physics, graph theory and computer science.

Dislocation structures. Part II. Slip system dependence

Abstract: Part I established, via extensive transmission electron microscopy investigations, that the type of dislocation structure formed in metals of medium-to-high stacking fault energy upon deformation in tension or rolling to moderate strain levels (≤0.8) depends strongly on crystallographic grain orientation. This paper analyzes the grain orientation-dependent structures in terms of the active slip systems, focusing on the crystallographic plane of extended planar boundaries (geometrically necessary boundaries). The analysis establishes slip systems as the factor controlling the dislocation structure. Five fundamental slip classes, consisting of one to three active slip systems, have been identified. Multiple activation of these slip classes is also considered. The slip classes give rise to different types of dislocation structure, of which all except one contains geometrically necessary planar boundaries aligning with unique crystallographic planes (not necessarily slip planes). A slip class leads to the sam...

Investigating carbonization and graphitization using electron energy loss spectroscopy (EELS) in the transmission electron microscope (TEM)

Abstract: Electron energy loss spectroscopy (EELS) in the transmission electron microscope (TEM) is explored as a useful characterization technique in the study of carbonization and graphitization of organic...

Ensemble averaging stress–strain fields in polycrystalline aggregates with a constrained surface microstructure – Part 2: crystal plasticity

Abstract: The effect of three-dimensional (3D) grain morphology on the deformation at a free surface in polycrystalline aggregates is investigated by means of a large-scale finite element and statistical approach. For a given two-dimensional surface at z = 0 containing 39 grains with given crystal orientations, eight 3D random polycrystalline aggregates are constructed having different 3D grain shapes and orientations except at z = 0, based on an original 3D image analysis procedure. They are subjected to overall tensile loading conditions. The continuum crystal plasticity framework is adopted and the resulting plastic strain fields at the free surface z = 0 are analyzed. Ensemble average and variance maps of the plastic strain field at the observed free surface are computed. In the case of elastoplastic copper grains, fluctuations ranging between 2% and 80% are found in the equivalent plastic slip level at a given material point of the observed surface from one realization of the microstructure to another. The obt...

An electrochemomechanical theory of defects in ionic solids. I. Theory

Abstract: Charged defects diffuse through an ionic solid under electrochemical driving forces. Such a diffusion process can be affected by mechanical stresses in the solid. A deviation of defect concentration from its stoichiometric value during diffusion can cause volumetric strains in the solid. Such strains will result in mechanical stresses if the ionic solid is under mechanical constraint, or if the defect distribution is non-uniform. We develop a framework to account for the coupling between mechanical stresses and diffusion of charged defects in ionic solids. The framework consists of a system of nonlinear differential/algebraic equations governing the defect concentrations, electrostatic potential and the mechanical stresses. It is believed that this framework is the first fully coupled theory for the interaction between mechanical stresses and electrochemical forces in ionic solids.

Does In form In-rich clusters in InGaN quantum wells?

Abstract: The reason the InGaN/GaN quantum well system emits intense light even though the dislocation density is high is assessed. First, the evidence from electron microscopy for nanometre-scale In-rich clusters in InGaN quantum wells is presented. Such clusters would localize the excitons away from the dislocations and hence the dislocations would not quench the light emission, consistent with observations. However, it is then shown that InGaN damages extremely rapidly under an electron beam, and that the damage causes In-rich clusters to form. No evidence is found of gross indium clustering at low beam currents and short exposure times in the electron microscope. However, at such low electron doses the image is noisy, and low-level indium clustering in InGaN quantum wells could possibly exist, but be masked by the noise. A different technique, 3-dimensional atom probe field ion microscopy, has therefore been used to image the InGaN quantum wells. This reveals that InGaN is a random alloy, with the local statist...

Correlation effects and energetics of point defects in uranium dioxide: a first principle investigation

Abstract: Density functional (DFT) calculations have been used to investigate the stability of point defects in uranium dioxide. Correlation effects are taken into account within the DFT + U approach as implemented in the Vienna ab initio simulation package (VASP). More particularly, the formation energies of both intrinsic and extrinsic point defects, i.e. vacancies, interstitials, Frenkel pairs and Schottky trio defects, are calculated. Our results are compared with available experimental data and are also discussed in relation to previous calculations based on conventional functionals, such as the local spin-density approximation and generalized gradient approximations.

