**Fundamentals and Theoretical Aspects**

**1** 

*Russia* 

Elena A. Chechetkina

**Crystallization in Glass Forming Substances:** 

*Institute of General and Inorganic Chemistry of Russian Academy of Sciences, Moscow,* 

Glassy materials are strongly connected with crystallization or - more strictly - with the ability to *avoid* crystallization when cooling a melt. The more stable a supercooled liquid against crystallization, the higher its glass forming ability. It should be noted that almost every substance can be preapared in the form of amorphous solid by means of special methods of fast melt cooling (in the form of ribbons), evaporating (films), deposition (layers) and sol-gel technique (initially porous samples), etc. [1]. The resulting materials are often named "glass" (e.g., "glassy metals"), but they are actually outside the scope of this chapter. Here, we consider only typical inorganic glasses, such as SiO2 and Se, which correspond to an understanding of glass as a *bulk non-crystalline solid prepared by melt cooling.* "Bulk" means a 3D sample having a size of 1 cm and larger, a condition that implies a cooling rate of about 10 K/s and lower. Despite the "technological" character of this definition, it is the most objective one, being free from the declared or hidden speculations about the nature of glass. The theoretical background is given in Section 2, beginning with classical notions about crystallization in ordinary melts and the concept of the *critical cooling rate* (CCR), understood as the minimal cooling rate that provides the solidification of a melt without its crystallization. As far as any crystallization ability is reciprocal to a glass forming ability, one can evaluate a *glass forming ability* in terms of the CCR, using both crystallization theory and experiments. It will demonstrate a principal inapplicability of classical crystallization theory in the case of glass forming substances. As an alternative, we have developed a new approach based on the hypothesis of *initial reorientation*, considered as a specific prenucleation stage in a non-crystalline network. The classical model of a *continuous random network* consisting of common covalent bonds (two-centre two-electron, 2c-2e) is modified in two respects. First, we introduce *hypervalent bonds* (HVB) in addition to covalent bonds (CB); and besides, such bonds can be transformed into one another (CB↔HVB). Second, the elementary acts of bond exchange are in spatiotemporal correlation, thus providing a *bond wave* by means of which the collective processes - including initial reorientation and further crystallization - may occur. A bond wave also means a non-crystalline order of hierarchical character: from the well known short-range order (it changes in the vicinity of HVB), through the well known but poorly-understood medium-range order (in the limits of the wavefronts populated with HVB), to the non-crystalline long-range order generated by bond waves. Insofar as bond waves change their parameters and direction during melt cooling, being frozen in solid glass, the non-crystalline long-range order depends not only

**1. Introduction** 

**The Chemical Bond Approach** 

## **Crystallization in Glass Forming Substances: The Chemical Bond Approach**

Elena A. Chechetkina

*Institute of General and Inorganic Chemistry of Russian Academy of Sciences, Moscow, Russia* 

## **1. Introduction**

Glassy materials are strongly connected with crystallization or - more strictly - with the ability to *avoid* crystallization when cooling a melt. The more stable a supercooled liquid against crystallization, the higher its glass forming ability. It should be noted that almost every substance can be preapared in the form of amorphous solid by means of special methods of fast melt cooling (in the form of ribbons), evaporating (films), deposition (layers) and sol-gel technique (initially porous samples), etc. [1]. The resulting materials are often named "glass" (e.g., "glassy metals"), but they are actually outside the scope of this chapter. Here, we consider only typical inorganic glasses, such as SiO2 and Se, which correspond to an understanding of glass as a *bulk non-crystalline solid prepared by melt cooling.* "Bulk" means a 3D sample having a size of 1 cm and larger, a condition that implies a cooling rate of about 10 K/s and lower. Despite the "technological" character of this definition, it is the most objective one, being free from the declared or hidden speculations about the nature of glass.

The theoretical background is given in Section 2, beginning with classical notions about crystallization in ordinary melts and the concept of the *critical cooling rate* (CCR), understood as the minimal cooling rate that provides the solidification of a melt without its crystallization. As far as any crystallization ability is reciprocal to a glass forming ability, one can evaluate a *glass forming ability* in terms of the CCR, using both crystallization theory and experiments. It will demonstrate a principal inapplicability of classical crystallization theory in the case of glass forming substances. As an alternative, we have developed a new approach based on the hypothesis of *initial reorientation*, considered as a specific prenucleation stage in a non-crystalline network. The classical model of a *continuous random network* consisting of common covalent bonds (two-centre two-electron, 2c-2e) is modified in two respects. First, we introduce *hypervalent bonds* (HVB) in addition to covalent bonds (CB); and besides, such bonds can be transformed into one another (CB↔HVB). Second, the elementary acts of bond exchange are in spatiotemporal correlation, thus providing a *bond wave* by means of which the collective processes - including initial reorientation and further crystallization - may occur. A bond wave also means a non-crystalline order of hierarchical character: from the well known short-range order (it changes in the vicinity of HVB), through the well known but poorly-understood medium-range order (in the limits of the wavefronts populated with HVB), to the non-crystalline long-range order generated by bond waves. Insofar as bond waves change their parameters and direction during melt cooling, being frozen in solid glass, the non-crystalline long-range order depends not only

Crystallization in Glass Forming Substances: The Chemical Bond Approach 5

greater the glass forming ability. Moreover, and to the contrary, for non-glass forming substances (*Qc*>102 K/s), there is no relation between *Qc* and its variation, the latter actually


+1,7

+9,1 +10

+8,0 +10 +8

Table 1. The calculated CCRs among the groups of oxides, substances with hydrogen bonding and/or organic glass formers, halides, sodium salts and metals. The "G" in

takes lg*Qc*=−14 after [4], the point falls far below the correlation line for glass formers.

One should note the absence of benzene (see C6H6 in Table 1) in Fig.1. This is a result of the outstanding disagreement in the evaluations of its CCR: benzene looks like an excellent glass former after [4], but after [5] it is a non-glass former. When lg*Qc*=+4,9 after [5], the point for difference Δ|lg*Qc*|=18,9 is dramatically lifted above the "amorphous line"; however, if one

The Qc interval in Fig.1 extends for 19 orders in magnitude. The practical interval is far narrower. For glasses, there are three useful CCR points: the glass definition limit *Qc*=102 K/s, which corresponds roughly to a quenching of a 1 cm3 sample in water with ice, the rate of 10 K/s corresponding to quenching in the air, and the rate of 1-10-2 K/s corresponding to free cooling in a furnace. Therefore, the extremely large negative values of lg*Qc*(K/s) in Table 1 - such as -16 for B2O3 or -23 for P2O5 [4] - only display an extremely high glass forming ability, i.e., the ability for very slow cooling in a furnace, a feature which provides the opportunity to obtain very massive articles, such as lenses for telescopes. For amorphous

+0,6 -3,7

+7,2 +7

[2] [3] [4] [5]


+12 -15 -2 -40 0 -14


+5 +3 +10 +7

+9 +2 +10

lg*Qc*(K/s) |Δlg*Qc*|

4,3 (G) 8,7 (G) 9,8 (G) 17,1 (G) 2,9

5,0 13,3 (G) 3,0 32,1 (G) 5,2 18,9 (G?)

1,0 (G) 16,6 (G) 1,5 3,3

1,6 2,0 2,0 2,6 1,6

1,3 0,5 5,9 1,3


+10,4

+1 -7,9 +5,2 +4,9


+8,4 +7,0 +5,0 +7,4 +8,4

+9,3 +8,5 +7,9 +8,7

being the same for all non-glass formers.

[4]

273 316.6 159 293 250,2 278,4

1234 1356 505,7 600,7

brackets indicates a well-recognized glass former.

Substance *Tm*, K

SiO2 (G) B2O3 (G) GeO2 (G) P2O5 (G) As2O3

H2O Salol (G) Ethanol Glycerin (G)

CCl4 C6H6 (G?)

NaCl NaNO3 NaCO3 NaSO4 NaWO4

Ag Cu Sn Pb

BeF2 (G) ZnCl2 (G) BiCl3 PbBr2

on the substance under consideration but also on the sample prehistory. This is a principal distinction from the ordinary long-range order in crystals, which is strictly determined for a given substance at a given temperature. As such, the problem of crystallization in glass can be considered as a competition between two types of order, and hypervalent bonds and their self-organization in the form of a bond wave play a central role in this process.

In Section 3 and Section 4, original crystallization experiments based on the above approach are described. In Section 3, we consider crystallization in **solid Se-X** glasses, in which additions of a different chemical nature (X = Cl, S, Te, As, Ge) are used. A set of properties, including the abilities of nucleation and surface crystallization, was investigated based upon the composition for each series. A strong *non-linearity* was found in the region of small additions *N<N\** (concentration *N\** depends on the nature of X), which is discussed in terms of the bond wave interaction with foreign atoms also existing in a hypervalent state. Finally, in Section 4, original experiments on the crystallization of **softening Se-X** glasses in an **ultrasonic field** are presented. In softening glass, bond waves are refrozen and an ultrasonic field can act as an *information field* for them; in our case, a US-field can give a predominant direction for the crystallization process. Actually, the resultant glasses become *anisotropic*, the anisotropy also being non-linear in respect to the composition.

Thus, starting from the classical approach operating with a 3-step process of crystallization (sub-critical unstable nucleation, stable nucleation, crystal growth), fluctuations, atomic jumps, atomic/geometrical structure, one-type bonding and one-type long-range order, we pass to a strange picture of mixed bonding, bond waves, "wavy" long-range order and glass as a self-organizing system.

The text is intended for a wide audience. The readers from the field of glass are invited to meet with new experiments and new ideas. For beginners – let us now enter into the mysterious world of glass!

## **2. Crystallization in supercooled glass forming liquids (T>Tg)**

## **2.1 Critical cooling rate**

The idea of the CCR as a measure of a glass forming ability seems to be trivial, but it remained in the shadows up until 1968, when *Sarjeant & Roy* [2] emphasized "the concept of '*critical quenching rate'*… defined as the cooling rate below which detectable crystalline phases are obtained from the melt" and proposed the first theoretical expression for the CCR in the form of:

$$Q\_{\varepsilon} = 2.0^{\ast}10^{\ast\_0} \, ^{\ast} (T\_m)^2 \, ^{\ast} R \slash V^{\ast} \eta \tag{1}$$

Here *Tm* is the melting point, *η* (poises) is the viscosity at *Tm* and *V* is the volume of the "diffusing species". The main problem is the evaluation of *V*, which needs "an extreme simplification and interaction of many parameters involved" [2]. Nevertheless, it is seen in **Table 1** that the values thus obtained are in agreement with the later estimations of the CCR [3-5] using the equations of classical crystallization kinetics.

Strictly specking, it is easy to fit into the "classical" gap [3-5] that extends for orders in magnitude. More interesting is the tendency for this gap, Δ|lg*Qc*|, to be especially wide just for glass-formers (G). It is seen in **Fig.1** that in glass forming substances (*Qc*<102 K/s by definition) the difference Δ|lg*Qc*| increases when *Qc* decreases, i.e., the larger the gap, the

on the substance under consideration but also on the sample prehistory. This is a principal distinction from the ordinary long-range order in crystals, which is strictly determined for a given substance at a given temperature. As such, the problem of crystallization in glass can be considered as a competition between two types of order, and hypervalent bonds and

In Section 3 and Section 4, original crystallization experiments based on the above approach are described. In Section 3, we consider crystallization in **solid Se-X** glasses, in which additions of a different chemical nature (X = Cl, S, Te, As, Ge) are used. A set of properties, including the abilities of nucleation and surface crystallization, was investigated based upon the composition for each series. A strong *non-linearity* was found in the region of small additions *N<N\** (concentration *N\** depends on the nature of X), which is discussed in terms of the bond wave interaction with foreign atoms also existing in a hypervalent state. Finally, in Section 4, original experiments on the crystallization of **softening Se-X** glasses in an **ultrasonic field** are presented. In softening glass, bond waves are refrozen and an ultrasonic field can act as an *information field* for them; in our case, a US-field can give a predominant direction for the crystallization process. Actually, the resultant glasses become *anisotropic*,

Thus, starting from the classical approach operating with a 3-step process of crystallization (sub-critical unstable nucleation, stable nucleation, crystal growth), fluctuations, atomic jumps, atomic/geometrical structure, one-type bonding and one-type long-range order, we pass to a strange picture of mixed bonding, bond waves, "wavy" long-range order and glass

The text is intended for a wide audience. The readers from the field of glass are invited to meet with new experiments and new ideas. For beginners – let us now enter into the

The idea of the CCR as a measure of a glass forming ability seems to be trivial, but it remained in the shadows up until 1968, when *Sarjeant & Roy* [2] emphasized "the concept of '*critical quenching rate'*… defined as the cooling rate below which detectable crystalline phases are obtained from the melt" and proposed the first theoretical expression for the

Here *Tm* is the melting point, *η* (poises) is the viscosity at *Tm* and *V* is the volume of the "diffusing species". The main problem is the evaluation of *V*, which needs "an extreme simplification and interaction of many parameters involved" [2]. Nevertheless, it is seen in **Table 1** that the values thus obtained are in agreement with the later estimations of the CCR

Strictly specking, it is easy to fit into the "classical" gap [3-5] that extends for orders in magnitude. More interesting is the tendency for this gap, Δ|lg*Qc*|, to be especially wide just for glass-formers (G). It is seen in **Fig.1** that in glass forming substances (*Qc*<102 K/s by definition) the difference Δ|lg*Qc*| increases when *Qc* decreases, i.e., the larger the gap, the

*Qc* = 2.0\*10-6 \*(*Tm*)2 \**R/V\*η* (1)

their self-organization in the form of a bond wave play a central role in this process.

the anisotropy also being non-linear in respect to the composition.

**2. Crystallization in supercooled glass forming liquids (T>Tg)** 

[3-5] using the equations of classical crystallization kinetics.

as a self-organizing system.

mysterious world of glass!

**2.1 Critical cooling rate** 

CCR in the form of:


greater the glass forming ability. Moreover, and to the contrary, for non-glass forming substances (*Qc*>102 K/s), there is no relation between *Qc* and its variation, the latter actually being the same for all non-glass formers.

Table 1. The calculated CCRs among the groups of oxides, substances with hydrogen bonding and/or organic glass formers, halides, sodium salts and metals. The "G" in brackets indicates a well-recognized glass former.

One should note the absence of benzene (see C6H6 in Table 1) in Fig.1. This is a result of the outstanding disagreement in the evaluations of its CCR: benzene looks like an excellent glass former after [4], but after [5] it is a non-glass former. When lg*Qc*=+4,9 after [5], the point for difference Δ|lg*Qc*|=18,9 is dramatically lifted above the "amorphous line"; however, if one takes lg*Qc*=−14 after [4], the point falls far below the correlation line for glass formers.

The Qc interval in Fig.1 extends for 19 orders in magnitude. The practical interval is far narrower. For glasses, there are three useful CCR points: the glass definition limit *Qc*=102 K/s, which corresponds roughly to a quenching of a 1 cm3 sample in water with ice, the rate of 10 K/s corresponding to quenching in the air, and the rate of 1-10-2 K/s corresponding to free cooling in a furnace. Therefore, the extremely large negative values of lg*Qc*(K/s) in Table 1 - such as -16 for B2O3 or -23 for P2O5 [4] - only display an extremely high glass forming ability, i.e., the ability for very slow cooling in a furnace, a feature which provides the opportunity to obtain very massive articles, such as lenses for telescopes. For amorphous

Crystallization in Glass Forming Substances: The Chemical Bond Approach 7

The general question is what is meant by the demand of melt cooling "without crystallization"? After *Vreeswijk et al.* [4] glass is considered as being partially crystallized even one critical nucleus is formed; this is a **theoretical** limit for estimation of the *Qc* value. Later, *Uhlmann* [3] proposed the crystalline fraction α=10-6 for the **experimentally** detected limit, and *Ruscenstein & Ihm* [5] proposed α=10-4 for a more convenient **practical** limit. Obviously, the lower α, the higher Qс for a given substance. However, when comparing the calculation results in Table 1 - we see that there is no a relation between the crystalline fraction permitted (α increases from [4] to [5]) and the corresponding *Qc* values. Thus, classical crystallization theory needs serious reconsideration for glass forming substances.

The critical nucleus - arising when overcoming the "thermodynamic" barrier W\* with the following steady-state nucleation of the *I(T)* frequency and the steady-state crystal growth of the *u(T)* rate - forms the basis of classical crystallization theory. The critical nucleus is considered as the cause of the *induction period* observed in the process of crystallization. It is unclear, however, to what extent the classical notions are true for glass forming substances. In order to test the assumption about steady-state nucleation, *Kelton & Greer* [18] have analysed the data for lithium disilicate (a typical glass former) and two metallic "glasses" ('amorphous metal' would be a more appropriate term). When the authors introduced the time-dependent nucleation frequency *I(τ,T)*, they revealed the so-called *transient effects* which occurred to be most significant only for typical glass. Transient effects, together with the fact that "*even small uncertainties of material parameters can introduce uncertainties of several orders in magnitude in the calculated nucleation frequency*" [18], make the application of classical

On the other hand, from the beginning of the 1980s onwards, we have developed an alternative approach for the prediction of glass forming ability in relation to crystallization ability. This approach is based on *Dembovsky's* "empirical theory of glass formation" [19], by which one can calculate the dimensionless glass forming ability, which was transformed further into an energetic barrier for crystallization *Ecr* [20]. The next stage was the transformation of the semi-empirical *Ecr* (enthalpy) to the barrier Δ*G#* (free energy), and finally Δ*G#* to the CCR measure *Vm* [21,22]. The reader can find a complete calculation scheme for *Vm* in [17; p.145] or [23; p.105]. Two points of this CCR are of special interest.

First, in order to calculate *Vm*, one needs *no kinetic parameters*, such as a coefficient of diffusion and/or viscosity. Instead, we use only the phase diagram of a system (in order to determine the liquidus temperature for a related unit: an element, a compound, an eutectic or else an intermediate), the first coordination number (presumably, in the liquid state at around *Tm*) and the averaged valence electron concentration (e.g., 6 for Se, 5.6 for As2Se3,

Second, the barrier Δ*G#* was attributed to a special pre-crystalline ordering named the *initial reorientation* (IReO), which is postulated as the stage after which crystal nucleation and growth is possible [21]. IReO means that neither nucleation nor crystal growth can proceed in a non-oriented - i.e., a truly "amorphous" - medium. Since IReO is a simple activation process with the Δ*G#* barrier, the reoriented fraction increases exponentially with time, as

**2.2 Induction period, transient effects and initial reorientation** 

crystallization theory problematic in the case of glass forming substances.

5.33 for SiO2, etc.).

the dotted line in Fig.2 shows.

materials (*Qc*>102 K/s), the experimental limit is about *Qc*=106 K/s, which corresponds with the preparation of amorphous ribbons, namely the 2D case. The calculated value of *Qc*=108- 1010 K/s displays only an extra low amorphization ability of a substance, which cannot be obtained by melt cooling at all. This is really the 1D case, when the amorphous film is prepared by evaporation onto a cold substrate.

Fig. 1. The absolute difference between the calculated values of the CCRs from [4,5] as a function of the CCR value after [5]. The data is from Table 1.

Based on the tendency in Fig.1, one can conclude that the greater the glass forming ability, the more problematic the application of the classical theory of crystallization. In reality, in the case of a lithium disilicate glass it was shown that "not only do all forms of classical theories predict nucleation rates many orders in magnitude smaller than those observed, but also the temperature dependence of the theoretical rate is quite different from that observed" [6]. Thus, crystallization in glasses needs special methods, both theoretical and experimental.

Although there are many experimental works on the crystallization of glass and glass forming liquids, only a few of them concern the CCR. The known evaluations of *Q*c in oxides [7-11] and chalcogenides [12-16] are made by either a direct method covering a relatively small lg*Qc* interval (e.g., from -2 to +4 after [7-10]), or indirect methods. By 'indirect' is meant that the crystallization data corresponding to some extent of crystallinity α~0,01-0,1 is recalculated to a much lower α corresponding to *Qc* (usually to α=10-6 [15-16]) using equations of the *Kolmogorov-Avrami* type. However, the applicability of such equations in an extra low α region is also under question. Thus, it is not surprising that interest in the investigation of the CCR is low at present. The reader can find a critical review of the "golden age" of the CCR in the last third of the Twentieth Century in our monograph [17]. The main result of this period seems to be the formulation of new problems concerning glass forming ability.

materials (*Qc*>102 K/s), the experimental limit is about *Qc*=106 K/s, which corresponds with the preparation of amorphous ribbons, namely the 2D case. The calculated value of *Qc*=108- 1010 K/s displays only an extra low amorphization ability of a substance, which cannot be obtained by melt cooling at all. This is really the 1D case, when the amorphous film is

> -8 -6 -4 -2 0 2 4 6 8 10 12 **lgQc**(K/s) [5]

Fig. 1. The absolute difference between the calculated values of the CCRs from [4,5] as a

crystallization in glasses needs special methods, both theoretical and experimental.

Based on the tendency in Fig.1, one can conclude that the greater the glass forming ability, the more problematic the application of the classical theory of crystallization. In reality, in the case of a lithium disilicate glass it was shown that "not only do all forms of classical theories predict nucleation rates many orders in magnitude smaller than those observed, but also the temperature dependence of the theoretical rate is quite different from that observed" [6]. Thus,

Although there are many experimental works on the crystallization of glass and glass forming liquids, only a few of them concern the CCR. The known evaluations of *Q*c in oxides [7-11] and chalcogenides [12-16] are made by either a direct method covering a relatively small lg*Qc* interval (e.g., from -2 to +4 after [7-10]), or indirect methods. By 'indirect' is meant that the crystallization data corresponding to some extent of crystallinity α~0,01-0,1 is recalculated to a much lower α corresponding to *Qc* (usually to α=10-6 [15-16]) using equations of the *Kolmogorov-Avrami* type. However, the applicability of such equations in an extra low α region is also under question. Thus, it is not surprising that interest in the investigation of the CCR is low at present. The reader can find a critical review of the "golden age" of the CCR in the last third of the Twentieth Century in our monograph [17]. The main result of this period seems to be the formulation of new problems concerning glass

**y = 0,0005x + 2,4563 R2 = 6E-07**

CCl4 Sn H2O

prepared by evaporation onto a cold substrate.

Glycerol

B2O3

function of the CCR value after [5]. The data is from Table 1.

0

forming ability.

5

10

15

P2O5

20

**lgQc|** [4,5]

25

30

35

**y = -3,7632x - 6,7631 R2 = 0,4993**

The general question is what is meant by the demand of melt cooling "without crystallization"? After *Vreeswijk et al.* [4] glass is considered as being partially crystallized even one critical nucleus is formed; this is a **theoretical** limit for estimation of the *Qc* value. Later, *Uhlmann* [3] proposed the crystalline fraction α=10-6 for the **experimentally** detected limit, and *Ruscenstein & Ihm* [5] proposed α=10-4 for a more convenient **practical** limit. Obviously, the lower α, the higher Qс for a given substance. However, when comparing the calculation results in Table 1 - we see that there is no a relation between the crystalline fraction permitted (α increases from [4] to [5]) and the corresponding *Qc* values. Thus, classical crystallization theory needs serious reconsideration for glass forming substances.

#### **2.2 Induction period, transient effects and initial reorientation**

The critical nucleus - arising when overcoming the "thermodynamic" barrier W\* with the following steady-state nucleation of the *I(T)* frequency and the steady-state crystal growth of the *u(T)* rate - forms the basis of classical crystallization theory. The critical nucleus is considered as the cause of the *induction period* observed in the process of crystallization. It is unclear, however, to what extent the classical notions are true for glass forming substances. In order to test the assumption about steady-state nucleation, *Kelton & Greer* [18] have analysed the data for lithium disilicate (a typical glass former) and two metallic "glasses" ('amorphous metal' would be a more appropriate term). When the authors introduced the time-dependent nucleation frequency *I(τ,T)*, they revealed the so-called *transient effects* which occurred to be most significant only for typical glass. Transient effects, together with the fact that "*even small uncertainties of material parameters can introduce uncertainties of several orders in magnitude in the calculated nucleation frequency*" [18], make the application of classical crystallization theory problematic in the case of glass forming substances.

On the other hand, from the beginning of the 1980s onwards, we have developed an alternative approach for the prediction of glass forming ability in relation to crystallization ability. This approach is based on *Dembovsky's* "empirical theory of glass formation" [19], by which one can calculate the dimensionless glass forming ability, which was transformed further into an energetic barrier for crystallization *Ecr* [20]. The next stage was the transformation of the semi-empirical *Ecr* (enthalpy) to the barrier Δ*G#* (free energy), and finally Δ*G#* to the CCR measure *Vm* [21,22]. The reader can find a complete calculation scheme for *Vm* in [17; p.145] or [23; p.105]. Two points of this CCR are of special interest.

First, in order to calculate *Vm*, one needs *no kinetic parameters*, such as a coefficient of diffusion and/or viscosity. Instead, we use only the phase diagram of a system (in order to determine the liquidus temperature for a related unit: an element, a compound, an eutectic or else an intermediate), the first coordination number (presumably, in the liquid state at around *Tm*) and the averaged valence electron concentration (e.g., 6 for Se, 5.6 for As2Se3, 5.33 for SiO2, etc.).

Second, the barrier Δ*G#* was attributed to a special pre-crystalline ordering named the *initial reorientation* (IReO), which is postulated as the stage after which crystal nucleation and growth is possible [21]. IReO means that neither nucleation nor crystal growth can proceed in a non-oriented - i.e., a truly "amorphous" - medium. Since IReO is a simple activation process with the Δ*G#* barrier, the reoriented fraction increases exponentially with time, as the dotted line in Fig.2 shows.

Crystallization in Glass Forming Substances: The Chemical Bond Approach 9

The theory of glass structure was actually formed at 1932 when, beginning with the words *"It must be frankly admitted that we know practically nothing about the atomic arrangement in glasses,"* the young scientist *Zachariasen* proposed his famous model of a **continuous random network** (CRN) [30]. This network consists of covalent bonds, - the same as that in a related crystal - and the only difference is the "random" arrangement of the bonds in CRN in contrast with their regular arrangement in a crystalline network. As far as the atomic arrangement in CRN is determined by the arrangement of chemical bonds, one might suppose that only chemists - who deal with chemical bonds - would play a leading role in the further development of the CRN model. However, the development was directed in another "physical" way. As a result, the chemical bond is present in contemporary glass theory in the form of "frustration", "elastic force" and rigid "sticks" connecting atoms-balls, etc., but not as a real object with its own specificity. We think that a pure "physical approach" opens a rich field for theoretical speculation, rather than clarifying the nature of glass. For example, to understand glass transition, the analogues with spin glasses, granular systems and colloids are used (e.g., see the excellent review [31]), although any similarity in

behaviour does not mean that the similarity is the reason behind that behaviour.

One can find, however, deviations from the main "physical" stream of glass science, even among physicians. As a fresh example, we may cite that: "*It is widely believed that crystallization in three dimensions is primary controlled by positional ordering, and not by bond orientational ordering. In other words, bond orientational ordering is usually considered to be merely a consequence of positional ordering and thus has often been ignored. … Here we proposed that bond orientational ordering can play a key role in (i) crystallization, (ii) the ordering to quasi-crystal and (iii) vitrification*" [29]. Although this is a rare case, it may be a sign that the need for a

Our goal was to return the chemical bond - as chemists know it - to the theory of glass. As far back as in 1981, *Dembovsky* [32] has connected the experimental fact of an *increased* coordination number in glass forming melts with the model of *quasimolecular defects* (QMD) following *Popov* [33] for the creation of the chemical-bond basis for glass theory. The principal feature of a QMD is its hyper-coordinated nature, a property that provides for the connectivity of a covalent network even at melt temperature, thus resulting in high melt viscosity, which is a characteristic property of a glass forming melt. QMDs in a noncrystalline network can also explain general features of glass [34-38]. The problem is that the concentration of QMDs should be high enough, especially in a glass forming liquid, to provide these features. Therefore, we have quickly substituted the initial term QMD (quasimolecular *defect*) for TCB (three-centre *bond*) and then, being based on a special quantum-chemical study (see the reviews [39] for chalcogenide glasses and [40] for oxide ones), TCB for HVB, **hypervalent bond.** TCB is only a particular case of HVB, which is rarely realizes in glass. For example, in the simplest case of Se not TCB, having two threecoordinated atoms (-Se<) and one "normal" two-coordinated atom (-Se-) [33], but a fourcoordinated atom (>Se< )was revealed [41]. Thus, we have introduced *alternative bonds* into CRN, the bonds whose concentration is commensurable with that of ordinary covalent

The term "hypervalent", which was introduced by *Muscher* in 1969 [42], has a long and controversial history, beginning from the 1920s up until the present (see [43] for an introduction). Currently, a large number of hypervalent molecules is known, and various methods of their theoretical description exist. For a long time, one of the most popular was

**2.3 Glass structure, hypervalent bonds and bond waves** 

chemical approach is now stronger.

bonds constituting classical CRN.

Fig. 2. Isothermal crystallization after [21]: the dotted curve describes the development of the reoriented non-crystalline fraction; the solid line is the model temporal dependence of the extent of crystallinity α; the pointed curve is a real crystallization process.

**Atomic states**: **A** is the initial truly "amorphous" state for a species in a supercooled liquid or glass; **A0** is the reoriented state; **ANu** is the state of the crystal nucleus; **ACr** is the state of the growing crystal.

**Barriers**: Δ*G*# is the initial orientation; *W\** is the critical nucleus; Δ*G'* is the steady-state nucleation; Δ*G''* is the crystal growth.

The three intervals in Fig.2 correspond to an *induction period* at 0<τ<τ\*, a *transient period* at τ\*<τ<τ0 and a *saturation* at τ>τ0. At first, the reoriented areas are relatively small and/or disconnected, such that only nucleation can proceed. Although nucleation is permitted, the extent of the crystalline fraction (the solid line) is negligible when compared with the reoriented fraction (the dotted line). At the moment τ\*, the reoriented areas become large enough and/or percolated so as to permit not only nucleation but also crystal growth, using both the arising and newly existing portions of the reoriented species. In this *transient* region τ\*<τ<τ0, crystallization slows gradually and at τ0 it becomes limited by the process of initial reorientation when each new portion of the reoriented species joins directly to the growing crystals.

One should note in Fig.2 the well known barriers *G'*, *G''* and *W\**. Fortunately, the problems with these barriers (e.g., varying *G'* after [18] or an anomalous increase of *W\** with a decreasing temperature at about *Tg* [24]) are outside of our method for the estimation of the CCR since *Vm* relates only to the initial reorientation stage with the Δ*G#* barrier which determines the boundary between glassy and crystalline states. Thus, *Vm* is the *upper* theoretical limit for the CCR and in this sense *Vm* satisfies the condition set by *Vreeswijk et al.* [4] concerning the absence of even one critical nucleus when determining the CCR.

By 'initial reorientation' is meant some kind of order. Interestingly, the notions concerning pre-crystalline ordering in glasses appear consistently in the literature. One can see the IReO in a real SEM image, which is interpreted by the authors as "a pre-crystallization stage, in which glass matrix becomes inhomogeneous, forming nano-sized volumes" [25]. Indirect evidence of the IReO is seen in the conclusion that "glass-transition kinetics can be treated as pre-crystallization kinetics" [26] as well as in the models of "dynamic heterogeneity" in an amorphous matrix before its crystallization [27-29]. The problem is that both "dynamic heterogeneity" and "initial reorientation" are the only terms that need decoding.

#### **2.3 Glass structure, hypervalent bonds and bond waves**

8 Crystallization – Science and Technology

Fig. 2. Isothermal crystallization after [21]: the dotted curve describes the development of the reoriented non-crystalline fraction; the solid line is the model temporal dependence of

**Atomic states**: **A** is the initial truly "amorphous" state for a species in a supercooled liquid or glass; **A0** is the reoriented state; **ANu** is the state of the crystal nucleus; **ACr** is the state of

The three intervals in Fig.2 correspond to an *induction period* at 0<τ<τ\*, a *transient period* at τ\*<τ<τ0 and a *saturation* at τ>τ0. At first, the reoriented areas are relatively small and/or disconnected, such that only nucleation can proceed. Although nucleation is permitted, the extent of the crystalline fraction (the solid line) is negligible when compared with the reoriented fraction (the dotted line). At the moment τ\*, the reoriented areas become large enough and/or percolated so as to permit not only nucleation but also crystal growth, using both the arising and newly existing portions of the reoriented species. In this *transient* region τ\*<τ<τ0, crystallization slows gradually and at τ0 it becomes limited by the process of initial reorientation when each new portion of the reoriented species joins directly to the growing

One should note in Fig.2 the well known barriers *G'*, *G''* and *W\**. Fortunately, the problems with these barriers (e.g., varying *G'* after [18] or an anomalous increase of *W\** with a decreasing temperature at about *Tg* [24]) are outside of our method for the estimation of the CCR since *Vm* relates only to the initial reorientation stage with the Δ*G#* barrier which determines the boundary between glassy and crystalline states. Thus, *Vm* is the *upper* theoretical limit for the CCR and in this sense *Vm* satisfies the condition set by *Vreeswijk et al.*

By 'initial reorientation' is meant some kind of order. Interestingly, the notions concerning pre-crystalline ordering in glasses appear consistently in the literature. One can see the IReO in a real SEM image, which is interpreted by the authors as "a pre-crystallization stage, in which glass matrix becomes inhomogeneous, forming nano-sized volumes" [25]. Indirect evidence of the IReO is seen in the conclusion that "glass-transition kinetics can be treated as pre-crystallization kinetics" [26] as well as in the models of "dynamic heterogeneity" in an amorphous matrix before its crystallization [27-29]. The problem is that both "dynamic

[4] concerning the absence of even one critical nucleus when determining the CCR.

heterogeneity" and "initial reorientation" are the only terms that need decoding.

**Barriers**: Δ*G*# is the initial orientation; *W\** is the critical nucleus; Δ*G'* is the steady-state

the extent of crystallinity α; the pointed curve is a real crystallization process.

the growing crystal.

crystals.

nucleation; Δ*G''* is the crystal growth.

The theory of glass structure was actually formed at 1932 when, beginning with the words *"It must be frankly admitted that we know practically nothing about the atomic arrangement in glasses,"* the young scientist *Zachariasen* proposed his famous model of a **continuous random network** (CRN) [30]. This network consists of covalent bonds, - the same as that in a related crystal - and the only difference is the "random" arrangement of the bonds in CRN in contrast with their regular arrangement in a crystalline network. As far as the atomic arrangement in CRN is determined by the arrangement of chemical bonds, one might suppose that only chemists - who deal with chemical bonds - would play a leading role in the further development of the CRN model. However, the development was directed in another "physical" way. As a result, the chemical bond is present in contemporary glass theory in the form of "frustration", "elastic force" and rigid "sticks" connecting atoms-balls, etc., but not as a real object with its own specificity. We think that a pure "physical approach" opens a rich field for theoretical speculation, rather than clarifying the nature of glass. For example, to understand glass transition, the analogues with spin glasses, granular systems and colloids are used (e.g., see the excellent review [31]), although any similarity in behaviour does not mean that the similarity is the reason behind that behaviour.

One can find, however, deviations from the main "physical" stream of glass science, even among physicians. As a fresh example, we may cite that: "*It is widely believed that crystallization in three dimensions is primary controlled by positional ordering, and not by bond orientational ordering. In other words, bond orientational ordering is usually considered to be merely a consequence of positional ordering and thus has often been ignored. … Here we proposed that bond orientational ordering can play a key role in (i) crystallization, (ii) the ordering to quasi-crystal and (iii) vitrification*" [29]. Although this is a rare case, it may be a sign that the need for a chemical approach is now stronger.

Our goal was to return the chemical bond - as chemists know it - to the theory of glass. As far back as in 1981, *Dembovsky* [32] has connected the experimental fact of an *increased* coordination number in glass forming melts with the model of *quasimolecular defects* (QMD) following *Popov* [33] for the creation of the chemical-bond basis for glass theory. The principal feature of a QMD is its hyper-coordinated nature, a property that provides for the connectivity of a covalent network even at melt temperature, thus resulting in high melt viscosity, which is a characteristic property of a glass forming melt. QMDs in a noncrystalline network can also explain general features of glass [34-38]. The problem is that the concentration of QMDs should be high enough, especially in a glass forming liquid, to provide these features. Therefore, we have quickly substituted the initial term QMD (quasimolecular *defect*) for TCB (three-centre *bond*) and then, being based on a special quantum-chemical study (see the reviews [39] for chalcogenide glasses and [40] for oxide ones), TCB for HVB, **hypervalent bond.** TCB is only a particular case of HVB, which is rarely realizes in glass. For example, in the simplest case of Se not TCB, having two threecoordinated atoms (-Se<) and one "normal" two-coordinated atom (-Se-) [33], but a fourcoordinated atom (>Se< )was revealed [41]. Thus, we have introduced *alternative bonds* into CRN, the bonds whose concentration is commensurable with that of ordinary covalent bonds constituting classical CRN.

The term "hypervalent", which was introduced by *Muscher* in 1969 [42], has a long and controversial history, beginning from the 1920s up until the present (see [43] for an introduction). Currently, a large number of hypervalent molecules is known, and various methods of their theoretical description exist. For a long time, one of the most popular was

Crystallization in Glass Forming Substances: The Chemical Bond Approach 11

Fig. 3. Bond waves after [46]. On the left: two adjacent wavefronts – layers "1" populated with a HVB (here modelled by a TCB: black atoms linked by spring) together with a CRN "2" between the layers (here an Se-like CRN, which consists of white atoms, each having two covalent bonds as shown by the lines). On the right: the intersection of three bond

A bond wave with a Λ wavelength represents a totality of d-layers, and the totality represents a periodic structure of the Λ period. Thus, there appears a Λ-lattice and a corresponding *longrange order* (LRO) in glass. This order is, in some respects, similar to the smectic-type LRO; however, in glass - in contrast with liquid crystal - there are two types of bonding species in the layer and the d-layers themselves are punctuated with thick pieces of the CRN. Such longrange periodicity is "invisible" to ordinary X-ray analysis, by which one can see a CRN (in the form of the peaks in the radial distribution function) and d-layers (in the form of a FSDP in an initial diffraction pattern), but not the Λ-lattice. The reflex from the Λ-lattice should be disposed in the 0.01-0.001Ǻ-1 range, which is inaccessible to the usual X-ray techniques. Thus, special techniques (e.g., those using synchrotron radiation) are needed to see a LRO in glass. One can note that the layer structure shown in the left side of Fig.3 is *anisotropic*, while glass is known to be an *isotropic* medium. Note, however, that anisotropic glasses can also be prepared (our own experiments of this sort will be considered below). The generally observed isotropic behaviour of glass means the *solitonic* behaviour of bond waves, i.e., the waves ability to intersect each other without distortion, as is shown on the right in Fig.3. One wave propagating through a CRN with the velocity *V1* forms the layer structure, two waves (*V1+V2*) create the columnar structure, and three waves (*V1+V2+V3*) correspond to the cellular structure, which is isotropic on the macroscopic scale. The factors which govern bond waves and, consequently, the structure that they form, can be divided into internal (chemical composition, temperature, pressure) and external (flows of energy and/or

As far as the elementary act CB↔HVB is considered to be a thermally activated process, both the wave frequency and the HVB concentration increase with temperature, while the wavelength *Λ* decreases [47]. Thus, when approaching the glass transition temperature *Tg*, the interlayer distance Λ becomes so large that the correlation between the layers/wavefronts becomes impossible, although the intimated HVBs within the layers continue to "feel" each

waves, which gives the 1-2-3 elementary cell.

information) categories.

the *Pimentel's* model [44] of the electron-rich three-centre four-electron (3c-4e) bond; this model was used by *Popov* to construct his "quasimolecular defect" in a covalent network of Se glass [33]. The principal step made by *Dembovsky* was in the understanding of QMD not as a "defect" but as the second type of bonding in a glassy network, the first being a common two-centre two-electron (2c-2e) covalent bond. The next step was made by means of *ab initio* quantum-chemical modelling, which reveals various metastable hypervalent configurations - configurations in which a central atom has more bonds with its neighbours than the "normal" surrounding covalent species - in glasses. We use the term *hypervalent bonds* in order to emphasize the additional bonding state in non-classical CRN, and so the additional structure possibilities in it. This network is not "random" now.

Three types of order can be realized in such a mixed network. When the diffraction pattern of glass is transformed into a radial distribution function one can observe the so-called *shortrange order* (SRO) extending to the limits of at least two coordination spheres around an arbitrary atom. The peak's position relates to the distances between the nearest neighbours and the peak square to the number of these neighbours. The SRO in glass is close (but not coinciding completely!) with the SRO in a counterpart crystal, with both SROs being determined by a bond length (first distance), valence (first coordination number) and valence angles (second coordination sphere). The SRO is the basis for a conventional continuous random network, but the problem is that a CRN consisting of covalent bonds cannot exist because of the rigid and directional character of the covalent bond. When moving from the first atom, the stress due to bond distortion accumulates rapidly up to a critical value above which the covalent bond should be destroyed (the covalent bond limits are known in the chemistry of related compounds). This means that a real CRN should contain either internal fractures (however, glass is known to be an optically transparent and mechanical stable material) or additional "soft" regions for relaxation. These regions are provided by HVB and are soft and flexible when compared with covalent bonds.

The next step of order which surely exists in a non-crystalline state is the so called *mediumrange order* (MRO), which is observed directly in the diffraction pattern of glass or glass forming liquids in the form of the well known *first sharp diffraction peak* (FSDP). It is a very narrow (for an amorphous state) peak located at about *Q1*=1-1,5Ǻ-1; the peak position and intensity depend upon the chemical composition, temperature and pressure (see [45,46] for a review). The two generally accepted parameters of the MRO are the correlation length *d=2π/Q1*≈6-4Ǻ and the coherence length *L= 2π/ΔQ1*≈10-20Ǻ, where *ΔQ1* is the FSDP half-width.

The nature of the FSDP/MRO is debatable at present. Our interpretation of the MRO is based on hypervalent bonds and their collective behaviour in the form of a *bond wave*, which is illustrated in **Fig.3**.

