Anisotropic Elastic Interactions Research Articles

Formula omitted] needle-shape precipitate formation in Al-Mg-Si alloy: Phase field simulation and experimental verification

β'' needle-shape precipitate structures significantly affect the mechanical properties of 6xxx aluminum alloys. In this study, a modified multi-phase field (MPF) method combined with the thermodynamic and kinetic databases provided with the CALPHAD (Computer Coupling of Phase Diagrams and Thermochemistry) method has been employed to investigate the morphological evolution of the monoclinic β'' precipitates in Al-Mg-Si alloy. Emphasis has been placed on understanding the influences of interfacial energy anisotropy and elastic interaction on the shape of β'' precipitates. It is shown that the high anisotropic strain alone cannot explain the needle shape of β'' precipitates. In the case of considering the contributions from both anisotropic interfacial energy and elastic interaction simultaneously, the simulations lead to the formation of unique micro-needle-like structures in age strengthening 6xxx Al alloy. The simulation results are compared with plate-shaped θ'(Al2Cu) precipitate in Al-Cu system, and verified with high-resolution transmission electron microscopy (HRTEM) images of needle-like β'' precipitates in Al-Mg-Si system.

Three-Dimensional Phase-Field Simulation and Experimental Validation of β-Mg17Al12 Phase Precipitation in Mg-Al-Based Alloys

A three-dimensional (3D) phase-field model has been developed to simulate the formation of lath-shaped β-Mg17Al12 phase during hcp→bcc transformation in Mg-Al-based alloys. The model considers the synergistic effects of the elastic strain energy associated with the lattice rearrangements that accompany the phase transformation, and the interface anisotropy (both in interfacial energy and interface mobility coefficient). By using the proposed model, the essential features of 3D morphology of the β phase precipitate have been successfully predicted and experimentally validated using high-resolution transmission electron microscopy and atomic force microscopy. Furthermore, the spatial distribution of anisotropic elastic interaction field around a pre-existing β precipitate has been quantitatively determined using 3D phase-field simulation, and the effects of the anisotropic elastic interaction energy on subsequent nucleation of β phase near a pre-existing precipitate have been revealed. The results suggest that the anisotropic elastic interaction energy can promote the formation of new nucleus near the lozenge ends of the pre-existing precipitate, as explicitly substantiated by the experimental observations. The influence of different combinations of interface anisotropy and elastic strain energy on the thickness of β phase precipitate has been elucidated. The correlation between microstructural design during precipitation and the alloy-strengthening mechanisms has also been discussed in terms of dislocation motion. Based on these results, possible strategies for strengthening Mg-Al-based alloys are proposed for magnesium alloy development and microstructural design.

Phase-Field Modeling of Nucleation in Solid-State Phase Transformations

Nucleation is a critically important process as the rate of nucleation determines the number density of new phase particles and thus microstructures of a material during phase transformations. Predicting and controlling nucleation rates in solids is one of the grand challenges in materials science because the spatial scale involved in nucleation is at the atomic/nanoscale, the rate of nucleation process is extremely temperature sensitive, and the morphology of a critical nucleus can be highly nonspherical and complex. In this article, we briefly review the recent advances in modeling and predicting nucleation during solid-phase transformations based on the diffuse-interface or nonclassical description of critical nucleus profiles. The focus is on predicting the critical nucleus morphology and nucleation free energy barrier under the influence of anisotropic interfacial energy and elastic interactions. Incorporation of nucleation events in phase-field modeling of solid-to-solid phase transformations and microstructure evolution is also discussed.

