**4. Kinetics of crystallization**

198 Advances in Crystallization Processes

amorphous alloy. Both *Tx*1 and *Tx*2 firstly decrease with the increase in pressure in the range of 0–1 GPa and then increase with pressure increasing up to 4 GPa. Such changes in crystallization temperature with pressure is related to the competing process between the thermodynamic potential barrier and the diffusion activation energy under pressure

Fig. 9. Pressure dependence of the crystallization temperatures for Al89La6Ni5 amorphous alloy. Reprinted from (Zhuang, et al., 2000), with permission from American Institute of Physics.

Crystallization of an amorphous alloy is normally regarded as a process proceeding by nucleation and subsequent growth of crystals. During the initial stage of nucleation of crystals in the amorphous phase, the effect of pressure on the crystallization kinetics is associated with the atomic diffusion process and the volume change effect. The crystallization temperature(s) of an amorphous alloy may be governed by the thermodynamic potential barrier of nucleation and diffusion activation energy. According to

where *I*0 is a constant, ∆G\* is the free energy required to form a nucleus of the critical size, i.e., the thermodynamic potential barrier of nucleation, *Q*n is the activation energy for the transport of an atom across the interface of an embryo, and *k* is the Boltzmann's constant.

In the Al89La6Ni5 alloy, ∆*G*\* is much larger than *Q*n and the dominant factor at low pressures (0–1 GPa). Thus, the nucleation work decreases with increasing pressure, leading to an enhancement of nucleation rate *I* and a reduction of the crystallization temperature with increasing pressure, as shown in Fig. 9. With increasing pressure, ∆*G*\* rapidly decreases while *Q*n increases, resulting in atomic diffusion a dominant factor in the nucleation process. Hence, the nucleation work ∆*G*\* + *Q*n increases with increasing pressure. Consequently, nucleation rate *I* decreases with the increase in pressure and an enhancement of

0exp *\* <sup>n</sup> I I [ ( G Q ) / kT ]* = −Δ + (1)

crystallization kinetics theory, the nucleation rate *I* can be written as,

The sum ∆G\* + *Q*n is called the nucleation work.

(Zhuang, et al., 2000).

The kinetics of crystallization of amorphous alloys has been extensively studied by using differential scanning calorimetry (DSC) or differential thermal analysis (DTA), as the structural change in a material upon heating or cooling is indicated by a defection or peak in the DSC/DTA curve. The kinetic behavior associated with a structural change leading to an alternative metastable state in an amorphous alloy above its glass transition is a key subject since it provides new opportunities for structural control by innovative design and processing strategies. *Section 5* will show some application examples by controlling crystallization from amorphous precursors in order to tailor microstructure for excellent properties. Such crystallization control requires fundamental understanding of the specific mechanisms influencing structural transformations.

In general, crystallization is a thermally activated reaction, either by isothermal or isochronal heating. The transformation rate during a reaction could be described as

$$\text{dax } \text{/ dt} = f\left(\alpha\right)\mathbf{k}\left(T\right) \tag{2}$$

where *α* is the fraction transformed. The temperature dependent function is generally assumed to follow an Arrehnius type dependency

$$k = k\_0 \exp(-E \text{ / RT})\tag{3}$$

where *k*<sup>0</sup> is the reaction constant, *R* is the gas constant and *E* is the activation energy. In general, the reaction function *f(*α*)* is unknown. From the above equations it follows that for transformation studies by performing studies at a constant temperature *T, E* can be obtained as below:

$$
\ln \left( t\_f \right) = \text{E} \ne RT + c\_i \tag{4}
$$

where *t*f is the time needed to reach a certain fraction transformed, and *c*<sup>i</sup> is a constant, which depends on the reaction stage and on the kinetic model. Thus, *E* can be obtained from two or more experiments at different *T*. For isothermal experiments *k(T)* is constant, the determination of *f(*α*)* is relatively straightforward, and is independent of *E*. For nonisothermal experiments, the reaction rate at all times depends on both *f (*α*)* and *k (T)*, and the determination of *f(*α*), k0* and *E* (the so-called kinetic triplet) is an interlinked problem. A deviation in the determination of any of the three parameters will cause a deviation in the other parameters of the triplet. Over the past decades a variety of non-isothermal methods have been proposed. Among them, the Kissinger method (Kissinger, 1957) is widely used in

