DISTRIBUTION OF THERMODYNAMIC PROCESSESCONTROLLING (NANO)CRYSTALLIZATION OF

IRON-BASED METALLIC GLASSES

K. Kristiakova and P. SvecInstitute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia

(Received August 21, 2000)(Accepted in revised form December 12, 2000)

Keywords:Transformation rates; Activation energy; Nanocrystallization; Nucleation

Introduction

Metallic glasses and related alloys prepared by rapid quenching are expected to exhibit several specificfeatures which concern the local arrangement of constituent atoms. The existence of local atomicordering in the amorphous matter necessarily has to have an influence on the complexity of theprocesses controlling crystallization in amorphous materials. The initial thermodynamic state of eventhe simplest amorphous system can be distributed over a range of possible local orderings and energeticstates around a certain mean value. Upon transformation to crystalline states, therefore, a distributionof processes controlling this transformation is to be expected, reflecting the same mechanism, yetdiffering in the initial local state and thus in the corresponding activation energy of the process. Our aimis to analyze theoretically the influence of the distribution of these activation energies and their eventualtemperature dependence on the transformation behaviour of metallic glasses and to apply the developedapproach to identify microprocesses controlling nanocrystallization in Fe73.5Cu1Nb3Si13.5B9, alloyusing isothermal changes of a suitable physical property which reflects this transformation process.

Distributions of Process Rates

In complex systems the existence of local heterogeneity is expected to generate distribution ofmicroprocesses with specific rates which take place in the course of the transition to a more stable state.Assuming continuous distribution of processes, under isothermal conditions the transition is generallyexponentially time-dependent, obeying a general formula [1] of the type

x~t! 5 1 2 E0

`

a~l!exp@2~lt!n#dl~1! (1)

with the normalization conditionE0

`a~l!dl 5 1 wherex(t) reflects the proceeding of the transition (e.

g. crystalline volume fraction) controlled by continuous distribution of processes with ratesle(0, `),a(l) is the probability density function of the process having the ratel andn is a parameter reflectingthe type and mechanism of the process (Kohlrausch exponent, Avrami exponent, etc.). From mathe-

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matical point of view the problem can be solved as the Fredholm integral equation of the first kind [2]with known kernel and the solution being equal toa(l). The time dependence ofx(t) is obtained viameasuring of the propertyP using a convenient instrumentation which has its own resolution function,yielding values ofP convoluted with the resolution function. The effect of measuring instrumentationand experimental noise is eliminated by the use of deconvolution procedure and inverse Laplacetransformation. The probability density function of transformation rates,pdf(l), is obtained withoutdirect determination of the instrument resolution function and without any assumptions about thedistribution of process rates [3].

Distributions of Activation Energies

The distribution of processes as a function ofEia can be obtained by transforming the process ratesl(T)expressed as

l~T! 5 l0exp~2Ea/RT! (2)

into activation energiesEa and by recalculating thepdf(lnl) into pdf(E) using the normalizationcondition forEa; l0 is the preexponential factor. We expressdE 5 1/(­ln l0/­E 2 1/RT)dlnl from eq.(2). The corresponding rates forE 5 0 and E5 ` arel0 and 0, respectively. PuttingL 5 ­ ln l0/­Ewe obtainpdf(E) 5 (1/RT2 L)lpdf(l) 5 (1/RT2 L)pdf(lnl), which can be used for transformationof pdf(l) to pdf(E). For thermodynamically simple casesl0 5 const., independent onT or on Ea andthe quantityL 5 ­ ln l0/­E 5 0. The transformation relation takes the simple formpdf(E 5 (1/RT)pdf(lnl). For themodynamically real cases where a real dependence of activation energy on tempera-ture, Ea 5 Ea(T), may be expected or interdependence ofEa andl0 is possible, i. e.l0 5 l0(Ea), adeviation from ideal Arrhenius case is obtained. ThusL 5 ­ ln l0/­E Þ 0, pdf(lnl) for differenttemperatures do not fall onto a single curve andL 5 L(E) has to be evaluated. In order to be able torescale (i. e. unify) the isothermal time dependencies of the measured property and the consequent ratedistributions over all annealing temperatures it is necessary to construct a matrix of suitable timescorresponding to the same processes active at these times. The criterion for their selection is such thatthe times must correspond to the specific amount of transformed volume at each annealing temperature(degree of crystallinity). At each of these times only certain part of wholepdf(lnl) is active in the timeinterval fromt to t1dt. The entire process can be divided into processes annealed out till timet (whichhave produced the corresponding crystallinity), processes active at timet and processes still awaitingactivation (after timet). The distribution of process rates active at timet can be obtained fromdistribution of processes annealed out,Panneal(t,T), given by the subintegral function ofx(t) asPactive(t,T) 5 (­Panneal /­ t)dt 5 la(l)n(lt)n[exp[-(lt)n]]dlnt, where Pactive(t, T) is the portion ofprocesses active betweenlnt andlnt1dlnt suitable for further calculations. For each subdistribution thei-th moment has been calculated as

Mi 5 Eln lmin

ln lmax

l i11 Pactive~t, T!d lnl. (3)

The average value,,l., of the subdistribution given by,l. 5 M0 /M-1 can be easily numericallyevaluated. Thus we obtain,l. of the subdistributions at the same degrees of crystallinity and thecorresponding values ofpdf(,l.). Applying l 5 l0 exp(2Ea/RT) the process rates can be transformedto the scale of activation energies, providing the values ofl0 and Ea for different fractions oftransformed volume. Then,pdf(ln l) at these values ofl can be converted topdf(E) using the valuesof L computed by a suitable minimalization procedure. Using the values ofL we can rewrite eq. (2)

