Effect Of Elastic Energy Research Articles

Isothermal martensitic transformation in Ti-Ni-Cu-Co shape memory alloy: Insight from a thermally activated kinetic model

Recently the isothermal martensitic transformation in shape memory alloys (SMAs) has been reported in many literatures, and several models have been proposed to interpret the isothermal and athermal kinetics. However, the underlying mechanisms remain inadequately understood. In this work, the isothermal transformation from B2 to B19 is confirmed in Ti-Ni-Cu-Co melt-spun ribbons at the temperature range between Ms and Mf. It reaches a saturation point at every isothermal temperature Tiso, and the saturation points correspond to the f−T curve at the cooling rate of 0.5 K/min. The experimental results indicate that the isothermal accumulation of martensite is a relaxation process from the transient state to the thermoelastic balance one. A thermally activated kinetic model is developed in this study to characterize the isothermal and athermal kinetics. The model is able to estimate the evolution of martensite volume fraction under any temperature path T(t) and it agrees with the experimental results well. According to the model, the effects of elastic energy, nucleation density, and activation energy on the kinetics are investigated. Among those, a small nucleation density ni as well as a large activation energy Q will result in a significant isothermal transition. In this work, the slighter isothermal effects originate from the higher value of ni. As for the non-stoichiometric SMAs, the higher value of Q is responsible for the accumulation of martensite at the isothermal process. Accordingly, the present work provides a novel view from a kinetic model to understand the isothermal martensitic transformation in SMAs.

The strong influence of internal stresses on the nucleation of a nanosized, deeply undercooled melt at a solid-solid phase interface.

The effect of elastic energy on nucleation and disappearance of a nanometer size intermediate melt (IM) region at a solid-solid (S1S2) phase interface at temperatures 120 K below the melting temperature is studied using a phase-field approach. Results are obtained for broad range of the ratios of S1S2 to solid-melt interface energies, k(E), and widths, k(δ). It is found that internal stresses only slightly promote barrierless IM nucleation but qualitatively alter the system behavior, allowing for the appearance of the IM when k(E) < 2 (thermodynamically impossible without mechanics) and elimination of what we termed the IM-free gap. Remarkably, when mechanics is included within this framework, there is a drastic (16 times for HMX energetic crystals) reduction in the activation energy of IM critical nucleus. After this inclusion, a kinetic nucleation criterion is met, and thermally activated melting occurs under conditions consistent with experiments for HMX, elucidating what had been to date mysterious behavior. Similar effects are expected to occur for other material systems where S1S2 phase transformations via IM take place, including electronic, geological, pharmaceutical, ferroelectric, colloidal, and superhard materials.

Effects of elastic energy on the spinodal decomposition in InAlGaN materials

AbstractThe purpose of our study is to determine the influence of elastic energy on the spinodal decomposition ranges of InAlGaN alloys lattice‐matched to GaN. The distinction in lattice constants of the binary compounds constituting these alloys gives rise to their deformation energies. Such energies provide the tendency to disintegration that can lead to an appearance of the thermodynamically unstable states with respect to the decomposition. The initial stage of spinodal decomposition of the InAlGaN quaternary alloys is accompanied by transfers of atoms over distances of a lattice parameter. Accordingly, these transfers produce a thin two‐layer region with negligibly small differences in their compositions. Formation of these layers causes the elastic energy. The alloys are represented as quaternary regular solutions. The internal deformation energy is calculated with the interaction parameters estimated by the valence force field model. The spinodal decomposition ranges of these materials are demonstrated up to 580 °C (© 2011 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Elastic fields generated by a semi-spherical hydride particle on a free surface of a metal and their effect on its growth

The formation of a metal hydride is associated with a large increase of volume relative to the parent metal and therefore in large strain energies. Effects of elastic energy on the hydriding of metals are revealed in the microstructural evolution and kinetics of hydride growth on free surfaces. In the present work, we study in detail the elastic fields set up by a semi-spherical hydride particle growing at a free surface of metal with cubic symmetry, with and without an oxide layer. These systems combine geometric (structural) and material anisotropies.Three stages along the microstructural evolution on the surface of some hydride forming metals exposed to hydrogen at constant pressure were described experimentally. For these stages along hydride growth, correlations with the elastic fields are suggested as follows. (a) A hydride particle at the free surface generates regions of tensile and compressive hydrostatic stress in the surrounding matrix. This may induce a preferred nucleation of new hydrides and formation of clusters of hydrides precipitates, which is indeed observed experimentally. (b) Clustering, on the other hand, may contribute to the cease of growth due to competition on hydrogen. In addition, as the particle grows, changes in the stress fields may retard further diffusion from the surface and be another contribution to the cease of growth. (c) A growing hydride increases the stress in the oxide layer and may finely break it. Then the elastic energy per unit volume drops to its minimum value and the growth may accelerate. The formation of such “growth center” is favored for that hydride precipitate that grow alone and not in a cluster.

