` PHASE FIELD MODELING A REPORT SUBMITTED ON THE COMPLETION OF SUMMER INTERNSHIP AT BHABHA ATOMIC RESEARCH CENTRE . UNDER THE GUIDANCE OF: SUBMITTD BY: DR. ASHOK ARYA NITIN SINGH MATERIALS SCIENCE DIVISION 09MT3010 BHABHA ATOMIC RESEARCH CENTRE IIT KHARAGPUR MUMBAI
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PHASE FIELD
MODELING A REPORT SUBMITTED ON THE COMPLETION OF SUMMER
INTERNSHIP AT BHABHA ATOMIC RESEARCH CENTRE .
UNDER THE GUIDANCE OF: SUBMITTD BY:
DR. ASHOK ARYA NITIN SINGH
MATERIALS SCIENCE DIVISION 09MT3010
BHABHA ATOMIC RESEARCH CENTRE IIT KHARAGPUR
MUMBAI
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INTRODUCTION: The phase-field method has recently
emerged as a powerful computational approach to modeling and
predicting mesoscale morphological and microstructure evolution
in materials. It describes a microstructure using a set of conserved
and non conserved field variables (or order parameters) that are
continuous across the interfacial regions. The temporal and
spatial evolution of the field variables is governed by the Cahn-
Hilliard nonlinear diffusion equation and the Allen-Cahn
relaxation equation. With the fundamental thermodynamic and
kinetic information as the input, the phase-field method is able to
predict the evolution of arbitrary morphologies and complex
microstructures without explicitly tracking the positions of
interfaces.
1 - WHY PHASE FIELD MODELLING IS IMPORTANT
IN MATERIAL SCIENCE? The properties of most engineered materials have a connection
with their underlying microstructure. For example, the crystal
structure and impurity content of silicon will determine its band
structure and its subsequent quality of performance in modern
electronics. Most large-scale civil engineering applications demand
high-strength steels containing a mix of refined crystal grains and
a dispersion of hard and soft phases throughout their
microstructure. For aerospace and automotive applications, where
weight to strength ratios are a paramount issue, lighter alloys are
strengthened by precipitating second-phase particles within the
original grain structure. The combination of grain boundaries,
precipitated particles, and the combination of soft and hard
regions allow metals to be very hard and still have room for
ductile deformation. It is notable that the lengthening of span
bridges in the world can be directly linked to the development of
pearlitic steels. In general, the technological advance of societies
has often been linked to their ability to exploit and engineer new
materials and their properties.
In most of the above examples, as well as a plethora of
untold others, microstructures are developed during the process of
solidification, solid-state precipitation, and thermomechanical
processing. All these processes are governed by the fundamental
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physics of free boundary dynamics and nonequilibrium phase
transformation kinetics. For example, in solidification and
recrystallization – both of which serve as a paradigm of a first-
order transformation – nucleation of crystal grains is followed by a
competitive growth of these grains under the drive to reduce the
overall free energy – bulk and surface – of the system, limited,
however, in their kinetics by the diffusion of heat and mass.
Thermodynamic driving forces can vary. For example,
solidification is driven by bulk free energy minimization, surface
energy and anisotropy. On the other hand, strain-induced
transformation must also incorporate elastic effects. These can
have profound effects on the morphologies and distribution of, for
example, second-phase precipitates during heat treatment of an
alloy.
The above raised arguments are quite sufficient to support
the cause of understanding and simulating the formation of
microstructure. Phase Field Modeling has emerged as a powerful
tool to simulate the evolution of microstructure which is much
easier than its predecessor („sharp interface approach‟) for such
work in terms of mathematics and its application. The following
section will make it much clearer.
1.2 - SHARP INTERFACE APPROACH:
In conventional modeling technique of for phase transformations
and microstructural evolution i.e the sharp interface approach,
the interfaces between different domains are considered to be
infinitely sharp, and a multi-domain structure is described by the
position of the interfacial boundaries. The kinetics of
microstructure formation is then modeled by a set of partial
differential equations that describe the release and diffusion of
heat, the transport of impurities, and the complex boundary
conditions that govern the thermodynamics at the interface for
each domain
. As a concrete example, in the solidification of a pure
material the advance of the solidification front is limited by the
diffusion of latent heat away from the solid–liquid interface, and
the ability of the interface to maintain two specific boundary
conditions; flux of heat toward one side of the interface is balanced
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by an equivalent flux away from the other side, and the
temperature at the interface undergoes a curvature correction
known as the Gibbs–Thomson condition. These conditions are
mathematically expressed in the following sharp interface model,
commonly known as the Stefan problem:
∂T/∂t = ∇.(k.∇T/б.cp) = ∇.(α ∇T)
бLf Vn = ks∇T. n|sint - kL∇T. n|L
int (1)
Tint = Tm-(γTM/Lf)κ – (Vn/μ)
where T =T(x, t) denotes temperature, k thermal conductivity
(which assumes values ks and kL in the solid and liquid,
respectively), б the density of the solid and liquid, cp the specific
heat at constant pressure, α the thermal diffusion coefficient, Lf
the latent heat of fusion for solidification, γ the solid–liquid
surface energy, TM the melting temperature, κ the local solid
liquid interface curvature, Vn the local normal velocity of the
interface, and μ the local atomic interface mobility. Finally, the
subscript “int” refers to interface and the superscripts “S” and “L”
refer to evaluation at the interface on the solid and liquid side,
respectively.
