Phase Separation in Ce-Based Metallic Glasses

Abstract

In this chapter, the results of our recent studies on the role of Ga substitution in place of Al in Ce75Al25 − xGax (x = 0, 0.01, 0.1, 0.5, 1, 2, 4, and 6) metallic glasses (MGs) have been discussed with the aim to understand the genesis of phase separation. X-ray diffraction (XRD) study reveals two broad diffuse peaks corresponding to the coexistence of two amorphous phases. In order to see any change in the behavior of 4f electron of Ce, X-ray absorption spectroscopy (XAS) has been carried out for Ce75Al25 − xGax MGs. From the XAS results, it is evident that for x = 0, the spectrum exhibits only a 4f1 component, which basically shows a pure localized configuration of electron. After the addition of Ga, 4f electrons of Ce atoms denoted by 4f0 are getting delocalized. Thus, the phase separation in Ce75Al25 − xGax is taking place, owing to the formation of two types of amorphous phases having localized and delocalized 4f electrons of Ce atoms, respectively. It has been discussed how change in the electronic structure of Ce atoms may lead to phase separation in Ce75Al25 − xGax alloys. Extensive TEM investigations have been done to study the phase separation in these alloys. The microstructural features have been compared with those obtained by phase field modeling.

1. Introduction

In the past decades, considerable research attention has been given to rare-earth (RE)-based metallic glasses (MGs) due to their novel physical properties such as glass-forming ability [1] and mechanical [2, 3], magnetic [4], superplastic [5], and thermoplastic properties [6]. Thus, these MGs hold potential in many applications in the future. Many novel RE-based MGs, e.g., Ce-, La-, Y-, Er-, and Sm-based MGs, have been synthesized [7]. Among RE-based MGs, Ce-based MGs are of special interest due to their unusual behavior linked to 4f electrons [8]. Ce is the most abundant RE metal on earth. It is also one of the most reactive RE metal and oxidizes very readily even at room temperature. One of the key features of Ce is its variable valance states and electronic structure [9, 10, 11]. Thus to change the relative occupancy of the electronic levels, only a small amount of energy is required, e.g., a volume change of approximately 10% results when Ce is subjected to high pressure or low temperatures [9, 11]. Therefore, Ce-based MGs may possess structural and physical properties which are different from other known MGs [12]. Recently, a pressure-induced devitrification behavior of Ce₇₅Al₂₅ MG ribbon has been reported [13, 14, 15]. Prior to our study, only few studies have been done on the substitution and mechanical behavior of Ce₇₅Al₂₅ glassy alloy [1, 16].

Any approach to the description of the amorphous structure suggests that it is a homogeneous isotropic structure. In fact, it turned out that the structure of amorphous phase in alloys cannot always be uniform and isotropic. One situation occurs in the case when the amorphous phase contains two or more metals with comparable scattering amplitude. In such systems, the appearance of inhomogeneity areas or two types of amorphous phases is much more pronounced, since the formation of regions with different chemical compositions leads to the appearance of at least two types of shortest distances between atoms, which naturally results in the phase separation and also affects various properties. The first report by Chen and Turnbull [17] on phase separation in Pd-Au-Si alloy has attracted considerable attention due to their unique microstructural variation of amorphous phases at different length scales. Following this, the possibility of phase separation in MG compositions has been investigated by many authors [18, 19, 20]. However, such a phase separation is incompatible with the glass-forming criteria of negative heat of mixing [21]. The models of MGs based on the nature of geometrical clusters [22] may be helpful in comprehending phase separation in these alloys. According to this model, the MGs have geometry incompatibility in main clusters with long-range translational orders and are joined by the cementing cluster known as glue cluster [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. Sohn et al. reported two general schemes for the design of phase-separating MGs [34]. The first scheme refers to the selection of atom pairs having positive enthalpy of mixing, and the second one refers to the selection of additional alloying element which can enhance glass-forming ability. In the case of ternary- and higher-component alloys, the opposite nature of enthalpy of mixing between the pairs of binaries is possible. In MG systems phase separation will be due to the complex interplay of positive and negative enthalpies of mixing, e.g., in Gd-Zr-Al-Ni Mg alloy system, the enthalpy of mixing is positive for Gd-Zr atom pairs, and other pairs consist of negative enthalpy of mixing [34]. That’s why phase separation is shown by MG system in amorphous state. Phase separation is exhibited by many alloy systems such as La-Zr-Al-Cu-Ni [35], Zr-Ti-Ni-Cu-Be [36], Zr-Gd-Co-Al [37], Cu-(Zr,Hf)-(Gd,Y)-Al [38], Cu-Zr-Al-Nb [39], and Gd-Hf-Co-Al [40]. However, there are very few ternary systems reported in literature which show phase separation. Wu et al. have studied ternary Pd-Ni-P alloy system and observed phase separation through spinodal decomposition [41]. It is worthwhile to mention here that so far no report is available prior to our present study where very sparse atomic percent (~ 0.01 at.%) addition of an element leads to phase separation in a binary system.

