Coherent precipitation of multi-variant Ti11Ni14 precipitates in TiNi alloys was investigated by employing a continuum field kinetic model. The structural difference between the precipitate phase and the matrix as well as the orientational differences between precipitate variants are distinguished by nonconserved structural field variables, whereas the compositional difference between the precipitate and matrix is described by a conserved field variable. The temporal evolution of the spatially dependent field variables is determined by numerically solving the time-dependent Ginzburg–Landau (TDGL) equations for the structural variables and the Cahn–Hilliard diffusion equation for the composition. In particular, the interaction between precipitates, and the growth morphology of Ti11Ni14 precipitates under strain-constraints were studied, without a priori assumptions on the precipitate shape and distribution. The predicted morphology and distribution of Ti11Ni14 variants were compared with experimental observations. Excellent agreement between the simulation and experimental observations was found.
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 Fe81.3Ga18.7 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.
The morphological evolution and coarsening kinetics of ordered intermetallic precipitates with coherency stress were studied using a diffuse-interface phase-field model in two dimensions (2D). The emphasis is on the effects of precipitate volume fraction. The average aspect ratio of the precipitates in the microstructure is found to increase with time and decrease with volume fraction. Contrary to all the existing coarsening theories but consistent with a number of experimental measurements on the coarsening kinetics of ordered γ′ precipitates in Ni-base superalloys, we found that the coarsening rate constant from the cubic growth law decreases as a function of volume fraction for small volume fractions (≲20%) and is constant for intermediate volume fractions (20–50%). From the simulation results, we infer that the two length scales in a stress-dominated coherent two-phase microstructure, the average precipitate size and average spacing between arrays of aligned precipitates, follow different growth exponents. It is demonstrated that as the volume fraction increases, the precipitate size distributions become broader and their skewness become increasingly positive.
Many structural transformations result in several orientation variants whose volume fractions and distributions can be controlled by applied stresses during nucleation, growth or coarsening. Depending on the type of stress and the coupling between the applied stress and the lattice misfit strain, the precipitate variants may be aligned parallel or perpendicular to the stress axis. This paper reports our studies on the effect of applied stresses on nucleation and growth of coherent θ′ precipitates in Al–Cu alloys using computer simulations based on a diffuse-interface phase-field kinetic model. In this model, the orientational differences among precipitate variants are distinguished by non-conserved structural field variables, whereas the compositional difference between the precipitate and matrix is described by a conserved field variable. The temporal evolution of the spatially dependent field variables is determined by numerically solving the time-dependent Ginzburg–Landau (TDGL) equations for the structural variables and the Cahn–Hilliard diffusion equation for composition. Random noises were introduced in both the composition and the structural order parameter fields to simulate the nucleation of θ′ precipitates. It is demonstrated that although an applied stress affects the microstructural development of a two-phase alloy during both the nucleation and growth stages, it is most effective to apply stresses during the initial nucleation stage for producing anisotropic precipitate alignment.
The shape of a coherent rhombohedral precipitate in a cubic matrix and its growth and dissolution during strain-constrained aging were investigated using a time-dependent Ginzburg-Landau kinetic model by taking into account the coupling between the constraint (applied) strain and the local strain. The effect of boundary conditions, constraint strain or constraint stress, has been discussed. A particular example of Ti11Ni14 precipitate growth in a TiNi shape memory alloy was considered without any a priori assumption about the particle shape. It was demonstrated that a Ti11Ni14 precipitate grown from a supersaturated cubic TiNi matrix has a lens-like shape, with its normal parallel to the B2 orientation of the matrix, in agreement with experimental observations. Precipitate growth and dissolution under various strain-constraint conditions have been discussed.
An efficient and accurate numerical method is implemented for solving the time-dependent Ginzburg—Landau equation and the Cahn—Hilliard equation. The time variable is discretized by using semi-implicit schemes which allow much larger time step sizes than explicit schemes; the space variables are discretized by using a Fourier-spectral method whose convergence rate is exponential in contrast to second order by a usual finite-difference method. We have applied our method to predict the equilibrium profiles of an order parameter across a stationary planar interface and the velocity of a moving interface by solving the time-dependent Ginzburg—Landau equation, and compared the accuracy and efficiency of our results with those obtained by others. We demonstrate that, for a specified accuracy of 0.5%, the speedup of using semi-implicit Fourier-spectral method, when compared with the explicit finite-difference schemes, is at least two orders of magnitude in two dimensions, and close to three orders of magnitude in three dimensions. The method is shown to be particularly powerful for systems in which the morphologies and microstructures are dominated by long-range elastic interactions.