Soft quasicrystals–Why are they stable?

Abstract: In the last two years we have witnessed the exciting experimental discovery of soft matter with nontrivial quasiperiodic long-range order–a new form of matter termed a soft quasicrystal. Two groups have independently discovered such order in soft matter: Zeng et al. in a system of dendrimer liquid crystals; and Takano et al. in a system of ABC star-shaped polymers. These newly discovered soft quasicrystals not only provide exciting platforms for the fundamental study of both quasicrystals and of soft matter, but also hold the promise for new applications based on self-assembled nanomaterials with unique physical properties that take advantage of the quasiperiodicity, such as complete and isotropic photonic band-gap materials. Here we provide a concise review of the emerging field of soft quasicrystals, suggesting that the existence of two natural length-scales, along with three-body interactions, may constitute the underlying source of their stability.

Selection of α variants during microstructural evolution in α/β titanium alloys

Abstract: The solid-state β→β + α transformation in titanium alloys leads to complex microstructures with feature spanning across a range of length scales. In order to develop a better understanding of the microstructural evolution process, a detailed characterization of the crystallography of α laths formed from the β phase in a candidate α/β Ti alloy, Timetal 550, has been carried out. Specifically, the influence of the orientation relationship (OR) between the grain boundary α (GB α) and the adjacent β grains on the microstructural evolution has been investigated in this alloy employing orientation imaging microscopy (OIM) studies in a high-resolution SEM. The results indicate that the colony microstructure (clustering of α laths belonging to the same variant) tends to develop in the β grain that exhibits the Burgers OR with the GB α allotriomorph, whereas the basketweave microstructure (clustering of multiple variants) develops in the adjacent β grain. Additionally, the multiple variants of α laths forming the ...

Methane hydrate crystal growth in a porous medium filled with methane-saturated liquid water

Abstract: The nucleation, growth and ageing of methane hydrate crystals were observed visually in a porous medium filled with liquid water presaturated with methane. The pore space dimensions of the porous medium were 1.0 × 102 µm. The pressure−temperature conditions at which hydrate formation was initiated corresponded to system subcoolings of 3.4 K, 6.7 K, 12.3 K and 14.1 K, respectively, where the system subcooling denotes the difference of the system temperature from the triple methane−hydrate−water equilibrium temperature under a given pressure. Faceted (skeletal) hydrate crystals grew and bridged the pore spaces without intervention of a liquid water layer when the subcoolings were equal or smaller than 6.7 K. The faceted crystals may form a physical bonding with the walls of the porous medium. At the higher subcoolings, the dispersive formation of dendritic crystals and subsequent morphological change into particulate crystals were observed. The bridging of the dendritic crystals is unlikely in the absence o...

Ensemble averaging stress–strain fields in polycrystalline aggregates with a constrained surface microstructure – Part 1: anisotropic elastic behaviour

Abstract: The effect of three-dimensional (3D) grain morphology on the deformation at a free surface in polycrystalline aggregates is investigated by means of a large-scale finite element and statistical approach. For a given two-dimensional surface at z = 0 containing 39 grains with given crystal orientations, 17 random 3D polycrystalline aggregates are constructed having different 3D grain shapes and orientations except at z = 0, based on an original 3D image analysis procedure. They are subjected to overall tensile loading conditions. The resulting stress–strain fields at the free surface z = 0 are analyzed. Ensemble average and variance maps of the stress field at the observed surface are computed. In the case of an anisotropic elastic behaviour of the grains, fluctuations ranging between 5% and 60% are found in the equivalent stress level at a given material point of the observed surface from one realization of the microstructure to another. These results have important implications in the way of comparing fin...

A study on the early-stage decomposition in the Al–Mg–Si–Cu alloy AA6111 by electrical resistivity and three-dimensional atom probe

Abstract: Electrical resistivity measurements and three-dimensional atom probe (3DAP) analysis were employed to investigate early-stage decomposition of the Al alloy AA6111 in the temperature range 60–180°C where electrical resistivity initially increased with ageing time. 3DAP measurements provided information on the shape, number density and solute content of the precipitates, as well as the solute concentration of the matrix, for the ageing conditions corresponding to the resistivity maxima. Using the 3DAP results, the precipitate size distributions for these ageing conditions were determined in terms of the measured number of solute atoms per precipitate. The number density and the Cu content of the precipitates decreased with increasing temperature, whereas the Mg/Si ratio increased. The size distribution of precipitates at the higher ageing temperatures showed the addition of larger size precipitates to the precipitate population. A modification to Matthiessen's law was employed to describe the anomalous resi...

