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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

properties of the material.

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3.1.1-Mechanism: In a binary

mixture spinodal decomposition

occurs when delocalized small

amplitude fluctuations

(concentration waves) grow

spontaneously (see figure) as the

time after the quenching proceeds.

These local concentration

fluctuations will lead to phase

change in the thermodynamically unstable state. The mechanism

should not occur in the whole two phase coexistence region of

mixture, but rather only inside a smaller region, the boundary of

which is given by the spinodal curve (or chemical spinodal).

Suppose the plot of free

energy vs composition for a binary

mixture appears as shown in the

figure, then the points of inflection

enclose that region in which

spinodal decomposition is going to

occur at that temperatur. Spinodal

curve is plotted by finding these

points of inflection at different

temperature and then joining

them. As shown in the figure, the

phase seperation inside the

spinodal curve takes place via

spinodal decomposition while in

between spinodal curve and

coexistence curve, it takes place

via nucleation mechanism. In this

region (outside the spinodal) , the

systen is stable against such weak

fluctuations and localised large

amplitude fluctuations must form

in order to start the

transformation, such as a formation of nucleus.

To be precisely correect, there is always a gradual

change from spinodal to nucleation mechanism.

Conc. Vs position

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3.1.2-Phase Field approach:

Let us consider a binary alloy of average composition Co occupying

the (2D) xy-plane. Let the alloy consist of two phases m and p,

that is, it is kept at a temperature (say T) that corresponds to the

two phase region in the phase diagram (Fig 3). We assume that

the temperature remains a constant, i.e., our present formulation

is an isothermal one.

Fig 3

The order parameter in this case is going to be the composition at

any point. The microstructure of the system will be completely

described by the composition field. Let the composition at any

point r in the xy-plane at time t be denoted by c(r, t).

Given an initial composition profile, say c(r,0),

the composition profile at any future time t can be obtained by

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solving the following (Cahn-Hilliard) non-linear diffusion

equation:

∂c /∂t = ∇. M∇μ (5)

where, M is the mobility, c is the (scaled) composition, t is the

time, and μ is the chemical potential, given by

μ = ∂F /∂c (6)

where ∂ /∂c denotes the variational derivative with respect to

composition, and F is the free energy functional.

F = ʃv [f ( c) + (Ɛ2c/2)*| ∇c|2] dv

or, F = ʃv [f ( c) + κ*| ∇c|2] dv (7)

where κ is the gradient energy coefficient, and f(c) is the bulk free

energy density, and is parameterized as a function of composition

as

f(c) = Ac2(1 - c)2 (8)

where A is a positive constant indicating the energy barrier

between the two equilibrium phases m and p (Fig 4)

Fig 4

Using the expressions (7) and (8) in the definition of the chemical

potential, we obtain

μ = h - 2κ∇2c (9)

where, h = ∂ f /∂c = 4Ac(1 - c)(1 - 2c) (10)

We assume the mobility M and the gradient energy coefficient κ to

be (scalar) constants: this amounts to assuming the interfacial

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energies and the diffusivities to be isotropic.

Using the Equation (9) above, and the fact that M is a constant,

we obtain the Cahn-Hilliard equation as follows:

∂c /∂t = M∇2(h - 2κ∇2c) (11)

3.1.3-Numerical Method to solve equation (11):

Since Cahn-Hilliard equation is nonlinear, it can only be solved

numerically through discretization in space and time. Due to its

simplicity and small memory requirement, most of the phase-field

simulations in the literature employed the explicit forward Euler

method in time and finite-difference in space. To maintain the

stability and to achieve high accuracy for the solutions, the time

step and spatial grid size have to be very small, which seriously

limits the system size and time duration of a simulation. The

ability to performing reliable long-time simulation

is critical in the fundamental understanding of the scaling

behavior of morphological pattern evolution.

In this report, we implement an accurate and

efficient semi-implicit Fourier-spectral method suggested by

L.Q.Chen, J.Shen for solving the phase-field equations. For the

time variable semi-implicit scheme is being employed. Thanks to

the exponential convergence of the Fourier-spectral discretization,

it requires a significantly smaller number of grid points to resolve

the solution to within a prescribed accuracy, say 1%. Moreover,

the semi-implicit Fourier spectral method is easy to implement

and can be extended for systems with position-dependent mobility.

Equation (11) is first non-dimensionalised and is then

discretised by employing Semi-Implicit Fourier-Spectral Method.

The discretised equation is:

Ĉ(k, t+Δt) = Ĉ(k, t) – ΔtMk2ĥ(k,t) (12) 1 + 2ΔtMk4 κ

In the above equation Ĉ and ĥ represent the fourier transform of

composition field and h field, where h is given by equation (10). k

= (k1, k2) is a vector in the Fourier space, k = (k12+k22)1/2 is the

magnitude of k. Other symbols have their usual meanings.