Like solidification, there are other diffusion-limited phase
transformations whose interface properties can, on large enough
length scales, be described by specific sharp interface kinetics.
Most of them can be described by sharp interface equations
analogous to those in Equation 1. Such models – often referred to
as sharp interface models – operate on scales much larger than
the solid–liquid interface width, itself of atomic dimensions. As a
result, they incorporate all information from the atomic scale
through effective constants such as the capillary length, which
depend on surface energy, the kinetic attachment coefficient, and
thermal impurity diffusion coefficient.
1.3 - SHARP INTERFACE MODELS VS DIFFUSE INTERFACE
MODELS (MORE GENERALLY REFERRED TO AS PHASE
FIELD MODELS):
A limitation encountered in modeling free boundary problems is
that the appropriate sharp interface model is often not known for
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many classes of phenomena. For example, the sharp interface
model for phase separation or particle coarsening, while easy to
formulate nominally , is unknown for the case when mobile
dislocations and their effect of domain coarsening are included. A
similar situation is encountered in the description of rapid
solidification when solute trapping and drag are relevant. There
are several sharp interface descriptions of this phenomenon, each
differing in the way they treat the phenomenological drag
parameters and trapping coefficients and lateral diffusion along
the interface.
Another drawback associated with sharp interface
models is that their numerical simulation also turns out to be
extremely difficult. The most challenging aspect is the complex
interactions between topologically complex interfaces that
undergo merging and pinch-off during the course of a phase
transformation. Such situations are often addressed by applying
somewhat arbitrary criteria for describing when interface merging
or pinch-off occurs and by manually adjusting the interface
topology. It is worth noting that numerical codes for sharp
interface models are very lengthy and complex, particularly in 3D.
Along with these two drawbacks of sharp
interface models, one would not be able to completely appreciate
the diffuse interface approach if the most important advantage of
the later over former is not mentioned here. Main advantage
gained by using phase-field method to model phase transitions,
compared to the sharp-interface method, is that the explicit
tracking of the moving surface, the liquid and solid interface, is
completely avoided. Instead, the phase of each point in the
simulated volume is computed at each time step. In classical
formulation the basic equations have to be written for each
medium and the interface boundary conditions must be explicitly
tracked. In diffuse-interface theory the basic equations, with
supplementary phase field terms, are deduced from a free energy
functional for the whole system and interface conditions do not
occur. In fact, they are replaced by a partial differential equation
for the phase field.
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2 – DETAILS OF PHASE FIELD MODELLING: As already mentioned, the phase field method has proved to be
extremely powerful in the visualization of the development of
microstructure without having to track the evolution of individual
interfaces, as is the case with sharp interface models. The method,
within the framework of irreversible thermodynamics, also allows
many physical phenomena to be treated simultaneously.
The primary purpose of this section is to present the general
concepts underlying the phase field modeling.
Imagine the growth of a
precipitate which is isolated from the matrix by an interface.
There are three distinct entities to consider: the precipitate,
matrix and interface. The interface can be described as an
evolving surface whose motion is controlled according to the
boundary conditions consistent with the mechanism of
transformation. The interface in this mathematical description is
simply a two dimensional surface; it is said to be a sharp interface
which is associated with an interfacial energy σ per unit area.
In the phase field method, the state of the entire
microstructure is represented continuously by a single variable
known as the order parameter ϕ. For example ϕ=1, ϕ=0 and 0<ϕ<1
represent the precipitate, matrix and interface respectively. The
latter is therefore located by the region over which ϕ changes from
its precipitate value to its matrix value, as shown in figure below.
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The range over which it changes is the width of the interface. The
set of values of the order parameter over the whole volume is the
phase field. The total free energy G of the volume is then
described in terms of the order parameter and its gradients, and
the rate at which the structure evolves with time is set in the
context of irreversible thermodynamics, and depends on how G
varies with ϕ. It is the gradients in thermodynamic variables that
drive the evolution of structure.
