**3.4 X-ray absorption spectroscopy (XAS) investigation of Ce75Al25 − xGax alloys**

**Figure 5** shows Ce L3 edge XAS spectra as a function of addition of Ga in Ce75Al25 alloy. The spectrum exhibited by Ce75Al25 alloy is having only 4*f* 1 component that gives a pure localized 4*f* 1 configuration. It can be seen that in the XAS spectra of Ce75Al25, the signature of 4*f* 0 electron is not present. The postedge feature

#### **Figure 4.**

*Energy dispersive spectra of the melt-spun Ce75Al25 − xGax alloys for (a) x = 0, (b) x = 0.5, (c) x = 1, and (d) x = 4 alloys (reprinted with kind permission from Reference [25], copyright 2016, Elsevier).*

**51**

Ga, 4*f* 1

**Table 1.**

*from Ref. [25], copyright 2016, Elsevier).*

represented by 4*f*

*0 and 4*f *1*

*[25], copyright 2016, Elsevier).*

**Figure 5.**

*points out the 4*f

0

addition, the delocalization of 4*f*

electron at 10 eV is higher than that of 4*f*

 *electronic states of Ce. The signature of 4f0*

1

explained on the basis of partial delocalization of 4*f*

stitution. The intensity increases with increase in the concentration of Ga. The XAS spectra are found to be in conformity with the completely itinerant state as available previous data in calculations and experiments of crystalline γ → α Ce transition and high pressure-induced polyamorphism by earlier workers [11, 43]. Thus, due to Ga

*In situ Ce L3-edge XAS spectra of Ce75Al25 − xGax metallic glass with x = 0, x = 2, x = 4, and x = 6. The arrow* 

*Upper inset shows the excursion of trivalent to tetravalent state (reprinted with kind permission from Reference* 

place. The current observation is also similar to the observation of chemical pressure effect made by Rueff et al. [55]. Based on this, it can be said that in the presence of

we discuss how the phase separation occurred in Ce75Al25 − xGax alloy due to the change in electronic structure of Ce. The XRD, TEM, and XAS observations can be

Thus, the short range ordering with Ce having localized and delocalized electrons

electrons are getting delocalized because of chemical pressure effect. Here

1

configuration of Ce in Ce75Al25 − xGax has taken

1

electron after Ga sub-

 *indicates delocalization of 4*f *electrons.* 

electron due to Ga substitution.

*Phase Separation in Ce-Based Metallic Glasses DOI: http://dx.doi.org/10.5772/intechopen.88028*

**S. No. x Nominal composition Average EDX composition\*** 0 Ce75Al25 Ce74.8 ± 1.5Al25.0 ± 0.8 0.1 Ce75Al24.9Ga0.1 Ce74.8 ± 1.5Al24.9 ± 1.7Ga0.1 ± 0.1 0.5 Ce75Al24.5Ga0.5 Ce75.1 ± 3.0Al24.2 ± 3.0Ga0.7 ± 0.3 1.0 Ce75Al24.0Ga1.0 Ce74.5 ± 1.7Al24.3 ± 0.9Ga1.2 ± 0.9 2.0 Ce75.0Al25.0Ga2.0 Ce74.2 ± 2.0Al23.7 ± 2.2Ga2.0 ± 1.3 4.0 Ce75.0Al21.0Ga4.0 Ce74.9 ± 2.0Al20.9 ± 1.7Ga4.2 ± 1.0 6.0 Ce75.0Al19.0Ga6.0 Ce75.0 ± 1.9Al19.2 ± 1.3Ga5.9 ± 1.7 *\*It can be seen that percentage error is higher for Ga. The reason behind this is there was variation in Ga while going from one area to another in the samples. The deviation in Ga is all calculated based on 4–6 readings for a given alloy.*

*Energy-dispersive spectra of the melt-spun Ce75Al25 − xGax (0* ≤ *x* ≤ *6) alloy (reprinted with kind permission* 

*Phase Separation in Ce-Based Metallic Glasses DOI: http://dx.doi.org/10.5772/intechopen.88028*