A bond wave is the spatiotemporal correlation of elementary acts of the reversible transformation of a covalent bond and a hypervalent bond: CB↔HVB. A snapshot of a bond wave, which spreads in the Se-like network, is shown in the left part of Fig.3. The wavefronts represent equidistant layers populated with hypervalent bonds (they are modelled by TCB in Fig.3), and so the correlation length *d=2π/Q1* roughly corresponds with the HVB length. The layers give a true Bragg reflex at *Q1=2π/d*, so the FSDP intensity and the FSDP width depend upon the number of these reflecting layers and their reflection ability, with both depending on the temperature.

the *Pimentel's* model [44] of the electron-rich three-centre four-electron (3c-4e) bond; this model was used by *Popov* to construct his "quasimolecular defect" in a covalent network of Se glass [33]. The principal step made by *Dembovsky* was in the understanding of QMD not as a "defect" but as the second type of bonding in a glassy network, the first being a common two-centre two-electron (2c-2e) covalent bond. The next step was made by means of *ab initio* quantum-chemical modelling, which reveals various metastable hypervalent configurations - configurations in which a central atom has more bonds with its neighbours than the "normal" surrounding covalent species - in glasses. We use the term *hypervalent bonds* in order to emphasize the additional bonding state in non-classical CRN, and so the

Three types of order can be realized in such a mixed network. When the diffraction pattern of glass is transformed into a radial distribution function one can observe the so-called *shortrange order* (SRO) extending to the limits of at least two coordination spheres around an arbitrary atom. The peak's position relates to the distances between the nearest neighbours and the peak square to the number of these neighbours. The SRO in glass is close (but not coinciding completely!) with the SRO in a counterpart crystal, with both SROs being determined by a bond length (first distance), valence (first coordination number) and valence angles (second coordination sphere). The SRO is the basis for a conventional continuous random network, but the problem is that a CRN consisting of covalent bonds cannot exist because of the rigid and directional character of the covalent bond. When moving from the first atom, the stress due to bond distortion accumulates rapidly up to a critical value above which the covalent bond should be destroyed (the covalent bond limits are known in the chemistry of related compounds). This means that a real CRN should contain either internal fractures (however, glass is known to be an optically transparent and mechanical stable material) or additional "soft" regions for relaxation. These regions are

additional structure possibilities in it. This network is not "random" now.

provided by HVB and are soft and flexible when compared with covalent bonds.

is illustrated in **Fig.3**.

with both depending on the temperature.

The next step of order which surely exists in a non-crystalline state is the so called *mediumrange order* (MRO), which is observed directly in the diffraction pattern of glass or glass forming liquids in the form of the well known *first sharp diffraction peak* (FSDP). It is a very narrow (for an amorphous state) peak located at about *Q1*=1-1,5Ǻ-1; the peak position and intensity depend upon the chemical composition, temperature and pressure (see [45,46] for a review). The two generally accepted parameters of the MRO are the correlation length *d=2π/Q1*≈6-4Ǻ and the coherence length *L= 2π/ΔQ1*≈10-20Ǻ, where *ΔQ1* is the FSDP half-width. The nature of the FSDP/MRO is debatable at present. Our interpretation of the MRO is based on hypervalent bonds and their collective behaviour in the form of a *bond wave*, which

A bond wave is the spatiotemporal correlation of elementary acts of the reversible transformation of a covalent bond and a hypervalent bond: CB↔HVB. A snapshot of a bond wave, which spreads in the Se-like network, is shown in the left part of Fig.3. The wavefronts represent equidistant layers populated with hypervalent bonds (they are modelled by TCB in Fig.3), and so the correlation length *d=2π/Q1* roughly corresponds with the HVB length. The layers give a true Bragg reflex at *Q1=2π/d*, so the FSDP intensity and the FSDP width depend upon the number of these reflecting layers and their reflection ability,

Fig. 3. Bond waves after [46]. On the left: two adjacent wavefronts – layers "1" populated with a HVB (here modelled by a TCB: black atoms linked by spring) together with a CRN "2" between the layers (here an Se-like CRN, which consists of white atoms, each having two covalent bonds as shown by the lines). On the right: the intersection of three bond waves, which gives the 1-2-3 elementary cell.

A bond wave with a Λ wavelength represents a totality of d-layers, and the totality represents a periodic structure of the Λ period. Thus, there appears a Λ-lattice and a corresponding *longrange order* (LRO) in glass. This order is, in some respects, similar to the smectic-type LRO; however, in glass - in contrast with liquid crystal - there are two types of bonding species in the layer and the d-layers themselves are punctuated with thick pieces of the CRN. Such longrange periodicity is "invisible" to ordinary X-ray analysis, by which one can see a CRN (in the form of the peaks in the radial distribution function) and d-layers (in the form of a FSDP in an initial diffraction pattern), but not the Λ-lattice. The reflex from the Λ-lattice should be disposed in the 0.01-0.001Ǻ-1 range, which is inaccessible to the usual X-ray techniques. Thus, special techniques (e.g., those using synchrotron radiation) are needed to see a LRO in glass.

One can note that the layer structure shown in the left side of Fig.3 is *anisotropic*, while glass is known to be an *isotropic* medium. Note, however, that anisotropic glasses can also be prepared (our own experiments of this sort will be considered below). The generally observed isotropic behaviour of glass means the *solitonic* behaviour of bond waves, i.e., the waves ability to intersect each other without distortion, as is shown on the right in Fig.3. One wave propagating through a CRN with the velocity *V1* forms the layer structure, two waves (*V1+V2*) create the columnar structure, and three waves (*V1+V2+V3*) correspond to the cellular structure, which is isotropic on the macroscopic scale. The factors which govern bond waves and, consequently, the structure that they form, can be divided into internal (chemical composition, temperature, pressure) and external (flows of energy and/or information) categories.

As far as the elementary act CB↔HVB is considered to be a thermally activated process, both the wave frequency and the HVB concentration increase with temperature, while the wavelength *Λ* decreases [47]. Thus, when approaching the glass transition temperature *Tg*, the interlayer distance Λ becomes so large that the correlation between the layers/wavefronts becomes impossible, although the intimated HVBs within the layers continue to "feel" each

Crystallization in Glass Forming Substances: The Chemical Bond Approach 13

In this way, not only is a principal ability to form a bond wave justified, so is the connection of a *structural order* due to a Λ-lattice with an "*energetic order*", which is based on the intuitive belief that the more structurally ordered substance has the lower energy (with other conditions being equal). As such, the processes of the crystallization of glass, both below and above *Tg*, can be considered as the competition of two types of order: a crystalline long-range order and a specific non-crystalline long-range order, provided by hypervalent bonds and bond waves. In what follows, these notions are tested by means of special crystallization experiments.

For these experiments, we chose selenium as the simplest one-element glass; the additions, with various valence abilities, were introduced into the Se matrix, giving five series of Se-X (X = S, Te, As, Ge, Cl) with a varied but relatively low concentration of the second component. The as-prepared samples were cylinders measuring 25 mm in diameter and 15 mm in height; the two cylinder ends were polished. The optical transmission and ultrasonic velocity were measured through the ends; the X-ray fluorescent spectra were measured

**The as-prepared samples** were quite transparent, actually having the same value of transmission at 1000 cm-1 (this value, which corresponds with entry into the so-called "window of transparency" for selenide glasses, we shall call *transparency*) of around 60%, a value that is typical for chalcogenide glasses of high quality for the given thickness of 15 mm. The two other properties (*V* and *r*) investigated in the fresh glasses, however, were strongly dependent upon composition, as is seen in **Fig.5**. Note that the both properties are *macroscopic* in character. The ultrasound velocity, *V*, characterizes the elastic ability of the Sebased network. The relative intensity of the X-ray fluorescence, *r*=Sval/Schar (Sval and Schar are the integral intensities of the Kβ2 and Kβ2 emission lines for Se, the first corresponding to the 4p→1s transition from a valence band and the second to the characteristic 3p→1s inner transition for Se), belongs to the totality of selenium atoms, reflecting the average valence

We see a strong non-linear character for all of the dependencies in Fig.5. The two upper cases of Se-Te and Se-S seem to be the most surprising because S and Te belong to the same VI group of the Periodic Table, having the same number of covalently bonded neighbours per atom. Additionally, the Se-Te phase diagram is a simple "fish", which corresponds with the discontinuous series of liquid and solid solutions. However, the *metastable phase diagram* (remember that glass is formed far below the melting point from a metastable liquid) has a "two-fish" form, with "two series of solid solutions meeting at 96.8 at%Se [i.e., 3.2 at%Te –

Note that the Se-Te glasses display extrema in the 1-4%Te range for the other compositionproperty dependences, e.g., for electrical and crystallization properties [56] and for the glass transition temperature [57]. *Dembovsky et al*. [58] have also revealed non-linearity when investigating crystallization kinetics in Se-Te glasses in the 0-5% range. In **Fig.6**, the data for CCRs calculated with the use of the data obtained is shown; one can see not only the conditional character of the CCR, the composition dependence of which depends on the chosen α, but also the non-linearity of *Q(N)* for every chosen extent of crystallinity with the extrema located in the region 1-5%Te. Note that our calculation of the CCR in the form of *Vm*

**3. Crystallization in solid glass (T<Tg)** 

from the end surface.

state for Se.

ECh] and temperature 180OC" [55].

**3.1 The composition dependent rate of nucleation** 

other. This means that a 3D bond wave (Fig.3, right side) stops its propagation through the structure and the 2D bond wave within the stopped d-layers (Fig.3, left side) remains mobile. Thus, the glass transition can be considered as a 3D→2D bond wave transition [48].

As far HVBs represent active sites in the mixed CB/HVB network, one should distinguish between the processes above and below *Tg*. More specifically, the abovementioned prenucleation stage in the form of initial reorientation (IReO) proceeds within the stopped dlayers below *Tg* and only after the reconstruction of the layers can it penetrate into the CRN: the process is slow and has an induction period (see τ<τ0 in Fig.2). Above *Tg*, where the 3D bond waves exist, the d-layers pass through every structure element: the induction period is short, if it even exists, and the process is much faster and homogeneous.

Crystallization is not the first property considered by means of the bond wave model. Earlier, this model was applied successively for the interpretation of the thermodynamic features of glass forming substances [49], characteristic glass fractures [50], the first sharp diffraction peaks [46] and the temperature dependence of viscosity [51]. Unfortunately, there are only interpretations and so far there is no direct evidence of bond waves at present, i.e., the direct observation of a Λ-lattice in a structural experiment. Instead, we performed a computational experiment [52], presented in **Fig.4**, in which we intended to investigate the ability of the HVB for association – this property alone is a necessary requirement for the existence of a bond wave.

In model clusters like those shown in Fig.4, a single HVB looks like a "defect" embedded into an ordinary continuous random network (CRN). A single HVB was shown to be a lowenergy "defect" (compare 0,3 eV for Se40 [41], with 2 eV needed to generate a broken bond 2Se2→2Se1 or with 1 eV proposed for the so called "valence alternation pair" Se3 +Se1- after [53], the index below is the coordination and the index above is the charge). Nevertheless, even such low-energy defects cannot ensure the above-mentioned over-coordination in glass (there needs to be ~1% for four-coordinated atoms) and especially so in a melt (ten%) [37]. From Fig.4, it follows that the associated HVBs are much more stable, so that even *negative energy* regions can arise in a CRN for a definite HVB arrangement (SS).

Fig. 4. Quantum-chemical modelling of the HVB interaction in Se after *Zyubin & Dembovsky* [52]. The cluster energies are **−0.16 eV** (SS), +0.08 eV (LS) and +0.09 eV (LL); the notations correspond with the mutual orientation of short (S) and long (L) pairs of bonds surrounding each Se4 0 atom (1 or 4). The white spheres are the hydrogen atoms terminating the clusters.

In this way, not only is a principal ability to form a bond wave justified, so is the connection of a *structural order* due to a Λ-lattice with an "*energetic order*", which is based on the intuitive belief that the more structurally ordered substance has the lower energy (with other conditions being equal). As such, the processes of the crystallization of glass, both below and above *Tg*, can be considered as the competition of two types of order: a crystalline long-range order and a specific non-crystalline long-range order, provided by hypervalent bonds and bond waves. In what follows, these notions are tested by means of special crystallization experiments.

## **3. Crystallization in solid glass (T<Tg)**

12 Crystallization – Science and Technology

other. This means that a 3D bond wave (Fig.3, right side) stops its propagation through the structure and the 2D bond wave within the stopped d-layers (Fig.3, left side) remains mobile.

As far HVBs represent active sites in the mixed CB/HVB network, one should distinguish between the processes above and below *Tg*. More specifically, the abovementioned prenucleation stage in the form of initial reorientation (IReO) proceeds within the stopped dlayers below *Tg* and only after the reconstruction of the layers can it penetrate into the CRN: the process is slow and has an induction period (see τ<τ0 in Fig.2). Above *Tg*, where the 3D bond waves exist, the d-layers pass through every structure element: the induction period is

Crystallization is not the first property considered by means of the bond wave model. Earlier, this model was applied successively for the interpretation of the thermodynamic features of glass forming substances [49], characteristic glass fractures [50], the first sharp diffraction peaks [46] and the temperature dependence of viscosity [51]. Unfortunately, there are only interpretations and so far there is no direct evidence of bond waves at present, i.e., the direct observation of a Λ-lattice in a structural experiment. Instead, we performed a computational experiment [52], presented in **Fig.4**, in which we intended to investigate the ability of the HVB for association – this property alone is a necessary

In model clusters like those shown in Fig.4, a single HVB looks like a "defect" embedded into an ordinary continuous random network (CRN). A single HVB was shown to be a lowenergy "defect" (compare 0,3 eV for Se40 [41], with 2 eV needed to generate a broken bond 2Se2→2Se1 or with 1 eV proposed for the so called "valence alternation pair" Se3+Se1- after [53], the index below is the coordination and the index above is the charge). Nevertheless, even such low-energy defects cannot ensure the above-mentioned over-coordination in glass (there needs to be ~1% for four-coordinated atoms) and especially so in a melt (ten%) [37]. From Fig.4, it follows that the associated HVBs are much more stable, so that even *negative* 

Fig. 4. Quantum-chemical modelling of the HVB interaction in Se after *Zyubin & Dembovsky* [52]. The cluster energies are **−0.16 eV** (SS), +0.08 eV (LS) and +0.09 eV (LL); the notations correspond with the mutual orientation of short (S) and long (L) pairs of bonds surrounding each Se40 atom (1 or 4). The white spheres are the hydrogen atoms terminating the clusters.

Thus, the glass transition can be considered as a 3D→2D bond wave transition [48].

short, if it even exists, and the process is much faster and homogeneous.

*energy* regions can arise in a CRN for a definite HVB arrangement (SS).

requirement for the existence of a bond wave.

## **3.1 The composition dependent rate of nucleation**

For these experiments, we chose selenium as the simplest one-element glass; the additions, with various valence abilities, were introduced into the Se matrix, giving five series of Se-X (X = S, Te, As, Ge, Cl) with a varied but relatively low concentration of the second component. The as-prepared samples were cylinders measuring 25 mm in diameter and 15 mm in height; the two cylinder ends were polished. The optical transmission and ultrasonic velocity were measured through the ends; the X-ray fluorescent spectra were measured from the end surface.

**The as-prepared samples** were quite transparent, actually having the same value of transmission at 1000 cm-1 (this value, which corresponds with entry into the so-called "window of transparency" for selenide glasses, we shall call *transparency*) of around 60%, a value that is typical for chalcogenide glasses of high quality for the given thickness of 15 mm. The two other properties (*V* and *r*) investigated in the fresh glasses, however, were strongly dependent upon composition, as is seen in **Fig.5**. Note that the both properties are *macroscopic* in character. The ultrasound velocity, *V*, characterizes the elastic ability of the Sebased network. The relative intensity of the X-ray fluorescence, *r*=Sval/Schar (Sval and Schar are the integral intensities of the Kβ2 and Kβ2 emission lines for Se, the first corresponding to the 4p→1s transition from a valence band and the second to the characteristic 3p→1s inner transition for Se), belongs to the totality of selenium atoms, reflecting the average valence state for Se.

We see a strong non-linear character for all of the dependencies in Fig.5. The two upper cases of Se-Te and Se-S seem to be the most surprising because S and Te belong to the same VI group of the Periodic Table, having the same number of covalently bonded neighbours per atom. Additionally, the Se-Te phase diagram is a simple "fish", which corresponds with the discontinuous series of liquid and solid solutions. However, the *metastable phase diagram* (remember that glass is formed far below the melting point from a metastable liquid) has a "two-fish" form, with "two series of solid solutions meeting at 96.8 at%Se [i.e., 3.2 at%Te – ECh] and temperature 180OC" [55].

Note that the Se-Te glasses display extrema in the 1-4%Te range for the other compositionproperty dependences, e.g., for electrical and crystallization properties [56] and for the glass transition temperature [57]. *Dembovsky et al*. [58] have also revealed non-linearity when investigating crystallization kinetics in Se-Te glasses in the 0-5% range. In **Fig.6**, the data for CCRs calculated with the use of the data obtained is shown; one can see not only the conditional character of the CCR, the composition dependence of which depends on the chosen α, but also the non-linearity of *Q(N)* for every chosen extent of crystallinity with the extrema located in the region 1-5%Te. Note that our calculation of the CCR in the form of *Vm*

Crystallization in Glass Forming Substances: The Chemical Bond Approach 15

4,9 5 5,1 5,2 5,3 5,4 5,5 5,6

*r*, %

Fig. 5. Longitudinal ultrasound velocity (*V*) and relative intensity of X-ray fluorescence from

[21,22], which corresponds to the pre-crystallization IReO stage, really represents the upper limit for the CCR, but only outside the non-linearity range. Remember, however, that when calculating *Vm* we only take into account the *equilibrium* fish-like phase diagram, whereas

Fig. 6. Critical cooling rates (CCRs) in the Se-Te system after [58]. *Qr, Q0.01* and *Qc* correspond to the crystalline fraction α=0,63; 0,01 and 10-6 respectively. *Vm* is our calculation after

The Se-S series in Fig.5 looks similar to Se-Te, although one can propose a more complex behaviour below 2%S, which can be compared with a more complex equilibrium Se-S phase

In the Se-As and Se-Ge series, the additions belong to other groups of Periodic Table (IV for Ge and V for As), so one can expect another property-composition behaviour. Accordingly,

real glass relates more to metastable diagram(s) with peculiarities at about 5%Te.

0 0,1 0,2 0,3 **X,** at%

**Cl**

**Cl**

*V,* 

м/с

0 0,1 0,2 0,3 **X**, at%

[21,22], without the use of crystallization data.

diagram [59] in contrast with the simple "fish" for Se-Te.

the Se valence band (*r*) in the as-prepared Se-X glasses after [54].

5,1

5

4 4,5 5 5,5 6 6,5 7

5,2

5,3

5,4

*r*, %

5,5

5,6

*r*, %

5,1

5,2

*r*, %

5,3

5,4

5,5

0 2 4 6 8 10

**S**

0 2 4 6 8 10

**As**

**Ge**

012345

0 2 4 6 8 10

**Te**

5,3

*r*, %

5,5

5,7

1850

*V* , м/с

*V*, м/с

*V*, м/с

0 2 4 6 8 10

**S**

0 2 4 6 8 10

**As**

**Ge**

012345

0 2 4 6 8 10

**Te**

1860

1870

*V* , м/с

1880

1890

Fig. 5. Longitudinal ultrasound velocity (*V*) and relative intensity of X-ray fluorescence from the Se valence band (*r*) in the as-prepared Se-X glasses after [54].

[21,22], which corresponds to the pre-crystallization IReO stage, really represents the upper limit for the CCR, but only outside the non-linearity range. Remember, however, that when calculating *Vm* we only take into account the *equilibrium* fish-like phase diagram, whereas real glass relates more to metastable diagram(s) with peculiarities at about 5%Te.

Fig. 6. Critical cooling rates (CCRs) in the Se-Te system after [58]. *Qr, Q0.01* and *Qc* correspond to the crystalline fraction α=0,63; 0,01 and 10-6 respectively. *Vm* is our calculation after [21,22], without the use of crystallization data.

The Se-S series in Fig.5 looks similar to Se-Te, although one can propose a more complex behaviour below 2%S, which can be compared with a more complex equilibrium Se-S phase diagram [59] in contrast with the simple "fish" for Se-Te.

In the Se-As and Se-Ge series, the additions belong to other groups of Periodic Table (IV for Ge and V for As), so one can expect another property-composition behaviour. Accordingly,

Crystallization in Glass Forming Substances: The Chemical Bond Approach 17

expected crystallization, the **aged glasses** became less transparent, but not to the same extent. It can be seen in **Fig 7** that the transparency shows a non-linear character, like that shown in Fig.5 for ultrasound velocity and electron emission from the valence Se band. Thus, one can suppose that even in fresh Se-X glasses the special regions of initial reorientation (IReO) for further crystallization have been developed. These regions manifested themselves in *V(N)* and *r(N)* in Fig.5, but not in the transparency because the IReO regions were congruently inserted into a glassy matrix. During the process of ageing, the IReO regions are transformed into crystalline regions and so become visible due to light

The question is: what stage of crystallization do we observe in the aged glasses? The X-ray pattern of the aged samples usually looked like a wide hill of a low intensity; only the darkest samples display very weak crystalline peaks on the hill. Thus, because of the low extent of crystallinity (usually lower than 1%) together with the considerable darkening, we can conclude that by the optical transmission method we can observe the *nucleation stage* of crystallization. Next, from Fig.7 one can conclude that compositions of 1%Te and 5%Te,

Nucleation in glass is known to be usually *heterogeneous*, i.e., the nuclei tend to appear on the surface of a sample rather than distribute evenly in the volume. Note that only the case of homogeneous nucleation is considered in classical crystallization kinetics, a fact that creates additional problems when comparing theory with experiment for such non-classical objects

In order to evaluate heterogeneity in a numerical way, we have elaborated a simple method which includes the removal of the surface layer [61]. The main effect of the removal is the resection of the surface nuclei with the consequent rise of transparency. Next, the extent of

where *T1* is transparency (i.e., the optical transmission at 1000 cm-1) for the as-prepared glass (of 15 mm thickness), *T2* is the transparency after ageing and *T3* is the transparency of the aged sample after the removal of the surface layer (0.5 mm from each side of the cylinder, a procedure that gives a sample of 14 mm thickness). The method is illustrated on the Se-Te

From Fig.8, it can be seen that the distribution of crystalline nuclei is most homogeneous in the Se-Ge system having G=0-0.2. The negative *G* for the 5%Ge sample may indicate the "inverse heterogeneity", a situation whereby nuclei prefer to burn within the volume rather than at the surface. The opposite deviation G>1, which is observed in the Se-Cl series for the 0.05%Cl sample when the volume becomes more transparent after ageing (T3>T1), may be a result of the sample thinning. Another possibility, of *non-crystalline ordering* during ageing, seems to be more interesting. The reason for this ordering may be the development of the

One can note a remarkable similarity in the *r(N)* graphs in Fig.5 for the as-prepared glasses and the *G(N)* graphs in Fig.8 for the aged glasses, and not only in the extrema location,

IReO regions in the glassy matrix (see the dotted line in Fig.2) without nucleation.

*G* = (*T3*-*T2*)/*T1* , (2)

5%S, 1%Ge, 2%As and (surprisingly!) 0.01%Cl are the most stable against nucleation.

scattering from new crystal/glass boundaries.

like glass.

heterogeneity *G* can be evaluated as:

**3.2 The composition dependent extent of heterogeneity** 

series as an example in the left-top and left-bottom (Te) of Fig.8.

the *V(N)* and *r(N)* dependencies change in the same way in Se-As glasses (maximum on the both), in contrast with the behaviour in Se-Te and Se-S glasses, which display a maximum on *V(N)* and a minimum on *r(N)*. Although Se-Ge glasses are similar to Se-Te as concerns the maximum on *V(N)* at 2%Ge and the minimum on *r(N)* at about 1%Ge, the extrema in Se-Ge are much sharper and there is different post-extreme behaviour in Se-Ge as compared with Se-S and Se-Te.

The Se-Cl glasses are of a special interest because, in contrast with the above four additions which belong to the well recognized glass forming Se-X systems, chlorine in *not* a glass forming addition for Se. It is usually assumed that Cl can only break the Se-Se bonds, thereby decreasing viscosity and thus the glass forming ability. In fact, Se-Cl glasses can be only prepared in the 0-0.3%Cl range, but in other respects the Se-Cl series demonstrates the same behaviour: the as-prepared Se-Cl glasses are quite transparent (about 60%) and their properties are strongly non-linear (see Fig 5, at the bottom).

Fig. 7. Darkening in Se-X glasses after 5-year ageing at room temperature [60]. Here, *T* is the transparency (optical transmission at 1000 cm-1); index "1" refers to the as-prepared glass and "2" to the aged glass. The unchanged transparency, i.e., the absence of darkening, corresponds to (T2/T1)=1 and complete darkening to (T2/T1)=0.

Next, the five series of glasses were stored for 5 years at room temperature, which is somewhat below the glass transition temperature (Tg≈35ºC for Se). In accordance with the

the *V(N)* and *r(N)* dependencies change in the same way in Se-As glasses (maximum on the both), in contrast with the behaviour in Se-Te and Se-S glasses, which display a maximum on *V(N)* and a minimum on *r(N)*. Although Se-Ge glasses are similar to Se-Te as concerns the maximum on *V(N)* at 2%Ge and the minimum on *r(N)* at about 1%Ge, the extrema in Se-Ge are much sharper and there is different post-extreme behaviour in Se-Ge as compared

The Se-Cl glasses are of a special interest because, in contrast with the above four additions which belong to the well recognized glass forming Se-X systems, chlorine in *not* a glass forming addition for Se. It is usually assumed that Cl can only break the Se-Se bonds, thereby decreasing viscosity and thus the glass forming ability. In fact, Se-Cl glasses can be only prepared in the 0-0.3%Cl range, but in other respects the Se-Cl series demonstrates the same behaviour: the as-prepared Se-Cl glasses are quite transparent (about 60%) and their

> 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

**T2 T/ 1**

0 0,05 0,1 0,15 0,2 0,25 0,3 **N(Cl),** at%

Fig. 7. Darkening in Se-X glasses after 5-year ageing at room temperature [60]. Here, *T* is the transparency (optical transmission at 1000 cm-1); index "1" refers to the as-prepared glass and "2" to the aged glass. The unchanged transparency, i.e., the absence of darkening,

Next, the five series of glasses were stored for 5 years at room temperature, which is somewhat below the glass transition temperature (Tg≈35ºC for Se). In accordance with the

012345 **N(As,Ge),** at%

**As**

**Ge**

properties are strongly non-linear (see Fig 5, at the bottom).

**Te**

0 1 2 3 4 5 6 7 8 9 10 **N(Te,S),** at%

0,0

corresponds to (T2/T1)=1 and complete darkening to (T2/T1)=0.

0,2

0,4

**T2 T/ 1** 0,6

0,8

**Cl**

1,0

**S**

with Se-S and Se-Te.

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

**T2 T/ 1** expected crystallization, the **aged glasses** became less transparent, but not to the same extent. It can be seen in **Fig 7** that the transparency shows a non-linear character, like that shown in Fig.5 for ultrasound velocity and electron emission from the valence Se band. Thus, one can suppose that even in fresh Se-X glasses the special regions of initial reorientation (IReO) for further crystallization have been developed. These regions manifested themselves in *V(N)* and *r(N)* in Fig.5, but not in the transparency because the IReO regions were congruently inserted into a glassy matrix. During the process of ageing, the IReO regions are transformed into crystalline regions and so become visible due to light scattering from new crystal/glass boundaries.

The question is: what stage of crystallization do we observe in the aged glasses? The X-ray pattern of the aged samples usually looked like a wide hill of a low intensity; only the darkest samples display very weak crystalline peaks on the hill. Thus, because of the low extent of crystallinity (usually lower than 1%) together with the considerable darkening, we can conclude that by the optical transmission method we can observe the *nucleation stage* of crystallization. Next, from Fig.7 one can conclude that compositions of 1%Te and 5%Te, 5%S, 1%Ge, 2%As and (surprisingly!) 0.01%Cl are the most stable against nucleation.

#### **3.2 The composition dependent extent of heterogeneity**

Nucleation in glass is known to be usually *heterogeneous*, i.e., the nuclei tend to appear on the surface of a sample rather than distribute evenly in the volume. Note that only the case of homogeneous nucleation is considered in classical crystallization kinetics, a fact that creates additional problems when comparing theory with experiment for such non-classical objects like glass.

In order to evaluate heterogeneity in a numerical way, we have elaborated a simple method which includes the removal of the surface layer [61]. The main effect of the removal is the resection of the surface nuclei with the consequent rise of transparency. Next, the extent of heterogeneity *G* can be evaluated as:

$$\mathbf{G} = (T\_3 \mathbf{-} T\_2) / T\_1 \,\tag{2}$$

where *T1* is transparency (i.e., the optical transmission at 1000 cm-1) for the as-prepared glass (of 15 mm thickness), *T2* is the transparency after ageing and *T3* is the transparency of the aged sample after the removal of the surface layer (0.5 mm from each side of the cylinder, a procedure that gives a sample of 14 mm thickness). The method is illustrated on the Se-Te series as an example in the left-top and left-bottom (Te) of Fig.8.

From Fig.8, it can be seen that the distribution of crystalline nuclei is most homogeneous in the Se-Ge system having G=0-0.2. The negative *G* for the 5%Ge sample may indicate the "inverse heterogeneity", a situation whereby nuclei prefer to burn within the volume rather than at the surface. The opposite deviation G>1, which is observed in the Se-Cl series for the 0.05%Cl sample when the volume becomes more transparent after ageing (T3>T1), may be a result of the sample thinning. Another possibility, of *non-crystalline ordering* during ageing, seems to be more interesting. The reason for this ordering may be the development of the IReO regions in the glassy matrix (see the dotted line in Fig.2) without nucleation.

One can note a remarkable similarity in the *r(N)* graphs in Fig.5 for the as-prepared glasses and the *G(N)* graphs in Fig.8 for the aged glasses, and not only in the extrema location,

Crystallization in Glass Forming Substances: The Chemical Bond Approach 19

bond waves which spread in non-crystalline network during the glass preparation (3D waves)

From a phenomenological point of view, one can distinguish between "*impurity*" (nonlinear) and "*component*" (smooth) concentration regions [54]. The boundary concentration, after which non-linearity begins to vanish, depends upon the system; it is about 2% for Se-S, Se-Te and Se-Ge glasses, about 1% for Se-As glasses, and about 0.05% for Se-Cl glasses (Figs.5-8). Let us discuss a possible nature of non-linearity in a low-concentration

Starting from classical CRN [30], one cannot wait for any singularities except for those stipulated by a phase diagram, and it is generally accepted that foreign atoms enter into a CRN with "normal" valence (e.g., Te forms the –Se-Se-**Te**-Se-Se- chains), thus causing a proportional change in glass properties. The above-demonstrated non-linearity means that this is not the case and, hence, that the valence might be unusual. We have a model of *hypervalent bonds* - HVBs - after Dembovsky (section 2.3) for just this situation. It is clear, however, that the HVBs in a concentration of about 1% cannot explain the *macroscopic* effects observed. This is the *bond wave* model for the case of the collective interaction between "normal" bonds and HVBs. Next, the question about the nature of non-linearity should be reformulated: how do selenium bond waves interact with foreign atoms in a selenium network? Let us begin with the quantum-chemical modelling of the simplest case of Se-Te.

Fig. 9. A quantum-chemical modelling of HVBs in a selenium clusters containing Te atoms after [62]. The energies below are compared with the selenium CRN of the same size.

It can be seen in **Fig.9** that even a simple hypervalent Te40 decreases the energy of the surrounding network (-0.23 eV). When the selenium wavefront (see d-layer in Fig.3, on the left) moves closer, a more stable Se40Te40 (-0.65 eV) is formed. Hence, a hypervalent Te atom represents an energetic trap for the selenium bond wave; however, this is so for only that local part of the wavefront, a part that interacts with the atom. When the Te concentration is high enough, the atoms can form hypervalent associates with the additional decrease in

and ageing (2D waves). What is a mechanism for such an interaction?

**3.3 Non-linearity and self-organization in glass** 

"impurity" region.

energy - see dimer 2(Te40) in Fig.9.

which coincides for every Se-X series, but also in the dependence shape which is qualitatively the same for the Se-Te, Se-As and Se-Cl series. For the Se-S and Se-Ge series, the shapes are somewhat different. One should note, however, that the Se-S glass is a special case because of the introduction of sulphur with *Tg*= -25oC, which would decrease the glass transition temperature in the series. Accordingly, Se-S glasses are especially inclined to crystallization when ageing at room temperature.

Fig. 8. The extent of heterogeneity, *G* by eq.(2), for spontaneous nucleation in Se-X glasses. *Ti* is the optical transmission at 1000 cm-1 for the as-prepared glass (1), the aged glass (2) and the aged glass without a surface layer (3).

In spite of these deviations, a non-trivial relation between the inclination to heterogeneous nucleation, *G(N)*, and a change in chemical bonding, reflecting by *r(N)*, is observed. Note that by means of the X-ray fluorescence method we probe only the *surface* layer of the sample; therefore, there exists a direct connection between *nucleation ability* and the *density of valence electrons*, with which the value of *r(N)* can be related. This is a direct route to hypervalent electron-rich bonds, which were discussed in section 2.3: the higher the electron density, the easier the formation of such HVBs representing active centres in covalent network, the centres which provide for processes including crystallization. The concentration of HVBs may be high owing to the decrease of energy when HVBs are associated (see Fig.4), probably in the form of bond waves. Thus, foreign atoms can interact strongly and non-linearly with the bond waves which spread in non-crystalline network during the glass preparation (3D waves) and ageing (2D waves). What is a mechanism for such an interaction?

#### **3.3 Non-linearity and self-organization in glass**

18 Crystallization – Science and Technology

which coincides for every Se-X series, but also in the dependence shape which is qualitatively the same for the Se-Te, Se-As and Se-Cl series. For the Se-S and Se-Ge series, the shapes are somewhat different. One should note, however, that the Se-S glass is a special case because of the introduction of sulphur with *Tg*= -25oC, which would decrease the glass transition temperature in the series. Accordingly, Se-S glasses are especially inclined to

> -0,2 0 0,2 0,4 0,6 0,8 1

0 0,2 0,4 0,6 0,8 1 1,2

**G**

Fig. 8. The extent of heterogeneity, *G* by eq.(2), for spontaneous nucleation in Se-X glasses. *Ti* is the optical transmission at 1000 cm-1 for the as-prepared glass (1), the aged glass (2) and

In spite of these deviations, a non-trivial relation between the inclination to heterogeneous nucleation, *G(N)*, and a change in chemical bonding, reflecting by *r(N)*, is observed. Note that by means of the X-ray fluorescence method we probe only the *surface* layer of the sample; therefore, there exists a direct connection between *nucleation ability* and the *density of valence electrons*, with which the value of *r(N)* can be related. This is a direct route to hypervalent electron-rich bonds, which were discussed in section 2.3: the higher the electron density, the easier the formation of such HVBs representing active centres in covalent network, the centres which provide for processes including crystallization. The concentration of HVBs may be high owing to the decrease of energy when HVBs are associated (see Fig.4), probably in the form of bond waves. Thus, foreign atoms can interact strongly and non-linearly with the

**G**

012345 **N(As,Ge)**, at%

0 0,1 0,2 0,3 **N(Cl)**, at%

**As**

**Ge**

**Cl**

crystallization when ageing at room temperature.

**2**

**1**

**3**

0 2 4 6 8 10 **N(Te),** at**%**

0 2 4 6 8 10 **N(Te,S)**, at%

the aged glass without a surface layer (3).

0

0,2

**Te**

**S**

0,4

0,6

**G**

0,8

1

**T,** 

%

From a phenomenological point of view, one can distinguish between "*impurity*" (nonlinear) and "*component*" (smooth) concentration regions [54]. The boundary concentration, after which non-linearity begins to vanish, depends upon the system; it is about 2% for Se-S, Se-Te and Se-Ge glasses, about 1% for Se-As glasses, and about 0.05% for Se-Cl glasses (Figs.5-8). Let us discuss a possible nature of non-linearity in a low-concentration "impurity" region.

Starting from classical CRN [30], one cannot wait for any singularities except for those stipulated by a phase diagram, and it is generally accepted that foreign atoms enter into a CRN with "normal" valence (e.g., Te forms the –Se-Se-**Te**-Se-Se- chains), thus causing a proportional change in glass properties. The above-demonstrated non-linearity means that this is not the case and, hence, that the valence might be unusual. We have a model of *hypervalent bonds* - HVBs - after Dembovsky (section 2.3) for just this situation. It is clear, however, that the HVBs in a concentration of about 1% cannot explain the *macroscopic* effects observed. This is the *bond wave* model for the case of the collective interaction between "normal" bonds and HVBs. Next, the question about the nature of non-linearity should be reformulated: how do selenium bond waves interact with foreign atoms in a selenium network? Let us begin with the quantum-chemical modelling of the simplest case of Se-Te.

Fig. 9. A quantum-chemical modelling of HVBs in a selenium clusters containing Te atoms after [62]. The energies below are compared with the selenium CRN of the same size.

It can be seen in **Fig.9** that even a simple hypervalent Te4 0 decreases the energy of the surrounding network (-0.23 eV). When the selenium wavefront (see d-layer in Fig.3, on the left) moves closer, a more stable Se40Te40 (-0.65 eV) is formed. Hence, a hypervalent Te atom represents an energetic trap for the selenium bond wave; however, this is so for only that local part of the wavefront, a part that interacts with the atom. When the Te concentration is high enough, the atoms can form hypervalent associates with the additional decrease in energy - see dimer 2(Te40) in Fig.9.

Crystallization in Glass Forming Substances: The Chemical Bond Approach 21

existing in the sample 0.01%Cl, which is most persistent for nucleation (Fig.7), and in the sample 0.05%Cl, which has the greatest capacity for heterogeneous nucleation (Fig.8). Interestingly, the same 792 cm-1 band also observed in the Se-As series for the 0.25%As sample (compared with the minimal heterogeneity in Fig.8) and unexpectedly at 5%As [65] (one can also see the extrema at the same concentrations of 0.01-0.05%Cl and 0.025%As in other properties for fresh glasses in Fig.5). Hence, the *ordering* which reveals itself as the 792 cm-1 narrow band can be related to the selenium matrix, the structure of which is modulated by means of the impurity introduced into it. Interestingly, the ordering can manifest in properties in the opposite manner, depending upon the impurity, as it is seen in Fig.5 for the

**0.01%Cl**

**0,05%Cl**

Fig. 11. IR spectra of Se-Cl glasses within the "phonon" range (400-1000 cm-1) after [60]. *T* is the transmission at a given frequency, *T*(1000) is the transmission at 1000 cm-1, i.e., the

When analysing the properties, a thoughtful reader can find out the disagreement in the *r*(0) values in Fig.5 and in the *T2/T1* values in Fig.7 for pure selenium glass (X=0) in different Se-X systems, a fact that may devaluate the data itself. The explanation is rather simple: different series were formed in different regimes: the first regime was applied for Se-Te and Se-Ge, the second one for Se-As and Se-S, and the third one for Se-Cl. Indeed, Se glasses have similar properties in the series of the same preparation. This is the *memory effect*, which is

The non-linearity, spontaneous ordering and the memory effect discussed above are signs of a *self-organizing* system, a system that possesses various scenarios for evolution, depending upon the system's nature and the information provided by external medium [66]. When glass is considered as a self-organizing system with bond waves as the basis for self-

700 720 740 760 780 800 **,** cm-1

0,1%Cl

extrema at 0.01-0.05%Cl and 0.025%As .

0

well known for anyone who works with glasses.

0,2

0,4

0,6

**T/T(1000)**

*transparency* (see Fig.7 and Fig.8).

0,8

1

1,2

The result of such an interaction is currently unclear. Given only the traps, foreign HVBs such as Te40 - should slow down or even destroy a bond wave, most probably to the extent that it is proportional to their concentration. However, we observe both maxima and minima in concentration dependencies in different Se-X series (Figs.5-8); hence, the interaction is not simple and depends on the concentration of foreign atoms. The reasons may be both in their bonding state and/or their arrangement – random or ordered. An ordered state can be realized by a directed diffusion of impurity under the action of any bond waves spreading through the melt, before the waves are frozen in solid glass.

The nature of the added atoms is also significant for their interaction with the bond wave. For example, a similar quantum chemical study for the Se-Cl system [63] leads to the same qualitative result as for with Se [52] and Se-Te [62], namely, the aggregation of HVBs with the participation of foreign atoms which leads to a lowering of energy – see **Fig.10** (chlorine atoms are chequered).

Fig. 10. Two of the lowest energy complex HVBs in Se-Cl after [63].

The configurations and energies in the Se-Cl case differ from those of Se-Te (Fig.9); in this way, the specificity of foreign atoms is revealed. Note that in the left configuration of the lowest energy in **Fig.10**, the Cl2Se5 fragment has not only over-coordinated Cl (a twocoordinated instead of the usual one-coordinated state) but also five-coordinated Se, i.e. chlorine induces the most coordinated Se state obtained by us so far (see four-coordinated Se in Fig.4, Fig.9 and Fig.10, as well as the configurations with three-coordinated Se in other chalcogenide glasses [39]). One should note, however, that the relation between coordination and valence is not strictly determined, especially in the hypervalent case [64]. Indeed, by use of only the geometric arrangement, it is impossible to conclude does a chemical bonding exist between the atoms under consideration – this is the question to the quantum-chemistry study of a concrete configuration.

The method of adding foreign atoms in a simple non-crystalline matrix with the further comparison of experimental features with quantum-chemical models seems to be a fruitful way for understanding what is the nature of chemical bonding in glass formers and how one can creates various bonding states and/or structures. As to the structures, let us consider an additional feature that we found when investigating Se-X glasses by means of IR spectroscopy.

In **Fig.11** one can see a *narrow* absorption band at 792 cm-1 on the high-frequency side of the third overtone for the 245 cm-1 band of Se: the line half width is one-third of that for the overtone. Note that the 792 cm-1 band develops only for a definite concentration of chlorine,

The result of such an interaction is currently unclear. Given only the traps, foreign HVBs such as Te40 - should slow down or even destroy a bond wave, most probably to the extent that it is proportional to their concentration. However, we observe both maxima and minima in concentration dependencies in different Se-X series (Figs.5-8); hence, the interaction is not simple and depends on the concentration of foreign atoms. The reasons may be both in their bonding state and/or their arrangement – random or ordered. An ordered state can be realized by a directed diffusion of impurity under the action of any

The nature of the added atoms is also significant for their interaction with the bond wave. For example, a similar quantum chemical study for the Se-Cl system [63] leads to the same qualitative result as for with Se [52] and Se-Te [62], namely, the aggregation of HVBs with the participation of foreign atoms which leads to a lowering of energy – see **Fig.10** (chlorine

The configurations and energies in the Se-Cl case differ from those of Se-Te (Fig.9); in this way, the specificity of foreign atoms is revealed. Note that in the left configuration of the lowest energy in **Fig.10**, the Cl2Se5 fragment has not only over-coordinated Cl (a twocoordinated instead of the usual one-coordinated state) but also five-coordinated Se, i.e. chlorine induces the most coordinated Se state obtained by us so far (see four-coordinated Se in Fig.4, Fig.9 and Fig.10, as well as the configurations with three-coordinated Se in other chalcogenide glasses [39]). One should note, however, that the relation between coordination and valence is not strictly determined, especially in the hypervalent case [64]. Indeed, by use of only the geometric arrangement, it is impossible to conclude does a chemical bonding exist between the atoms under consideration – this is the question to the

The method of adding foreign atoms in a simple non-crystalline matrix with the further comparison of experimental features with quantum-chemical models seems to be a fruitful way for understanding what is the nature of chemical bonding in glass formers and how one can creates various bonding states and/or structures. As to the structures, let us consider an additional feature that we found when investigating Se-X glasses by means of IR spectroscopy. In **Fig.11** one can see a *narrow* absorption band at 792 cm-1 on the high-frequency side of the third overtone for the 245 cm-1 band of Se: the line half width is one-third of that for the overtone. Note that the 792 cm-1 band develops only for a definite concentration of chlorine,

bond waves spreading through the melt, before the waves are frozen in solid glass.