Ginzburg–Landau modelling of precursor nanoscale textures in ferroelastic materials

A Ginzburg–Landau free-energy model is proposed to study spatially inhomogeneous states that often occur as precursors of ferroelastic/martensitic transitions. Disorder is included in the harmonic coefficient of the free-energy density which gives rise to a spatial distribution of transition temperatures, and lattice integrity is imposed through Saint-Venant compatibility conditions which lead to a long-range anisotropic elastic interaction. We show that precursor textures are a result of the competition between elastic anisotropy and disorder. Cross-hatched modulations (tweed patterns) take place for temperatures above the martensitic phase in the limit of high anisotropy and/or low disorder while a nano-cluster phase-separated state occurs at low anisotropies or high disorder. In the latter case, nanoscale inhomogeneities give rise to glassy behaviour while the structural transition is inhibited. Interestingly, in this case, the ferroelastic system also displays a large thermo-mechanical response so that the low-symmetry structure can be easily induced by the application of relatively small stresses within a broad temperature range.

Microstructure from ferroelastic transitions using strain pseudospin clock models in two and three dimensions: A local mean-field analysis

We show how microstructure can arise in first-order ferroelastic structural transitions, in two and three spatial dimensions, through a local meanfield approximation of their pseudospin hamiltonians, that include anisotropic elastic interactions. Such transitions have symmetry-selected physical strains as their $N_{OP}$-component order parameters, with Landau free energies that have a single zero-strain 'austenite' minimum at high temperatures, and spontaneous-strain 'martensite' minima of $N_V$ structural variants at low temperatures. In a reduced description, the strains at Landau minima induce temperature-dependent, clock-like $\mathbb{Z}_{N_V +1}$ hamiltonians, with $N_{OP}$-component strain-pseudospin vectors ${\vec S}$ pointing to $N_V + 1$ discrete values (including zero). We study elastic texturing in five such first-order structural transitions through a local meanfield approximation of their pseudospin hamiltonians, that include the powerlaw interactions. As a prototype, we consider the two-variant square/rectangle transition, with a one-component, pseudospin taking $N_V +1 =3$ values of $S= 0, \pm 1$, as in a generalized Blume-Capel model. We then consider transitions with two-component ($N_{OP} = 2$) pseudospins: the equilateral to centred-rectangle ($N_V =3$); the square to oblique polygon ($N_V =4$); the triangle to oblique ($N_V =6$) transitions; and finally the 3D cubic to tetragonal transition ($ N_V =3$). The local meanfield solutions in 2D and 3D yield oriented domain-walls patterns as from continuous-variable strain dynamics, showing the discrete-variable models capture the essential ferroelastic texturings. Other related hamiltonians illustrate that structural-transitions in materials science can be the source of interesting spin models in statistical mechanics.

Mathematical and Numerical Aspects of a Phase-field Approach to Critical Nuclei Morphology in Solids

We investigate a phase-field model for homogeneous nucleation and critical nucleus morphology in solids. We analyze the mathematical properties of a free energy functional that includes the long-range, anisotropic elastic interactions. We describe the numerical algorithms used to search for the saddle points of such a free energy functional based on a minimax technique and the Fourier spectral implementation. It is demonstrated that the phase-field model is mathematically well defined and is able to efficiently predict the critical nucleus morphology in elastically anisotropic solids without making a priori assumptions.

Diffuse-interface description of strain-dominated morphology of critical nuclei in phase transformations

A diffuse interface model combined with the minimax technique is implemented to predict the morphology of critical nuclei during solid to solid phase transformations in both two and three dimensions. It takes into account the anisotropic interfacial energy as well as the anisotropic long-range elastic interactions. It is demonstrated that the morphology of critical nuclei in cubically anisotropic solids can be efficiently predicted by the computational model without a priori assumptions. A particular example of cubic to cubic transformation within the homogeneous modulus approximation is considered. The effect of elastic energy contribution on the size and shape of a critical nucleus is studied. It is shown that strong elastic energy interactions may lead to critical nuclei with a wide variety of shapes, including plates, needles and cuboids with non-convex interfaces.

