Materials Science and Engineering, B6 (1990) 265-271 265

Growth of Crystalline Phase in Amorphous Alloys

R gVEC and P. DUHAJ

Institute of Physics SAS, Dt~bravskd cesta 9, 842 28 Bratislava (Czechoslovakia)

(Received February 1, 1990)

Abstract

Starting from the concept that the onset of crys- tallization of metallic glasses is determined by the presence of quenched-in nuclei-like associations, the mechanism and kinetics of growth of crystal- line phase from the amorphous matrix is described in terms of short-range ordering of metal and metalloid atoms and their chemical bonding. This mechanism does not require any pronounced changes in local chemical ordering or substantial rearrangement of associations upon crystallization that wouM require the disruption of chemical bonds existing in these associations. A model oJ crystal growth based on the motion of stable asso- ciations is proposed. The influence of associations on processes taking place at temperatures below crystallization is discussed.

1. Introduction

Crystallization in metallic glasses is a nuclea- tion-and-growth process. Both the formation of crystalline embryos and their growth to critical size, i.e. nucleation, and subsequent growth of such nuclei in glasses are expected to be con- trolled by the same mechanism of transfer of the matter, which ought to be very sensitive to the structure of the original matrix.

Concerning nucleation in metallic glasses, nowadays it is generally accepted that nuclei of the crystalline phase are induced already in the melt upon its cooling [1]. Several studies have been made which show that such "quenched-in nuclei" depend on the quenching rate of the melt and on its chemical composition. In our study of crystallization in the Fe-B system [2, 3] we have proposed the mechanism of crystallization to be influenced by the presence of so-called bridging complexes in the amorphous matrix; these are groups of atoms where boron is surrounded by

and chemically strongly bonded with iron atoms only and upon crystallization occupies octahedral positions in the a-Fe lattice. This seems to be the reason for the observed supersaturation of the a-Fe phase formed during the initial stages of crystallization. We believe that the presence of such complexes (associations) with internal arrangement, depending on chemical composi- tion, will significantly influence the whole process and, above all, the mechanism of crystallization.

2. The role of associations in crystallization

Originally the assumption about the existence and importance of associations has been pro- posed when formulating theories dealing with glass-forming ability, i.e. with the conditions necessary for the formation of amorphous phase from the melt. In an ideal melt, by definition, an entirely random distribution of different kinds of atoms should exist. As, however, has been shown [4-6], real melt systems are far from homogene- ous and evidence has been given which allows us to suppose that there exists in the melt molecule- like species which are composed of different kinds of atoms. It can further be supposed that the interatomic interactions within the associated volume parts are stronger than the interactions between atoms which are not contained in these associations. It is generally accepted that the necessary condition for easy glass formation is the existence of associations. These have to have a definite short-range ordering, i.e. atomic con- figuration, which must be as different as possible from the structures of the corresponding equilib- rium crystalline phases. Then, however, it logic- ally follows that the arrangement of atoms in the melt, which is preserved in glass after quenching, will be of substantial importance for the forma- tion of crystalline nuclei and their growth. It is possible to infer about this arrangement in-

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266

directly, for example by microscopic observation of crystallization of glasses. A detailed analysis of crystallization in hypoeutectic alloy Fe84Bl6 by HR transmission electron microscopy, performed in ref. 7, has shown that the growth of a-grains takes place not by single-atom jumps across the crystal-glass interface, but rather by inclusion of neighbouring associated domains.

3. The role of associations in grain growth from the amorphous matrix

If we accept the possibility of motion of single atoms (above all, of smaller metalloid atoms) in glass over longer or shorter distances, the clas- sification of crystallization reactions [8] into poly- morphous, primary and eutectic holds. This concept is supported experimentally for example by measuring reasonable values of activation energies for growth rates and by very correct low values of the Avrami parameter n ~ 1.5, cor- responding to growth governed by long-range motion of atoms. This is further supported by the apparent validity of the parabolic growth law for time dependence of crystal radii R in primary reactions. In eutectic reactions, however, when the grain growth rate u(T) is approximated by the usual Arrhenian formula, the preexponential factor u 0 is found to be larger by six to nine orders of magnitude than expected [ 1].

