The phase field method is a versatile technique for simulating
microstructural evolution. Amongst others, it has been applied to
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
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.
The temporal evolution of the phase field variables is described by
a set of partial differential equations, which are solved
are derived based on general thermodynamic and kinetic principles
contain a number of phenomenological parameters related
to the physical properties of the material.
parameters are determined based on experimental and
Different thermodynamic driving forces for microstructure
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.
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
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,
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 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.
Surface tension has an important effect on
grain growth in thin films.
for example formed
at the film surface
to equilibrate surface and grain boundary
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 (Rostel al.
2003) and an image of a 3D phase field simulation of grain growth and
grooving in a thin
Experimental measurements and atomistic simulations have shown that grain
boundary energy and mobility are not uniform, but highly depend on the
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
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.
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.
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
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).
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.
- 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
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
Phase field representation of a Cu-substrate/Ag-Cu-Sn solder joint
(courtesy of A. Serbruyns).
Phase field simulation of the coarsening of a
joint with Cu6Sn5-precipitates. Colors: Cu(fcc)=blue,
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.