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Algorithm for microstructural evolution:

Given a composition profile at time t = 0, we calculate the h

and its Fourier transform as well as the Fourier transform of

C.

Using Ĉ and ĥ in equation (12), we calculate the composition

profile at some future time t + Δt.

The inverse Fourier transform of Ĉ (t+Δt) gives the

composition profile at time t + Δt.

Repeat the above three steps to march in time for the given

number of time steps.

A code was developed in MATLAB using the above mentioned

algorithm. Periodic boundary conditions were also used. The

MATLAB code is being provided in APPENDIX A. The inputs

needed for the simulation are as follows:

N, M - size of the mesh

dx, dy - distance between the nodes in x & y direction

dt - length of time step

timesteps - total number of timesteps

A - free energy barrier

Mob - Mobility

Kappa - gradient energy coefficient

C(N,M) - Initial composition field information

Note: At every node a very small noise is added to its

concentration value for starting the simulation. Because this noise

is going to imitate the „concentration wave‟ happening in the real

process. Only those changes (or evolutions) in concentration at the

nodes will „live‟ which decrease the value of free energy functional

equation (7). Hence the evolution of the composition profile will

occur.

3.1.4-Observations:

(1)-Effect of Δx, Δy and Δt on the stability

and accuracy of the solution obtained by the Semi-Implicit

Fourier Spectral Method to solve Cahn-Hilliard equation

for Spinodal Decomposition:

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All the results correspond to a double well free energy potential

function with minima at 0 and 1. And the value of initial

composition i.e c_0 = 0.5.

Effect of Δx and Δy with Δt = 1.0 - Keeping all the

thermodynamic parameters i.e A, kappa, mobility equal to one

and a 256x256 mesh size, it was observed that stable solution was

obtained in the range of Δx = Δy = [0.45, 2.5]. The accuracy of

solution increased as the value was increased from 1.0. The

increase in accuracy can be attributed to the fact that all the

numerical methods to solve a Partial Differential Equation suffer

inaccuracy and anomaly when the value of Δt is larger than Δx ( or

Δy) beyond a certain value.

Effect of Δt with and Δy = 1.0 - Keeping all the

thermodynamic parameters i.e A, kappa, mobility equal to one

and a 256x256 mesh size, it was observed that stable solution was

obtained with Δt as large as 6.0 for the problem being examined.

The accuracy of the solution was of course better with smaller

value of Δt. None of the earlier proposed method to solve the

corresponding discretized equations could be able to handle such a

large value of time step. This is surely one of the major

advantages of Semi-Implicit Fourier Spectral Method where the

emphasis may be on quantitative analysis rather than precise

accuracy.

However, to be on the safer side for all the discussions that

follow the values chosen are Δx = Δy = 1.0, Δt = 2.5.

(2)- Effect of Initial Concentration on the solvability of

Cahn-Hilliard equation for Spinodal Decomposition via

Semi-Implicit Fourier Spectral method for a double free

energy well potential function:

All the results correspond to a double well free energy potential

function with minima at 0 and 1.

When all the thermodynamic parameters were set to one,

the composition segregation occurred perfectly fine for initial

concentration value c_0 in the range (0.3, 0.7).

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However, when the initial composition value was chosen in

the range of (0, 0.3] or [0.7, 1), the composition segregation did not

occur and the concentration values at all points moved to the

nearest free energy double well potential minima in each case.

One possible explanation to this observation

can be that when the value of c_0 is too small (or too large) to

jump the free energy barrier, the composition segregation becomes

impossible to be achieved. As a solution to this problem, the value

of A was increased. It was observed that below c_0 = 0.3, value of

A=1.3(or greater) was able to show composition segregation.

However, the value of A has to be progressively increased as we

approach c_0 nearer to 0.0. The same holds true when the value of

c_0 is greater than or equal to 0.7

For all the discussions that follow, c_0 = 0.5 will be used in

computations.

(3) Effect of noise strength on the microstructure evolution

during Spinodal Decomposition:

Why the noise was introduced into the system?

Spinodal Decomposition occurs when in a thermodynamically

unstable initial state, long wavelength small amplitude stastical

fluctuations grow spontaneously in amplitude as time proceeds. To

simulate, these „concentration waves‟ (or simply concentration

fluctuations), randomness about average composition is

introduced by imparting a small noise to the system.

