Nanostructured precipitates: Experimental versus exact theoretical saxs profiles

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NanoStrucNred Materials, Vol. 12, pp. 649-652, 1999 Elsevier Science Ltd 0 1999 Acta Metallurgica Inc. Printed in the USA. All tights reserved 0965-9773/99/S-see front maner

PI1 SO9659773(99)00208-l

NANOSTRUCTURED PRECIPITATES: EXPERIMENTAL VERSUS EXACT THEORETICAL SAXS PROFILES V. Garrido’, D. Crespo’, E. Pineda’>‘,T. Pradell’ & M. CapitainJ ’ Dept. Ffsic-a A licada Univ. Politecnica de Catahmya, Campus Nord, Modul B4, 08034Barcelona, SPAl h , e-mail: [email protected].

: ESAB, Univ. Politecnica de Catalunya, Urge11187,08036-Barcelona, SPAIN. European !2ynchrotron Research Facility, BP220 - F - 38043 Grenoble Cedex, FRANCE. Abstract -- Small Angle X-Ray Scattering is one of the few techniques suitable for the determination of the grain size distribution of nanostructured materials. A theoretical developmem has allowed us to determine the spatial self-correlationfunction of such systems and thus the scattering profile associated with the grain size distribution in the SAXS spectra without supposing any specific shape in the precipitates and taking into account their impingement. It has been tested in the primary crystallization of the amorphous Fe 73.sCu,Nb,Si,7,$5 giving excellent agreement with the experimental data. @1999Acta Metallurgica 1~.

INTRODUCTION Microstructural characterization by Small Angle X-Ray Scattering (SAXS) is a powerful analysis technique for the description of the macroscopical properties of nanostructured materials. It is well known that the SAXS intensity of a structured system can be exactly computed (l), provided that the complete description of the system geometry (composition, shape and size of the particles) is available. The inverse process, that is, the description of the microstructure of an unknown system, cannot be performed in general due to the fact that the scattering intensity function cannot be deconvolvecl uniquely. The usual deconvolution is performed by assuming that the system is composed of self-similar particles of known shape (2). The scattering function of the characteristic particle is then computed and the grain size distribution is optimized by fitting the theoretical SAXS spectrum to the experimental data. The above procedure cannot be used with materials obtained by phase transformations driven by nucleation and growth kinetics, where the shape of the grown particles is unknown. In a recent paper (3,4) Garrido and Crespo have shown that the exact characteristic function of such a system can be exactly computed, provided that the grain size distribution is known, without making any assumption about the shape of the precipitates. Once the characteristic function has been obtained, the computation of the exact SAXS profile is straightforward, and comparison with the experimental data can be performed easily.

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FOURTH INTERNATIONAL CONFERENCEON NANOSTRUCTURED MATERIALS

In this paper we show the application of this procedure to the primary crystallization of a FINEMET amorphous alloy of composition Fe,,,, Cu,Nb,Si,,,,B, by isothermal annealing. Experimental SAXS data, obtained at several stages of the transformation, is compared to the theoretical SAXS profile.

GENERALIZED SCATTERING INTENSITY FUNCTION The procedure used in the calculation of the characteristic function of a nanostructured material is summarized in (4), and so we will only recall here the final results. Let F(?,r) be the system global averaged self-correlation function and X(t) the transformed volume fraction at time t. We have shown (3,4) that in the isotropic case the actual form of F(T,t) with impingement is

where Va=~ri

is the volume of an sphere of radius r,, n,(t) is the grain density of particles of is the scaling length, V(t) is the total volume of the transformed

volume V,, L(t)=

phase, N(t) is the total number of particles and f.

k/rJ = [ l-tk/ri)+$(?YrJ3

PI

] 0(2r,-r)

is the self-correlation function of a sphere, 0 (x) being the Heaviside function. The scattering intensity of a grain structure can be expressed (5) in terms of the scattering field function Y(Y,t) =

& &$JFJ)

131

i=O

where &i is the scattering amplitude of the i-th. phase and gi(T,t) is a bi-valued function which describes the geometry of the microstructure, being 1 inside the particles of the i-th. phase and 0 elsewhere. In terms of the characteristic function above defined, the scattering intensity is Z(q’,t)=

‘wIq?,t)Y(~,t)> /dTe

= 1[l -X(t)]Eo + X(t)il”

= (2n>36(q3 + {[l-X(t)]

F(q’,t) = J’dJe ‘FF(T,t) = x-*(t)

IEo-iI*

+ .-j-g)

X(t) F(fj,t)

[41

L(t) 1

-JET c c n,(t) n,(t) v, VP 5;* 4rg’y ( q3 Y P

where the scattering amplitudes are thermally averaged, 6 is the Dirac-deltadistribution and J3,*(x) is the Bessel function of fractional order 3/2.

