Institute of Fundamental Technological Research

This report presents the modeling of pressure-assisted sintering within the framework of a multiscale approach. Three individual numerical methods have been collectively applied to predict the behavior of a sintering body at three different scales. The appropriate solutions to connect each model/scale have been proposed. Molecular dynamics have been employed to evaluate the grain boundary diffusion coefficient at the atomistic scale. The obtained results of diffusive parameters have been transferred to the micromechanical model of sintering. Here, the discrete element method was used to represent the sintered material properties at the microscopic scale. Micromechanical based results have been validated by own experimental data of material density evolution, indicating the required coincidence. The transfer from micro- to the macroscopic model has been realized by determining the macroscopic viscous moduli from discrete element simulations and subsequently applying them to the continuum model of sintering. The numerical results from finite element simulations at the macroscopic scale have been compared with discrete element ones.

sintering, multiscale modeling, discrete element method, molecular dynamics, finite element method

The robust and simple optimization method of functionally graded material (FGM) for combined cyclic thermal and mechanical loading with application to valve de sign is proposed.

The optimization procedure starts from the homogeneous ceramic material distribution and after thermomechanical analysis of the whole process, the new distribution of material is determined by reducing concentration of the ceramic phase at places of high tensile stresses and by increasing ceramic contents at places of high effective stresses. The optimal distribution of ceramic phase is found through iterations. We have shown the numerical example of the proposed method for optimization of a composite exhaust valve of combustion engine. The example illustrates the optimal density distribution of ceramic phase of Al 2 O 3 within NiAl matrix. In the design study we have used the transient analysis of stress and temperature fields.

The proposed method shares merits of standard optimization and topology optimization, it allows for creating one phase of material inside the other. It can be especially useful to problems of structural elements subjected to thermomechanical loading histories.

thermomechanical analysis, high frequency cyclic loading, coating, design, FGM, optimization

The interfacial energy due to elastic micro-strains at the austenite–twinned martensite interface is calculated for the NiTi shape memory alloy undergoing cubic-to-monoclinic B2 ↔ B19' transformation. For each crystallographically distinct microstructure, an energetically favourable local shape of the interface is determined. The approach employs finite element computations and energy minimization with respect to shape parameters, taking into account elastic anisotropy of the phases and finite-strain kinematics. The effect of atomic-scale interfacial energy is studied.

microstructure, martensitic phase transformation, shape memory alloys (SMA), interface structure, micromechanical modelling

A model of evolving martensitic microstructures is formulated that incorporates the interfacial energy and dissipation on three different scales corresponding to the grain boundaries attained by martensite plates, the interfaces between austenite and martensite plates, and the twin interfaces within martensite plates. Three different time scales are also considered in order to clarify the meaning of rate-independent dissipation related to instabilities at more refined temporal and spatial scales. On the slowest time scale, the process of deformation and martensitic phase transformation is investigated as being composed of segments of smooth quasi-static evolution separated by sudden jumps associated with creation or annihilation of interfaces. A general evolution rule is used in the form of minimization of the incremental energy supply to the whole compound thermodynamic system, including the rate-independent dissipation. Close relationship is shown between the evolution rule and the thermodynamic condition for stability of equilibrium, with no a priori assumption on convexity of the dissipation function. It is demonstrated that the extension of the minimum principle from the first-order rates to small but finite increments requires a separate symmetry restriction imposed on the state derivative of the dissipation function. Formulae for the dissipation associated with annihilation of interfaces are proposed that exhibit limited path-independence and satisfy that symmetry requirement. By exploiting the incremental energy minimization rule with the help of the transport theorems, local propagation conditions are derived for both planar and curved phase transformation fronts. The theory serves as a basis for the algorithm for calculation of the stress-induced evolution of martensitic microstructures along with their characteristic dimensions and related hysteresis loops in shape memory alloys; the examples are given in Part II of the paper.

Microstructures, Phase transformation, Grain boundaries, Energy methods, Stability

Spatially resolved X-ray diffraction (SRXRD) is used for micro-imaging of strain in GaAs:Si layers grown by liquid phase epitaxial lateral overgrowth (ELO) on SiO2-masked GaAs substrates. We show that laterally overgrown parts of the layers (wings) are tilted towards the underlying mask. By SRXRD mapping local wing tilt is easily distinguished from macroscopic sample curvature. The direction of the tilt and distribution of tilt magnitude across the width of each layer can also be readily determined. This allows measuring of the shape of the lattice planes in individual ELO stripes. Downward wing tilt disappears completely when the mask is removed by selective etching. Then residual strain in ELO layers is exposed. In particular, upward tilt is found in free-standing ELO wings. Numerical simulations show that this phenomenon is caused by different concentrations of silicon dopant in vertically and laterally grown parts of the layer.