In this chapter, we present extensive investigations of amorphous phase formation in Ce₇₅Al_25 − xGa_x alloys with a wide range of concentration of Ga (x = 0, 0.01, 0.1, 0.5, 1, 2, 4, and 6). Both Al and Ga are having the same valency (+3), comparable atomic radii (Ga, 1.41 Å; Al, 1.43 Å), and lying in the same group of the periodic table. Thus, the substitution of Al by Ga does not change the e/a ratio of Ce-Al alloy system (e/a = 1.39). It has been undertaken with a view to understanding the genesis of phase separation in this alloy system. The microstructural features arise due to phase separation which has been studied by transmission electron microscopy (TEM) and compared with those obtained by phase field modeling. The role of Ce electronic structure in phase separation has been discussed. It is important to mention that due to change in the electronic states of Ce, 4f electrons under high pressure, Ce₇₅Al₂₅ alloy undergoes polyamorphic transition [13, 42, 43]. One may expect that chemical pressure effect of Ga substitution in Ce₇₅Al₂₅ MG leads to change in the electronic structure of the Ce in this alloy [44]. Chemical pressure effect basically deals with the change in the electronic structure of atoms due to pressure, temperature, or alloying addition. Keeping these facts in view, extensive use of X-ray absorption spectroscopy (XAS) has been done to investigate Ce₇₅Al_25 − xGa_x alloys. Our investigations have clearly demonstrated that two types of short range order (SRO) may set in Ce₇₅Al_25 − xGa_x amorphous alloys [23]. This is due to delocalization of 4f electron with addition of Ga. The change in the electronic structure of Ce is considered as one of the important reasons for the phase separation in Ce-Al-Ga MG alloy system. The remarkable change in the behavior of glass transition with Ga substitution has been observed through DSC investigation [25, 26, 27, 28, 29, 30]. The thermal stability of the studied materials has been discussed elsewhere, and for this we refer the readers to reference [27].

In this chapter, the effect of Ga substitution (with x as low as 0.01 at.%) on the phase separation has been discussed. The substitution of Ga at place of Al in various alloy systems has been extensively studied by our group [45, 46, 47, 48, 49, 50]. The Ce-Al [51] and Ce-Ga [52] binaries have negative heat of mixing, while Ga-Al pair has very low positive heat of mixing, i.e., 0.7 KJ/mol [53]. It seems unlikely that the phase separation is caused by Ga-Al which has a very small positive heat of mixing. Hence, the alternative explanation for this has been called for. One may thus expect that the substitution of Ga on Al sites may lead to change in the electronic behavior of Ce 4f electrons (owing to chemical pressure effect) [54]. We have also discussed the effect of Ga substitution on the formation of nanoamorphous domains as well as on the nature of Ce 4f electronic states. It should be pointed out that pressure-induced delocalization of 4f electron (using XAS studies) has also been reported by other researchers [13, 42]. However, the partial delocalization of 4f electron of Ce atoms in Ce₇₅Al_25 − xGa_x alloys due to Ga substitution has been pointed out for the first time based on XAS studies.

2. Materials and experimental procedure

The details of the preparation methods of Ce₇₅Al_25 − xGa_x melt-spun alloys are reported elsewhere [2, 21]. The structural characterization has been carried out using X–ray diffractometer (X’Pert Pro PANalytical diffractometer) with CuK_α radiation. The electrolyte with 70% methanol and 30% nitric acid at 253 K has been used to thin the ribbons for TEM characterization. The TEM using FEI: Tecnai 20G² electron microscope has been used to observe the thinned samples. Energy-dispersive X-ray analysis (EDX) attached to the TEM Tecnai 20 G² is obtained at 200 keV using 100 seconds exposure time and 4 μA beam current. The X-ray absorption spectroscopy (XAS) measurements on these samples at Ce L₃ edge were carried out in fluorescence mode with beamline (BL-9), INDUS-2 synchrotron source (2.5 GeV, 100 mA), at RRCAT, India.

