Crystallization of glasses: the Avrami theory revisitedK. Shimakawa
Gifu University, Gifu
Nagoya Industrial Research Institute, Nagoya
Although dynamics of crystallization under isothermal conditions in glassy
materials is fairly well described by the Kolmogorov-Johnson-Mehl-Avrami (KJMA)
formalism, there are still unsolved problems with the KJMA equation. By introducing afractal structural concepts and the Meyer-Neldel rule on thermally activated processes,
an extended KJMA model is proposed that allows one to get a better insight into thephase changes in glasses. The phase changes of Ge2Sb2Te5 (GST) system in steady andnanosecond-scale pulse-heating conditions are also discussed.
1. Introduction
Phase transitions are one of the most important topics in material science, since they
are usually encountered in the production cycle of materials. During the last decade,there has been great interest in amorphous chalcogenides such as Ge2Sb2Te5 (GST) forapplications in optical data storage [1]. Random nucleation and isotropic growth in an
infinite specimen is well described by the Kolmogorov [2], Johnson and Mehl [3], andAvrami [4] models (KJMA model). The KJMA model gives the macroscopic evolutionof the transformed phase under isothermal conditions [5-7].
Dynamics of crystallization in GST materials has been fairly well interpreted interms of the KJMA model [8,9]. At the same time, while its applicability is criticized bySenkader and Write [10]. The KJMA formalism may have some difficulties, which havenot been well understood so far; e.g. the Avrami exponent and the reaction rate [10]. On
the other hand, crystallization kinetics under pulse-heating conditions has been
discussed in GST [11,12]. It is also unclear whether or not the KJMA model can beapplied to the phase transitions under pulse-heating conditions.
In the present study, we concentrate a macroscopic property which is based on theKJMA model. The unclear growth kinetics (the Avrami exponent and reaction rate) ofphase changes will be discussed by introducing a fractal structural concept [13] and by
the Meyer-Neldel rule [14] for the growth rate. A modified, or extended KJMA equation
proposed here may be useful to get proper understandings of phase changes in GST andother glassy systems, without losing simplicity of the KJMA formalism. It should bealso noted that microscopic view is still a matter of debate [1,15-17], which will be notdiscussed in the present paper.

2. KJMA formalism
We briefly discuss the KJMA theory. This theory
requires that the nucleation sites should be randomly
distributed over the volume V. In Fig.1, three particles havegrown together and growth ceases where the particles
impinge, as indicated by the heavy lines [6]. The
particle-overlap in the absence of impingement is shown aswell (extended volume Vex).
The actual (transformed) volume Vtr is given as  V tr   exdV tr  1    dV .
V

Fig.1 Particle growth(1)

Note that the extended volume Vex is a virtual volume. Equation (1) is written asf  1  exp  f ex ,(2)

where f = Vtr/V , fex = Vex/V. Here, f is the transformed volume fraction. In the case of
instantaneous nucleation of spherical grains with number density N, the volume growthrate with radius r for an interface-controlled reaction can be expressed asdV exdr 4 r 2 NV.
(3)
dt
dt
Here the effective growth rate is given by G = dr/dt and then fex at time t is expressed byf ex 4
4
Nr 3 N (Gt )3 .
33(4)We then get well known KJMA equation
1  f  exp  (kt ) n  ,(5)with n = 3. Here, n is called the Avrami exponent (coefficient), and k is an effective rate
constant (growth rate). The Avrami exponent should be an integer providing informationon the dimensionality of the crystallization process. Actually, however, the values of nin the range from 1 to 5.8 in GST [10]. The effective rate constant k istemperature-dependent and is given by E k (T )   exp   A  ,(6) kT 
where  is the frequency factor and EA the activation energy. Note that  takes
unreasonably large values (1017-1024 s-1) [10]. These will be discussed in the following
section together with non-integer Avrami coefficient.