*\*It can be seen that percentage error is higher for Ga. The reason behind this is there was variation in Ga while going from one area to another in the samples. The deviation in Ga is all calculated based on 4–6 readings for a given alloy.*

#### **Table 1.**

*Metallic Glasses*

**X-ray analysis**

gives a pure localized 4*f*

of Ce75Al25, the signature of 4*f*

also show two diffuse halos from the matrix of one amorphous phase and dispersed (secondary) amorphous phase. The clear variation in the microstructure (**Figure 3**) due to Ga addition can be seen. However, in the XRD patterns, not much variation in the intensities of two humps is found. It can be said that the two humps are due to the presence of two types of "short range order" in coexisting amorphous phases.

**3.3 Compositional analysis of Ce75Al25 − xGax alloys through energy-dispersive** 

The EDX spectra of Ce75Al25 − xGax alloys (x = 0, 0.5, 1, and 4) are shown in **Figure 4(a–d)**. **Table 1** represents the average and nominal composition variations for the alloys with x = 0–6. The deviation reported is on the basis of measurements taken from four to six regions of the sample. The percentage experimental error in the case of Ga is found to be highest. The analysis shows Ga is responsible for contrast variation because of two kinds of amorphous domains in Ce75Al25 − xGax alloys. Within the traceable limit of EDX, the presence of silicon (Si) could not be found. Because of very fine droplet-like features (<7 nm), it is not possible to characterize the variation of Ga in amorphous matrix as well as droplet-like structure. For compositional analysis in TEM, the probe size is ~50 μm at magnification of 13.5 k.

**3.4 X-ray absorption spectroscopy (XAS) investigation of Ce75Al25 − xGax alloys**

*Energy dispersive spectra of the melt-spun Ce75Al25 − xGax alloys for (a) x = 0, (b) x = 0.5, (c) x = 1, and (d) x = 4 alloys (reprinted with kind permission from Reference [25], copyright 2016, Elsevier).*

**Figure 5** shows Ce L3 edge XAS spectra as a function of addition of Ga in Ce75Al25

configuration. It can be seen that in the XAS spectra

electron is not present. The postedge feature

1

component that

That's why only nominal and average composition of Ga is shown.

alloy. The spectrum exhibited by Ce75Al25 alloy is having only 4*f*

0

1

**50**

**Figure 4.**

*Energy-dispersive spectra of the melt-spun Ce75Al25 − xGax (0* ≤ *x* ≤ *6) alloy (reprinted with kind permission from Ref. [25], copyright 2016, Elsevier).*

#### **Figure 5.**

*In situ Ce L3-edge XAS spectra of Ce75Al25 − xGax metallic glass with x = 0, x = 2, x = 4, and x = 6. The arrow points out the 4*f *0 and 4*f *1 electronic states of Ce. The signature of 4f0 indicates delocalization of 4*f *electrons. Upper inset shows the excursion of trivalent to tetravalent state (reprinted with kind permission from Reference [25], copyright 2016, Elsevier).*

represented by 4*f* 0 electron at 10 eV is higher than that of 4*f* 1 electron after Ga substitution. The intensity increases with increase in the concentration of Ga. The XAS spectra are found to be in conformity with the completely itinerant state as available previous data in calculations and experiments of crystalline γ → α Ce transition and high pressure-induced polyamorphism by earlier workers [11, 43]. Thus, due to Ga addition, the delocalization of 4*f* 1 configuration of Ce in Ce75Al25 − xGax has taken place. The current observation is also similar to the observation of chemical pressure effect made by Rueff et al. [55]. Based on this, it can be said that in the presence of Ga, 4*f* 1 electrons are getting delocalized because of chemical pressure effect. Here we discuss how the phase separation occurred in Ce75Al25 − xGax alloy due to the change in electronic structure of Ce. The XRD, TEM, and XAS observations can be explained on the basis of partial delocalization of 4*f* 1 electron due to Ga substitution. Thus, the short range ordering with Ce having localized and delocalized electrons

