NanoSTRUCTURED MATERIALS VOL. 1, PP. 59-64, 1992 0965-9773/92 $5.00 + .00 COPYRIGHT ©1992 PERGAMON PRESS plc ALL RIGHTS RESERVED PRINTED IN THE USA

GRAIN GROWTH IN MICROCRYSTALLINE MATERIALS STUDIED BY CALORIMETRY

L.C. Chen* and F. Spaepen Division of Applied Sciences, Harvard University

Cambridge, Massachusetts 02138 *General Electric Research and Development Center

Schenectady, NY 12301

Introduction

Materials with a fine scale microstructure can have unusual physical and chemical properties [I]. The use and processing of these materials for scientific studies or technological applications requires stability of the microstructure, which is often limited by the phenomenon of grain growth.

Differential calorimetry can be used very effectively in studying the nature of nanoscale structures and their stability. For example, a grain growth process, in which the larger grains in a population grow, on the average, at the expense of the smaller ones, is usually studied by direct observation in the optical or electron microscope. If the grain size is less than about lOnm, precise direct measurements become difficult. The large interface-to-volume-ratio in nanoscale materials, on the other hand, makes it possible to monitor grain growth calorimetrically, at least if the grain boundary mobility is relatively high, as in metals.

Another incentive for the use of calorimetry on these systems has been the difficulty of distinguishing whether a material that exhibits broad diffraction halos and appears featureless in dark-field transmission electron microscopy is microcrystalline (i.e. an assembly of randomly oriented fragments of bulk crystalline phase) or truly amorphous (i.e. one with no translational symmetry at all, as in liquids and glasses). The nature of the transformation of both structures to one with sharp diffraction rings is fundamentally different, however, and calorimetry can often distinguish these transformations clearly by their kinetics. Observation of the characteristic increase in specific heat [2] upon heating through the glass transition temperature is the traditional method for identifying a liquid-like structure [3]. The absence of such evidence for a glass transition does not necessarily mean that the structure is not amorphous. The fact that it is not often observable is sometimes because the change is imperceptibly small, or is obscured by the onset of crystallization. Observation of a grain growth process upon heating, on the other hand, is clear evidence for a nanocrystalline structure.

Differential calorimetry is a convenient and widely used method for studying the kinetics of phase transformations. This can be done in either the isothermal or linear heating (scanning) mode. In isothermal experiments the transformed fraction, proportional to the integrated signal, is obtained as a function of time, and is usually fit to the Johnson-Mehl-Avrami {JMA) formalism [4] to determine the mechanisms that govern nucleation and growth. In scanning experiments, similar kinetic parameters can also be obtained. The best known method is the Klssinger analysis of the shift of the transformation peaks as a function of heating rate [5,6].

This paper is a brief review of our work on the anlaysis of calorimetric data obtained from grain growth processes. Both the isothermal and scanning signals have characteristic shapes, and can be analyzed to yield the grain growth parameters: grain growth exponent, mobility, activation enthalpy. At the same time, the absolute average interfacial enthalpy can also be determined directly from such studies. Furthermore, the qualitative differences between the signals from nucleation-and-growth and grain growth are pointed out. These are useful, in combination with diffraction, to distinguish truly amorphous from microcrystalline structures. As an example, grain growth in microquasicrystalline sputtered AI-Mn thin films will be discussed. A more extensive discussion of this work can be found in some of our earlier papers [7,8,9,10].