Reorientation of planets with lithospheres: The effect of elastic energy

It is commonly assumed that internal energy dissipation will ultimately drive planets to principal axis rotation, i.e., where the rotation vector is aligned with the maximum principle axis, since this situation corresponds to the minimum rotational energy state. This assumption simplifies long-term true polar wander (TPW) studies since the rotation pole can then be found by diagonalizing the appropriate (non-equilibrium) inertia tensor. We show that for planets with elastic lithospheres the minimum energy state does not correspond to principal axis rotation. As the planet undergoes reorientation elastic energy is stored in the deforming lithosphere, and the state of minimum total energy is achieved before principal axis rotation. We find solutions for the TPW of planets that include this effect by calculating the elastic stresses associated with deformation, and then minimizing the total (rotational and elastic) energy. These expressions indicate that the stored elastic energy acts to reduce the effective size of the driving load (relative to predictions which do not include this energy term). Our derivation also yields expressions for the TPW-induced stress field that generalizes several earlier results. As an illustration of the new theory, we consider TPW driven by the development of the Tharsis volcanic province on Mars. Once the size of the Tharsis load and the Mars model is specified, the extended theory yields a more limited range on the possible TPW.

Phase-field microelasticity theory and micromagnetic simulations of domain structures in giant magnetostrictive materials

A computational model is proposed to predict the stability of magnetic domain structures and their temporal evolution in giant magnetostrictive materials by combining a micromagnetic model with the phase-field microelasticity theory of Khachaturyan. The model includes all the important energetic contributions, including the magnetocrystalline anisotropy energy, exchange energy, magnetostatic energy, external field energy, and elastic energy. While the elastic energy of an arbitrary magnetic domain structure is obtained analytically in Fourier space, the Landau–Liftshitz–Gilbert equation is solved using the efficient Gauss–Seidel projection method. Both Fe 81.3Ga 18.7 and Terfenol-D are considered as examples. The effects of elastic energy and magnetostatic energy on domain structures are studied. The magnetostriction and associated domain structure evolution under an applied field are modeled under different pre-stress conditions. It is shown that a compressive pre-stress can efficiently increase the overall magnetostrictive effect. The results are compared with existing experiment measurements and observations.

Effects of elastic energy of thin films on bending of a cantilevered magnetostrictive film-substrate system

In this paper, effects of elastic energy of magnetostrictive film on the deflection of a cantilevered film-substrate system are investigated. The total energy including the elastic energy of magnetostrictive film is formulated. And it is minimized to give the curvatures and the position of neutral axis of the cantilevered system. To discuss the effects of the elastic energy of film in a measured system, three magnetostrictive unimorph cantilevers and a bimorph cantilever reported elsewhere are reviewed. It is shown that the assumption, since the thickness of film is much smaller than that of substrate the film elastic energy is negligible, can cause considerable error in evaluating magnetostrictive coefficients. Not the ratio of thicknesses but elastic energies between film and substrate is also shown to play important role in making decision whether the assumption is valid or not.

Computer simulation of spinodal decomposition in constrained films

The morphological evolution during spinodal decomposition of a binary alloy thin film elastically constrained by a substrate is studied. Elastic solutions, derived for elastically anisotropic thin films subject to the mixed stress-free and constraint boundary conditions, are employed in a three-dimensional phase-field model. The Cahn–Hilliard diffusion equation for a thin film boundary condition is solved using a semi-implicit Fourier-spectral method. The effect of composition, coherency strain, film thickness and substrate constraint on the microstructure evolution was studied. Numerical simulations show that in the absence of coherency strain and substrate constraint, the morphology of decomposed phases depends on the film thickness and the composition. For a certain range of compositions, phase separation with coherency strain in an elastically anisotropic film shows the behavior of surface-directed spinodal decomposition driven by the elastic energy effect. Similar to bulk systems, the negative elastic anisotropy in the cubic alloy results in the alignment of phases along 〈1 0 0〉 elastically soft directions.

Three-dimensional phase-field modeling of spinodal decomposition in constrained films

The morphological evolution during spinodal decomposition in binary alloy thin films elastically constrained by substrates is studied. Elastic solutions, derived for both elastically isotropic and anisotropic thin films subject to mixed stress-free and constraint boundary conditions, are employed in a three-dimensional phasefield model to investigate the effect of coherency strain and substrate constraint on microstructural evolution. The temporal evolution of the Cahn-Hilliard equation under thin film boundary conditions is solved with a semi-implicit Fourier-spectral method. The phase separation with coherency strain in an elastically anisotropic film shows the behavior of surface-directed spinodal decomposition driven by the elastic energy effect. Negative elastic anisotropy in the cubic alloy causes the alignment of the phases along <100> elastically soft directions.