Effect of surface on phase modulation of light in a nematic layer

The nanorelief of orienting surfaces in a nematic layer is studied experimentally. The initial inclination angle of the director and the phase retardation of light in the crystal are determined, and the director reorientation dynamics in the crystal under SB deformation in an electric field is analyzed. It is shown that a thin layer of amorphous hydrogenated carbon (a-C: H) deposited on a GeO monoxide layer with an anisotropic nanorelief produced by the inclined deposition method smoothens the surface topography without changing the surface structure. Modification of the structure and physicochemical properties of the GeO surface alters the conditions of the anisotropic-elastic interaction at the interface with the liquid crystal, as evidenced by an increase in the S-effect threshold and a decrease in the initial inclination of the director from 22° (on the GeO surface) to 0–6°. Strong influence of the surface nanostructure on the dynamics of the director reorientation in the electric field and on the phase modulation of light is experimentally demonstrated. It is shown that the phase retardation of light in the GeO layer covered by an a-C: H film is twice as large as in the layer of the same thickness with a virgin surface.

Simulation of the coarsening behavior of Intermetallic precipitates in Ni 75Al x V 25- x alloys

The coarsening behavior of L1 2 and DQ 22 in Ni 75Al x V 25- x ( x, at.%) alloys including coherent strain was investigated using the microscopic phase-field model. The simulation results indicate that the shape transition and spatial correlation of L1 2 and D0 22 are caused by the morphological-dependent anisotropic elastic interactions in the system. The coarsening process of the particles is by means of neighbor particles impingement and aggregation into larger ones. For the strain-induced interactions between the precipitates, the LSW theory is altered for the coarsening behavior of L1 2 and D0 22. In addition, the simulation reveals that the growth and coarsening of D0 22 present two obvious stages at lower Al concentration regions and proceed simultaneously at high Al concentration regions. The growth and coarsening processes of L1 2 at the same regions is reverse to those of D0 22.

Depth dependence of elastic grain interaction and mechanical stress: Analysis by x-ray diffraction measurements at fixed penetration/information depths

Gradients of the type of elastic grain interaction and the (residual) internal state of stress were determined. This was possible by the application of x-ray diffraction stress measurements at various fixed penetration/information depths. The analysis was applied to nickel films of thicknesses 2 and 4μm. Surface anisotropy was considered as source of direction-dependent (anisotropic) elastic grain interaction. It was found that only a small gradient of the state of stress, but a pronounced gradient of the grain interaction constraints, prevails in the investigated specimens. Thereby the evidence for the depth dependence of the so-called surface anisotropy was obtained.

Effect of elastic interaction on nucleation: I. Calculation of the strain energy of nucleus formation in an elastically anisotropic crystal of arbitrary microstructure

We analyze the anisotropic elastic interaction of a nucleating particle with an arbitrary pre-existing coherent microstructure. Under the assumption that the length scale of the microstructure is considerably larger than the size of the nucleus, their elastic interaction energy can be expressed as a linear function of the nucleus’s volume, and combined directly with the chemical nucleation driving force in the classical nucleation theory. Using cubic → cubic and cubic → tetragonal transformations as examples, we evaluate the elastic energy associated with the formation of a nucleus in a pre-existing microstructure. It is found that, similar to other stress-generating crystalline defects, coherent precipitates could have a significant effect on the spatial location of nuclei, resulting in correlated nucleation where the existing particles dictate where the new particles appear. This effect seems to be generic for nucleation in coherent solids and it could be responsible for the formation of self-organized morphological patterns during coherent transformations.

Creep and dimensional stability of high purity niobium electron-beam welds

A study was conducted to characterize the microstructure of electron beam welds in high purity niobium and its effect on creep behaviour at room temperature. The parent material was 2 mm sheet with a 50 μm grain size. The weld fusion zone had ∼1 mm grains, implying that these grains all intersected the free surface. The parent material showed no room temperature creep deformation below the yield stress, but room temperature creep of weld specimens caused up to 10% strain in the weld region at ∼75% of the yield strength, over 1–2 months. Creep deformation was not smooth or continuous; the strain saturated at some value, and then after an incubation time, the strain increased and saturated again several times over 1–2 months. The magnitude of the strain for several specimens was similar but the creep deformation behavior was highly dependent on the actual microstructure and loading history. An initial prestrain with unloading shut down the creep deformation mechanism at the prior stress due to a dislocation-locking effect. The local stresses in the weld fusion zone arose from anisotropic elastic interactions due to different crystal orientations that caused local regions to exceed the yield strength.

