Variational Problems in Materials Science

Acta Numerica ◽  
2016 ◽  
Vol 25 ◽  
pp. 567-679 ◽  
Author(s):  
Ulrik S. Fjordholm ◽  
Siddhartha Mishra ◽  
Eitan Tadmor

A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multi-dimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures – parametrized probability measures which can describe the limits of such oscillatory sequences – offer the more general paradigm of measure-valued solutions for these problems.We present viable numerical algorithms to compute approximate measure-valued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. We also discuss the use of these algorithms, and their extensions, in uncertainty quantification and contexts other than fluid dynamics, such as non-convex variational problems in materials science.


2016 ◽  
Vol 26 (07) ◽  
pp. 1319-1355 ◽  
Author(s):  
Sergio Conti ◽  
Barbara Zwicknagl

We study microstructure formation in two nonconvex singularly-perturbed variational problems from materials science, one modeling austenite–martensite interfaces in shape-memory alloys, the other one slip structures in the plastic deformation of crystals. For both functionals we determine the scaling of the optimal energy in terms of the parameters of the problem, leading to a characterization of the mesoscopic phase diagram. Our results identify the presence of a new phase, which is intermediate between the classical laminar microstructures and branching patterns. The new phase, characterized by partial branching, appears for both problems in the limit of small volume fraction, that is, if one of the variants (or of the slip systems) dominates the picture and the volume fraction of the other one is small.


Author(s):  
C. Colliex ◽  
P. Trebbia

The physical foundations for the use of electron energy loss spectroscopy towards analytical purposes, seem now rather well established and have been extensively discussed through recent publications. In this brief review we intend only to mention most recent developments in this field, which became available to our knowledge. We derive also some lines of discussion to define more clearly the limits of this analytical technique in materials science problems.The spectral information carried in both low ( 0<ΔE<100eV ) and high ( >100eV ) energy regions of the loss spectrum, is capable to provide quantitative results. Spectrometers have therefore been designed to work with all kinds of electron microscopes and to cover large energy ranges for the detection of inelastically scattered electrons (for instance the L-edge of molybdenum at 2500eV has been measured by van Zuylen with primary electrons of 80 kV). It is rather easy to fix a post-specimen magnetic optics on a STEM, but Crewe has recently underlined that great care should be devoted to optimize the collecting power and the energy resolution of the whole system.


Author(s):  
Hannes Lichte ◽  
Edgar Voelkl

The object wave o(x,y) = a(x,y)exp(iφ(x,y)) at the exit face of the specimen is described by two real functions, i.e. amplitude a(x,y) and phase φ(x,y). In stead of o(x,y), however, in conventional transmission electron microscopy one records only the real intensity I(x,y) of the image wave b(x,y) loosing the image phase. In addition, referred to the object wave, b(x,y) is heavily distorted by the aberrations of the microscope giving rise to loss of resolution. Dealing with strong objects, a unique interpretation of the micrograph in terms of amplitude and phase of the object is not possible. According to Gabor, holography helps in that it records the image wave completely by both amplitude and phase. Subsequently, by means of a numerical reconstruction procedure, b(x,y) is deconvoluted from aberrations to retrieve o(x,y). Likewise, the Fourier spectrum of the object wave is at hand. Without the restrictions sketched above, the investigation of the object can be performed by different reconstruction procedures on one hologram. The holograms were taken by means of a Philips EM420-FEG with an electron biprism at 100 kV.


Author(s):  
J.C.H. Spence ◽  
J. Mayer

The Zeiss 912 is a new fully digital, side-entry, 120 Kv TEM/STEM instrument for materials science, fitted with an omega magnetic imaging energy filter. Pumping is by turbopump and ion pump. The magnetic imaging filter allows energy-filtered images or diffraction patterns to be recorded without scanning using efficient parallel (area) detection. The energy loss intensity distribution may also be displayed on the screen, and recorded by scanning it over the PMT supplied. If a CCD camera is fitted and suitable new software developed, “parallel ELS” recording results. For large fields of view, filtered images can be recorded much more efficiently than by Scanning Reflection Electron Microscopy, and the large background of inelastic scattering removed. We have therefore evaluated the 912 for REM and RHEED applications. Causes of streaking and resonance in RHEED patterns are being studied, and a more quantitative analysis of CBRED patterns may be possible. Dark field band-gap REM imaging of surface states may also be possible.


