Grain boundary faceting and abnormal grain growth in BaTiO3

When 0.1 mol% TiO₂–excess BaTiO₃ was sintered below the eutectic temperature in air, abnormally large grains formed in the fine matrix BaTiO₃ grains. The abnormal grains contained {111} twin lamellae, while the matrix grains did not. A TEM observation revealed that almost all the grain boundaries were faceted. On the other hand, however, when the air-sintered sample with faceted grain boundaries was annealed in H₂, the faceted boundaries became defaceted, and the growth of abnormal grains was suppressed while the growth of the matrix grains was enhanced, showing normal grain growth behavior. In addition, the abnormal grains that had been elongated along their twin lamella grew rather isotropically, irrespective of the presence of {111} twins. It appears therefore that {111} twins appear to enhance the growth of the abnormal grains along the twin lamellae only when the grain boundary is faceted. After re-annealing the H₂-annealed sample in air, however, the grain growth behavior and grain boundary structure were found to recover those observed in the air-sintered sample. From these observations, it is concluded that abnormal growth of BaTiO₃ grains observed is related to grain boundary faceting and that boundary faceting is a necessary condition for abnormal grain growth.     

Phase field modeling of grain boundary migration with solute drag

The grain boundary (GB) motion in the presence of GB segregation is investigated by means of phase field simulations. It is found that the solute concentration at the moving GB may increase with increasing velocity and becomes larger than the equilibrium value, which is unexpected according to the solute drag theory proposed by Cahn, but has been observed in some experiments. A non-linear relation between the driving force (curvature) and the GB velocity is found in two cases: (1) the GB motion undergoes a transition from the low-velocity extreme to the high-velocity extreme; (2) the GB migrates slowly in a strongly segregating system. The first case is consistent with the solute drag theory of Cahn. As for the second case, which is unexpected according to solute drag theory, the non-linear relation between the GB velocity and curvature comes from two sources: the non-linear relation of the solute drag force with GB velocity, and the variation in GB energy with curvature. It is also found that, when the diffusivity is spatially inhomogeneous, the kinetics of GB motion is different from that with a constant diffusivity.     

Grain coarsening inhibited by solute segregation

It will be shown that grain boundaries are in a metastable thermodynamic equilibrium in the presence of solute atoms and, therefore, grain coarsening is stopped as there is no driving force. This is in contradiction to a generally accepted interpretation, where solute drag, i.e. zero mobility of the boundaries, stops grain coarsening. Based on the empirical relation between terminal solubility of a solute and its grain boundary segregation it can be shown that a two-phase mixture with solute atoms agglomerating in a precipitated phase will be the stable thermodynamic equilibrium state. However, if precipitation is kinetically hindered, the metastable equilibrium with a certain grain boundary area and a zero grain boundary energy is attained. Changes in this grain boundary area or grain size respectively are calculated as a function of temperature and compared with experimental findings.     

Reducing grain boundary,dislocation line and vacancy formation energies by solute segregation. I. Theoretical background

The Gibbs adsorption isotherm and Wagner’s definition of excess solute at surfaces and grain boundaries are both extended to include other crystalline defects, like dislocations and vacancies. By using a thermodynamic state function which is suitable for a partially open system, open with respect to solvent and closed with respect to solute atoms, a generalized Gibbs adsorption isotherm can be derived. Thus solute segregation to dislocations and vacancies gives rise to a reduction of their formation energies, too. The results of the presented treatment are compared with results stemming from statistical mechanics or computer simulations. Special attention is paid to the question whether defect energies might become zero or negative.     

Solute segregation transition and drag force on grain boundaries

We investigate solute segregation and transition at grain boundaries and the corresponding drag effect on grain boundary migration. A continuum model of grain boundary segregation based on gradient thermodynamics and its discrete counterpart (discrete lattice model) are formulated. The model differs from much previous work because it takes into account several physically distinctive terms, including concentration gradient, spatial variation of gradient-energy coefficient and concentration dependence of solute–grain boundary interactions. Their effects on the equilibrium and steady-state solute concentration profiles across the grain boundary, the segregation transition temperature and the corresponding drag forces are characterized for a prototype planar grain boundary in a regular solution. It is found that omission of these terms could result in a significant overestimate or underestimate (depending on the boundary velocity) of the enhancement of solute segregation and drag force for systems of a positive mixing energy. Without considering these terms, much higher transition temperatures are predicted and the critical point is displaced towards much higher bulk solute concentration and temperature. The model predicts a sharp transition of grain boundary mobility as a function of temperature, which is related to the sharp transition of solute concentration of grain boundary as a function of temperature. The transition temperatures obtained during heating and cooling are different from each other, leading to a hysteresis loop in both the concentration–temperature plot and the mobility–temperature plot. These predictions agree well with experimental observations.     