will be different. The short range ordering of amorphous phase of Ce with localized 4*f* 1 electron (with Al and Ga) will be the same as that of pristine Ce75Al25 composition. In recent years, the analysis of atomic level structure of amorphous alloys has been done in terms of Kasper polyhedron built up of local packing of atoms [56, 57]. In terms of topology and coordination number (CN), many types of local coordination polyhedra are not geometrically the same for each MG. They are considered to be quasi-equivalent for a given glass. The topology and coordination number of cluster-like units will change in the presence and absence of 4*f* 0 delocalized electrons in Ce75Al25 − xGax alloys. The amorphous state containing Ce with localized 4*f* 1 electrons along with Al will have short range ordering like Ce75Al25 composition, while the other amorphous state containing Ce with delocalized 4*f* 0 electron (along with Al and Ga) will have different SRO. Because of the presence of both types of amorphous phases, the diffuse peak in the XRD may be shifted. They refer to the volume collapse of Ce atoms due to delocalization of 4*f* electrons (the shorter the effective atomic radii of Ce atoms) as well as change in the SRO. The two effects must be the main reason in the Ga-rich-dispersed amorphous domain. Also, the weak peak detected in XAS at ~5732 eV may be due to the excursion of 4*f* electrons leading to transformation from trivalent to tetravalent states of Ce atoms [58]. Thus, it may be concluded that the substitution of Ga changes the chemical environment and its valence states from trivalent to tetravalent states are altered by Ce.

As discussed above, the 4*f* electrons in some of Ce atoms are delocalized due to Ga substitution in Ce75Al25 alloys. Hence, glassy Ce75Al25 − xGax may exhibit two types of SRO. The Ce atoms having 4*f* 1 localized electron will have pristine SRO in the alloy without Ga. The Ce atoms with 4*f* 0 electrons may have a different type of cluster-like units with Al and Ga, and these are arranged differently in 3D space. Based on this model, one can understand the presence and formation of two coexisting amorphous phases which are simultaneously present in this alloy. It may be emphasized that the volume collapse resulting due to shrinkage of effective atomic radii of Ce atoms and delocalization of 4*f* 1 electron of Ce may not be the sufficient reason for the formation of new peak around 44° in XRD since the shift in angle will be less than the observed value. The new type of cluster units are formed because of the delocalization of Ce 4*f* electrons, and their arrangements in 3D space will make such a change in the angle value in XRD corresponding to second amorphous phase.

### **3.5 Plausible mechanism for phase separation in Ce75Al25 − xGax alloys**

A schematic diagram of effective atomic radii of Ce atoms in Ce75Al25 − xGax alloys to understand the effect of 4*f* electron is shown in **Figure 6**. For x = 0, Ce atoms are having localized 4*f* electrons, while for the alloy with x = 4, the partial delocalization of 4*f* electrons has taken place. Because of delocalization of 4*f* electrons, the effective atomic radius of Ce atoms decreases.

The partial delocalization of 4*f* electrons has led to decrement of week Ce-Ce bonds among the neighboring atoms and intercluster Ce-Ce bonds causing the considerable shrinkage and distortion of the clusters. Thus, the densification nature of certain clusters has increased (as shown on the right side of **Figure 6**). Subsequently, the alloy with delocalized 4*f* electrons of Ce atoms may form two kinds of density clusters which are low-density clusters (LDC) with localized 4*f* electrons and highdensity clusters (HDC) with delocalized 4*f* electrons for Ce atoms. The nanoamorphous domains with different SRO are formed due to the presence of two types of density clusters in alloy with x = 4. The formation of two types of amorphous domains due to Ga substitution and its link with 4*f* electrons offers a fascinating opportunity to investigate the microstructural effect on the various properties as glass-forming ability and mechanical and transport properties of Ce75Al25 − xGax alloys.

**53**

eter with time:

*Phase Separation in Ce-Based Metallic Glasses DOI: http://dx.doi.org/10.5772/intechopen.88028*

**using MATLAB**

**Figure 6.**

for spinodal decomposition).