3. Investigation of Ce₇₅Al_{25 − x}Ga_x (x = 0, 0.01, 0.1, 0.5, 1, 2, 4, and 6) alloys

3.1 A comparative X-ray diffraction investigation of Ce₇₅Al_{25 − x}Ga_x alloys

Figures 1 and 2 show the XRD patterns of Ce₇₅Al_25 − xGa_x alloys at different Ga concentrations. For the alloy with x = 0, the broad halo peak is found within the angular range 28–35°. This indicates the formation of homogenous glassy phase in Ce₇₅Al₂₅ alloy. While for the alloys with x = 2–6, broad halo peak is found within the angular range 39–50°. The unusual effect was seen in the XRD pattern on substitution of 0.01 at.% Ga. The second diffuse peak with higher intensity can be seen at higher-angle side. With increase in the quantity of Ga (x = 0.1, 0.5, 1, 2, 4, and 6), the positions and intensities of the higher-angle diffuse peak remains almost the same for different concentrations of Ga. The formation of additional diffuse halo peak on the higher-angle side in the XRD pattern due to addition of such sparse amount of Ga refers to unusual effect.

Figure 1.

XRD patterns of as-synthesized ribbons of Ce₇₅Al_25 − xGa_x alloys (x = 0, 0.01, 0.1, 0.5, and 1) (reprinted with kind permission from Ref. [25], copyright 2016, Elsevier).

Figure 2.

XRD patterns of as-synthesized ribbons of Ce₇₅Al_25 − xGa_x alloys (x = 0, 2, and 6) (reprinted with kind permission from Reference [27], copyright 2014, Elsevier).

The prominent low-angle peak (~32°) with low intensity has been observed for x = 0 with respect to Ga addition. The formation of two amorphous phases for the alloys with x = 0.01–6 has been depicted from the two diffuse peaks with different intensities in the XRD patterns of Ce₇₅Al_25 − xGa_x alloys. It can be noticed that one hump is at its original position which indicates that the nature of short range order has not changed for pristine phase. The second diffuse peak appears at ~44^o which indicates the significant change in the short range order. It may be pointed out that usually the hump in the XRD patterns for the large number of MGs occurs in the range of 26–38°. In the present case, the second hump is lying in the same range indicating that the SRO is very similar to the most common type of MGs. Similar observation of two humps has also been reported by Kim et al. for phase separation in Ti₄₅Y₁₁Al₂₄Co₂₀ metallic glass [54].

3.2 Comparative electron microscopic (TEM) investigation of Ce₇₅Al_{25 − x}Ga_x alloys

The TEM image of Ce₇₅Al₂₅ depicts homogenous contrast, and its corresponding selected area diffraction (SAD) shows single diffuse halo ring (c.f. Figure 3(a)). After Ga substitution, the presence of two different amorphous phases having two different contrasts can be seen in Figure 3(b–f). There is one type of amorphous phase which is dispersed in the matrix of other amorphous phase. Figure 3(b–f) displays SAD patterns with two diffuse halos after Ga substitution. The analysis of domain size dispersed in the amorphous matrix has been carried out, and the domain size variation with Ga addition has been done using IMAGE J software. The value domain size (in nanometer) increases linearly with Ga addition and then obtains a saturation value, i.e., ~7 nm at x = 4 and beyond. In Figure 3(b–f), insets 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.

Figure 3.

Bright-field TEM microstructures and the corresponding selected area diffraction patterns (shown in inset) of Ce₇₅Al_25 − xGa_x alloys with (a) x = 0, (b) x = 0.1, (c) x = 0.5, (d) x = 1, (e) x = 2, and (f) x = 4 (reprinted with kind permission from Reference [25], copyright 2016, Elsevier).

3.3 Compositional analysis of Ce₇₅Al_{25 − x}Ga_x alloys through energy-dispersive X-ray analysis

The EDX spectra of Ce₇₅Al_25 − xGa_x 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 Ce₇₅Al_25 − xGa_x 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. That’s why only nominal and average composition of Ga is shown.

Figure 4.

Energy dispersive spectra of the melt-spun Ce₇₅Al_25 − xGa_x 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).