Shape evolution of InAs quantum dots during overgrowth

The effects of strain and thickness of an In x Ga 1− x As ( x=0−0.2) cap layer grown at low temperature on large low-growth-rate InAs quantum dots (QDs) are systematically studied by atomic force microscopy. The dot height drastically reduces and the dot shape transforms into an elongated ridge-valley structure at the early stage of GaAs overgrowth, while the dots tend to preserve their shape during InGaAs capping. The effects of elastic energy and surface energy included in the surface chemical potential can qualitatively explain the observed surface evolution. The increase of the elastic energy of the QDs and surface energy of the cap drives the In atoms away from the InAs QDs to the GaAs surface. The In atom migration away from the InAs QDs and the unfavorable growth of GaAs on top of the QDs results in a collapse of the InAs QDs and a valley formation at the beginning of the GaAs overgrowth. In case of the In x Ga 1− x As cap layer, an increased In content decreases the elastic energy and also the cap layer surface energy, resulting in a reduction of the chemical potential gradient and consequently a preservation of the QD shape. At high overgrowth temperature, the material transport from the dots to the cap surface is less pronounced due to increased In–Ga intermixing.

The nucleation of coherent semiconductor islands during the Stranski-Krastanov growth induced by elastic strains

The Ge island growth on the Si(100) and Si(111) surfaces was investigated through spectral ellipsometry in real time. It is found that both cases correspond to Stranski-Krastanov growth; i.e., a Ge wetting layer is initially formed, and only then do the islands of the new phase grow on the surface of this layer. However, the island nucleation on the (100) surface is accompanied by a substantial decrease in the wetting layer thickness, whereas, on the (111) surface, the islands nucleate and grow on the wetting layer of constant thickness. The Ge atoms on the (100) surface transfer from the wetting layer to islands, thus substantially decreasing the elastic energy of the system, but increasing the surface energy. For this reason, it is concluded that, in this case, it is the elastic energy which represents the fundamental driving force of the island nucleation. Thermodynamic and kinetic theories of island nucleation from the wetting layer under the effect of elastic energy are developed. A new notion of overstress is introduced by analogy with supersaturation and overcooling. The time evolution of the wetting layer thickness, the nucleation rate, and the island surface density of the new phase is described. The theoretical results are compared to experimental data obtained through ellipsometric simulation, and it is found that the theory and experiment are in good agreement.

Elastic energy anisotropy effects on antiphase boundaries

The effect of elastic energy on the anisotropy of antiphase boundary (APB) interfacial energy has been studied theoretically. The model used, a generalization of one developed by Kikuchi and Cahn, treats composition and atomic ordering as local variables defined on each of the atomic planes parallel to the APB. The equilibrium APB structure is found as the set of planar variables that minimizes the free energy. A strain energy term involving an orientation-dependent stiffness parameter and a phenomenological law giving lattice constant as a function of local composition and order parameter is included in the definition of the free energy. Elastic effects in APBs in B2 ordered FeAl alloys with 0.2 – 0.3 atomic fraction Al were investigated. It was found that, despite the high elastic anisotropy of FeAl alloys, elastic effects would not result in any substantial anisotropy in APB interfacial energy. This is because in FeAl alloys the variation in lattice constant due to order parameter variation at the APB is almost completely cancelled out by the variation in lattice constant due to compositional variation. In a hypothetical model with physically reasonable parameters, APB energies were found to vary by up to 9% with orientation, purely as a result of elastic effects. It was concluded that elastically induced APB anisotropy would be observable in alloys such as FeSi in which lattice constant variations due to order parameter variation are cooperative with those due to compositional variation.

A computer simulation technique for spinodal decomposition and ordering in ternary systems

To the author's knowledge, the nonlinear dynamics of inhomogeneous morphological evolution during spinodal decomposition and ordering including the subsequent coarsening process in a ternary system has not been extensively investigated. To predict morphological evolution in an inhomogeneous system without a priori assumptions on the transformation sequence and morphologies, they have to resort to computer simulations. The main objective of this work is, therefore, to develop a computer simulation technique which can predict microstructural evolution during ordering and spinodal decomposition in ternary alloys. Since the present kinetic model is microscopic in nature, it can be applied to model both ordering and phase separation as well as simultaneous ordering and phase separation. The elastic energy effect on the decomposition kinetics can also be easily incorporated into the model. As an example, a computer simulation of isostructural spinodal decomposition in a ternary alloy was performed in a simple two-dimensional square lattice.

Computer Simulation of the Elastic Energy Contribution to Antiphase Boundary Energy Anisotropy

ABSTRACTThe results of computer simulations on the effect of elastic energy on APB anisotropy are presented. The model used treats composition and order parameter as local variables defined at crystal planes parallel to the APB. The set that minimizes the free energy, including elastic energy terms, defines the equilibrium APB structure. It was found that, despite their high elastic anisotropy, elastic effects would not result in any substantial APB anisotropy in Fe-Al alloys. However, a model with different parameters did show a high degree of anisotropy.

Some new consequences of the order parameter description of glassy states

Abstract By extending the classical order parameter theory of glassy states to the case of inhomogeneous glassy states, some new consequences of the theory are derived. It is shown that the phase boundaries, critical points, and the kinetics of phase transition differ in the annealed and the quenched states. For example, the critical point of phase separation is always suppressed in the quenched states. Cahn's analysis of elastic energy effects is shown to be a special case of the present formalism.