59

60 LC CHEN AND F SPAEPEN

The Enthaloy Evolution in a Grain Growth Process

The driving force for grain growth is the decrease in interracial free energy. The rate of increase of the average grain radius, r, is generally written as dr/dt = cM~ny/r n'', where c is a numerical factor, M is the mobility, y is the interfacial free energy, ~ is the interatomic distance, and n is an exponent, empirically between 1.5 and 4 [ I l l . Most theoretical models give n=2 [12,13]. The average grain radius as a function of time is then

c

rn(C) = r"(O) + F k ( ~ dC , where k(T) = ncM~ny. The temperature dependence of this rate 0

constant can be written as k(T) - (ko/T) exp(-Oj'ksT ) [9]. This factor is somethimes assumed to have a simple Arrhenius-type temperature dependence:

k(T) = koexp(-O/keT). (I)

In the original Kissinger analysis the rate constant is treated in the lat ter way. In an isothermal experiment the evolution of the grain radius is:

rn(t) = rn(o) + k(T)t. (2)

In a scanning experiment, at constant heating rate b - dT/dt, the grain size evolution is:

T

_r"(t:.) = rn(O) + / k---~-dT = rn(O) + I (T ) I b . o

(3)

The total interracial enthalpy of a system with volume V and average grain radius r is:

H = gyHV/r, (4)

where y~ is the enthalpic part of the interfacial free energy, and g is a geometric factor, on the order of unity. As the grains grow, the total interfacial enthalpy decreases:

H(t) = HorJr(t), (5)

where H o and r R are, respectively, the in i t i a l enthalpy and grain radius. The calorimetric signal is the rate, H, at which the enthalpy of the sample changes ( a negative quantity for exothermal processes):

(6)

12 dt

By combining equations (2), (3 ) and (6), both the isothermal and scanning signals can be calculated.

As i l l us t ra ted in Fig. 1, the isothermal signal in a grain growth process is always monotonically decreasing. In the case of c rys ta l l i za t ion , on the other hand, a f i rs t -order t ransi t ion from amorphous to a new phase by nucleation-and-growth, the isothermal signal usually has a character ist ic peak shape. In some unusual cases, such as thickening of needles or plates originating from pre-exist ing nuclei, i t is possible to have monotonically decreasing calorimeter signals. However, such conditions are immediately apparent in microscopic observation. Therefore, i f when materials with broad d i f f rac t ion halos upon heating transform to polycrystal l ine phases with sharp d i f f rac t ion peaks, a monotonically decreasing isothermal transformation signal is observed, and the product does not show any of the unusual morphologies described above, the transformation is one of grain growth, and the original phase is microcrystal l ine [7,10]. I t should be kept in mind that the reverse is not necessarily true: observation of a peak in the isothermal transformation signal does not prove that the structure is amorphous. A microcrystalline structure could transform by abnormal grain growth or by nucleation and growth of a new crystall ine phase, both of which would also give a peak in the isothermal experiment.

Typical shapes of the scanning signals at various heating rates are shown in Figure 2. Figure 3 shows the change in peak shape for variations in the i n i t i a l grain radius, r o, the activation enthalpy Q, and the prefactor of the rate constant, k o. Notice that the leading edge of the grain growth peak is steeper than the t ra i l ing edge. A comparison of these curves with the analogous ones for nucleation-and-growth reveals some interesting differences and similarit ies

MICROCRYSTALLINE MATERIALS GRAIN GROWTH 61

[]0,14]. For example, preanneallng of a grain growth sample increases the i n i t i a l grain radius, r o, which shifts the onset of the peak to higher temperatures and leads to a narrower transformation range. Preannealing of a nucleatlon-and-growth sample results in partial transformation of the material. Subsequent scanning then gives a broader peak that is shifted to lower temperatures.

Determination of the Grain Growth Parameters

The grain growth exponent, n, can be obtained most accurately from the integrated signal in C

an isothermal experiment, i .e . , /Y(t) = H o + f ~ d t . Using equations (2) and (5), the evolution 0

of the total interfacial enthalpy in a grain growth process can be most conveniently expressed by:

(7)

---#~-o ) r~+k(~ t t+---~ '

where t = r ~ l k ( ~ . To obtain the grain growth exponent, n, the parameter f is determined by adjusting a plot of ln( t+r) vs ln(H/Ho) to give a straight l ine. The slope of that optimized l ine corresponds to -n.