Consider a more complex example, the growth of a grain
within a binary liquid (Fig.2). In the absence of fluid flow, in the
sharp interface method, this requires the solution of seven
equations involving heat and solute diffusion in the solid, the
corresponding processes in the liquid, energy conservation at the
interface and the Gibbs–Thomson capillarity equation to allow for
the effect of interface curvature on local equilibrium. The number
of equations to be solved increases with the number of domains
separated by interfaces and the location of each interface must be
tracked during transformation. This may make the computational
task prohibitive. The phase field method clearly has an advantage
in this respect, with a single functional to describe the evolution of
the phase field, coupled with equations for mass and heat
conduction, i.e. three equations in total, irrespective of the number
of particles in the system. The interface illustrated in Fig. 2b
simply becomes a region over which the order parameter varies
between the values specified for the phases on either side. The
locations of the interfaces no longer need to be tracked but can be
inferred from the field parameters during the calculation.
Fig. 2 (a) sharp interface (b) diffuse interface
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Notice that the interface in Fig. 2b is drawn as a region with finite
width, because it is defined by a smooth variation in ϕ between ϕ=
0(solid) and ϕ=1(liquid). The order parameter does not change
discontinuously during the traverse from the solid to the liquid.
The position of the interface is fixed by the surface where ϕ=0.5.
2.1 - ORDER PARAMETER:
The order parameter in phase field modeling is a function
of space and time which may or may not have macroscopic
physical interpretations. For two-phase materials, ϕ is typically
set to 0 and 1 for the individual phases, and the interface is the
domain where 0<ϕ<1. For the general case of N phases present in
a matrix, there will be a corresponding number of phase field
order parameters ϕi with i=1 to N. ϕi=1 then represents the
domain where phase i exists, ϕi=0 where it is absent and 0<ϕi<1
its bounding interfaces. Suppose that the matrix is represented by
ϕo then it is necessary that at any location:
N
Σ ϕi = 1
i=0
It follows that the interface between phases 1 and 2, where 0<ϕ1<1
and 0<ϕ2<1 is given by ϕ1+ ϕ2 =1; similarly, for a triple junction
between three phases where 0<ϕi<1 for i=1,2,3, the junction is the
domain where ϕ1+ ϕ2+ ϕ3 = 1.
The order parameters in phase field modeling can be the
either of two types:
Conserved Order Parameters – Conserved quantities as quite
clearly decipherable, are the ones which remain unchanged during
the process to be studied. Example can be the concentration of an
element or alloy undergoing solidification, because the average
concentration is never going to change. The change in conserved
order parameters with time is governed by Cahn-Hilliard
equation.
Non-conserved Order Parameters – Non-conserved quantities
are the ones that change during a process. Example can be spin
or crystalline order. The change in non-conserved order
parameters with time is governed by Cahn-Allen equation.
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2.2 – GOVERNING EQUATIONS AND MATHEMATICS OF
PHASE FIELD MODELLING:
“Derivations of the important expressions are given in full, on the
premise that it is easier for a reader to skip a step than it is for
another to bridge the algebraic gap between it is easily shown that
and the ensuing equation” – J.E Hilliard (on the mathematics of
their phase field model for spinodal decomposition)
As a first requirement for any problem to be modeled by phase
field modeling, a free energy functional (for isothermal cases
and for non-isothermal cases free entropy functional) has to
be defined as a function of order parameter. The general
expression of a free energy functional is shown below:
F = ʃv [f (ϕ, c, T) + (Ɛ2c/2)*| ∇c|2 + (Ɛ2
ϕ/2)*| ∇ϕ|2] dv (2)
The first term in the left hand side of the equation is free energy
density of the bulk phase as a function of concentration, order
parameter and temperature. The second and the third term
denote the energy of the interface. The second term denotes the
energy due to the gradient present in the concentration and the
third term denotes the energy due to the gradient present in the
order parameter.
After doing a little bit of mathematics (which is intentionally
ignored here, considering the point that only the application of
these equations shall be sufficient at undergraduate level study),
one arrives at two kinds of equation. The first one is for conserved
order parameters and the second one is for non-conserved order
parameters.
Cahn-Hilliard Equation – Cahn-Hilliard equation gives the rate
of change of conserved order parameter with time.
∂ ϕ /∂t = M.∇2[∂f/∂ϕ - Ɛ2ϕ ∇2ϕ] (3)
The above equation is for constant (position-independent) mobility
M. ϕ is order parameter, ∇ is divergence, f is free energy of the
bulk, Ɛϕ is gradient energy coefficient. As one can quite clearly
notice that Cahn-Hilliard equation is nothing but modified form of
Fick‟s second law for transient diffusion.