Crystallization Behavior and Control of Amorphous Alloys 201

proportional to the fractional areas of the exothermic peak. Hence, the crystallized volume fraction during the isothermal annealing process can be accurately determined by measuring the area of the exothermic peak. The crystallized fractions *x*c(*t*,*T*) after time *t* at a certain temperature *T* for amorphous phase could be derived from the isothermal DSC

> () () () 0 0

where *h(t,T)* is the enthalpy release. Using the JMA equation, the reaction rate as well as the parameters governing the nucleation rate and/or the growth morphology can be obtained. As shown before, it is inappropriate to describe the crystallization mechanism by using the average Avrami exponent derived from the non-linear JMA plot in the whole range of volume fraction. An alternative method of examining the isothermal DSC results is to evaluate the local value of the Avrami exponent, *N*loc, which is defined as (Calka &

*N ln ln x t,T / (t ) loc* =∂ − − ∂ − ( ) 1 l *<sup>c</sup>* ( ) n

as a function of crystallized volume fraction *x*c(*t*,*T*). Such a differential Avrami plot can

The isothermal activation energy for the crystallization process can also be determined in terms of the incubation period *τ* at different temperatures during isothermal annealing,

0 *iso*

where *τ<sup>0</sup>* is a constant and *E*iso is the activation energy for crystallization. The plot of ln*τ* vs. 1/*T* yields a straight line. From the slope, the activation energy *E*iso for crystallization of an

Ti50Cu18Ni22Al4Sn6 735-755 2.5-3.3 0.05-0.60 399±55 392±17 with 10 vol.% TiC 723-750 2.1-2.8 0.05-0.60a 384±10 382±22

Table 2 compares the active energy of crstallization estimated by the aforementioned two methods for the ball-milled amorphous Ti50Cu18Ni22Al4Sn6 alloy and its composite containing 10 vol.% TiC (Zhang, et al., 2006a). As seen from Table 2, there are no essential differences in the activation energies between those evaluated using the Arrhenius equation

Table 2. Avrami exponent (*n*) and activation energy of crystallization (*E*xτ) in terms of incubation time during isothermal annealing and the activation energy of crystallization (*E*iso) determined from a Kissinger plot for the ball-milled amorphous Ti50Cu18Ni22Al4Sn6 alloy and its composite containing 10 vol.% TiC. Reprinted from (Zhang, et al., 2006a), with

highlight changes in reaction kinetics during the progress of crystallization.

τ τ

range (K) *<sup>n</sup>*

using the Arrhenius equation for a thermally activated process (Luborsky, 1977):

∞

<sup>=</sup> (8)

τ(9)

= −*( E / RT )* (10)

*E*iso (kJ/mol)

*E*x (kJ/mol)

*xc*(*t*,*T*) range

*<sup>c</sup> x t,T h t,T dt / h t,T dt*

curves by assuming that *x*c(*t*,*T*) is proportional to the integrated enthalpy

Radlinski, 1988)

amorphous phase is calculated.

Samples Temperature

a 0.05-0.40 was used for the composite at 723 K.

permission from American Institute of Physics.

*t*

the isochronal method for the calculation of the activation energy for the crystallization. A higher value of the activation energy is generally interpreted as a measure of the high stability and resistance of the amorphous phase towards crystallization. The activation energy for crystallization could be determined using

$$\ln\left(\beta \;/\; T\_p^2\right) = -\mathcal{E} \;/\; RT\_p + \mathcal{C} \tag{5}$$

where *β* is the heat rates that used to heating amorphous samples in DSC, *T*p is the temperature corresponding to the peak of the crystallization event (exothermic peak), *R* is the gas constant and *C* is a constant. Thus, by plotting ( ) <sup>2</sup> *<sup>p</sup> ln / T* β against 1/*T*p, one could obtain a straight line whose slope is −*E/R*, from which the activation energy for the transformation, *E,* can be calculated.

