Effect of finite boundary junction mobility on the growth rate of grains in 3D polycrystals

With decreasing grain size, grain boundary junctions become increasingly important for microstructure evolution. We show that the effect of a limited mobility of triple junctions on the growth rate of polycrystals can be implemented in theories of three-dimensional (3D) grain growth. Respective analytical relations are derived on the basis of the average n-hedra approach introduced by Glicksman to describe the volume rate of change of 3D grains in a polycrystalline aggregate under the impact of a limited triple junction mobility. The theoretical predictions were compared to network-model computer simulations, and good agreement was obtained.     

Grain boundary curvature and grain growth kinetics with particle pinning

Second-phase particles are used extensively in design of polycrystalline materials to control the grain size. According to Zener’s theory, a distribution of particles creates a pinning pressure on a moving grain boundary. As a result, a limiting grain size is observed, but the effect of pinning on the detail of grain growth kinetics is less known. The influence of the particles on the microstructure occurs in multiple length scales, established by particle radius and the grain size. In this article, we use a meso-scale phase-field model that simulates grain growth in the presence of a uniform pinning pressure. The curvature of the grain boundary network is measured to determine the driving pressure of grain growth in 2D and 3D systems. It was observed that the grain growth continues, even under conditions where the average driving pressure is smaller than the pinning pressure. The limiting grain size is reached when the maximum of driving pressure distribution in the structure is equal to the pinning pressure. This results in a limiting grain size, larger than the one predicted by conventional models, and further analysis shows consistency with experimental observations. A physical model is proposed for the kinetics of grain growth using parameters based on the curvature analysis of the grain boundaries. This model can describe the simulated grain growth kinetics.     

Effect of thermal fluctuations on the coarsening dynamics of 2D hexagonal system

The dynamics of ordering in a 2D hexagonal system was investigated through the Cahn-Hilliard-Cook model. At low thermal noise amplitudes, pinning forces acting on grain boundaries dominate the dynamics and the coarsening evolves logarithmically in time. As noise amplitude increases, fluctuations becomes large enough to unlock dislocations located along grain boundaries and the grain boundary motion is driven by curvature. The grain boundary relaxation leads to a grain structure with Lifshitz's configurations. In this case the dynamic is also logarithmic as a consequence of the pinning of triple points.     

Perturbation treatment of non-linear transport via the Robertson statistical formalism

A perturbation solution is found for the differential equation defining an operator Tˆ used by Robertson to relate the information-theoretic phase-space distribution σ to the solution ρ of the classical Liouville equation. This relation provides a closure, used in obtaining an exact equation for σ. Multiplying the latter equation by F, a phase-space function odd under momentum reversal, of which heat and diffusion fluxes are among the examples, one gets an exact equation for ∂〈F〉/∂t. 〈F〉 is the phase space integral of ρF. The dissipative terms in ∂〈F〉/∂t can be expanded, like Tˆ, in successive orders O(〈F〉ⁿ). For a model in which equilibrium ensemble fluctuations relax exponentially, terms linear and O(〈F〉³) are calculated. The non-linear terms exhibit an explicit time-dependence for short times. In a steady state induced by external driving forces, the explicit time-dependence disappears, in agreement with existing phenomenology. For simplicity, spatial uniformity is assumed. A generalization is required for large temperature or velocity gradients.     

Theory of Triple Junctions Mobility in Crystals with Impurities

We consider the steady state migration of the triple junction in the tricrystal with impurities which segregate strongly at the grain boundaries. If the mobility of impurities inside grain boundaries is much higher than the rate of impurity atoms jumps from the grain boundary into the bulk, the triple junction migration causes the divergence of the impurity content at the triple point. We show that this divergence can be relaxed either by the non-equilibrium segregation at the growing grain boundary or by the formation of the inclusion of the impurity-rich phase at the triple point. In the former case the dihedral angle at the triple point differs considerably from its equilibrium value and is strongly temperature-dependent. However, the triple junction cannot be described as an individual object with its own mobility. In the latter case of the cavity formation at the triple point the triple junction can be characterized by its own mobility. It is shown that the dependence of the triple junction migration rate on the driving force is approximately linear at the low migration rates and highly nonlinear at high migration rates. Moreover, there is the maximal allowable steady-state migration rate of the system triple junction-inclusion. For the higher migration rates the jerky motion of the triple junction occurs. Both models are in a good agreement with the experimental data.     