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Cahn-Allen Equation - Cahn-Allen equation gives the rate of
change of non-conserved order parameter with time.
∂ϕ/∂t = -M [∂f/∂ϕ – Ɛ2ϕ ∇2ϕ] (4)
Cahn-Allen equation is also known as time-dependent Ginsburg-
Landau equation.
Note: While deriving the Cahn-Hilliard and Cahn-Allen equation,
an important expression ∂ϕ/∂t = M*∂F/∂ϕ was used, which clearly
does makes sense because the change in free energy functional
with respect to change in order parameter ϕ, must be in some
relation with change in order parameter ϕ with respect to time.
2.3 – PSEUDO ALGORITHM TO MODEL A PROCESS VIA
PHASE FIELD MODELING:
The “diffuse interface‟ idea can be extended to almost all systems
with an evolution of microstructure or interfacial boundary
involved in it. Here is the Pseudo algorithm to approach a
problem:
Describe the microstructure using a suitable set of field
variables (commonly referred as order parameters), some are
conserved variables and others are non-conserved.
Write the energy of a configuration consistent with the
system‟s thermodynamics. It will have both bulk and
gradient energy terms, similar to as equation (2)
Write the evolution equations: Cahn-Hilliard equation for
conserved variables, and Cahn-Allen equation for non-
conserved variables.
Discretize the evolution equations via a suitable scheme (a
variety of schemes can be found in the literature, each
having their own advantages over others) to solve it
numerically.
Provide the system inputs as well as spacial and time
information and then numerically march in time in order to
evolve the order parameters.
Hope (pray?) that your model will behave nicely!
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3 – EXAMPLES: As already explained, a diffuse interface approach is very much
capable to model almost any kind of problem in the field of
material science. The most common problems that are solved are:
Spinodal decomposition.
Isothermal and non-isothermal solidification.
Disorder – order phase transformation.
Grain growth during recrystallization.
Dislocation dynamics.
Dendritic growth.
A problem involving a combination of above mentioned and many
a others to count.
In the present report, three kinds of problem i.e Spinodal
Decomposition, Isotermal Solidification for pure substance
and Dendritic Growth for pure substance are studied, their
theory and a detailed method to solve them via phase field
modeling is presented
3.1 – SPINODAL DECOMPOSITION:
Spinodal decomposition is a mechanism by which a solution of two
or more components can separate into distinct regions (or phases)
with distinctly different chemical compositions and physical
properties. This mechanism differs from classical nucleation in
that phase separation due to spinodal decomposition is much more
subtle, and occurs uniformly throughout the material and not just
at discrete nucleation sites.
Spinodal decomposition is of interest for two
primary reasons. In the first place, it is one of the few phase
transformations in solids for which there is any plausible
quantitative theory. The reason for this is the inherent simplicity
of the reaction. Since there is no thermodynamic barrier to the
reaction inside of the spinodal region, the decomposition is
determined solely by diffusion. Thus, it can be treated purely as a
diffusional problem.
From a more practical standpoint, spinodal decomposition
provides a means of producing a very finely dispersed
microstructure that can significantly enhance the physical
for index = 1:ntmax %calculate g, g is parameterised
%as 2Ac(1-c)(1-2c) for i = 1:N for j = 1:M g(j+M*(i-1)) =
2*A*comp(j+M*(i-1))*(1-
comp(j+M*(i-1)))*(1-
2*comp(j+M*(i-1))); end end
%calculate the fourier transform
%of composition and g field f_comp = fft(comp); f_g = fft(g);
%Next step is to evolve the
&composition profile for i1 = 1:N if i1 < half_N kx = i1*del_kx; else kx = (i1-N-2)*del_kx; end kx2 = kx*kx; for i2 = 1:M if i2 < half_M ky = i2*del_ky; else ky = (i2-M-
2)*del_ky; end ky2 = ky*ky;
k2 = kx2 + ky2;
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k4 = k2*k2; denom = 1.0 +
2.0*kappa*Mob*k4*del_t; f_comp(i2+M*(i1-1)) =
(f_comp(i2+M*(i1-1))-
k2*del_t*Mob*f_g(i2+M*(i1-
1)))/denom;
end end
%Let us get the composition back
%to real space comp = real(ifft(f_comp));
disp(comp);
disp(index);
%for graphical display of the
%microstructure evolution, %lets store the composition
%field into a 256x256 2-d
%Matrix. for i = 1:N for j = 1:M U(i,j) = comp(j+M*(i-
1)); end end %visualization of the output
figure(1) image(U*55) colormap(Jet) end disp('done');
APPENDIX B MATLAB code for isothermal solidification with no anisotropy:
% Isothermal solidifation of a
%single component liquid phase
%using explicit finite %difference scheme without any