 **E= −0,57 eV E= −0,37 eV** 

Fig. 10. Two of the lowest energy complex HVBs in Se-Cl after [63].

quantum-chemistry study of a concrete configuration.

atoms are chequered).

existing in the sample 0.01%Cl, which is most persistent for nucleation (Fig.7), and in the sample 0.05%Cl, which has the greatest capacity for heterogeneous nucleation (Fig.8). Interestingly, the same 792 cm-1 band also observed in the Se-As series for the 0.25%As sample (compared with the minimal heterogeneity in Fig.8) and unexpectedly at 5%As [65] (one can also see the extrema at the same concentrations of 0.01-0.05%Cl and 0.025%As in other properties for fresh glasses in Fig.5). Hence, the *ordering* which reveals itself as the 792 cm-1 narrow band can be related to the selenium matrix, the structure of which is modulated by means of the impurity introduced into it. Interestingly, the ordering can manifest in properties in the opposite manner, depending upon the impurity, as it is seen in Fig.5 for the extrema at 0.01-0.05%Cl and 0.025%As .

Fig. 11. IR spectra of Se-Cl glasses within the "phonon" range (400-1000 cm-1) after [60]. *T* is the transmission at a given frequency, *T*(1000) is the transmission at 1000 cm-1, i.e., the *transparency* (see Fig.7 and Fig.8).

When analysing the properties, a thoughtful reader can find out the disagreement in the *r*(0) values in Fig.5 and in the *T2/T1* values in Fig.7 for pure selenium glass (X=0) in different Se-X systems, a fact that may devaluate the data itself. The explanation is rather simple: different series were formed in different regimes: the first regime was applied for Se-Te and Se-Ge, the second one for Se-As and Se-S, and the third one for Se-Cl. Indeed, Se glasses have similar properties in the series of the same preparation. This is the *memory effect*, which is well known for anyone who works with glasses.

The non-linearity, spontaneous ordering and the memory effect discussed above are signs of a *self-organizing* system, a system that possesses various scenarios for evolution, depending upon the system's nature and the information provided by external medium [66]. When glass is considered as a self-organizing system with bond waves as the basis for self-

Crystallization in Glass Forming Substances: The Chemical Bond Approach 23

role of an *information field* that can orient bond waves. Finally, *anisotropy* can arise after the

The experimental equipment shown in **Fig.12** is rather simple: a glass with water (the excited medium for ultrasonic cavitation), in which the sample is placed in the holder. The sample has two pairs of perpendicular grains - A-A and B-B - the distance between like grains being equal: dA-A=dB-B. The US-emitter (its own frequency is 24 kHz and its vibration amplitude is 3 μ) is disposed at 10 mm above the upper grain A. The temperature varied from 50oC to 72oC and the time of treatment from 10 to 5 min. Note that the field is *weak*, having an intensity of about 0.2-0.3 W/cm2 (the cavitation threshold is 0.1 W/cm2). The vibration frequency in the excited medium is within the range of 1 kHz - 1 GHz, with a maximum at 5-10 MHz. At the end of treatment, the US-input is turned off, the emitter is lifted and the sample together with the holder is taken out of the glass with further cooling in the air down to room temperature. Then all the samples of a given Se-X series are

The optical measurements were made in two directions: in the A-A direction along the axis of the US-emitter (i.e., through two "frontal" A grains, which are parallel to the emitter end in Fig.12) and in the B-B direction perpendicular to the emitter axis (one can see one of the lateral grains B facets in face in Fig.12). These directions are easily distinguished in the samples owing to the curve sides remaining after transformation of initial cylinders/tablets, which were used in the previous experiments at *T<Tg*, into the blocks using in the present experiments; the A-A grains correspond to the ends of the initial cylinders. The spectra were measured within the 400-5000 cm-1 range and the data up to 1000 cm-1 are discussed here. It may be desirable to catch the eye of future investigators to the 1000-5000 cm-1 interval: although we do not discuss the related data, it is a quite sensitive area for the observation of

This study is currently in progress, although two articles describing experiments on the Se-As series [65] and Se-Te series [67] have just become available in English. Here, only basic experimental features are discussed; for additional information the reader is invited to look at the originals. Note that we use the optical transmission method for watching the crystallization process, which is again attributed primarily to the nucleation stage, a process that decreases transmission due to light scattering from a well-developed internal glass/crystal surface.

The first feature is a very fast darkening (for minutes) as compared with the previously considered darkening due to ageing (years). Thus, the first question is: what causes such

It can be seen in **Fig.13** that even at 72oC - the highest temperature which was applied - the temperature itself has only a low effectiveness (compare the spectra 12 to 13) and only the US treatment strongly accelerates nucleation in softening glass (see spectrum 5 of a very low

The second feature is optical *anisotropy*, which is very weak in the initial glasses but can develop after US treatment. The anisotropy value in a given series depends on composition, as can be seen in **Fig.14** with Se-As as an example. Despite the series investigated, anisotropy always arises as a result of a strong darkening in the A-A direction together with a negligible change in transparency in the B-B direction. As to the anisotropy peak position,

subjected to optical investigation, which is performed at the same day.

the crystallization process in the scale from 10μ to 2μ, respectively.

**4.2 Basic experimental features** 

rapid darkening? Temperature or cavitation?

intensity).

treatment.

organization, one acquires an appropriate tool not only for the explanation of the abovedescribed phenomena but also for the planning of special experiments. In the following experiments, we use an external *information field* for governing the development of a glass structure by means of the directed spreading of bond waves in the region where they are most effective, i.e., at *T>Tg* where 3D bond waves exist. Note that the above-described experiments concern nucleation in solid glass (*T>Tg*), a process that is provided by 2D bond waves spreading in the limits of frozen wavefronts. Thus, let us defreeze them.

## **4. The crystallization of softening glass in an ultrasonic field**

## **4.1 Experimental conditions**

The same Se-X series of glasses is used and optical transmission is also used as the general method for the observation of the crystallization process. The distinctions from the above experiments are as follows. First, the *temperature* range of the treatment 50-72ºC is somewhat *above* the glass transition temperature *Tg* (which is about 35ºC for Se), although the samples retain their form (in general). Second, the *time* of the treatment was 5-10 min instead of 5 years, as in the previous case. Third, the *ultrasonic field* is applied during treatment. Finally, the optical transmission was measured in *two directions* perpendicular to one another, before and after treatment.

Fig. 12. The treatment of a Se-X sample in an ultrasonic field in a cavitation regime. Note the glass with water, the sample in the holder near the 100 ml mark and the end of a US-emitter at the 200 ml mark, as well as the large bubbles at the water's surface and near the sample.

In the framework of our understanding of glass as a self-organizing system, owing to the collective behaviour of hypervalent bonds in a covalent network, these conditions mean, first, the existence of 3D bond waves which activate *all of the volume* of a substance. Second, the crystallization process *accelerates* owing to not only the increase of the bond wave dimensionality but also of the bond wave velocity. Third, an ultrasonic field can play the role of an *information field* that can orient bond waves. Finally, *anisotropy* can arise after the treatment.

The experimental equipment shown in **Fig.12** is rather simple: a glass with water (the excited medium for ultrasonic cavitation), in which the sample is placed in the holder. The sample has two pairs of perpendicular grains - A-A and B-B - the distance between like grains being equal: dA-A=dB-B. The US-emitter (its own frequency is 24 kHz and its vibration amplitude is 3 μ) is disposed at 10 mm above the upper grain A. The temperature varied from 50oC to 72oC and the time of treatment from 10 to 5 min. Note that the field is *weak*, having an intensity of about 0.2-0.3 W/cm2 (the cavitation threshold is 0.1 W/cm2). The vibration frequency in the excited medium is within the range of 1 kHz - 1 GHz, with a maximum at 5-10 MHz. At the end of treatment, the US-input is turned off, the emitter is lifted and the sample together with the holder is taken out of the glass with further cooling in the air down to room temperature. Then all the samples of a given Se-X series are subjected to optical investigation, which is performed at the same day.

The optical measurements were made in two directions: in the A-A direction along the axis of the US-emitter (i.e., through two "frontal" A grains, which are parallel to the emitter end in Fig.12) and in the B-B direction perpendicular to the emitter axis (one can see one of the lateral grains B facets in face in Fig.12). These directions are easily distinguished in the samples owing to the curve sides remaining after transformation of initial cylinders/tablets, which were used in the previous experiments at *T<Tg*, into the blocks using in the present experiments; the A-A grains correspond to the ends of the initial cylinders. The spectra were measured within the 400-5000 cm-1 range and the data up to 1000 cm-1 are discussed here. It may be desirable to catch the eye of future investigators to the 1000-5000 cm-1 interval: although we do not discuss the related data, it is a quite sensitive area for the observation of the crystallization process in the scale from 10μ to 2μ, respectively.

#### **4.2 Basic experimental features**

22 Crystallization – Science and Technology

organization, one acquires an appropriate tool not only for the explanation of the abovedescribed phenomena but also for the planning of special experiments. In the following experiments, we use an external *information field* for governing the development of a glass structure by means of the directed spreading of bond waves in the region where they are most effective, i.e., at *T>Tg* where 3D bond waves exist. Note that the above-described experiments concern nucleation in solid glass (*T>Tg*), a process that is provided by 2D bond

The same Se-X series of glasses is used and optical transmission is also used as the general method for the observation of the crystallization process. The distinctions from the above experiments are as follows. First, the *temperature* range of the treatment 50-72ºC is somewhat *above* the glass transition temperature *Tg* (which is about 35ºC for Se), although the samples retain their form (in general). Second, the *time* of the treatment was 5-10 min instead of 5 years, as in the previous case. Third, the *ultrasonic field* is applied during treatment. Finally, the optical transmission was measured in *two directions* perpendicular to one another, before

Fig. 12. The treatment of a Se-X sample in an ultrasonic field in a cavitation regime. Note the glass with water, the sample in the holder near the 100 ml mark and the end of a US-emitter at the 200 ml mark, as well as the large bubbles at the water's surface and near the sample.

In the framework of our understanding of glass as a self-organizing system, owing to the collective behaviour of hypervalent bonds in a covalent network, these conditions mean, first, the existence of 3D bond waves which activate *all of the volume* of a substance. Second, the crystallization process *accelerates* owing to not only the increase of the bond wave dimensionality but also of the bond wave velocity. Third, an ultrasonic field can play the

waves spreading in the limits of frozen wavefronts. Thus, let us defreeze them.

**4. The crystallization of softening glass in an ultrasonic field** 

**4.1 Experimental conditions** 

and after treatment.

This study is currently in progress, although two articles describing experiments on the Se-As series [65] and Se-Te series [67] have just become available in English. Here, only basic experimental features are discussed; for additional information the reader is invited to look at the originals. Note that we use the optical transmission method for watching the crystallization process, which is again attributed primarily to the nucleation stage, a process that decreases transmission due to light scattering from a well-developed internal glass/crystal surface.

The first feature is a very fast darkening (for minutes) as compared with the previously considered darkening due to ageing (years). Thus, the first question is: what causes such rapid darkening? Temperature or cavitation?

It can be seen in **Fig.13** that even at 72oC - the highest temperature which was applied - the temperature itself has only a low effectiveness (compare the spectra 12 to 13) and only the US treatment strongly accelerates nucleation in softening glass (see spectrum 5 of a very low intensity).

The second feature is optical *anisotropy*, which is very weak in the initial glasses but can develop after US treatment. The anisotropy value in a given series depends on composition, as can be seen in **Fig.14** with Se-As as an example. Despite the series investigated, anisotropy always arises as a result of a strong darkening in the A-A direction together with a negligible change in transparency in the B-B direction. As to the anisotropy peak position,

Crystallization in Glass Forming Substances: The Chemical Bond Approach 25

it probes in the A-A direction (the beam falls onto the nucleated layers) and scatter much less in the B-B direction (the beam spreads within transparent glass-like regions between the

> 012345 **As**, ат%

Fig. 14. The anisotropy in transparency (transmission at 1000 cm-1 measured in the A-A and B-B directions) for the Se-As series. The two lines correspond with the measurements before

In an attempt to see the proposed internal structure, we used electron microscopy; the two images shown in **Fig.15** correspond to fresh fractures made in the two perpendicular directions.

Fig. 15. A SEM image of the fracture made parallel to the A grain (on the left) and the B

grain (on the right) for a selenium sample after US treatment at 72oС.

layers, as like along the optical wire).

0

(4) and after (5) US treatment (720C, 5 min).

**4**

1

2

3

**A\***=T\*B/T\*

A

4

5

6

**5**

7

a special concentration of 0.25%As in Fig.14 corresponds with the above-mentioned appearance of a narrow 792 cm-1 band - the same as that which is shown in Fig.11 for the Se-Cl series. This fact establishes a connection between the ability of pre-crystalline ordering in an as-prepared glass and the ability for the development of gigantic optical anisotropy in the glass after its further high-temperature treatment in an ultrasonic field.

In the framework of the bond wave model, anisotropy arises due to the definite location of an excitation source (see Fig.12): although one can suppose a nearly spherical distribution for the dissipation of the input energy in a cavitating medium, the input itself violates the symmetry. The excited medium, in addition, has a temperature above the glass transition temperature for the Se-X glasses investigated. Therefore, when an initially isotropic glass is placed into the cell (Fig.12), the bond waves - which were initially frozen in different directions, as in the case shown in the right-hand part of Fig.3 - begin to refreeze (2D→3D), and the refrozen bond wave will move in a definite direction, remaining this direction after glass cooling.

Fig. 13. Optical transmission spectra of the aged Se glass (11), the aged glass after polishing (12) and the polished glass after a 5 min treatment at 72oC (13). Curve 5 corresponds with the polished sample (an analogue of 12) after a 5 min treatment in a cavitation US field at the same temperature, 72oC. The measurements are made in the A-A direction.

The question is: in what direction will the refrozen bond waves move? It seems likely that the layers/wavefronts will be oriented in the A-A direction, parallel to the vibrating end of the US-emitter (see Fig.13). Insofar as crystallization begins within the layers containing active hypervalent bonds, then one obtains a system of partially-crystallized layers divided by periodic glass-like regions. This system is frozen in the sample after its removal from the cell with further quenching in the air. The sample reveals optical anisotropy (see line 5 in Fig.14) because in such a layered structure a measuring beam should scatter strongly when

a special concentration of 0.25%As in Fig.14 corresponds with the above-mentioned appearance of a narrow 792 cm-1 band - the same as that which is shown in Fig.11 for the Se-Cl series. This fact establishes a connection between the ability of pre-crystalline ordering in an as-prepared glass and the ability for the development of gigantic optical anisotropy in the

In the framework of the bond wave model, anisotropy arises due to the definite location of an excitation source (see Fig.12): although one can suppose a nearly spherical distribution for the dissipation of the input energy in a cavitating medium, the input itself violates the symmetry. The excited medium, in addition, has a temperature above the glass transition temperature for the Se-X glasses investigated. Therefore, when an initially isotropic glass is placed into the cell (Fig.12), the bond waves - which were initially frozen in different directions, as in the case shown in the right-hand part of Fig.3 - begin to refreeze (2D→3D), and the refrozen bond wave will move in a definite direction, remaining this direction after

> 400 500 600 700 800 900 1000 **,** cm-1

Fig. 13. Optical transmission spectra of the aged Se glass (11), the aged glass after polishing (12) and the polished glass after a 5 min treatment at 72oC (13). Curve 5 corresponds with the polished sample (an analogue of 12) after a 5 min treatment in a cavitation US field at the

The question is: in what direction will the refrozen bond waves move? It seems likely that the layers/wavefronts will be oriented in the A-A direction, parallel to the vibrating end of the US-emitter (see Fig.13). Insofar as crystallization begins within the layers containing active hypervalent bonds, then one obtains a system of partially-crystallized layers divided by periodic glass-like regions. This system is frozen in the sample after its removal from the cell with further quenching in the air. The sample reveals optical anisotropy (see line 5 in Fig.14) because in such a layered structure a measuring beam should scatter strongly when

same temperature, 72oC. The measurements are made in the A-A direction.

**11**

**5**

**12**

**13**

glass after its further high-temperature treatment in an ultrasonic field.

**Se (0%Te)**

glass cooling.

0

0.1

0.2

0.3

**T(**

0.4

0.5

0.6

0.7

it probes in the A-A direction (the beam falls onto the nucleated layers) and scatter much less in the B-B direction (the beam spreads within transparent glass-like regions between the layers, as like along the optical wire).

Fig. 14. The anisotropy in transparency (transmission at 1000 cm-1 measured in the A-A and B-B directions) for the Se-As series. The two lines correspond with the measurements before (4) and after (5) US treatment (720C, 5 min).

In an attempt to see the proposed internal structure, we used electron microscopy; the two images shown in **Fig.15** correspond to fresh fractures made in the two perpendicular directions.

Fig. 15. A SEM image of the fracture made parallel to the A grain (on the left) and the B grain (on the right) for a selenium sample after US treatment at 72oС.

Crystallization in Glass Forming Substances: The Chemical Bond Approach 27

[4] J.C.A. Vreeswijk, R.G. Gossink, J.M. Stevels. J. Non-Cryst. Solids, 16, 15 (1974). [5] E. Ruckenstein, S.K. Ihm. J. Chem. Soc. Faraday Trans. I, 72, 764 (1976). [6] G.F. Neilson, M.C. Weinberg. J. Non-Cryst. Solids, 34, 137 (1979).

[8] J.C.Th.G.M. Van der Wielen, H.N. Stein, J.M. Stevels. J. Non-Cryst. Solids, 1, 18 (1968).

[15] S.A. Dembovsky, L.M. Ilizarov, E.A. Chechetkina, A.Yu. Khar'kovsky. In: Proc. "Amorphous Semiconductors – 84". Gabrovo, Bulgaria, 1984. p.39. [16] S.A. Dembovsky, L.M. Ilizarov, A.Yu. Khar'kovsky. Mater. Res. Bull. 21, 1277 (1986). [17] S.A. Dembovsky, E.A. Chechetkina. Glass Formation (in Russ.). Nauka, Moscow, 1990.

[20] S.A. Dembovsky. Proc. "Amorphous Semiconductors – 80". Kishinev, USSR, 1980. p.22.

[24] V.M. Fokin, E.D. Zanotto, W.P. Schmelzer, O.V. Potapov. J. Non-Cryst. Solids, 351, 1491

[25] J.L. Nowinski, M. Mroczkowska, J.E. Garbarczyk, M. Wasiucionek. Materials Science-

[40] S.A. Dembovsky, A.S. Zyubin, O.A. Kondakova. Mendeleev Chemistry Journal

[7] R.J.H. Gelsing, H.N. Stein, J.M. Stevels. Phys. Chem. Glasses, 7, 185 (1966).

[9] A.C.J. Havermans, H.N. Stein, J.M. Stevels. J. Non-Cryst. Solids, 5, 66 (1970). [10] M.H.C. Baeten, H.N. Stein, J.M. Stevels. Silicates Industriels, 37, 33 (1972).

[19] S.A. Dembovsky. Zh. Neorg. Khim. (Russ. J. Inorg. Chem.), 22, 3187 (1977).

[21] S.A. Dembovsky, E.A. Chechetkina. Mater. Res. Bull., 16, 606 (1981). [22] S.A. Dembovsky, E.A. Chechetkina. Mater. Res. Bull., 16, 723 (1981). [23] S.A. Dembovsky, E.A. Chechetkina. J. Non-Cryst. Solids 64, 95 (1984).

[26] A.H. Moharram, A.A. Abu-sehly et al. Physica B, 324, 344 (2002).

[29] H. Tanaka. J. Phys.: Condens. Matter, 23, 284115 (2011). [30] W.H. Zachariasen. J. Amer. Ceram. Soc., 54, 3841(1932).

[37] S.A. Dembovsky. J. Non-Cryst. Solids, 353, 2944 (2007). [38] E.A. Chechetkina. J. Optoel. Adv. Mater., 13, 1385 (2011).

[42] J.L. Muscher. Angew. Chem. Int. Ed., 8, 54 (1969). [43] W.B. Jensen. J. Chem. Educ., 83, 1751 (2006).

[32] S.A. Dembovsky. Mater. Res. Bull., 16, 1331 (1981).

[31] G. Biroli. Sèminaire Poincarè XII, 37 (2009).

[33] N.A. Popov. JETP Lett., 31, 409 (1980).

[28] P.N. Pusley, E. Zaccarelli et al. Phil. Trans. R. Soc. A, 367, 4993 (2009).

[34] S.A. Dembovsky, E.A. Chechetkina. J. Non-Cryst. Solids, 85, 346 (1986). [35] S.A. Dembovsky, E.A. Chechetkina. Philos. Mag., B53, 367 (1986). [36] S.A. Dembovsky, E.A. Chechetkina. J. Optoel. Adv. Mater., 3, 3 (2001).

[39] S.A. Dembovsky, A.S. Zyubin. Russ. J. Inorg. Chem., 46, 121 (2001).

(Zhurnal Ross. Khim. Ob-va im.D.I.Mendeleeva), 45, 92 (2001). [41] S. Zyubin, F.V. Grigoriev, S.A. Dembovsky. Russ. J. Inorg. Chem., 46, 1350 (2001).

[2] P.T. Sarjeant, R. Roy. Mater. Res. Bull., 3, 265 (1968). [3] D.R. Uhlmann. J. Non-Cryst. Solids, 7, 337 (1972).

[11] G. Whichard, D.E. Day. J. Non-Cryst. Solids, 66, 477 (1984). [12] W. Huang, C.S. Ray, D.E. Day. J. Non-Cryst. Solids, 86, 204 (1986). [13] M.D. Mikhailov, A.C. Tver'janovich. Glass Phys. Chem., 6, No.5 (1980). [14] M.D. Mikhailov, A.C. Tver'janovich. Glass Phys. Chem., 12, No.3 (1986).

[18] K.F. Kelton, A.L. Greer. J. Non-Cryst. Solids, 79, 295 (1986).

(2005).

Poland 24, 161 (2006).

[27] L. Berthier. Physics 4, 42 (2011).

In order to exclude the effects of many-component crystallization, we made the fractures using pure Se glass subjected to US treatment. This is that glass belonging to the Se-As series with the induced anisotropy of A\*=2 (see 0% in Fig.14): after the US treatment, the transparency in the A-A direction was decreased doubly, while the transparency in the B-B direction was unchanged. In **Fig.15**, we present two images for the two fractures. The left image probably corresponds to the crystallized *d*-layer, along which a fracture developed. The right image looks like ordinary glass, probably corresponding to the glass-like region between the two adjacent *d*-layers.

For the crystallized layer (Fig.15, on the left) it is interesting to note the unusual *needle* form of nucleation in the selenium, which is known to be inclined to form the spheric-like crystallites. Of course, a much more intensive SEM investigation, as well as a wide set of other methods, is needed for a more adequate characterization of the materials obtained and, consequently, a deeper understanding of the processes involved in their formation. We hope that the described experiments of crystallization in glass under the action of impurities and/or ultrasonic field will help the coming investigators of glass and glass-ceramics in obtaining the new materials using non-traditional ways.

## **5. Conclusions**

The structure of glass, considered from the chemical bond point of view using the bond wave model, acquires specific elements of order which develop during cooling (open system) owing to the alternation of chemical bonds (two-stable element) in the form of a bond wave (collective feedback between the elements). In the brackets, there are three general conditions for self-organization; they are naturally fulfilled in glass forming substances owing to their ability to form alternative hypervalent bonds. The processes in a self-organizing system proceed in a specific manner, and this is the reason why classic notions about crystallization works poorly in glass formers. On the other hand, selforganization opens new possibilities for both the understanding and management of related materials. One should note that although "self-organization in glass" becomes a hot topic (see [68] as an example), a real chemical bond is actually absent in the related works. We hope that our approach, connecting contemporary notions about chemical bonding - from the one side - and the self-organization theory (synergetics) - from the other side - will help to fill a significant gap in glass theory, including the understanding of the mechanism of glass-crystal transition and, in glass practice, the elaboration of new principles for the formation of glass-ceramic materials.

## **6. Acknowledgements**

Dedicated to Prof. S. A. Dembovsky (1932-2010); his ideas and our long-term collaboration actually make him the co-author of this work.

This work was supported by the Russian Foundation of Basic Research (Grant No. 09-03- 01158).

## **7. References**

[1] J. Zarzycki. Glasses and the vitreous state. Cambridge University Press, Cambridge-New-York-Melbourne, 1991.


In order to exclude the effects of many-component crystallization, we made the fractures using pure Se glass subjected to US treatment. This is that glass belonging to the Se-As series with the induced anisotropy of A\*=2 (see 0% in Fig.14): after the US treatment, the transparency in the A-A direction was decreased doubly, while the transparency in the B-B direction was unchanged. In **Fig.15**, we present two images for the two fractures. The left image probably corresponds to the crystallized *d*-layer, along which a fracture developed. The right image looks like ordinary glass, probably corresponding to the glass-like region

For the crystallized layer (Fig.15, on the left) it is interesting to note the unusual *needle* form of nucleation in the selenium, which is known to be inclined to form the spheric-like crystallites. Of course, a much more intensive SEM investigation, as well as a wide set of other methods, is needed for a more adequate characterization of the materials obtained and, consequently, a deeper understanding of the processes involved in their formation. We hope that the described experiments of crystallization in glass under the action of impurities and/or ultrasonic field will help the coming investigators of glass and glass-ceramics in

The structure of glass, considered from the chemical bond point of view using the bond wave model, acquires specific elements of order which develop during cooling (open system) owing to the alternation of chemical bonds (two-stable element) in the form of a bond wave (collective feedback between the elements). In the brackets, there are three general conditions for self-organization; they are naturally fulfilled in glass forming substances owing to their ability to form alternative hypervalent bonds. The processes in a self-organizing system proceed in a specific manner, and this is the reason why classic notions about crystallization works poorly in glass formers. On the other hand, selforganization opens new possibilities for both the understanding and management of related materials. One should note that although "self-organization in glass" becomes a hot topic (see [68] as an example), a real chemical bond is actually absent in the related works. We hope that our approach, connecting contemporary notions about chemical bonding - from the one side - and the self-organization theory (synergetics) - from the other side - will help to fill a significant gap in glass theory, including the understanding of the mechanism of glass-crystal transition and, in glass practice, the elaboration of new principles for the

Dedicated to Prof. S. A. Dembovsky (1932-2010); his ideas and our long-term collaboration

This work was supported by the Russian Foundation of Basic Research (Grant No. 09-03-

[1] J. Zarzycki. Glasses and the vitreous state. Cambridge University Press, Cambridge-

between the two adjacent *d*-layers.

formation of glass-ceramic materials.

actually make him the co-author of this work.

New-York-Melbourne, 1991.

**6. Acknowledgements** 

01158).

**7. References** 

**5. Conclusions** 

obtaining the new materials using non-traditional ways.


**2** 

 *India* 

**Crystallization Kinetics of** 

*Department of Physics, Banaras Hindu University, Varanasi,* 

Chalcogenide glasses are disordered non crystalline materials which have pronounced tendency their atoms to link together to form link chain. Chalcogenide glasses can be obtained by mixing the chalcogen elements, viz, S, Se and Te with elements of the periodic table such as Ga, In, Si, Ge, Sn, As, Sb and Bi, Ag, Cd, Zn etc. In these glasses, short-range inter-atomic forces are predominantly covalent: strong in magnitude and highly directional, whereas weak van der Waals' forces contribute significantly to the medium-range order. The atomic bonding structure is, in general more rigid than that of organic polymers and more flexible than that of oxide glasses. Accordingly, the glass-transition temperatures and elastic properties lay in between those of these materials. Some metallic element containing chalcogenide glasses behave as (super) ionic conductors. These glasses also behave as semiconductors or, more strictly, they are a kind of amorphous semi-conductors with band gap energies of 1±3eV (Fritzsche, 1971). Commonly, chalcogenide glasses have much lower mechanical strength and thermal stability as compared to existing oxide glasses, but they have higher thermal expansion, refractive index, larger range of infrared transparency and

It is difficult to define with accuracy when mankind first fabricated its own glass but sources demonstrate that it discovered 10,000 years back in time. It is also difficult to point in time, when the field of chalcogenide glasses started. The vast majority of time the vitreous glassy state was limited to oxygen compounds and their derivatives. Schulz-Sellack was the first to report data on oxygen-free glass in 1870 (Sellack, 1870). Investigation of chalcogenide glasses as optoelectronics materials in infra-red systems began with the rediscovery of arsenic trisulfide glass (Frerichs, 1950, 1953) when R. Frerichs was reported his work. Development of the glasses as a practical optoelectronic materials were continued by W. A. Fraser and J. Jerger in 1953 (Fraser et. al., 1953). During the 1950-1970 periods (Hilton, 2010) the glasses ware made in ton quantities by several companies and it frequently used in commercial devices. As an example, devices were made to detect the overheated bearings in the railroad cars. Hot objects could be detected by the radiation transmitted through the 3- to 5 μm atmospheric window, for this transparent arsenic trisulfide glass was used. While, to make chalcogenide glass compositions which capable in transmitting longer wavelengths arose the concept of passive

**1. Introduction** 

**1.1 Background of chalcogenides** 

higher order of optical non-linearity.

thermal optical systems was adopted.

**Chalcogenide Glasses** 

Abhay Kumar Singh


## **Crystallization Kinetics of Chalcogenide Glasses**

Abhay Kumar Singh *Department of Physics, Banaras Hindu University, Varanasi, India* 

## **1. Introduction**

28 Crystallization – Science and Technology

[45] S.C. Moss, D.L. Price. In: Physics of Disordered Materials (Eds. D. Adler, H. Fritzsche,

[50] E.A. Сhechetkina. In: Fractal Concepts in Materials Science. MRS Proc., vol.367 (1995).

[54] S.A. Dembovsky, E.A. Chechetkina, T.A. Kupriyanova. Materialovedenie (Mater. Sci.

[55] M.F. Kotkata, E.A. Mahmoud, M.K. El-Mously. Acta Phys. Acad. Sci. Hung., 50, 61

[58] S.A. Dembovsky, L.M. Ilizarov, A.Yu. Khar'kovsky. Mater. Res. Bull., 21, 1277 (1986).

[60] E.A. Chechetkina, A.B. Vargunin, V.A. Kuznetzov, S.A. Dembovsky. Materialovedenie

[61] E.A. Chechetkina, E.B. Kryukova, A.B. Vargunin. Materialovedenie (Mater. Sci. Trans.),

[65] E.A. Chechetkina, E.V. Kisterev, E.B. Kryukova, A.I. Vargunin. Inorg. Mater.: Appl.

[66] H. Haken. Information and Self-Organization. A macroscopic Approach to Complex

[67] E.A. Chechetkina, E.V. Kisterev, E.B. Kryukova, A.I. Vargunin. J. Optoel. Adv. Mater.,

[68] P. Boolchand, G. Lucovsky, J.C. Phillips, M.F. Thorpe. Philos. Mag., 85, 3823 (2005).

[62] A.S. Zyubin, E.A. Chechetkina, S.A. Dembovsky. Russ. J. Inorg. Chem., 57, 913 (2012).

[59] M. Hansen, K. Anderko. Constitution of Binary Alloys. McGraw-Hill, 1958.

[48] E.A. Сhechetkina. In: Proc XVII Inter. Congr. Glass, Beijing, 1995. vol.2, p.285.

[44] G. Pimentel. J. Chem. Phys., 19, 446 (1981).

Trans.), No.4, 37 (2004).

(1981).

No.6, 29 (2007).

Res., 2, 360 (2011).

11, 2034 (2009).

S.R. Ovshinsky), Plenum, New York, 1985. p.77. [46] E.A. Сhechetkina. J. Phys.: Condes. Matter, 7, 3099 (1995). [47] E.A. Сhechetkina. Solid State Commun., 87, 171 (1993).

[52] S.A. Dembovsky, A.S. Zyubin. Russ. J. Inorg. Chem. 54, 455 (2009). [53] M. Kastner, D. Adler, H. Fritzsche. Phys. Rev. Lett. 37, 1504 (1976).

[56] M.F. Kotkata, M.K. El-Mously. Acta Phys. Hung., 54, 303 (1983). [57] G. Parthasarathy, K.J. Rao, E.S.R. Gopal. Philos. Mag., B50, 335 (1984).

[63] S. Zyubin, S.A. Dembovsky. Russ. J. Inorg. Chem., 56, 329 (2011).

[49] E.A. Сhechetkina. J. Non-Cryst. Solids, 128, 30 (1991).

[51] E.A. Сhechetkina. J. Non-Cryst. Solids, 201, 146 (1996).

(Mater. Sci. Trans.), No.7, 28 (2006).

[64] D.W. Smith. J. Chem. Educ., 82, 1202 (2005).

Systems. Springer, 1988, 2000, 2006.

## **1.1 Background of chalcogenides**

Chalcogenide glasses are disordered non crystalline materials which have pronounced tendency their atoms to link together to form link chain. Chalcogenide glasses can be obtained by mixing the chalcogen elements, viz, S, Se and Te with elements of the periodic table such as Ga, In, Si, Ge, Sn, As, Sb and Bi, Ag, Cd, Zn etc. In these glasses, short-range inter-atomic forces are predominantly covalent: strong in magnitude and highly directional, whereas weak van der Waals' forces contribute significantly to the medium-range order. The atomic bonding structure is, in general more rigid than that of organic polymers and more flexible than that of oxide glasses. Accordingly, the glass-transition temperatures and elastic properties lay in between those of these materials. Some metallic element containing chalcogenide glasses behave as (super) ionic conductors. These glasses also behave as semiconductors or, more strictly, they are a kind of amorphous semi-conductors with band gap energies of 1±3eV (Fritzsche, 1971). Commonly, chalcogenide glasses have much lower mechanical strength and thermal stability as compared to existing oxide glasses, but they have higher thermal expansion, refractive index, larger range of infrared transparency and higher order of optical non-linearity.

It is difficult to define with accuracy when mankind first fabricated its own glass but sources demonstrate that it discovered 10,000 years back in time. It is also difficult to point in time, when the field of chalcogenide glasses started. The vast majority of time the vitreous glassy state was limited to oxygen compounds and their derivatives. Schulz-Sellack was the first to report data on oxygen-free glass in 1870 (Sellack, 1870). Investigation of chalcogenide glasses as optoelectronics materials in infra-red systems began with the rediscovery of arsenic trisulfide glass (Frerichs, 1950, 1953) when R. Frerichs was reported his work. Development of the glasses as a practical optoelectronic materials were continued by W. A. Fraser and J. Jerger in 1953 (Fraser et. al., 1953). During the 1950-1970 periods (Hilton, 2010) the glasses ware made in ton quantities by several companies and it frequently used in commercial devices. As an example, devices were made to detect the overheated bearings in the railroad cars. Hot objects could be detected by the radiation transmitted through the 3- to 5 μm atmospheric window, for this transparent arsenic trisulfide glass was used. While, to make chalcogenide glass compositions which capable in transmitting longer wavelengths arose the concept of passive thermal optical systems was adopted.

Crystallization Kinetics of Chalcogenide Glasses 31

theory on the electronic processes in non-crystalline chalcogenide glasses (Mott et. al., 1979), and Kawamura (Kawamura et. al., 1983) was discovered xerography. Applications of solar cells were developed by Ciureanu and Middehoek (Ciureanu et. al., 1992) and Robert and his coworkers (Robert et.al., 1998). Infrared optics applications were studied by Quiroga and Leng and their coworkers (Quiroga et al., 1996, Leng et. al., 2000 ). The switching device applications were introduced by Bicerono and Ovshinsky (Bicerono et. al.,1985) and Ovshinsky (Ovshinsky, 1994). P. Boolchand and his coworkers (Boolchand et.al., 2001) was discovered intermediate phase in chalcogenide glasses. In this order several investigators have been also reported that the useful optoelectronics applications in infrared transmission and detection, threshold and memory switching (Selvaraju et al., 2003), optical fibers (Bowden et al., 2009, Shportko et al., 2008, Milliron et al.,2007) functional elements in integrated-optic circuits (Pelusi et al.2009) non-linear optics (Dudley et al., 2009), holographic & memory storage media (Vassilev et al., 2009, Wuttig et al.,2007), chemical and bio-sensors (Anne et al.,2009, Schubert et at., 2001), infrared photovoltaics (Sargent, 2009), microsphere laser (Elliott, 2010), active plasmonics (Samson,2010), microlenses in inkjet printing (Sanchez, 2011) and other photonics (Eggleton, 2011) applications. In this respect, the analysis of the composition dependence of their thermal properties was an important aspect for the study (Singh et al.,

Subsequently, several review books were published on chalcogenide glasses e. g. "The Chemistry of Glasses" by A Paul in 1982, "The Physics of Amorphous Solids" by R.Zallen in 1983 and "Physics of Amorphous Materials" by S.R.Elliott in 1983. However, first book entirely dedicated to chalcogenide glassy materials entitled "Chalcogenide Semiconducting Glasses" was published in 1983 by Z.U.Borisova. In this order, G.Z. Vinogradova was published her monograph "Glass formation and Phase Equilibrium in Chalcogenide Systems" in 1984. M.A. Andriesh dedicates a book to some specific applications of chalcogenide glasses entitled "Glassy Semiconductors in Photo-electric Systems for Optical Recording of Information". M.A. Popescu gave large and detailed account on physical and technological aspect of chalcogenide systems in his book "Non-Crystalline Chalcogenides". The compendium of monographs on the subject of photo-induced processes in chalcogenide glasses entitled "Photo-induced Metastability in Amorphous Semiconductors" was compiled by A. Colobov-2003. Robert Fairman and Boris Ushkov-2004 described physical properties in "Semiconducting Chalcogenide Glass I: Glass formation, structure, and simulated transformations in Chalcogenide Glass". Finally, A. Zakery and S.R. Elliott demonstrated the

"Optical Nonlinearities in Chalcogenide Glasses and their Applications" in 2007.

Structure of chalcogenide glasses have been extensively studied in binary compositions considering both the bulk and thin film forms. Chalcogens can form alloys together, Se-S, Se-Te (Gill, 1973) and S-Te amorphous (Sarrach et. al, 1976, Hawes, 1963) compounds were identified; however the scientific community seems to have, for the moment at least, left these glasses aside. Many binary compounds can be synthesized by associating one of the chalcogen with another element of the periodic table like, indium, antimony, copper, germanium, phosphorus, silicon and tin. A few other compounds based on heavy or light elements and alkali atoms have also been investigated. Abrikosov and his co-workers in 1969 (Lopez, 2004) were first reported the molecular structures of most extensively studied As-S, As-Se binary chalcogenide alloys in their monograph, the phase diagrams for the As- S

2009, 2010, 2011).

**1.1.1 Binary chalcogenides** 

Jerger, Billian, and Sherwood (Hilton, 1966 & 2010) extended their investigation on arsenic glasses containing selenium and tellurium and later adding germanium as a third constituent. The goal was to use chalcogen elements heavier than sulfur to extend longwavelength transmission to cover the 8- to 12 μm window with improve physical properties. A subsequent work also in Ioffee Institute, in Lenin-grad under the direction of Boris Kolomiets was also reported in 1959 (Hilton, 2010). Work along the same line was begun in the United Kingdom by Nielsen and Savage (Nielsen, 1962, Savage et. al., 1964, Savage et. al., 1966) as well as work also at Texas Instruments (TI) began as an outgrowth of the thermoelectric materials program. The glass forming region for the silicon-arsenictellurium system was planed by Hilton and Brau (Hilton et. al., 1963). This development led to an exploratory DARPA- ONR program from 1962 to 1965 (Hilton, 2010). The ultimate goal of the program was to find infrared transmitting chalcogenide glasses with physical properties comparable to those of oxide optical glasses and a softening point of 5000C. Futher, Hilton (Hilton, 1974) also worked on sulfur-based glasses in 1973 to 1974. The exploratory programs resulted the eight chalcogenide glass U.S. patents and a number of research paper published in an international journal detailing the results (Hilton et. al., 1966). After that scientific community made serious effort in development of chalcogenide glasses and organized symposia and meeting in All-Union Symposium on the Vitreous Chalcogenide Semiconductors held in May 1967 (Kokorina, 1966) in Leningrad (now called St. Petersburg). In that symposium Stanley Ovshinsky (founder of Electron Energy Conversion Devices in Troy, Michigan) was presented his first paper dealing switching devices based on the electronic properties of chalcogenide glasses. Similar work was reported by A. D. Pearson (Pearson, 1962) in United States, thus started a great world wide effort to investigate chalcogenide glasses and their electronic properties. The purpose was to pursue a new family of inexpensive electronic devices based on amorphous semiconductors. The effort in this field far exceeded the effort directed toward optoelectronics applications. Some of the results of the efforts in the United States were reported in a symposium (Doremus, 1969) and in another symposium (Cohen et. al.,1971). In this order Robert Patterson mapped out the glass forming region for the germanium-antimony-selenium system and granted a U.S. patent (Patterson, 1966-67) covering the best composition selection.

In 1967 Harold Hafner was made many important contributions including a glass casting process and a glass tempering process in Semiconductor Production Division under the direction of Charlie Jones. There work concentrated the efforts on a glass from the germaniumarsenic-selenium system and outcomes (Jones et. al., 1968) agreed with the conclusions of the Russian, U.K. Alternatively, Servo group efforts that the germanium-arsenic-selenium system produced the best glasses for infrared system applications. Don Weirauch (Hilton, 2010) was conducted a crystallization study on the germanium-arsenic selenium family of glasses and identified a composition in which crystallites would not form. In 1972 a commercial group was successfully cast (12 in 24 in 0.5 in) a window which flat polished, parallel and antireflectioncoated (Hafner, 1972). In the late 1960s and early 1970s, passive 8- to 12μ m systems began to be produced in small numbers mostly for the defense uses.