Case1 - Noise = 0.5*10-1 Case2 - Noise = 0.5* 10-2

t = 2.5 t = 2.5

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t = 62.5 t = 62.5

t = 150 t = 150

Conclusions drawn from the above results:

Before we proceed, we must make a point clear that a noise value

of 0.5*10-1 is quite large and is rarely experienced by any system,

however just for the sake of comparison we have chosen it against

the one that is having far more possibility of occurring.

t = 2.5 - As the value of noise in increased, after the first time

step, larger concentration segregation is observed in Case1. In

Case1 the range of concentration was [0.4854, 0.5149], whereas in

the Case2 the range was [0.498, 0.501].

t = 62.5 – The concentration segregation was greater in Case1

than Case2 as it can be clearly seen in the images shown above as

well. The area covered by the diffuse interface was also lesser for

the Case1 then Case2.

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t = 450 – At this time, the shape and size of the region occupied by

two phases in Case1 became more or less stable and the system

seemed to have attained the final state with well segregated

regions. However in Case2, the decomposition was not stabilized

and the coarsening of the regions would have continued had the

system been allowed more time to stand.

From the above discussion, few conclusions that can be

made are that with larger noise the segregation becomes easy and

that system attains equilibrium state relatively quickly. Moreover

at any time, the area covered by the diffuse interface in the

microstructure is lesser. For the proceeding discussions, the

value of noise strength will be 0.5*10-2 unless otherwise

stated.

(4) Effect of Mobility on the microstructure evolution

during Spinodal Decomposition.

Since Cahn-Hilliard equation is nothing but a modified form of

Fick‟s law for transient diffusion equation. Thus, the mobility (M)

in Cahn-Hilliard equation can be realized as a corresponding for

Diffusivity(D) in the diffusion equation. Thus, the value of M shall

make a difference in the time taken by the system to achieve

equilibrium state. The results obtained after solving the Cahn-

Hilliard equation with increasing values of M were in good

agreement as well.

The following images will show the state of the microstructures

with increasing values of M at t = 62.5.

M = 0.5 M = 1.0

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M = 3.0 M = 5.0

Conclusions :

The larger the value of M, the lesser the time is taken by the

system to move towards equilibrium. As it can be clearly seen

from the above images that coarsening of the precipitates is larger

for M = 5.0 than other values of M and that too at just t = 62.5.

Accuracy – It was observed that the accuracy of the results

suffered when the value of M was increased above 3. For the case

with M = 5, the concentration values at various nodes lies in the

range [-0.066, 1.1512]. This clearly shows that the concentration

values shooted up from their respective limits.

(5) Effect of thermodynamic barrier (A) on the

microstructure evolution during Spinodal Decomposition.

The change in the value of A led to some interesting results that

are summarized as below:

A = 0.25 - The concentration segregation could not be

achieved. However when a large noise of 0.05 was provided (which

is of course too large to occur in any physical example), the

composition segregation began to occur.

A = 0.50 – The concentration segregation could not be

achieved until 70 time steps and the concentration value at

all the points remained more or less around 0.50. However,

the segregation began to occur after 70 time steps and

achieved its equilibrium state after number of time steps.

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A = 1.0 - The segregation in this case occurred just fine

right from the start and achieved equilibrium after around 300

time steps.

A = 2.5 – The segregation in this case occurred in almost no

time. Actually it was so fast that that both the concentration

limits were exceeded and just after 30 time steps, the range of

concentration at different mesh nodes were [-0.4506, 1.4828]. This

made worm like fluctuations to occur inside the microstructure

(image shown below). Thus, much higher values of A are of course

going to affect the accuracy.

This completes our discussion on spinodal decomposition.

3.2 - ISOTHERMAL SOLIDIFICATION FOR SINGLE

COMPONENT WITH SURFACE ENTROPY ISOTROPY: For modeling crystal growth from an undercooled pure substance,

the system of variables consists of one pure and constant

component (c = 1), of the inner energy e, and of an order

parameter ϕ(x, t), called the phase-field variable. The value of

ϕ(x, t) characterizes the phase state of the system and its volume

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fraction in space at time t. In contrast to classical sharp interface

models, the interfaces are represented by thin diffuse regions in

which ϕ(x, t) smoothly varies between the values of φ associated

with the adjoining bulk phases. For a solid–liquid phase system, a

phase-field model may be scaled such that ϕ(x, t) = 1 characterizes

the region of the solid phase and ϕ(x, t) = 0 the region of the liquid

phase. The diffuse boundary layer, where 0< ϕ(x, t) < 1, and the

profile across the interface are schematically drawn in Fig. 5. The

darker region is liquid.

Fig. 5

Since the phase field variable in this case is non-conserved, the

Cahn-Allen equation will be solved. However, the reader must be

informed that the final differential equation will not be same as

equation (4) but instead it would be of the same form as (4). This

is because phase field modeling is too complex to generalize

equations for different processes.