3. Extended KJMA model
The KJMA formalism is a very simple and is useful to understand the time evolutionof crystallization of glasses. However, at least, two problems on the KJMA should beovercome without losing its simplicity. The first problem is the Avrami exponent n. The
theory predicts that n represents the Euclid dimensions (n = 1, 2, 3). However, thevalues of n are reported to be non-integer and sometimes n is bigger than 3 as alreadystated. Non-integer values of n can be attributed to violation of the fundamental
assumptions (strict conditions) of KJMA formula which requires (i) a random
distribution of potential site of nucleation, (ii) instantaneous nucleation, (iii)interface-controlled grain growth, and time (or grain size ) independent growth rate k[10,12].
Here, we introduce a fractal concept on space dimension, which may be useful for
discussing the phase changes, since the surfaces of most materials are expected to be
fractals [13,18]. The volume growth rate, eq.(3), can then be modified as
dV ex
dr Cr D NV,(7)dtdt
where C is a constant and D the fractal dimension of the surface of grains. Finally, weget f as1  f  exp  (kt ) D+1  .

(8)

It is therefore suggested that various values of the Avrami exponent n observed can be
attributed to complex shape of the crystalline grain surfaces with fractal structure. This
fractal-surface-controlled (FSC) crystallization may be supported by the following
reports, e.g., that D-value takes  3.0 for silica gels [19] and 2.6-2.9 for sandstones[20].
Turn to the second problem: We discuss the temperature-dependent rate constantk(T) [eq.(6)]. The reported activation energies EA are around 2 eV (1-3 eV) in GST
alloys [10], which are deduced also from the other thermal analysis such as theDifferential Scanning Calorimetry (DSC) [21]. The problem is that unreasonably large
frequency factor  (1017-1024 s-1) is deduced [10,22]. It is known that thermally
activated phenomena, such as electronic and ionic transports, and structural relaxationin disordered matters, obey the Meyer-Neldel compensation rule [14,23]. For examplek(T) in eq.(6) is modified to E  E k (T )   0 exp  A  exp   A  , kT  EMN 

where the frequency factor  experimentally deduced is(9) EA ,
 EMN    0 exp (10)and EMN is called the Meyer-Neldel energy and 0 is a constant. Equation (9) is called
the Meyer-Neldel (MN) rule or compensation law [23]. In temperature-activatedelectronic transport in amorphous chalcogenides, the conductivity pre-factor also
involves the term of exp(EA/EMN) with EMA= 25-60 meV [24].
Recent reports on structural relaxation andphase change in GST also follow the MN rule,as shown in Fig. 2 [22]. Open circles show thestructural relaxation (SR), i.e. the conductivity
change under thermal annealing, and othersymbols are for the crystallization in GST. The
both phenomena fall into similar MN rule.
EMN and 0 here are estimated to be 66 meVand 106-107 s-1, respectively. Thus theexperimentally deduced unphysical frequency
factor  can be explained in the frame work ofFig.2 Correlation between EA andThe MN rule. The Origin of the MN rule is still in GST.
not clear [22], although many-body thermal excitation in activated processes issuggested [22,23].
4. Phase changes under pulse-heating (non-isothermal condition)
The KJMA theory is applied to the phase changes under isothermal conditions. It isof interest to examine whether or not the KJMA formalism applies to the phase changes
under pulse heating (non-isothermal conditions). Weidenhof et al. [12] have reported thephase changes, f vs. t, in GST under isothermal and non-isothermal (pulse heating)conditions. Results are shown in Fig. 3 (a) and (b). The Avrami exponent n is reported tobe 2.5 under isothermal condition. n under the pulse-heating condition (severalhundreds nanoseconds) has two components, 2.2 for short time and 1.3 for longer time
ranges. The buckling of the KJMA plot after approximately 220 ns is suggested to be
due to saturation of the growth in vertical direction [12]. These results suggest that theisothermal conditions may be hold if the phase changes proceed during pulse heating.
Finally, it should be noted that the phase change processes in nano- and pico-seconds
time scales have been reported [25,26]. It is not clear whether or not the KJMA works
for such short time scale.

(a)(b)Fig. 3 The KJMA plots in GST. (a) under isothermal condition, (b) pulse-heating
condition [12]..
5. Summary
An extended KJMA theory has been proposed. The classical KJMA theory withsome modifications (extended KJMA theory) can be applied to explain the dynamics ofcrystallization in GST. Some difficulties underlying the KJMA formalism wereovercome by introducing fractal structural concept on crystal growth. Unphysicalfrequency factor in the KJMA can also be explained in the frame work of theMeyer-Neldel rule. It is still not clear whether or not the extended KJMA applies to thephase change processes in nano- and pico- seconds regions.
Acknowledgement
The author would like to thank Alex Kolobov and Krisztian Kohary for useful
discussions.
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