On the other hand, kinetic data on first-order transformations are often obtained by isothermal analysis. The isothermal crystallization kinetics of the amorphous phase can be usually analyzed in terms of the generalized theory of the well-known Kolmogorov-Johnson-Mehl-Avrami (JMA) equation (Christian, 2002) for a phase transition:

$$\max\_{t} \left( t, T \right) = 1 - \exp\left[ -k(t - \tau)^{\mu} \right] \tag{6}$$

or

$$
\ln\left[-\ln\left(1-\mathbf{x}\_c\left(t,T\right)\right)\right] = n\ln k + n\ln(t-\tau) \tag{7}
$$

where *x*c(*t*,*T*) is the volume fraction of crystallized phases after annealing time *t*, *τ* is the incubation period of transient nucleation, which is the time period that must elapse prior to formation of nuclei, *k* is a temperature-dependent kinetic parameter and *n* is the Avrami exponent, which is a significant parameter to describe the crystallization mechanism, such as nucleation and growth behavior, and varies from 1 to 4 (Doherty, 1996). For diffusioncontrolled growth, one may have the following cases: 1 < *n* < 1.5 indicates growth of particles with an appreciable initial volume; *n* = 1.5 indicates growth of particles with a nucleation rate close to zero; 1.5 < *n* < 2.5 reflects growth of particles with decreasing nucleation rate; *n* = 2.5 reflects growth of particles with constant nucleation rate, and *n* > 2.5 pertains to the growth of small particles with an increasing nucleation rate (Doherty, 1996). A JMA plot of ln[-ln(1- *x*c(*t*,*T*))] vs. ln(*t*-τ) yields a straight line with slop *n* and intercept *n*ln*k*. Using a DSC, operated under isothermal mode, phase transformations can be distinguished unambiguously in terms of those occurring only by growth of existing nuclei or those occurring by nucleation and growth. For a transformation resulting in grain growth or structural relaxation results in a monotonically decreasing signal, a "bell-shape" exothermic curve is the classical signature for a nucleation-and growth transformation (Chen & Spaepen, 1991).

The transformed volume fraction, *x*, during the isothermal process at a particular temperature, *T*, can then be determined by measuring the area under the exothermic curve. It is assumed that the volume fraction transformed, *x*, up to any time, *t*, is proportional to the fractional area of the exothermic peak or the integrated enthalpy. Therefore, in the isothermal DSC scans, the transformed volume fraction, *x*c(*t*,*T*), up to any time *t* is

the isochronal method for the calculation of the activation energy for the crystallization. A higher value of the activation energy is generally interpreted as a measure of the high stability and resistance of the amorphous phase towards crystallization. The activation

*p p ln / T E / RT C*

where *β* is the heat rates that used to heating amorphous samples in DSC, *T*p is the temperature corresponding to the peak of the crystallization event (exothermic peak), *R* is

obtain a straight line whose slope is −*E/R*, from which the activation energy for the

On the other hand, kinetic data on first-order transformations are often obtained by isothermal analysis. The isothermal crystallization kinetics of the amorphous phase can be usually analyzed in terms of the generalized theory of the well-known Kolmogorov-