Cahn-Hilliard equation

expression of a free energy functional is shown below:

*kind permission from Reference [26], copyright 2016, Elsevier).*

**4. Understanding of microstructural evolution due to phase separation** 

*The effective atomic radii of Ce atoms showing low-density cluster (LDC) and high-density cluster (HDC) with (a) localized 4*f *electrons and (b) delocalized 4*f *electrons for Ce75Al25 − xGax MGs (right side). The Ce atoms with localized 4*f *electrons are shown by red balls, the Ce atoms with delocalized 4*f *electron state are shown by the medium-sized green balls, and the smallest blue ball represents the Al/Ga atoms (reprinted with* 

A phase field modeling of the microstructure based on Cahn-Hilliard equation has been carried out in order to understand the nature of microstructure evolution due to Ga substitution in Ce75Al25 − xGax amorphous alloy [59]. The isotropic properties applicable for phase separation glasses as well as polymers at different length scales are shown by numerical simulation model. "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

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

F = ʃ*v* [*f* (ϕ,*c*,*T*) + (Ɛ2c/2) ∗| ∇c|2 + (Ɛ2ϕ/2) ∗| ∇ϕ|2] dv.

parameters, and the second one is for non-conserved order parameters.

The first term in the left-hand side of the equation is a free energy density of the bulk phase as a function of concentration, order parameter, and temperature. The second and the third terms 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), one arrives at two kinds of equation. The first one is for conserved order

The Cahn-Hilliard equation gives the rate of change of conserved order param-

*Phase Separation in Ce-Based Metallic Glasses DOI: http://dx.doi.org/10.5772/intechopen.88028*

#### **Figure 6.**

*Metallic Glasses*

4*f* 1

will be different. The short range ordering of amorphous phase of Ce with localized

 electron (with Al and Ga) will be the same as that of pristine Ce75Al25 composition. In recent years, the analysis of atomic level structure of amorphous alloys has been done in terms of Kasper polyhedron built up of local packing of atoms [56, 57]. In terms of topology and coordination number (CN), many types of local coordination polyhedra are not geometrically the same for each MG. They are considered to be quasi-equivalent for a given glass. The topology and coordination number of

0

0

localized electron will have pristine SRO in

electrons may have a different type of

electron of Ce may not be the sufficient

in XRD since the shift in angle will

delocalized electrons

electron (along with Al

1 elec-

cluster-like units will change in the presence and absence of 4*f*

the other amorphous state containing Ce with delocalized 4*f*

from trivalent to tetravalent states are altered by Ce.

types of SRO. The Ce atoms having 4*f*

the alloy without Ga. The Ce atoms with 4*f*

radii of Ce atoms and delocalization of 4*f*

reason for the formation of new peak around 44°

in Ce75Al25 − xGax alloys. The amorphous state containing Ce with localized 4*f*

trons along with Al will have short range ordering like Ce75Al25 composition, while

and Ga) will have different SRO. Because of the presence of both types of amorphous phases, the diffuse peak in the XRD may be shifted. They refer to the volume collapse of Ce atoms due to delocalization of 4*f* electrons (the shorter the effective atomic radii of Ce atoms) as well as change in the SRO. The two effects must be the main reason in the Ga-rich-dispersed amorphous domain. Also, the weak peak detected in XAS at ~5732 eV may be due to the excursion of 4*f* electrons leading to transformation from trivalent to tetravalent states of Ce atoms [58]. Thus, it may be concluded that the substitution of Ga changes the chemical environment and its valence states

As discussed above, the 4*f* electrons in some of Ce atoms are delocalized due to Ga substitution in Ce75Al25 alloys. Hence, glassy Ce75Al25 − xGax may exhibit two

cluster-like units with Al and Ga, and these are arranged differently in 3D space. Based on this model, one can understand the presence and formation of two coexisting amorphous phases which are simultaneously present in this alloy. It may be emphasized that the volume collapse resulting due to shrinkage of effective atomic

1

be less than the observed value. The new type of cluster units are formed because of the delocalization of Ce 4*f* electrons, and their arrangements in 3D space will make such a change in the angle value in XRD corresponding to second amorphous phase.