A diffuse-interface field model is proposed for describing diffusional processes in coherent systems with arbitrary microstructures and arbitrary spatial distribution of structural defects such as dislocations. It takes into account the effect of both the coherency elastic energy of a microstructure and the elastic coupling between the coherency strains and defect strains. In this model, any arbitrary spatial distribution of defects is described using the micromechanics concept of space-dependent “stress-free” or “eigen” strains. As examples, the solute segregation as well as the nucleation and diffusional growth of a coherent precipitate around an edge dislocation are considered. It is shown that coherent nucleation may become barrierless under the influence of the local elastic field of a dislocation.
A coherent precipitate phase usually has a number of variants that are oriented in different but equivalent crystallographic directions. The distribution of the variants can be changed by applying stress during ageing. Under the stress constraint, the growth of differently oriented variants becomes selective, and this, in turn, varies the materials microstructure. The number of precipitate variants may be determined by decomposing the space group of the parent phase into the coset of the space group of the coherent precipitate phase; and which variants grow selectively is, however, dependent upon the coupling between the applied stress and the local strain of the variants. Selective variant growth of Ti11Ni14 precipitates in Ti-51.5 at% Ni alloy was investigated. A pole projection method was proposed and used to predict the selective variant growth of Ti11Ni14 precipitates under the stress constraint. TEM observation was conducted to corroborate the prediction. A positive correlation between the theoretical analysis and the experiment was established.
A deformation processed 20 vol.% Nb/Al metal–matrix composite sheet was fabricated with powder metallurgical (PM) route, i.e. by hot extrusion and cold rolling techniques. A comparative study was made on the mechanical properties for this rolled in situ 20 vol.% Nb/Al metal–metal composites and a powder metallurgy processed pure Al sheet specimens at room and elevated temperatures up to 773 K. The results showed that the Nb particulates developed an elongated ribbon-like morphology after severe cold rolling reduction (CRR) and the mechanical properties of this kind material were largely influenced by the cold rolling reductions. The tensile test results at both room and elevated temperatures and strengthening mechanisms were also discussed.
An efficient phase-field model is proposed to study the coherent microstructure evolution in elastically anisotropic systems with significant elastic modulus inhomogeneity. It combines an iterative approach for obtaining the elastic displacement fields and a semi-implicit Fourier–spectral method for solving the time-dependent Cahn–Hilliard equation. Each iteration in our iterative numerical simulation has a one-to-one correspondence to a given order of approximation in Khachatuyrans perturbation method. A unique feature of this approach is its ability to control the accuracy by choosing the appropriate order of approximation. We examine shape dependence of isolated particles as well as the morphological dependence of a phase-separated multi-particle system on the degree of elastic inhomogeneity in elastically anisotropic systems. It is shown that although prior calculations using first-order approximations correctly predicted the qualitative dependence of a two-phase morphology on elastic inhomogeneity, the local stress distributions and thus the driving force for microstructure evolution such as coarsening were in serious error quantitatively for systems with strong elastic inhomogeneity.
Morphological evolution and splitting of coherent precipitate particles under applied stresses were investigated using a diffuse-interface field kinetic approach. A particular example, γ′ precipitates in Ni-based superalloys, was studied. In the absence of externally applied stress, a coherent γ′ particle exhibits a cuboidal shape, as a result of the competition between anisotropic coherency strain energy and nearly isotropic specific interfacial energy. It was demonstrated that under a uniaxial applied constraint strain, growth of the γ′ particle became tetragonal, resulting in a shape transformation from being cuboidal to tetragonal. As the magnitude of the applied strain was further increased, it is interesting to observe that the γ′ particle became unstable and split into two or more parallel plates. The influences of stress magnitude, precipitate volume fraction, and interfacial energy on the splitting process are discussed.
Phase field approach is applied to modeling the spinodal decomposition process in a thin film with periodically distributed arrays of interfacial dislocations. The elastic stress field in the simultaneous presence of interfacial dislocations, substrate constraint, and compositional strains is obtained by solving the mechanical equilibrium equations using an iteration method. It is shown that the periodic stress field associated with the array of interfacial dislocations leads to a directional phase separation and the formation of ordered mesoscale microstructures. It is demonstrated that when the periodicity of the dislocation is small, the wavelength of the ordered microstructure tends to be the same periodicity as the dislocation array. The results have important practical implications that an ordered nanostructures could be produced by controlling the interfacial dislocation distribution.