The free energy functional is given by:

F(φ) =ʃv (f(φ) –(Ɛa(∇φ) +(w(φ)/Ɛ))) dv (13)

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This equation is similar to the free energy functional shown

previously. The second term in the integral (enclosed in the

brackets) reflects the thermodynamics of the interface Ɛ is a small

length scale parameter related to the thickness of the diffuse

interface.

FREE ENERGY FUNCTION: The free energy density uses two

functions: a double well function and an interpolating function.

Here we chose the two functions w(ϕ) = γ ϕ2(1 − ϕ)2 and

f(ϕ)= ( L(T – TM)/TM)* ϕ2(3−2 ϕ), respectively. This assumption will

make ∂(f+w)/∂ ϕ = 0 at both ϕ =0 and 1 for all temperatures as

shown in Fig. 6.

Fig. 6

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After applying the equation for the energy conservation, a phase

field equation Ɛτ∂ϕ/∂t = M*∂F/∂ϕ (where τ is kinetic mobility)

and a little bit of mathematics, the details of which are

intentionally avoided here, one arrives at the final partial

differential equation for the evolution of the phase field variable ϕ.

Ɛτ∂ϕ/∂t = ε∇ · a,∇ϕ(∇ϕ) - w,ϕ(ϕ)/Ɛ - f,ϕ(T,ϕ)/T (14)

where a,∇ϕ, w,ϕ, and f, ϕ denote the partial derivative with respect

to ∇ ϕ and ϕ, respectively.

f(T,ϕ) =( L(T – TM)/TM)* ϕ2(3 − 2ϕ) bulk free energy density

w(ϕ) = γ ϕ2(1 − ϕ)2 double well potential

a(∇ϕ) = γ ac2 (∇ ϕ)|∇ ϕ|2 gradient entropy density

where TM is the melting temperature and γ defines the surface

entropy density of the solid– liquid interface.

For the system of two phases, the gradient entropy

density reads a(∇ϕ) = γ|∇ϕ|2. The double well potential is w(ϕ) = γ

ϕ2(1 − ϕ)2. And f(T,ϕ) =m*ϕ2(3 − 2ϕ), where m is a constant bulk

energy density related to the driving force of the process, for

example, to the isothermal undercooling ΔT, for example, m =

m(ΔT). By utilizing the phase field equation and by inserting the

values of above mentioned term in equation (14), the modified

form of the partial differential equation becomes:

τƐ∂tϕ = Ɛ(2 γ) Δϕ − (18 γ (2ϕ3 − 3ϕ2 + ϕ)/Ɛ) − 6mϕ(1 −ϕ) (15)

Δϕ is the laplacian of the ϕ matrix.

3.2.1 – Numerical method to solve equation (15):

The most straightforward discretization of finite differences with

an explicit time marching scheme is used here by writing the

phase-field equation (15) in discrete form and by defining suitable

boundary conditions.

ϕi,jn+1 = ϕi,j +∂t/τ{2γ((ϕi+1,jn-2ϕi,j

n + ϕi-1,jn)/∂x2 + (ϕi,j+1

n- 2ϕi,jn+ ϕi,j1

n)/∂y2)

- A*18γ/Ɛ2(2(ϕi,jn)3-3)( ϕi,j

n)2+ ϕi,jn) – B*6m/Ɛ(ϕi,j

n(1- ϕi,jn))} (16)

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Here, A,B ∈ {0, 1} are introduced in order to switch on or off the

corresponding terms in the phase-field equation for our later case

study. Two types of boundary conditions are used in the

simulation.

Periodic boundary condition: A periodic boundary condition

mimics an infinite domain size with a periodicity in the structure.

The values of the boundary are set to the value of the neighboring

cell from the opposite side of the domain.

ϕ1,j = ϕNx-1,j , ϕNx,j = ϕ2,j with j = 1, . . . , Ny

ϕi,1 = ϕi,Ny-1 , ϕi,Ny = ϕi,2 with 1 = 1, . . . , Nx

Neumann Boundary condition: The component of ∇ϕ normal to

the domain boundary should be zero. In our rectangular domain,

this can be realized by copying the ϕ value of the neighboring

(interior) cell to the boundary cell:

ϕ1,j = ϕ2,j , ϕNx,j = ϕNx-1,j with j = 1, . . . , Ny

ϕi,1 = ϕi,2 , ϕi,Ny = ϕi,Ny-1 with 1 = 1, . . . , Nx

Stability Condition: To ensure the stability of the explicit

numerical method and to avoid generating oscillations, the

following condition for the time step ∂t depending on the spatial

discretizations ∂x and ∂y must be fulfilled:

