Bravais lattice | EPFL Graph Search

In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3, where the ni are any integers, and ai are primitive translation vectors, or primitive vectors, which lie in different directions (not necessarily mutually perpendicular) and span the lattice. The choice of primitive vectors for a given Bravais lattice is not unique. A fundamental aspect of any Bravais lattice is that, for any choice of direction, the lattice appears exactly the same from each of the discrete lattice points when looking in that chosen direction. The Bravais lattice concept is used to formally define a crystalline arrangement and its (finite) frontiers. A crystal is made up of one or more atoms, called the basis or motif, at each lattice point. The basis may consist of atoms, molecules,

Grain Boundary Relaxation in 18-carat Gold Alloys

Ann-Kathrin Audren

Grain boundaries (GBs) play an important role for the mechanical properties of metals. In addition to dislocation motion inside the grains, some metal alloys deform along the GBs at high temperatures. GBs can also be the cause of a brittle behaviour of some metallic alloys, where crack propagation along the GBs is observed. This thesis aims to understand the role of GBs on the thermo-mechanical behaviour of metals and especially to define the microscopic mechanism, which leads to a deformation at the GBs. In this thesis, polycrystals, single crystals and bi-crystals of a yellow gold alloy are studied in detail, primarily by mechanical spectroscopy. The analysis and interpretation of the experimental data identifies different anelastic relaxations (internal energy dissipation processes), that produce peaks in the mechanical loss spectrum. The mechanical loss spectrum of polycrystals shows a relaxation peak P2 at about 780K, which is absent in single crystals made from the same alloy. Stepwise deformation of a single crystal causes an increase of the high temperature mechanical loss background and the appearance of a high temperature peak (P3). Above a critical deformation, P3 disappears and the peak P2, which is normally observed in polycrystals, appears. The increase of the exponential background is interpreted as due to the introduction of new dislocations whereas the peak P3 is attributed to a dislocation relaxation mechanism in the sub-grain boundaries. The peak P2 located at intermediate temperatures depends on the grain size: with grain growth, the peak position shifts to higher temperatures. It is shown that the relaxation time is proportional to the grain size d. Such a grains size dependence is in agreement with the Zener model based on geometrical considerations of an assembly of elastic grains, which can slide against each other. The investigations on bi-crystals containing a single GB with a specific misorientation between crystal lattices and a specific boundary plane orientation show that the relaxation peak P2 is closely related to a mechanism taking place at the boundary. The peak height is proportional to the GB density in a bi-crystal, whereas a single crystalline part cut from a bi-crystal does not show a relaxation peak. Molecular dynamics simulations are performed in order to illustrate the potential microscopic mechanisms responsible for the stress relaxation peak P2 in polycrystals. A Sigma5 grain boundary is submitted to a shear deformation parallel to the boundary plane. The grain boundary shows a migration perpendicular to the boundary plane coupled to shear for temperatures below 700K. Above 1000K, only grain boundary sliding is observed. Two models are developed that provide expressions for the relaxation strength and the relaxation time that are compared to experimental measurements performed on polycrystals. The observed grain size dependence favours the sliding model over the migration model. Measurements as a function of stress allow a refinement of the sliding model. The stress amplitude dependence of the peak P2 indicates a depinning mechanism, which is interpreted as due to GB dislocations between zones where sliding of different amount occurred. Obstacles to the sliding movement like steps or extended coincidence sites in the GB plane act as pinning points. The pinning points can be overcome at high temperatures close to the melting point, which results in the onset of local microscopic creep.