We present transmission electron microscopy (TEM) and x-ray quantitative studies of the indium distribution in InxGa1−xN/GaN multiple quantum wells (MQWs) with x = 0.1 and 0.18. The quantum wells were grown by low-pressure metalorganic chemical vapour deposition (LP-MOCVD) on a bulk, dislocation-free, mono-crystalline GaN substrate. By using the quantitative TEM methodology the absolute indium concentration was determined from the 0002 lattice fringe images by the strain measurement coupled with finite element (FE) simulations of surface relaxation of the TEM sample. In the x-ray diffraction (XRD) investigation, a new simulation program was applied to monitor the indium content and lateral composition gradients. We found a very high quality of the multiple quantum wells with lateral indium fluctuations no higher than ΔxL = 0.025. The individual wells have very similar indium concentration and widths over the whole multiple quantum well (MQW) stack. We also show that the formation of 'false clusters' is not a limiting factor in indium distribution measurements. We interpreted the 'false clusters' as small In-rich islands formed on a sample surface during electron-beam exposure.

A micromechanical scheme is developed for predicting the morphology and interfacial energy of the interface layer between the parent phase and internally twinned martensite. Low-energy morphologies are determined by minimizing, with respect to shape parameters, the elastic microstrain energy associated with local incompatibility of transformation strains. The computational scheme involves a finite element solution to a problem of non-linear elasticity with eigenstrains, shape sensitivity analysis with respect to general shape parametrization and minimization employing a gradient-based algorithm. As an application, low-energy morphologies are studied for the austenite–martensite interface in the cubic-to-orthorhombic transformation in a CuAlNi shape memory alloy. Discussion of the results of the analysis includes comparison to alternative simplified methods in terms of the predicted morphologies and the corresponding interfacial energies.

Microstructure, Martensitic phase transformation, Shape memory alloys (SMA), Interface structure, Micromechanical modelling

A micromechanical scheme is developed for the analysis of elastic micro-strains induced by local incompatibilities at the austenite-twinned martensite interface. The aim of the paper is to estimate the elastic micro-strain energy which is an important factor in the formation of microstructures during the martensitic transformation. The finite deformation framework is applied, consistent with the crystallographic theory of martensite, and full account is taken for elastic anisotropy of the phases. As an example, the microstructures in the cubic-to-orthorhombic transformation in CuAlNi shape memory alloy are analyzed by the finite element method for the assumed class of zigzag shapes of the austenite-martensite interface at the micro-level. Finally, the effect of the interphase boundary energy on the microstructure of the transition layer is studied.

microstructure, martensitic phase transformation, shape memory alloys (SMA), interface structure, micromechanical modelling

Chemical composition in a ternary alloy is examined using a quantitative high resolution transmission electron microscopy, finite element modelling of the thin foil relaxation phenomena and microscopy image simulation. The measurement of local lattice distortion on transmission electron microscopy images is a powerful tool for chemical composition determination. However, for the correct interpretation of the results, one needs to take into account the inhomogeneous relaxation of the sample and the strain averaging across the sample. The 3D finite element modelling of such phenomena have been performed as a function of chemical composition and geometry of an indium rich cluster in a MOCVD InxGa1−xN/GaN quantum well. Lattice distortion field measured on: experimental transmission electron microscopy image and simulated one, obtained on the basis of finite element simulation, are compared. This procedure allows an accurate determination of chemical composition in such heterostructures.

Indium clusters, Vapour deposition, Transmissionelectron microscopy, Elasticity, Finite element method, Lattice distortion, Image simulation

In this paper the field theory of dislocations is used in the finite element analysis of residual stresses in epitaxial layers. By digital processing of the HRTEM image of a GaAs/ZnTe/CdTe system the tensor maps of dislocation distribution are extracted. Such obtained maps are used as the input data to the finite element code. The mathematical foundations of this code are based on the compatibility equations for lattice distortions. The surface tension induced by misfit dislocations is considered here in terms of a 3D boundary-value problem for stress equilibrium in the interfacial zone. The numerical results show how strongly the surface tension depends on the nonlinear elastic behaviour of the crystal structure.

Microscopy and microanalysis techniques, Nonlinear elasticity, Dislocation structure, Finite element analysis, Residual stresses, Layered structures

A finite element algorithm for the determination of both residual stresses and lattice orientation in dislocated crystals is presented. The applied theoretical approach is based on the continuum theory of dislocations. Due to the compatibility condition, an initial plastic distortion field has been used to introduce an elastic incompatibility corresponding to the assumed dislocation density tensor field. The presented method gives the possibility to develop a numerical code suitable for determination of the stress field and lattice orientation. Two numerical examples are given: a low-angle dislocation wall and a subboundary structure. The method can be applied especially to dislocation sets composed of a few groups of monomial dislocations.