( ) 1 exp *<sup>n</sup> <sup>c</sup> x t,T [ k(t ) ]* =− − −

( ) 1 ( ) *<sup>c</sup> ln ln x t,T n* −− = + − *lnk nln(t )*

where *x*c(*t*,*T*) is the volume fraction of crystallized phases after annealing time *t*, *τ* is the incubation period of transient nucleation, which is the time period that must elapse prior to formation of nuclei, *k* is a temperature-dependent kinetic parameter and *n* is the Avrami exponent, which is a significant parameter to describe the crystallization mechanism, such as nucleation and growth behavior, and varies from 1 to 4 (Doherty, 1996). For diffusioncontrolled growth, one may have the following cases: 1 < *n* < 1.5 indicates growth of particles with an appreciable initial volume; *n* = 1.5 indicates growth of particles with a nucleation rate close to zero; 1.5 < *n* < 2.5 reflects growth of particles with decreasing nucleation rate; *n* = 2.5 reflects growth of particles with constant nucleation rate, and *n* > 2.5 pertains to the growth of small particles with an increasing nucleation rate (Doherty, 1996). A JMA plot of ln[-ln(1- *x*c(*t*,*T*))] vs. ln(*t*-τ) yields a straight line with slop *n* and intercept *n*ln*k*. Using a DSC, operated under isothermal mode, phase transformations can be distinguished unambiguously in terms of those occurring only by growth of existing nuclei or those occurring by nucleation and growth. For a transformation resulting in grain growth or structural relaxation results in a monotonically decreasing signal, a "bell-shape" exothermic curve is the classical signature for a nucleation-and growth transformation (Chen &

The transformed volume fraction, *x*, during the isothermal process at a particular temperature, *T*, can then be determined by measuring the area under the exothermic curve. It is assumed that the volume fraction transformed, *x*, up to any time, *t*, is proportional to the fractional area of the exothermic peak or the integrated enthalpy. Therefore, in the isothermal DSC scans, the transformed volume fraction, *x*c(*t*,*T*), up to any time *t* is

=− + (5)

against 1/*T*p, one could

(6)

*<sup>p</sup> ln / T* β

τ

τ(7)

β

Johnson-Mehl-Avrami (JMA) equation (Christian, 2002) for a phase transition:

the gas constant and *C* is a constant. Thus, by plotting ( ) <sup>2</sup>

energy for crystallization could be determined using

( ) <sup>2</sup>

transformation, *E,* can be calculated.

or

Spaepen, 1991).

proportional to the fractional areas of the exothermic peak. Hence, the crystallized volume fraction during the isothermal annealing process can be accurately determined by measuring the area of the exothermic peak. The crystallized fractions *x*c(*t*,*T*) after time *t* at a certain temperature *T* for amorphous phase could be derived from the isothermal DSC curves by assuming that *x*c(*t*,*T*) is proportional to the integrated enthalpy

$$\mathbf{x}\_c(t, T) = \bigwedge\_{0}^{t} \mathbf{h}(t, T) \, dt \, / \, \underset{\mathbf{0}}{\text{in}} \Big|\, \mathbf{h}(t, T) \, dt \, \tag{8}$$

where *h(t,T)* is the enthalpy release. Using the JMA equation, the reaction rate as well as the parameters governing the nucleation rate and/or the growth morphology can be obtained. As shown before, it is inappropriate to describe the crystallization mechanism by using the average Avrami exponent derived from the non-linear JMA plot in the whole range of volume fraction. An alternative method of examining the isothermal DSC results is to evaluate the local value of the Avrami exponent, *N*loc, which is defined as (Calka & Radlinski, 1988)

$$N\_{loc} = \partial \ln \left[ -\ln \left( 1 - \mathbf{x}\_c(t, T) \right) \right] / \partial \ln(t - \tau) \tag{9}$$

as a function of crystallized volume fraction *x*c(*t*,*T*). Such a differential Avrami plot can highlight changes in reaction kinetics during the progress of crystallization.

The isothermal activation energy for the crystallization process can also be determined in terms of the incubation period *τ* at different temperatures during isothermal annealing, using the Arrhenius equation for a thermally activated process (Luborsky, 1977):

$$
\pi = \pi\_0(-E\_{iso} \text{ / RT)} \tag{10}
$$

where *τ<sup>0</sup>* is a constant and *E*iso is the activation energy for crystallization. The plot of ln*τ* vs. 1/*T* yields a straight line. From the slope, the activation energy *E*iso for crystallization of an amorphous phase is calculated.


a 0.05-0.40 was used for the composite at 723 K.

Table 2. Avrami exponent (*n*) and activation energy of crystallization (*E*xτ) in terms of incubation time during isothermal annealing and the activation energy of crystallization (*E*iso) determined from a Kissinger plot for the ball-milled amorphous Ti50Cu18Ni22Al4Sn6 alloy and its composite containing 10 vol.% TiC. Reprinted from (Zhang, et al., 2006a), with permission from American Institute of Physics.