In the standard Kissinger analysis of a nucleation-and-growth process the activation enthalpy can be determined by p lot t ing, for a number of scan rates b, ln(b/To) vs (1/To), where Tp is the peak temperature (maximum) of the transformation signal. The slope 6f a straight l ine f i t through the plot is then -q/k B. For the temperature range of interest, Q>>keT, and assuming Arrhenius-type temperature dependence of k(T), the same analysis applies to a grain growth process as well. I f a correct kinet ic factor is used instead, -0j'k e is the slope of a plot of ln(b/To) vs (1/T,). However, for the usual conditions in calorimetric experiments, the values of Q obtained by b6th methods are identical to within less than a tenth of an eV [9].

The activation enthalpy of the rate constant can also be determined by performing a series of isothermal experiments at d i f ferent temperatures. Since a d i f ferent amount of grain growth occurs during heat-up to these di f ferent temperatures, the i n i t i a l inter facia l enthalpy Ho(T) depends on the holding temperature. This quantity is d i rect ly measurable by integrating the entire isothermal signal, since R(t) becomes negl ig ibly small for r>>r o. The time t l . . is defined as the

• . . !

time required to lower the total in ter fac ia l enthalpy to hal f i ts i n i t i a l value h(twz,T) = Ho(T)/2. Since the grain size doubles over that time, the following condition is obtained fr6~equation (2):

n (2"-I) to (T) = k(T) tz/2. Using equation (5), we can write ~o(T) X(T) tz/2 = C, where C is a temperature-independent constant. Depending on the form of the temperature dependence of k(T), the slope of a straight-l ine f i t of in[t~/zI-I~o(T)/T] or In[tz/z/f~o(~] vs ]/T corresponds to Q/k R. Again, the difference between the two determinations is negligibly small.

In the calorimetric analysis, the grain radius, r, and the kinetic parameter, k, always occur together, as in the measurable parameter • = r ~ / k ( ~ , for example. The value of r o, or r at any other time, must therefore be obtained from other observations. The most direct technique is high- resolution electron microscopy. I t is, however, often d i f f i cu l t to observe the in i t i a l grain size directly, since even in the thinnest samples many nanometer-size grains overlap []5]. Measurements of the grain size after some growth are somewhat more feasible. A second technique is based on the broadening of the dif fract ion peaks, but this is to some degree model-dependent. Although a precise determination of the absolute grain size is therefore d i f f i cu l t , i t is possible to get a reasonable estimate i f the two methods give similar results.

Once r o is known, the average enthalpic part of the interfacial free energy, YH, can be estimated from the area under the transformation peak in a scanning experiment, i .e . , H o, and equation (4). The main uncertainty here is then the value of the geometrical factor g which depends on the shape and size distribution of the grains. For equiaxed grains g is estimated at 1.3 ± 0.2 [g], which gives an uncertainty of about 30% for y,.

Case Study

Sputtered Al~Mn~7 thin films were mechanically removed from the substrates and studied in a Perkin Elmer DSCTI calorimeter. In the scanning calorimetry traces, two transformations are observed (Fig. 4). X-ray and electron microscope observations [8] show that the as-deposited

62 LC CHEN AND F SPAEPEN

materials have broad, amorphous-like d i f f rac t ion peaks. During the f i r s t transformation the d i f f rac t ion pattern sharpens up continuously to a sharp, icosahedral one, and after the second scanning peak, the mtcrostructure consists of somewhat larger, equiaxed periodic crystals. The enthalptes released from the f i r s t and second transformations are ],050 J/mole and 3,822 J/mole, respectively. Notice that the total heat given of f from the f i r s t transformation is considerably smaller than in the second one. A Kissinger plot (shown in Fig. 5) of the peak temperatures for the f i r s t transformation of Fig. 4 gives Q - 243 kJ/mole. The act ivat ion enthalpy observed here is much higher than in pure A1 [16]. This is not ent i re ly surprising since solute segregation is l i ke l y to occur tn the al loy. A small addition of Sn to Pb, for example, has been reported to have increased the act ivat ion energy of the grain boundary veloci ty dramatically [17].

