A phase field model of deformation twinning: Nonlinear theory and numerical simulations

A continuum phase field theory and corresponding numerical solution methods are developed to describe deformation twinning in crystalline solids. An order parameter is associated with the magnitude of twinning shear, i.e., the lattice transformation associated with twinning. The general theory addresses the following physics: large deformations, nonlinear anisotropic elastic behavior, and anisotropic phase boundary energy. The theory is applied towards prediction of equilibrium phenomena in the athermal and non-dissipative limit, whereby equilibrium configurations of an externally stressed crystal are obtained via incremental minimization of a free energy functional. Outcomes of such calculations are elastic fields (e.g., displacement, strain, stress, and strain energy density) and the order parameter field that describes the size and shape of energetically stable twin(s). Numerical simulations of homogeneous twin nucleation in magnesium single crystals demonstrate fair agreement between phase field solutions and available analytical elasticity solutions. Results suggest that critical far-field displacement gradients associated with nucleation of a twin embryo of minimum realistic size are 4.5%–5.0%, with particular values of applied shear strain and equilibrium shapes of the twin somewhat sensitive to far-field boundary conditions and anisotropy of twin boundary surface energy.     

Boundary conditions to the Ginzburg-Landau equations at the twinning plane in a (<Emphasis Type="Italic">d</Emphasis>+<Emphasis Type="Italic">s</Emphasis>) superconductor

Effective boundary conditions to the Ginzburg-Landau equations at the twinning plane of an orthorhombic (d+s) superconductor are obtained on the basis of microscopic BCS theory. The range of the parameters of orthorhombic strain is found where the order parameter behaves variously on both sides of the twinning boundary.     

Effect of Hydrogen Charging on Mechanical Twinning,Strain Hardening,and Fracture of ‹111› and ‹144› Hadfield Steel Single Crystals

This paper studies the effect of electrolytic hydrogen charging on the plastic deformation and fracture of Hadfield steel single crystals oriented for tension along the ?111? and ?144? directions, which the major deformation mechanism is mechanical twinning. Electrolytic hydrogen charging for five hours at a current density of 100 A/m² slightly affects the stages of plastic flow, deformation mechanism, and the value of uniform elongation of ?111? and ?144? single clreystals. Hydrogen saturation causes shear microlocalization and a decrease of the strain hardening coefficient in twinning in one system, but slightly affects the strain hardening characteristics in multiple twinning. Hydrogen charging increases the fraction of the brittle component on fracture surfaces and leads to microand macrocracking near the fracture zone on the lateral surface of deformed specimens. It has been found experimentally that the stress relaxation rate in loaded ?111? single clreystals after hydrogen saturation decreases. Mechanisms of describing this phenomenon have been proposed.     

Elastic field of cylindrical inclusion

The elastic field of a cylindrical inclusion with plane or antiplane transformation strain is found under assumptions of linear isotropic homogeneous theory of elasticity. Different integral expressions are obtained from expressions known for the three-dimensional case and interpreted in terms of the theory of dislocations. The surface integrals correspond to a volume distribution of infinitesimal dislocation dipoles inside the inclusion, the line integrals to a surface distribution of dislocations in the boundary between the inclusion and matrix. The case of a shear transformation in a rectangular region is discussed in more detail in connection with twinning.     

Brittle-ductile behavior of a nanocrack in nanocrystalline Ni: A quasicontinuum study

The effects of stacking fault energy, unstable stacking fault energy, and unstable twinning fault energy on the fracture behavior of nanocrystalline Ni are studied via quasicontinuum simulations. Two semi-empirical potentials for Ni are used to vary the values of these generalized planar fault energies. When the above three energies are reduced, a brittle-to-ductile transition of the fracture behavior is observed. In the model with higher generalized planar fault energies, a nanocrack proceeds along a grain boundary, while in the model with lower energies, the tip of the nanocrack becomes blunt. A greater twinning tendency is also observed in the more ductile model. These results indicate that the fracture toughness of nanocrystalline face-centered-cubic metals and alloys might be efficiently improved by controlling the generalized planar fault energies.     