Al–Mg–Si alloy plates friction stir welded at a tool traverse speed of 400 mm/min exhibited higher tensile strength with 45° shear fracture, whereas lower tensile strengths with nearly vertical fractures were observed for samples welded at a lower speed of 100 mm/min. The fracture paths corresponded well with the lowest hardness distribution profiles in the joints. The heat indexes cannot be used as parameters to evaluate the thermal input, mechanical properties and fracture mode.
The response of several ceramic high-Tc superconductors to NO and CO in Taguchi-type gas sensors has been investigated. YBa2Cu3O7-δ decomposes in the presence of NO. Ba1.5La1.5Cu3Ox and La1.85Sr0.15CuOx do not show any sensitivity or selectivity of interest. Nd1.85Ce0.15CuO4-x exhibits sensitivity to CO, but no sensitivity to NO. These materials are not suitable for NO gas sensors. Bi2Sr2CuOsu6+x, Bi2Sr2CaCu2O8+x and Bi1.8Pb0.2Sr2Ca2Cu3O10+x show high selectivity to No against CO. They are promising materials for Taguchi-type NO sensors. Bi2Sr2CaCu2O8+x exhibits the best properties. The sensitivities of composites of Bi2Sr2CaCu2O8+x with 0.5, 2 and 10 wt.% A12O3, Bi2Sr2CaCu2O8+x with 20 wt.% Fe2O3, Bi2Sr2Ca Cu2O8+x with 20 wt.% NiO, Bi2Sr2CaCu2O8+x with 20 wt% ZrO2, Bi2Sr2CaCu2O8+x with 50 wt.% Sb2O5 and the lithium-intercalated compound Li0.112Sr2CaCu2O8x to NO and CO at 300 and 350 °C have been studied. While additions of A12O3, ZrO2 and Sb2O5 to Bi2Sr2CaCu2O8+x do not improve its selectivity to NO CO, additions of Fe2O3 and NiO and lithium intercalation do improve the selectivity.
A new experimental procedure—stir-in-plate welding was adopted to eliminate the initial butt surface of two plates to be joined and examine the effect of the initial butt surface on the formation of the zigzag line and the tensile properties of the welds. The comparison between the butt and stir-in-plate welds indicated that under as-welded condition the zigzag line did not show up in the welds, and two welds exhibited similar tensile properties and fracture characteristics. After post-weld T6-treatment, the zigzag line appeared on the butt weld as zigzag micro-crack at the root tip and discontinuously-distributed cavities of 50–200 μm throughout the weld, which were verified to be associated with the oxide particles. This resulted in the reduced tensile strength and significantly deteriorated ductility with the fracture initiating and propagating along the zigzag line. No zigzag line was discernible on the T6-treated stir-in-plate weld.
In this paper double-hit compression processes are utilized to study the effects of cooling rate from β region to α+β region and deformation amount on the microstructures of near-α titanium alloys containing two different contents of carbon. It is found that during double-hit compression, the flow stress of first step in the β region is comparable to that of the single-pass deformation, and that after second compression in the α+β region is obviously higher than that in single-pass compression. The higher cooling rate and larger deformation amount in the α+β region resulted in recrystallization and globularization of α platelets for both alloys because of inhibition of β phase dynamic recovery caused by dislocation annihilation. Under the same conditions the amount of globularized α was much higher in the 0.3% C alloy than that in the 0.06% C alloy, which can be ascribed in carbide dissolution in solid solution as an α stabilizer during deformation in α+β region and accelerated globularization of the α platelets.
The coarsening kinetics of self-accommodating coherent domain structures is investigated using computer simulations based on a continuum phase-field model. The domain structures are produced from coherent hexagonal to orthorhombic phase transformations. It is found that the long-range elastic interactions arising from the lattice accommodation among different orientation domains of the orthorhombic phase dominate the domain morphologies and the kinetics of domain coarsening. It is shown that the long-range elastic interactions result in several new features for the domain coarsening as compared to normal grain growth. For example, the domain growth rate is reduced significantly and the growth exponent becomes a function of the relative contribution of the elastic energy reduction to the total driving force. In general, the elastic interaction is in favor of fine domains. Although triple junctions are dominant in the microstructure, a significant amount of quadrojunctions exist throughout the domain coarsening process. The average number of sides of the domain is also reduced.