The cover picture of this issue depicts indium composition fluctuations in InGaN/GaN multi quantum wells. The coded color strain distribution (left) was derived from finite element method calculations of the strain relaxation process and high‐resolution transmission electron microscopy (HRTEM) image simulations, superimposed on the HRTEM image of the quantum wells. The possible corresponding shape and εxx strain profiles in the indium rich clusters (right) hint at a concentration close to pure InN in their core. The paper by Pierre Ruterana et al. [1] was presented at the 5th International Symposium on Blue Laser and Light Emitting Diodes (ISBLLED‐2004), held in Gyeongju, Korea, 15–19 March 2004.

HRTEM, quantum well, composition fluctuation, strain distribution

A nonlinear finite element approach presented here is based on the constitutive equations for anisotropic hyperelatic materials. By digital image processing the elastic incompatibilities (lattice mismatch) are extracted from the HRTEM image of GaN epilayer. Such obtained tensorial field of dislocation distribution is used next as the input data to the FE code. This approach is developed to study the stress distribution associated with lattice defects in highly mismatched heterostructures applied as buffer layers for the optically active structures.

Dislocations, Anisotropic Hyperelasticity, Residual Stresses

The robust and simple optimization method of FGM is proposed for combined cyclic thermal and mechanical loading.

The optimization procedure starts from the homogeneous ceramic material distribution and after thermomechanical analysis of the whole process, the new distribution of material is determined by reducing concentration of the ceramic phase at places of high tensile stresses and by increasing ceramic contents at places of high effective stresses. The optimal distribution of ceramic phase is found through iterations. The proposed method shares merits of standard optimization and topology optimization because it allows for creation of one phase of material inside the other. The numerical examples show application of the proposed method to optimization of composite exhaust valve of combustion engine. Examples provided illustrate the optimal density distribution of ceramic phase of Al 2 O 3 within NiAl matrix. The transient analysis of stress and temperature fields is used in the design study. The proposed method can be especially useful to problems of structural elements subjected to thermo-mechanical loading histories.

thermo-mechanical analysis, high-cycle loading, coating, FGM, optimization

We present results of theoretical studies of the pressure and temperature tuned laser diodes (LDs) based on InGaP/AlGaInP heterostructures taking into account mounting‐induced strains. Our studies reveal that mounting‐induced strains play an important role in the quantitative description of these LDs. We determine their influence on the laser wave‐length tuning by hydrostatic pressure.

Laser diodes, III-V semiconductors, Heterojunctions, Hydrostatics, Laser theory

We present the differential photocurrent (DPC) measurements made on set of identical (Al)InGaP red emitting lasers under pressures up to 2GPa. We find that the strains applied by the submount change the pressure tuning rate from 80meV/GPa for the samples mounted on compressibility matched submounts down to about 20–50meV/GPa for the ones mounted on hard A1N or diamond submounts. The findings are discussed with the help of finite element (FE) calculations of the strains in the active layers.

Finite element methods, Active layer, Diamond, Laser diodes, Photoelectric conversion

It is well known that classical Stoney's formula for radius of thin film bowing fails when thin film is dislocated. Thereby, a method of determination lattice parameters for each of heterostructure epilayers is needed when inelastic relaxation takes place. In this article, we have developed a method of determination lattice parameters of a heterostructure from radius curvature measurement data. Description of deformation when dislocations are nucleated is carefully analyzed. It is shown that in order to take into account nucleated dislocations in thin film an additional term responsible for increased volume of layers should be included in the analysis. The proposed method is based on the finite deformation elasticity and uses the finite element method and bowing radius measurement by the laser beam reflection method. As an example the nitride alloy heterostructure with GaN as a substrate is analyzed. The verification is performed using X-ray measurement of lattice parameters. A very good correspondence between numerically determined lattice parameters and XRD measurements data are observed.

Following the need to accurately understand the In composition fluctuations and their role on the optical properties of the GaN based heterostructures, an investigation of MOCVD InGaN/GaN quantum wells is carried out. To this end, quantitative High Resolution Transmission Electron Microscopy (HRTEM) is coupled with image simulation and Finite Element Method (FEM) for the thin foil relaxation modelling. The results show that the indium content can reach x = 1 in the clusters inside the core. In these MOCVD QWs, we attempt to connect the Quantum dot density, composition, and shape to the growth conditions, in order to help the engineering process of highly efficient devices.

Stress-induced formation and growth of internally twinned martensite plates within the austenite matrix in crystals of shape memory alloys is investigated. The recently developed model of evolving laminates is extended by taking into account the interfacial energy at three different scales. The interfacial energy density due to elastic strains at a twinned martensite boundary is calculated by the finite element method. Analytic relationships between characteristic dimensions of the rank-two laminate are derived by minimizing the size-dependent contribution of the interfaces to the total free energy. Specific examples have been calculated for the cubic-to-orthorhombic (β1 to γ1) stress-induced transformation in a CuAlNi shape memory alloy.

FGM, optimization, thermo-mechanical analysis, coating, high-cycle loading