Triple Junction Mobility: A Molecular Dynamics Study

We present a molecular dynamics simulation study of the migration of individual grain boundary triple junctions. The simulation cell was designed to achieve steady state migration. Observations of the triple junction angle and grain boundary profiles confirm that steady state was achieved. The static, equilibrium grain boundary triple junction angles and the dynamic triple junction angles were measured as a function of grain size and grain boundary misorientation. In most cases, the static and dynamic triple junction angles are nearly identical, while substantial deviations were observed for low boundary misorientations. The intrinsic, steady-state triple junction mobilities were extracted from measurements of the rate of change of grain boundary area in simulations with and without triple junctions. The triple junction velocity is found to be inversely proportional to the grain size width. The normalized triple junction mobility exhibits strong variations with boundary misorientation, with strong minima at misorientations corresponding to orientations corresponding to low values of . The triple junctions create substantial drag on grain boundary migration at these low mobility misorientations.     

Effects of the triple junction types on the grain boundary carbide precipitation in a nickel-based superalloy,a statistical analysis

The morphology evolution of carbide precipitated on grain boundary nearby different triple junctions in grain boundary engineering (GBE) treated nickel-based Inconel Alloy 690 aged at 715°C for different time was investigated by scanning electron microscopy and electron backscatter diffraction. The results show that, the diversity of triple junction types was increased by GBE significantly. The size and morphology of grain boundary carbide were not only affected by the grain boundary character, but also the nearby grain boundary character at the triple junction. The higher Σ values of the nearby grain boundaries, the larger carbide precipitated on the other grain boundary. Based on the experimental results, the effects of grain boundary characters and triple junction types on the carbide precipitation behaviours are discussed.     

The Effect of Triple Junctions on Grain Boundary Motion and Grain Microstructure Evolution

The theory of steady state motion of grain boundary sytems with triple junctions and the main features of such systems are considered. A special technique of in-situ observations and recording of triple junction motion is introduced, and the results of experimental measurements on Zn tricrystals are discussed. It is shown, in particular, that the described method makes it possible to measure the triple junction mobility. It was found that the measured shape of a moving half-loop with a triple junction agrees with theoretical predictions. A transition from triple junction kinetics to grain boundary kinetics was observed. This means that triple junctions can drag boundary motion. It is demonstrated that the microstructural (granular) evolution is slowed down by triple junction drag for any n-sided grain. The second consequence pertains to six-sided grains. For a boundary system with dragging triple junctions there is no unique dividing line between vanishing and growing grains with respect to their topological class anymore, like n = 6 in the Von Neumann-Mullins relation.     

An ignition-temperature model with two free interfaces in premixed flames†

In this paper we consider an ignition-temperature zero-order reaction model of thermo-diffusive combustion. This model describes the dynamics of thick flames, which have recently received considerable attention in the physical and engineering literature. The model admits a unique (up to translations) planar travelling wave solution. This travelling wave solution is quite different from those usually studied in combustion theory. The main qualitative feature of this travelling wave is that it has two interfaces: the ignition interface where the ignition temperature is attained and the trailing interface where the concentration of deficient reactants reaches zero. We give a new mathematical framework for studying the cellular instability of such travelling front solutions. Our approach allows the analysis of a free boundary problem to be converted into the analysis of a boundary value problem having a fully nonlinear system of parabolic equations. The latter is very suitable for both mathematical and numerical analysis. We prove the existence of a critical Lewis number such that the travelling wave solution is stable for values of Lewis number below the critical one and is unstable for Lewis numbers that exceed this critical value. Finally, we discuss the results of numerical simulations of a fully nonlinear system that describes the perturbation dynamics of planar fronts. These simulations reveal, in particular, some very interesting ‘two-cell’ steady patterns of curved combustion fronts.     