In the isothermal experiments, samples were heated up at a rate of 20K/min to the transformation temperature, and held unt i l the signal was no longer measurable. After cooling to room temperature, the sample was run again under identtca] conditions to establish the baseline, which gave a perfect ly horizontal isothermal l ine, ru l ing out the poss ib i l i t y that the signal is an instrumental a r t i fac t . That the signals of Figure 6 a l l decay monotonica]ly demonstrates unambiguously that the transformation process is one of grain growth. The total enthalpy released in the isothermal experiments, Ho(T), is 932, 857, and 777 J/mole, respectively. The difference between these values and that measured in the scanning experiment is due to grain growth during heat-up. Figure 7 shows a plot of the isothermal data according to equation (7), using f = 3, 2.24, and 0.65 minutes, respectively. The corresponding grain growth exponents, n, derived according to the procedures outl ined above, are 1.9 ± 0.2, c]ose to the va]ue n=2 used in most theoretical models. Figure 8 shows a plot of l n [ t l / 2h~o(~ ] vs ]/T for the three isothermals. The slope of the s t ra igh t - l ine f i t corresponds to Q - 212 kJ/mole, which is s imi lar to the value obtained by the Ktsstnger method. That the kinet ic factors obtained in isotherma] and scanning experiments are not precisely the same is most l i ke l y due to some temperature dependence of some of the parameters assumed to be constant in the theory, such as the in ter fac ia l free energy.

The grain radius in the as-deposited material, to, was determined from the broadening of the d i f f rac t ion peaks to be I6A, consistent with htgh-resoTution electron microscopy observations [8]. This, taken as the i n i t i a l grain radius at the lowest isothermal testing temperature (563K), corresponds to a kinet ic coef f ic ient k(563K) = ~ / ~ (5630 = 1.078Al 'gs -~. Since the activation enthalpy, Q, ts known from the Kissinger plot~th)~p~efactor of the kinet ic coef f ic ient can be determined from equation (1): k o - 4.316 x 10"" A ' " s " . The i n i t i a l grain radius of the higher temperature tsothermals can be calculated based on equation (5): ro T1/ro Tz " Ho T2/Ho T~ Using these values, the isothermal runs can be simulated with equat ion ' (5) . ' FiguFe 6'shows the comparison with the data. That the discrepancy between data and simulation is greatest in the i n i t i a l growth stage is to be expected, since deviations from the macroscopic grain growth model are most l i ke l y to occur in th is regime.

The average enthalpic part of the inter facia l tension, y,, is determined from the total transformation heat, 1050 J/mole, measured in the scanning experiment and equation (4). Using a geometrical factor g- l .3 ± 0.2, as discussed above, and vlg.184 cm~/mole [18], gives y, - 0.14 ± 0.03 N/m. Compared to pure aluminum, th is value is about hal f of the average grain boundary energy ~19], and only two times larger than the twin boundary energy [20]. Recent study indeed suggests that polycrystal l tne aggregates with an overall icosahedral symmetry have a very low energy [21].

Conclusion Di f ferent ia l calorimetry is a re la t i ve ly simple and powerful technique in the study of the

grain growth process of nanometer-scale materials where i t is almost impossible to monitor the evolution of the grain size accurately by di rect observation methods. The high interface-to- volume-ratio of the nanophase materials makes i t possible to determine the average interfactal enthalpy, which is not known in many materials. A monotonically decreasing isotherma] signal associated with the transformation behavior is unambiguous evidence for grain growth in a mtcrocrystalltne structure. Scanning peaks that are low and wide with a long high-temperature t a i l , and which sh i f t to higher temperatures after pre-annealing are also an indication of grain growth. The technique described here, applied to sputtered Al-Mn f i lms, both to demonstrate that they are micro-quastcrystal l ine and to extract the kinet ic parameters of grain growth can obviously be applied to many of the nanophase structures of current interest.