If we, however, accept the existence of poly- atomic associations in the amorphous matrix and if we assume no necessity for changes in local chemical composition upon crystallization, the following question arises: How would the pre- sence of such relatively stable associations acting as quenched-in embryos influence growth rate u(T) of crystalline grains in such a matrix with associated structure? Growth in this case could then take place not by the classical motion of single, thermally activated atoms, but again, as Hirotsu et al. [7] have directly observed, by inclu- sion of neighbouring associations and, possibly, by their slight internal rearrangement. Let us investigate such a growth mechanism a little closer.

In unassociated glass made up of atoms with radius q , the growth rate determined by motion, or inclusion, of single atoms will be given [9] by the usual formula which is the product of atomic volume v (q )= 4/3 n r 13, number of atoms per unit area of the interface N l , their vibration frequency v, probability of correct vibration direction P~

and the exponential term with activation energy E~ per mole of atoms necessary to overcome the energetic barrier of the interface, thus

i -t;'j) 1',/1 =l,t(r[)=V(F 1 ) N 1 vP, exp,[ . . . . 1)

' ' ' \ R T /

This formula can be rewritten as the classical Arrhenian relation ul = u~,~exp( - EE/RT) with preexponential factor u0j and activation energy El, where the indices "1" emphasize single-atom (unassociated) motion.

Let us now presume that in an associated glass the atoms will be arranged in spherical associ- ations with radii r,, ranging from atomary r I to a certain maximal value r ....... and will thus contain from one to nma x atoms. Let us also suppose that only the volume v(r,,)and the activation energy E,,(r,,) per mole of associations of size r,, to cross (or be included by) the interface are size depen- dent and let E,, be proportional to the cross-sec- tion of associate, E,,(r,,)~ r,,=. Then we can obtain a formula for the activation energy E(r,,) per mole of atoms in associations of size r,, as E(r,,)=Ei.rl/r,,. The contribution to overall growth rate from associations of size r,,, will be given as

- { r " l s { t~T t (2) u ( r , , ) - uo ,~ ) exp '-hE-'}Tr-"!!

The overall growth rate in associated glass will be the weighted sum of contributions from associa- tions of all permitted sizes

/ ,: , i , b ,

u= V p(r,,) u(r,,) (3) r = r I

where the probability of finding an atom in asso- ciation of size r,, p(r,,), plays the role of the weighting function. The function p(r,,) should reflect, besides the chemical interaction of atoms. the thermal history of the glass as well. For an ideal case of a single-component metallic glass prepared without any quenched-in associations. the Volmer-Becker-Doring theory of nucleation in undercooled liquids can be used to determine the function p [9].

Applying this theory we can find in the tem- perature dependence of crystal growth rate u(T) temperature intervals with various maximal per- missible dimensions of associations r ....... . The value r ..... is, according to the theory [9], equal to

the radius of the critical nucleus r~,~t, which is an increasing function of temperature starting with q (region without associations) at a certain tem- perature and increasing without limits as T approaches the melting temperature T M. The value rcrit , as well as the distribution function p, are also functions of the enthalpy of fusion AH and of the mean interracial energy of surface ten- sion X. For small enough temperatures, when r~r~t is less or equal to the atomic radius r I , no associa- tion process is allowed and growth takes place by motion of single atoms. With increasing tem- perature, associations containing two, three and more atoms can appear and the total growth rate is given by their corresponding contributions to the motion of the crystal-glass interface.

It can be shown that for temperatures not too close to T M the associated growth rate u(T) is well described by only the last term in the sum, thus

u = constant(rc,it)6exp E l rcrit

where W is the energy of formation of critical embryo. The growth rate u(T) due to participa- tion of associations can be approximated, for the sake of comparison with experiment, by the Arrhenian relation UA = U0Aexp( - EA/RT) and compared with the growth rate due to motion of single atoms in unassociated glass ul. The indices "1" and 'W' indicate single atom and associated growth motion respectively.

The model has two parameters which are in general difficult to determine experimentally, namely the enthalpy of fusion AH and the mean surface energy y. Numerical as well as theoretical analyses of the model have been performed. The results of one such numerical comparison of tem- perature dependences of growth rates u A and u~ for a system of iron-like atoms are shown in Fig. 1. For input thermodynamic parameters as observed in metallic glasses, u(T) is charac- terized by the values of pre-exponential factor UoA/Uoj = 1 0 6 - 1 0 9 a n d a c t i v a t i o n e n e r g i e s EA/ E j = I - 2 where E A is weakly temperature dependent. This is in a good agreement with experimental observations on crystallization of metallic glasses.