∂t < (1/4γ)*(1/∂x2 + 1/∂y2)-1

A code in Matlab was developed to evolve the phase field variable

using equation (16) and suitable boundary conditions. The code is

being provided in APPENDIX B. The inputs needed for the

simulation are as follows:

N, M - size of the mesh

dx, dy - distance between the nodes in x & y direction

dt - length of time step

timesteps - total number of timesteps

p(N,M) - Initial phase field variable information

epsilon - thickness of the interface

m - driving force

Mob - Kinetic Mobility

Gamma - Surface isotropy density

A, B - coefficients to switch on or off the respective

. terms

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3.2.2 – Observations:

To investigate the phase-field equation, we switch on the potential

entropy contribution wϕ (ϕ) and the bulk driving force fϕ (ϕ) by

setting the coefficients A = 1 and B = 1. The colorscale indicates

ϕ = 1(solid) in yellow, ϕ = 0(liquid) in black, and the diffuse

interface region in varying colors.

(1)Diffuse Interface Thickness: A planar solid–liquid front is

placed in the center of the domain at Nx/2 with a sharp interface

profile, with zero driving force m = 0 and with Neumann boundary

conditions on each side. The effect of different values of the small

length scale parameter: Ɛ = 1 and Ɛ = 10 responsible for the

thickness of the diffuse interface is shown in Fig. 7a, Fig 7.b

Fig 7.a

Fig 7.b

(2)Driving Force: As a next configuration, the three simulations

shown in Fig. 8a–c were performed with Ɛ = 1 with different

values of the driving force and with the initial configuration of

Fig. (a). For m = 0, the initial planar front remains stable, for

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m = −1 the solid phase (yellow color) grows, whereas for m = 1 the

solid phase shrinks.

Fig. 8a

Fig. 8b

Fig. 8c

Fig. 8d

(3)Phase-Field Simulation of growing nucleus: For the

simulation in Fig. 9a, a solid nucleus is set in a 2D domain of

Nx×Ny = 100×100 grid points, with A = 1, B = 1, with driving force

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m = −2.5 and with Neumann boundary condition. Due to periodic

boundary conditions in Fig. 9b the particle grows across the lower

boundary and appears at the top boundary.

Fig. 9a

Fig. 9b

3.3 – DEDRITIC SOLIDIFICATION:

Dendrites are formed when surface anisotropy is included in the

system, which means that now there are going to be some

preferred directions for solidification. Though there are many

models to simulate dendritic growth in the literature but in the

present report, a phase field model suggested by Ryo Kobayashi is

being studied.

3.3.1 – Phase Field Model:

The model includes two variables; one is a phase field ϕ(r, t) and

the other is a temperature field T(r, t). The variable ϕ(r, t) is an

ordering parameter at the position r and the time t, ϕ = 0 means

being liquid and ϕ = 1 solid. And the solid/liquid interface is

expressed by the steep layer of ϕ connecting the values 0 and 1.

Fig. 5 shows how the shape of crystal is described by the phase

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field ϕ. In order to keep the profile of ϕ such form and to move it

reasonably, we consider the following Ginzburg-Landau type free

energy functional F similar to equation (2) including m as a

parameter:

F = ʃv [f (ϕ, m) + (Ɛ2ϕ/2)*| ∇ϕ|2] dv (17)

where Ɛ is a small parameter which determines the thickness of

the layer. It is a microscopic interaction length and it also controls

the mobility of the interface. f is a double-well potential which has

local minimums at ϕ = 0 and 1 for each m. Here we take the

specific form of f as follows:

f(ϕ, m) = 1/4ϕ4 - (1/2 – 1/3*m)ϕ3 + (1/4 -1/2*m)ϕ2 (18)

Anisotropy can be introduced by assuming that Ɛ depends on the

direction of the outer normal vector at the interface. So Ɛ is

represented as a function of the vector v = vi satisfying

Ɛ(λ,v) = Ɛ(v) for λ>0. The outer normal vector is represented by -∇ϕ at the interface. Thus, we consider:

F = ʃv [f (ϕ, m) + (Ɛ(-∇ϕ )2/2)*| ∇ϕ|2] dv (19)

From the formula τ ∂ϕ/∂t = ∂f/ϕ and further simplifying, we have

the following evolution equation:

τ ∂ϕ/∂t = -∂/∂x(ƐƐ’∂ϕ/∂y)+ ∂/∂y(ƐƐ’∂ϕ/∂x) + ∇.(Ɛ2∇ϕ) + ϕ(1-ϕ)(ϕ-0.5+m)

(20)

where τ is a small positive constant and ∂Ɛ/∂v=(∂Ɛ/∂vi)i. The

parameter m gives a thermodynamical driving force. Especially in

two dimensional space, we can take Ɛ = Ɛ(θ) where θ is an angle

between v and a certain direction (for example the positive

direction of the x-axis). Ɛ’ means derivative with respect to θ.