On the Origin and Energy of Triple Junction Defects Due to the Finite Length of Grain Boundaries

King [1] established that due to the discrete nature of their dislocation structure, finite length grain boundaries (GBs) in polycrystalline materials possess discrete values of misorientation angle. For a GB with a length that is not a multiple of the GB period, this leads to the formation of specific disclinations at their junctions with neighboring GBs, which compensate the difference between the misorientations of finite and infinite boundaries. In the present paper the origin of these compensating disclinations within GB triple junctions is elucidated and their strength is calculated using the disclination-structural unit model. It is shown that for a GB with length of about 10 nm the junction disclinations can have a strength value not more than 1°, in contrast to King's calculations that indicate much larger values. Elastic energies of triple junctions due to compensating disclinations are calculated for both equilibrium and non-equilibrium structures of a finite length GB, which differ by the position of the grain boundary dislocation network with respect to the junctions. The calculations show that triple junction energies are comparable to dislocation energies, and that compensating disclinations can play a significant role in the properties of nanocrystalline metals with grain sizes less than about 10 nm.     

Grain Boundary Migration in Metals: Recent Developments

Current research on grain boundary migration in metals is reviewed. For individual grain boundaries the dependence of grain boundary migration on misorientation and impurity content are addressed. Impurity drag theory, extended to include the interaction of adsorbed impurities in the boundary, reasonably accounts quantitatively for the observed concentration dependence of grain boundary mobility. For the first time an experimental study of triple junction motion is presented. The kinetics are quantitatively discussed in terms of a triple junction mobility. Their impact on the kinetics of microstructure evolution during grain growth is outlined.     

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.     

Travelling waves in a radiation-combustion free-boundary model for flame propagation

We study a combustion-radiation model which models premixed flames propagating in a gaseous mixture with inert dust. This model combines diffusion of mass and temperature with reaction at the flame front. We choose a free boundary model to describe the propagating flames and take a linearized approximation to model the radiation, but we keep a nonlinear reaction term which is temperature dependent. The radiative transfer of thermal energy emitted and absorbed by dust is modelled using the Eddington equation. We analyse the bifurcation diagram of the travelling wave solution curve. In a specific parameter plane, travelling waves are given by a single smooth curve which is parameterized by the flame temperature.     

Interaction of Solute Impurity with Grain Boundary: The Impurity Drag Effect

An equation of grain boundary motion in a binary polycrystal is derived. The derivation is based on minimization of free energy of the total systems. The equation takes into account an impurity segregation at the grain boundary, grain boundary curvature and energy.As an example, we apply this equation to the analysis of the impurity drag effect problem. It is shown, that the sign of the impurity effect on grain boundary velocity (delay or acceleration) does not depend on kinetic coefficients. The sign of the effect is determined by a thermodynamic function which combines the grain boundary segregation coefficient, the derivative of grain boundary energy with respect to absorbed impurity concentration, and the derivative of bulk free energy with respect to bulk impurity concentration.     

Evaluating Anisotropic Surface Energies Using the Capillarity Vector Reconstruction Method

By measuring the geometry and crystallography of the three interfaces that meet at grain boundary thermal grooves, it is possible to determine the anisotropy of the surface free energy. Previously, the surface energy of MgO at 1400°C in air was approximated by a truncated double Fourier series with coefficients that were determined by fitting the observations to Herring's condition for local equilibrium at a triple junction. The purpose of this paper is to describe an alternative analysis of the same data set that is not limited by an assumed functional form of the surface energy. In this case, the space of surface characters is discretized and each orientation is associated with a capillarity vector (according to the Cahn–Hoffmann definition). The set of capillarity vectors that most closely satisfies the condition for local equilibrium at each triple junction is then determined by an iterative method. The relative surface free energies derived from this analysis are more anisotropic than those derived from the series fit and more consistent with the observed faceting of MgO in air at 1400°C. The relative surface energies of the low index planes are ₁₁₀/₁₀₀ = 1.07±0.04 and ₁₁₁/₁₀₀ = 1.17±0.04.     

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.     

Study of the motion of individual triple junctions in aluminum

The mobility of individual triple junctions in aluminum is studied. Triple junctions with 〈111〉, 〈100〉, and 〈110〉 tilt boundaries are studied. The data obtained show that, at low temperatures, the mobility of the system of grain boundaries with a triple junction is controlled by the mobility of the triple junction (the junction kinetics). At high temperatures, the system mobility is determined by the mobility of the grain boundaries (the boundary kinetics). There is a temperature at which the transition from the junction kinetics to the boundary kinetics occurs; this temperature is determined by the crystallographic parameters of the sample.     