Molecular dynamics simulation of twinning in devitrite,Na2Ca3Si6O16

Reports of Type II twins are quite rare for most crystal structures. When they do occur, they are usually one of a number of possible twinning modes observed in a particular material. However, for the triclinic phase devitrite, Na₂Ca₃Si₆O₁₆, which nucleates from commercial soda?lime?silica float glass subjected to suitable heat treatments, the only reported twinning mode to date is a Type II twinning mode. In this study, this Type II twinning mode is first examined by molecular dynamics simulation to determine the lowest energy configuration of perfect twin boundaries for the twin mode. This is then compared with the lowest energy configurations of perfect twin boundaries found for six possible Type I twinning modes for devitrite for which the formal deformation twinning shear is less than 0.6. The most favourable twin plane configuration for the Type II twinning crystallography is shown to produce reasonably low twin boundary energies and sensible predictions for the optimum locations of the twin plane, K ₁, and the [1?0?0] rotation axis, η ₁, about which the 180° Type II twinning operation takes place. By comparison, all the Type I twinning modes were found to have very energetically unstable atomic configurations, and for each of these twinning modes, the lowest energy configurations found all led to high effective K ₁ twin boundary energies relative to perfect crystal. These results therefore provide a rationale for the experimental observation of the particular Type II twinning mode seen in devitrite.     

剪切应变下刃型位错的滑移机理的晶体相场模拟

针对刃型位错的滑移运动, 构建包含外力场与晶格原子密度耦合作用的体系自由能密度函数, 建立剪切应变作用体系的晶体相场模型. 模拟了双相双晶体系的位错攀移和滑移运动, 计算了位错滑移的Peierls势垒和滑移速度. 结果表明: 施加较大的剪切应变率作用, 体系能量变化为单调光滑曲线, 位错以恒定速度做连续运动, 具有刚性运动特征; 剪切应变率较小时, 体系能量变化出现周期波动特征, 位错运动是处于低速不连续运动状态, 运动出现周期“颠簸”式滑移运动, 具有黏滞运动特征; 位错启动运动, 存在临界的势垒. 位错启动攀移运动的Peierls势垒要比启动滑移Peierls势垒大几倍. 位错攀移和滑移运动特征与实验结果相符合.     

人工水晶-x面上巴西双晶的形成机制

本文以人工水晶-x面上的巴西双晶作为研究对象,在实验的基础上,利用X射线衍射形貌术和电子探针显微分析等手段,研究了双晶的形成机制,同时分析了Kern的双晶成核动力学理论的不足之处。作者提出了双晶界应变能势垒的概念,认为由于晶体的不完整性提高了晶体本身的能量状态,结果相对降低了应变能势垒的高度,使得双晶易于发生。本文结合晶体生长实验和晶体缺陷的测试,分析了双晶形成与籽晶取向、杂质、晶体缺陷的相互关系,着重指出缺陷对双晶形成的影响。本文还以凹入角生长机制讨论了双晶的发育。 关键词：     

Spinodal twinning of a deformed crystal

We propose the possibility of a spinodal mechanism for deformation twinning in addition to the nucleation and growth mechanism assumed in all existing studies of twinning, using the thermodynamic stability analysis of a homogeneously deformed crystal by examining its energy landscape as a function of strain along the twinning direction obtained from first-principles calculations. Twinning occurs continuously owing to thermodynamic instability with respect to twinning at large shear strains, whereas it can only take place through the nucleation and growth mechanism at small shear strains.     

Giant strain associated with microstructure control by magnetic field in Fe-31.2Pd, CoO and Nd0.5Sr0.5MnO3

We have made in situ optical microscope observation for the microstructure control driven by magnetic field in Fe-31.2Pd (at%), CoO and Nd_0.5Sr_0.5MnO₃. These materials exhibit structural transitions, and their low-temperature phases are composed of several crystallographic domains (variants), which are separated by twinning planes. In the case of ferromagnetic Fe-31.2Pd and antiferromagnetic CoO, the magnetic field promotes the twinning plane movement. This movement gives a large strain of several percent and is essentially explained by the fact that the magnetic shear stress, which corresponds to the magnetic anisotropy energy divided by the twinning shear, is larger than the twinning stress. In the case of Nd_0.5Sr_0.5MnO₃, the twinned microstructure of the charge-ordered phase disappears under a magnetic field in association with the melting of the charge-ordered phase.     

