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OVERVIEW

 

PHASE FIELD MODELING

Introduction: Phase field method (PFM) for modeling microstructural evolution

The phase field method is a versatile technique for simulating microstructural evolution. Amongst others, it has been applied to describe dendritic growth in solidification, the formation and coarsening of precipitates, martensitic and other solid-state phase transitions, grain growth and dislocation dynamics.

The microstructures considered in phase field simulations typically consist of a number of grains. The shape and mutual distribution of the grains, is represented by functions that are continuous in space and time, the phase field variables. Within the grains, the phase field variables have nearly constant values, which are related to the structure, orientation and composition of the grains. The interface between two grains is defined as a narrow region where the phase field variables gradually vary between their values in the neighboring grains. This modeling approach is called a diffuse interface description. The evolution of the shape of the grains, or in other words the position of the interfaces, as a function of time, is implicitly given by the evolution of the phase field variables. An important advantage of the phase field method is that, thanks to the diffuse interface description, there is no need to track the interfaces (to follow explicitly the position of the interfaces by means of mathematical equations) during microstructural evolution. Therefore, the evolution of complex grain morphologies, typically observed in technical alloys, can be predicted without making any a priori assumption on the shape of the grains.

Illustration of a a) diffuse and b) sharp interface.

Illustration of a a) diffuse and b) sharp interface.

The temporal evolution of the phase field variables is described by a set of partial differential equations, which are solved numerically. The equations are derived based on general thermodynamic and kinetic principles and contain a number of phenomenological parameters related to the physical properties of the material. These parameters are determined based on experimental and theoretical information. Different thermodynamic driving forces for microstructure evolution, such as chemical bulk free energy, chemical interfacial energy and elastic strain energy, and different transport processes, such as heat and mass diffusion, can be considered at the same time.

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Effect of Zener pinning and thermal grooving on grain growth

Grain growth occurs in all polycrystalline materials to reduce the free energy associated with grain boundaries. Especially at elevated temperature, where the mobility of the grain boundaries is high, considerable grain growth may occur. Characteristic for the proces is that the microstructure coarsens by shrinkage and disappearance of the smaller grains and growth of the larger grains.

The pinning effect of small second-phase particles on grain boundaries, also called Zener pinning, is an important mechanism for controlling the amount of grain growth. The second-phase particles retard grain growth by pinning the grain boundaries and eventually, when a critical grain size is reached, arrest grain growth.

The pinning effect is of great practical importance in tailoring material properties. For example, in the case of steels for structural applications, a small grain size is required for good strength, toughness and cold deformability. Grain growth during thermomechanical processing or welding of the steel mostly results in inferior mechanical properties and is therefore highly unwanted. Therefore small amounts of niobium, titanium, vanadium or aluminum are added to the steel. The elements react with carbon and nitrogen present in the steel and form finely dispersed precipitates, like NbC-, AlN-, and TiN-precipitates. During thermomechanical processing or welding of the steel, the precipitates pin the grain boundaries so that the amount of grain growth is reduced and a fine-grained steel is obtained. Moreover, in the production of thin films, the pinning effect of precipitates is often used to promote abnormal grain growth. In this way, films with a huge grain size are obtained which is desirable for good electrical properties.

A phase field model for grain growth in materials containing second-phase particles has been worked out and implemented. The pinning effect of finely dispersed second-phase particles on grain growth was studied by means of phase field simulations.


SEM image of a grain boundary pinned by a MnS
precipitate in low C steel and a cross-sectional image of a 3D  phase field simulations
of a spherical grain boundary
passing by a particle.

SEM image of a grain boundary pinned by a MnS precipitate in low C steel and a phase field simulations image of a spherical grain boundary passing by a particle.


Left: In-situ TEM image of a grain structure pinnend by CuAl2 precipitates in an
Al alloy film (Longworth and Thompson 1991). Right:  Image from a 3D
simulation of grain growth in a thin film, assuming the same number,
size and initial distribution of the precipitates and the 
same film thickness as for the Al alloy
film on the left. The section is taken through the middle of the film.

Left: In-situ TEM image of a grain structure pinnend by CuAl2 precipitates in an Al alloy film (Longworth and Thompson 1991). Right: Image from a 3D simulation of grain growth in a thin film, assuming the same number, size and initial distribution of the precipitates and the same film thickness as for the Al alloy film on the left. The section is taken through the middle of the film.


Surface tension has an important effect on grain growth in thin films. Grooves are for example formed at the film surface to equilibrate surface and grain boundary tension. They exert a drag effect on migrating grain boundaries. Furthermore, grains with an orientation with low surface energy will preferentially grow and consume grains with high surface energy. This behavior results in texture strengthening and may lead to abnormal grain growth. A 3D phase field model that accounts for the effects of orientation dependent surface tension on grain growht in thin films has been developed.

In-situ STM image of grooving at the surface of a gold film (Rost
el al. 2003) and an image of a 3D phase field simulation of grain growth and grooving in a thin
film.

In-situ STM image of grooving at the surface of a gold film (Rostel al. 2003) and an image of a 3D phase field simulation of grain growth and grooving in a thin film.


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Grain growth in anisotropic systems

Experimental measurements and atomistic simulations have shown that grain boundary energy and mobility are not uniform, but highly depend on the misorientation between the adjacent grains and the inclination of the grain boundary with respect to the crystal lattice of one of the grains. Furthermore, it is believed that industrially important phenomena like abnormal grain growht and texture strengthening require particular spatial distributions of the grain boundary properties. However, it is not yet understood how the properties of individual boundaries and their interactions at triple and quadruple junctions affect the evolution of a large network of grain boundaries. Mesoscale simulation techniques such as the phase field method have the capacity to bring new insights into this subject.

