Ernst Gamsjäger

Kinetics of interfaces during diffusional transformations

At high temperatures diffusion of components and migration of interfaces are activated in solid state systems. The microstructure evolves due to these processes, and both the volume fraction and chemical composition of the individual phases change. In this paper we formulate a boundary value problem for the microstructural evolution during a diffusional transformation in a binary alloy. We clarify the chemical and mechanical contributions to the thermodynamic driving force on the interface and derive the boundary condition that is implied by interface kinetics for solute diffusion in the bulk. By applying this framework, the consequences of (non-equilibrium) interface kinetics on the microstructure evolution during the ferrite transformation in low-carbon steels is predicted using both a novel analytical technique and a finite difference numerical method.

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.

Modelling the role of surface stress on the kinetics of tissue growth in confined geometries

In a previous paper we presented a theoretical framework to describe tissue growth in confined geometries based on the work of Ambrosi and Guillou [Ambrosi D, Guillou A. Growth and dissipation in biological tissues. Cont Mech Thermodyn 2007;19:245-51]. A thermodynamically consistent eigenstrain rate for growth was derived using the concept of configurational forces and used to investigate growth in holes of cylindrical geometries. Tissue growing from concave surfaces can be described by a model based on this theory. However, an apparently asymmetric behaviour between growth from convex and concave surfaces has been observed experimentally, but is not predicted by this model. This contradiction is likely to be due to the presence of contractile tensile stresses produced by cells near the tissue surface. In this contribution we extend the model in order to couple tissue growth to the presence of a surface stress. This refined growth model is solved for two geometries, within a cylindrical hole and on the outer surface of a cylinder, thus demonstrating how surface stress may indeed inhibit growth on convex substrates.

Diffusional phase transformation and deformation in steels

Abstract During diffusional phase transformations at high temperatures (e.g. γ–α transformation in low-alloy steels) the interface migrates due to a chemical and/or mechanical driving force. A low solubility in the product phase forces the solute atoms (C in α) to move by diffusion, and, therefore, the mole fraction of the solute depends on time and spatial position. In order to evaluate the transformation kinetics numerically, carbon diffusion and interface migration, coupled by appropriate boundary conditions, are simulated. The grain size, the mobility, and the diffusion coefficient determine the growth kinetics. A micromechanical study has been performed in order to investigate the influence of an external uniaxially applied load on the transformation kinetics.

A concise derivation of the contact conditions at a migrating sharp interface

A sharp interface model can be used to simulate the kinetics of diffusional phase transformations provided the transformation process is controlled by migration of the interface and by diffusion of components in the bulk phases only. The contact conditions at the migrating sharp interface can be derived by applying the principles of mass balance and maximum dissipation. A concise derivation of the contact conditions on this basis is presented in this work.

Relaxation of the elastic strain energy of misfitting inclusions due to diffusion of vacancies

Vacancy diffusion is investigated as a mechanism for relaxation of the elastic strain energy caused by a misfitting inclusion. The kinetics of reduction of the total eigenstrain by the deposition or removal of an atom layer along the interface is derived. The time evolution, as well as an estimate for the characteristic time of the relaxation process, is presented. The relaxation times are compared with recent in situ measurements of stress relaxation times in aluminum with small lead-alloy inclusions after their solidification. Experimentally observed relaxation times and those theoretically predicted agree very well.

Modeling of kinetics of diffusive phase transformation in binary systems with multiple stoichiometric phases

The thermodynamic extremal principle is used for the treatment of the evolution of a binary system under the assumption that all phases in the system are nearly stoichiometric with no sources and sinks for vacancies in the bulk. The interfaces between the individual phases are assumed to act as ideal sources and sinks for vacancies, and to have an infinite mobility. Furthermore, it is assumed that several phases are nucleated in the contact plane of the diffusion couple at the beginning of the computer experiment. Then, it is shown that the number of newly nucleated phases determines the maximum number of polyfurcations (i.e., branching of a single configuration into several distinct configurations) of the initial contact (Kirkendall) plane. The model is demonstrated on a hypothetical binary system with four stoichiometric phases. The inverse problem, namely, the determination of the tracer diffusion coefficients in newly nucleated phases from the thicknesses of new phases and the positions of polyfurcated Kirkendall planes, is treated too.