Grain-boundary segregation and dynamic solute drag theory—A phase-field approach

We propose a model based on a phase-field approach to study grain-boundary segregation and solute drag. We will show that it is possible to model the dynamics of grain-boundary segregation to a stationary boundary as well as solute drag on a moving boundary with the same phase-field model. We shall achieve this by introducing a concentration dependency in the height of the double-well potential in the Gibbs-energy expression. As the model then will be able to treat the build-up of a concentration spike in the boundary as well as its disappearance we shall term this treatment dynamic solute-drag theory.     

Influence of solute segregation and drag on properties of migrating interfaces

Recent models on solute segregation and drag search for steady-state solutions of the diffusion equation in the region of a migrating interface and adjacent semi-infinite grains. Such solutions are limited to model massive transformations, for which the chemical composition of the parent grains and that of the product grains are the same. Simulation of diffusive transformations in transient systems with grains of different chemical composition is a hot issue. In the present model the steady-state solution of the diffusion equation is investigated only in the interface. The coupling with the diffusion equation in the adjacent grains is ensured by proper boundary conditions. For simulation of transient diffusive phase transformations it is very convenient if the interface can be treated as a sharp interface with a known effective mobility and prescribed boundary conditions at the interface. In this paper it is shown that solute segregation and drag can be taken into consideration by the effective mobility of a sharp interface and by proper boundary conditions at the sharp interface. The effective interface mobility reflects Gibbs energy dissipation due to rearrangement of solvent atoms and to trans-interface diffusion and drag of solute atoms in the migrating interface. Trans-interface diffusion and drag of solute atoms are also reflected by the jump of the chemical potential of the solute across the interface, which represents a boundary condition for coupling with the diffusion equation in the adjacent grains. Finally, it is shown that the incorporation of the effects of trans-interface diffusion and drag of solute atoms does not make the simulation more complicated provided that some necessary calculations are performed in preprocessing.     

Reducing grain boundary,dislocation line and vacancy formation energies by solute segregation: II. Experimental evidence and consequences

The extended version of the Gibbs adsorption isotherm including dislocations and vacancies is used to analyse existing experimental data. Thus phenomena and models like solid solution softening, hydrogen-enhanced local plasticity, brittleness of hydrides and superabundant vacancies could be interpreted on the basis of thermodynamics as caused by changing the defect energy by solute segregation; like in Gibbs’ original work, surface and grain boundary energies are reduced by excess solute. In addition, the analysis of experimental results addresses the question whether zero or negative defect energies are feasible and how this will affect materials behaviour.     

Effect of grain boundary energy on surface-energy induced abnormal grain growth in columnar-grained film

In columnar-grained film with anisotropy in the surface energy, abnormal grain growth occurs to minimize the overall energy. We studied the effect of grain boundary energy on the surface-energy driven abnormal grain growth in thin films using the Monte Carlo computer simulation. Our simulation results show that the growth speed of abnormal growth slows down when grain boundary energy increases. It is because the speed of normal growth driven by the grain boundary energy minimization gets faster with increasing grain boundary energy, while that of the abnormal growth induced by the fixed surface energy is consistent. The growth kinetics was monitored based on the growth velocity of the abnormal grains relative to the average growth velocity of all the grains. It was also found that the abnormal growth kinetics was retarded by impingements among abnormally growing grains when there was more than one abnormal grain.     

Modelling the influence of grain-size-dependent solute drag on the kinetics of grain growth in nanocrystalline materials

The large relative change in total grain-boundary area that accompanies grain growth in a nanocrystalline material has a potentially strong influence on the kinetics of grain growth whenever grain-boundary migration is controlled by solute (impurity) drag. As the grain-boundary area decreases, the concentration of solute or impurity atoms segregated to the boundaries is expected to increase rapidly, introducing a grain-size dependence to the retarding force on boundary migration. We have modified the Burke equation—which assumes the drag force to be independent of the average grain size—to take into account a linear dependence of grain-boundary pinning on grain size. The form of the resulting grain-growth curve is surprisingly similar to Burke's solution; in fact, a constant rescaling of the boundary mobility parameter is sufficient to map one solution approximately onto the other. The activation energies for grain-boundary motion calculated from the temperature dependence of the mobility parameter are therefore identical for both models. This fact provides an explanation for the success of Burke's solution in fitting grain-growth data obtained in systems, such as nanocrystalline materials, for which the assumption of grain-size-independent solute drag is incorrect.     