A quantitative phase-field approach for simulating grain growth in anisotropic systems with arbitrary inclination and misorientation dependence has been worked out, together with a procedure to determine the model parameters. The methodology allows us to account correctly for grain boundary mobility and stiffness data -- for example obtained from molecular dynamics simulations -- in mesoscale simulations, and gives high controllability of the numerical accuracy.

Misorientation dependence of the grain boundary energy assumed for
the system shown in the figures below.
Misorientation dependence of the grain boundary energy assumed for the system shown in the figures below.


Evolution of a 3D structure with misorientation dependent grain
boundary energy.   Grains with similar orientation have a similar color. 
Boundaries with low misorientation  are
white (1.5° misorientation) and gray (3°), special high angle
boundaries
(37.5° misorientation) are red and the other boundaries are black.
Evolution of a 3D structure with misorientation dependent grain boundary energy. Grains with similar orientation have a similar color. Boundaries with low misorientation are white (1.5° misorientation) and gray (3°), special high angle boundaries (37.5° misorientation) are red and the other boundaries are black.

Cluster of grains with a similar orientation.
Cluster of grains with a similar orientation.

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Diffusion and coarsening in multi-phase and multi-component systems

Most engineering materials are polycrystalline and multi-phase on the microscopic scale. The average grain size and grain size distribution and the fraction, grain shape and spatial arrangement of the different phases largely correlate with the macroscpic material properties and behavior. By combining the phase-field method with CALPHAD models to obtain Gibbs energy descriptions as a function of composition and temperature for the multi-phase systems, microstructure evolution simulations can be performed for realistic engineering materials. Our model developments in this area aim to make both approaches fully compatible.

  • A new form of interpolation functions for multi-phase systems is introduced that allows for a thermodynamically consistent and quantitative formulation of the bulk energy contribution in phase- field models. It is compatible with most multi-phase phase-field formulations currently existing in literature.

    Free energy density landscape of the interfacial and total energy
of a 2-phase system represented by 2 order-parameter fields using the
new interpolation functions. Also after addition of the bulk energy
contribution, the minima of the total free energy density remain at 
(1,0) and (0,1).
    Free energy density landscape of the interfacial and total energy of a 2-phase system represented by 2 order-parameter fields using the new interpolation functions. Also after addition of the bulk energy contribution, the minima of the total free energy density remain at (1,0) and (0,1).

  • Microstructure simulation models for the growth of intermetallic phases with low solubility are developed and validated.

    Reactive growth of intermetallic phases with low solubility is important for many applications, such as soldering, growth of silicide layers and oxidation. In CALPHAD Gibbs energy models, phases with low solubility are often treated as stoichiometric or with a very limited composition domain. Microstructure and diffusion simulations however require information on the composition dependence of the Gibbs energy and diffusion mobilities, also outside there stability region. Different forms (e.g. parabolic, sublattice and order-disorder) of composition dependence are compared. The choice of Gibbs energy and diffusion mobility model has a great effect on the growth behavior of the intermetallic phases and diffusion paths in multi-component systems.

    Gibbs energy expressions as a function of composition for different phases of the Cu-Sn system
at 450 K, assuming a) no solubility and b) parabolic dependence, c)
sublattice and d) order-disorder composition dependence.

    Gibbs energy expressions as a function of composition for different phases of the Cu-Sn system at 450 K, assuming a) no solubility and b) parabolic dependence, c) sublattice and d) order-disorder composition dependence.

  • Within the framework of the European Concerted Action (COST) on Advanced Solder Materials for High Temperature Application -- HISOLD, COST MP0602
  • the new model developments were applied to microstructure evolution in leadfree solder joints.
    • Group Project on Complex Modelling of the Microstructural Changes in the Interdiffusion Zone for Ag-Cu-Sn and Cu-Ni-sn Leadfree Solder Joints

      The purpose of the group project was to develop theoretical models and simulation techniques that predict the microstructural changes in the interdiffusion zone in lead-free solder joints. Different modeling techniques such as the phase field method, finite element modeling, the CALPHAD-method and ab-initio calculations, are combined to describe processes at different length scales. The modeling is supported by experimental studies (diffusion couple experiments, annealing, surface tension and contact angle measurements and microstructure characterization) to obtain the necessary information for the input parameters, information on the sequence of phase formation, and for validation of the modelling approaches. Different phenomena important in the life cycle of a solder joint are considered, such as nucleation, solidification of the solder, growth of the intermetallic layer and precipitates, Kirkendal voiding, and crack formation. Also the effect of oxygen on the morphological evolution is studied. The focus is on the systems Ag-Cu-Sn and Ni-Cu-Sn for which the thermodynamic properties have been studied extensively. There were approximately 12 universities or research institutions from 8 European countries involved in the group project. Read more in the project reports.

Phase field representation of a Cu-substrate/Ag-Cu-Sn solder joint.
Phase field representation of a Cu-substrate/Ag-Cu-Sn solder joint (courtesy of A. Serbruyns).

Phase field simulation of the coarsening of a Cu-Cu6Sn5-Sn(Cu)
joint with Cu6Sn5-precipitates.
Phase field simulation of the coarsening of a Cu-Cu6Sn5-Sn(Cu) joint with Cu6Sn5-precipitates. Colors: Cu(fcc)=blue, Cu6Sn5=green, Sn(bct)=red

3D microstructure.
grain boundary diffusion flux.
Three dimensional simulations of the growth of the intermetallic phases Cu3Sn and Cu6Sn5 at the interface in a Cu-Sn solder joint. The evolution of the grains and grain boundary diffusion in the intermetallic phases are considered. The grain structure is shown on the left. The vertical component of the diffusion flux of Sn is shown on the right.

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