A systematic investigation has been carried out on the time dependence of the response to NO and CO of a gas sensor with a thick porous film of Bi2Sr2CaCu2O8+x (BSCCO). The oxidation of reducing gases such as NO and CO by lattice oxygen occurs in parallel with desorption of NO2 and CO2, respectively, in the whole range of temperatures investigated. An extension of a reported model has been proposed in order to describe the response of the sensor. The time dependence of the electrical resistivity can be interpreted by assuming a conductivity dominated by Schottky barriers at grain boundaries. A kinetic model underlying the temperature dependence of the sensitivity to NO and the selectivity against CO is discussed. The high sensitivity to NO as compared to that to CO is attributed to the high adsorption rate of NO at low temperatures. The good selectivity at high temperatures is attributed to the rapid increase of the desorption rate of CO and/or CO2 with increasing temperature. The response time increases with increasing equilibrium sensitivity S0. The recovery time decreases with increasing S0. They both decrease with increase of the desorption rate of CO(NO) and CO2(NO2).
DOI : 10.1016/0925-4005(94)87026-8 Anahtar Kelimeler :
BSCCO, Gas sensors
Cilt: 22 Sayı: 3 Sayfa: 227-236 ISSN: 0925-4005
We investigate the influence of an applied homogeneous strain on the coherent α2 (DO19) to O-phase transformation in Ti–Al–Nb alloys using a phase-field approach. The emphasis is on the effect of the applied strain on the two-phase morphology, as well as the equilibrium volume fractions of different orientation domains of the O-phase. It is found that the applied homogeneous strain field does not change the essential features of the morphological patterns, but does alter the relative amount of each orientation domain and the equilibrium volume fraction of the O-phase. When the applied strain is of the same order of magnitude as the stress-free transformation strain, the initial two-phase mixture becomes unstable and transforms into a single O-phase.
In this paper supercritical equilibria and critical speeds of axially moving beams constrained by sleeves with torsion springs are deduced. Transverse vibration of the beams is governed by a nonlinear integro-partial-differential equation. In the supercritical regime, the corresponding static equilibrium equation for the hybrid boundary conditions is analytically solved for the equilibria and the critical speeds. In the view of the non-trivial equilibrium, comparisons are made among the integro-partial-differential equation, a nonlinear partial-differential equation for transverse vibration, and coupled equations for planar motion under the hybrid boundary conditions.
Natural frequencies of nonlinear coupled planar vibration are investigated for axially moving beams in the supercritical transport speed ranges. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. The finite difference scheme is developed to calculate the non-trivial static equilibrium. The equations are cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. Under fixed boundary conditions, time series are calculated via the finite difference method. Based on the time series, the natural frequencies of nonlinear planar vibration, which are determined via discrete Fourier transform (DFT), are compared with the results of the Galerkin method for the corresponding governing equations without nonlinear parts. The effects of material parameters and vibration amplitude on the natural frequencies are investigated through parametric studies. The model of coupled planar vibration can reduce to two nonlinear models of transverse vibration. For the transverse integro-partial-differential equation, the equilibrium solutions are performed analytically under the fixed boundary conditions. Numerical examples indicate that the integro-partial-differential equation yields natural frequencies closer to those of the coupled planar equation.
We present a multiscale model for studying the growth and coarsening of θ′ precipitates in Al–Cu alloys. Our approach utilizes a novel combination of the mesoscale phase-field method with atomistic approaches such as first-principles total energy and linear response calculations, as well as a mixed-space cluster expansion coupled with Monte Carlo simulations. We give quantitative first-principles predictions of: (i) bulk energetics of the Al–Cu solid solution and θ′ precipitate phases, (ii) interfacial energies of the coherent and semi-coherent θ′/Al interfaces, and (iii) stress-free misfit strains and coherency strain energies of the θ′/Al system. These first-principles data comprise all the necessary energetic information to construct our phase-field model of microstructural evolution. Using our multiscale approach, we elucidate the effects of various energetic contributions on the equilibrium shape of θ′ precipitates, finding that both the elastic energy and interfacial energy anisotropy contributions play critical roles in determining the aspect ratio of θ′ precipitates. Additionally, we have performed a quantitative study of the morphology of two-dimensional multi-precipitate microstructures during growth and coarsening, and compared the calculated results with experimentally observed morphologies. Our multiscale first-principles/phase-field method is completely general and should therefore be applicable to a wide variety of problems in microstructural evolution.