Using the outlined association approach, the experimentally observed time dependence of growth rate can be explained as well. Time dependence of growth due to motion of single

267

--q

z<z,

2 -

0 0 7 5 ~

[- 0.125 ~ / ~a" ~' , o15 ,,"- z ' " _ /

~ 2 0 125

J. . . . . . J

200

0 075 01

015

I L

600 1000 T [ K ]

Fig. 1. Associated growth rate uA(T ) as compared with an unassociated (single-atom) growth rate u,(T). Input values are atomic radius r, = 2 x 10-~0 m, molar heat of fusion AH = - 12 kJ mo l -L melt temperature T M = 1500 K, single atom molar volume vm = 8 x 10 - 6 m 3 mol ~ and surface free energy per unit area 7 = 7, T/TM. The parameter of the curves 7~ is in joules per square metre.

atoms over longer distances, such as presumed in growth governed by the primary reaction mechanism, is described by the parabolic growth law

/DIt/2 .t,l=tW Time dependence of associated growth rate

with limited number of associations can be approximated [10], similarly as heterogeneous nucleation with exhaustion of nucleation sites, by a function decreasing exponentially with time

u(t, T)= u0exp(~TT A) exp{-tI(T)} (6a)

I(T) = I0 exp { - ( W+EA)RT (6b)

with I being the temperature-dependent part of such nucleation. Then for the grain size time dependence we obtain by integration from u(t) that R(t) increases exponentially from zero to a maximum value Rmax, which satisfies the follow- ing temperature dependence:

R(t, T)=Rmax(T)[1-exp{-tI(T)} ] (7a)

268

"7'

N Nto t ~ . r r 4 n C°11 Fe88B15

t

0.1 L 0 - t _ L i

i 10 rain

0.1 [

~Rma× 0.4 100 rain

°2

0 i ~ i _ L i i i ,

0.05 0.1 0.135 R [~m]

(b)

Fig. 2. (a) Crystalline grains of cx-(FeCo) formed m Fen sCo~TB~5 annealed at 670 K. Annealing times (from top to bottom) 1, 2, 4, 10, 100 min. (b) Distribution of grain sizes in Fe~CowBl~ from (a).

(7b) R

[m]

1.10 -7

Thus this approach predicts the exponential time dependence for grain sizes R(t) formed at t-~ 0, the existence of maximum attainable size of crystal grains Rmax( T ) at a given temperature and its temperature dependence.

4. Experimental details

Kinetics and mechanism of crystallization have been investigated in the system Fe85_xCoxB~5 [11, 12]. The alloy F%sCo17B~5 has been selected to test our hypothesis about the proposed grain growth law (eqns. (7a) and (7b)). Samples of Co17Fe68Bt5 10 mm wide have been annealed at 670 K for times from the onset till the end of the first crystallization reaction to obtain the time dependence R(t) and at five temperatures from 650 to 760 K again until the end of the first reac- tion to obtain the temperature dependence Rmax(T). It is to be understood that the values R(t) are maximal grain dimensions observed at time t, while the values Rmax( T ) are grain dimen- sions observed as t--- oo, i.e. at the end of crystal- lization determined by kinetic analysis methods. Grain dimensions have been determined from a series of 10 transmission electron micrographs to obtain satisfactory statistics; the accuracy of R(t) and Rmax(T) determination has been better than _+ 10%.

The shape and size distribution of crystalline grains is shown in Fig. 2. The time dependence R(t) is shown in Fig. 3. It is clearly visible that the parabolic growth law (full circles) seems to hold for small times only, while the exponential law is well fulfilled until the end of the reaction as it is well demonstrated in Fig. 3(b). It is noteworthy that the diffusion coefficient D determined from the parabolic growth law fits well the reported values of diffusion at this temperature [1].

For annealing temperatures as indicated (from 650 to 760 K) the maximal grain sizes ranged from 0.15 to 0.08 /~m. The values of In Rma x plotted against 1/T (Fig. 4) again fit a straight line well with W = 18 kJ mol- ~.

Making further use of eqn. (4) we can write, assuming that for a narrow temperature interval rcrit = ~Vcrit ( T )

E A "~- E l Fl - - + w ( 8 ) Fcr i t

269

0.5404

0

(a)

In Rma x

[m]

-16.3

% , - - . . . . . .