Control of grain boundary microstructures in molybdenum polycrystals by thermomechanical processing of single crystals

The formation of grain structures and grain boundary microstructures in polycrystalline molybdenum, produced by thermomechanical processing from cylindrical single crystals with different initial surface normal orientations of ?110?, ?111? and ?112?, were investigated with the objective of controlling grain boundary microstructures. The polycrystalline specimens displayed different microstructures depending on the initial orientation of the single crystal and the deformed microstructure. The recrystallized microstructure was composed of some oriented-grain clusters, in which grains possessing a similar orientation were assembled. The frequency of low-angle boundaries was very high in the oriented-grain clusters. A close relationship also existed between the grain boundary character distribution (GBCD) and the triple junction distribution. Grain boundary microstructures were compared of bcc molybdenum and fcc polycrystalline materials with reference to path-dependent percolation resistance.     

Velocity selection problem in the presence of the triple junction

Melting of a bicrystal along the grain boundary is discussed. A triple junction plays a crucial role in the velocity selection problem in this case. In some range of the parameters an entirely analytical solution of this problem is given. This allows us to present a transparent picture of the structure of the selection theory. We also discuss the selection problem in the case of the growth of a "eutectoid dendrite."     

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.     

A new model for electromigration grain boundary noise based on free volume

Grain boundary plays a key role in electromigration process of polycrystal interconnection. We take a free volume to represent a 'vacancy--ion complex' as a function of grain boundary specific resistivity, and develop a new characterisation model for grain boundary noise. This model reveals the internal relation between the boundary scattering section and electromigration noise. Comparing the simulation result with our experimental result, we find the source as well as the form of noise change in the electromigration process. In order to describe the noise enhancement at grain boundary quantitatively, we propose a new parameter-grain boundary noise enhancement factor, which reflects that the grain boundary noise can characterise the electromigration damage sensitively.     

Stress distribution and hillock formation in Au/Pd thin films as a function of aging treatment in capacitor applications

Effect of quenching in different media on hillock formation and electrical resistivity has been studied in the Au–Pd layers. Oxygen was released from substrate due to substrate relaxation process. It was suggested that hillocks appear on the triple junction grain boundaries. However, lower electrical resistivity has been seen in the sample which quenched in the air. It was concluded that grain boundary scattering decreases the conductivity of the quenched films due to higher density of dislocations.     

Exact boundary condition for time-dependent wave equation based on boundary integral

An exact non-reflecting boundary conditions based on a boundary integral equation or a modified Kirchhoff-type formula is derived for exterior three-dimensional wave equations. The Kirchhoff-type non-reflecting boundary condition is originally proposed by L. Ting and M.J. Miksis [J. Acoust. Soc. Am. 80 (1986) 1825] and numerically tested by D. Givoli and D. Cohen [J. Comput. Phys. 117 (1995) 102] for a spherically symmetric problem. The computational advantage of Ting–Miksis boundary condition is that its temporal non-locality is limited to a fixed amount of past information. However, a long-time instability is exhibited in testing numerical solutions by using a standard non-dissipative finite-difference scheme. The main purpose of this work is to present a new exact boundary condition and to eliminate the long-time instability. The proposed exact boundary condition can be considered as a limit case of Ting–Miksis boundary condition when the two artificial boundaries used in their method approach each other. Our boundary condition is actually a boundary integral equation on a single artificial boundary for wave equations, which is to be solved in conjunction with the interior wave equation. The new boundary condition needs only one artificial boundary, which can be of any shape, i.e., sphere, cubic surface, etc. It keeps all merits of the original Kirchhoff boundary condition such as restricting the temporal non-locality, free of numerical evaluation of any special functions and so on. Numerical approximation to the artificial boundary condition on cubic surface is derived and three-dimensional numerical tests are carried out on the cubic computational domain.