Author(s):  
Robert M. Glaeser ◽  
Bing K. Jap

The dynamical scattering effect, which can be described as the failure of the first Born approximation, is perhaps the most important factor that has prevented the widespread use of electron diffraction intensities for crystallographic structure determination. It would seem to be quite certain that dynamical effects will also interfere with structure analysis based upon electron microscope image data, whenever the dynamical effect seriously perturbs the diffracted wave. While it is normally taken for granted that the dynamical effect must be taken into consideration in materials science applications of electron microscopy, very little attention has been given to this problem in the biological sciences.


Author(s):  
M. Rühle ◽  
J. Mayer ◽  
J.C.H. Spence ◽  
J. Bihr ◽  
W. Probst ◽  
...  

A new Zeiss TEM with an imaging Omega filter is a fully digitized, side-entry, 120 kV TEM/STEM instrument for materials science. The machine possesses an Omega magnetic imaging energy filter (see Fig. 1) placed between the third and fourth projector lens. Lanio designed the filter and a prototype was built at the Fritz-Haber-Institut in Berlin, Germany. The imaging magnetic filter allows energy-filtered images or diffraction patterns to be recorded without scanning using efficient area detection. The energy dispersion at the exit slit (Fig. 1) results in ∼ 1.5 μm/eV which allows imaging with energy windows of ≤ 10 eV. The smallest probe size of the microscope is 1.6 nm and the Koehler illumination system is used for the first time in a TEM. Serial recording of EELS spectra with a resolution < 1 eV is possible. The digital control allows X,Y,Z coordinates and tilt settings to be stored and later recalled.


Author(s):  
Vinayak P. Dravid ◽  
M.R. Notis ◽  
C.E. Lyman

The concept of interfacial width is often invoked in many materials science phenomena which relate to the structure and properties of internal interfaces. The numerical value of interface width is an important input parameter in diffusion equations, sintering theories as well as in many electronic devices/processes. Most often, however, this value is guessed rather than determined or even estimated. In this paper we present a method of determining the effective structural and electronic- structural width of interphase interfaces using low- and core loss fine structure effects in EELS spectra.The specimens used in the study were directionally solidified eutectics (DSEs) in the system; NiO-ZrO2(CaO), NiO-Y2O3 and MnO-ZrO2(ss). EELS experiments were carried out using a VG HB-501 FE STEM and a Hitachi HF-2000 FE TEM.


Author(s):  
J. Bentley ◽  
E. A. Kenik

Instruments combining a 100 kV transmission electron microscope (TEM) with scanning transmission (STEM), secondary electron (SEM) and x-ray energy dispersive spectrometer (EDS) attachments to give analytical capabilities are becoming increasingly available and useful. Some typical applications in the field of materials science which make use of the small probe size and thin specimen geometry are the chemical analysis of small precipitates contained within a thin foil and the measurement of chemical concentration profiles near microstructural features such as grain boundaries, point defect clusters, dislocations, or precipitates. Quantitative x-ray analysis of bulk samples using EDS on a conventional SEM is reasonably well established, but much less work has been performed on thin metal foils using the higher accelerating voltages available in TEM based instruments.


Author(s):  
John Silcox

Determination of the microstructure and microchemistry of small features often provides the insight needed for the understanding of processes in real materials. In many cases, it is not adequate to use microscopy alone. Microdiffraction and microspectroscopic information such as EELS, X-ray microprobe analysis and Auger spectroscopy can all contribute vital parts of the picture. For a number of reasons, dedicated STEM offers considerable promise as a quantitative instrument. In this paper, we review progress towards effective quantitative use of STEM with illustrations drawn from studies of high Tc superconductors, compound semiconductors and metallization of H-terminated silicon.Intrinsically, STEM is a quantitative instrument. Images are acquired directly by detectors in serial mode which is particularly convenient for digital image acquisition, control and display. The VG HB501A at Cornell has been installed in a particularly stable electromagnetic, vibration and acoustic environment. Care has been paid to achieving UHV conditions (i.e., 10-10 Torr). Finally, it has been interfaced with a VAX 3200 work station by Kirkland. This permits, for example, the acquisition of bright field (or energy loss) images and dark field images simultaneously as quantitative arrays in perfect registration.


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