A simple model for the checkerboard pattern of modulated hole densities in underdoped cuprates

A simple model is proposed as a possible explanation for the checkerboard pattern of modulations in the hole density observed in recent tunneling experiments on underdoped cuprates. Two assumptions are made; first, an enhanced hole density near the acceptor dopants and secondly short range correlations in the positions of these dopants caused by their electrostatic and anisotropic elastic interactions. Together these can lead to a structure factor in qualitative agreement with experiment.

The analysis of homogeneously and inhomogeneously anisotropic microstructures by X-ray diffraction

The microstructure of materials is generally, macroscopically, anisotropic and/or inhomogeneous. Traditional diffraction analyses do not take into account this anisotropy and/or inhomogeneity of microstructural features. Thus obtained results can be incomplete, ambiguous, or even erroneous. In this work instrumental requirements (application of parallel beam diffractometers with X-ray lenses or X-ray mirrors and parallel-plate collimators in the laboratory and at synchrotron beam lines) and methodological approaches for the diffraction analysis of anisotropic and inhomogeneous microstructures have been discussed and have been illustrated on the basis of two experimental examples: analysis of the anisotropic nature of the structural imperfection of a sputterdeposited Ti3Al layer and analysis of the anisotropic and inhomogeneous elastic grain interaction in a sputter-deposited Ni layer.

Effect of substrate constraint on spinodal decomposition in an elastically inhomogeneous thin film

Spinodal decomposition in a binary thin film is studied by a three-dimensional (3D) phase field model. A cubic thin film with an (001) orientation is considered. The focus is on the effect of the types of substrate constraint on the morphological evolution during spinodal decomposition as compared to the corresponding bulk. The elastic strain effect is incorporated by solving the elasticity equations for an elastically inhomogeneous thin film with a free surface and constrained by a substrate. Temporal evolution for the composition field, and thus the morphological evolution, is obtained by solving the Cahn-Hilliard equation using a semi-implicit Fourier-spectral method. It is shown that a biaxial substrate constraint has essentially no effect on the spinodally decomposed two-phase morphology which is primarily controlled by the cubically anisotropic elastic interactions. The asymmetry in the strain components along the [100] and [010] directions from the substrate constraint results in the preferential alignment along one of the two directions. In the particular case of a harder phase whose lattice parameter increases with composition, a tensile substrate constraint along the film plane leads to the alignment of two-phase microstructures parallel to the tensile direction.

Phase-field simulation of domain structure evolution during a coherent hexagonal-to-orthorhombic transformation

Abstract The formation and temporal evolution of domain structures during a hexagonal → orthorhombic transformation is studied using computer simulations based on a continuum diffuse-interface phase-field approach. All the essential driving forces for the domain formation and evolution are taken into account, including the bulk chemical free energy, the domain wall energy and the elastic strain energy. The various domain configurations observed from the computer simulations show excellent agreement with existing experimental observations in a number of systems undergoing hexagonal → orthorhombic or hexagonal → monoclinic transformations. It is shown that, even with the assumption of isotropic domain wall energy and isotropic elastic modulus, the anisotropic elastic interactions alone caused by the non-dilatational strains can reproduce all the interesting domain structures observed experimentally. It is also demonstrated that many of the specific domain configurations are actually formed during the domain...

Role of elastic compatibility in martensitic texture evolution

We employ the time-dependent Ginzburg-Landau (TDGL) method to analyze the time evolution of strain fields in a model for materials with martensitic phase transformations. The free energy functional is expressed in terms of the components of the strain tensor, and its functional derivatives with respect to these components give their rate of change. However, the components of the strain tensor are not independent fields; rather, they are related by the Saint-Venant compatibility condition. This condition imposes constraints on the variations of the strain tensor components needed to obtain the equations of motion. Incorporating these constraints in the TDGL procedure introduces extra terms that effectively act as long-range, anisotropic elastic interactions. The latter govern the types of elastic textures that may emerge during a martensitic transformation. The results from the numerical solution of these evolution equations exhibit fine and coarse tweed, twinning, and tip-splitting.