In 1968, Ovshinsky and his co-workers was discovered (Stocker, 1969) the some chalcogenide glasses exhibited memory and switching effects. After this discovery it became clear that the electric pulses could be switch the phases in chalcogenide glasses back and forth between amorphous and crystalline state.Around the same period in 1970's, Sir N. F. Mott (a former Noble Prices winner in Physics-1977) and E.A. Davis were developed the

Jerger, Billian, and Sherwood (Hilton, 1966 & 2010) extended their investigation on arsenic glasses containing selenium and tellurium and later adding germanium as a third constituent. The goal was to use chalcogen elements heavier than sulfur to extend longwavelength transmission to cover the 8- to 12 μm window with improve physical properties. A subsequent work also in Ioffee Institute, in Lenin-grad under the direction of Boris Kolomiets was also reported in 1959 (Hilton, 2010). Work along the same line was begun in the United Kingdom by Nielsen and Savage (Nielsen, 1962, Savage et. al., 1964, Savage et. al., 1966) as well as work also at Texas Instruments (TI) began as an outgrowth of the thermoelectric materials program. The glass forming region for the silicon-arsenictellurium system was planed by Hilton and Brau (Hilton et. al., 1963). This development led to an exploratory DARPA- ONR program from 1962 to 1965 (Hilton, 2010). The ultimate goal of the program was to find infrared transmitting chalcogenide glasses with physical properties comparable to those of oxide optical glasses and a softening point of 5000C. Futher, Hilton (Hilton, 1974) also worked on sulfur-based glasses in 1973 to 1974. The exploratory programs resulted the eight chalcogenide glass U.S. patents and a number of research paper published in an international journal detailing the results (Hilton et. al., 1966). After that scientific community made serious effort in development of chalcogenide glasses and organized symposia and meeting in All-Union Symposium on the Vitreous Chalcogenide Semiconductors held in May 1967 (Kokorina, 1966) in Leningrad (now called St. Petersburg). In that symposium Stanley Ovshinsky (founder of Electron Energy Conversion Devices in Troy, Michigan) was presented his first paper dealing switching devices based on the electronic properties of chalcogenide glasses. Similar work was reported by A. D. Pearson (Pearson, 1962) in United States, thus started a great world wide effort to investigate chalcogenide glasses and their electronic properties. The purpose was to pursue a new family of inexpensive electronic devices based on amorphous semiconductors. The effort in this field far exceeded the effort directed toward optoelectronics applications. Some of the results of the efforts in the United States were reported in a symposium (Doremus, 1969) and in another symposium (Cohen et. al.,1971). In this order Robert Patterson mapped out the glass forming region for the germanium-antimony-selenium system and granted a U.S.

patent (Patterson, 1966-67) covering the best composition selection.

be produced in small numbers mostly for the defense uses.

In 1967 Harold Hafner was made many important contributions including a glass casting process and a glass tempering process in Semiconductor Production Division under the direction of Charlie Jones. There work concentrated the efforts on a glass from the germaniumarsenic-selenium system and outcomes (Jones et. al., 1968) agreed with the conclusions of the Russian, U.K. Alternatively, Servo group efforts that the germanium-arsenic-selenium system produced the best glasses for infrared system applications. Don Weirauch (Hilton, 2010) was conducted a crystallization study on the germanium-arsenic selenium family of glasses and identified a composition in which crystallites would not form. In 1972 a commercial group was successfully cast (12 in 24 in 0.5 in) a window which flat polished, parallel and antireflectioncoated (Hafner, 1972). In the late 1960s and early 1970s, passive 8- to 12μ m systems began to

In 1968, Ovshinsky and his co-workers was discovered (Stocker, 1969) the some chalcogenide glasses exhibited memory and switching effects. After this discovery it became clear that the electric pulses could be switch the phases in chalcogenide glasses back and forth between amorphous and crystalline state.Around the same period in 1970's, Sir N. F. Mott (a former Noble Prices winner in Physics-1977) and E.A. Davis were developed the theory on the electronic processes in non-crystalline chalcogenide glasses (Mott et. al., 1979), and Kawamura (Kawamura et. al., 1983) was discovered xerography. Applications of solar cells were developed by Ciureanu and Middehoek (Ciureanu et. al., 1992) and Robert and his coworkers (Robert et.al., 1998). Infrared optics applications were studied by Quiroga and Leng and their coworkers (Quiroga et al., 1996, Leng et. al., 2000 ). The switching device applications were introduced by Bicerono and Ovshinsky (Bicerono et. al.,1985) and Ovshinsky (Ovshinsky, 1994). P. Boolchand and his coworkers (Boolchand et.al., 2001) was discovered intermediate phase in chalcogenide glasses. In this order several investigators have been also reported that the useful optoelectronics applications in infrared transmission and detection, threshold and memory switching (Selvaraju et al., 2003), optical fibers (Bowden et al., 2009, Shportko et al., 2008, Milliron et al.,2007) functional elements in integrated-optic circuits (Pelusi et al.2009) non-linear optics (Dudley et al., 2009), holographic & memory storage media (Vassilev et al., 2009, Wuttig et al.,2007), chemical and bio-sensors (Anne et al.,2009, Schubert et at., 2001), infrared photovoltaics (Sargent, 2009), microsphere laser (Elliott, 2010), active plasmonics (Samson,2010), microlenses in inkjet printing (Sanchez, 2011) and other photonics (Eggleton, 2011) applications. In this respect, the analysis of the composition dependence of their thermal properties was an important aspect for the study (Singh et al., 2009, 2010, 2011).

Subsequently, several review books were published on chalcogenide glasses e. g. "The Chemistry of Glasses" by A Paul in 1982, "The Physics of Amorphous Solids" by R.Zallen in 1983 and "Physics of Amorphous Materials" by S.R.Elliott in 1983. However, first book entirely dedicated to chalcogenide glassy materials entitled "Chalcogenide Semiconducting Glasses" was published in 1983 by Z.U.Borisova. In this order, G.Z. Vinogradova was published her monograph "Glass formation and Phase Equilibrium in Chalcogenide Systems" in 1984. M.A. Andriesh dedicates a book to some specific applications of chalcogenide glasses entitled "Glassy Semiconductors in Photo-electric Systems for Optical Recording of Information". M.A. Popescu gave large and detailed account on physical and technological aspect of chalcogenide systems in his book "Non-Crystalline Chalcogenides". The compendium of monographs on the subject of photo-induced processes in chalcogenide glasses entitled "Photo-induced Metastability in Amorphous Semiconductors" was compiled by A. Colobov-2003. Robert Fairman and Boris Ushkov-2004 described physical properties in "Semiconducting Chalcogenide Glass I: Glass formation, structure, and simulated transformations in Chalcogenide Glass". Finally, A. Zakery and S.R. Elliott demonstrated the "Optical Nonlinearities in Chalcogenide Glasses and their Applications" in 2007.

#### **1.1.1 Binary chalcogenides**

Structure of chalcogenide glasses have been extensively studied in binary compositions considering both the bulk and thin film forms. Chalcogens can form alloys together, Se-S, Se-Te (Gill, 1973) and S-Te amorphous (Sarrach et. al, 1976, Hawes, 1963) compounds were identified; however the scientific community seems to have, for the moment at least, left these glasses aside. Many binary compounds can be synthesized by associating one of the chalcogen with another element of the periodic table like, indium, antimony, copper, germanium, phosphorus, silicon and tin. A few other compounds based on heavy or light elements and alkali atoms have also been investigated. Abrikosov and his co-workers in 1969 (Lopez, 2004) were first reported the molecular structures of most extensively studied As-S, As-Se binary chalcogenide alloys in their monograph, the phase diagrams for the As- S

Crystallization Kinetics of Chalcogenide Glasses 33

new binary matrix. Most extensively studied ternary As-S-Se system was shown a very wide glass-forming region (Flaschen et. al., 1959). The solid solutions can be formed along the line As2 S3 –As2 Se3 which proved via IR spectra and X-ray analysis by Velinov and his coworkers (Velinov et. al., 1997). The Covalent Random Network (CRN) and the Chemically Ordered Network (CON) models both satisfy the 8-N rule under the distribution of bond types in a covalent network with multi elements. As- rich glasses can be formed As–As, As– Se, and As–S bonds; thus Se-rich glasses have As–Se, As–S, and Se–Se bonds and S-rich glasses As–Se, As–S, and S–S bonds. The relative weight of each of the above units is expected to be proportionate to the overall composition of the glass itself (Yang et. al., 1989). In recent years Zn containing ternary chalcogenide glasses attracted much attention due to higher melting point, metallic nature and advanced scientific interest (Boo et. al., 2007). Crystalline state zinc has hexagonal close-packed crystal structure with average coordination number four. While, in amorphous structure it is expected to metallic Zn dissolve in Se chains and makes homopolar and heteropolar bonds. Addition of third element concentration in binary alloy affects the chemical equilibrium of exiting bonds, therefore newly form ternary glass stoichiometry would heavily cross-linked, and makes homopolar and heteropolar bonds in respect of alloying elements. Specifically, Se-Zn-In ternary chalcogenide glasses can form Se-In heteronuclear bonds with strong fixed metallic Zn-In, Zn-Se bonds. Incorporation of indium concentration as-cost of selenium amount, the Se-Zn-In became heavily cross-linked results the steric hindrance increases at the threshold compositional concentration and beyond the threshold concentration a drastic change in

Addition of more than three elements in chalcogenide alloys refers as multicomponent alloys. In recent years there is an intensive interest made on study of new multi-component chalcogenide glasses to make sophisticated device technology as well as from the point of view of basic physics. Although Se rich binary and ternary chalcogenide glasses exhibit high resistivity, greater hardness, lower aging effect, enhanced electrical and optical properties with good working performance. But ternary glasses have certain drawbacks which implying the limitation in applications. It is worth then to add more than two components into selenium matrix can produce considerable changes in the properties complex glasses. Predominantly, metal and semimetal containing multi-component amorphous semiconductors promising materials to investigations such as; Ge-Bi-Se-Te, Al-(Ge-Se-Y), Ge–As–Se–Te, Cd (Zn)- Ge(As), GeSe2– Sb2Se3–PbSe, Cu2ZnSnSe4 etc. (Thingamajig et. al., 2000, Petkov, 2002, Vassilev, 2006, Wibowo et. al. 2007). More specifically Se-Zn-Te-In multi-component chalcogenide glasses make Se-In heteronuclear bonds with other possible bonds Zn-In and Te-In. Due to the addition of Indium in quaternary glassy matrix, the structures become heavily cross-linked and steric hindrance increases. Therefore, at the expanse of Se chains and replacement of weak Se-Se bonds by Se-In bonds results the

increase and decrease in their associative physical properties of Se-Zn-Te-In glasses.

Therefore, the thermal, electrical and optical properties of chalcogenide glasses widely depend on alloying concentration and intrinsic structural changes make them a chemical threshold a particular concentration of alloy. In view of these basic property several past research work in chalcogenide glasses were reported on binary, ternary and very few on

physical properties has been observed.

**1.1.3 Multicomponent chalcogenides** 

and As-Se systems. As-S alloys can be formed with an As content up to 46%, while in As-Se this maximum content can be raised to almost 60%. Glasses with low As content can easily crystallize (e.g. for a content of 6% As the glass crystallizes at room temperature in one day) in the range 5-16 weight %, in a couple of days at 60ºC while it takes 30 days for As S at 280ºC (Lopez, 2004). As-Se alloys can crystallize along the all composition range, however this was to be done under pressure and at elevated temperatures. The typical As2 S3 structure has usually pictured as an assembly of six AsS pyramids (the As atom being the top of the pyramid while three S atoms form the base). Goriunova and Kolomiets in 1958 (Lopez, 2004) were pointed out that the importance of covalent bonding in chalcogenide glasses as the most important property to make stability of these glasses. As opposed to metallic bonding, covalent bonding ensures easier preparation of the glasses. Thus, the crosslinking initiated by the As atoms should reduce the freedom for disorder in which bonds are covalent. Further Vaipolin and Porai-Koshits reported X-ray studies in beginning of the 1960's (Lopez, 2004), for the vitreous As2 S3 and As2Se3 and a number of binary glass compositions based on these two compounds. These glasses were shown to contain corrugated layers, which deformed with increasing size of the chalcogens and arsenic atoms became octohedrally coordinated. The character of the bonds was also found become more ionic when at equimolecular compositions. At the beginning of the 1980's, Tanaka was also characterized the chalcogenide glasses as a phase change materials and demonstrated that they structurally rigid and not having long-range ordering.

Alternatively, Se-S, Se-Te and S-Te (Hamada et. al., 1968, Bohmer & Angell, 1993) were extensively studied binary chalcogens alloys in which Se and S taken as host material. Amorphous selenium and sulfer molecular structure become the mixture of chain and rings which bridging the gap between molecular glasses and polymers. They covalently bonded with two coordination number. The most stable trigonal structural phase a-Se consists of parallel helical chains and two monoclinic bonds forms the composed of rings of eight atoms. These polymorphs distinguished by the correlation between neighboring dihedral angles. The amorphous selenium has relatively low molecular weight polymer with low concentration of rings (Bichara & pellegatti, 1993, Caprion & Schober, 2000 & 2002, Echeveria et. al., 2003, Malek et. al., 2009, etc).

Particularly Se-In binary chalcogenide compositions were getting much attention due to their versatile technical applications. VI- III family compounds Se- In form layered structures with strong covalent bonds. Basically VI-III group Se- In compounds have hexagonal symmetry structure. It consists of two layers which separated by tetrahedrally or pentagonally coordinated Se and In (JablÇonska et. al., 2001 & Pena et al., 2004). Amorphous Se-In compounds contains *a*-In2Se3, *a*-InSe, and *a*-In4Se3 binary phases. The number of In-Se nearest neighbor heteropolar bonds considerably larger than the homopolar bonds. The In-In and Se-Se homopolar bonds contribute mainly to the left- and right-hand side of the first peak in the radial distribution function, but they do not influence original position. The number of nearest-neighbor Se-Se bonds in *a*-InSe and *a*-In4Se3 structures is generally negligible (Kohary et. al., 2005).

#### **1.1.2 Ternary chalcogenides**

Ternary chalcogenide glasses also broadly studied from more than three decades. Ternary chalcogenides can be prepared by introducing a suitable additive element in well known or

and As-Se systems. As-S alloys can be formed with an As content up to 46%, while in As-Se this maximum content can be raised to almost 60%. Glasses with low As content can easily crystallize (e.g. for a content of 6% As the glass crystallizes at room temperature in one day) in the range 5-16 weight %, in a couple of days at 60ºC while it takes 30 days for As S at 280ºC (Lopez, 2004). As-Se alloys can crystallize along the all composition range, however this was to be done under pressure and at elevated temperatures. The typical As2 S3 structure has usually pictured as an assembly of six AsS pyramids (the As atom being the top of the pyramid while three S atoms form the base). Goriunova and Kolomiets in 1958 (Lopez, 2004) were pointed out that the importance of covalent bonding in chalcogenide glasses as the most important property to make stability of these glasses. As opposed to metallic bonding, covalent bonding ensures easier preparation of the glasses. Thus, the crosslinking initiated by the As atoms should reduce the freedom for disorder in which bonds are covalent. Further Vaipolin and Porai-Koshits reported X-ray studies in beginning of the 1960's (Lopez, 2004), for the vitreous As2 S3 and As2Se3 and a number of binary glass compositions based on these two compounds. These glasses were shown to contain corrugated layers, which deformed with increasing size of the chalcogens and arsenic atoms became octohedrally coordinated. The character of the bonds was also found become more ionic when at equimolecular compositions. At the beginning of the 1980's, Tanaka was also characterized the chalcogenide glasses as a phase change materials and demonstrated that

Alternatively, Se-S, Se-Te and S-Te (Hamada et. al., 1968, Bohmer & Angell, 1993) were extensively studied binary chalcogens alloys in which Se and S taken as host material. Amorphous selenium and sulfer molecular structure become the mixture of chain and rings which bridging the gap between molecular glasses and polymers. They covalently bonded with two coordination number. The most stable trigonal structural phase a-Se consists of parallel helical chains and two monoclinic bonds forms the composed of rings of eight atoms. These polymorphs distinguished by the correlation between neighboring dihedral angles. The amorphous selenium has relatively low molecular weight polymer with low concentration of rings (Bichara & pellegatti, 1993, Caprion & Schober, 2000 & 2002,

Particularly Se-In binary chalcogenide compositions were getting much attention due to their versatile technical applications. VI- III family compounds Se- In form layered structures with strong covalent bonds. Basically VI-III group Se- In compounds have hexagonal symmetry structure. It consists of two layers which separated by tetrahedrally or pentagonally coordinated Se and In (JablÇonska et. al., 2001 & Pena et al., 2004). Amorphous Se-In compounds contains *a*-In2Se3, *a*-InSe, and *a*-In4Se3 binary phases. The number of In-Se nearest neighbor heteropolar bonds considerably larger than the homopolar bonds. The In-In and Se-Se homopolar bonds contribute mainly to the left- and right-hand side of the first peak in the radial distribution function, but they do not influence original position. The number of nearest-neighbor Se-Se bonds in *a*-InSe and *a*-In4Se3 structures is generally

Ternary chalcogenide glasses also broadly studied from more than three decades. Ternary chalcogenides can be prepared by introducing a suitable additive element in well known or

they structurally rigid and not having long-range ordering.

Echeveria et. al., 2003, Malek et. al., 2009, etc).

negligible (Kohary et. al., 2005).

**1.1.2 Ternary chalcogenides** 

new binary matrix. Most extensively studied ternary As-S-Se system was shown a very wide glass-forming region (Flaschen et. al., 1959). The solid solutions can be formed along the line As2 S3 –As2 Se3 which proved via IR spectra and X-ray analysis by Velinov and his coworkers (Velinov et. al., 1997). The Covalent Random Network (CRN) and the Chemically Ordered Network (CON) models both satisfy the 8-N rule under the distribution of bond types in a covalent network with multi elements. As- rich glasses can be formed As–As, As– Se, and As–S bonds; thus Se-rich glasses have As–Se, As–S, and Se–Se bonds and S-rich glasses As–Se, As–S, and S–S bonds. The relative weight of each of the above units is expected to be proportionate to the overall composition of the glass itself (Yang et. al., 1989).

In recent years Zn containing ternary chalcogenide glasses attracted much attention due to higher melting point, metallic nature and advanced scientific interest (Boo et. al., 2007). Crystalline state zinc has hexagonal close-packed crystal structure with average coordination number four. While, in amorphous structure it is expected to metallic Zn dissolve in Se chains and makes homopolar and heteropolar bonds. Addition of third element concentration in binary alloy affects the chemical equilibrium of exiting bonds, therefore newly form ternary glass stoichiometry would heavily cross-linked, and makes homopolar and heteropolar bonds in respect of alloying elements. Specifically, Se-Zn-In ternary chalcogenide glasses can form Se-In heteronuclear bonds with strong fixed metallic Zn-In, Zn-Se bonds. Incorporation of indium concentration as-cost of selenium amount, the Se-Zn-In became heavily cross-linked results the steric hindrance increases at the threshold compositional concentration and beyond the threshold concentration a drastic change in physical properties has been observed.

#### **1.1.3 Multicomponent chalcogenides**

Addition of more than three elements in chalcogenide alloys refers as multicomponent alloys. In recent years there is an intensive interest made on study of new multi-component chalcogenide glasses to make sophisticated device technology as well as from the point of view of basic physics. Although Se rich binary and ternary chalcogenide glasses exhibit high resistivity, greater hardness, lower aging effect, enhanced electrical and optical properties with good working performance. But ternary glasses have certain drawbacks which implying the limitation in applications. It is worth then to add more than two components into selenium matrix can produce considerable changes in the properties complex glasses. Predominantly, metal and semimetal containing multi-component amorphous semiconductors promising materials to investigations such as; Ge-Bi-Se-Te, Al-(Ge-Se-Y), Ge–As–Se–Te, Cd (Zn)- Ge(As), GeSe2– Sb2Se3–PbSe, Cu2ZnSnSe4 etc. (Thingamajig et. al., 2000, Petkov, 2002, Vassilev, 2006, Wibowo et. al. 2007). More specifically Se-Zn-Te-In multi-component chalcogenide glasses make Se-In heteronuclear bonds with other possible bonds Zn-In and Te-In. Due to the addition of Indium in quaternary glassy matrix, the structures become heavily cross-linked and steric hindrance increases. Therefore, at the expanse of Se chains and replacement of weak Se-Se bonds by Se-In bonds results the increase and decrease in their associative physical properties of Se-Zn-Te-In glasses.

Therefore, the thermal, electrical and optical properties of chalcogenide glasses widely depend on alloying concentration and intrinsic structural changes make them a chemical threshold a particular concentration of alloy. In view of these basic property several past research work in chalcogenide glasses were reported on binary, ternary and very few on

Crystallization Kinetics of Chalcogenide Glasses 35

corresponding to coordination number and demonstrate the structural motifs change in such materials. Subsequently, Agarwal & Sanghera (Agarwal & Sanghera, 2002) were discussed the development and application of chalcogenide glass optical fibers in near scanning field microscopy/spectroscopy and Jackson & Srinivas (Jackson & Srinivas, 2002) demonstrated the modeling of metallic chalcogenide glasses using density function theory calculations. Tanaka (Tanaka, 2003) reviewed the nanoscale structures of chalcogenide glasses and inspect surface modifications at nanometer resolution and Micoulaut & Phillips (Micoulaut & Phillips ,2003) were shown the three elastic phases of covalent networks ( (I) floppy, (II) isostatically rigid, and (III) stressed-rigid) depend on the degree freedom of material. They were also suggested that the ring factor is responsible for high crystallization temperature in metallic/ semi-metallic chalcogenide glasses. Lezal and his coworkers (Lezal et. al., 2004) were reviewed the chalcogenide glasses for optical and photonics applications. Vassilev & Boycheva (Vassilev & Boycheva, 2005) were critically reviewed the achievements in application of chalcogenide glasses as membrane materials. They were also demonstrated that the advantages and disadvantages in analytical performance and compared with the corresponding polycrystalline analogous. Emin (Emin, 2006) was explained the polaron conduction machines in amorphous semiconductors and Kokenyesi (Kokenyesi, 2006) reviewed the amorphous chalcogenide nano-multilayers: research and developments. While Phillips (Phillips, 2006) demonstrated the, ideally glassy materials have hydrogen-bonded networks. Bosch and his coworkers (Bosch et. al. 2007) were critically reviewed the last decade developments in optical fibers in bio sensing and Vassilev and his coworkers (Vassilev et. al.,2007) introduced the new Se- based multicomponent chalcogenide glasses and studied their composition dependence physical properties. Furthermore, Wachter and Taeed their coworkers (Wachter et. al., 2007, Taeed et. al., 2007) were demonstrated the composition dependence reversible and tunable glass-crystal-glass phase transition properties in new class of multicomponent chalcogenide glasses and show chalcogenide glasses useful for all-optical signal processing devices due to their large ultrafast third-order nonlinearities, low two-photon absorption and the absence of free carrier absorption in a photosensitive medium. Dahshan and Lousteau and their coworkers (Dahshan et. al.,2008, Lousteau et. al.,2008) were demonstrated the thermal stability and activation energy of some Cu doped chalcogenide glasses and the fabrication of heavy metal fluoride glass to explore the optical planar waveguides by hot-spin casting. Further, Klokishner (Klokishner et. al., 2008) were studied the concentration effects on the photoluminescence band centers in multicomponent metallic chalcogenide glasses and Ielmini and his coworkers (Ielmini et. al., 2008) demonstrated the threshold switching mechanism by high-field energy gain in the hopping transport of chalcogenide glasses. Mehta and his coworkers (Mehta et. al., 2009) were studied the effect of metallic and non metallic additive elements on Se-Te based chalcogenide glasses. In the same year Turek and Anne their coworkers (Turek et. al.,2009, Anne et. al.,2009) were demonstrated the artificial intelligence/fuzzy logic method for analysis of combined signals from heavy metal chemical sensors and commented on, due to the remarkable properties of chalcogenide glasses can be used as a biosensor which can collect the information on whole metabolism alterations rapidly. Further, Khan and his coworkers (Khan et. al., 2009) were demonstrated the composition dependence electrical transport and optical properties of metallic element doped Se based chalcogenide glasses. In order to this Snopatin and his coworkers (Snopatin et. al., 2009) were demonstrated the

multicomponent systems (Tonchev et. al., 1999, Wagner et. al., 1998, Mehta et. al., 2008, Patial et. al., 2011, Malek et. al., 2003, Soltan et. al., 2003, Song et. al., 1997, Usuki et. al., 2001, Fayek et. al., 2001, Wang et. al., 2007, Vassilev et. al., 2007, Eggleton et. al., 2011, Prashanth, et. al., 2008, Vassilev et. al.,2007, Othman et. al., 2006, Zhang et. al.,2004, Hegab et. al., 2007, Narayanan et. al., 2001, etc). Scientific and technological drawbacks, like low thermal stability, low crystallization temperature and aging effects (Guo et. al., 2007, Boycheva et. al., 2002, Ivanova et. al., 2003, Vassilev et. al., 2005, Xu et. al., 2008, Troles et. al., 2008) of non-metallic binary and ternary alloys motivates to investigators to make metallic multicomponent chalcogen alloys to achieve high thermal stability and harder chalcogenide glasses (Pungor, 1997, Demarco et. al., 1999, Kobelke et. al.,1999, Zhang et. al., 2005, Singh, 2011).

Extensive research on metal containing multicomponent chalcogenide alloys was begin nearly end of nineties when Kikineshy and Sterr 1989 & 1990 (Kokenyesi, et. al.,2007 & Ivan & Kikineshi, 2002) were demonstrated the multilayer of chalcogen alloys simply nanostructures materials which can be rather easily produced with controlled geometrical parameters. In this order Ionov and his coworkers (Ionov et. al.,1991) were demonstrated the electrical and electrophotographic properties of selenium based metal containing multicomponent chalcogenide glass and outlined these materials would be useful for electrophotographic and laser printer photoreceptors. Saleh and his coworkers (Saleh et. al., 1993) were studied the nuclear magnetic resonance relaxation of Cu containing chalcogenide glasses. Carthy & Kanatzidis (Carthy & Kanatzidis, 1996) were introduced the bismuth and antimony containing new class of multicomponent chalcogenide glasses. In the same year Natale and his coworkers (Natale et. al.,1996) were demonstrated the heavy metal multicomponent glasses useful for array sensors. Further, Nesheva and his coworkers (Nesheva et. al., 1997) were studied the amorphous pure and allying selenium based multilayers and demonstrated the photoreceptor properties at room temperature unaltered throughout in a year. Efimov (Efimov,1999) was described the mechanism of formation of the vibrational spectra of glasses such as quasi-molecular model, central force model and its recent refinements (model of phonon localization regions) and deduce the trends in the IR and Raman band assignments in inorganic systems. Goetzberger & Hebling (Goetzberger & Hebling, 2000) were commented on the present, past and future of photovoltaic materials. In the same year Naumis ( Naumis, 2000) demonstrated the jump of the heat capacity in chalcogenide glasses during glass transition and show change in glass fragility and excess thermal expansivity is a function of average coordination number. While Mortensen and his coworkers (Mortensen et. al., 2000) were used heavy metals based chalcogenides sensors in detection of flow injection. Mourizina and his coworkers (Mourizina et. al., 2001) were demonstrated the ion selective light addressable poentiometric senor based on metal containing chalcogenide glass film. In the same year Rau and his coworkers (Rau et. al., 2001) were studied the effect of the mixed cation in chalcogenide glasses and reported that the non-linear structural changes in Raman and infrared spectra. Further, Messaddeq and his coworkers (Messaddeq et. al., 2001) were demonstrated the light induced volume expansion in chalcogenide glasses under the irradiation UV light. Moreover, Hsu and Narayanan and their coworkers (Hsu et. al., 2001, Narayanan et. al., 2001) were studied the near field microscopic properties of electronic & photonic materials and devices and large switching fields in metal containing chalcogenide glasses owing to chemical disordering. Salmon & Xin (Salmon & Xin, 2002) were studied the effect of high modifier content

multicomponent systems (Tonchev et. al., 1999, Wagner et. al., 1998, Mehta et. al., 2008, Patial et. al., 2011, Malek et. al., 2003, Soltan et. al., 2003, Song et. al., 1997, Usuki et. al., 2001, Fayek et. al., 2001, Wang et. al., 2007, Vassilev et. al., 2007, Eggleton et. al., 2011, Prashanth, et. al., 2008, Vassilev et. al.,2007, Othman et. al., 2006, Zhang et. al.,2004, Hegab et. al., 2007, Narayanan et. al., 2001, etc). Scientific and technological drawbacks, like low thermal stability, low crystallization temperature and aging effects (Guo et. al., 2007, Boycheva et. al., 2002, Ivanova et. al., 2003, Vassilev et. al., 2005, Xu et. al., 2008, Troles et. al., 2008) of non-metallic binary and ternary alloys motivates to investigators to make metallic multicomponent chalcogen alloys to achieve high thermal stability and harder chalcogenide glasses (Pungor,

Extensive research on metal containing multicomponent chalcogenide alloys was begin nearly end of nineties when Kikineshy and Sterr 1989 & 1990 (Kokenyesi, et. al.,2007 & Ivan & Kikineshi, 2002) were demonstrated the multilayer of chalcogen alloys simply nanostructures materials which can be rather easily produced with controlled geometrical parameters. In this order Ionov and his coworkers (Ionov et. al.,1991) were demonstrated the electrical and electrophotographic properties of selenium based metal containing multicomponent chalcogenide glass and outlined these materials would be useful for electrophotographic and laser printer photoreceptors. Saleh and his coworkers (Saleh et. al., 1993) were studied the nuclear magnetic resonance relaxation of Cu containing chalcogenide glasses. Carthy & Kanatzidis (Carthy & Kanatzidis, 1996) were introduced the bismuth and antimony containing new class of multicomponent chalcogenide glasses. In the same year Natale and his coworkers (Natale et. al.,1996) were demonstrated the heavy metal multicomponent glasses useful for array sensors. Further, Nesheva and his coworkers (Nesheva et. al., 1997) were studied the amorphous pure and allying selenium based multilayers and demonstrated the photoreceptor properties at room temperature unaltered throughout in a year. Efimov (Efimov,1999) was described the mechanism of formation of the vibrational spectra of glasses such as quasi-molecular model, central force model and its recent refinements (model of phonon localization regions) and deduce the trends in the IR and Raman band assignments in inorganic systems. Goetzberger & Hebling (Goetzberger & Hebling, 2000) were commented on the present, past and future of photovoltaic materials. In the same year Naumis ( Naumis, 2000) demonstrated the jump of the heat capacity in chalcogenide glasses during glass transition and show change in glass fragility and excess thermal expansivity is a function of average coordination number. While Mortensen and his coworkers (Mortensen et. al., 2000) were used heavy metals based chalcogenides sensors in detection of flow injection. Mourizina and his coworkers (Mourizina et. al., 2001) were demonstrated the ion selective light addressable poentiometric senor based on metal containing chalcogenide glass film. In the same year Rau and his coworkers (Rau et. al., 2001) were studied the effect of the mixed cation in chalcogenide glasses and reported that the non-linear structural changes in Raman and infrared spectra. Further, Messaddeq and his coworkers (Messaddeq et. al., 2001) were demonstrated the light induced volume expansion in chalcogenide glasses under the irradiation UV light. Moreover, Hsu and Narayanan and their coworkers (Hsu et. al., 2001, Narayanan et. al., 2001) were studied the near field microscopic properties of electronic & photonic materials and devices and large switching fields in metal containing chalcogenide glasses owing to chemical disordering. Salmon & Xin (Salmon & Xin, 2002) were studied the effect of high modifier content

1997, Demarco et. al., 1999, Kobelke et. al.,1999, Zhang et. al., 2005, Singh, 2011).

corresponding to coordination number and demonstrate the structural motifs change in such materials. Subsequently, Agarwal & Sanghera (Agarwal & Sanghera, 2002) were discussed the development and application of chalcogenide glass optical fibers in near scanning field microscopy/spectroscopy and Jackson & Srinivas (Jackson & Srinivas, 2002) demonstrated the modeling of metallic chalcogenide glasses using density function theory calculations. Tanaka (Tanaka, 2003) reviewed the nanoscale structures of chalcogenide glasses and inspect surface modifications at nanometer resolution and Micoulaut & Phillips (Micoulaut & Phillips ,2003) were shown the three elastic phases of covalent networks ( (I) floppy, (II) isostatically rigid, and (III) stressed-rigid) depend on the degree freedom of material. They were also suggested that the ring factor is responsible for high crystallization temperature in metallic/ semi-metallic chalcogenide glasses. Lezal and his coworkers (Lezal et. al., 2004) were reviewed the chalcogenide glasses for optical and photonics applications. Vassilev & Boycheva (Vassilev & Boycheva, 2005) were critically reviewed the achievements in application of chalcogenide glasses as membrane materials. They were also demonstrated that the advantages and disadvantages in analytical performance and compared with the corresponding polycrystalline analogous. Emin (Emin, 2006) was explained the polaron conduction machines in amorphous semiconductors and Kokenyesi (Kokenyesi, 2006) reviewed the amorphous chalcogenide nano-multilayers: research and developments. While Phillips (Phillips, 2006) demonstrated the, ideally glassy materials have hydrogen-bonded networks. Bosch and his coworkers (Bosch et. al. 2007) were critically reviewed the last decade developments in optical fibers in bio sensing and Vassilev and his coworkers (Vassilev et. al.,2007) introduced the new Se- based multicomponent chalcogenide glasses and studied their composition dependence physical properties. Furthermore, Wachter and Taeed their coworkers (Wachter et. al., 2007, Taeed et. al., 2007) were demonstrated the composition dependence reversible and tunable glass-crystal-glass phase transition properties in new class of multicomponent chalcogenide glasses and show chalcogenide glasses useful for all-optical signal processing devices due to their large ultrafast third-order nonlinearities, low two-photon absorption and the absence of free carrier absorption in a photosensitive medium. Dahshan and Lousteau and their coworkers (Dahshan et. al.,2008, Lousteau et. al.,2008) were demonstrated the thermal stability and activation energy of some Cu doped chalcogenide glasses and the fabrication of heavy metal fluoride glass to explore the optical planar waveguides by hot-spin casting. Further, Klokishner (Klokishner et. al., 2008) were studied the concentration effects on the photoluminescence band centers in multicomponent metallic chalcogenide glasses and Ielmini and his coworkers (Ielmini et. al., 2008) demonstrated the threshold switching mechanism by high-field energy gain in the hopping transport of chalcogenide glasses. Mehta and his coworkers (Mehta et. al., 2009) were studied the effect of metallic and non metallic additive elements on Se-Te based chalcogenide glasses. In the same year Turek and Anne their coworkers (Turek et. al.,2009, Anne et. al.,2009) were demonstrated the artificial intelligence/fuzzy logic method for analysis of combined signals from heavy metal chemical sensors and commented on, due to the remarkable properties of chalcogenide glasses can be used as a biosensor which can collect the information on whole metabolism alterations rapidly. Further, Khan and his coworkers (Khan et. al., 2009) were demonstrated the composition dependence electrical transport and optical properties of metallic element doped Se based chalcogenide glasses. In order to this Snopatin and his coworkers (Snopatin et. al., 2009) were demonstrated the

Crystallization Kinetics of Chalcogenide Glasses 37

Subsequently it is easier to a perfect crystal in a molten solid than to grow again a good crystal from the resulting solution. Hence the nucleation and growth process of a crystal are

Nucleation of the substance reflects the initiation of a phase change in a small region cause the formation of a solid crystal from a liquid solution. It is a consequence of rapid local fluctuations on a molecular scale in a homogeneous phase which define as a metastable equilibrium state. The whole nucleation process of a substance is the sum of heterogeneous (nucleation that occurs in the absence of a second phase) and heterogeneous (nucleation that occurs in the presence of a second, foreign phase) category of nucleation. Homogeneous nucleation due to clustering of molecules (embryos) in a supersaturated environment, in which a process began, combines two or more than molecules. In the reversible clustering process a few molecules grew at the same time and others dissolving. Once embryos attained a certain critical size then it decrease its total free energy by growing and becomes stable (Reid et. al., 1970). But, in practice it is difficult to find complete homogeneous nucleation owing to presence of insoluble amounts of matter even in pure material. Therefore, heterogeneous nucleation is always associated with homogeneous nucleation due to presence of second phase in bulk molten material. The heterogeneous nucleation occurs

As earlier mentioned crystal growth is the successive process of nucleation in which the critical nuclei of microscopic size form a crystal. Crystal growth in crystallization process takes place by fusion and re-solidification of the material. In this process within a solid material constituent of molecules (embryos) are arranged in an orderly repeating pattern extending in all three spatial dimensions. Crystal growth is a major stage of a crystallization process which consists the addition of new molecules (embryos) strings into the characteristic arrangement of a crystalline lattice. The growth typically follows an initial stage of either homogeneous or heterogeneous nucleation. The crystal growth process yields a crystalline solid whose molecules are typically close packed with fixed positions in space relative to each other. In general crystalline solids are typically formed by cooling and solidification from the molten (or liquid) state. As per the Ehrenfest classification it is firstorder phase transitions with a discontinuous change in volume (and thus a discontinuity in the slope or first derivative with respect to temperature, dV/dT) at the melting point. Hence, the crystal and melt are distinct phases with an interfacial discontinuity having a surface of tension with a positive surface energy. Thus, a metastable parent phase represents it always stable with respect to the nucleation of small embryos from a daughter phase with a positive surface of tension. Hence, crystal growth process is first-order transitions consist advancement of an interfacial region whose structure and properties vary discontinuously from the parent phase. In the crystal growth process stiochiometry of glass compositions do not undergo in compositional changes during crystallization, mean, no need to long-range diffusion (Swanson, 1977) for crystal growth in chalcogenide glasses; thus, interfacial rearrangements are likely to control the crystal growth process. This (melt quenched) type of

well control under the thermodynamic kinetic.

in a random fashion at various sites in a matter.

**1.3 Nucleation and growth** 

**1.3.1 Nucleation** 

**1.3.2 Crystal growth** 

some high purity multicomponent chalcogenide glasses for fiber optics. Kumar and his coworkers (Kumar et. al., 2010) were demonstrated the calorimetric studies of Se-based metal containing multicomponent chalcogenide glasses. Peng & Liu (Peng & Liu ,2010) were reviewed the advances and achievements in SPM-based data storage in viewpoint of recording techniques including electrical bistability, photoelectrochemical conversion, fieldinduced charge storage, atomic manipulation or deposition, local oxidation, magneto-optical or magnetic recording, thermally induced physical deformation or phase change, and so forth as well as achievements in design and synthesis of organic charge-transfer (CT) complexes towards thermochemical-hole-burning memory, the correlation between holeburning performances and physicochemical properties of CT complexes.

Story of the investigations will be remain continue in field of metallic chalcogenides (not limited to above outlined the major events) to deduce the new future prospective multicomponent chalcogenide glassy alloys. Amorphous chalcogenide alloys which full fill the essential requirement of modern optoelectronics. So, it can be outlined potential field of optoelectronics and advanced material is rapidly growing owing to their possible uses. Therefore, it is important to have an understanding regarding on crystallization process of chalcogenide glasses (predominately in metal containing alloys).

#### **1.2 Crystallization**

Crystallization is a natural process of formation of solid crystals from a solution/- melt. Crystallization of a substance can also achieve from the chemical solid-liquid separation technique, in which mass transfer from the liquid solution to a pure solid crystalline phase. The crystallization process of a substance mainly consists of two major events nucleation and crystal growth. In nucleation process the molten molecules dispersed in solid solution and begin to formation of clusters at the nanometer scale. Crystal growth is the subsequent growth of the nuclei which develop critical size of the formed clusters (because size of clusters plays an important role in the application of the material). Hence, the nucleation and growth are the continuous process which occurs simultaneously when supercooling exists in a system. Thus, system supercooling state acts as a driving force for the crystallization process. The supercooling driving force depending upon the conditions, either nucleation or growth may be predominant over to other and outcomes can be formed crystals with different sizes and shapes. Once the supercooling is established in a solidliquid system and reached at equilibrium then crystallization process is completed (Mersmann, 2001).

In general supercooled materials/ or alloys have ability to crystallize with different crystal structures, this process is known as polymorphism. Each polymorph is in fact a different thermodynamic solid state and crystal polymorphs of the same alloy/-compound which exhibited different physical properties, such as dissolution rate, shape, melting point, etc. Thus the crystallization process of a substance is governed by both thermodynamic and kinetic factors which highly variable and difficult to control. Factors those affect the crystallization process of a substance/alloy are the impurity level, mixing regime, vessel design, and cooling profile and shape of crystals. Usually in those materials crystallization process occurs at lower temperatures, in supercooling situation they obey the law of thermodynamics. Its literal meaning a crystal can be more easily destroyed than it is formed. Subsequently it is easier to a perfect crystal in a molten solid than to grow again a good crystal from the resulting solution. Hence the nucleation and growth process of a crystal are well control under the thermodynamic kinetic.

#### **1.3 Nucleation and growth**

#### **1.3.1 Nucleation**

36 Crystallization – Science and Technology

some high purity multicomponent chalcogenide glasses for fiber optics. Kumar and his coworkers (Kumar et. al., 2010) were demonstrated the calorimetric studies of Se-based metal containing multicomponent chalcogenide glasses. Peng & Liu (Peng & Liu ,2010) were reviewed the advances and achievements in SPM-based data storage in viewpoint of recording techniques including electrical bistability, photoelectrochemical conversion, fieldinduced charge storage, atomic manipulation or deposition, local oxidation, magneto-optical or magnetic recording, thermally induced physical deformation or phase change, and so forth as well as achievements in design and synthesis of organic charge-transfer (CT) complexes towards thermochemical-hole-burning memory, the correlation between hole-

Story of the investigations will be remain continue in field of metallic chalcogenides (not limited to above outlined the major events) to deduce the new future prospective multicomponent chalcogenide glassy alloys. Amorphous chalcogenide alloys which full fill the essential requirement of modern optoelectronics. So, it can be outlined potential field of optoelectronics and advanced material is rapidly growing owing to their possible uses. Therefore, it is important to have an understanding regarding on crystallization process of

Crystallization is a natural process of formation of solid crystals from a solution/- melt. Crystallization of a substance can also achieve from the chemical solid-liquid separation technique, in which mass transfer from the liquid solution to a pure solid crystalline phase. The crystallization process of a substance mainly consists of two major events nucleation and crystal growth. In nucleation process the molten molecules dispersed in solid solution and begin to formation of clusters at the nanometer scale. Crystal growth is the subsequent growth of the nuclei which develop critical size of the formed clusters (because size of clusters plays an important role in the application of the material). Hence, the nucleation and growth are the continuous process which occurs simultaneously when supercooling exists in a system. Thus, system supercooling state acts as a driving force for the crystallization process. The supercooling driving force depending upon the conditions, either nucleation or growth may be predominant over to other and outcomes can be formed crystals with different sizes and shapes. Once the supercooling is established in a solidliquid system and reached at equilibrium then crystallization process is completed

In general supercooled materials/ or alloys have ability to crystallize with different crystal structures, this process is known as polymorphism. Each polymorph is in fact a different thermodynamic solid state and crystal polymorphs of the same alloy/-compound which exhibited different physical properties, such as dissolution rate, shape, melting point, etc. Thus the crystallization process of a substance is governed by both thermodynamic and kinetic factors which highly variable and difficult to control. Factors those affect the crystallization process of a substance/alloy are the impurity level, mixing regime, vessel design, and cooling profile and shape of crystals. Usually in those materials crystallization process occurs at lower temperatures, in supercooling situation they obey the law of thermodynamics. Its literal meaning a crystal can be more easily destroyed than it is formed.

burning performances and physicochemical properties of CT complexes.

chalcogenide glasses (predominately in metal containing alloys).

**1.2 Crystallization** 

(Mersmann, 2001).