Table 2 compares the active energy of crstallization estimated by the aforementioned two methods for the ball-milled amorphous Ti50Cu18Ni22Al4Sn6 alloy and its composite containing 10 vol.% TiC (Zhang, et al., 2006a). As seen from Table 2, there are no essential differences in the activation energies between those evaluated using the Arrhenius equation

Crystallization Behavior and Control of Amorphous Alloys 203

1000 MPa is achieved by the formation of an amorphous phase. The bulk nanocrystalline alloys, which contain a mixed structure of intermetallic compounds embedded fcc-Al matrix by the crystallization of Al-based amorphous phase, exhibit high mechanical strength of 700–1000 MPa and have been commercialized as a commercial name of GIGAS. By controlling the crystallization of Al-based amorphous alloys, the tensile strength of the Albased amorphous alloys increases to 1560 MPa by the homogeneous precipitation of nanoscale fcc-Al particles into an amorphous phase, which is higher than the strength of

> Constructional (Al-based)

GIGAS®

*V*cr 70–75% *λ*<sup>s</sup> ≈ 0 ≤ 40% ductility ≤ 100% *D* ≤ 15 nm <*K*> ≈ 0 *V*cr ↑, *D* ↓ *σ*<sup>f</sup> ↑ < 25 nm

Table 3. General characteristics of the three main groups of nanocrystalline materials produced by devitrification of amorphous alloys (*V*cr – volume fraction of crystalline phase, *D* – diameter of nanocrystals, *λ*s – saturation magnetostriction constant, <*K*> – averaged magnetocrystalline anisotropy, *σ*f – fracture strength). Reprinted from (Kulik, 2001), with

Although amorphous alloys have exhibited unique properties compared the conventional polycrystalline materials, the metastable nature of amorphous phase has imposed a barrier to broad commercial adoption, particularly where the processing requirements of these alloys conflict with conventional metal processing methods. In general, amorphous alloys are super-strong with compressive yield strengths of approximately 2 GPa and even up to 5 GPa for some exotic bulk glass-forming alloys, as has already shown in Fig. 1. However, amorphous alloys suffer from a strong tendency toward shear localization upon yielding, which results in macroscopically brittle failure at ambient temperatures. Therefore, processing of amorphous alloys at ambient temperatures is extremely hard. When an amorphous solid is continuously heated in the supercooled liquid region the viscosity decreases dramatically as the alloy relaxes into the metastable equilibrium state of the supercooled liquid and the large viscous flowability is obtained (Bakke et al., 1995; Volkert

Al–RE–TM (RE=Y, Ce, Nd, Sm; TM=Ni, Co, Fe, Cu)

Amorphous matrix + Nanocrystals (fcc-Al)

High specific strength at high temperatures

e.g. Fe82.3Nd11.8B5.9

Magnetically hard

(Fe-based)

Fe88Nb2Pr5B5

Nanocrystals Fe14Nd2B (+Fe3B, bcc-Fe, Am)

High coercivity, high remanence

Fe–RE–B

1260 MPa by the formation of an amorphous single phase.

(Fe73.5Cu1Nb3Si13.5B9)

(Fe84Zr3.5Nb3.5B8Cu1)

(Fe44Co44Zr7B4Cu1)

+ Amorphous matrix

low magnetic losses

**5.2 Net-shape (micro-)forming in supercooled liquid region** 

Magnetically soft (Fe-based)

Nanocrystalline materials

Sructural parameters

Alloys Finemet®

Hitperm

Nanoperm®

Structure Nanocrystals (bcc-Fe)

Properties High permeability,

permission from Elsevier.

in isothermal annealing and those obtained by isochronal annealing as revealed by Kissinger analysis for the Ti-based amorphous alloy with and without TiC particles. The activation energy of crystallization determined from the Kissinger analysis and the Arrhenius equationfor both powders show that the composite has slightly lower activation energy. The addition of 10 vol.% TiC particles into the Ti-based amorphous alloy may slightly affect the crystallization kinetics of the amorphous phase and the TiC particles may act as potential heterogeneous nucleation sites.