Grain boundary grooving in bi-crystal thin films induced by surface drift-diffusion driven by capillary forces and applied uniaxial tensile stresses

Grain boundary (GB) grooving, induced by surface drift-diffusion and driven by the combined actions of capillary forces and applied uniaxial tensile stresses, is investigated in bi-crystal thin films using self-consistent dynamical computer simulations. A physico-mathematical model, based on the irreversible thermodynamics treatment of surfaces and interfaces with singularities allowed auto-control of the otherwise free-motion of the triple junction at the intersection of the grooving surface and the GB, without having any a priori assumption on the equilibrium dihedral angles. In the present theory, the generalised driving forces for stress-induced surface drift-diffusion arise not only from the usual elastic strain energy density (ESED), but also much stronger elastic dipole tensor interactions (EDTI) between the applied stress field and the mobile atomic species situated at the surface layer and in the GB regions. Accelerated groove-deepening kinetics shows that the surface drift-diffusion enhanced by the applied uniaxial tensile stresses through EDTI is dominant over the GB flux leakage at the triple junction. At high uniaxial stress levels (≥500?MPa for a 100-nm thick copper film), a sequential time-frame for micro-crack nucleation and growth is recorded just before specimen failure took place. These non-equilibrium thermokinetics discoveries (kinetics and energetics) contradict or at least do not support the hypothesis of the steady-state diffusive GB micro-crack formation and propagation due to ‘constant’ flux drainage through GB enhanced by tensile stresses acting normal to it.     

Grain Growth in Polycrystalline Thin Films of Semiconductors

Grain growth in thin films is usually abnormal, leading not only to an increase in the average grain size, but also to an evolution in the shape of the grain size distribution and to an evolution in the distribution of grain orientations. The latter can be driven by surface, interface or strain energy minimization, depending on film and substrate properties and on deposition conditions, and can lead to different final textures depending on which energy dominates.In semiconductor films, as in other materials, grain growth stagnation coupled with texture-selective driving forces leads to secondary grain growth, the rate of which is higher in thinner films. Self ion-bombardment enhances the rate of pre-stagnation grain growth, and doping of Si with electron donor leads to enhanced pre-stagnation grain growth as well as surface-energy-driven secondary grain growth. The effects of ion-bombardment and dopants on grain growth in Si can be understood in terms of associated increases in point defect concentrations and the effects of point defects on grain boundary mobilities.     

Axiomatic Geometric Formulation of Electromagnetism with Only One Axiom: The Field Equation for the Bivector Field <Emphasis Type="Italic">F</Emphasis> with an Explanation of the Trouton-Noble Experiment

In this paper we present an axiomatic, geometric, formulation of electromagnetism with only one axiom: the field equation for the Faraday bivector field F. This formulation with F field is a self-contained, complete and consistent formulation that dispenses with either electric and magnetic fields or the electromagnetic potentials. All physical quantities are defined without reference frames, the absolute quantities, i.e., they are geometric four-dimensional (4D) quantities or, when some basis is introduced, every quantity is represented as a 4D coordinate-based geometric quantity comprising both components and a basis. The new observer-independent expressions for the stress-energy vector T(n) (1-vector), the energy density U (scalar), the Poynting vector S and the momentum density g (1-vectors), the angular momentum density M (bivector) and the Lorentz force K ((1-vector) are directly derived from the field equation for F. The local conservation laws are also directly derived from that field equation. The 1-vector Lagrangian with the F field as a 4D absolute quantity is presented; the interaction term is written in terms of F and not, as usual, in terms of A. It is shown that this geometric formulation is in a full agreement with the Trouton-Noble experiment.     

Relations between the Coulomb gas picture and conformal invariance of two-dimensional critical models

Partition functions of critical 2D models on a torus can be derived from their microscopic formulation and their free field representation in the continuum limit. This is worked out explicitly for theO(n) andQ-state Potts model. Forn orQ integer we recover results obtained from conformal invariance, but our procedure also extends to nonintegral values. In the latter case the expansion on characters of the Virasoro algebra involves real coefficients of either sign. The operator content of both models is discussed in detail.     

Exact probability distributions for noncorrelated random walk models

A stochastic model for the idealized locomotion of cells is studied. The cell is assumed to cover a polygonal line in ⁿ , the times between turns are exponentially distributed and independent of the directions, and the density of thenth directione does not depend on the (n–1)th directione. The resulting Markov process (X(t), D(t)) for position and direction of the motion at timet is studied by using the integrodifferential equation for the transition function. For example, the joint distribution of (X(t), D(t)) is derived in closed form ifn=2 orn=3 and all chosen directions (including the initial one) are uniformly distributed. For higher dimensions the combined Fourier-Laplace transform ofX(t) is given. The case of a fixed initial direction is also considered.     