Energy of a wedge-shaped nanotwin calculated in terms of a dislocation mesoscopic model

The total energy of a wedge-shaped micro- and nanotwin is calculated in terms of a dislocation mesoscopic model. The total energy of the twin is represented as a sum of the elastic energy, energy of interaction between twinning dislocations, and stacking-fault energy of partial dislocations of the wedge-shaped twin. It is found that the evolution of the twin is controlled by the energy of interaction between twinning dislocations: in the case of a microtwin, it is five orders of magnitude higher than the elastic energy and six orders of magnitude higher than the stacking-fault energy. In the case of a nanotwin with the number of twinning dislocations at the twin boundary less than 20, all the three energies listed above are of the same order of magnitude. Therefore, all the components of the total energy contribute to the origination of a wedge-shaped twin. As the length of the twin increases with its width and the number of twinning dislocations at twin boundaries fixed, the total energy modulo grows although the density of twinning dislocations at twin boundaries decreases. This indicates that long-range stress fields due to twinning dislocations play an important part in the evolution of a wedge-shaped twin.     

Discovery,characterization and modelling of novel shape memory behaviour of fcc metal nanowires

Novel shape memory behaviour was discovered recently in single-crystalline fcc nanowires of Cu, Ni and Au with lateral dimensions below 5?nm. Under proper thermomechanical conditions, these wires can recover elongations up to 50%. This phenomenon only exists at the nanoscale and is associated with reversible lattice reorientations within the fcc lattice structure driven by surface stresses. Whereas the propagation of partial dislocations and twin planes specific to fcc metals are the required mechanism, only materials with higher propensities for twinning (e.g. Cu and Ni) show this behaviour and those with lower propensities for twinning (e.g. Al) do not. This paper provides an overview of this novel behaviour with a focus on the transformation mechanism, driving force, reversible strain, size and temperature effects and energy dissipation. A mechanism-based micromechanical continuum model for the tensile behaviour is developed. This model uses a decomposition of the lattice reorientation process into a reversible, smooth transition between a series of phase-equilibrium states and a superimposed irreversible, dissipative propagation of a twin boundary. The reversible part is associated with strain energy functions with multiple local minima and quantifies the energy conversion process between the twinning phases. The irreversible part is due to the ruggedness of the strain energy landscape, associated with dislocation nucleation, gliding and annihilation, and characterizes the dissipation during the transformation. This model captures all major characteristics of the behaviour, quantifies the size and temperature effects and yields results which are in excellent agreement with data from molecular dynamics simulations.     

High resolution transmission electron microscope observation of zero-strain deformation twinning mechanisms in Ag

We have observed a new deformation-twinning mechanism using the high resolution transmission electron microscope in polycrystalline Ag films, zero-strain twinning via nucleation, and the migration of a Σ3{112} incoherent twin boundary (ITB). This twinning mechanism produces a near zero macroscopic strain because the net Burgers vectors either equal zero or are equivalent to a Shockley partial dislocation. This observation provides new insight into the understanding of deformation twinning and confirms a previous hypothesis: detwinning could be accomplished via the nucleation and migration of Σ3{112} ITBs. The zero-strain twinning mechanism may be unique to low staking fault energy metals with implications for their deformation behavior.     