Analysis of the mobility of migrating austenite–ferrite interfaces

Dilatometric studies assisted by high-temperature laser scanning confocal microscopy provide a comprehensive experimental picture with regard to cyclic austenite-to-ferrite transformations in Fe–C alloys. The validity range for the sharp interface and effective mobility approach is identified by comparing modelling results with calculations based on experiments. The interface velocity for the austenite-to-ferrite transformation in pure iron is exclusively controlled by the intrinsic interface mobility conforming to the upper boundary of mobilities. The austenite-to-ferrite transformation in Fe–C alloys under conventional cooling and heating conditions is primarily controlled by carbon diffusion in austenite. The lower boundary of the temperature-dependent interface mobility has been established for an Fe–C alloy over a wide range of temperatures during cycling transformation. Austenite-to-ferrite transformations in Fe–C–X alloys are characterized by still lower effective mobilities depending on both temperature and composition, because substitutional elements X give rise to a solute drag effect. An estimate for the effective mobility valid for the austenite-to-ferrite transformation in lean Fe–C–Mn alloys is provided.

Semiconductor properties of thin and thick film Ga2O3 ceramic layers

Abstract The semiconductor behavior of thin and thick film β Ga2O3 layers is studied by measuring the resistivity as a function of oxygen partial pressure and temperature in the range up to 900°C. As for ZnO and SnO2 a relatively high initial oxygen vacancy defect concentration has to be assumed for β Ga2O3. However, the conductivity is by many orders of magnitude lower and the activation energy by about 1 order of magnitude higher. With increasing temperature a change at about 810±50°C from a lower value of the activation energy EA(2)=1·6 ± 0·1 eV to a higher one EA(1)=2·4 ± 0·1 eV is observed at thin film ceramic layers thus leading to the assumption that oxygen cleavage in contact with the atmosphere is achieved in the upper range. Contrary to the band model which is convincingly founded for ZnO and SnO2 in the literature, polaron hopping seems to be the more suitable model for analysis of the conductivity data of β Ga2O3. The lower value EA(2) is interpreted as the polaron hopping energy at approximately constant charge carrier concentration. On the other hand, in the high temperature range above Tch the charge carrier density is varying. However, at the applied measuring conditions, this variation remains below the initial oxygen vacancy defect concentration. Corresponding to formula I GaIII2-2xGaII2x′O3-xV {O,x} and formula II GaIII2-xGaIx″O3-xV {O,x} two different structures for the oxygen vacancy defects in β Ga2O3 are discussed. The measurements seem to confirm formula I. However, provided that there is an equilibrium between states corresponding to formula II and I, the assumption of double occupied GaI states is also consistent with the experimental results.

Modelling of diffusive and massive phase transformations in binary systems – thick interface parametric model

Abstract In general three dissipative processes have to be considered during solid/solid diffusive phase transformations; diffusion in the bulk phases, rearrangement of the crystal lattice and diffusion in the interfacial region. It is evident that different length scales (the thickness of the interface is small compared to the grain size) and time scales (the diffusion across the interface is fast compared to the diffusion in the bulk materials) make it difficult to solve this problem numerically. Instead of solving the coupled interface migration and diffusion equations it is advantageous to set up appropriate, parameter dependent shape functions describing mole fraction profiles. The time dependent evolution of the parameters of these shape functions can be derived by means of the thermodynamic extremal principle. Four independent parameters suffice to describe the transformation kinetics in the case that diffusion of the components is fast in the product phase compared to diffusion of components in th...