200 400 600 t l s J

m 2

[rn z ]

I × 10 "14

0.5 ~ 1044

3

J L

200 40O 600

(b) t [s 1

Fig. 3. Time dependence of grain sizes in Fe6sCot7Bi5 annealed at 670 K shown: (a) in linear (open circles) and parabolic (full circles) coordinates; (b) in logarithmic coordinates.

w In RmQ , = in Ro~ax + R-~

' ~ x ~ W = 18 k J / m o l e

tn Romo× =-19.2

R O = 5 . 1 0 "o m

-16.1

-159

-15.7 I I

1.3 1.4 1.9 1000 [K-11

T

Fig. 4. Temperature dependence of maximal grain sizes Rm,~x in Fe6eCo17Bi5 glass.

Knowing W, determining E A as the apparent activation energy of the whole process, for example from isothermal resistivity kinetic analy- sis of this alloy [11] to be EA = 120 kJ mol-~ and

270

presuming E~ to be the activation energy of diffu- sion of iron in this system, E~ = 200 kJ mol- ~, as published elsewhere [1, 13], we obtain rcr~/r ~ = 2 o r ncrit = (rcrit/r I )3= 8, w h i c h agrees well with the proposed size of the "bridging complex" [2, 3]. Furthermore, the Avrami n for growth rate described as above fits well both experimental data [11] and the average value n = 1.5 observed elsewhere and attributed to a diffusion-controlled growth mechanism. Further support for this con- sideration is in the fact that the apparently para- bolic R ( t ) has been observed mostly in glasses with relatively lower metalloid content where association formation and nucleation with an exhaustion effect is to be expected.

5. Associations and processes prior to crystallization

Detailed analysis of the distribution function p(r,,) in terms of the Volmer-Becker-Doring theory allows the interpretation of certain pheno- mena observed in metallic glasses prior to crystal- lization. Let us include a time dependence [14] into the function p(r,). The function p(r,) is a steady state distribution of atoms in associations reached as t -~ ~ . Let us start with an ideal un- associated glass at t = O . Then until a certain induction time rR when the associations of size rc~it appear (Fig. 5), the process of formation and disintegration of associations is reversible and the processes influenced by the presence of associ- ations will also be reversible. For times t > rR tWO cases Can occur. At higher temperatures nuclea- tion can take place, thus associations with dimen- sions r~rit are removed from the assembly, making the process irreversible. At lower temperatures, when the associations remain in the assembly, only their size distribution changes with time and the processes can be almost reversible.

Let us attempt to apply our considerations to optimization of thermal treatment of metallic glasses in order to obtain maximum values of desired properties. Let us assume that with increasing size and number of associations the properties increase to a certain maximal size and decrease with further growth of associations and/ or crystallization. Then using an optimal quench- ing rate ~opt (Fig. 6) the optimal state with the maximum of desired properties is obtained. For quenching rates ~ < ~opt, glass with irreparably lower values of properties or partially crystallized glass is obtained. For higher quenching rates glass

p(r~)

11

Fig. 5. Size distribution function p(r,,) with time parameter.

a s

o_

~oDt.

i : tanneal, opt

J i !

J i

number of associations crystattization

annealing time

Fig. 6. Qualitative scheme describing the optimization of properties of metallic glasses.

J J I

J

with a higher degree of disorder although with properties other than optimal can be obtained. These properties, however, can be optimized by a suitable choice of annealing time and temperature

/anneal . opt.( T~ ~ ).

6. Conclusions

Following the experimental observations and the considerations presented it can be supposed that the formation of a critical nucleus of crystal- line phase and its simultaneous growth in the amorphous matrix will be determined by the type, size and number of associations in the matrix.

These, in their turn, will be given by the chemical composition and thermal history of the material in the molten state and during quenching. Then, in the course of crystallization, the formation of nuclei and their growth will be governed by a unique mechanism without the necessity of pro- nounced local change in chemical composition and without substantial rearrangement of associa- tions requesting the disruption of their chemical bonding. Heterogeneous nucleation should not change the proposed mechanism and should only facilitate the formation of relatively stable nuclei. The grains formed during the early stages of crys- tallization in the metal-metalloid glassy systems will be supersaturated and/or metastable and the metalloid from these supersaturated phases will migrate to the interface with the amorphous matrix where a subsequent crystallization reac- tion will take place. The kinetics of this reaction will be governed by diffusion processes on the interface and will, in their turn, determine the morphology of subsequent growth.

271

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