S. No.	x	Nominal composition	Average EDX composition*
1	0	Ce75Al25	Ce74.8 ± 1.5Al25.0 ± 0.8
2	0.1	Ce75Al24.9Ga0.1	Ce74.8 ± 1.5Al24.9 ± 1.7Ga0.1 ± 0.1
3	0.5	Ce75Al24.5Ga0.5	Ce75.1 ± 3.0Al24.2 ± 3.0Ga0.7 ± 0.3
4	1.0	Ce75Al24.0Ga1.0	Ce74.5 ± 1.7Al24.3 ± 0.9Ga1.2 ± 0.9
5	2.0	Ce75.0Al25.0Ga2.0	Ce74.2 ± 2.0Al23.7 ± 2.2Ga2.0 ± 1.3
6	4.0	Ce75.0Al21.0Ga4.0	Ce74.9 ± 2.0Al20.9 ± 1.7Ga4.2 ± 1.0
7	6.0	Ce75.0Al19.0Ga6.0	Ce75.0 ± 1.9Al19.2 ± 1.3Ga5.9 ± 1.7

Table 1.

Energy-dispersive spectra of the melt-spun Ce₇₅Al_25 − xGa_x (0 ≤ x ≤ 6) alloy (reprinted with kind permission from Ref. [25], copyright 2016, Elsevier).

*

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.

3.4 X-ray absorption spectroscopy (XAS) investigation of Ce₇₅Al_{25 − x}Ga_x 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 4f¹ component that gives a pure localized 4f¹ configuration. It can be seen that in the XAS spectra of Ce₇₅Al₂₅, the signature of 4f⁰ electron is not present. The postedge feature represented by 4f⁰ electron at 10 eV is higher than that of 4f¹ 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 4f¹ configuration of Ce in Ce₇₅Al_25 − xGa_x 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, 4f¹ electrons are getting delocalized because of chemical pressure effect. Here we discuss how the phase separation occurred in Ce₇₅Al_25 − xGa_x 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 4f¹ 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 4f¹ 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 4f⁰ delocalized electrons in Ce₇₅Al_25 − xGa_x alloys. The amorphous state containing Ce with localized 4f¹ electrons along with Al will have short range ordering like Ce₇₅Al₂₅ composition, while the other amorphous state containing Ce with delocalized 4f⁰ 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 4f 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 4f 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.

Figure 5.

In situ Ce L₃-edge XAS spectra of Ce₇₅Al_25 − xGa_x metallic glass with x = 0, x = 2, x = 4, and x = 6. The arrow points out the 4f⁰ and 4f¹ electronic states of Ce. The signature of 4f⁰ indicates delocalization of 4f electrons. Upper inset shows the excursion of trivalent to tetravalent state (reprinted with kind permission from Reference [25], copyright 2016, Elsevier).

As discussed above, the 4f electrons in some of Ce atoms are delocalized due to Ga substitution in Ce₇₅Al₂₅ alloys. Hence, glassy Ce₇₅Al_25 − xGa_x may exhibit two types of SRO. The Ce atoms having 4f¹ localized electron will have pristine SRO in the alloy without Ga. The Ce atoms with 4f⁰ 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 4f¹ 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 4f 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 Ce₇₅Al_{25 − x}Ga_x alloys

A schematic diagram of effective atomic radii of Ce atoms in Ce₇₅Al_25 − xGa_x alloys to understand the effect of 4f electron is shown in Figure 6. For x = 0, Ce atoms are having localized 4f electrons, while for the alloy with x = 4, the partial delocalization of 4f electrons has taken place. Because of delocalization of 4f electrons, the effective atomic radius of Ce atoms decreases.

Figure 6.

The effective atomic radii of Ce atoms showing low-density cluster (LDC) and high-density cluster (HDC) with (a) localized 4f electrons and (b) delocalized 4f electrons for Ce₇₅Al_25 − xGa_x MGs (right side). The Ce atoms with localized 4f electrons are shown by red balls, the Ce atoms with delocalized 4f 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).

The partial delocalization of 4f 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 4f electrons of Ce atoms may form two kinds of density clusters which are low-density clusters (LDC) with localized 4f electrons and high-density clusters (HDC) with delocalized 4f 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 4f electrons offers a fascinating opportunity to investigate the microstructural effect on the various properties as glass-forming ability and mechanical and transport properties of Ce₇₅Al_25 − xGa_x alloys.