EPFL2014

In-situ Laue diffraction to follow dislocation patterning during fatigue

Ainara Irastorza Landa

The macroscopic strength of metals is determined by the dislocation arrangements that are formed when dislocations slip in the crystal lattice in response to the applied stress. Despite the extensive research carried out, the transition from uniform to non-uniform dislocation structures is not yet fully understood. This information is however essential to support the development of computational models that aim to predict dislocation patterning. Lattice rotation caused by the presence of dislocations is, for instance, a parameter that is often not taken into account in computational models although it plays a role in the development of dislocation ensembles. Experimentally following in-situ dislocation patterning including involved lattice rotations at lengthscales comparable to those in simulations is not straightforward. In fact, the current available techniques that have the necessary spatial and angular resolution are either destructive (3D-EBSD) or very time consuming (3D X-ray Laue diffraction). In this dissertation we present a new experimental approach that allows following lattice rotation and dislocation ensembles time-resolved during deformation. The technique is based on X-ray Laue diffraction scanning in transmission mode, which provides a sub-micron spatial resolution in 2D and statistical information on orientation spread in the third dimension. The in-situ mechanical tests are performed at the microXAS beamline of the Swiss Light Source Synchrotron. The method has been applied to understand dislocation pattering of copper single crystals occurring in the earlier phases of low cycle fatigue. Samples with different crystal orientations (single crystal, coplanar and collinear) have been cyclically deformed up to a maximum of 120 cycles. A dedicated miniaturized shear-fatigue device compatible with Laue diffraction has been constructed for that purpose and a sample preparation based on picosecond laser ablation has been developed. The developed procedure analyzes the evolving dislocation microstructures in terms of lattice rotation, lattice curvature and geometrically necessary dislocation densities at lengthscales similar to those addressable in computational models. It sacrifices on the fully 3D aspect but it brings the time resolution required to validate ongoing simulations. It has been shown that the error made by integration depends on the dislocation network expected. For instance, the technique can be used to validate 3DDD models or density based dislocation dynamics models when applied in deformation geometries where the formation of 2D dislocation patterns are expected (e.g. vein-channel structure in single slip oriented fatigued samples). The greatest advantage of this technique is that the evolving regions are easy to follow due to the high rotational sensitivity and time resolution of the method. What is more, the influence of the initial microstructure can be evaluated and quantitative values can be provided. On the other hand, the principal drawbacks are related to the gauge thickness of the samples. Even if the technique can estimate the orientation spread along the third direction, the approach cannot physically resolve the integrated signal along the thickness. This is source of uncertainty when providing actual values of lattice curvatures and geometrically necessary dislocation densities. Another shortcoming is the sensitivity of the method to the energy distribution of the X-ray beam provided in the beamline, which determines the collected signal - the principal basis of the subsequent analysis and interpretation.

EPFL2016

Experimental and first-principles study of point defects, domain walls, and point-defect/domain-wall interactions in ferroelectric oxides

Anand Chandrasekaran

Ferroelectric oxides, such as lead zirconate titanate, have proved invaluable due to their excellent dielectric and piezoelectric properties. These classes of materials possess a large electric polarization below the Curie temperature. Regions of the crystal lattice having different directions of polarization are separated by nano-scale interfaces known as domain walls. These interfaces, which can be moved by electric and mechanical fields, are not only important for electromechanical properties but they also exhibit unique structural and electronic properties that may be exploited through novel application in nanoelectronics and photovoltaics. The mobility of domain walls is strongly dependent on the defect chemistry of the material due to defect-domain wall interactions. Indeed, aliovalent doping of these materials has been used to tune the properties of ferroelectric oxides for specific applications. In this work, we investigate the origin of the so-called âhardâ and âsoftâ behavior in lead zirconate titanate. Experimental evidence suggests that small concentrations of donor dopants (such as niobium) results in enhanced domain wall motion while doping with acceptor dopants (such as iron) leads to an inhibited domain wall response. The microscopic origin of this phenomena is best investigated using first-principles simulations of the atomistic properties of defects and domain wall in these materials to bring to light novel structural and electronic properties. First, we show that polar defect complexes are likely to exist in both acceptor-doped and undoped PbTiO3 . These defects and defect associates are attracted to 180o domain walls and cause pinning of such interfaces. Donor- doped materials, on the other hand, do not show the presence of polar defect complexes. A closer investigation of the 180o domain wall in PbTiO3 reveals the presence of a âBlochâ component of polarization at the domain wall. In other words, due to polarization rotation, there exists a ferroelectric phase within the domain wall itself. We characterize the strain dependence of this phenomena and calculate the piezoelectric properties of such domain walls. A complete study of domain walls in PbTiO3 also entails looking closer at the properties of the ferroelastic âhead-to-tailâ 90o domain boundary. We show the presence of an asymmetry in the variation of the lattice parameter across the domain wall. This asymmetry is verified using high resolution aberration corrected electron microscopy. We look at the energy landscape of oxygen vacancies in the vicinity of these walls to explain the pinning effect in terms of random-bond and random-field defects. Next we look at the electronic properties of the recently discovered charged domain walls. We characterize the band bending phenomena in 90o head-to-head and tail-to-tail domain wall in PbTiO3 . We then look at the long range effects of clusters of oxygen vacancies in BaTiO3; especially in terms of the formation of 180o tail-to-tail domain wall. In parallel to the first priciples work, we also prepared high quality samples of PZT 50/50 with different concentrations of donor dopant. We characterized the microstructure and hysteresis loops as a function of donor dopant concentration. Based on experimental observations and ab initio calculations we propose ideas on the origins of hardening and softening in PZT.

EPFL2016

Grain Boundary Relaxation in 18-carat Gold Alloys

Ann-Kathrin Audren

EPFL2014

Experimental and first-principles study of point defects, domain walls, and point-defect/domain-wall interactions in ferroelectric oxides

Anand Chandrasekaran

EPFL2016

In-situ Laue diffraction to follow dislocation patterning during fatigue

Ainara Irastorza Landa

EPFL2016