FOURTH INTERNATIONAL CONFERENCEON NANOSTRUCTURED MATERIALS

RESULTS

651

I

Isotherm (490°C) The isothermal nanocrystallization of -----------CUclustering Finemet exhibits two overlapped processes (6), 2 -----F&i crystallization as shown in Fig. 1. On the one hand a decaying g 200calorimetric signal corresponding to the g formation of Cu clusters.There is no temperature 2 100dependence on this signal in the range 43O”C- $! 55O”C, indic,ating a very fast process of growth. 0_ II I ,tl Direct evidences from TEM images were also 03 10 30 60 given by Hono (7). t(min) On the other hand, following this first Figure 1 Isothermal DSC data. an exothermic process but overlapped, asymmetric calorimetric peak appears. This peak is related to the nanocrystalline precipitation of a Si depleted FeSi BCC phase with DO, ordering. The asymmetry is related to a diffusion controlled growth process with decreasing growth consequence of the accumulation of diffusing species in the matrix and known as soft impingement. The growth rate is given by (8) 300-

dr -=-dt

WT) Y-W) r

1 -X(t)

PI

where y accounts for the transformed fraction at the end of the primary precipitation and D(T) is the diffusion coefficient of the slowest diffusing species, which in FINEMET corresponds to Nb. Nb is a well-known stabilizer of the amorphous phase and therefore a reduction of the nucleation rate should also be expected as transformation proceeds. We have taken (8) IV,@ =~,ux 1-x(w)

[61

where T is the time at which the nucleus forms, and I,,(T) is the nucleation rate obtained for an undercooled liquid. The grain size distribution along the transformation has been evaluated from these kinetic parameters b:y using the PKJMA model (9) giving the final distribution shown in figure 2. From this data the exact correlation functions are evaluated by using Equation (1). The addition of the SAXS profile corresponding to the nanocrystalline precipitates and the amorphous matrix are also shown in figure 2 after 10 and 60 minutes isothermal annealing at 490°C. It can be clearly seen that a second1grain size distribution of smaller sizes has to be added in order to fit the SAXS profile at large q values. While introducing an extra grain size distribution with average size of about 2 nm, we can obtain a good fit of the experimental profile. The sizes of this phase agree with the measured sizes of the segregated Cu-clusters (7). Very small changes in the size of the clusters are found between the first minutes of isothermal annealing and the end of the nanocrystallization. However, although the relative intensities are obtained from the SAXS profile, it is not directly related to the number of clusters by unit volume because of the different absorption coefficients between the nanocrystalline precipitate and the Cu clusters.

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FWRTH INTERNATIONAL CONFERENCEON NANOSTRUCTURED MATERIALS

5x10”

-

F&i

crystals

+

Exp%ime”tal

c”c,“~s ~~~~~~~~~ W

0

2

4 * Mm)

6

8

0.4

0.8

qhm-’

1.2

0.4

0.8

1.2

qhm-’

Figure 2 Computed grain size distribution at the end of the isothermal annealing at 490°C (left). Experimental and computed SAXS profile after 10 (center) and 60 (right) minutes. CONCLUSIONS In this paper we have presented a new method of calculating the scattering intensity function in systems underlying first order phase transitions. The grain size distributions used are obtained by the PKJMA model. The procedure has been applied to determine the SAXS profile associated with the grain size distribution appearing in the primary cristallization of the amorphous Fe,,,Cu,Nb,Si,,,B,. Comparison with the experimental profile shows a discrepancy corresponding to an additional grain size distribution of smaller size. According to the DSC data and also to microstructural analysis, this grain population corresponds to the clusterization of the Cu present in the amorphous, which occurs previously to the primary crystallization. This work was partially financed by DGICYT, grant PB94-1209, Generalitat de Catalunya 1997SGR 00039 and UPC, grant PR9505. SAXS measurements were performed at the European Synchrotron Research Facility at Grenoble, experiment HS-447.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

Guinier, A. and Fournet, G., Small-angle scattering of X-rays, John Wiley and Sons, New York, 1955. Glatter, O., J. Appl. Cry&, 1980, 13,7. Garrido, V. and Crespo, D., Phys. Rev. E, 1997,56,2781. Garrido, V., and Crespo, D., “Characteristicfunctions of nunostructured materials”, in this volume. Ohta, S., Ohta, T. and Kawasaki, K., Physica, 1987, 140A, 478. Pradell, T., Clavaguera, N., Zhu, J. and Clavaguera-Mora, M.T., J. Phys.: Condens. Matter 7 1995,4129. Hono, K., Hiraga, K., Wang,Q. and Sakurai, T., Sulface.Sci. 266 1992,385. Pradell, T., Crespo, D., Clavaguera, N. and Clavaguera-Mora, M.T. J. ofPhys: Condensed Matter 10 1998,3833. Crespo, D. , Pradell, T. , Clavaguera-Mora, M. T. and Clavaguera, N. , Phys. Rev. B, 1997, 55, 3435.

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