Grain Boundary Dynamics: A Novel Tool for Microstructure Control

The reaction of grain boundaries to a wide spectrum of forces is reviewed. Curvature, volume energy and mechanical forces are considered. The boundary mobility is strongly dependent on misorientation, which is attributed to both grain boundary structure and segregation. In magnetically anisotropic materials grain boundaries can be moved by magnetic forces. For the first time a directionality of boundary mobility is reported. Flat boundaries can also be moved by mechanical forces, which sheds new light on microstructure evolution during elevated temperature deformation. Curvature driven and mechanically moved boundaries can behave differently. A sharp transition between the small and large angle boundary regime is observed. It is shown that grain boundary triple junctions have a finite mobility and thus, may have a serious impact on grain growth in fine grained materials. The various dependencies can be utilized to influence grain boundary motion and thus, microstructure evolution during recrystallization and grain growth.     

Stability and dissociation of quadruple junctions in polycrystals

Junctions of four grain boundaries (quadruple junctions) may form during grain growth. An analysis of the stability/unstability of quadruple junctions in 2D polycrystals is presented based on simple thermodynamic arguments. Depending on the angles between the four grain boundaries, their junction may dissociate into two triple junctions or be stable. In the first case there are two possible dissociations, but in general one is favored. A stable quadruple junction moves as a whole to a new position before it dissociates. These alternatives in the behavior of a quadruple junction are conveniently displayed in maps which indicate the regions where each alternative occurs.     

On generalized Lax equation of the Lax triple of KP Hierarchy

In terms of the operator Nambu 3-bracket and the Lax pair (L, B_n) of the KP hierarchy, we propose the generalized Lax equation with respect to the Lax triple (L, B_n, B_m). The intriguing results are that we derive the KP equation and another integrable equation in the KP hierarchy from the generalized Lax equation with the different Lax triples (L, B_n, B_m). Furthermore we derive some no integrable evolution equations and present their single soliton solutions.     

Quantifying cooperativity in water clusters: an attempt towards obtaining a generalised equation

The current study describes the development of a general equation of cooperativity energy (CE) for an n-mer system based on the difference between interaction energy of the system and the sum of interaction energy of its dimeric subunits. The counterpoise corrected generalised CE equation of n-mer system has also been developed. In order to validate these two generalised equations, we have considered four different kinds of water clusters, viz. W1D, W2D, W2D-H and W3D, ((H₂O)_n: n = 3–10), with the increasing cluster size as well as changing arrangements. The CE obtained from the two generalised equations follow the trend W3D > W2D-H > W2D > W1D. The deviation at n = 6 for W3D clusters in the plot of CE with cluster size has also been observed for O–H stretching frequencies, electron density and the Laplacian of the electron density plots. The contrastingly higher CE for cyclic pentamer compared to hexamer is in accordance with earlier observations as well as computed average hydrogen bond lengths. We have observed that the average charge value is more on oxygen atoms of pentamer than hexamer resulting in the shortening of hydrogen bond in pentamer and hence the CE values are more for pentamer than hexamer. We have shown that CE has the capability to quantify the cooperativity effect in water clusters.     

On Shtelen’s Solution of the Free Linear Schrödinger Equation

Abstract

The solution of the three-dimensional free Schrödinger equation due to W.M. Shtelen based on the invariance of this equation under the Lorentz Lie algebra so(1,3) of nonlocal transformations is considered. Various properties of this solution are examined, including its extension to n ≥ 3 spatial dimensions and its time decay; which is shown to be slower than that of the usual solution of this equation. These new solutions are then used to define certain mappings, F _n, on L ²(?ⁿ) and a number of their properties are studied; in particular, their global smoothing properties are considered. The differences between the behavior of F _n and that of analogous mappings constructed from usual solutions of the free Schrödinger equation are discussed.     