Al晶界的第一性原理拉伸试验

基于密度泛函理论和局域密度近似的第一性原理方法，进行了Al晶界的第一性原理拉伸试验.得到Al晶界的理论拉伸强度为9.5 GPa，对应的应变为16%.根据价电荷密度、键长和原子构型随应变的变化，我们证实断裂发生在晶界面，其特征是所有界面键的断裂.同时还发现在周围原子键的数目减少的情况下，界面重构的Al-Al原子键具有共价键的性质.因此Al晶界依然保持着较高的界面强度. 关键词： Al晶界 第一性原理拉伸试验 理论拉伸强度     

Rearrangement of crystallographic domains driven by magnetic field in antiferromagnetic CoO

The rearrangement was investigated of crystallographic domains in the antiferromagnetic pseudo-tetragonal phase in CoO (Néel temperature: 293 K) when the domains were driven by a magnetic field. A rearrangement is generally observed in ferromagnetic shape-memory alloys. The rearrangement was found to occur at temperatures between 170 K and 293 K, but not at temperatures below 170 K. In order to determine the reason for such a difference, the shear stress driven by a magnetic field, τ _mag, was calculated and compared with the shear stress required for twinning plane movement, τ _req. It was found that τ _mag is equal to or larger than τ _req whenever the rearrangement of crystallographic domains occurs due to the application of a magnetic field, and vice versa. This observation is similar to past observations in the case of many ferromagnetic shape-memory alloys.     

Diamagnetic length scales of the Condon domain phase in Lifschitz–Kosevich–Shoenberg approximation

Equilibrium properties of the non-uniform diamagnetic phase in normal metals (Condon domains) are studied theoretically in the framework of Lifschitz–Kosevich–Shoenberg (LKS) approximation. It is found that characteristic diamagnetic lengths of the phase, e.g. a period of the domain structure and width of interface boundary between domains, as well as specific surface energy of domain wall, are strongly affected by electron correlations and depend on temperature, magnetic field and purity of the sample. The developed theory is in a good agreement with existent experiment data.     

Surface strain due to embedded epitaxial islands

We present a simple methodology for the evaluation of strain on the free surface above an embedded island. This work is motivated by the technological importance of self-organization of nanostructures. We first obtain the exact solution of the problem of a particle embedded in a half-space, using continuum theory. Then, we obtain a simple mathematical expression for the strain on the surface. In order to account for the influence of surface energy at the nanoscale, surface and interface stress are considered through the continuum Gurtin formulation. The results are compared with the classical approach, which typically ignores surface stress. This comparison illustrates a very dramatic (even qualitative) difference from the classical elasticity based prediction. It proves that surface stress must be taken into account when islands are small and softer than the substrate material. The procedure illustrated here should be useful to those who need usable expressions for surface strain and have no interest in solving the corresponding boundary value problem.     

In-plane structural order of domain engineered La0.7Sr0.3MnO3 thin films

We present a detailed structural study of tensile-strained La_0.7Sr_0.3MnO₃ thin films. We use the substrate miscut to control the number of rhombohedral variants in the films and study the in-plane order and structural distortions. Using high-resolution X-ray diffraction, we demonstrate that step-edge induced lattice modulations occur in 4-variant films, whereas periodic twinning is the dominant in-plane order for 2-variant films. We show that the in-plane twinning angle is almost completely relaxed. However, the relaxation of shear strain by the out-of-plane twinning angle and the monoclinic distortion is only partial. Furthermore, the film thickness dependence of the domain width reveals that domain formation is a universal mechanism for shear strain relaxation. Finally, we show that the structural response to the transition from the paramagnetic to the ferromagnetic phase of La_0.7Sr_0.3MnO₃ at 345?K is smaller in 4-variant films compared to 2-variant films.     

Effect of dipole structures on field emission of wide-gap semiconductor emitters

It is shown that dipole structures placed in a thin (less than 1 nm) near-surface layer of a high-resistivity field emitter produce small domains on the emitting surface in which the electric field may exceed 10⁸ V/cm. In these domains, the emitter surface potential is positive, providing effective electron transport from inside the emitter to the emission boundary. Optimal dipole orientations ensuring maximal electric fields at the surface are found. When the surface density of dipoles localized in the near-surface layer is on the order of 10⁶ cm⁻², one can expect an emitter-averaged emission current density of higher than 1 A/cm². The dipole structures in the near-surface layer may persist owing to incorporated impurity molecules having a dipole moment or result from a random combination of positively charged ionized impurities and electrons captured by deep traps. Trap charging/discharging asymmetry accounts for the hysteresis of the emission I–V characteristics.