Equation (20) gives the evolution of the order parameter or phase

field variable ϕ with time.

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Here we assume that m is a function of the temperature T, for

example, m(T) = γ(Te - T) where Te is an equilibrium temperature,

which means that the driving force of interfacial motion is

proportional to the supercooling there. But in the following

simulations, we used the form m(T)=(α -l[γ(Te - T)] where α

and γ are positive constants; α < 1, since this assures |m(T)| < ½

for all values of T. Also m(T) is almost linear for T near Te. To

take anisotropy into account, let us specify Ɛ to be:

Ɛ = 1 + δcos(j(θ-θo)) (21)

The parameter δ means the strength of anisotropy and j is a mode

number of anisotropy. Side branching can be stimulated in the

dendrites by adding a small random noise in the equation (20).

The noise can be of the form aϕ(1-ϕ)x, where a is the strength of

noise and x is a random number in the range[-0.5,0.5].

The equation for T is derived from the conservation

law of enthalpy as:

∂T/∂t = ∇2T + K∂p/∂t (22)

T is non-dimensionalized so that the characteristic cooling

temperature is 0 and the equilibrium temperature is 1. K is a

dimensionless latent heat which is proportional to the latent heat

and inversely proportional to the strength of the cooling. For

simplicity, the diffusion constant is set to be identical in both of

solid and liquid regions. (22) is a heat conduction equation having

a heat source along the moving interface, since K∂p/∂t has

non-zero value only when the interface passes through the point.

3.3.2 – Numerical method to simulate the Dendritic growth:

The simplest finite difference scheme with a nine point laplacian

is used to solve equations (20) and (22). First, the new value of

phase field variable is calculated to substitute it into the

temperature field to get the new value of temperature field. Nine

point laplacian is required for the stability of the solution.

A code in Matlab was developed to evolve the phase

field variable and temperature field using equations (20) and (22)

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respectively and periodic boundary conditions. The code is

attached along with this report in the CD-Drive. The inputs

needed for the simulation are as follows:

N, M - size of the mesh

dx, dy - distance between the nodes in x & y direction

dt - length of time step

timesteps - total number of timesteps

p(N,M) - initial phase field variable information

T(N,M) - initial temperature field information

K - latent Heat

TAU - phase field relaxation time

EPS - interfacial width

DELTA - strength of anisotropy

ANISO - mode number of anisotropy

ALPHA - positive constant

GAMMA - positive constant

3.3.2 – Observations:

The code written in Matlab was run using the following inputs:

N= 500; M=500; dx=0.03; dy=0.03;dt=0.0003; K=4; TAU=0.0003,

EPS= 0.01; GAMMA=10.0; DELTA=0.02; ANISO=4.0;ALPHA=0.9;

Noise was not added to the system. The following was the

structure of dendrites obtained.

The yellow region indicates solid, black region indicates liquid

while the rest in indicated by the interface. Notice that there is

directional solidification in four directions because ANISO = 4.0 .

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Effect of latent heat of solidification K:

The below shown four images correspond to a dimensionless latent

heat K =0.8, 1.0, 1.2 and 1.8 from left to right and top to bottom.

Notice that as the value of latent heat is increased, the dendritic

structure becomes much defined because release of latent heat is

the driving force for the process. Also noise is added to stimulate

side branching.

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(4) CONCLUSION Phase-field models have been successfully applied to various

materials processes including solidification, solid-state phase

transformations, coarsening, and growth. With the phase-field

approach, one can deal with the evolution of arbitrary

morphologies and complex microstructures without explicitly

tracking the positions of interfaces. This approach can describe

different processes such as phase transformations and particle

coarsening within the same formulation, and it is rather

straightfoward to incorporate the effect of coherency and applied

stresses, as well as electrical and magnetic fields. Efforts are

being made in Material Science to focus on the exploration of

novel applications of the phase-field method to various materials

problems, e.g., problems involving simultaneous long-range elastic

and electric or magnetic dipole-dipole interactions, low-

dimensional systems such as thin films and multilayer structures,

and interactions between phase and defect microstructures such

as random defects and dislocations. There will also be increasing

efforts in establishing schemes to obtain the phase-field

parameters directly from more fundamental first-principles

electronic structure or atomic calculations. For practical

applications, significant additional efforts are required to develop

approaches for connecting phase-field models with existing or

future thermodynamic, kinetic, and crystallographic databases.