三维溶质枝晶生长数值模拟

建立了在低Péclet数条件下三维溶质枝晶生长的数值模拟模型.该模型采用Zhu和Stefanescu 提出的溶质平衡方法,即根据固/液界面的平衡浓度和实际浓度之差计算固/液界面演化的驱动力.界面的平衡浓度由界面温度和曲率所确定,实际浓度通过采用有限差分法对溶质扩散控制方程进行数值求解而获得.该方法能够合理定量地描述枝晶从初始的非稳态到稳态的生长过程,并且具有较高的计算效率.为了描述具有不同晶体学取向的三维枝晶生长,提出了一种权值平均曲率算法用于计算固/液界面的曲率,在权值平均曲率的算法中耦合了界面能各向异性的因素.该算法简单易实现,并易于从二维推广到三维系统.为了对模型进行验证,将模拟的枝晶尖端稳态生长数据和理论模型的预测结果进行了比较.结果表明,模拟的Al-2wt%Cu合金枝晶尖端稳态生长速率和半径随过冷度的变化接近于Lipton-Glicksman-Kurz解析模型的预测结果.模拟分析了稳态枝晶尖端的形貌,发现三维枝晶尖端是非轴对称的,以四次对称的方式偏离旋转抛物面.最后,应用所建立的模型模拟出具有发达分枝和不同晶体学取向的三维等轴多枝晶生长形貌. 关键词： 微观组织模拟 溶质枝晶生长 权值平均曲率 三维     

MacPherson-Srolovitz晶粒长大速率方程的仿真验证

2007年MacPherson和Srolovitz联合推导出一个三维个体晶粒长大的准确速率方程,但并未给出实验或计算机仿真的验证.采用Potts模型Monte Carlo方法对该速率方程进行了大尺度仿真验证.结果表明,仿真数据与MacPherson-Srolovitz速率方程符合很好,从而初步证实了该速率方程,即三维晶粒长大速率是晶粒棱长和晶粒平均宽度的函数. 关键词： 三维晶粒长大 速率方程 Monte Carlo仿真     

The stability of triple junctions during the grain boundary diffusion process

The grain boundary diffusion in a system with triple junctions is considered in such a geometry, in which the flows of diffusing atoms meet at the triple line. The solutions of the diffusion equation is given in the frameworks of Fisher's model and under the assumption of quasi-stationary distribution of the diffusing atoms along the grain boundaries. The change of the mechanical equilibrium at the triple junction due to the increase of the concentration of solute atoms is considered. It is shown that under some circumstances the triple junction looses its stability with respect to migration in the direction to the diffusion source. The stability diagrams in the segregation-diffusivity parameter space are plotted.     

Symmetry Analysis of the Multidimensional Polywave Equation

Abstract

We present symmetry classification of the polywave equation □ ^l u = F (u). It is established that the equation in question is invariant under the conformal group C(1, n) iff F (u) = λe ^u, n + 1 = 2l or F (u) = λu ^{(n+1+2l)/(n+1?2l)}, n + 1 6= 2l. Symmetry reduction for the biwave equation □ ² u = λu ^k is carried out. Some exact solutions are obtained.     

Σ3 CSL boundary distributions in an austenitic stainless steel subjected to multidirectional forging followed by annealing

The effect of processing and annealing temperatures on the grain boundary characters in the ultrafine-grained structure of a 304-type austenitic stainless steel was studied. An S304H steel was subjected to multidirectional forging (MDF) at 500–800°C to total strains of ~4, followed by annealing at 800–1,000°C for 30 min. The MDF resulted in the formation of ultrafine-grained microstructures with mean grain sizes of 0.28–0.85 μm depending on the processing temperature. The annealing behaviour of the ultrafine-grained steel was characterized by the development of continuous post-dynamic recrystallization including a rapid recovery followed by a gradual grain growth. The post-dynamically recrystallized grain size depended on both the deformation temperature and the annealing temperature. The recrystallization kinetics was reduced with an increase in the temperature of the preceding deformation. The grain growth during post-dynamic recrystallization was accompanied by an increase in the fraction of Σ3ⁿ CSL boundaries, which was defined by a relative change in the grain size, i.e. a ratio of the annealed grain size to that evolved by preceding warm working (D/D₀). The fraction of Σ3ⁿ CSL boundaries sharply rose to approximately 0.5 in the range of D/D₀ from 1 to 5, which can be considered as early stage of continuous post-dynamic recrystallization. Then, the rate of increase in the fraction of Σ3ⁿ CSL boundaries slowed down significantly in the range of D/D₀ > 5. A fivefold increase in the grain size by annealing is a necessary condition to obtain approximately 50% Σ3ⁿ CSL boundaries in the recrystallized microstructure.