Constant-angle surfaces in liquid crystals

Abstract: We discuss some properties of surfaces in whose unit normal vector has constant angle with an assigned direction field. The constant-angle condition can be rewritten as a Hamilton–Jacobi equation correlating the surface and the direction field. We focus on examples motivated by the physics of interfaces in liquid crystals and of layered fluids, and examine some properties of the constant-angle surfaces when the direction field is singular along a line (disclination) or at a point (hedgehog). We also show how our results may be used to study the shape of disclination cores.

Role of twinning on texture evolution of silver during equal channel angular extrusion

Abstract: The role of deformation twinning in the texture evolution of pure polycrystalline silver subjected to equal channel angular extrusion (Route A, three passes) has been examined. Microstructural characterization using electron backscattering diffraction and transmission electron microscopy revealed high twinning activity in every pass, as well as significant grain refinement. Polycrystal modelling combined with experimental analysis shows that texture evolution is a result of slip and deformation twinning that occurs in every pass. It is shown that the primary consequence of twinning is the reorientation of the $A_1$ ideal component into the $A_2$ orientation. This process results in a weak $A_1$ and a strong $A_2$ component. This twinning mechanism is repeated in each pass aided by the strain path changes associated with Route $A$ and an apparent regeneration of the microstructure. As a result, with each pass the $A_1$ and C ideal shear components weaken, whereas the B/B components strengthen, tendencies that are distinct from those of high stacking fault fcc metals like Al, Cu and Ni.

Nucleation of misfit dislocations and plastic deformation in core/shell nanowires

Abstract: During fabrication of metal nanowires, an oxide layer (shell) that surrounds the metal (core) may form. Such an oxide-covered nanowire can be viewed as a cylindrical core/shell nanostructure, possessing a crystal lattice mismatch between the core and shell. Experimental evidence has shown that, in response to this mismatch, mechanical stresses induce plastic deformation in the shell and misfit dislocations nucleate at the core/shell interface. As a result, the mechanical, electrical and optoelectronic properties of the nanowire are affected. It is therefore essential to be able to predict the critical conditions at which misfit dislocation nucleation at the nanowire interface takes place and the critical applied load at which the interface begins deforming plastically. Two approaches are explored in order to analyze the stress relaxation processes in these oxide-covered nanowires: (i) energy considerations are carried out within a classical elasticity framework to predict the critical radii (of the core a...

Spin ordering in three-dimensional crystals with strong competing exchange interactions

Abstract: In this paper we discuss magnetic ordering in three-dimensional crystals in which Heisenberg exchange interactions appear to dominate. Particular emphasis is placed on systems with strong competing exchange interactions, called frustrated spin systems. In the absence of such competition, one finds collinear spin ordering. However, strong competing interactions lead to non-collinear structures, often spirals and many variations thereof. The problem of understanding the origin and physical properties of such spin states within the classical Heisenberg model has been intensively addressed by a large community of researchers over the last 2–3 decades. The study of such problems actually began about five decades ago, and led to a large body of literature that has been overlooked in the publications of the last 2–3 decades (with two very recent exceptions). The early work established important fundamental concepts, including an unconventional theoretical approach, the generalized Luttinger–Tisza method and the ...

Jump condition for GND evolution as a constraint on slip transmission at grain boundaries

Abstract: The jump condition at a possibly non-material interface for geometrically-necessary dislocation density in field dislocation mechanics (FDM) and its averaged approximation, phenomenological mesoscopic FDM (PMFDM), is derived. In the context of grain boundaries, the condition implies a tensorial constraint on all five grain boundary parameters, slip transmission at the boundary, possible grain boundary motion and dislocation nucleation/annihilation at the grain boundary. The jump condition is physically interpreted in special cases, and the importance of understanding dislocation motion at a boundary/interface as a flux across curves, and not surfaces, is emphasized.