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APPENDIX A MATLAB code for spinodal decomposition:

%Numerical solution of Cahn-

%Hilliard Equation via Semi-

%implicit Fourier Spectral %method for conserved

%quantities. %Non-dimensionalisation is done

%and isothermal conditions are

%considered

%For further details, refer to

%the research paper"Applications

%of semi-implicit Fourier-

%spectral method to phase field

%equations"-L.Q. Chen, Jie Shen

clear clc format long

%spatial dimensions -- adjust N

%and M to increase or decrease %the size of the computed

%solution.

N =256; M = 256; del_x = 1.0; del_y = 1.0;

%time parameters -- adjust ntmax

%to take more time steps, and %del_t to take longer time

%steps. del_t = 2.5; ntmax = 500;

%thermodynamic parameters A = 1.0; Mob = 1.0; kappa = 1.0;

%initial composition and noise

%strenght information c_0 = 0.50; noise_str = 0.5*(10^-2);

%composition used in

%calculations with a noise for i = 1:N for j = 1:M

comp(j+M*(i-1)) = c_0 +

noise_str*(0.5-rand); end end

%The half_N and half_M are

%needed for imposing the

%periodic boundary conditions half_N = N/2; half_M = M/2;

del_kx = (2.0*pi)/(N*del_x); del_ky = (2.0*pi)/(M*del_y);

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

%surface aniostropy.

% For more details, refer to

%chapter 7-Phase Field

%Modelling,Britta Nestler % -Computational Materials

%Engineering, Bernaad Jansenns.

clear clc %input parameters epsilon = 1.0; %thickness

%of the interface gamma = 1.0; %surface

%anistropy density m = -2.5; %driving

%force Mob = 1.0; %kinetic

%mobility Nx = 100; Ny = 100; %mesh size dx = 0.1; dy = 0.1; %distance

%between two consecutive nodes dt = 0.001; %length of

%time step timesteps = 100; %total

%number of time steps A = 1.0; %A and B

%are the coefiecients to switch

B = 1.0; %on and

%off the corresponding terms.

%Lets generate the initial

%picture p=zeros(Nx,Ny); for i1=Nx/2-10:Nx/2+10 for i2=Ny/2-10:Ny/2+10 p(i1,i2)=1; end end

%evolution of the order

%parameter for index=1:timesteps Lap = (4*del2(p))/(dx^2);

%Laplacian of the matrix p

for i1=1:Nx for i2=1:Ny p(i1,i2) = p(i1,i2) +

(dt/Mob)*((2*gamma*Lap(i1,i2))-

A*(18*gamma*((2*(p(i1,i2)^3))-

(3*(p(i1,i2)^2))+p(i1,i2))/(epsi

lon^2))-B*(6*m*(p(i1,i2)*(1-

p(i1,i2)))/epsilon)); end end %periodic boundary condition for i1=1:Nx p(i1,1) = p(i1,Ny-1); p(i1,Ny)= p(i1,2);

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end for i2=1:Ny p(1,i2) = p(Nx-1,i2); p(Nx,i2)= p(2,i2); end

%neumann boundary condition %for i1=1:Nx % p(i1,1) = p(i1,2); % p(i1,Ny)= p(i1,Ny-1); %end %for i2=1:Ny % p(1,i2) = p(2,i2); % p(Nx,i2)= p(Nx-1,i2); %end %visualization of the output

disp(p) figure(1) image(p*50) colormap('hot')

end

APPENDIX C MATLAB code for dendritic solidification based on Kobayashi‟s

Model: %Phase Field Modelling of

%dendritic growth as suggested

%by Ryo Kobayashi %For more details refer to

%"Modeling and numerical

%simulationsof dendritic crystal

%growth-Ryo Kobayashi

clear clc

K = 4.0; %Latent

%heat TAU =0.0003; %PF

%relaxation time EPS =0.01; %interfacial

%width DELTA= 0.02; %modulation

%of the interfacial width ANGLEO =0.0; %orientation

%of the anisotropy axis ANISO= 4.0; %anisotropy

%2*PI/ANISO ALPHA =0.9; %m(T) =

%ALPHA/PI * atan(GAMMA*(TEQ-T)) GAMMA= 10.0; TEQ =1.0; %melting

%temperature NX= 500; %size of the

%mesh NX*NY NY =500;

H= 0.03; %spatial

%resolution DT =0.0003; %temporal

%resolution timesteps =2000000; %number of

%time steps pi =3.14159265358;

%intial temperature and phase

%field information T = zeros(NY,NX); p = zeros(NY,NX); for i1=1:NY for i2=1:NX if ((i1-NY/2)*(i1-

NY/2)+(i2-NX/2)*(i2-NX/2)<100) p(i1,i2) = 1.0; else p(i1,i2) = 0.0; end end end

%preallocation of matrices for

%faster calculations grad_p_X = zeros(NY,NX); grad_p_Y = zeros(NY,NX); aX = zeros(NY,NX); aY = zeros(NY,NX); eps2 = zeros(NY,NX);