Nucleation of the substance reflects the initiation of a phase change in a small region cause the formation of a solid crystal from a liquid solution. It is a consequence of rapid local fluctuations on a molecular scale in a homogeneous phase which define as a metastable equilibrium state. The whole nucleation process of a substance is the sum of heterogeneous (nucleation that occurs in the absence of a second phase) and heterogeneous (nucleation that occurs in the presence of a second, foreign phase) category of nucleation. Homogeneous nucleation due to clustering of molecules (embryos) in a supersaturated environment, in which a process began, combines two or more than molecules. In the reversible clustering process a few molecules grew at the same time and others dissolving. Once embryos attained a certain critical size then it decrease its total free energy by growing and becomes stable (Reid et. al., 1970). But, in practice it is difficult to find complete homogeneous nucleation owing to presence of insoluble amounts of matter even in pure material. Therefore, heterogeneous nucleation is always associated with homogeneous nucleation due to presence of second phase in bulk molten material. The heterogeneous nucleation occurs in a random fashion at various sites in a matter.

#### **1.3.2 Crystal growth**

As earlier mentioned crystal growth is the successive process of nucleation in which the critical nuclei of microscopic size form a crystal. Crystal growth in crystallization process takes place by fusion and re-solidification of the material. In this process within a solid material constituent of molecules (embryos) are arranged in an orderly repeating pattern extending in all three spatial dimensions. Crystal growth is a major stage of a crystallization process which consists the addition of new molecules (embryos) strings into the characteristic arrangement of a crystalline lattice. The growth typically follows an initial stage of either homogeneous or heterogeneous nucleation. The crystal growth process yields a crystalline solid whose molecules are typically close packed with fixed positions in space relative to each other. In general crystalline solids are typically formed by cooling and solidification from the molten (or liquid) state. As per the Ehrenfest classification it is firstorder phase transitions with a discontinuous change in volume (and thus a discontinuity in the slope or first derivative with respect to temperature, dV/dT) at the melting point. Hence, the crystal and melt are distinct phases with an interfacial discontinuity having a surface of tension with a positive surface energy. Thus, a metastable parent phase represents it always stable with respect to the nucleation of small embryos from a daughter phase with a positive surface of tension. Hence, crystal growth process is first-order transitions consist advancement of an interfacial region whose structure and properties vary discontinuously from the parent phase. In the crystal growth process stiochiometry of glass compositions do not undergo in compositional changes during crystallization, mean, no need to long-range diffusion (Swanson, 1977) for crystal growth in chalcogenide glasses; thus, interfacial rearrangements are likely to control the crystal growth process. This (melt quenched) type of

Crystallization Kinetics of Chalcogenide Glasses 39

∆G is positive when Te > T and it depends on the latent heat of transition. The change in free energy can also represent as the product of change in entropy and super cooling

Although equation (5) representing melt growth, in which one may depend on concentration rather than supercooling for solution growth and vapour growth. Equation (5)

where R is the Rydberg constant, C, C0 are the concentration of solid solution and concentration at critical transition, P, P0 are the pressers in vapour phase and S is the

Thus the equations (4) and (6) explain how the free energy changes depend on the supercooling parameters which are decisive in the process of crystallization. The rate of growth of a crystal can be expressed as a monotonically increasing function of Gibbs free

Since in nucleation process in the supercooled solution forms the small clusters of molecules (Joseph, 2010), therefore, free energy change between the solid and liquid can be expressed

The surface energy increase in term of r2 and volume energy decreases in term of r3. The

Further, in formation of critical nucleus size free energy **(∆G\*)** change can be calculated as;

where σ- is the interfacial energy and r is the spherical radius of the molecule

Hence the critical size **(r\*)** of the nucleus decreases with increasing cooling rate.

Equation (11) in terms of Gibbs thermodynamical relation can be written as;

Where ∆T=Te-T

temperature ∆T**.**

In general form

supercooling ratio.

in more convent form can be expressed as;

energy, when other parameters remain the same.

critical size of nucleus can be obtained by the equation (9)

as ∆Gv and Gibbs equation can be written as;

where Ω is the molecular volume

∆G=∆H. ∆T/Te (4)

∆G=∆S. ∆T (5)

∆G~RT ln (C/C0) (6)

∆G~RT ln (P/P0) (7)

∆G~RT ln S (8)

∆G = 4πr2σ – 4/3 πr3 ∆Gv (9)

r\* = 2 σ/ ∆Gv (10)

∆G\* = 16πσ3/ 3∆Gv2 (11)

∆G\* = 16πσ3Ω/ 3(kT ln.S) 2 (12)

crystal growth is generally described from these three basic standard models: (i) the screw dislocation model; (ii) the normal or continuous growth model; and (iii) the twodimensional surface nucleation growth.

Theoretically, nucleation and crystal growth process of molten solids first reported by Volmer and Weber (Volmer and Weber, 1925) and later on it explained in a large amount of literature. Tammann (1925) discussed the theory of nucleation and crystal growth and outlined the various parameters involved in terms of a probabilistic model involving a functional relation to pressure, temperature, and time. Many later studies deal with the nucleation and growth of crystals on an atomistic level. Nucleation of crystals from a melt is the mobility of atoms and molecules in the melt as measured by the diffusion coefficient. In glass-forming systems, liquid diffusion coefficients drop markedly with decreasing temperature (Towers and Chipman, 1957). When temperature drops below the liquidus, nucleation will increase from zero to a maximum at some undercooling. Diffusion rates then very low and nucleation decreases with further decrease in temperature. This explains the pattern of nucleation in liquids where the liquid diffusivity decreases with temperature and finally it grow a crystal. Rate of growth is thus a function of mobility of crystal-forming species within the melt. Mobility, can be measured by the diffusion coefficient, as drops with decreasing temperature and growth rate, like nucleation.

Thermodynamics of crystal growth process in molten solids can be expressed as (Warghese, 2010); the thermodynamical equilibrium between solid and liquid phases occur when the free energy of the two phases are equal

$$\mathbf{G}\_{\mathrm{I}} = \mathbf{G}\_{\mathrm{S}} \tag{1}$$

here GL, and GS are representing solid and liquid phases free energies

Free energy, internal energy and entropy of a system can be related from the Gibbs equation

$$\mathbf{G} \mathbf{=} \mathbf{H} \text{-TS} \tag{2}$$

here G is the Gibbs free energy, H is the enthalpy, S is the entropy and T is the temperature.

Formation of a crystal can be considered as a controlled change of phase to the solid state. Therefore, the driving force for crystallization comes from the lowering of the free energy of the system during this phase transformation. Change in free energy in such transition can be related as;

$$
\Delta \mathbf{G} = \Delta \mathbf{H} \text{-T} \Delta \mathbf{S} \tag{3}
$$

Where ∆H=HL-HS

$$
\Delta \mathbf{S} \mathbf{=} \mathbf{S}\_{\mathbb{L}} \mathbf{S}\_{\mathbb{S}}
$$

$$
\Delta \mathbf{G} \mathbf{=} \mathbf{G}\_{\mathbb{L}} \mathbf{G}\_{\mathbb{S}}
$$

For the equilibrium ∆G= 0

∆H=Te .∆S

Where, Te is the equilibrium temperature

$$
\Delta \mathbf{G} = \Delta \mathbf{H}. \,\Delta \mathbf{T}/\mathbf{T\_c} \tag{4}
$$

Where ∆T=Te-T

38 Crystallization – Science and Technology

crystal growth is generally described from these three basic standard models: (i) the screw dislocation model; (ii) the normal or continuous growth model; and (iii) the two-

Theoretically, nucleation and crystal growth process of molten solids first reported by Volmer and Weber (Volmer and Weber, 1925) and later on it explained in a large amount of literature. Tammann (1925) discussed the theory of nucleation and crystal growth and outlined the various parameters involved in terms of a probabilistic model involving a functional relation to pressure, temperature, and time. Many later studies deal with the nucleation and growth of crystals on an atomistic level. Nucleation of crystals from a melt is the mobility of atoms and molecules in the melt as measured by the diffusion coefficient. In glass-forming systems, liquid diffusion coefficients drop markedly with decreasing temperature (Towers and Chipman, 1957). When temperature drops below the liquidus, nucleation will increase from zero to a maximum at some undercooling. Diffusion rates then very low and nucleation decreases with further decrease in temperature. This explains the pattern of nucleation in liquids where the liquid diffusivity decreases with temperature and finally it grow a crystal. Rate of growth is thus a function of mobility of crystal-forming species within the melt. Mobility, can be measured by the diffusion coefficient, as drops

Thermodynamics of crystal growth process in molten solids can be expressed as (Warghese, 2010); the thermodynamical equilibrium between solid and liquid phases occur when the

GL=GS (1)

Free energy, internal energy and entropy of a system can be related from the Gibbs equation

here G is the Gibbs free energy, H is the enthalpy, S is the entropy and T is the temperature. Formation of a crystal can be considered as a controlled change of phase to the solid state. Therefore, the driving force for crystallization comes from the lowering of the free energy of the system during this phase transformation. Change in free energy in such transition can be

∆S=SL-SS

∆G=GL-GS

∆H=Te .∆S

G=H-TS (2)

∆G= ∆H-T∆S (3)

dimensional surface nucleation growth.

free energy of the two phases are equal

related as;

Where ∆H=HL-HS

For the equilibrium ∆G= 0

Where, Te is the equilibrium temperature

with decreasing temperature and growth rate, like nucleation.

here GL, and GS are representing solid and liquid phases free energies

∆G is positive when Te > T and it depends on the latent heat of transition. The change in free energy can also represent as the product of change in entropy and super cooling temperature ∆T**.**

$$
\Delta \mathbf{G} = \Delta \mathbf{S}. \,\Delta \mathbf{T} \tag{5}
$$

Although equation (5) representing melt growth, in which one may depend on concentration rather than supercooling for solution growth and vapour growth. Equation (5) in more convent form can be expressed as;

$$
\Delta \text{G} \sim \text{RT} \ln \text{ (C/C}\_0\text{)}\tag{6}
$$

$$
\Delta \mathbf{G} \sim \text{RT} \ln \text{ (P/P}\_0\text{)}\tag{7}
$$

In general form

$$
\Delta \text{G} \because \text{RT} \ln \text{S} \tag{8}
$$

where R is the Rydberg constant, C, C0 are the concentration of solid solution and concentration at critical transition, P, P0 are the pressers in vapour phase and S is the supercooling ratio.

Thus the equations (4) and (6) explain how the free energy changes depend on the supercooling parameters which are decisive in the process of crystallization. The rate of growth of a crystal can be expressed as a monotonically increasing function of Gibbs free energy, when other parameters remain the same.

Since in nucleation process in the supercooled solution forms the small clusters of molecules (Joseph, 2010), therefore, free energy change between the solid and liquid can be expressed as ∆Gv and Gibbs equation can be written as;

$$
\Delta \mathbf{G} = 4 \text{nr} \mathbf{2} \mathbf{o} - 4/3 \text{ nr}^3 \,\Delta \mathbf{G}\_\mathbf{v} \tag{9}
$$

where σ- is the interfacial energy and r is the spherical radius of the molecule

The surface energy increase in term of r2 and volume energy decreases in term of r3. The critical size of nucleus can be obtained by the equation (9)

$$\mathbf{r}^\* = \mathbf{2} \,\mathrm{\sigma/\ } \Delta \mathbf{G}\_{\mathrm{v}} \tag{10}$$

Hence the critical size **(r\*)** of the nucleus decreases with increasing cooling rate.

Further, in formation of critical nucleus size free energy **(∆G\*)** change can be calculated as;

$$
\Delta \mathbf{G}^\* = 16 \text{n} \mathbf{o}^3 / \, 3 \Delta \mathbf{G}\_\mathbf{v}^2 \tag{11}
$$

Equation (11) in terms of Gibbs thermodynamical relation can be written as;

$$
\Delta \mathbf{G}^\* = 16 \text{nɔyì} \,\Omega/\,\, \Im(\mathbf{kT} \,\ln \mathbf{S})^2 \tag{12}
$$

where Ω is the molecular volume

Crystallization Kinetics of Chalcogenide Glasses 41

of amorphous or amorphous glassy materials is a thermally activated process.

Here Ec is the activation energy of crystallization, K0 is the pre-exponential factor, R is the universal gas constant and T is the temperature. For isothermal condition parameters Ec and K0 in Eq. (17) can be assumed practically independent of the temperature (at least in the

In non-isothermal crystallization, it is assumed that the constant heating rate during the experiment. The relation between the sample temperature T and the heating rate β can be

 T=Ti+βt (18) Here Ti is the initial temperature. The crystallization rate is obtained by taking the derivative of expression (1) with respect to time, keeping in mind that the reaction rate constant is a

The derivative of K with respect to time can be obtained from Eqs. (17) and (18), which

(dK/dt) = (dK / dT) (dT/dt) = (βEc/RT2) K (20)

Using Eq. (21) Augis and Bennett (Augis & Bennett, 1978) have developed a crystallization kinetic method. They have taken proper account of the temperature dependence crystallization reaction rate. Their approximation results have also been verified the linear relation between ln (Tc − Ti)/β versus 1/Tc ( here Tc is the onset crystallization or critical transition temperature). This can be deduced by substituting the u for Kt in Eq. (21),

(d2α/dt2) = [(d2u/dt2) u - (du/dt)2 × (nu2- n + 1)] nu (n-2) (1- α) = 0 (24)

(d2u/dt2) = (du/dt) [(1/ t) + a] + u [(-1/ t2)] + (da/ dt) (25)

temperature interval accessible in the calorimetric measurements).

time function which represents Arrhenius temperature dependence.

K = K0 exp (-Ec / RT) (17)

(dα/dt) = n (Kt)n-1[K+(dK / dt) t] (1-α) (19)

(dα/dt) = nKntn-1[1+at].(1-α) (21)

(dα/dt) = n (du/dt) u (n-1) (1- α) (22)

(du/dt) = u [(1/t) + a] (23)

Mathematically it can be expressed as

written as

follows as:

Where

From Eq. (19) we obtained

accordingly the rate of reaction can be expressed as

Second derivatives of Eqs. (22) and (23) are given as:

In Eq. (25) substituting for (da/dt) = −(2β/T)a, then it can be written as

where a = (βEc/RT2).

The rate of nucleation **(J)** can be expressed as;

$$\mathbf{J} = \mathbf{J}\_0 \exp\left[\mathbf{-}\Delta \mathbf{G}^\*/\,\mathbf{kT}\right] \tag{13}$$

or in terms of themodynamical parameters

$$\mathbf{J} \equiv \mathbf{J}\_0 \exp\left[-16\pi\mathbf{n}\mathbf{o}^\mathbf{j}\mathbf{Q}^\mathbf{2}/3\mathbf{k}^\mathbf{i}\mathbf{T}^3 (\ln\mathbf{S})^2\right] \tag{14}$$

where J0 is the pre-exponential factor

For critical supercooling condition J = 1, so that ln.J = 0, then expression can be expressed as;

$$\mathbf{S}\_{\rm ri} = \exp\left[16\,\text{n}\nu\Omega^2/\,3k^3\Gamma^3\ln\text{I}\_0\right]\mathbf{l}/2\tag{15}$$

#### **1.4 Crystallization kinetics**

Study of the crystallization of the amorphous materials with respect to time and temperature is called crystallization kinetics. Crystallization kinetic study of the materials can be performed in either isothermal or non-isothermal mode of Differential Scanning Calorimetry (DSC). In the isothermal method, the sample is brought near to the crystallization temperature very quickly and the physical quantities, which change drastically are measured as function of time. In the non-isothermal method the sample is heated at a fixed rate and physical parameters recorded as a function of temperature. Investigators (Sbirrazzuoli, 1999) preferred to perform DSC measurements in nonisothermal mode. Owing to fact, it is not possible to ensure the homogeneity (or constant) of DSC furnace temperature in isothermal mode during the injection of material sample.

Crystallization kinetics parameters of the materials generally interpreted at glass transition temperature (Tg), crystallization temperature (Tc) and peak crystallization temperature (Tp) with help well defined statistical approximations (such as Hurby, Ozawa, Augis and Bentt, Moynihan and Kissinger) (Hruby,1972, Ozawa, 1970, Augis & Bennett, 1978, Moynihan et al., 1974, Kissinger,1957). All the existing approximations described on the basis of JMA (Johnson,1939, Avrami, 1939& 1940) model statics, although in recent years investigators also reported the (Sanchez-Jimenez, 2009) JMA model not a universal model to explain the crystallization kinetics of the materials, because it has few limitations. Despite of this majority view of investigators toward to kinetic methods based on JMA model are more reliable to explain the crystallization of chalcogenide glasses, polymers, metallic and oxide glasses.

#### **1.5 Theoretical basis of crystallization kinetics**

In isothermal phase transformation the extent of crystallization (α) of a certain material can be represent from the Avrami's equation (Moynihan et al., 1974, Kissinger,1957, Johnson,1939)

$$\mathbf{a(t)} \equiv \mathbf{1} \cdot \exp\left[ (\mathbf{-Kt})^\mathbf{n} \right] \tag{16}$$

where K is the crystallization rate constant and n is the order parameter which depends upon the mechanism of crystal growth.

In general the value of crystallization rate constant K increases exponentially with temperature. The temperature dependence behaviour of K indicates that the crystallization

For critical supercooling condition J = 1, so that ln.J = 0, then expression can be expressed as;

Scri = exp [16πσ3Ω2/ 3k3T3 lnJ0]1/2 (15)

Study of the crystallization of the amorphous materials with respect to time and temperature is called crystallization kinetics. Crystallization kinetic study of the materials can be performed in either isothermal or non-isothermal mode of Differential Scanning Calorimetry (DSC). In the isothermal method, the sample is brought near to the crystallization temperature very quickly and the physical quantities, which change drastically are measured as function of time. In the non-isothermal method the sample is heated at a fixed rate and physical parameters recorded as a function of temperature. Investigators (Sbirrazzuoli, 1999) preferred to perform DSC measurements in nonisothermal mode. Owing to fact, it is not possible to ensure the homogeneity (or constant) of DSC furnace temperature in isothermal mode during the injection of material sample.

Crystallization kinetics parameters of the materials generally interpreted at glass transition temperature (Tg), crystallization temperature (Tc) and peak crystallization temperature (Tp) with help well defined statistical approximations (such as Hurby, Ozawa, Augis and Bentt, Moynihan and Kissinger) (Hruby,1972, Ozawa, 1970, Augis & Bennett, 1978, Moynihan et al., 1974, Kissinger,1957). All the existing approximations described on the basis of JMA (Johnson,1939, Avrami, 1939& 1940) model statics, although in recent years investigators also reported the (Sanchez-Jimenez, 2009) JMA model not a universal model to explain the crystallization kinetics of the materials, because it has few limitations. Despite of this majority view of investigators toward to kinetic methods based on JMA model are more reliable to explain the crystallization of chalcogenide glasses, polymers, metallic and oxide glasses.

In isothermal phase transformation the extent of crystallization (α) of a certain material can be represent from the Avrami's equation (Moynihan et al., 1974, Kissinger,1957,

where K is the crystallization rate constant and n is the order parameter which depends

In general the value of crystallization rate constant K increases exponentially with temperature. The temperature dependence behaviour of K indicates that the crystallization

α(t)=1-exp[(-Kt)n] (16)

J = J0 exp [-∆G\*/ kT] (13)

J = J0 exp [-16πσ3Ω2/ 3k3T3 (lnS) 2] (14)

The rate of nucleation **(J)** can be expressed as;

or in terms of themodynamical parameters

**1.5 Theoretical basis of crystallization kinetics** 

upon the mechanism of crystal growth.

Johnson,1939)

where J0 is the pre-exponential factor

**1.4 Crystallization kinetics** 

of amorphous or amorphous glassy materials is a thermally activated process. Mathematically it can be expressed as

$$\mathbf{K} = \mathbf{K}\_0 \exp\left(\mathbf{-E}\_\epsilon / \,\mathrm{RT}\right) \tag{17}$$

Here Ec is the activation energy of crystallization, K0 is the pre-exponential factor, R is the universal gas constant and T is the temperature. For isothermal condition parameters Ec and K0 in Eq. (17) can be assumed practically independent of the temperature (at least in the temperature interval accessible in the calorimetric measurements).

In non-isothermal crystallization, it is assumed that the constant heating rate during the experiment. The relation between the sample temperature T and the heating rate β can be written as

$$\mathbf{T} = \mathbf{T}\_i + \mathbf{\dot{\beta}t} \tag{18}$$

Here Ti is the initial temperature. The crystallization rate is obtained by taking the derivative of expression (1) with respect to time, keeping in mind that the reaction rate constant is a time function which represents Arrhenius temperature dependence.

$$\text{(dq/dt)} = \text{n (Kt)}^{n+1} [\text{K} + (\text{dK} \text{ / dt}) \text{ t]} \,\text{(1-a)} \tag{19}$$

The derivative of K with respect to time can be obtained from Eqs. (17) and (18), which follows as:

$$\text{(dK/dt)} = \text{(dK / dT)} \text{ (dT/dt)} = \text{([\$\text{E}\_c/\text{RT}\$}\$ ) K} \tag{20}$$

From Eq. (19) we obtained

$$\mathbf{u} \cdot (\mathbf{d}\mathbf{a}/\mathbf{d}\mathbf{t}) = \mathbf{n}K^{\mathbf{n}}\mathbf{t}^{\mathbf{n}-1} [1+\mathbf{a}\mathbf{t}]. (1\mathbf{-a}) \tag{21}$$

where a = (βEc/RT2).

Using Eq. (21) Augis and Bennett (Augis & Bennett, 1978) have developed a crystallization kinetic method. They have taken proper account of the temperature dependence crystallization reaction rate. Their approximation results have also been verified the linear relation between ln (Tc − Ti)/β versus 1/Tc ( here Tc is the onset crystallization or critical transition temperature). This can be deduced by substituting the u for Kt in Eq. (21), accordingly the rate of reaction can be expressed as

$$\mathbf{u} \text{ (dɔ/dɛt) = n (dɔ/dɛt) u \text{ ( $n$ -1) ( $1-a$ ) ( $1-a$ ) }}\tag{22}$$

Where

$$\mathbf{u} \cdot (\mathbf{du}/\mathbf{dt}) = \mathbf{u} \left[ (1/\mathbf{t}) + \mathbf{a} \right] \tag{23}$$

Second derivatives of Eqs. (22) and (23) are given as:

$$\left(\text{d2}\,\text{u/dt}\right) = \left[\left(\text{d2}\,\text{u/dt}\right)\,\text{u} \cdot \left(\text{du/dt}\right)\,\text{2} \times \left(\text{nu2-n} + 1\right)\right] \,\text{nu}\,\left(\text{u}\cdot 2\right) \left(1\cdot \text{u}\right) = 0 \tag{24}$$

$$(\text{d}2\text{u/dt}2) = (\text{du/dt})\left[(1/\text{ t}) + \text{a}\right] + \text{u}\left[(\text{-1/ t2})\right] + (\text{da/dt})\tag{25}$$

In Eq. (25) substituting for (da/dt) = −(2β/T)a, then it can be written as

Crystallization Kinetics of Chalcogenide Glasses 43

ampoules (length of ampoules 8 cm and diameter 14 mm). All the ampoules were evacuated and sealed under at a vacuum of 10-5 Torr to avoid the reaction of glasses with oxygen at high temperature. A bunch of sealed ampoules was heated in electric furnace up to 1173K at a rate of 5-6 K/min and held at that temperature for 10-11 h. During the melting process ampoules were frequently rocked to ensure the homogeneity of molten materials. After achieving desired melting time, the ampoules with molten materials were frequently quenched into ice cooled water. Finally ingots of glassy materials were obtained by breaking the ampoules. The preparation and characterizations technique of the test materials also

Fig. 1. DSC patterns of Se93-×-Zn2-Te5-In× (0≤ X ≤ 10) chalcogenide glasses at heating rate

5 K/min

outlined in our past [Singh & Singh, 2010] research work.

$$\mathbf{(d^2u/dt^2) = u \ [a^2 + (2aT\_i/tT)]}\tag{26}$$

The last term in the above equation was omitted in the original derivation of Augis and Bennett [33] (Ti T) and resulted in the simple form:

$$(\mathbf{d}^2 \mathbf{u} / \mathbf{d} \mathbf{t}^2) = \mathbf{a}^2 \mathbf{u} \tag{27}$$

Substitution of (du/dt) and (d2u/dt2) in Eqs. (23), (27), and (24) gives the following expression:

$$\mathbf{u} \cdot (\mathbf{n} \mathbf{u}^\mathbf{n} \mathbf{-n} + \mathbf{1}) = [\mathbf{a} \mathbf{t} / (\mathbf{1} \mathbf{+} \mathbf{t} \mathbf{t})]^2. \tag{28}$$

For E/RT >> 1, the right-hand bracket approaches its maximum limit and consequently u (at the peak) = 1, or

$$\mathbf{u} = (\mathbf{K}\mathbf{t})\_\circ = \mathbf{K}\_0 \exp\left(\mathbf{-}\mathbf{E}\_\circ \;/\; \mathbf{R}\mathbf{T}\_\circ\right) \left[\left(\mathbf{T}\_\circ \mathbf{-}\mathbf{T}\_\circ\right) / \; \middle|\; \beta\right] \approx 1\tag{29}$$

In logarithm form, for Ti << Tc

$$\ln\left(\left\|\left<\mathbf{T\_{c}}\right>\mathbf{T\_{c}}\right>\text{-}\left(\mathbf{-E\_{c}}\right>\,\mathrm{RT\_{c}}\right) + \ln\mathrm{K}\_{0}\tag{30}$$

Value of Ec and K(T) can be obatined from the equation (30) by using the plots of ln β/Tc against 1/Tc. Further, by using the Eq (17) Hu et.al. (Sbirrazzuoli, 1999) have introduced the crystallization rate constant stability criterion corresponding to Tc.

$$\mathbf{K}\left(\mathbf{T}\_{\mathrm{c}}\right) = \mathbf{K}\_{0} \exp\left(-\mathbf{E}\_{\mathrm{c}}/\mathbf{RT}\_{\mathrm{c}}\right) \tag{31}$$

#### **1.6 Differential Scanning Calorimetry (DSC) thermograms**

Endothermic and exothermic peaks in amorphous glassy materials arise due to thermal relaxation from a state of higher enthalpy toward to metastable equilibrium states of lower enthalpy. The process of the thermal relaxation depends on temperature and may quite fast near the glass-transition temperature. The glass transition peak in DSC measurement represents the abrupt change in specific heat and decrease in viscosity (Matusita, 1984), while the crystallization peak demonstrate to the production of excess free-volume, and endothermic peak at Tm reflects the amount of energy which liberate owing to complete destroy the solid phase structure cause braking of all type existing bonds in solid alloy. Hence, the materials crystallizations temperatures as well as mode of crystallizations extensively depend on the compositions of alloys.

In general DSC thermograms of amorphous glassy (i.e. chalcogenide glasses) materials have exhibited a considerable shifts in endothermic glass-transition and exothermic crystallization temperatures with increasing heating rates. But in recent investigations (Singh & Singh, 2009) investigators have also been reported vary small or negligible endothermic glass transitions shifts in metal, semi-metal and non metal containing multicomponent chalcogenide glasses. In order to this, we have performed the DSC measurements on recent developed Se93-xZn2Te5Inx (0≤x ≤10) chalcogenide glasses.

These materials could be prepared by the well known most convenient melt quenched method. The high purity elements Selenium, Zinc, Tellurium and Indium were used. The suitable amounts of elements were weighed by electronic balance and put into clean quartz

The last term in the above equation was omitted in the original derivation of Augis and

Substitution of (du/dt) and (d2u/dt2) in Eqs. (23), (27), and (24) gives the following

For E/RT >> 1, the right-hand bracket approaches its maximum limit and consequently u (at

Value of Ec and K(T) can be obatined from the equation (30) by using the plots of ln β/Tc against 1/Tc. Further, by using the Eq (17) Hu et.al. (Sbirrazzuoli, 1999) have introduced the

Endothermic and exothermic peaks in amorphous glassy materials arise due to thermal relaxation from a state of higher enthalpy toward to metastable equilibrium states of lower enthalpy. The process of the thermal relaxation depends on temperature and may quite fast near the glass-transition temperature. The glass transition peak in DSC measurement represents the abrupt change in specific heat and decrease in viscosity (Matusita, 1984), while the crystallization peak demonstrate to the production of excess free-volume, and endothermic peak at Tm reflects the amount of energy which liberate owing to complete destroy the solid phase structure cause braking of all type existing bonds in solid alloy. Hence, the materials crystallizations temperatures as well as mode of crystallizations

In general DSC thermograms of amorphous glassy (i.e. chalcogenide glasses) materials have exhibited a considerable shifts in endothermic glass-transition and exothermic crystallization temperatures with increasing heating rates. But in recent investigations (Singh & Singh, 2009) investigators have also been reported vary small or negligible endothermic glass transitions shifts in metal, semi-metal and non metal containing multicomponent chalcogenide glasses. In order to this, we have performed the DSC

These materials could be prepared by the well known most convenient melt quenched method. The high purity elements Selenium, Zinc, Tellurium and Indium were used. The suitable amounts of elements were weighed by electronic balance and put into clean quartz

measurements on recent developed Se93-xZn2Te5Inx (0≤x ≤10) chalcogenide glasses.

crystallization rate constant stability criterion corresponding to Tc.

**1.6 Differential Scanning Calorimetry (DSC) thermograms** 

extensively depend on the compositions of alloys.

Bennett [33] (Ti T) and resulted in the simple form:

expression:

the peak) = 1, or

In logarithm form, for Ti << Tc

(d2u/dt2) = u [a2 + (2a Ti/ tT)] (26)

(d2u/dt2) = a2 u (27)

(nun-n+1) = [at/(1+at)]2. (28)

ln (β/Tc) ≈ (-Ec / RTc) + lnK0 (30)

K (Tc) = K0 exp (- Ec /RTc ) (31)

u = (Kt)c = K0 exp (-Ec / RTc) [ (Tc-Ti) / β] ≈ 1 (29)

ampoules (length of ampoules 8 cm and diameter 14 mm). All the ampoules were evacuated and sealed under at a vacuum of 10-5 Torr to avoid the reaction of glasses with oxygen at high temperature. A bunch of sealed ampoules was heated in electric furnace up to 1173K at a rate of 5-6 K/min and held at that temperature for 10-11 h. During the melting process ampoules were frequently rocked to ensure the homogeneity of molten materials. After achieving desired melting time, the ampoules with molten materials were frequently quenched into ice cooled water. Finally ingots of glassy materials were obtained by breaking the ampoules. The preparation and characterizations technique of the test materials also outlined in our past [Singh & Singh, 2010] research work.

Fig. 1. DSC patterns of Se93-×-Zn2-Te5-In× (0≤ X ≤ 10) chalcogenide glasses at heating rate 5 K/min

Crystallization Kinetics of Chalcogenide Glasses 45

Fig. 2. Variations of Tc-Tg with Indium atomic percentage at 5, 10, 15 and 20 K/min heating

5 36 51 53 55 51 10 39 54 59 60 56 15 41 59 61 64 60 20 44 61 64 65 62

5 14 15 21 21 16 10 15 16 21 22 16 15 15 16 22 23 17 20 14 17 22 23 19

5 147 132 131 126 134 10 143 127 123 119 126 15 140 122 119 114 121 20 136 119 115 110 119

Table 2. Evaluated values of Tc-Tg, Tp-Tc and Tm-Tc at different heating rates

Se93Zn2Te5 Se91Zn2Te5In2 Se89Zn2Te5In4 Se87Zn2Te5In6 Se83Zn2Te5In10

rates

Tc-Tg

Tp-Tc

Tm-Tc

 Heating rate

DSC patterns of Se93-xZn2Te5Inx (0≤x ≤10) glasses at heating rate of 5 K (min)-1 is given in Figure.1. DSC traces clearly show the endothermic and exothermic phase reversal peaks at the glass transitions and crystallizations temperatures. Obtained values of the glass transitions temperatures (Tg), onset crystallizations temperatures (Tc), peak crystallizations temperatures (Tp) and melting temperatures (Tm) at heating rates of 5, 10, 15 and 20 K (min)- 1 is listed in Table 1.


**Se93-×-Zn2-Te5-In× (X=0, 2, 4, 6 and 10)** 

Table. 1 Obatined values of Tg, Tc, Tp and Tm at heating rates 5, 10, 15 and 20 K/min

Outcome demonstrates, a very small glass-transitions temperatures shifts and a considerable shifts in onset and peak crystallizations temperatures, the corresponding Tc-Tg result is given in Figure.2 and their values listed in Table.2. This result revealed the values of Tc and Tp crystallizations temperatures increases upto 6 at. wt. % indium and beyond this decreased for 10 atomic percentage composition glass.

DSC patterns of Se93-xZn2Te5Inx (0≤x ≤10) glasses at heating rate of 5 K (min)-1 is given in Figure.1. DSC traces clearly show the endothermic and exothermic phase reversal peaks at the glass transitions and crystallizations temperatures. Obtained values of the glass transitions temperatures (Tg), onset crystallizations temperatures (Tc), peak crystallizations temperatures (Tp) and melting temperatures (Tm) at heating rates of 5, 10, 15 and 20 K (min)-

**Se93-×-Zn2-Te5-In× (X=0, 2, 4, 6 and 10)** 

Heating rate Tg (K) Tc (K) Tp (K) Tm(K) 5 318 354 368 501 10 320 359 374 502 15 322 363 378 503 20 323 367 381 503

5 319 370 385 502 10 322 376 392 503 15 323 382 398 504 20 324 385 402 504

5 319 372 393 503 10 322 381 402 504 15 324 385 407 504 20 325 389 411 504

5 320 375 396 504 10 323 383 405 504 15 325 389 412 506 20 327 392 415 506

5 319 370 386 504 10 322 378 394 504 15 323 383 400 504 20 324 386 405 505

Table. 1 Obatined values of Tg, Tc, Tp and Tm at heating rates 5, 10, 15 and 20 K/min

decreased for 10 atomic percentage composition glass.

Outcome demonstrates, a very small glass-transitions temperatures shifts and a considerable shifts in onset and peak crystallizations temperatures, the corresponding Tc-Tg result is given in Figure.2 and their values listed in Table.2. This result revealed the values of Tc and Tp crystallizations temperatures increases upto 6 at. wt. % indium and beyond this

1 is listed in Table 1.

Se93-Zn2-Te5

Se91-Zn2-Te5-In2

Se89-Zn2-Te5-In4

Se87-Zn2-Te5-In6

Se83-Zn2-Te5-In10

Fig. 2. Variations of Tc-Tg with Indium atomic percentage at 5, 10, 15 and 20 K/min heating rates


Table 2. Evaluated values of Tc-Tg, Tp-Tc and Tm-Tc at different heating rates

Crystallization Kinetics of Chalcogenide Glasses 47

Fig. 3. Plots of GFA parameter with indium atomic percentage at 5, 10, 15 and 20 K/min

The glass transition activation reflects the endothermic energy of the material which produced due to unsaturated or hydrogen like bond braking at the pre-crystallization critical temperature. Glass transitions activations energies of the under test glasses can be

> *g g*

where β is the heating rate, Eg is the glass transition activation energy, C is the constant in usual meaning. Obtained Ozawa plots, ln β vs 1000/Tg, for these materials is given in

Outcomes revel the Eg values have a maxima and a minima for 0 and 6 percentage indium compositions glasses. Thus the Eg values have very small increasing and decreasing trend in Tg values (see Table 3&Table 1) with increasing DSC heating rates in these metal, semi-metal and non-metal containing multicomponent chalcogenide glasses. The high activations energies at pre-crystallizations reflect the materials rigidity. While, normally reported Tg values for non-metallic chalcogenides compositions has show a considerable shifts with

*RT*

*C*

(33)

*E*

ln

Figure.4 and their corresponding Eg values is listed in Table 3.

heating rates

**1.8.1 Glass transition activation energy** 

defined by using Ozawa method (Ozawa, 1970).

#### **1.7 Glass forming ability (GFA)**

GFA of material describes the relative ability to a set of compound adopt the amorphous structure (Mehta et al., 2006, Jain et al., 2009). In practice criteria to establish the GFA of vitreous materials are based on DSC measurements. Usually, unstable glass has show a crystallization peak near to the glass transition temperature while stable glass peak close to melting temperature. GFA can be evaluated by meen of the difference between the crystallization temperature (peak temperature Tp and/or onset temperature Tc) and the glass transition temperature (Tg). This difference varies with alloys concentrations and higher and lower for a certain composition. To evaluate the GFA of glassy alloys several quantitative methods have been introduced from the investigators. Most of the methods (Saad & Poulin,1987, Dietzel,1968 ) based on characteristics temperatures of glassy alloys. Dietzel (Dietzel,1968) has introduced the first GFA criterion DT = Tc- Tg. Further Hruby (Hruby,1972) developed the HR GFA criterion [HR = Tc-Tg / Tm-Tc].This method has additional advantage to describe the thermal stability of amorphous materials.

To applying the GFA criterion method, obtained critical characteristics temperatures difference Tc-Tg, Tp-Tc and Tm-Tc (Here Tg is glass transition temperature, Tp is peak crystallization temperature and Tm is the melting temperature) values of under examine materials is listed in Table 2. Using these values HR parameter of GFA can be described as:

$$H\_{\mathcal{R}} = \left(\frac{T\_c - T\_{\mathcal{g}}}{T\_m - T\_c}\right) \tag{32}$$

GFA variation with indium atomic weight percentage at heating rates 5, 10, 15 and 20 is given in Figure. 3 and their corresponding average values listed in Table 3. The higher GFA value is obtained for threshold indium concentration glass. High GFA value of threshold composition also reflects their high order thermal stability as compare to other glasses of this series.

#### **1.8 Activation energy**

Activation energy reflects the involvement of molecular motions and rearrangements of the atoms around the critical transitions temperatures (Suri et al., 2006). In DSC measurement atoms undergo infrequent transitions between the local (or metastable state) potential minima which separated from different energy barriers in the configuration space, where each local minima represent a different structure. The most stable configuration has local minima structure in glassy region. This literal meaning a glass atoms possessing minimum activation energy have a higher probability to jump in metastable state of lower internal energy configuration. This local minima configuration occurs at particular composition of alloy which refers as a most stable glass (Imran et al., 2001). The activations energies of chalcogenide glasses at the critical temperatures can be interpreted in these words: the glass transition activation energy (Eg), onset crystallization activation energy (Ec) and peak crystallization activation energy (Ep) are the amount of energies which absorbed by a group of atoms for a jump from one metastable state to another state (Imran et al., 2001, Agarwal et al., 1991

GFA of material describes the relative ability to a set of compound adopt the amorphous structure (Mehta et al., 2006, Jain et al., 2009). In practice criteria to establish the GFA of vitreous materials are based on DSC measurements. Usually, unstable glass has show a crystallization peak near to the glass transition temperature while stable glass peak close to melting temperature. GFA can be evaluated by meen of the difference between the crystallization temperature (peak temperature Tp and/or onset temperature Tc) and the glass transition temperature (Tg). This difference varies with alloys concentrations and higher and lower for a certain composition. To evaluate the GFA of glassy alloys several quantitative methods have been introduced from the investigators. Most of the methods (Saad & Poulin,1987, Dietzel,1968 ) based on characteristics temperatures of glassy alloys. Dietzel (Dietzel,1968) has introduced the first GFA criterion DT = Tc- Tg. Further Hruby (Hruby,1972) developed the HR GFA criterion [HR = Tc-Tg / Tm-Tc].This method has

additional advantage to describe the thermal stability of amorphous materials.

*R*

*H*

To applying the GFA criterion method, obtained critical characteristics temperatures difference Tc-Tg, Tp-Tc and Tm-Tc (Here Tg is glass transition temperature, Tp is peak crystallization temperature and Tm is the melting temperature) values of under examine materials is listed in Table 2. Using these values HR parameter of GFA can be described as:

> *c g*

GFA variation with indium atomic weight percentage at heating rates 5, 10, 15 and 20 is given in Figure. 3 and their corresponding average values listed in Table 3. The higher GFA value is obtained for threshold indium concentration glass. High GFA value of threshold composition also reflects their high order thermal stability as compare to other glasses of

Activation energy reflects the involvement of molecular motions and rearrangements of the atoms around the critical transitions temperatures (Suri et al., 2006). In DSC measurement atoms undergo infrequent transitions between the local (or metastable state) potential minima which separated from different energy barriers in the configuration space, where each local minima represent a different structure. The most stable configuration has local minima structure in glassy region. This literal meaning a glass atoms possessing minimum activation energy have a higher probability to jump in metastable state of lower internal energy configuration. This local minima configuration occurs at particular composition of alloy which refers as a most stable glass (Imran et al., 2001). The activations energies of chalcogenide glasses at the critical temperatures can be interpreted in these words: the glass transition activation energy (Eg), onset crystallization activation energy (Ec) and peak crystallization activation energy (Ep) are the amount of energies which absorbed by a group of atoms for a jump from one metastable state to another state (Imran et al., 2001, Agarwal et

*m c T T*

*T T* (32)

**1.7 Glass forming ability (GFA)** 

this series.

al., 1991

**1.8 Activation energy** 

Fig. 3. Plots of GFA parameter with indium atomic percentage at 5, 10, 15 and 20 K/min heating rates

#### **1.8.1 Glass transition activation energy**

The glass transition activation reflects the endothermic energy of the material which produced due to unsaturated or hydrogen like bond braking at the pre-crystallization critical temperature. Glass transitions activations energies of the under test glasses can be defined by using Ozawa method (Ozawa, 1970).

$$
\ln \alpha \mathcal{J} = -\left(\frac{E\_{\mathcal{g}}}{RT\_{\mathcal{g}}}\right) + \mathcal{C} \tag{33}
$$

where β is the heating rate, Eg is the glass transition activation energy, C is the constant in usual meaning. Obtained Ozawa plots, ln β vs 1000/Tg, for these materials is given in Figure.4 and their corresponding Eg values is listed in Table 3.

Outcomes revel the Eg values have a maxima and a minima for 0 and 6 percentage indium compositions glasses. Thus the Eg values have very small increasing and decreasing trend in Tg values (see Table 3&Table 1) with increasing DSC heating rates in these metal, semi-metal and non-metal containing multicomponent chalcogenide glasses. The high activations energies at pre-crystallizations reflect the materials rigidity. While, normally reported Tg values for non-metallic chalcogenides compositions has show a considerable shifts with

Crystallization Kinetics of Chalcogenide Glasses 49

semi-metal and non-metal containing multicomponent chalcogenide glasses have exhibited either very small or negligible glass transitions temperatures shifts with increasing DSC

Onset crystallization activation energy (Ec) is the amount of thermal energy which requires to begin the phase transformation from glassy to crystallization state. Quantitive knowledge of onset crystallization activation energy at Tc defines the heat/energy storage capability of the material which useful for different physical applications. The exothermic onset crystallization activation energy at Tc arises due to barking of existing covalent bonds in glassy configuration. In case of complex metallic multicomponent chalcogenide glasses high energy homopolar and heteropolar covalent bonds formed as compare to metallic binary and ternary compositions. Due to this the critical onset crystallization temperatures of the complex metallic glasses (see Table 1&Table 3) increases and their corresponding activation energies tend to be decrease upto threshold composition then visa-verse direction. While in case of non-metallic binary, ternary and multicomponet chalcogenide alloys reports demonstrated they have lower values of onset crystallizations temperatures owing to

Onset crystallizations activations energies of under examine complex metallic multicomponent chalcogenide glasses described by employing the Ozawa method (Ozawa,

> ln *c c <sup>E</sup> <sup>C</sup> RT*

Here symbols (β is heating rate, Ec is the onset crystallization activation energy, Tc is the onset crystallization temperature and R & C are the constant) are in usual meaning. Obtained Ozawa plots ln β vs 1000/Tc is given in Figure. 5 and their corresponding Ec values is listed in Table 3. Outcomes show a phase reversal in Ec values which have a maxima and minima corresponding to 0 and 6 atomic weight percentage of indium glasses.