Anisotropy of grain boundary energies as cause of abnormal grain growth in electroplated copper films

The dihedral angle distribution of electroplated (EP) Cu films is significantly broader than those of vapor-deposited films, which may indicate anisotropy of the grain boundary energies. Additives incorporated during EP are presumed to cause anisotropy in the grain boundary energy and abnormal grain growth during self-annealing.     

On precipitation-controlled grain size in the presence of solute segregation

This paper proposes an approach to determine the smallest volume fraction of second-phase particles that is necessary to arrest grain growth and to stabilize grain microstructure.     

Grain boundary diffusion and segregation of Ni in Cu

Grain boundary (GB) diffusion of ⁶³Ni in polycrystalline Cu was investigated by the radiotracer technique in an extended temperature interval from 476 to 1156 K. The independent measurements in Harrison’s C and B kinetic regimes resulted in direct data of the GB diffusivity D_gb and of the so-called triple product P = s · δ · D_gb (s and δ are the segregation factor and the diffusional GB width, respectively). Arrhenius-type temperature dependencies for both the D_gb and P values were measured, resulting in the pre-exponential factors $D_{\mathrm{gb}}^0=6.93\times {10}^{-7}$ m² s⁻¹ and P₀ = 1.89 × 10⁻¹⁶ m³ s⁻¹ and the activation enthalpies of 90.4 and 73.8 kJ mol⁻¹, respectively. Although Ni is completely soluble in Cu, it reveals a distinct but still moderate ability to segregate copper GBs with a segregation enthalpy of about −17 kJ mol⁻¹.     

Grain boundary segregation of phosphorus in iron-vanadium alloys

Grain boundary segregation of phosphorus has been studied in Fe---V---P and Fe---V---P---C alloys through fracture experiments in a scanning Auger microprobe with the objective of examining the effects of vanadium on the interaction processes operative under circumstances when the structure in the interior of the grain (in the present case carbide formation) and grain boundary segregation occur simultaneously. It is understood that to predict and analyse the behaviour of an alloy, it is pertinent to consider the atomic interactions both at the grain boundaries and in the grain interior and that between the constituents and the grain boundaries. The study suggests that the determining factor for suppression or decrease in the migration of phosphorus to the grain boundaries is whether vanadium is present in the combined form (say, carbide) or is available in solid solution form. When vanadium is present in solid solution form, grain boundary segregation of phosphorus is low because of the chemical interaction of vanadium and phosphorus. However, as carbon is increasingly introduced in the alloy, vanadium now preferentially interacts with carbon in view of a higher interaction for carbon as compared to that of phosphorus. A consequence of this is an increase in the grain boundary concentration of phosphorus. In such a situation, the presence of excess carbon in addition to what is stoichiometrically required to precipitate the entire vanadium as vanadium carbides serves as a palliative with regard to the reduction in the intergranular concentration of phosphorus. This palliative behaviour is explained in terms of the site-competition model. An effort is also made to examine the behaviour of segregating elements in terms of a whole range of probable interactions (both at the grain boundaries and in the grain interior) and chemical interaction energies.     

Analysis of grain growth process in melt spun Fe-B alloys under the initial saturated grain boundary segregation condition

A grain growth process in the melt spun low-solid-solubility Fe-B alloys was analyzed under the initial saturated grain boundary (GB) segregation condition. Applying melt spinning technique, single-phase supersaturated nanograins were prepared. Grain growth behavior of the single-phase supersaturated nanograins was investigated by performing isothermal annealing at 700 °C. Combined with the effect of GB segregation on the initial GB excess amount, the thermo-kinetic model [Chen et al., Acta Mater. 57 (2009) 1466] was extended to describe the initial GB segregation condition of nanoscale Fe-B alloys. In comparison of pure kinetic model, pure thermodynamic model and the extended thermo-kinetic model, an initial saturated GB segregation condition was determined. The controlled-mechanism of grain growth under initial saturated GB segregation condition was proposed using two characteristic annealing times (t₁ and t₂), which included a mainly kinetic-controlled process (t ≤ t₁), a transition from kinetic-mechanism to thermodynamic-mechanism (t₁ < t < t₂) and pure thermodynamic-controlled process (t ≥ t₂).