Kinetic Instability of Semiconductor Alloy Growth

AbstractA kinetic theory of the instability of homogeneous alloy growth with respect to fluctuations of alloy composition is developed. The growth mechanism studied is the step-flow growth of an alloy from the vapor on a surface vicinal to the (001) surface of a cubic substrate. The epitaxial growth implies that the adsorbed atoms migrate on the surface during growth of each monolayer, and that their motion is “frozen” after the completion of the monolayer. Frozen fluctuations in all completed monolayers create, via the composition-dependent lattice parameter, an effective potential which influences the surface migration of adatoms. The migration consists of diffusion and strain-induced drift in the effective potential. For temperatures lower than a certain critical temperature Tc, strain-induced drift dominates diffusion and results in the kinetic instability of the homogeneous alloy growth. In the linear approximation in the fluctuation amplitude, the instability means the exponential increase of the fluctuation amplitude with the thickness of the epitaxial film. It is shown that the critical temperature of kinetic instability Tc, increases with the increase of elastic effects. The wave vector kc of the most unstable mode of composition fluctuations is determined by the interplay of anisotropic elastic interaction and anisotropic diffusion on a stepped vicinal surface. The direction of the wave vector kc differs from the lowest-stiffness direction of the crystal, and any direction of kc is possible.

Kinetic instability of semiconductor alloy growth

A kinetic theory of the instability of homogeneous alloy growth with respect to fluctuations of alloy composition is developed. The growth mechanism studied is the step-flow growth of an alloy from the vapor on a surface vicinal to the (001) surface of a cubic substrate. The epitaxial growth implies that the adsorbed atoms migrate on the surface during the growth of each monolayer, and that their motion is ``frozen'' after the completion of the monolayer. ``Frozen'' fluctuations of alloy composition in all completed monolayers create, via a composition-dependent lattice parameter, an elastic strain that influences the migration of adatoms of the growing monolayer. The migration consists of diffusion- and strain-induced drift in an effective potential. For temperatures lower than a certain critical temperature ${T}_{c},$ strain-induced drift dominates diffusion and results in the kinetic instability of the homogeneous alloy growth. In an approximation linear in the fluctuation amplitude, the instability means the exponential increase of the fluctuation amplitude with the thickness of the epitaxial film. It is shown that the critical temperature of the kinetic instability ${T}_{c}$ increases with the increase of elastic effects. The wave vector ${\mathbf{k}}_{c}$ of the most unstable mode of composition fluctuations is determined by the interplay of the anisotropic elastic interaction and the anisotropic diffusion of the adatoms on a stepped vicinal surface. The direction of the wave vector ${\mathbf{k}}_{c}$ differs from the lowest-stiffness direction of the crystal. Regions in k space of both stable and unstable modes are found by model calculations.

Ising model for phase separation in alloys with anisotropic elastic interaction—II. A computer experiment

When a metallic alloy is quenched into a miscibility gap, a mixture of two phases develops, whose domain structure then coarsens because of the interfacial energy between the two phases. This spatial arrangement of the domains and the rate at which they evolve may be strongly influenced by elastic interactions. In a recent paper, the authors described a method for simulating the effect of anisotropic elastic interactions in a two-dimensional Ising model of a cubic alloy, using Kawasaki dynamics with the elastic interactions represented by a long-range two-body interaction potential. Here we present the results of such simulations at various temperatures, alloy compositions and misfits (by “misfit” we mean the difference in size between the two kinds of atom), exhibiting snapshots both of the microscopic configurations (corresponding to experimental measurements using transmission electron microscopy) and of their squared Fourier transforms (corresponding to measurements using small-angle X-ray or neutron scattering). The anisotropy of the two-phase structure increased with time, misfit and composition but decreased with increasing temperature. The rate of coarsening of the domains was found, surprisingly, to be roughly independent of the misfit at a given value of temperature relative to the critical phase transformation temperature.