Peak crystallization activation energy (Ep) of a glass expresses the amount of heat energy which requires for utmost crystallization. By mean at peak crystallization point almost all the existing heteropolar covalent bonds have to be broken and material achieve to maximum crystallization i.e. a glassy phase material completely transform to crystalline phase and relax toward to original state. The Ep values of examined materials can be

<sup>2</sup> ln

Obtained Ep values from Kissinger plots (see Figure. 6) are listed in Table.3.

*T RT*

*P P*

Here symbols (β is heating rate, Ep is the peak crystallization activation energy, Tp is the paek crystallization temperature and R & C are the constant) are in usual meaning.

*P*

*<sup>E</sup> <sup>C</sup>*

(34)

(35)

existence of week homopolar and heteropolar bonds in glassy configuration.

heating rates.

1970).

**1.8.2 Onset crystallization activation energy** 

**1.8.3 Peak crystallization activation energy** 

described by using the Kissinger method (Kissinger,1957).

DSC heating rates. Hence the obtained Tg values results in increasing DSC heating rates for metal, semi-metal and non-metal containing multicomponent chalcogenide glasses are not in good agreement with previous reported non metallic compositions. Deviations in the results arise due to existence relatively hard metallic, semi-metallic characters unsaturated bonds with hydrogen like week bonds in the alloys stoichiometrics. Further, it is quite possible to large amount of metallic, semi-metallic characters unsaturated bonds sustain over to Tg critical transition temperature of the materials owing to requirement greater amount of energy to bark the heteropolar unsaturated bonds. As consequence the metal,

Fig. 4. Ozwa polts of Se93-×-Zn2-Te5-In× (0≤ X ≤ 10) glasses to obtain Eg


Table 3. Eg, Ec, Ep and GFA values of Se93-×-Zn2-Te5-In×(0≤ x ≤10) chalcogenide glasses

DSC heating rates. Hence the obtained Tg values results in increasing DSC heating rates for metal, semi-metal and non-metal containing multicomponent chalcogenide glasses are not in good agreement with previous reported non metallic compositions. Deviations in the results arise due to existence relatively hard metallic, semi-metallic characters unsaturated bonds with hydrogen like week bonds in the alloys stoichiometrics. Further, it is quite possible to large amount of metallic, semi-metallic characters unsaturated bonds sustain over to Tg critical transition temperature of the materials owing to requirement greater amount of energy to bark the heteropolar unsaturated bonds. As consequence the metal,

Fig. 4. Ozwa polts of Se93-×-Zn2-Te5-In× (0≤ X ≤ 10) glasses to obtain Eg

Glass transition activation energy Eg (KJ/mol)

Alloy

compositions

**Se93-×-Zn2-Te5-In× (X=0, 2, 4, 6 and 10)** 

Peak crystallization activation energy

Average (GFA)

Ep (KJ/mol)

crystallization activation energy Ec (KJ/mol)

Se93Zn2Te5 229.94 115.64 118.23 0.282 Se91Zn2Te5In2 205.02 106.11 96.40 0.451 Se89Zn2Te5In4 195.47 99.02 91.09 0.487 Se87Zn2Te5In6 174.23 93.66 82.77 0.548 Se83Zn2Te5In10 199.88 101.84 92.70 0.460 Table 3. Eg, Ec, Ep and GFA values of Se93-×-Zn2-Te5-In×(0≤ x ≤10) chalcogenide glasses

Onset

semi-metal and non-metal containing multicomponent chalcogenide glasses have exhibited either very small or negligible glass transitions temperatures shifts with increasing DSC heating rates.

#### **1.8.2 Onset crystallization activation energy**

Onset crystallization activation energy (Ec) is the amount of thermal energy which requires to begin the phase transformation from glassy to crystallization state. Quantitive knowledge of onset crystallization activation energy at Tc defines the heat/energy storage capability of the material which useful for different physical applications. The exothermic onset crystallization activation energy at Tc arises due to barking of existing covalent bonds in glassy configuration. In case of complex metallic multicomponent chalcogenide glasses high energy homopolar and heteropolar covalent bonds formed as compare to metallic binary and ternary compositions. Due to this the critical onset crystallization temperatures of the complex metallic glasses (see Table 1&Table 3) increases and their corresponding activation energies tend to be decrease upto threshold composition then visa-verse direction. While in case of non-metallic binary, ternary and multicomponet chalcogenide alloys reports demonstrated they have lower values of onset crystallizations temperatures owing to existence of week homopolar and heteropolar bonds in glassy configuration.

Onset crystallizations activations energies of under examine complex metallic multicomponent chalcogenide glasses described by employing the Ozawa method (Ozawa, 1970).

$$
\Delta \ln \mathcal{J} = -\left(\frac{E\_c}{RT\_c}\right) + \mathbf{C} \tag{34}
$$

Here symbols (β is heating rate, Ec is the onset crystallization activation energy, Tc is the onset crystallization temperature and R & C are the constant) are in usual meaning. Obtained Ozawa plots ln β vs 1000/Tc is given in Figure. 5 and their corresponding Ec values is listed in Table 3. Outcomes show a phase reversal in Ec values which have a maxima and minima corresponding to 0 and 6 atomic weight percentage of indium glasses.

#### **1.8.3 Peak crystallization activation energy**

Peak crystallization activation energy (Ep) of a glass expresses the amount of heat energy which requires for utmost crystallization. By mean at peak crystallization point almost all the existing heteropolar covalent bonds have to be broken and material achieve to maximum crystallization i.e. a glassy phase material completely transform to crystalline phase and relax toward to original state. The Ep values of examined materials can be described by using the Kissinger method (Kissinger,1957).

$$\ln\left(\frac{\mathcal{J}}{T\_P^2}\right) = -\frac{E\_P}{RT\_P} + \mathcal{C} \tag{35}$$

Here symbols (β is heating rate, Ep is the peak crystallization activation energy, Tp is the paek crystallization temperature and R & C are the constant) are in usual meaning. Obtained Ep values from Kissinger plots (see Figure. 6) are listed in Table.3.

Crystallization Kinetics of Chalcogenide Glasses 51

Values of Ep also show a phase reversal with alloying compositions and have a maxima and minima respectively for 0 and 6 atomic weight of indium. Commonly, in metal, semimetal and non-metallic elements containing multicomponent chalcogenides show a sharp and continuous crystallization process (exception is also reported in few composition of chalcogenide glasses) with lower Ep values. The sharp crystallization prevail between Tc(where crystallization began) and Tp (where crystallization completed) owing to continuous braking of rigid heteropolar bonds cause generation of greater amount of heat

Melting temperature of amorphous glassy materials defines as; temperature at which solid state materials destroy all the existing homopolar and heteropolar bonds and alloying elements separated. Melting temperatures of amorphous glassy materials extensively depend on the constituent of the alloys. Technologically kinetics at Tm have less impotence, therefore investigators interest to provide only introductory information regarding to phase

Crystallizations kinetics variations in under test metal, semi-metal and non-metal containing multicomponent chalcogenide glasses can be interpreted in term of bond formation in solids. It is expected to Zn and Te dissolved in Se chains and makes Zn-Zn, Te-Te, Se-Se, Se-Zn, Se-Te, Se-Zn-Te homopolar and heteropolar bonds. Essentially ternary Se-Zn-Te glass can forms cross-link heteropolar metastable state structure. The heteropolar bonds will be produced the defects in density of localized state owing to existence of dangling bonds in alloy configuration (Maharjan et al., 2000, Saffarini, 2002, Abdel Latif, 1998). Further incorporation of foreign element Te concentration in ternary configuration transforms the whole stoichiometry into quaternary or multicomponent system. The metal, semi-metal and non-metal multicomponent glassy configuration possibly makes them dominating Se-In heteropolar bonds with other metallic character bonds Zn-In, Te-In. The Se-In heteropolar bonds play an important role in crystallization kinetics variations due to fixed amounts of Zn and Te. Addition of additional indium concentration has produced the heavily crosslinked structure in which steric hindrance increases. Therefore the expanse of Se chains and replacement of weak Se-Se bonds by Se-In bonds results the increase and decrease in associative activations energies. A chemical threshold has established at critical composition (6 at wt % of In). At this concentration glassy structure become more chemically ordered and contains large number of Se-In bonds (Singh, 2011). As consequence a significant

change is appeared in crystallization parameters of threshold composition alloy.

Furthermore, incorporation of indium concentration beyond the threshold composition reduced the Se-In bonds and increases the In-In bond strength in glassy configuration. The increase and decrease bonds strengths of Se-In and In-In influenced the defects / dangling bonds concentrations in the glassy stoichiometry. Owing to alternation in dangling bonds densities the GFA, activations energies Eg, Ec and Ep of the corresponding glass show a

energy in the specimen.

**2. Discussions** 

**1.9 Melting temperature (Tm)** 

transformation at Tm in amorphous glassy materials.

significant change in kinetic parameters.

Fig. 5. Ozwa polts of Se93-×-Zn2-Te5-In× (0≤ X ≤ 10) glasses to obtain Ec

Fig. 6. Kissinger polts of Se93-×-Zn2-Te5-In× (0≤ X ≤ 10) glasses to obtain Ep

Values of Ep also show a phase reversal with alloying compositions and have a maxima and minima respectively for 0 and 6 atomic weight of indium. Commonly, in metal, semimetal and non-metallic elements containing multicomponent chalcogenides show a sharp and continuous crystallization process (exception is also reported in few composition of chalcogenide glasses) with lower Ep values. The sharp crystallization prevail between Tc(where crystallization began) and Tp (where crystallization completed) owing to continuous braking of rigid heteropolar bonds cause generation of greater amount of heat energy in the specimen.

### **1.9 Melting temperature (Tm)**

Melting temperature of amorphous glassy materials defines as; temperature at which solid state materials destroy all the existing homopolar and heteropolar bonds and alloying elements separated. Melting temperatures of amorphous glassy materials extensively depend on the constituent of the alloys. Technologically kinetics at Tm have less impotence, therefore investigators interest to provide only introductory information regarding to phase transformation at Tm in amorphous glassy materials.

## **2. Discussions**

50 Crystallization – Science and Technology

In=0 In=2 In=4 In=6 In=10

**In=0**

**In=2**

**In=4**

**In=6**

**In=10**

**2.45 2.55 2.65 2.75 2.85**

**1000/Tc (K-1)**

**2.3 2.4 2.5 2.6 2.7 2.8**

**1000/Tp(K-1)**

Fig. 6. Kissinger polts of Se93-×-Zn2-Te5-In× (0≤ X ≤ 10) glasses to obtain Ep

Fig. 5. Ozwa polts of Se93-×-Zn2-Te5-In× (0≤ X ≤ 10) glasses to obtain Ec

**1.5**

**-10.6**

**-10.2**

**-9.8**

**ln(**

**β/T**

**p**

**)**

**2**

**-9.4**

**-9**

**1.9**

**ln** 

**β (K-min-1)**

**2.3**

**2.7**

**3.1**

Crystallizations kinetics variations in under test metal, semi-metal and non-metal containing multicomponent chalcogenide glasses can be interpreted in term of bond formation in solids. It is expected to Zn and Te dissolved in Se chains and makes Zn-Zn, Te-Te, Se-Se, Se-Zn, Se-Te, Se-Zn-Te homopolar and heteropolar bonds. Essentially ternary Se-Zn-Te glass can forms cross-link heteropolar metastable state structure. The heteropolar bonds will be produced the defects in density of localized state owing to existence of dangling bonds in alloy configuration (Maharjan et al., 2000, Saffarini, 2002, Abdel Latif, 1998). Further incorporation of foreign element Te concentration in ternary configuration transforms the whole stoichiometry into quaternary or multicomponent system. The metal, semi-metal and non-metal multicomponent glassy configuration possibly makes them dominating Se-In heteropolar bonds with other metallic character bonds Zn-In, Te-In. The Se-In heteropolar bonds play an important role in crystallization kinetics variations due to fixed amounts of Zn and Te. Addition of additional indium concentration has produced the heavily crosslinked structure in which steric hindrance increases. Therefore the expanse of Se chains and replacement of weak Se-Se bonds by Se-In bonds results the increase and decrease in associative activations energies. A chemical threshold has established at critical composition (6 at wt % of In). At this concentration glassy structure become more chemically ordered and contains large number of Se-In bonds (Singh, 2011). As consequence a significant change is appeared in crystallization parameters of threshold composition alloy.

Furthermore, incorporation of indium concentration beyond the threshold composition reduced the Se-In bonds and increases the In-In bond strength in glassy configuration. The increase and decrease bonds strengths of Se-In and In-In influenced the defects / dangling bonds concentrations in the glassy stoichiometry. Owing to alternation in dangling bonds densities the GFA, activations energies Eg, Ec and Ep of the corresponding glass show a significant change in kinetic parameters.

Crystallization Kinetics of Chalcogenide Glasses 53

applications are also identified as a prominent alternative of conventional energy which can provides terawatts capacity at cheaper cost. But improvement in low efficiency chalcogenide based photovoltaics is challenging in future. Thus, in view of author glassy and nano embedded (glassy) chalcogenides would be matter of future research to undersatnd the molecular /- or nano-phase photonics of the materials, particularly for thrust areas in PCM memory and photovoltaics applications. In author opinion chalcogenide glasses (bulk or

Author thankful to Dr.Kedar Singh and senior faculty members of Department of Physics, Banaras Hindu University, Varanasi-221005, India, for their kind support to carry out this

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**4. Acknowledgment** 

work.

**5. References** 

## **3. Summary**

In summary, in this work an effort is made to present the fundamentals (in short form) of nucleation and growth processes and crystallization in amorphous glassy (chalcogenide glasses) materials. Further, project a clear view on natural crystallization and process of non–isothermal crystallization kinetics of chalcogenide glasses. Subsequently a concrete explanation on origin of endothermic and exothermic peaks in DSC measurements is also discussed. Glass forming ability and crystallizations kinetics of recent developed Se93 xZn2Te5Inx (0≤x ≤10) metal, semi-metal and non-metal containing multicomponent chalcogenide glasses have also been taken under discussion. Outcomes revealed such combinations glasses have high order GFA and thermal stability and low Ec, Ep activations energies as compare to previous reported non-metallic compositions. In subsequent it has found these materials critical kinetic parameters extensively depend on their alloys constituents and have a maxima and a minima in respective manor for threshold composition glass. It is also concluded that the crystallizations kinetics variations in these glasses can be occur owing to fluctuations in solid state bonds densities in localized states. In case of metallic multicomponent chalcogenide glasses heteropolar unsaturated and covalent bonds may play an important role in crystallization kinetics variations.

Indeed, chalcogenide glasses are the potential materials which used in various optoelectronics applications, but still have plenty of room for their applications in different areas which have either less studied or undiscovered. A few thrust areas of these materials are outlined here in which they have (or may have) potential applications. Surface plasmon resonance (SPR) is a very versatile and accurate technique to determining small changes in optoelectronics parameters like, refractive index at the interface of a metal layer and the adjacent dielectric medium. The SPR detection mechanism has secured a very important place among several sensing techniques due to its better performance and reliable procedure. Chalcogenide materials can also perform as infrared sensing for an efficient, nondestructive and highly selective technique in detection of organic and biological species. This technique has combined benefits of ATR spectroscopy with the flexibility of using a fiber as the transmission line of the optical signal, which allows for remote analysis during field measurements or in clinical environments. Sensing mechanism based on absorption of the evanescent electric field, which propagates outside the surface of the fiber and interacts with any absorbing species at the fiber interface. However, their efficiency controlled multilayer optical filters by periodically switching and evaporation angle, leading to periodic dielectric structure makes them a potential candidate for chemical sensing application. Further, their larger refractive index amongst the glasses makes them to made chalcogenide based ultra-low loss waveguides devices. Chalcogenide materials have most promising applications in area of phase change memory (PCM) and in photonics. Their unique physical characteristic the reversible amorphous to crystalline phase change which can be induced by controlled thermal cycling (through laser absorption or current flow) in certain chalcogenide alloys. Phase-change materials have always technological importance to make read–write storage device (commercially rewritable CD/DVD), because they can be switched (in nanoseconds) rapidly back and forth between amorphous and crystalline phases by applying appropriate laser heat pulses. Although optical phase-change storage is a widespread and successful technology, further advances in areal densities will be very challenging. Moreover, chalcogenide (in glassy or nano embedded) based photovoltaic cells applications are also identified as a prominent alternative of conventional energy which can provides terawatts capacity at cheaper cost. But improvement in low efficiency chalcogenide based photovoltaics is challenging in future. Thus, in view of author glassy and nano embedded (glassy) chalcogenides would be matter of future research to undersatnd the molecular /- or nano-phase photonics of the materials, particularly for thrust areas in PCM memory and photovoltaics applications. In author opinion chalcogenide glasses (bulk or nano phase) has a bright future and it is still open for further inventions.

## **4. Acknowledgment**

Author thankful to Dr.Kedar Singh and senior faculty members of Department of Physics, Banaras Hindu University, Varanasi-221005, India, for their kind support to carry out this work.

## **5. References**

52 Crystallization – Science and Technology

In summary, in this work an effort is made to present the fundamentals (in short form) of nucleation and growth processes and crystallization in amorphous glassy (chalcogenide glasses) materials. Further, project a clear view on natural crystallization and process of non–isothermal crystallization kinetics of chalcogenide glasses. Subsequently a concrete explanation on origin of endothermic and exothermic peaks in DSC measurements is also discussed. Glass forming ability and crystallizations kinetics of recent developed Se93 xZn2Te5Inx (0≤x ≤10) metal, semi-metal and non-metal containing multicomponent chalcogenide glasses have also been taken under discussion. Outcomes revealed such combinations glasses have high order GFA and thermal stability and low Ec, Ep activations energies as compare to previous reported non-metallic compositions. In subsequent it has found these materials critical kinetic parameters extensively depend on their alloys constituents and have a maxima and a minima in respective manor for threshold composition glass. It is also concluded that the crystallizations kinetics variations in these glasses can be occur owing to fluctuations in solid state bonds densities in localized states. In case of metallic multicomponent chalcogenide glasses heteropolar unsaturated and

covalent bonds may play an important role in crystallization kinetics variations.

Indeed, chalcogenide glasses are the potential materials which used in various optoelectronics applications, but still have plenty of room for their applications in different areas which have either less studied or undiscovered. A few thrust areas of these materials are outlined here in which they have (or may have) potential applications. Surface plasmon resonance (SPR) is a very versatile and accurate technique to determining small changes in optoelectronics parameters like, refractive index at the interface of a metal layer and the adjacent dielectric medium. The SPR detection mechanism has secured a very important place among several sensing techniques due to its better performance and reliable procedure. Chalcogenide materials can also perform as infrared sensing for an efficient, nondestructive and highly selective technique in detection of organic and biological species. This technique has combined benefits of ATR spectroscopy with the flexibility of using a fiber as the transmission line of the optical signal, which allows for remote analysis during field measurements or in clinical environments. Sensing mechanism based on absorption of the evanescent electric field, which propagates outside the surface of the fiber and interacts with any absorbing species at the fiber interface. However, their efficiency controlled multilayer optical filters by periodically switching and evaporation angle, leading to periodic dielectric structure makes them a potential candidate for chemical sensing application. Further, their larger refractive index amongst the glasses makes them to made chalcogenide based ultra-low loss waveguides devices. Chalcogenide materials have most promising applications in area of phase change memory (PCM) and in photonics. Their unique physical characteristic the reversible amorphous to crystalline phase change which can be induced by controlled thermal cycling (through laser absorption or current flow) in certain chalcogenide alloys. Phase-change materials have always technological importance to make read–write storage device (commercially rewritable CD/DVD), because they can be switched (in nanoseconds) rapidly back and forth between amorphous and crystalline phases by applying appropriate laser heat pulses. Although optical phase-change storage is a widespread and successful technology, further advances in areal densities will be very challenging. Moreover, chalcogenide (in glassy or nano embedded) based photovoltaic cells

**3. Summary** 


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**3** 

**Numerical Models of Crystallization and Its** 

**Direction for Metal and Ceramic Materials in** 

Structure of metallic and also majority of ceramic alloys is one of the factors, which significantly influence their physical and mechanical properties. Formation of structure is strongly affected by production technology, casting and solidification of these alloys. Solidification is a critical factor in the materials industry, e.g. (Chvorinov, 1954). Solute segregation either on the macro- or micro-scale is sometimes the cause of unacceptable products due to poor mechanical properties of the resulting non-equilibrium phases. In the areas of more important solute segregation there occurs weakening of bonds between atoms and mechanical properties of material degrade. Heterogeneity of distribution of components is a function of solubility in solid and liquid phases. During solidification a solute can concentrate in inter-dendritic areas above the value of its maximum solubility in solid phase. Solute diffusion in solid phase is a limiting factor for this process, since diffusion coefficient in solid phase is lower by three up to five orders than in the melt (Smrha, 1983). When analysing solidification of these alloys so far no unified theoretical model was created, which would describe this complex heterogeneous process as a whole. During the last fifty years many approaches with more or less limiting assumptions were developed. Solidification models and simulations have been carried out for both macroscopic and microscopic scales. The most elaborate numerical models can predict micro-segregation with comparatively high precision. The main limiting factor of all existing mathematical micro-segregation models consists in lack of available thermodynamic and kinetic data, especially for systems of higher orders. There is also little experimental data to check the

Many authors deal with issues related to modelling of a non-equilibrium crystallisation of alloys. However, majority of the presented works concentrates mainly on investigation of modelling of micro-segregation of binary alloys, or on segregation of elements for special cases of crystallisation – directional solidification, zonal melting, one-dimensional thermal field, etc. Moreover these models work with highly limiting assumption concerning phase diagrams (constant distribution coefficients) and development of dendritic morphology

**1. Introduction** 

models (Kraft & Chang, 1997).

Frantisek Kavicka1, Karel Stransky1, Jana Dobrovska2, Bohumil Sekanina1, Josef Stetina1 and Jaromir Heger1

**Technical Application** 

*1Brno University of Technology,* 

*2VSB TU Ostrava Czech Republic* 

Zhang, X. H., Ma, H., & Lucas, J. (2004). Evaluation of glass fibers from the Ga–Ge–Sb–Se system for infrared applications, *Optical Materials*, Vol. 25, pp.85-89, ISSN 0925- 3467

## **Numerical Models of Crystallization and Its Direction for Metal and Ceramic Materials in Technical Application**

Frantisek Kavicka1, Karel Stransky1, Jana Dobrovska2, Bohumil Sekanina1, Josef Stetina1 and Jaromir Heger1 *1Brno University of Technology, 2VSB TU Ostrava Czech Republic* 

## **1. Introduction**

64 Crystallization – Science and Technology

Zhang, X. H., Ma, H., & Lucas, J. (2004). Evaluation of glass fibers from the Ga–Ge–Sb–Se

3467

system for infrared applications, *Optical Materials*, Vol. 25, pp.85-89, ISSN 0925-

Structure of metallic and also majority of ceramic alloys is one of the factors, which significantly influence their physical and mechanical properties. Formation of structure is strongly affected by production technology, casting and solidification of these alloys. Solidification is a critical factor in the materials industry, e.g. (Chvorinov, 1954). Solute segregation either on the macro- or micro-scale is sometimes the cause of unacceptable products due to poor mechanical properties of the resulting non-equilibrium phases. In the areas of more important solute segregation there occurs weakening of bonds between atoms and mechanical properties of material degrade. Heterogeneity of distribution of components is a function of solubility in solid and liquid phases. During solidification a solute can concentrate in inter-dendritic areas above the value of its maximum solubility in solid phase. Solute diffusion in solid phase is a limiting factor for this process, since diffusion coefficient in solid phase is lower by three up to five orders than in the melt (Smrha, 1983). When analysing solidification of these alloys so far no unified theoretical model was created, which would describe this complex heterogeneous process as a whole. During the last fifty years many approaches with more or less limiting assumptions were developed. Solidification models and simulations have been carried out for both macroscopic and microscopic scales. The most elaborate numerical models can predict micro-segregation with comparatively high precision. The main limiting factor of all existing mathematical micro-segregation models consists in lack of available thermodynamic and kinetic data, especially for systems of higher orders. There is also little experimental data to check the models (Kraft & Chang, 1997).

Many authors deal with issues related to modelling of a non-equilibrium crystallisation of alloys. However, majority of the presented works concentrates mainly on investigation of modelling of micro-segregation of binary alloys, or on segregation of elements for special cases of crystallisation – directional solidification, zonal melting, one-dimensional thermal field, etc. Moreover these models work with highly limiting assumption concerning phase diagrams (constant distribution coefficients) and development of dendritic morphology

Numerical Models of Crystallization

Kirchoff's equation (2).

change [W.m-3], x,y,z axes in given directions [m].

called pre-processing and post-processing.

 

In equation (2) wx, wy, wz are velocity in given directions [m.s-1 ]

suitably chosen samples from continuously crystallised blanks.

and Its Direction for Metal and Ceramic Materials in Technical Application 67

222 <sup>T</sup> <sup>λ</sup> TTT <sup>Q</sup> = ++ + τ ρc xyz ρc

In equation (1) are T temperature [K], τ time [s], λ heat conductivity [W.m-1.K-1], ρ density [kg.m-3], c specific heat capacity [J.kg-1.K-1], QSOURCE latent heat of the phase or structural

Implementation of continuously casting (shortly concasting), which considerably increased rate of melt cooling between temperatures of liquidus and solidus brought about time necessary for crystallic structure homogenisation. 3D transient temperature field of the of the system of concast blank-mould or concast blank-ambient is described by Fourier-

> <sup>T</sup> <sup>λ</sup> TTT T T T <sup>Q</sup> = + + + w +w +w + τ ρc xyz xyz ρc

These equations are solvable only by means of modern numerical methods. Therefore original models of the transient temperature field (models **A**) of both systems of gravitational casting or continuous casting were developed. Both models are based on the 1st and 2nd Fourier's laws on transient heat conduction, and the 1st and 2nd law of thermodynamics. They are based on the numerical method of finite differences with explicit formula for the unknown temperature of the mesh node in the next time step, which is a function of temperatures of the same node and six adjacent nodes in Cartesian coordinate system in the previous time step. Models take into account non-linearity of the task, it means dependence of thermo-physical properties of all materials of the systems on temperature and dependence of heat transfer coefficients on temperature of all external surfaces. Models are equipped with and interactive graphical environment for automatic generation of a mesh, and for evaluation of results, it means by so

Another model, which has also been already mastered is model of chemical heterogeneity of chemical elements (model **B)**, enables description and measurement of dendritic segregation of constitutive elements and admixtures in crystallising and cooling blank (casting or concasting). This model is based on the 1st and 2nd Fick's laws of diffusion and it comprises implicitly also the law of conservation of mass. The solution itself is based on the Nernst distribution law, which quantifies at crystallisation distribution of chemical elements between liquid and solid phases of currently crystallising material in the so called mushy zone (i.e. in the zone lying between the temperature of liquidus and solidus). Majority of parameters necessary for application of the models **A** is known, but parameters necessary for use of the model **B** had to be determined by measurements on the work itself, i.e. on

Measurement was realised in the following manner: at selected segments of the cast blank concentration of main constitutive, additive and admixture elements was determined from the samples taken in regular steps. In dependence on chemical heterogeneity and structure of a blank the segments with length of 500 to 1000 m were selected, and total number of

xyz

SOURCE

SOURCE

(2)

(1)

2 2

(mostly one-dimensional models of dendrites); e.g. overview works (Boettingen, 2000; Rappaz, 1989; Stefanescu, 1995). Comprehensive studies of solidification for higher order real alloys are rarer. Nevertheless, there is a strong industrial need to investigate and simulate more complex alloys because nearly all current commercial alloys have many components often exceeding ten elements. Moreover, computer simulation have shown that even minute amounts of alloying elements can significantly influence microstructure and micro-segregation and cannot be neglected (Kraft & Chang, 1997). Dendritic crystallisation is general form of crystallisation of salts, metals and alloys. At crystallisation of salts from solutions a dendritic growth of crystals occurs at high crystallisation rate, which requires high degree of over-saturation. Findings acquired at investigation of crystallisation of salts were confirmed at investigation of crystallisation of metals. If negative temperature gradient is present in the melt before the solidification front, this leads to a disintegration of the crystallisation front and to formation of dendritic crystallisation (Davies, 1973). High crystallisation rate is characteristic feature of dendritic crystallisation. Solutes have principal influence on the crystallisation character, as they are the cause of melt supercooling before the crystallisation front and formation of the negative temperature gradient. This kind of supercooling is called constitutional supercooling. For example a layer of supercooled melt is formed in a steel ingot in the immediate vicinity of the interface melt-solid, in principle at the very beginning of crystallisation as a result of segregation of solutes, which causes decrease in solidification temperature of this enriched steel. Increased concentration of solutes creates soon a broad zone of constitutionally supercooled steel, in which the crystallisation rate is high. During subsequent solidification, when the crystallisation rate is low, the value of temperature gradient is also low, which means that conditions for dendritic crystallisation are fulfilled again (Šmrha, 1983). More detailed information on dendritic crystallisation – see classical works (Chalmers, 1940), (Flemings, 1973), (Kurz &Fisher, 1986). According Chvorinov (1954), Smrha (1983) and others metallic alloys and also majority of ceramic alloys in technical application are always characterised by their dendritic crystallisation. It is therefore of utmost importance that their final desirable dendritic structure has appropriate properties that can be used in technical practice. These properties depend of the kind of practical use and they comprise flexibility, elasticity, tensile strength, hardness, but also for example toughness. In the case of ceramic materials the properties of importance are brittleness, fragility and very often also refractoriness or resistance to wear. This chapter presents numerical models of crystallisation of steel, ductile cast iron and ceramics EUCOR aimed at optimisation of their production and properties after casting.

#### **2. Numerical models of the temperature field and heterogeneity**

Crystallization and dendritic segregation of constitutive elements and admixtures in solidifying (crystallising) and cooling gravitationally cast casting or continuously cast blank (shortly concast blank) is directly dependent on character of formation of its temperature field. Especially rate and duration of the running crystallisation at any place of the blank, so called local solidification time, is important. Solidification and cooling-down of a gravitationally cast casting as well as the simultaneous heating of a metal or non-metal mould is a rather complex problem of transient heat and mass transfer. This process in a system casting- mould-ambient can be described by the Fourier's equation (1) of 3D transient conduction of heat.

(mostly one-dimensional models of dendrites); e.g. overview works (Boettingen, 2000; Rappaz, 1989; Stefanescu, 1995). Comprehensive studies of solidification for higher order real alloys are rarer. Nevertheless, there is a strong industrial need to investigate and simulate more complex alloys because nearly all current commercial alloys have many components often exceeding ten elements. Moreover, computer simulation have shown that even minute amounts of alloying elements can significantly influence microstructure and micro-segregation and cannot be neglected (Kraft & Chang, 1997). Dendritic crystallisation is general form of crystallisation of salts, metals and alloys. At crystallisation of salts from solutions a dendritic growth of crystals occurs at high crystallisation rate, which requires high degree of over-saturation. Findings acquired at investigation of crystallisation of salts were confirmed at investigation of crystallisation of metals. If negative temperature gradient is present in the melt before the solidification front, this leads to a disintegration of the crystallisation front and to formation of dendritic crystallisation (Davies, 1973). High crystallisation rate is characteristic feature of dendritic crystallisation. Solutes have principal influence on the crystallisation character, as they are the cause of melt supercooling before the crystallisation front and formation of the negative temperature gradient. This kind of supercooling is called constitutional supercooling. For example a layer of supercooled melt is formed in a steel ingot in the immediate vicinity of the interface melt-solid, in principle at the very beginning of crystallisation as a result of segregation of solutes, which causes decrease in solidification temperature of this enriched steel. Increased concentration of solutes creates soon a broad zone of constitutionally supercooled steel, in which the crystallisation rate is high. During subsequent solidification, when the crystallisation rate is low, the value of temperature gradient is also low, which means that conditions for dendritic crystallisation are fulfilled again (Šmrha, 1983). More detailed information on dendritic crystallisation – see classical works (Chalmers, 1940), (Flemings, 1973), (Kurz &Fisher, 1986). According Chvorinov (1954), Smrha (1983) and others metallic alloys and also majority of ceramic alloys in technical application are always characterised by their dendritic crystallisation. It is therefore of utmost importance that their final desirable dendritic structure has appropriate properties that can be used in technical practice. These properties depend of the kind of practical use and they comprise flexibility, elasticity, tensile strength, hardness, but also for example toughness. In the case of ceramic materials the properties of importance are brittleness, fragility and very often also refractoriness or resistance to wear. This chapter presents numerical models of crystallisation of steel, ductile cast iron and ceramics EUCOR aimed at optimisation of their production and properties

**2. Numerical models of the temperature field and heterogeneity** 

Crystallization and dendritic segregation of constitutive elements and admixtures in solidifying (crystallising) and cooling gravitationally cast casting or continuously cast blank (shortly concast blank) is directly dependent on character of formation of its temperature field. Especially rate and duration of the running crystallisation at any place of the blank, so called local solidification time, is important. Solidification and cooling-down of a gravitationally cast casting as well as the simultaneous heating of a metal or non-metal mould is a rather complex problem of transient heat and mass transfer. This process in a system casting- mould-ambient can be described by the Fourier's equation (1) of 3D

after casting.

transient conduction of heat.

$$\frac{\partial \mathbf{T}}{\partial \mathbf{r}} = \frac{\lambda}{\rho \mathbf{c}} \left( \frac{\hat{\mathbf{c}}^2 \mathbf{T}}{\partial \mathbf{x}} + \frac{\hat{\mathbf{c}}^2 \mathbf{T}}{\partial \mathbf{y}} + \frac{\hat{\mathbf{c}}^2 \mathbf{T}}{\partial \mathbf{z}^2} \right) + \frac{\mathbf{Q}\_{\text{SOURCRE}}}{\rho \mathbf{c}} \tag{1}$$

In equation (1) are T temperature [K], τ time [s], λ heat conductivity [W.m-1.K-1], ρ density [kg.m-3], c specific heat capacity [J.kg-1.K-1], QSOURCE latent heat of the phase or structural change [W.m-3], x,y,z axes in given directions [m].

Implementation of continuously casting (shortly concasting), which considerably increased rate of melt cooling between temperatures of liquidus and solidus brought about time necessary for crystallic structure homogenisation. 3D transient temperature field of the of the system of concast blank-mould or concast blank-ambient is described by Fourier-Kirchoff's equation (2).

$$\frac{\partial \mathbf{T}}{\partial \mathbf{r}} = \frac{\lambda}{\rho \mathbf{c}} \left( \frac{\partial^2 \mathbf{T}}{\partial \mathbf{x}^2} + \frac{\partial^2 \mathbf{T}}{\partial \mathbf{y}^2} + \frac{\partial^2 \mathbf{T}}{\partial \mathbf{z}^2} \right) + \left( \mathbf{w}\_x \frac{\partial \mathbf{T}}{\partial \mathbf{x}} + \mathbf{w}\_y \frac{\partial \mathbf{T}}{\partial \mathbf{y}} + \mathbf{w}\_z \frac{\partial \mathbf{T}}{\partial \mathbf{z}} \right) + \frac{\mathbf{Q}\_{\text{SOURCE}}}{\rho \mathbf{c}} \tag{2}$$

In equation (2) wx, wy, wz are velocity in given directions [m.s-1 ]

These equations are solvable only by means of modern numerical methods. Therefore original models of the transient temperature field (models **A**) of both systems of gravitational casting or continuous casting were developed. Both models are based on the 1st and 2nd Fourier's laws on transient heat conduction, and the 1st and 2nd law of thermodynamics. They are based on the numerical method of finite differences with explicit formula for the unknown temperature of the mesh node in the next time step, which is a function of temperatures of the same node and six adjacent nodes in Cartesian coordinate system in the previous time step. Models take into account non-linearity of the task, it means dependence of thermo-physical properties of all materials of the systems on temperature and dependence of heat transfer coefficients on temperature of all external surfaces. Models are equipped with and interactive graphical environment for automatic generation of a mesh, and for evaluation of results, it means by so called pre-processing and post-processing.

Another model, which has also been already mastered is model of chemical heterogeneity of chemical elements (model **B)**, enables description and measurement of dendritic segregation of constitutive elements and admixtures in crystallising and cooling blank (casting or concasting). This model is based on the 1st and 2nd Fick's laws of diffusion and it comprises implicitly also the law of conservation of mass. The solution itself is based on the Nernst distribution law, which quantifies at crystallisation distribution of chemical elements between liquid and solid phases of currently crystallising material in the so called mushy zone (i.e. in the zone lying between the temperature of liquidus and solidus). Majority of parameters necessary for application of the models **A** is known, but parameters necessary for use of the model **B** had to be determined by measurements on the work itself, i.e. on suitably chosen samples from continuously crystallised blanks.

Measurement was realised in the following manner: at selected segments of the cast blank concentration of main constitutive, additive and admixture elements was determined from the samples taken in regular steps. In dependence on chemical heterogeneity and structure of a blank the segments with length of 500 to 1000 m were selected, and total number of

Numerical Models of Crystallization

*lS* and *L* for the criterion

**3. Gravitational casting** 

chemical and structural heterogeneity.

investigation are presented on following technical applications.

**3.1 Solidification of massive casting of ductile cast iron** 

**3.1.1 Calculation and meassurement of the temperature field** 

Fig. 1. The forming of casting no. 1 with chills on one side

and Its Direction for Metal and Ceramic Materials in Technical Application 69

results of experimental metallographic analysis, the mean value of distances between branches of secondary dendrites *L* was determined for 9 samples of the blank. The values

determined for individual measured elements in each sample of the blank. It is possible to make from the equation (4) an estimate of the diffusion coefficient of each analysed element in individual samples of the blank. At the moment, when temperature of any point of the mesh drops below the liquidus temperature, it is valid that the share of the forming solid phase *gs* grows till its limit value *gs* = 1 (i.e. in solid phase). In this case segregation of the

The combination of mentioned models and methodology of chemical heterogeneity

The quality of a massive casting of cast iron with spheroidal graphite is determined by all the parameters and factors that affect the metallographic process and also others. This means the factors from sorting, melting in, modification and inoculation, casting, solidification and cooling inside the mould and heat treatment. The centre of focus were not only the purely practical questions relating to metallurgy and foundry technology, but mainly the verification of the possibility of applying two original models – the 3D model of transient solidification and the cooling of a massive cast-iron casting and the model of

The application of the 3D numerical model on a transient temperature field requires systematic experimentation, including the relevant measurement of the operational parameters directly in the foundry. A real 5001000500 mm ductile cast-iron block had been used for the numerical calculation and the experiment. They were cast into sand moulds with various arrangements of steel chills of cylindrical shape. The dimensions of the selectid casting, the mould, the chills and their arrangements are illustrated in Figure 1. The

investigated element achieves in the residual inter-dendritic melt its maximum.

, are calculated from the model for each sample, which were

steps, in which concentration of elements was measured, was set to 101. Measurement of concentration of elements was performed by methods of quantitative energy dispersive analysis (EDA) and wave dispersive analysis (WDA) of X-ray spectral microanalysis, for which special software and special measurement device was developed for use in combination with the analytical complex JEOL JXA 8600/KEVEX.

After completion of measurement the sample surface was etched in order to make visible the original contamination of surface by electron beam, and the measured traces were photographically documented, including the mean distance of dendrites axes within the measured segment. It was verified that the basic set of measured concentration data of elements (8 to 11 elements) makes it possible to obtain a semi-quantitative to quantitative information on chemical heterogeneity of the blank, and that it is possible to apply at the same time for evaluation of distribution of elements in the blank structure the methods of mathematical- statistical analysis. It is possible to determine the distribution curve of the element concentration in the measured segment of the analysed blank and their effective distribution coefficient between the solid and liquid phase during crystallisation. In this way the crucial verified data necessary for creation of the conjugated model (**AB**) of crystallising, solidifying and cooling down blank were obtained. It was verified that re-distribution of constitutive, additive elements and admixtures can be described by effective distribution coefficient, which had been derived for parabolic growth by Brody and Flemings (Brody&Flemings, 1966). At the moment of completed crystallisation, at surpassing of an isosolidic curve in the blank, it is possible to express the ratio of concentration of dendritically segregating element *CS* to the mean concentration of the same element at the given point of the blank *C0* by the relation

$$\mathbf{C\_{S}}/\mathbf{C\_{0}} = \mathbf{k\_{e}}[1 - (1 - 2\alpha \mathbf{k\_{e}})] \mathbf{g\_{S}}[^{(\mathbf{k\_{e}} \cdot 1)} / (1 \cdot 2\mathbf{a})] \tag{3}$$

where *kef* is effective distribution coefficient, *gS* is mass share of the solidified phase, and is dimensionless Fourier's number of the 2nd kind for mass transfer. This number is given by the relation

$$
\alpha = \mathbf{D}\_{\mathbb{S}} \theta\_{\mathbb{IS}} / \mathrm{L}^2 \tag{4}
$$

in which *DS* is diffusion coefficient of the segregating element in solid phase, *lS* is local time of crystallisation (i.e. time of persistence of the assumed dendrite between the temperature of liquidus and solidus) and *L* is mean half distance of dendritic axes (namely of axes of secondary dendrites). In the next step it is necessary to express the ratio of concentrations *CS/C0* express as a function of concrete heterogeneity index *IH* and of statistical distribution of the measured element, expressed by distributive curve of crystallization segregation. In this manner the following equation is available for each measured element:

$$\mathbf{C}\_{\rm S}/\mathbf{C}\_{\rm 0} = \mathbf{I}\_{\rm H} \tag{5}$$

which expresses by concrete numbers the parameters, defined by the equation (3). By solving equations (3, 4) it is then possible to determine for each analysed element (i.e. for its measured index of dendritic heterogenity, effective distribution coefficient and distribution curve of dendritic segregation, i.e. for the established statistic character of distribution of the analysed element in structure of the blank) certain values of dimensionless criterion . Afterwards on the basis of semi-empiric relations and rates of movement of the crystallisation front, calculated from the thermal field model in confrontation with the results of experimental metallographic analysis, the mean value of distances between branches of secondary dendrites *L* was determined for 9 samples of the blank. The values *lS* and *L* for the criterion , are calculated from the model for each sample, which were determined for individual measured elements in each sample of the blank. It is possible to make from the equation (4) an estimate of the diffusion coefficient of each analysed element in individual samples of the blank. At the moment, when temperature of any point of the mesh drops below the liquidus temperature, it is valid that the share of the forming solid phase *gs* grows till its limit value *gs* = 1 (i.e. in solid phase). In this case segregation of the investigated element achieves in the residual inter-dendritic melt its maximum.