IRON-BASED METALLIC GLASSES1276 Vol. 44, Nos. 8/9

more realistically asl(T) 5 l00 exp(2Ea/RT), where l00 is truly constant andE(T) reflects thetemperature dependence of activation energy. ExpandingE(T) around mean annealing temperatureT0

into Taylor series asE(T) 5 E(T0)1E9(T-T0) and puttingE(T0) 5 E0 and E9 5 dE/dT at T0, wesubstitute forE(T) to getlnl 5 ln l00 2 E9/R 2 (E0 2 E9 T0)/RT. By comparison with eq. (2) we obtainln l0 5 ln l00 2 E9/R, Ea 5 E0 2 E9 T0, thus L(E a) 5 2(dE9(Ea)/dEa)/R. Knowing L(Ea) theequation above may be considered as differential equation which can be solved by numerical integrationfor E9(Ea)

E9~Ea! 5 REEa

Ea,L50

L~Ea!dEa (4)

From E9(Ea) the values ofl00 andEa can be immediately computed. In order to obtainpdf(E0) frompdf(Ea) one must again use the normalization condition to getpdf(E0) 5 pdf(Ea)/[1 - RT0L(Ea)]. To beable to compute such distributions for other temperatures the process described above has to berepeated, transforming fromE0 to trueE(T), yielding

pdf~E! 5 pdf~E0!@1 2 R T0L~E0!#/@1 2 R T L~E0!#. (5)

Results and Discussion

The model-independent method for obtaining the distribution of transformation rates has been appliedto the isothermal crystallization of the Fe73.5Cu1Nb3Si13.5B9, using time dependencies of electricalresistivity (Fig. 1) measured from 752 to 800K, assuming nucleation and growth mechanism. Thedependencies, while being sigmoidal in shape, exhibit a peculiar behavior in the initial stages ofannealing. This is evident especially at lower annealing temperatures and may be attributed toCu-clustering observed by APFIM [4]. Pronounced increase in volume content of the nanocrystallinephase corresponds to sharper decrease of resistivity which ends when this value reaches;50% of theentire volume, the grain sizes being;10–12nm. Further (slight) grain size increase up to;15nm isobserved with increasing temperature and time of annealing [5]. The first two processes are wellreflected by the rate distributions (Fig. 2) obtained using the above-described approach. The small broadpeaks in Figs. 2 and 3 which exhibit a specific evolution with temperature reflect the effect ofCu-clustering [4] observed in the early stages of transformation prior to the nucleation of thenanocrystalline phase. The increase of nanocrystalline volume content is reflected by subsequentsubstantial increase of the probability density of transformation rates. The distribution of true activation

Figure 1. Experimental time dependence of the isothermal electrical resistivity of Fe73.5Cu1Nb3Si13.5B9 in the course ofnanocrystallization.

IRON-BASED METALLIC GLASSES 1277Vol. 44, Nos. 8/9

energies for all annealing temperatures (Fig. 3) shows two intervals of activation energies. The first isclearly temperature dependent and shifts to higher values of activation energies with increasingannealing temperature, which is in accord with the general temperature behaviour of nucleationprocesses. The second is quite stable at;430kJ/mol, as witnessed by low values of the temperaturecoefficient of activation energy, E9, and corresponds to the formation ofa-Fe(Si) nanograins [5]. Thedensity of processes with this energy relatively increases with annealing temperature. For higher valuesof activation energies the temperature dependence vanishes, as may be expected for the simple diffusionmechanisms responsible for the slight size increase of nanograins. The value of the preexponentialfactor l00 has been found to be strictly constant (1.260.2z1028 s21). The obtained temperaturedependence ofpdf(E(T))allows to extrapolate the annealing behaviour to low temperatures.

Conclusions

Using a novel model-independent continuous rate distribution approach to analysis of isothermaltransformations in Fe73.5Cu1Nb3Si13.5B9 alloy it has been possible to determine the distribution of trueactivation energies and their temperature dependence as well as the value of preexponential factor, toidentify different stages of nanocrystallization, namely the effect of Cu precipitation in the initial stagesleading to the subsequent nucleation and growth of nanograins and to quantify the energetics of theobserved stages.

Figure 2. Normalized transformation rate probability density functions for the isotherms from Fig. 1.

Figure 3. The dependence of temperature coefficient of activation energy, E9 5 dE/dt at T0 and probability density functions(distributions) of activation energies, pdf(E(T)) vs. E(T). Dotted arrows indicate temperature evolution of different processescontrolling nanocrystal formation.

IRON-BASED METALLIC GLASSES1278 Vol. 44, Nos. 8/9

Acknowledgment

The authors express their thanks to the Grant Agency for Science of Slovakia and NATO Science forPeace for the support of this research (Grants No. 2/6064/99 and SfP-973649).

References

1. W. Primak, Phys. Rev. 100, 1677 (1955).2. A. N. Tichonov and A. A. Samarskij, Equations of Mathematical Physics, Nauka, Moscow (1951) (in Russian).3. K. Kristiakova and P. Svec, JMNM. accepted.4. K. Hono, D. H. Ping, M. Ohnuma, and H. Onodera, Acta Mater. 47, 997 (1999).5. I. Matko, P. Duhaj, P. Svec, and D. Janickovic, Mater. Sci. Eng. 179–180, 557 (1994).

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