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angle = zeros(NY,NX); epsilon_prime = zeros(NY,NX); epsilon = zeros(NY,NX); dXdY = zeros(NY,NX); dYdX = zeros(NY,NX); grad_eps2_X = zeros(NY,NX); grad_eps2_Y = zeros(NY,NX); lap_p = zeros(NY,NX); lap_T = zeros(NY,NX);

for index=1:timesteps %calculation of all the relevant

%matrices for i1=1:NY for i2=1:NX ip = mod(i2,NX)+1; im = mod((NX+i2-

2),NX)+1; jp = mod(i1,NY)+1; jm = mod((NY+i1-

2),NY)+1; grad_p_X(i1,i2) =

((p(i1,ip) - p(i1,im))/H); grad_p_Y(i1,i2) =

((p(jp,i2) - p(jm,i2))/H); lap_p(i1,i2) =

(2.0*(p(i1,ip)+p(i1,im)+p(jp,i2)

+p(jm,i2))+p(jp,ip)+p(jm,im)+p(j

p,im)+p(jm,ip)-

12.0*p(i1,i2))/(3.0*H*H); lap_T(i1,i2) =

(2.0*(T(i1,ip)+T(i1,im)+T(jp,i2)

+T(jm,i2))+T(jp,ip)+T(jm,im)+T(j

p,im)+T(jm,ip)-

12.0*T(i1,i2))/(3.0*H*H); end end

for i1 = 1:NY for i2=1:NX if (grad_p_X(i1,i2)==0.0

&& grad_p_Y(i1,i2)>0.0) angle(i1,i2) =

0.5*pi; end if (grad_p_X(i1,i2)==0.0

&& grad_p_Y(i1,i2)<=0.0) angle(i1,i2) = -

0.5*pi; end

if (grad_p_X(i1,i2)>0.0

&& grad_p_Y(i1,i2)>0.0)

angle(i1,i2) =

atan(grad_p_Y(i1,i2)/grad_p_X(i1

,i2)); end

if (grad_p_X(i1,i2)>0.0

&& grad_p_Y(i1,i2)<=0.0) angle(i1,i2) = 2.0*pi

+

atan(grad_p_Y(i1,i2)/grad_p_X(i1

,i2)); end

if(grad_p_X(i1,i2)<0.0) angle(i1,i2) = pi +

atan(grad_p_Y(i1,i2)/grad_p_X(i1

,i2)); end end end

for i1 =1:NY for i2=1:NX epsilon(i1,i2) =

EPS*(1.0 +

DELTA*cos(ANISO*(angle(i1,i2)-

ANGLEO))); epsilon_prime(i1,i2) = -

EPS*ANISO*DELTA*sin(ANISO*(angle

(i1,i2)-ANGLEO)); end end

for i1 = 1:NY for i2=1:NX aY(i1,i2) = -

epsilon(i1,i2)*epsilon_prime(i1,

i2) * grad_p_Y(i1,i2); aX(i1,i2) =

epsilon(i1,i2)*epsilon_prime(i1,

i2) * grad_p_X(i1,i2); eps2(i1,i2) =

epsilon(i1,i2)*epsilon(i1,i2); end end

for i1=1:NY for i2=1:NX ip = mod(i2,NX)+1; im = mod((NX+i2-

2),NX)+1; jp = mod(i1,NY)+1; jm = mod((NY+i1-

2),NY)+1; dXdY(i1,i2) = (aY(i1,ip)

- aY(i1,im))/H;

[Pick the date]

2

dYdX(i1,i2) = (aX(jp,i2)

- aX(jm,i2))/H; grad_eps2_X(i1,i2) =

(eps2(i1,ip) - eps2(i1,im))/H; grad_eps2_Y(i1,i2) =

(eps2(jp,i2) - eps2(jm,i2))/H; end end

for i1=1:NY for i2=1:NX po = p(i1,i2); m = (ALPHA/pi) *

atan(GAMMA*(TEQ-T(i1,i2))); scal=

grad_eps2_X(i1,i2)*grad_p_X(i1,i

2)+

grad_eps2_Y(i1,i2)*grad_p_Y(i1,i

2); %evolution of the phase field

variable

p(i1,i2) =

p(i1,i2)+((dXdY(i1,i2)+dYdX(i1,i

2) + eps2(i1,i2)*lap_p(i1,i2)+

scal + po*(1.0-po)*(po-

0.5+m))*DT/TAU); %evolution of temperature field T(i1,i2) =

T(i1,i2)+(lap_T(i1,i2)*DT) +

(K*(p(i1,i2) - po)); end end

%visualization of the output disp(p) figure(1) image(p*50) colormap('hot')

end