The combination of mentioned models and methodology of chemical heterogeneity investigation are presented on following technical applications.

## **3. Gravitational casting**

68 Crystallization – Science and Technology

steps, in which concentration of elements was measured, was set to 101. Measurement of concentration of elements was performed by methods of quantitative energy dispersive analysis (EDA) and wave dispersive analysis (WDA) of X-ray spectral microanalysis, for which special software and special measurement device was developed for use in

After completion of measurement the sample surface was etched in order to make visible the original contamination of surface by electron beam, and the measured traces were photographically documented, including the mean distance of dendrites axes within the measured segment. It was verified that the basic set of measured concentration data of elements (8 to 11 elements) makes it possible to obtain a semi-quantitative to quantitative information on chemical heterogeneity of the blank, and that it is possible to apply at the same time for evaluation of distribution of elements in the blank structure the methods of mathematical- statistical analysis. It is possible to determine the distribution curve of the element concentration in the measured segment of the analysed blank and their effective distribution coefficient between the solid and liquid phase during crystallisation. In this way the crucial verified data necessary for creation of the conjugated model (**AB**) of crystallising, solidifying and cooling down blank were obtained. It was verified that re-distribution of constitutive, additive elements and admixtures can be described by effective distribution coefficient, which had been derived for parabolic growth by Brody and Flemings (Brody&Flemings, 1966). At the moment of completed crystallisation, at surpassing of an isosolidic curve in the blank, it is possible to express the ratio of concentration of dendritically segregating element *CS* to the mean concentration of the same element at the

CS/C0 = kef[1 (1 2αkef)gS] (kef - 1) / (1 - 2α) (3)

.

where *kef* is effective distribution coefficient, *gS* is mass share of the solidified phase, and

in which *DS* is diffusion coefficient of the segregating element in solid phase,

manner the following equation is available for each measured element:

is dimensionless Fourier's number of the 2nd kind for mass transfer. This number is given by

of crystallisation (i.e. time of persistence of the assumed dendrite between the temperature of liquidus and solidus) and *L* is mean half distance of dendritic axes (namely of axes of secondary dendrites). In the next step it is necessary to express the ratio of concentrations *CS/C0* express as a function of concrete heterogeneity index *IH* and of statistical distribution of the measured element, expressed by distributive curve of crystallization segregation. In this

 CS/C0 = IH (5) which expresses by concrete numbers the parameters, defined by the equation (3). By solving equations (3, 4) it is then possible to determine for each analysed element (i.e. for its measured index of dendritic heterogenity, effective distribution coefficient and distribution curve of dendritic segregation, i.e. for the established statistic character of distribution of the analysed element in structure of the blank) certain values of dimensionless criterion

Afterwards on the basis of semi-empiric relations and rates of movement of the crystallisation front, calculated from the thermal field model in confrontation with the

= DS lS /L2 (4)

*lS* is local time

combination with the analytical complex JEOL JXA 8600/KEVEX.

given point of the blank *C0* by the relation

the relation

## **3.1 Solidification of massive casting of ductile cast iron**

The quality of a massive casting of cast iron with spheroidal graphite is determined by all the parameters and factors that affect the metallographic process and also others. This means the factors from sorting, melting in, modification and inoculation, casting, solidification and cooling inside the mould and heat treatment. The centre of focus were not only the purely practical questions relating to metallurgy and foundry technology, but mainly the verification of the possibility of applying two original models – the 3D model of transient solidification and the cooling of a massive cast-iron casting and the model of chemical and structural heterogeneity.

## **3.1.1 Calculation and meassurement of the temperature field**

The application of the 3D numerical model on a transient temperature field requires systematic experimentation, including the relevant measurement of the operational parameters directly in the foundry. A real 5001000500 mm ductile cast-iron block had been used for the numerical calculation and the experiment. They were cast into sand moulds with various arrangements of steel chills of cylindrical shape. The dimensions of the selectid casting, the mould, the chills and their arrangements are illustrated in Figure 1. The

Fig. 1. The forming of casting no. 1 with chills on one side

Numerical Models of Crystallization

and Its Direction for Metal and Ceramic Materials in Technical Application 71

A 50050040 mm plate had been mechanically cut out of the middle of the length by two parallel transversal cuts. Then, further samples were taken from exactly defined points and tested in terms of their structural parameters and chemical heterogeneity. Samples in the form of testing test-samples for ductility testing, with threaded ends, were taken from the bottom part of the casting (A), from the middle part (C) and from the upper part (G). The 15 mm in diameter and 12 mm high cylindrical samples served the actual measurements in order to determine the structural parameters and chemical heterogeneity. In the points of the defined positions of the samples prepared in this way, the quantitative metallographic analysis was used to establish the structural parameters of cast-iron, the in-line point analysis to establish the chemical composition of selected elements and numerical calculation using the 3D model to establish the local solidification time. Quantitative analyses of the basic micro-structural parameters in the samples have been the subject of a special study. On each sample a total number of 49 views were evaluated. On the basis of average values of these results the structural parameters of graphite, i.e. the radius of the spheroids of graphite – *Rg*, the distances between the edges of graphite particles – *Lg* and the radius of the graphite cells – *Rc* have been determined for each sample. The concentration of selected elements in each of the samples was measured on the line of *Lg* between the edges of two particles of spheroidal graphite. The actual measurements of concentrations of ten elements – Mg, Al, Si, P, S, Ti, Cr, Mn, Fe, Ni – was carried out. On each of the samples, the concentrations of all ten elements had been measured in three intervals with each individual step being 3 m. Before the actual measurement, the regions were selected on an unetched part of the surface and marked with a micro-hardness tester. After the micro-analysis, the samples were etched with nitric acid in alcohol in order to make the contamination of the ground surface visible using an electron beam. Then, using a Neophot light microscope, the interval within which the concentrations were measured was documented. The method of

Element C Mn Si P S Ti Al Cr Ni Mg wt.% 3.75 0.12 2.15 0.039 0.004 0.01 0.013 0.07 0.03 0.045

Table 1. Chemical composition of ductile cast-iron (casting No. 1)

selection of the measurement points is illustrated in Figures 3a,b.

(a) (b)

a) *Lg* = 165 m, b) *Lg* = 167 m ( in Fig. *Lg* ≡ z)

Fig. 3. An example of the chemical micro-heterogeneity measurement of ductile cast-iron

courses of the temperatures on casting No. 1 were measured for 19 hours 11 min after pouring. The iso-zones, calculated in castings No.1 and in the chills in the total solidification time after casting, are illustrated in Figure 2 (Dobrovska et al., 2010).

Fig. 2. The calculated iso-zones in casting No. 1 (5 hours)

#### **3.1.2 The relation between the model of the temperature field and the model of structural and chemical heterogeneity**

The 3D numerical model of the temperature field of a system comprising the casting-moldambient is based on the numerical finite-element method. The software ANSYS had been chosen for this computation because it enables the application of the most convenient method of numerical simulation of the release of latent heats of phase and structural changes using the thermodynamic enthalpy function. The software also considers the nonlinearity of the task, i.e. the dependence of the thermophysical properties (of all materials entering the system) on the temperature, and the dependence of the heat-transfer coefficients (on all boundaries of the system) on the temperature of the surface—of the casting and mold. The original numerical model had been developed and used for estimation of structural and chemical heterogeneity. Initial and boundary conditions had been defined by means of theory of similarity. The verifying numerical calculation of the local solidification times *θ* – conducted according to the 3D model proved that, along the height, width and length of these massive castings, there are various points with differences in the solidification times of up to two orders. The aim was to verify the extent to which the revealed differences in the local solidification times affect the following parameters (Stransky et al., 2010):


The relationships – among the given four parameters and the corresponding local solidification times – were determined in the series of samples that had been selected from defined positions of the massive casting. The bottom part of its sand mould was lined with (a total number of) 15 cylindrical chills of a diameter of 150 mm and a height of 200 mm. The upper part of the mould was not lined with any chills. The average chemical composition of the cast-iron before casting is given in Table 1.

courses of the temperatures on casting No. 1 were measured for 19 hours 11 min after pouring. The iso-zones, calculated in castings No.1 and in the chills in the total

solidification time after casting, are illustrated in Figure 2 (Dobrovska et al., 2010).

**3.1.2 The relation between the model of the temperature field and the model of** 

The 3D numerical model of the temperature field of a system comprising the casting-moldambient is based on the numerical finite-element method. The software ANSYS had been chosen for this computation because it enables the application of the most convenient method of numerical simulation of the release of latent heats of phase and structural changes using the thermodynamic enthalpy function. The software also considers the nonlinearity of the task, i.e. the dependence of the thermophysical properties (of all materials entering the system) on the temperature, and the dependence of the heat-transfer coefficients (on all boundaries of the system) on the temperature of the surface—of the casting and mold. The original numerical model had been developed and used for estimation of structural and chemical heterogeneity. Initial and boundary conditions had been defined by means of theory of similarity. The verifying numerical calculation of the local solidification times *θ* – conducted according to the 3D model proved that, along the height, width and length of these massive castings, there are various points with differences in the solidification times of up to two orders. The aim was to verify the extent to which the revealed differences in the local solidification times affect the following parameters

d. The chemical heterogeneity of the elements in the cross-sections of individual graphite cells. The relationships – among the given four parameters and the corresponding local solidification times – were determined in the series of samples that had been selected from defined positions of the massive casting. The bottom part of its sand mould was lined with (a total number of) 15 cylindrical chills of a diameter of 150 mm and a height of 200 mm. The upper part of the mould was not lined with any chills. The average chemical composition of

Fig. 2. The calculated iso-zones in casting No. 1 (5 hours)

a. The average size of the spheroidal graphite particles; b. The average density of the spheroidal graphite particles; c. The average dimensions of the graphite cells, and

the cast-iron before casting is given in Table 1.

**structural and chemical heterogeneity** 

(Stransky et al., 2010):


Table 1. Chemical composition of ductile cast-iron (casting No. 1)

A 50050040 mm plate had been mechanically cut out of the middle of the length by two parallel transversal cuts. Then, further samples were taken from exactly defined points and tested in terms of their structural parameters and chemical heterogeneity. Samples in the form of testing test-samples for ductility testing, with threaded ends, were taken from the bottom part of the casting (A), from the middle part (C) and from the upper part (G). The 15 mm in diameter and 12 mm high cylindrical samples served the actual measurements in order to determine the structural parameters and chemical heterogeneity. In the points of the defined positions of the samples prepared in this way, the quantitative metallographic analysis was used to establish the structural parameters of cast-iron, the in-line point analysis to establish the chemical composition of selected elements and numerical calculation using the 3D model to establish the local solidification time. Quantitative analyses of the basic micro-structural parameters in the samples have been the subject of a special study. On each sample a total number of 49 views were evaluated. On the basis of average values of these results the structural parameters of graphite, i.e. the radius of the spheroids of graphite – *Rg*, the distances between the edges of graphite particles – *Lg* and the radius of the graphite cells – *Rc* have been determined for each sample. The concentration of selected elements in each of the samples was measured on the line of *Lg* between the edges of two particles of spheroidal graphite. The actual measurements of concentrations of ten elements – Mg, Al, Si, P, S, Ti, Cr, Mn, Fe, Ni – was carried out. On each of the samples, the concentrations of all ten elements had been measured in three intervals with each individual step being 3 m. Before the actual measurement, the regions were selected on an unetched part of the surface and marked with a micro-hardness tester. After the micro-analysis, the samples were etched with nitric acid in alcohol in order to make the contamination of the ground surface visible using an electron beam. Then, using a Neophot light microscope, the interval within which the concentrations were measured was documented. The method of selection of the measurement points is illustrated in Figures 3a,b.

Fig. 3. An example of the chemical micro-heterogeneity measurement of ductile cast-iron a) *Lg* = 165 m, b) *Lg* = 167 m ( in Fig. *Lg* ≡ z)

Numerical Models of Crystallization

product.

quadrant only.

the square of the local solidification time.

**3.2 Casting of corundo-baddeleyit ceramic material** 

for castings (Chvorinov, 1954; Smrha, 1983 and others).

**3.2.1 Measurements and computation results (the original riser)** 

and Its Direction for Metal and Ceramic Materials in Technical Application 73

taken from above. The testing indicates that the local solidification time *θ* has significant influence on the ductility *A5*. The relationship between the ductility and the local solidification time (equation 10) indicates that the reduction in the ductility of cast-iron in the state immediately after pouring is – in the first approximation – directly proportional to

It can be seen from previous experimentation and the evaluations of the results that – in the general case of the solidification of ductile cast-iron – there could be a dependence of the size of the spheroids of graphite, the size of the graphite cells and therefore even the distance among the graphite particles on the local solidification time. The described connection with the 3D model of a transient temperature field, which makes it possible to determine the local solidification time, seems to be the means via which it is possible to estimate the differences in structural characteristics of graphite in cast-iron and also the effect of the local solidification time on ductility in the poured casting. The main economic goal observed is the saving of liquid material, moulding and insulation materials, the saving of energy and the already mentioned optimization of pouring and the properties of the cast

The corundo-baddeleyit material (CBM) belongs to the not too well known system of the Al2O3-SiO2-ZrO2 oxide ceramics. Throughout the world, it is produced only in several plants, in the Czech Republic under the name of EUCOR. This production process entails solely the utilisation of waste material from relined furnaces from glass-manufacturing plants. EUCOR is heat resistant, wear resistant even at extreme temperatures and it is also resistant to corrosion. It was shown that from the foundry property viewpoint, EUCOR has certain characteristics that are similar to the behavior of cast metal materials, especially steel

The assignment focussed on the investigation of the transient 3D temperature field of a system comprising a casting-and-riser, the mold and ambient, using a numerical model (Heger et al., 2002; Kavicka et. al., 2010) . The dimensions of the casting — the so-called *"*stone" — were 400 x 350 x 200 mm (Figure 4). The results attained from the numerical analysis of the temperature field of a solidifying casting and the heating of the mold represent only one quadrant of the system in question. Figure 5 shows the 3D temperature field of the casting with the original riser and the mold at two times after pouring. The risermold interface is an interesting place for monitoring. Once this point solidifies, the riser can no longer affect the process inside the casting. The initial temperature of the mold was 20C. The pouring temperature of the melt was 1800C. That was approximately 300C higher, when compared with, for example, the steel pouring temperature. The temperature of the liquidus was 1775C and the solidus 1765C. The temperature field was symmetrical along the axes, i.e. it was sufficient for the investigation of the temperature field of a single

<sup>2</sup> A5 23.399-8.1703 %,hr (10)

The results of the measurements of the chemical heterogeneity were evaluated statistically. The element heterogeneity index *IH*(*i*) is defined by the quotient of standard deviation of element concentration σ*<sup>c</sup>*(*<sup>i</sup>*) and average element concentration *Ci av* in the analysed area, i.e. *IH*(*i*) = σ*<sup>c</sup>*(*<sup>i</sup>*)/*Ci av*. The element segregation index *IS*(*<sup>i</sup>*) is defined by the quotient of element maximum concentration *Ci max* and average element concentration *Ci av* in the analysed area, i.e. *IS*(*i*) *= Ci max /Ci av* . The local solidification times of the selected samples of known coordinates within the massive experimental casting were calculated by the 3D model. The calculation of the temperature of the liquidus and solidus for a melt with a composition according to the data in Table 1 was performed using special software with the values: 1130 ºC and 1110 ºC (the liquidus and solidus temperature). If the local solidification time is known, then it is also possible to determine the average rate of cooling *w* of the mushy zone as a quotient of the temperature interval and the local solidification time *w* = *T/θ* ºC/s. It is obvious from the results that in the vertical direction *y* from the bottom of the massive casting (sample A: *y* = 50 mm) to the top (gradually samples C: *y* = 210 mm and G: *y* = 450 mm) the characteristic and significant relations are as follows:


The relationships between the structural characteristics of graphite in the casting and the local solidification time were expressed quantitatively using a semi-logarithmic dependence:

$$\mathbf{R\_{g}} = 19.08 + 2.274 \ln \theta \text{ , } \mathbf{R\_{c}} = 61.33 + 5.567 \ln \theta \text{ , } \mathbf{L\_{g}} = 84.50 + 6.586 \ln \theta \text{ [μm.s] } \tag{6.7.8}$$

As far as chemical heterogenity of the measured elements is concerned, the analogous relation was established only for the dependence of the segregation index of titanium on the local solidification time, which has a steadily increasing course from the bottom of the casting all the way up to the top. The relevant relation was expressed in the form of a logarithmic equation

$$\ln\text{I}\_{\mathbb{S}} \text{\AA} = \text{1.201} + 0.1410 \ln \theta \,\,\_{\text{l}\text{\nu}} \qquad \text{[s]}\tag{9}$$

The local solidification time *θ* naturally affects the mechanical properties of cast-iron however with regard to the dimensions of the test pieces; it is not possible to assign the entire body a single local solidification time. To assess relationship among structural parameters, chemical microheterogenity and mechanical properties of analysed cast-iron casting, the selected mechanical properties have been measured. The samples for testing of the tensile strength were taken from the test-sample of the experimental casting in such a way that one had been taken from under the metallographic sample and the second was

The results of the measurements of the chemical heterogeneity were evaluated statistically. The element heterogeneity index *IH*(*i*) is defined by the quotient of standard deviation of

*max* and average element concentration *Ci*

*T/θ* ºC/s. It is obvious from the results that in the vertical direction *y* from the bottom of the massive casting (sample A: *y* = 50 mm) to the top (gradually samples C: *y* = 210 mm

a. The average size of the spheroids of graphite *Rg*, the average size of the cells of graphite *Rc* and also the average distance between the individual particles of the graphite *Lg* are all increasing. This relation was confirmed by the quantitative metallographic analysis. b. The chemical heterogeneity within the individual graphite cells is also changing. The increase in the chemical heterogeneity is reflected most significantly in the increase in the indexes of segregation *IS* for titanium which are increasing in the direction from the bottom of the massive casting to the top in the following order: *IS*(*Ti*) *=* 5.79-to-9.39-to-

c. The local solidification time increases very significantly – from the bottom of the casting to the top – from 48 s more than 50 (near the centre of the casting) and 95 (at the top

The relationships between the structural characteristics of graphite in the casting and the local solidification time were expressed quantitatively using a semi-logarithmic dependence:

As far as chemical heterogenity of the measured elements is concerned, the analogous relation was established only for the dependence of the segregation index of titanium on the local solidification time, which has a steadily increasing course from the bottom of the casting all the way up to the top. The relevant relation was expressed in the form of a

The local solidification time *θ* naturally affects the mechanical properties of cast-iron however with regard to the dimensions of the test pieces; it is not possible to assign the entire body a single local solidification time. To assess relationship among structural parameters, chemical microheterogenity and mechanical properties of analysed cast-iron casting, the selected mechanical properties have been measured. The samples for testing of the tensile strength were taken from the test-sample of the experimental casting in such a way that one had been taken from under the metallographic sample and the second was

R 19.08 2.274ln <sup>g</sup> , R 61.33 5.567 ln <sup>c</sup> , L 84.50 6.586ln <sup>g</sup> [μm,s] (6, 7, 8)

ln ISTi = 1.201 + 0.1410ln ls, s (9)

and G: *y* = 450 mm) the characteristic and significant relations are as follows:

coordinates within the massive experimental casting were calculated by the 3D model. The calculation of the temperature of the liquidus and solidus for a melt with a composition according to the data in Table 1 was performed using special software with the values: 1130 ºC and 1110 ºC (the liquidus and solidus temperature). If the local solidification time is known, then it is also possible to determine the average rate of cooling *w* of the mushy zone as a quotient of the temperature interval and the local solidification time

*av*. The element segregation index *IS*(*<sup>i</sup>*) is defined by the quotient of element

*av* . The local solidification times of the selected samples of known

*av* in the analysed area, i.e.

*av* in the analysed area,

element concentration σ*<sup>c</sup>*(*<sup>i</sup>*) and average element concentration *Ci*

*IH*(*i*) = σ*<sup>c</sup>*(*<sup>i</sup>*)/*Ci*

i.e. *IS*(*i*) *= Ci*

*w* = 

11.62

logarithmic equation

of the massive casting).

maximum concentration *Ci*

*max /Ci*

taken from above. The testing indicates that the local solidification time *θ* has significant influence on the ductility *A5*. The relationship between the ductility and the local solidification time (equation 10) indicates that the reduction in the ductility of cast-iron in the state immediately after pouring is – in the first approximation – directly proportional to the square of the local solidification time.

$$\mathbf{A\_5 = 23.399\text{-}8.1703\theta^2} \qquad \left[\%, \text{hr}\right] \tag{10}$$

It can be seen from previous experimentation and the evaluations of the results that – in the general case of the solidification of ductile cast-iron – there could be a dependence of the size of the spheroids of graphite, the size of the graphite cells and therefore even the distance among the graphite particles on the local solidification time. The described connection with the 3D model of a transient temperature field, which makes it possible to determine the local solidification time, seems to be the means via which it is possible to estimate the differences in structural characteristics of graphite in cast-iron and also the effect of the local solidification time on ductility in the poured casting. The main economic goal observed is the saving of liquid material, moulding and insulation materials, the saving of energy and the already mentioned optimization of pouring and the properties of the cast product.

## **3.2 Casting of corundo-baddeleyit ceramic material**

The corundo-baddeleyit material (CBM) belongs to the not too well known system of the Al2O3-SiO2-ZrO2 oxide ceramics. Throughout the world, it is produced only in several plants, in the Czech Republic under the name of EUCOR. This production process entails solely the utilisation of waste material from relined furnaces from glass-manufacturing plants. EUCOR is heat resistant, wear resistant even at extreme temperatures and it is also resistant to corrosion. It was shown that from the foundry property viewpoint, EUCOR has certain characteristics that are similar to the behavior of cast metal materials, especially steel for castings (Chvorinov, 1954; Smrha, 1983 and others).

## **3.2.1 Measurements and computation results (the original riser)**

The assignment focussed on the investigation of the transient 3D temperature field of a system comprising a casting-and-riser, the mold and ambient, using a numerical model (Heger et al., 2002; Kavicka et. al., 2010) . The dimensions of the casting — the so-called *"*stone" — were 400 x 350 x 200 mm (Figure 4). The results attained from the numerical analysis of the temperature field of a solidifying casting and the heating of the mold represent only one quadrant of the system in question. Figure 5 shows the 3D temperature field of the casting with the original riser and the mold at two times after pouring. The risermold interface is an interesting place for monitoring. Once this point solidifies, the riser can no longer affect the process inside the casting. The initial temperature of the mold was 20C. The pouring temperature of the melt was 1800C. That was approximately 300C higher, when compared with, for example, the steel pouring temperature. The temperature of the liquidus was 1775C and the solidus 1765C. The temperature field was symmetrical along the axes, i.e. it was sufficient for the investigation of the temperature field of a single quadrant only.

Numerical Models of Crystallization

parameter *α* using the equation

the right-hand side of which

parameter *α* is

on the EUCOR material are as follows:

and Its Direction for Metal and Ceramic Materials in Technical Application 75

solidification. The preconditions for the application of the model of chemical heterogeneity

If the analytically expressed distribution of micro-heterogeneity of the oxides of the ceramic material is available, if their effective distribution coefficient is known, and if it is assumed that it is possible to describe the solidification of the ceramic material via analogical models as with the solidification of metal alloys, then it is possible to conduct the experiment on the mutual combination of the calculation of the temperature field of a solidifying ceramic

If the Brody-Flemings Model (Brody & Flemings, 1966) is applied for the description of the segregation of oxides of the solidifying ceramic material and if an analogy with metal alloys is assumed, then it is possible to express the relationship between the heterogeneity index *IH* of the relevant oxide, its effective distribution coefficient *kef* and the dimensionless

heterogeneity, is already known and through whose solution it is possible to determine the parameter *α*, which is also on the right-hand side of the equation in *2αkef = X*. The quantity *n* has a statistical nature and expresses what percentage of the measured values could be found within the interval *xs ± nsx* (where *xs* is the arithmetic mean and sx is the standard deviation of the set of values of the measured quantity). If *n* = 2, then 95% of all measured values can be found within this interval. If the dimensionless parameter *α* is known for each oxide, then a key to the clarification of the relationship exists between the local EUCOR solidification time *θ*, the diffusion coefficient *D* of the relevant oxide within the solidifying phase and the structure parameter *L*, which characterizes the distances between individual dendrites in metalic and ceremic alloys (Figure 6). The equation of the dimensionless

It is possible to take the dimension of a structure cell as the structure parameter for the EUCOR material. The verification of the possibility of combining both methods was conducted on samples taken from the EUCOR blocks – from the edge (sample B) – and from the centre underneath the riser (sample C). Both the measured and the computed parameters of chemical micro-heterogeneity and the computed parameters of the local solidification time *θ* (according to the temperature-field model) were calculated. The local solidification time of the sample B was *θB* = 112.18 s and of the sample C was *θC* = 283.30 s. The computed values of parameter *α* and the local solidification time *θ* determine, via their ratio, the quotient of the diffusion coefficient *D* and the square of the structure parameter *L*,

 α/θ = D/L2 [s-1] (13) The calculated values of relation (13) for oxides of the samples B and C are arranged in

[ln(2kef*)*/(1 - 2kef) = ln[(1 + nIH(m*)*)/kef/(kef - 1) (11)

α = Dθ/L2 [ - ], [m2s-1, s, m ] (12)

*/(kef - 1)*, based on the measurement of micro-

casting with the model describing the chemical heterogeneity of the oxides.

*ln[(1+ nIH(m))/kef*

which means that the following relation applies:

Table 2 together with the parameters *α*

Fig. 5. a) The 3D teperature field of one quadrant of the casting wirh riser- mold, b) The 2D temperature field on the riser mold and riser casting interferace

#### **3.2.2 The model of the chemical heterogeneity and its application**

The concentration distribution of individual oxides, making up the composition of the ceramic material EUCOR, was determined using an original method (Dobrovska, et al., 2009) and applied in the process of measuring the macro- and micro-heterogeneity of elements within ferrous alloys. This method was initially modified with respect to the differences occurring during solidification of the ceramic material, when compared to ferrous alloys. It was presumed that within EUCOR, the elements had been already distributed, together with oxygen, at the stoichiometric ratio (i.e. the chemical equation), which characterized the resulting composition of the oxides of individual elements after

Fig. 4. The casting-riser-mold system

(a) (b)

**3.2.2 The model of the chemical heterogeneity and its application** 

temperature field on the riser mold and riser casting interferace

Fig. 5. a) The 3D teperature field of one quadrant of the casting wirh riser- mold, b) The 2D

The concentration distribution of individual oxides, making up the composition of the ceramic material EUCOR, was determined using an original method (Dobrovska, et al., 2009) and applied in the process of measuring the macro- and micro-heterogeneity of elements within ferrous alloys. This method was initially modified with respect to the differences occurring during solidification of the ceramic material, when compared to ferrous alloys. It was presumed that within EUCOR, the elements had been already distributed, together with oxygen, at the stoichiometric ratio (i.e. the chemical equation), which characterized the resulting composition of the oxides of individual elements after solidification. The preconditions for the application of the model of chemical heterogeneity on the EUCOR material are as follows:

If the analytically expressed distribution of micro-heterogeneity of the oxides of the ceramic material is available, if their effective distribution coefficient is known, and if it is assumed that it is possible to describe the solidification of the ceramic material via analogical models as with the solidification of metal alloys, then it is possible to conduct the experiment on the mutual combination of the calculation of the temperature field of a solidifying ceramic casting with the model describing the chemical heterogeneity of the oxides.

If the Brody-Flemings Model (Brody & Flemings, 1966) is applied for the description of the segregation of oxides of the solidifying ceramic material and if an analogy with metal alloys is assumed, then it is possible to express the relationship between the heterogeneity index *IH* of the relevant oxide, its effective distribution coefficient *kef* and the dimensionless parameter *α* using the equation

$$\left(\ln(2\alpha \mathbf{k}\_{\rm ef})\right)/(1 - 2\alpha \mathbf{k}\_{\rm ef}) = \left(\ln[(1 + \mathbf{n}\mathbf{I}\_{\rm l}(\mathbf{m})/\mathbf{k}\_{\rm ef})]/(\mathbf{k}\_{\rm ef} - 1)\right) / \left(\mathbf{k}\_{\rm ef} - 1\right) \tag{11}$$

the right-hand side of which *ln[(1+ nIH(m))/kef/(kef - 1)*, based on the measurement of microheterogeneity, is already known and through whose solution it is possible to determine the parameter *α*, which is also on the right-hand side of the equation in *2αkef = X*. The quantity *n* has a statistical nature and expresses what percentage of the measured values could be found within the interval *xs ± nsx* (where *xs* is the arithmetic mean and sx is the standard deviation of the set of values of the measured quantity). If *n* = 2, then 95% of all measured values can be found within this interval. If the dimensionless parameter *α* is known for each oxide, then a key to the clarification of the relationship exists between the local EUCOR solidification time *θ*, the diffusion coefficient *D* of the relevant oxide within the solidifying phase and the structure parameter *L*, which characterizes the distances between individual dendrites in metalic and ceremic alloys (Figure 6). The equation of the dimensionless parameter *α* is

$$\mathbf{a} = \mathbf{D}\mathbf{\hat{o}}/\mathbf{L}^2 \qquad \left[ \mathbf{-} \right] / \left[ \mathbf{m}^2 \mathbf{s}^{-1}, \mathbf{s}, \mathbf{m} \right] \tag{12}$$

It is possible to take the dimension of a structure cell as the structure parameter for the EUCOR material. The verification of the possibility of combining both methods was conducted on samples taken from the EUCOR blocks – from the edge (sample B) – and from the centre underneath the riser (sample C). Both the measured and the computed parameters of chemical micro-heterogeneity and the computed parameters of the local solidification time *θ* (according to the temperature-field model) were calculated. The local solidification time of the sample B was *θB* = 112.18 s and of the sample C was *θC* = 283.30 s. The computed values of parameter *α* and the local solidification time *θ* determine, via their ratio, the quotient of the diffusion coefficient *D* and the square of the structure parameter *L*, which means that the following relation applies:

$$\mathbf{a}/\theta = \mathbf{D}/\mathbf{L}^2 \qquad \left[\mathbf{s}^{\mathbf{1}}\right] \tag{13}$$

The calculated values of relation (13) for oxides of the samples B and C are arranged in Table 2 together with the parameters *α*

Numerical Models of Crystallization

(*L*c = 919 μm)

**4. Continuously casting** 

and Its Direction for Metal and Ceramic Materials in Technical Application 77

(a) (b)

**4.1 Chemical microheterogeneity of continuously cast steel slab** 

Fig. 6. a)The structure of the sample B (Lb = 564 μm), b) The structure of the ample C

Structure of metallic alloys is one of the factors, which significantly influence their physical and mechanical properties. Formation of structure is strongly affected by production technology, casting and solidification of these alloys. Solidification is a critical factor in the materials industry (Kavicka et al., 2007). Solute segregation either on the macro- or microscale is sometimes the cause of unacceptable products due to poor mechanical properties of the resulting non-equilibrium phases. In the areas of more important solute segregation there occurs weakening of bonds between atoms and mechanical properties of material degrade. Heterogeneity of distribution of components is a function of solubility in solid and liquid phases. During solidification a solute can concentrate in inter-dendritic areas above the value of its maximum solubility in solid phase. Solute diffusion in solid phase is a limiting factor for this process, since diffusion coefficient in solid phase is lower by three up to five orders than in the melt. When analysing solidification of steel so far no unified theoretical model was created, which would describe this complex heterogeneous process as a whole. During the last fifty years many approaches with more or less limiting assumptions were developed. Solidification models and simulations have been carried out for both macroscopic and microscopic scales. The most elaborate numerical models can predict micro-segregation with comparatively high precision. The main limiting factor of all existing mathematical micro-segregation models consists in lack of available thermodynamic and kinetic data, especially for systems of higher orders. There is also little experimental data to check the models. Many authors deal with issues related to modelling of a non-equilibrium crystallisation of alloys. However, majority of the presented works concentrates mainly on investigation of modelling of micro-segregation of binary alloys, or on segregation of elements for special cases of crystallisation – directional solidification, zonal melting, onedimensional thermal field, etc. Moreover these models work with highly limiting assumption concerning phase diagrams (constant distribution coefficients) and development of dendritic morphology (mostly one-dimensional models of dendrites. Comprehensive studies of solidification for higher order real alloys are rarer. Nevertheless, there is a strong industrial need to investigate and simulate more complex alloys because


Table 2. Calculated values of the equation (13)

It comes as a surprise that the values of the parameter *α/θ = D/L*2 of the oxides of elements Na, Al, Si, K, Ca, Ti, and Fe differed by as much as an order from the value of the same parameter of the oxide of zirconium and hafnium. This could be explained by the fact that zirconium contains hafnium as an additive and, therefore, they segregate together and the forming oxides of zirconium and hafnium show the highest melting temperatures. From the melt, both oxides segregated first, already in their solid states. Further redistribution of the oxides of both elements ran on the interface of the remaining melt and the successive segregation of other oxides only to a very limited extent. It was therefore possible to count on the fact that the real diffusion coefficients of zirconium and hafnium in the successively forming crystallites were very small (i.e. *D*Zr *→* 0 and *D*Hf *→* 0). On the other hand, the very close values of the parameters *α/θ = D/L*2 of the remaining seven analyzed oxides:

$$\text{D/L}\_{\text{B}} = (6.51 \pm 0.25).104 \quad \text{and} \quad \text{D/L}\_{\text{C}} = (2.45 \pm 0.12).104 \quad \text{[s}^{\text{-1}}\text{]}\tag{14.15}$$

indicated that the redistribution of these oxides between the melt and the solid state ran in a way, similar to that within metal alloys, namely steels.

It would be possible to count – in the first approximation – the diffusion coefficients of the oxides in the slag having the temperatures of 1765ºC (solidus) and 1775º C (liquidus), the average value of *D* = (2.07±0.11) x 10-6 cm2/s (the data referred to the diffusion of aluminum in the slag of a composition of 39% CaO-20% Al2O3-41% SiO2). For these cases, and using Equation (13), it was possible to get the magnitude of the structure parameters that governed the chemical heterogeneity of the values:

$$\begin{aligned} \text{L}\_{\text{B}} &= \sqrt{\left[ \left( 2.07 \times 10^{-6} \right) / \left( 6.51 \times 10^{-4} \right) \right]} = 0.05639\\ \text{L}\_{\text{C}} &= \sqrt{\left[ \left( 2.07 \times 10^{-6} \right) / \left( 2.45 \times 10^{-4} \right) \right]} = 0.09192 \end{aligned} \tag{16.17}$$

It corresponded to 564 μm in the sample B (which was taken from the edge of the casting block) and 919 μm in the sample C (which was taken from underneath the riser of the same casting block). The comparison of the micro-structures of the analyses samples B and C (Figures 6a,b) has clearly shown that the sample B micro-structure (*L*B) was significantly finer than the micro-structure of the sample C (*L*C), which semi-quantitatively corresponded to the qualified estimate of the structure parameters *L*, conducted on the basis of calculations using the data obtained from both models.

Fig. 6. a)The structure of the sample B (Lb = 564 μm), b) The structure of the ample C (*L*c = 919 μm)

## **4. Continuously casting**

76 Crystallization – Science and Technology

Oxide Sample B: **α** α/θ<sup>B</sup> · 104 [1/s] Sample C: **α** α/θ<sup>C</sup> · 104 [1/s] Na2O 0.0732 6.53 0.0691 2.44 Al2O3 0.0674 6.01 0.0662 2.34 SiO2 0.0741 6.61 0.0663 2.34 ZrO2 0.00035 0.0312 0.00008 0.0028 K2O 0.0721 6.43 0.0665 2.35 CaO 0.075 6.69 0.0703 2.48 TiO2 0.0759 6.77 0.0757 2.67 Fe2O3 0.0732 6.53 0.0711 2.51 HfO2 0.0165 1.47 0.00017 0.006

It comes as a surprise that the values of the parameter *α/θ = D/L*2 of the oxides of elements Na, Al, Si, K, Ca, Ti, and Fe differed by as much as an order from the value of the same parameter of the oxide of zirconium and hafnium. This could be explained by the fact that zirconium contains hafnium as an additive and, therefore, they segregate together and the forming oxides of zirconium and hafnium show the highest melting temperatures. From the melt, both oxides segregated first, already in their solid states. Further redistribution of the oxides of both elements ran on the interface of the remaining melt and the successive segregation of other oxides only to a very limited extent. It was therefore possible to count on the fact that the real diffusion coefficients of zirconium and hafnium in the successively forming crystallites were very small (i.e. *D*Zr *→* 0 and *D*Hf *→* 0). On the other hand, the very

 D/LB2 = (6.51±0.25).10-4 and D/LC2 = (2.45±0.12).10-4 [s-1] (14,15) indicated that the redistribution of these oxides between the melt and the solid state ran in a

It would be possible to count – in the first approximation – the diffusion coefficients of the

average value of *D* = (2.07±0.11) x 10-6 cm2/s (the data referred to the diffusion of aluminum in the slag of a composition of 39% CaO-20% Al2O3-41% SiO2). For these cases, and using Equation (13), it was possible to get the magnitude of the structure parameters that

C (liquidus), the

[cm] (16,17)

close values of the parameters *α/θ = D/L*2 of the remaining seven analyzed oxides:

oxides in the slag having the temperatures of 1765ºC (solidus) and 1775º

 

L 2.07 x10 / 6.51x10 0.05639

L 2,07 x10 / 2.45x10 0.09192

6 4

6 4

It corresponded to 564 μm in the sample B (which was taken from the edge of the casting block) and 919 μm in the sample C (which was taken from underneath the riser of the same casting block). The comparison of the micro-structures of the analyses samples B and C (Figures 6a,b) has clearly shown that the sample B micro-structure (*L*B) was significantly finer than the micro-structure of the sample C (*L*C), which semi-quantitatively corresponded to the qualified estimate of the structure parameters *L*, conducted on the basis of calculations

Table 2. Calculated values of the equation (13)

way, similar to that within metal alloys, namely steels.

governed the chemical heterogeneity of the values:

B

C

using the data obtained from both models.

#### **4.1 Chemical microheterogeneity of continuously cast steel slab**

Structure of metallic alloys is one of the factors, which significantly influence their physical and mechanical properties. Formation of structure is strongly affected by production technology, casting and solidification of these alloys. Solidification is a critical factor in the materials industry (Kavicka et al., 2007). Solute segregation either on the macro- or microscale is sometimes the cause of unacceptable products due to poor mechanical properties of the resulting non-equilibrium phases. In the areas of more important solute segregation there occurs weakening of bonds between atoms and mechanical properties of material degrade. Heterogeneity of distribution of components is a function of solubility in solid and liquid phases. During solidification a solute can concentrate in inter-dendritic areas above the value of its maximum solubility in solid phase. Solute diffusion in solid phase is a limiting factor for this process, since diffusion coefficient in solid phase is lower by three up to five orders than in the melt. When analysing solidification of steel so far no unified theoretical model was created, which would describe this complex heterogeneous process as a whole. During the last fifty years many approaches with more or less limiting assumptions were developed. Solidification models and simulations have been carried out for both macroscopic and microscopic scales. The most elaborate numerical models can predict micro-segregation with comparatively high precision. The main limiting factor of all existing mathematical micro-segregation models consists in lack of available thermodynamic and kinetic data, especially for systems of higher orders. There is also little experimental data to check the models. Many authors deal with issues related to modelling of a non-equilibrium crystallisation of alloys. However, majority of the presented works concentrates mainly on investigation of modelling of micro-segregation of binary alloys, or on segregation of elements for special cases of crystallisation – directional solidification, zonal melting, onedimensional thermal field, etc. Moreover these models work with highly limiting assumption concerning phase diagrams (constant distribution coefficients) and development of dendritic morphology (mostly one-dimensional models of dendrites. Comprehensive studies of solidification for higher order real alloys are rarer. Nevertheless, there is a strong industrial need to investigate and simulate more complex alloys because

Numerical Models of Crystallization

and Its Direction for Metal and Ceramic Materials in Technical Application 79

Fig. 9. Example of structure of the sample 21 with a microscopic trace of 1000 µm long

Analytical complex unit JEOL JXA 8600/KEVEX Delta V Sesame was used for determination of concentration distribution of elements, and concentration was determined by method of energy dispersive X-ray spectral micro-analysis. As an example, Figures 10 a,b

Chemical micro-heterogeneity, i.e. segregation of individual elements at distances, order of which is comparable to dendrite arms spacing, can be quantitatively evaluated from the

Fig. 8. Scheme of sampling from a slab and marking of samples

present t*he basic concentration spectrum* of Mn, Si, P and S.

 (a) (b) Fig. 10. Sample 21. Basic concentration spectrum a) of Mn and Si, b) of S and P

nearly all current commercial alloys have many components often exceeding ten elements. Moreover, computer simulation have shown that even minute amounts of alloying elements can significantly influence microstructure and micro-segregation and cannot be neglected.

#### **4.1.1 Methodology of chemical heterogeneity investigation**

Original approach to determination of chemical heterogeneity in structure of polycomponent system is based on experimental measurements made on samples taken from characteristic places of the casting, which were specified in advance. Next procedure is based on statistical processing of concentration data sets and application of the original mathematical model for determination of distribution curves of dendritic segregation of elements, characterising the most probable distribution of concentration of element in the frame of dendrite (Dobrovska et al., 2009), and the original mathematical model for determination of effective distribution coefficients of these elements in the analysed alloy.

#### **4.1.2 Application of methodology of chemical heterogeneity investigation – investigation into chemical micro-heterogeneity of CC steel slab**

A continuously cast steel slab (CC steel slab, Figure 7 ) with dimensions 1530x250 mm was chosen for presentation of results, with the following chemical composition in (wt. %): 0.14C; 0.75Mn; 0.23Si; 0.016P; 0.010S; 0.10Cr; 0.050Cu; 0.033Altotal.

After solidification and cooling of the cast slab a transversal band was cut out, which was then axially divided into halves. Nine samples were taken from one half for determination of chemical heterogeneity according to the diagram in Figure 8. The samples had a form of a cube with an edge of approx. 20 mm, with recorded orientation of its original position in the CC slab. Figure 9 shows an example of microstructure of the analysed slab. On each sample a concentration of seven elements (*aluminium, silicon, phosphor, sulphur, titanium, chromium and manganese*) were measured along the line segment long 1000 μm. The distance between the measured points was 10 μm.

Fig. 7. The steel slab caster

nearly all current commercial alloys have many components often exceeding ten elements. Moreover, computer simulation have shown that even minute amounts of alloying elements can significantly influence microstructure and micro-segregation and cannot be neglected.

Original approach to determination of chemical heterogeneity in structure of polycomponent system is based on experimental measurements made on samples taken from characteristic places of the casting, which were specified in advance. Next procedure is based on statistical processing of concentration data sets and application of the original mathematical model for determination of distribution curves of dendritic segregation of elements, characterising the most probable distribution of concentration of element in the frame of dendrite (Dobrovska et al., 2009), and the original mathematical model for determination of effective distribution coefficients of these elements in the analysed alloy.