A schematic diagram of effective atomic radii of Ce atoms in Ce75Al25 − xGax alloys to understand the effect of 4*f* electron is shown in **Figure 6**. For x = 0, Ce atoms are having localized 4*f* electrons, while for the alloy with x = 4, the partial delocalization of 4*f* electrons has taken place. Because of delocalization of 4*f*

The partial delocalization of 4*f* electrons has led to decrement of week Ce-Ce bonds among the neighboring atoms and intercluster Ce-Ce bonds causing the considerable shrinkage and distortion of the clusters. Thus, the densification nature of certain clusters has increased (as shown on the right side of **Figure 6**). Subsequently, the alloy with delocalized 4*f* electrons of Ce atoms may form two kinds of density clusters which are low-density clusters (LDC) with localized 4*f* electrons and highdensity clusters (HDC) with delocalized 4*f* electrons for Ce atoms. The nanoamorphous domains with different SRO are formed due to the presence of two types of density clusters in alloy with x = 4. The formation of two types of amorphous domains due to Ga substitution and its link with 4*f* electrons offers a fascinating opportunity to investigate the microstructural effect on the various properties as glass-forming

**3.5 Plausible mechanism for phase separation in Ce75Al25 − xGax alloys**

ability and mechanical and transport properties of Ce75Al25 − xGax alloys.

electrons, the effective atomic radius of Ce atoms decreases.

0

1

**52**

*The effective atomic radii of Ce atoms showing low-density cluster (LDC) and high-density cluster (HDC) with (a) localized 4*f *electrons and (b) delocalized 4*f *electrons for Ce75Al25 − xGax MGs (right side). The Ce atoms with localized 4*f *electrons are shown by red balls, the Ce atoms with delocalized 4*f *electron state are shown by the medium-sized green balls, and the smallest blue ball represents the Al/Ga atoms (reprinted with kind permission from Reference [26], copyright 2016, Elsevier).*

## **4. Understanding of microstructural evolution due to phase separation using MATLAB**

A phase field modeling of the microstructure based on Cahn-Hilliard equation has been carried out in order to understand the nature of microstructure evolution due to Ga substitution in Ce75Al25 − xGax amorphous alloy [59]. The isotropic properties applicable for phase separation glasses as well as polymers at different length scales are shown by numerical simulation model. "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:

$$\mathbf{F} = \mathfrak{f}\nu \left[ f \left( \phi, c, T \right) + \left( \mathfrak{E} \mathfrak{L} \mathfrak{L} / 2 \right) \* \right] \nabla \mathfrak{c} \left[ 2 + \left( \mathfrak{E} \mathfrak{L} \mathfrak{d} / 2 \right) \* \right] \nabla \phi \left[ 2 \right] \text{ dvs.}$$

The first term in the left-hand side of the equation is a free energy density of the bulk phase as a function of concentration, order parameter, and temperature. The second and the third terms 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), 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

The Cahn-Hilliard equation gives the rate of change of conserved order parameter with time:

∂ ϕ/∂t = M.∇2[∂*f*/∂ϕ − Ɛ2ϕ ∇2ϕ].

The above equation is for constant (position-independent) mobility M, where ϕ is the order parameter, ∇ is the divergence, *f* is the free energy of the bulk, and Ɛϕ is the gradient energy coefficient. As one can quite clearly notice, Cahn-Hilliard equation is nothing but modified form of Fick's second law for transient diffusion.

## **Programming formulism**

A code was developed in MATLAB [60] using the abovementioned algorithm. Periodic boundary conditions were also used. The MATLAB code is being provided below. The inputs needed for the simulation are as follows:

N, M—size of the mesh

dx, dy—distance between the nodes in x and y directions

dt—length of time step

Time steps—total number of time steps

A—free energy barrier

Mob—mobility

Kappa—gradient energy coefficient

C (N, M)—initial composition field information

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. Hence, the evolution of the composition profile will occur.

clear

```
clc
format long
%spatial dimensions -- adjust N %and M to increase or decrease
%the size of the computed %solution.
```
N = 100; M = 100;

del\_x = 1.5;

del\_y = 1.5;

%time parameters -- adjust ntmax %to take more time steps, and %del\_t to take longer time %steps.

```
del_t = 10;
   ntmax = 500;
   %thermodynamic parameters
   A = 1.0;
   Mob = 1.0;
   kappa = 1.0;
   %initial composition and noise %strenght information
   c_0 = 0.5;
   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*(0.5-2);
   end
   end
   %The half_N and half_M are %needed for imposing the %periodic boundary 
conditions.
   half_N = N/2;
```
**55**