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 Ce₇₅Al_25 − xGa_x 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:

$$\mathrm{F}=\mathrm{ʃ}v\left[f\left(\mathrm{\varphi },c,T\right)+\left(\mathrm{Ɛ}\text{2c}/2\right)\ast \left|\text{∇}\mathrm{c}\right|2+\left(\mathrm{Ɛ}\text{2ϕ}/2\right)\ast \left|\text{∇}\mathrm{\varphi }\right|2\right]\mathrm{dv}.$$

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:

$$\mathrm{\partial }\mathrm{\varphi }/\mathrm{\partial t}=\mathrm{M}.\text{∇}2\left[\mathrm{\partial }f/\mathrm{\partial \varphi }-\mathrm{Ɛ}\text{2ϕ}\text{∇}\text{2ϕ}\right].$$

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;

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;

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’);

4.1 Effect of initial composition

Figure 7 shows the phase separation patterns with different initial average concentrations during time steps 200, without considering the fluid flow. It has been suggested that there are two phases, namely, B and C, in the evaluated microstructures. In Figure 7 the red region and blue region show the B-rich and C-rich phase, respectively. The volume fraction of the C phase has been shown in Figure 7. As we can see, when the volume fraction of the B and C phases is around 0.7 and 0.3, respectively, droplet-like structure has been formed (Figure 7(a)). When the volume fraction of the C phase increases from 0.3 to 0.4, an interconnected structure will form at the initial stage (Figure 7(c)). Figure 7(e) shows the equal volume fraction of both initial average concentrations with 0.5. It has been shown that at equal initial average concentration, spinodal- or interconnected-type microstructure has grown completely. Figure 7(f–i) shows the spinodal or interconnected to droplet-like microstructures, when it is subjected to increasing the initial average concentration of phase C from 0.5 to 0.7.

Figure 7.

Evolution of microstructure based on phase field modeling with different amounts of phase-separating domains from the homogenous matrix phase.

4.2 Comparison of experimental and evaluated microstructures by phase field modeling

In this section we have compared the experimental microstructure with numerical simulation microstructure based on the Cahn-Hilliard equation of phase separation and conjecture the experimental environments or synthesis parameter (Figure 8). Figure 8(a) shows the numerical simulated microstructure with the following parameters:

1. Initial composition b = 0.43 and c = 0.57

2. Cooling rate Δt_max = 300

Figure 8.

Comparison of experimental and theoretical phase field model of phase separation in spinodal decomposition (a) numerical simulated microstructure with 43% and 57% phase fraction and (b) experimental microstructure of Ce₇₅Al₂₁Ga₄ alloy.

Figure 8(b) shows the phase-separated Ce₇₅Al₂₁Ga₄ metallic glass. There are so many parameters which have also been calibrated like thermal mobility, gradient of energy coefficient, and noise string, which play an important role in numerical simulation. It can be seen that both microstructures are about the same features like spinodal decomposition phases. Figure 8(b) shows the experimental bright-field TEM microstructure of Ce₇₅Al₂₁Ga₄ metallic glass. After comparing both images, one can notice that the evaluated microstructures are in good agreements with experimental results. It has been found that the numerical simulations are in good agreement with the experimental findings.

5. Conclusions

Based on the results described and discussed in this chapter, the following conclusions can be drawn:

1. The substitution of Ga results in the formation of additional strong diffuse peak in XRD at the higher diffraction angle indicating the formation of two types of amorphous phases in Ce₇₅Al_25 − xGa_x alloys. The present investigation clearly demonstrates the formation of nanoamorphous domains in melt-spun ribbons of Ce₇₅Al_25 − xGa_x alloys even at very low concentration of Ga (0.01 at.%).

2. After Ga substitution, the phase separation in this case is related to change in the electronic state of Ce-4f electron. The study of Ce L₃ edge XAS spectra of as-synthesized ribbons suggest that the Ga substitution partially given rise to Ce-4f⁰ delocalized state. This study therefore opens up a new direction of investigation, delineating issues related to the formation of two types of amorphous phases.

3. The microstructure evaluated after solving the Cahn-Hilliard equation of phase separation using phase field modeling. It has been found that both droplet-like structure and interconnected structure appear in phase field modeling, when the phase fraction of the dispersed phase is increased from 30 to 45% and the size of each amorphous domain has increased with increasing cooling rate.

4. A comparison of microstructure of phase-separated nanoamorphous domains has been made with computer simulations using phase field modeling. It can be concluded that phase fraction may be 43 and 57%.

Acknowledgments

The authors would like to thank Prof. O.N. Srivastava and Prof. R.K. Mandal for providing lab facilities to carry out experiments. One of the authors (Dharmendra Singh) thankfully acknowledges the financial support by UGC under the scheme RGNF [2013-2014-36995], and Devinder Singh is grateful to DST, New Delhi, India, for financial support in the form of INSPIRE Faculty Award [IFA12-PH-39].