A continuously cast steel slab (CC steel slab, Figure 7 ) with dimensions 1530x250 mm was chosen for presentation of results, with the following chemical composition in (wt. %):

After solidification and cooling of the cast slab a transversal band was cut out, which was then axially divided into halves. Nine samples were taken from one half for determination of chemical heterogeneity according to the diagram in Figure 8. The samples had a form of a cube with an edge of approx. 20 mm, with recorded orientation of its original position in the CC slab. Figure 9 shows an example of microstructure of the analysed slab. On each sample a concentration of seven elements (*aluminium, silicon, phosphor, sulphur, titanium, chromium and manganese*) were measured along the line segment long 1000 μm. The distance between

**4.1.2 Application of methodology of chemical heterogeneity investigation –** 

**investigation into chemical micro-heterogeneity of CC steel slab** 

0.14C; 0.75Mn; 0.23Si; 0.016P; 0.010S; 0.10Cr; 0.050Cu; 0.033Altotal.

the measured points was 10 μm.

Fig. 7. The steel slab caster

**4.1.1 Methodology of chemical heterogeneity investigation** 

Fig. 8. Scheme of sampling from a slab and marking of samples

Fig. 9. Example of structure of the sample 21 with a microscopic trace of 1000 µm long

Analytical complex unit JEOL JXA 8600/KEVEX Delta V Sesame was used for determination of concentration distribution of elements, and concentration was determined by method of energy dispersive X-ray spectral micro-analysis. As an example, Figures 10 a,b present t*he basic concentration spectrum* of Mn, Si, P and S.

Fig. 10. Sample 21. Basic concentration spectrum a) of Mn and Si, b) of S and P

Chemical micro-heterogeneity, i.e. segregation of individual elements at distances, order of which is comparable to dendrite arms spacing, can be quantitatively evaluated from the

Numerical Models of Crystallization

ability.

**<sup>11</sup>***IH*

**<sup>12</sup>***IH*

**<sup>13</sup>***IH*

**<sup>21</sup>***IH*

**<sup>22</sup>***IH*

**<sup>23</sup>***IH*

**<sup>31</sup>***IH*

**<sup>32</sup>***IH*

**<sup>33</sup>***IH*

*kef*

*kef*

*kef*

*kef*

*kef*

*kef*

*kef*

*kef*

*kef*

and Its Direction for Metal and Ceramic Materials in Technical Application 81

where *Ci* is the concentration in *i*-th point of the sequence (i.e. in the *i*-th point of the curve in Figures 11 a, b) and *CR*(*i*) is the average concentration of the element in the residual part of

> <sup>n</sup> R j

C (i) 1 /(n i 1) C

where *n* was the number of the measured points. In this way it was possible to determine the values of effective distribution coefficients for all *i* 1, *n*, i.e. for the entire curve characterising the segregation during solidification. The effective distribution coefficients of all the analysed elements were calculated by this original method. The average values of determined effective distribution coefficients are listed in Table 3. No segregation occurs when *k*ef = 1; the higher is the deviation from the number 1, the higher is the segregation

The effective distribution coefficients calculated in this way inherently include in themselves both the effect of segregation in the course of alloy solidification and the effect of homogenisation, occurring during the solidification as well as during the cooling of alloy. Average values of measured and calculated quantities in the set of samples are in Table 4.

> 1.22 0.33

> 1.12 0.36

> 1.25 0.32

> 1.58 0.24

> 1.31 0.30

> 1.34 0.29

> 1.22 0.33

> 1.16 0.34

> 1.24 0.32

Table 3. The average values of the heterogeneity index *IH* and the effective distribution

j 1

**Al Si P S Ti Cr Mn**

1.45 0.26

1.74 0.20

1.48 0.26

1.49 0.25

1.41 0.27

1.86 0.18

2.34 0.18

1.49 0.25

1.64 0.22 0.30 0.76

0.29 0.78

0.30 0.77

0.31 0.76

0.30 0.77

0.26 0.80

0.31 0.76

0.34 0.74

0.35 0.74 0.22 0.83

0.27 0.79

0.29 0.78

0.24 0.81

0.26 0.80

0.28 0.78

0.23 0.82

0.25 0.80

0.26 0.80 0.14 0.88

0.15 0.88

0.15 0.88

0.13 0.89

0.14 0.88

0.13 0.89

0.16 0.87

0.14 0.88

0.13 0.89

k (i) C /C (i) ef i R (19)

(20)

It was therefore possible to substitute the equation (18) by the formula

the curve (i.e. for *f*<sup>S</sup> *i*, 1), expressed by the relation:

Sample Element

0.28 0.78

0.30 0.77

0.30 0.78

0.29 0.78

0.28 0.78

0.29 0.78

0.28 0.78

0.27 0.79

0.29 0.78

1.24 0.32

1.54 0.24

1.44 0.27

1.33 0.29

1.14 0.35

1.56 0.24

1.11 0.37

1.44 0.27

1.32 0.30

coefficient *kef* of elements in the individual samples

basic statistical parameters of the measured concentrations of elements in individual samples. These parameters comprise: *Cx* average concentration of element (arithmetic average) in the selected section, *sx* standard deviation of the measured concentration of element, *Cmin* minimum concentration of element and *Cmax* maximum concentration of element measured always on the selected section of the sample. It is possible to calculate from these data moreover *indexes of dendritic heterogeneity IH* of elements in the measured section of individual samples as ratio of standard deviation *sx* and average concentration *Cx* of the element. Then the element distribution profiles can be plotted according to the Gungor's method (Gungor,1989) from the concentration data sets measured by the method ED along the line segment 1000 m long. Data plotted as the measured weight percent composition versus number of data (Figures 10 a,b) were put in an ascending or descending order and *x*-axis was converted to the fraction solid (*fS≈gs* in Equation 3) by dividing each measured data number by total measured data number. The element composition versus fraction solid, i.e. element distribution profile (*distribution curve of dendritic segregation*) was then plotted; Figures 11 a,b represent such dependences for manganese, silicon, phosphor and sulphur. The slope of such curve (ascending or descending) depended on the fact, whether the element in question enriched the dendrite core or the inter-dendritic area in the course of solidification.

From these statistical data it is also possible to determine with use of original mathematical model for each analysed element from the given set of samples the values of *effective distribution coefficients kef*. The procedure of the effective distribution coefficient calculation will be outlined here as follows:

The sequence of such arranged concentrations (Figures 11 a,b) was seen as a distribution of concentrations of the measured element in the direction from the axis (*f*S = 0) to the boundary (*f*S = 1) of one average dendrite.

Fig. 11. Experimentally determined distribution curve of dendritic segregation (sample 21) a) for Mn and Si ,b) for P and S

The effective distribution coefficient *k*ef was in this case defined by the relation

$$\mathbf{k}\_{\rm ef}(\mathbf{f}\_{\rm s}) = \mathbf{C}\_{\rm s}(\mathbf{f}\_{\rm s}) / \,\mathbf{C}\_{\rm L}(\mathbf{f}\_{\rm s}) \tag{18}$$

where *CS* is the solute concentration in the solidus and *CL* is its concentration in liquidus and argument (*fS*) expressed the dependence of both concentrations on the fraction solid. A perfect mixing of an element in the interdendritic melt was then assumed (this assumption.

basic statistical parameters of the measured concentrations of elements in individual samples. These parameters comprise: *Cx* average concentration of element (arithmetic average) in the selected section, *sx* standard deviation of the measured concentration of element, *Cmin* minimum concentration of element and *Cmax* maximum concentration of element measured always on the selected section of the sample. It is possible to calculate from these data moreover *indexes of dendritic heterogeneity IH* of elements in the measured section of individual samples as ratio of standard deviation *sx* and average concentration *Cx* of the element. Then the element distribution profiles can be plotted according to the Gungor's method (Gungor,1989) from the concentration data sets measured by the method ED along the line segment 1000 m long. Data plotted as the measured weight percent composition versus number of data (Figures 10 a,b) were put in an ascending or descending order and *x*-axis was converted to the fraction solid (*fS≈gs* in Equation 3) by dividing each measured data number by total measured data number. The element composition versus fraction solid, i.e. element distribution profile (*distribution curve of dendritic segregation*) was then plotted; Figures 11 a,b represent such dependences for manganese, silicon, phosphor and sulphur. The slope of such curve (ascending or descending) depended on the fact, whether the element in question

enriched the dendrite core or the inter-dendritic area in the course of solidification.

(a) (b)

The effective distribution coefficient *k*ef was in this case defined by the relation

will be outlined here as follows:

a) for Mn and Si ,b) for P and S

boundary (*f*S = 1) of one average dendrite.

From these statistical data it is also possible to determine with use of original mathematical model for each analysed element from the given set of samples the values of *effective distribution coefficients kef*. The procedure of the effective distribution coefficient calculation

The sequence of such arranged concentrations (Figures 11 a,b) was seen as a distribution of concentrations of the measured element in the direction from the axis (*f*S = 0) to the

Fig. 11. Experimentally determined distribution curve of dendritic segregation (sample 21)

where *CS* is the solute concentration in the solidus and *CL* is its concentration in liquidus and argument (*fS*) expressed the dependence of both concentrations on the fraction solid. A perfect mixing of an element in the interdendritic melt was then assumed (this assumption.

k (f ) C (f )/C (f ) ef s s s L s (18)

It was therefore possible to substitute the equation (18) by the formula

$$\mathbf{k}\_{\text{ef}}(\mathbf{i}) = \mathbf{C}\_{\text{i}} \;/\,\mathbf{C}\_{\text{R}}(\mathbf{i})\tag{19}$$

where *Ci* is the concentration in *i*-th point of the sequence (i.e. in the *i*-th point of the curve in Figures 11 a, b) and *CR*(*i*) is the average concentration of the element in the residual part of the curve (i.e. for *f*<sup>S</sup> *i*, 1), expressed by the relation:

$$\mathbf{C}\_{\rm R}(\mathbf{i}) = \left[ \mathbf{1} \;/\left(\mathbf{n} - \mathbf{i} + \mathbf{1}\right) \right] \sum\_{j=1}^{n} \mathbf{C}\_{j} \tag{20}$$

where *n* was the number of the measured points. In this way it was possible to determine the values of effective distribution coefficients for all *i* 1, *n*, i.e. for the entire curve characterising the segregation during solidification. The effective distribution coefficients of all the analysed elements were calculated by this original method. The average values of determined effective distribution coefficients are listed in Table 3. No segregation occurs when *k*ef = 1; the higher is the deviation from the number 1, the higher is the segregation ability.

The effective distribution coefficients calculated in this way inherently include in themselves both the effect of segregation in the course of alloy solidification and the effect of homogenisation, occurring during the solidification as well as during the cooling of alloy.

Average values of measured and calculated quantities in the set of samples are in Table 4.


Table 3. The average values of the heterogeneity index *IH* and the effective distribution coefficient *kef* of elements in the individual samples

Numerical Models of Crystallization

samples in Table 4.

cooling of alloy.

billet casting the frequency from 5 to 50 Hz.

and Its Direction for Metal and Ceramic Materials in Technical Application 83

Average value of this coefficient for all the analysed elements and the whole set of nine samples is given in Table 4. It follows from this table that dendritic heterogeneity of slab decreases in this order of elements: sulphur, aluminium, phosphor, titanium, silicon, chromium and manganese, which has the lowest index of heterogeneity. Dendritic heterogeneity of the analysed elements is expressed also by the values of their effective distribution coefficients, arranged for individual samples in Table 3 and for the set of

It is obvious from the tables that pair values of the index of dendritic heterogeneity and effective distribution coefficient for the same element do mutually correspond. The higher the value of the heterogeneity index, the lower the value of effective distribution coefficient and vice versa. The lowest value of the effective distribution coefficient is found in sulphur and the highest value is found in manganese. It follows from the Table 4, that effective distribution coefficient increases in this order of elements: sulphur, aluminium, phosphor, titanium, silicon, chromium and manganese. All the analysed elements segregate during solidification into an inter-dendritic melt, and their distribution coefficient is smaller than one. For comparison, the Table 4 contains also the values of distribution coefficients found in literature. It is obvious that our values of effective distribution coefficients, calculated according to the original model, are in good agreement with the data from literature, only with the exception of sulphur (and titanium). The reason for this difference is probably the means of calculation of the effective distribution coefficient – the value of this parameter is calculated from concentration data set measured on solidified and cooled casting. Consequently, the effective distribution coefficients calculated in this way inherently include in themselves both the effect of segregation in the course of alloy solidification and the effect of homogenisation, occurring during the solidification as well as during the

**4.2 Effect of elelectromagnetic stirring on the dendritic structure of steel billets** 

Currently, casters use rotating stators of electromagnetic melt-stirring systems. These stators create a rotating magnetic induction field with an induction of **B**, which induces eddycurrent **J** in a direction perpendicular to **B**, whose velocity is **ν**. Induction **B** and current **J** create an electromagnetic force, which works on every unit of volume of steel and brings about a stirring motion in the melt. The vector product (**ν B**) demonstrates a connection between the electromagnetic field and the flow of the melt. The speeds of the liquid steel caused by the elelectromagnetic stirring is somewhere from 0.1-to-1.0 m/s. The stirring parameters are within a broad range of values, depending on the construction and technological application of the stirrer. The power output is mostly between 100 and 800 kW, the electric current between 300 and 1000 A, the voltage up to 400 V and with

The elelectromagnetic stirring applied on the steel caster is basically a magneto-hydraulic process together with crystallisation processes and solidification of billet steel. The complexity of the entire process is enhanced further by the fact that the temperatures are higher than the casting temperatures of concast steel. The temperature of the billet gradually decreases as it passes through the caster down to a temperature lying far below


Table 4. Average values of the measured and calculated quantities in the set of all samples

Data represented in Table 3 and Table 4 make it possible to evaluate dendritic heterogeneity (micro-heterogeneity) of elements, as well as their effective distribution coefficients in individual samples, and also in the frame of the whole analysed half of the slab crosssection. It is obvious from these tables that dendritic heterogeneity of accompanying elements and impurities is comparatively high. This is demonstrated by the index of dendritic heterogeneity *IH*. It follows from Table 3, that distinct differences exist between micro-heterogeneity of individual elements. Figures 12 a, b show distribution of indexes of micro-heterogeneity of sulphur (the most segregating element) and manganese (the least segregating element) on slab cross-section.

Fig. 12. a) Differences in sulphur micro-heterogeneity in samples taken from one-half of slab cross-section. b) Differences in manganese micro-heterogeneity in samples taken from onehalf of slab cross-section.

*k*(ref)

0.046 0.12 –0.92

0.005 0.66 –0.91

0.035 0.06 –0.50

0.035 0.02 –0.10

0.019 0.05 –0.60

0.017 0.30 –0.97

0.033 0.72 –0.90

according to

Dobrovska et al., 2009

*kef*  ± *sk*

0.294

0.781

0.314

0.232

0.765

0.799

0.873

Table 4. Average values of the measured and calculated quantities in the set of all samples

Data represented in Table 3 and Table 4 make it possible to evaluate dendritic heterogeneity (micro-heterogeneity) of elements, as well as their effective distribution coefficients in individual samples, and also in the frame of the whole analysed half of the slab crosssection. It is obvious from these tables that dendritic heterogeneity of accompanying elements and impurities is comparatively high. This is demonstrated by the index of dendritic heterogeneity *IH*. It follows from Table 3, that distinct differences exist between micro-heterogeneity of individual elements. Figures 12 a, b show distribution of indexes of micro-heterogeneity of sulphur (the most segregating element) and manganese (the least

*IH*  ± *sI*

1.352 0.162

0.285 0.011

1.270 0.133

1.657 0.297

0.306 0.027

0.255 0.023

0.143 0.009

(a) (b)

Fig. 12. a) Differences in sulphur micro-heterogeneity in samples taken from one-half of slab cross-section. b) Differences in manganese micro-heterogeneity in samples taken from one-

*cx*  ± *sx*

0.0029

0.0068

0.0023

0.0030

0.0032

0.0076

0.0169

half of slab cross-section.

segregating element) on slab cross-section.

Al 0.0136

Si 0.1910

<sup>P</sup>0.0141

<sup>S</sup>0.0136

Ti 0.0951

Cr 0.1758

Mn 0.8232

Average value of this coefficient for all the analysed elements and the whole set of nine samples is given in Table 4. It follows from this table that dendritic heterogeneity of slab decreases in this order of elements: sulphur, aluminium, phosphor, titanium, silicon, chromium and manganese, which has the lowest index of heterogeneity. Dendritic heterogeneity of the analysed elements is expressed also by the values of their effective distribution coefficients, arranged for individual samples in Table 3 and for the set of samples in Table 4.

It is obvious from the tables that pair values of the index of dendritic heterogeneity and effective distribution coefficient for the same element do mutually correspond. The higher the value of the heterogeneity index, the lower the value of effective distribution coefficient and vice versa. The lowest value of the effective distribution coefficient is found in sulphur and the highest value is found in manganese. It follows from the Table 4, that effective distribution coefficient increases in this order of elements: sulphur, aluminium, phosphor, titanium, silicon, chromium and manganese. All the analysed elements segregate during solidification into an inter-dendritic melt, and their distribution coefficient is smaller than one. For comparison, the Table 4 contains also the values of distribution coefficients found in literature. It is obvious that our values of effective distribution coefficients, calculated according to the original model, are in good agreement with the data from literature, only with the exception of sulphur (and titanium). The reason for this difference is probably the means of calculation of the effective distribution coefficient – the value of this parameter is calculated from concentration data set measured on solidified and cooled casting. Consequently, the effective distribution coefficients calculated in this way inherently include in themselves both the effect of segregation in the course of alloy solidification and the effect of homogenisation, occurring during the solidification as well as during the cooling of alloy.

## **4.2 Effect of elelectromagnetic stirring on the dendritic structure of steel billets**

Currently, casters use rotating stators of electromagnetic melt-stirring systems. These stators create a rotating magnetic induction field with an induction of **B**, which induces eddycurrent **J** in a direction perpendicular to **B**, whose velocity is **ν**. Induction **B** and current **J** create an electromagnetic force, which works on every unit of volume of steel and brings about a stirring motion in the melt. The vector product (**ν B**) demonstrates a connection between the electromagnetic field and the flow of the melt. The speeds of the liquid steel caused by the elelectromagnetic stirring is somewhere from 0.1-to-1.0 m/s. The stirring parameters are within a broad range of values, depending on the construction and technological application of the stirrer. The power output is mostly between 100 and 800 kW, the electric current between 300 and 1000 A, the voltage up to 400 V and with billet casting the frequency from 5 to 50 Hz.

The elelectromagnetic stirring applied on the steel caster is basically a magneto-hydraulic process together with crystallisation processes and solidification of billet steel. The complexity of the entire process is enhanced further by the fact that the temperatures are higher than the casting temperatures of concast steel. The temperature of the billet gradually decreases as it passes through the caster down to a temperature lying far below

Numerical Models of Crystallization

150x150 mm

**4.2.2 The experiment** 

chemical composition (Table 5).

and Its Direction for Metal and Ceramic Materials in Technical Application 85

Fig. 14. The temperature history of marked points of the cross-section of the steel billet

Fig. 15. Computed iso-solidus and iso-liquidus curves in the axial longitudinal section

The first stirrer (MEMS) stirs the melt still in the mould while the billet is undergoing crystallization and solidification. The second stirrer (SEMS) works at a time when the melt is already enclosed by a shell of crystallites around the perimeter of the billet and inside the billet there is less melt than above in the active zone of the first stirrer. When both stirrers were switched off, the crystallisation and solidification continued in the normal way, i.e. the solidifying melt did not undergo a forced rotational movement. Samples were taken throughout the course of the experiment – from parts of the billet cast using the MEMS and SEMS and without and also using either one. The samples were taken in the form of crosssections (i.e. perpendicular to the billet axis). The samples were fine-ground and etched with the aim of making visible the dendritic structure which is characteristic for individual variants of the solidification of the billet. The verification of the influence of MEMS and SEMS on the macrostructure of the billet was carried out on two melts of almost the same

the solidus temperature. From the viewpoint of physics and chemistry, the course of the process is co-determined by a number of relevant material, physical and thermokinetic characteristics of the concast steel and also electrical and magnetic quantities. There is also a wide range of construction and function parameters pertaining to the caster and elelectromagnetic stirring as well as parameters relating to their mutual arrangement and synchronisation. Numerous works from recent years relate that exact mathematical modelling of elelectromagnetic stirring on a caster is still unsolvable (Stransky et al., 2009).

The basic elelectromagnetic stirring experiment was conducted on a continuously steel billet caster where two individual mixers were working (Figure 13). The first stirrer, entitled MEMS (Mould Electromagnetic Stirring), is mounted directly on the mould and the second stirrer, entitled SEMS (Strand Electromagnetic Stirring), is mounted at the beginning of the flow directly after the first cooling zones but in the secondary-cooling zone. Here the outer structure of the billet is already created by a compact layer of crystallites, however, in the centre of the billet there is still a significant amount of melt that is mixed by the SEMS.

### **4.2.1 The temperature field of a billet**

The temperature field of the billet of 150x150 mm computed via original numerical model (Stransky et al., 2009,2011) is in Figures 14-15.

Fig. 14. The temperature history of marked points of the cross-section of the steel billet 150x150 mm

Fig. 15. Computed iso-solidus and iso-liquidus curves in the axial longitudinal section

## **4.2.2 The experiment**

84 Crystallization – Science and Technology

the solidus temperature. From the viewpoint of physics and chemistry, the course of the process is co-determined by a number of relevant material, physical and thermokinetic characteristics of the concast steel and also electrical and magnetic quantities. There is also a wide range of construction and function parameters pertaining to the caster and elelectromagnetic stirring as well as parameters relating to their mutual arrangement and synchronisation. Numerous works from recent years relate that exact mathematical modelling of elelectromagnetic stirring on a caster is still unsolvable (Stransky et al.,

The basic elelectromagnetic stirring experiment was conducted on a continuously steel billet caster where two individual mixers were working (Figure 13). The first stirrer, entitled MEMS (Mould Electromagnetic Stirring), is mounted directly on the mould and the second stirrer, entitled SEMS (Strand Electromagnetic Stirring), is mounted at the beginning of the flow directly after the first cooling zones but in the secondary-cooling zone. Here the outer structure of the billet is already created by a compact layer of crystallites, however, in the centre of the billet there is still a significant amount of melt

Fig. 13. The steel billet caster of 150x150 mm. The positions of the MEMS and SEMS stirrers

The temperature field of the billet of 150x150 mm computed via original numerical model

2009).

that is mixed by the SEMS.

**4.2.1 The temperature field of a billet** 

(Stransky et al., 2009,2011) is in Figures 14-15.

The first stirrer (MEMS) stirs the melt still in the mould while the billet is undergoing crystallization and solidification. The second stirrer (SEMS) works at a time when the melt is already enclosed by a shell of crystallites around the perimeter of the billet and inside the billet there is less melt than above in the active zone of the first stirrer. When both stirrers were switched off, the crystallisation and solidification continued in the normal way, i.e. the solidifying melt did not undergo a forced rotational movement. Samples were taken throughout the course of the experiment – from parts of the billet cast using the MEMS and SEMS and without and also using either one. The samples were taken in the form of crosssections (i.e. perpendicular to the billet axis). The samples were fine-ground and etched with the aim of making visible the dendritic structure which is characteristic for individual variants of the solidification of the billet. The verification of the influence of MEMS and SEMS on the macrostructure of the billet was carried out on two melts of almost the same chemical composition (Table 5).

Numerical Models of Crystallization

mode 4A (Table 6)

and 2B).

and Its Direction for Metal and Ceramic Materials in Technical Application 87

(a) (b)

Fig. 16. a) Dendrite growth in the concasting structure without elelectromagnetic stirring – mode 2A, b) The growth of dendrites in the billet structure using the MEMS and SEMS –

The above-described mechanism of dendrite growth during concasting without stirring is frequently the object of interest (Figure 16a). Inside the billets, when using the MEMS stirrer (or both MEMS and SEMS), the kinetics of solidification and dendrite growth is initially the same as without stirring. This also creates columnar dendrites which touch along the diagonals, however, soon their growth ceases still near the surface. Dendrites, which are called equiaxed dendrites continue to grow – their orientation is more random and only partly directed towards the centre of the billet (Figure 16b). It appears that this dendrite growth mechanism manifests itself the most when both stirrers are working simultaneously (Table 6: 4A, 4B and 5B). If MEMS and SEMS are working simultaneously, the stirring effect significantly destroys the formation of columnar crystals. If only MEMS is working and SEMS is switched off (1A and 1B), then the desctruction of columnar crystals is less evident. The working mode of SEMS alone (modes 3A and 3B) cannot be clearly differentiated from the changes in the dendritic structure in relation to the structure formed without stirring (2A

Figure 16b (the macro-ground dendritic structure) shows the depth of the columnar band of dendrites in the direction away from the surface of the billet (Figure 16b – see arrows) and its value, which (with the simultaneous stirring of MEMS and SEMS) is 23.41.8 mm. The same qualified guess was made for ordinary billet casting (i.e. without stirring). Here, the depth of the dendrites can be guessed almost all the way to the central shrinkage at 70 mm (Figure 16a – see arrows). It is known that additives and impurities during solidification are often concentrated in points of contact of the growing dendrites, where the maximum of segregated additives and impurities and the greatest probability of technological defects occurs. In the given case, this undesirable effect can be expected along the diagonals which have a length of up to 100-to-103 mm towards the central shrinkage. This point of contact of the dendrites during the simultaneous working of SEMS and MEMS is only 29.81.9 mm,


Table 5. Chemical composition of experimental melts [wt.%]

The timing of the concasting process of the billets – without the involvement of the stirrers and with the working of the elelectromagnetic stirring of individual variants of stirrers (MEMS and SEMS) – is given in Table 6. The speed of the concasting (i.e. the movement, the proceeding of the billet through the mould) of the billet was maintained constant during the experimentation at a value of 2.7 m/min. Table 6 shows that as many as nine concasting variants were verified. The lengths of individual experimental billets – from which samples had been taken – were always a multiple of the metallurgical length. The average superheating of the steel above the liquidus was 32.8 ± 3.1 °C in melt A and 28.0 ± 4.6 °C in melt B, which lies within the standard deviation of the temperature measurements.


Table 6. The billet concasting modes and sampling

Note: Detailed records of the experimental verification of the effects of MEMS and SEMS during concasting on the relevant device pertain to Table 6. The data are appended with a time history of the MEMS and SEMS connection and with information relating to the lengths of individual billets and the points from which the actual samples had been taken (i.e. the cross-sections from which the dendritic structures had been created). Evaluation of all nine variants of concasting (Table 6) indicates that the arrangement of dendrites in the crosssection follow the same tendency in the first phase of crystallization. The structure is created by columnar crystals – dendrites – perpendicular to the walls of the billet (Figure 16a).

In the billets that were not stirred the dendrites gradually touch one another on the diagonals of the cross-section. Here their growth either ceases, or the dendrites bend in the directions of the diagonals and their growth continues all the way to the centre of the billet. The columnar dendrites that grow from the middle part of the surface maintain their basic orientation – perpendicular to the surface – almost all the way to the centre of the billet. In the central part of the cross-section there is an obvious hollow on all nine macroscopic images. This is most probably a shrinkage.

The timing of the concasting process of the billets – without the involvement of the stirrers and with the working of the elelectromagnetic stirring of individual variants of stirrers (MEMS and SEMS) – is given in Table 6. The speed of the concasting (i.e. the movement, the proceeding of the billet through the mould) of the billet was maintained constant during the experimentation at a value of 2.7 m/min. Table 6 shows that as many as nine concasting variants were verified. The lengths of individual experimental billets – from which samples had been taken – were always a multiple of the metallurgical length. The average superheating of the steel above the liquidus was 32.8 ± 3.1 °C in melt A and 28.0 ± 4.6 °C in

> MEMS stirring [Amperes]

SEMS stirring [Amperes] Fig.

melt B, which lies within the standard deviation of the temperature measurements.

2A 31 0 0 Fig. 16a

4A 30 210 57 Fig. 16b

Note: Detailed records of the experimental verification of the effects of MEMS and SEMS during concasting on the relevant device pertain to Table 6. The data are appended with a time history of the MEMS and SEMS connection and with information relating to the lengths of individual billets and the points from which the actual samples had been taken (i.e. the cross-sections from which the dendritic structures had been created). Evaluation of all nine variants of concasting (Table 6) indicates that the arrangement of dendrites in the crosssection follow the same tendency in the first phase of crystallization. The structure is created by columnar crystals – dendrites – perpendicular to the walls of the billet (Figure 16a).

In the billets that were not stirred the dendrites gradually touch one another on the diagonals of the cross-section. Here their growth either ceases, or the dendrites bend in the directions of the diagonals and their growth continues all the way to the centre of the billet. The columnar dendrites that grow from the middle part of the surface maintain their basic orientation – perpendicular to the surface – almost all the way to the centre of the billet. In the central part of the cross-section there is an obvious hollow on all nine macroscopic

Superheating of steel above liquidus [°C]

A 1A 37 210 0

3A 33 0 29

B 1B 35 210 0 2B 30 0 0 3B 27 0 57 4B 24 210 57 5B 24 210 29

Table 6. The billet concasting modes and sampling

images. This is most probably a shrinkage.

Melt C Mn Si P S Cu Cr Ni Al Ti A 0.14 0.31 0.22 0.014 0.009 0.03 0.05 0.02 0.02 0.002 B 0.13 0.32 0.22 0.018 0.012 0.09 0.06 0.04 0.02 0.002

Table 5. Chemical composition of experimental melts [wt.%]

Melt Concasting mode – sampling

Fig. 16. a) Dendrite growth in the concasting structure without elelectromagnetic stirring – mode 2A, b) The growth of dendrites in the billet structure using the MEMS and SEMS – mode 4A (Table 6)

The above-described mechanism of dendrite growth during concasting without stirring is frequently the object of interest (Figure 16a). Inside the billets, when using the MEMS stirrer (or both MEMS and SEMS), the kinetics of solidification and dendrite growth is initially the same as without stirring. This also creates columnar dendrites which touch along the diagonals, however, soon their growth ceases still near the surface. Dendrites, which are called equiaxed dendrites continue to grow – their orientation is more random and only partly directed towards the centre of the billet (Figure 16b). It appears that this dendrite growth mechanism manifests itself the most when both stirrers are working simultaneously (Table 6: 4A, 4B and 5B). If MEMS and SEMS are working simultaneously, the stirring effect significantly destroys the formation of columnar crystals. If only MEMS is working and SEMS is switched off (1A and 1B), then the desctruction of columnar crystals is less evident. The working mode of SEMS alone (modes 3A and 3B) cannot be clearly differentiated from the changes in the dendritic structure in relation to the structure formed without stirring (2A and 2B).

Figure 16b (the macro-ground dendritic structure) shows the depth of the columnar band of dendrites in the direction away from the surface of the billet (Figure 16b – see arrows) and its value, which (with the simultaneous stirring of MEMS and SEMS) is 23.41.8 mm. The same qualified guess was made for ordinary billet casting (i.e. without stirring). Here, the depth of the dendrites can be guessed almost all the way to the central shrinkage at 70 mm (Figure 16a – see arrows). It is known that additives and impurities during solidification are often concentrated in points of contact of the growing dendrites, where the maximum of segregated additives and impurities and the greatest probability of technological defects occurs. In the given case, this undesirable effect can be expected along the diagonals which have a length of up to 100-to-103 mm towards the central shrinkage. This point of contact of the dendrites during the simultaneous working of SEMS and MEMS is only 29.81.9 mm,

Numerical Models of Crystallization

data on de-oxidising elements (Al, Ca, etc.).

number

**6. Acknowledgment** 

**7. References** 

dendritic axes

and Its Direction for Metal and Ceramic Materials in Technical Application 89

which necessarily requires respecting reality of poly-component crystallising metallic system, formed usually by eight to eleven constitutive elements. Constitutive elements forming conjugated model have during crystallisation completely different physicalchemical properties in dependence of temperature. Their redistribution in the volume of crystallising tangible macroscopic system is governed by the 2nd Fick's law. Mutual functional connection of both models A and B into one mutually cooperating conjugated model AB represents a completely new step resting on real crystallising poly-component system. This connection of two models AB necessarily requires large amount of consistent concentration data of constitutive elements forming real crystallising tangible macroscopic poly-component system. These data concern alloying elements (e.g. Ni, Cr), basic tramp elements (Mn, Si, Ti, V, Mo), data on admixture elements and impurities (S, P), as well as

The most complicated conjugated model will be the model for continuous casting. The authors have prepared for its creation 50,000 experimentally verified and mutually consistent data on elements. These data, make it possible to express concentrations *C*, effective distribution coefficient of elements between melt and crystallising solid phase *kef*, diffusion coefficients *Ds* in the melt of all segregating elements of dendritically crystallising system in the sense of the equation (3), express also the degree of heterogeneity and shares of solidified phase *gs*. It contains also the equation (4), which is dimensionless Fourier's

diffusion coefficient also share of local solidification time and squares of half distance of

the melt and solid phase (dendrite) the degree of heterogeneity – these mutually consistent and already verified data on elements form the basic starting point for progressive

This research was conducted using a program devised within the framework of the GA CR projects GA CR projects No. 106/08/0606, 106/09/0370, 106/09/0940, 106/09/0969,

Boettinger, W.J. et al.(2000). Solidification microstructures: recent development, future

Brody, H.D. & Flemings, M.C. (1966). Solute redistribution in dendritic solidification, *Trans.* 

Chvorinov, N. (1954). *Krystalisace a nestejnorodost oceli [Crystallization and heterogeneity of* 

Dobrovska, J., Stransky, K., Dobrovska, V. & Kavicka, F. (2009).Characterization of

Continuously Cast Steel Slab Solidification by Means of Chemical Microheterogeneity Assessment, *Hutnicke listy* No.5, LXII , pp. 4-9, ISSN 0018-8069 Dobrovska, J., Kavicka F., Stransky, K., et al. (2010). Numerical Optimization of the Method

of Cooling of a Massive Casting of Ductile Cast-Iron , *NUMIFORM 2010*, Vols 1 and

functional creation of the above mentioned conjugated model AB.

directions, *Acta Mater*., 48 pp. 43-70 ISSN 1359-6457

Chalmers, B. (1964). *Principles of Solidification*. John Wiley & Sons, New York.

Davies, G.J. (1973). *Solidification and Casting*. Applied Science Pub., London.

P107/11/1566 and MSMT CR- MSM6198910013.

*AIME*, 236 (1966), pp. 615-624

*steel]*. Nakladatelstvi CSAV, Praha

of the second kind for mass transfer, which contains implicitly, apart from the

*lS/L2* . Equation (5) postulates by share of concentrations at the interface of

i.e. 3.4 less. The central area of the billet containing a hollow as a result of a shrinkage is then filled with dendrites growing into a vacuum (i.e. underpressure) (Figure 17).

Fig. 17. Dendrites in the centre of the billet

Under the assumption that the maximum of defects (i.e. vacancies, impurities, additives and micro-shrinkages) are formed along the diagonals it is possible to expect that in the areas of the corners – specifically on the edges – the nucleation of cracks will be higher than on the walls of the billet. If the first approximation of the fracture toughness of the relevant billet made from low-carbon steel is *KIC* ~ 75.0 MPa.m1/2, then in the ordinary concasting process it can be assumed that the length of the contact of columnar dendrites along the diagonal will be approximately *Δlnormal* ≈ 101.5 mm (Figure 16a). On the other hand, if both electromagnetic stirrers (MEMS and SEMS) are engaged simultaneously, the contact length of the columnar dendrites along the diagonal decreases to *Δlel.magnl* ≈ 29.8 mm (Figure 16b). Along these lengths (i.e. the areas) it could be expected that during concasting the concentration of the primary defects will increase.

A comparison of limit stresses and strains in the area of the edges of the billets during concasting without elelectromagnetic stirring and if both MEMS and SEMS stirrers are engaged indicates that the billets (otherwise cast under the same conditions) cast without stirring are almost twice as susceptible to cracking along the edges as billets cast using both stirrers. A similar assumption can be made even in the case of assessing the effect of columnar dendrites in the central part of the surface of the billet where, without stirring, their length grows from the surface of the wall all the way to the central shrinkage (Figure 16a), while with the stirrers the dendrites are significantly shorter. The boundaries of the dendrites are however much less damaged by technological defects (vacancies, etc.) than the areas of their touching – of the peaks along the diagonals. Long-term statistical monitoring of the quality of 150150 mm billets and the chemical composition has proven that the application of elelectromagnetic stirring has significantly reduced the occurrence of defects (in this case cracks).

## **5. Conclusion**

Progressive creation of numeric model of unsteady thermal field A connected with the model of chemical heterogeneity B, leads to a completely novel conjugated numeric model, which necessarily requires respecting reality of poly-component crystallising metallic system, formed usually by eight to eleven constitutive elements. Constitutive elements forming conjugated model have during crystallisation completely different physicalchemical properties in dependence of temperature. Their redistribution in the volume of crystallising tangible macroscopic system is governed by the 2nd Fick's law. Mutual functional connection of both models A and B into one mutually cooperating conjugated model AB represents a completely new step resting on real crystallising poly-component system. This connection of two models AB necessarily requires large amount of consistent concentration data of constitutive elements forming real crystallising tangible macroscopic poly-component system. These data concern alloying elements (e.g. Ni, Cr), basic tramp elements (Mn, Si, Ti, V, Mo), data on admixture elements and impurities (S, P), as well as data on de-oxidising elements (Al, Ca, etc.).

The most complicated conjugated model will be the model for continuous casting. The authors have prepared for its creation 50,000 experimentally verified and mutually consistent data on elements. These data, make it possible to express concentrations *C*, effective distribution coefficient of elements between melt and crystallising solid phase *kef*, diffusion coefficients *Ds* in the melt of all segregating elements of dendritically crystallising system in the sense of the equation (3), express also the degree of heterogeneity and shares of solidified phase *gs*. It contains also the equation (4), which is dimensionless Fourier's number of the second kind for mass transfer, which contains implicitly, apart from the diffusion coefficient also share of local solidification time and squares of half distance of dendritic axes *lS/L2* . Equation (5) postulates by share of concentrations at the interface of the melt and solid phase (dendrite) the degree of heterogeneity – these mutually consistent and already verified data on elements form the basic starting point for progressive functional creation of the above mentioned conjugated model AB.

## **6. Acknowledgment**

This research was conducted using a program devised within the framework of the GA CR projects GA CR projects No. 106/08/0606, 106/09/0370, 106/09/0940, 106/09/0969, P107/11/1566 and MSMT CR- MSM6198910013.

## **7. References**

88 Crystallization – Science and Technology

i.e. 3.4 less. The central area of the billet containing a hollow as a result of a shrinkage is

Under the assumption that the maximum of defects (i.e. vacancies, impurities, additives and micro-shrinkages) are formed along the diagonals it is possible to expect that in the areas of the corners – specifically on the edges – the nucleation of cracks will be higher than on the walls of the billet. If the first approximation of the fracture toughness of the relevant billet made from low-carbon steel is *KIC* ~ 75.0 MPa.m1/2, then in the ordinary concasting process it can be assumed that the length of the contact of columnar dendrites along the diagonal will be approximately *Δlnormal* ≈ 101.5 mm (Figure 16a). On the other hand, if both electromagnetic stirrers (MEMS and SEMS) are engaged simultaneously, the contact length of the columnar dendrites along the diagonal decreases to *Δlel.magnl* ≈ 29.8 mm (Figure 16b). Along these lengths (i.e. the areas) it could be expected that during concasting the

A comparison of limit stresses and strains in the area of the edges of the billets during concasting without elelectromagnetic stirring and if both MEMS and SEMS stirrers are engaged indicates that the billets (otherwise cast under the same conditions) cast without stirring are almost twice as susceptible to cracking along the edges as billets cast using both stirrers. A similar assumption can be made even in the case of assessing the effect of columnar dendrites in the central part of the surface of the billet where, without stirring, their length grows from the surface of the wall all the way to the central shrinkage (Figure 16a), while with the stirrers the dendrites are significantly shorter. The boundaries of the dendrites are however much less damaged by technological defects (vacancies, etc.) than the areas of their touching – of the peaks along the diagonals. Long-term statistical monitoring of the quality of 150150 mm billets and the chemical composition has proven that the application of elelectromagnetic stirring has significantly reduced the occurrence of

Progressive creation of numeric model of unsteady thermal field A connected with the model of chemical heterogeneity B, leads to a completely novel conjugated numeric model,

then filled with dendrites growing into a vacuum (i.e. underpressure) (Figure 17).

Fig. 17. Dendrites in the centre of the billet

concentration of the primary defects will increase.

defects (in this case cracks).

**5. Conclusion** 


**4** 

*Romania* 

**A Mathematical Model for Single Crystal** 

**Film-Fed Growth (EFG) Technique** 

Loredana Tanasie and Stefan Balint

*West University of Timisoara* 

**Cylindrical Tube Growth by the Edge-Defined** 

Modern engineering does not only need crystals of arbitrary shapes but also plate, rod and tube-shaped crystals, i.e., crystals of shapes that allow their use as final products without additional machining. Therefore, the growth of crystals of specified sizes and shapes with controlled defect and impurity structures are required. In the case of crystals grown from the melt, this problem appears to be solved by profiled-container crystallization as in the case of casting. However, this solution is not always possible, for example growing very thin plate-shaped crystals from the melt (to say nothing of more complicated shapes), excludes

The techniques which allow the shaping of the lateral crystal surface without contact with the container walls are appropriate for the above purpose. In the case of these techniques the shapes and the dimensions of the grown crystals are controlled by the interface and meniscus-shaping capillary force and by the heat- and mass-exchange conditions in the crystal-melt system. The edge-defined film-fed growth (EFG) technique is of this type. Whenever the E.F.G. technique is employed, a shaping device is used (Fig. 1). In the device a capillary channel is manufactured (Fig. 1) in which the melt raises and feeds the growth process. Frequently, a wettable solid body is used to raise the melt column above the shaper, where a thin film is formed. When a wettable body is in contact with the melt, an equilibrium liquid column embracing the surface of the body is formed. The column formation is caused by the capillary forces being present. Such liquid configuration is usually called a meniscus (Fig. 1) and in the E.F.G. technique, its lower boundary (Fig. 1 –

Let be the temperature of the meniscus upper horizontal section (Fig. 1 – *AB* ) the temperature of the liquid crystallization. So, above the plane of this section, the melt transforms in solid phase. Now set the liquid phase into upward motion with the constant rate, *v*, keeping the position of the phase-transition plane invariable by selection of the heat conditions. When the motion starts, the crystallized position of the meniscus will

**1. Introduction** 

**1.1 Crystal growth from the melt by E.F.G. technique** 

container application completely.[ Tatarchenko, 1993]

point C) is attached to the sharp edge of the shaper.

2 – Dedicated to professor O. C. Zienkiewicz (1921-2009) Book Series: AIP Conference Proceedings Vol. 1252 , pp. 578-585 , ISBN 978-0-7354-0799-2


Flemings, M. C. (1974). *Solidification Processing*. McGraw-Hill, New York.