*Phase Separation in Ce-Based Metallic Glasses DOI: http://dx.doi.org/10.5772/intechopen.88028*

%calculate g, g is parameterised %as 2Ac(1-c)(1-2c)

%Next step is to evolve the &composition profile

g(j + M\*(i-1)) = 2\*A\*comp(j + M\*(i-1))\*(1-comp(j + M\*(i-1)))\*

%calculate the fourier transform %of composition and g field

del\_kx = (2.0\*pi)/(N\*del\_x); del\_ky = (2.0\*pi)/(M\*del\_y);

for index = 1:ntmax

(1-2\*comp(j + M\*(i-1)));

f\_comp = fft(comp);

kx = (i1-N-2)\*del\_kx;

ky = (i2-M-2)\*del\_ky;

comp = real(ifft(f\_comp));

U(i,j) = comp(j + M\*(i-1));

%visualization of the output

denom = 1.0 + 2.0\*kappa\*Mob\*k4\*del\_t;

%Let us get the composition back %to real space

%for graphical display of the %microstructure evolution, %lets store the composition %field into a 256x256 2-d %Matrix.

f\_comp(i2 + M\*(i1-1)) = (f\_comp(i2 + M\*(i1-1))-k2\*del\_t\*Mob\*f\_g(i2 + M\*

f\_g = fft(g);

for i1 = 1:N if i1 < half\_N kx = i1\*del\_kx;

kx2 = kx\*kx; for i2 = 1:M if i2 < half\_M ky = i2\*del\_ky;

ky2 = ky\*ky; k2 = kx2 + ky2; k4 = k2\*k2;

(i1-1)))/denom; end end

> disp(comp); disp(index);

for i = 1:N for j = 1:M

figure(1) image(U\*55) colormap(Jet) colorbar; end

disp('done');

end end

for i = 1:N for j = 1:M

end end

else

end

else

end

```
half_M = M/2;
```
*Phase Separation in Ce-Based Metallic Glasses DOI: http://dx.doi.org/10.5772/intechopen.88028*

*Metallic Glasses*

**Programming formulism**

N, M—size of the mesh

dt—length of time step

A—free energy barrier Mob—mobility

clear clc

format long

longer time %steps. del\_t = 10; ntmax = 500;

> A = 1.0; Mob = 1.0; kappa = 1.0;

c\_0 = 0.5;

for i = 1:N for j = 1:M

end end

conditions.

half\_N = N/2; half\_M = M/2;

N = 100; M = 100; del\_x = 1.5; del\_y = 1.5;

∂ ϕ/∂t = M.∇2[∂*f*/∂ϕ − Ɛ2ϕ ∇2ϕ].

below. The inputs needed for the simulation are as follows:

C (N, M)—initial composition field information

Hence, the evolution of the composition profile will occur.

%initial composition and noise %strenght information

%composition used in %calculations with a noise

comp(j + M\*(i-1)) = c\_0 + noise\_str\*(0.5-2);

%spatial dimensions -- adjust N %and M to increase or decrease

Time steps—total number of time steps

Kappa—gradient energy coefficient

%the size of the computed %solution.

%thermodynamic parameters

noise\_str = 0.5\*(10^-2);

dx, dy—distance between the nodes in x and y directions

The above equation is for constant (position-independent) mobility M, where ϕ is the order parameter, ∇ is the divergence, *f* is the free energy of the bulk, and Ɛϕ is the gradient energy coefficient. As one can quite clearly notice, Cahn-Hilliard equation is nothing but modified form of Fick's second law for transient diffusion.

A code was developed in MATLAB [60] using the abovementioned algorithm. Periodic boundary conditions were also used. The MATLAB code is being provided

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.

%time parameters -- adjust ntmax %to take more time steps, and %del\_t to take

%The half\_N and half\_M are %needed for imposing the %periodic boundary

**54**

```
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;
   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)
   colorbar;
   end
   disp('done');
```