Ultrafine-grained (0.7 μm) Al–Mg–Sc alloy with an approximately random misorientation distribution and predominantly high-angle boundaries of 97% was produced by friction stir processing. A ductility of 235% was obtained at 200 °C. Increasing temperature from 200 to 300 °C resulted in an increase in superplasticity, optimum strain rate and strain rate sensitivity. Low temperature and high strain rate superplasticity with a ductility of 620% was achieved at 300 °C and 3 × 10−2 s−1. Abnormal grain growth occurred at 350 °C, resulting in the disappearance of superplasticity.
The effect of elastic interaction on the formation and dynamic evolution of multi-domain microstructures during a hexagonal to orthorhombic transformation in the absence and presence of an externally applied strain field is investigated numerically using the phase field model. In particular, three cases are considered, which include a single variant, two variants, and all three variants of the orthorhombic phase produced by the transformation. In each case, the morphology and spatial distribution of the orientation variants are characterized. It is shown that nucleation and growth of a single variant produces thin plates of the orthorhombic domains with definite habit planes. In the case of two variants, the domains developed at the initial stages are also platelets of well-defined habit planes, which is similar to the case of a single variant. However, the impingement and intersection of the platelets of different variants results in the formation of twin boundaries and “zig-zag” patterns. The overlap regions of the “zig-zag” cross sections remain untransformed which agrees very well with experimental observations. If all three variants are present, the hexagonal to orthorhombic transformation results in a number of unique multi-domain structures such as the star patterns, compound star patterns, fan patterns, etc., which have been frequently observed experimentally in systems undergoing hexagonal to orthorhombic or similar transformations. It is found that if the boundary of the system is constrained, e.g. a grain embedded in a polycrystalline material, the transformation can go to completion only when all three variants are present. In the presence of external strain field, the coupling between the applied strain field and the stress-free transformation strain associated with the domain formation leads to selective growth of variants.
A Taguchi-type sensor with a porous polycrystalline Bi2Sr2CaCu2O8+x (BSCCO) film as the active element has been developed. The electrical conductance of the film is “Schottky-barrier limited”. Its resistivity has beeen measured as a function of temperature in different CO/air and NOx/air gas mixtures. The power law that is valid for most Taguchi-type sensors is not suitable for fitting the experimental data. The linear relation between lnln S+ln S (where S is the sensitivity of the sensor) and ln P (where P is the partial pressure of the gas in air) is interpreted using a Schottky-barrier-limited model. In this model, the lattice oxygen in the surface layer of this material participates in the sensing reaction. The adsorption of reducing gases decreases the concentration of elecrtron holes in the surface layer. The increase in resistivity is attributed to the rise in height of the Schottky barrier between the primary particles resulting from the lower concentration of the charge carriers. The sensing reaction for CO is proposed to be CO(g)+OxO+2h·+ad⇌(CO2ad+Vö and that for NO NO(g)+Oxo+2h·+ad⇌(NO2) ad+Vö The model fits the experimental data for CO sensing, and satisfactory fits the data for NO sensing at high temperatures. The sensitivity to NO2 is ascribed to the NO formed by its decomposition.
The response of several ceramic oxide high Tc superconductors to NOx and COx in Taguchi-type sensors has been investigated. YBa2Cu3O7−δ decomposes in the presence of NO.Nd1.85Ce0.15CuO4−x has sensitivity to CO and no sensitivity to NOx.Bi2CuO6+x and Bi2Sr2CaCu2O8+x show high selectivity against CO and CO2. They are promising materials for Taguchi-type NOx sensors, the latter exhibiting the better response properties.
The morphology evolution of γ-hydride precipitation and growth in a zirconium bi-crystal was simulated using a phase field kinetic model. The effects of grain boundary and uniformly applied load were studied. The temporal evolution of the spatially dependent field variables is determined by numerically solving the time-dependent Ginzburg–Landau equations for the structural variables and the Cahn–Hilliard diffusion equation for the concentration variable. It is demonstrated that nucleation density of the hydride at the grain boundary increases as compared to the bulk and favorable hydride precipitation at the grain boundary become weaker when an external load is applied. The result also showed that hydrides will grow in those habit planes that are near the perpendicular direction of the applied tensile load.
Zirconium and its alloys are primary structure materials in nuclear industry. They often absorb hydrogen from environment and form hydrides. This will cause degradation of the materials. In this work, the hydrogen diffusion and hydride formation process near a blunt notch in a continuum media have been simulated by phase field models. The results agree with the experiments.