MICROCRYSTALLINE MATERIALS GRAIN GROWTH 63

Acknowledgements This work has been supported bythe National Science Foundation through the Harvard Materials

Research Laboratory under contract number DMR-86-14003 and by the Office of Naval Research under contract number NOOO14-85-K-O023. LCC has been supported by a predoctoral fellowship from IBM Corporation.

References I. R.W. Siegel, S. Ramaswamy, H. Hahn, Li Zongquan, Lu Ting, and R. Gronsky, J. Mat. Res. 3,

1367 (1988). 2. W. Kauzmann, Chem. Rev. 43, 219 (1948). 3. H.S. Chen and D. Turnbull, J. Chem. Phys. 48, 2560 (1968). 4. J.W. Christian, The Theory of Transformations in Metals and Alloys, 2nd edition, Pergamon,

Oxford (1975). 5. H.E. Kisstnger, Anal. Chem. 29, 1702 (1957). 6. D.W. Henderson, J. Non-Crystl. Solids 30, 301 (lgTg). 7. L.C. Chen and F. Spaepen, Nature 336, 366 (1988). 8. L.C. Chen, F. Spaepen, J.L. Robertson, S.C. Moss and K. Hiraga, J. Hat. Res. 9, 1871 (1990). 9. L.C. Chen and F. Spaepen, J. Appl. Phys., 69 679 (1991). 10. L.C. Chen and F. Spaepen, Mat. Sci. Eng. A133, 342 (]ggl) . 11. H.V. Atktnson, Acta Metall. 36, 469 (1988). 12. i .E. Burke and D. Turnbul], Prog. Metal Phys. 3, 220 (1952). 13. C.V. Thompson, H.J. Frost, and F. Spaepen, Acta Metall. 35, 887 (1987). 14. A.L. Greer, Acta Hetal l . 30, 171 (1982). 15. See, for example, the ultrahigh resolution images at 400 keV of Al-Hn fi lms (Figure 13 in

Ref. 8). 16. H. Schmttten, P. Haasen, and F. Haeszner, Z. f . Metallkde. 51, 101 (1960). 17. J.W. Rutter and K.T. Aust, Acta Met. 13, 181 (1965). 18. D. Turnbull, Acta Metall. 38, 243 (1990). 19. H. Gletter and B. Chalmers, Prog. Hat. Sci. 16, 13 (1972). 20. L.E. Murr, Acta Met. 21, 791 (1973). 21. L. Bendersky, J.W. Cahn, and D. Gratias, Phi]. Mag., [360, 837 (1989).

4 nm

2

rll I I I I

dH/~ I

o I J 10 20 3(} 40 50

~me rain

Fig. 1. Isothermal transformation of a micro- crystall ine material by grain growth. (top) Evolution of the average grain radius. (bottom) Corresponding enthalpy release. [From ref. 7].

12! 10

mW/J 8

6

4

dH/dt I 2 0 5O0

40K/rnin

550 600 650 700 750 Temperature K

800

Fig. 2. Simulation of the transformation of a mi crocrystal Iine sample by non-i sothermal grain growth at various heating ratesao~he parameters are: n=2, r^=lOA, k^=I3x10:°AL-K/ sec, Q=241kJ/mole (or ~.SeV). ~he units of dH/dt are normalized with respect to H o.

64 LC CHEN AND F SPAEPEN

12

a

105 r°=

mW/J 1 ~

'2 4~

2

I 0

10 Inko = 58.5 56.2 b

F A6 °

50O 550 600 650 700 750 800 Temperature K

I,

Fig. 3. The change in scanning calorimeter signals of grain growth for variations in the (a) i n i t i a l grain radius, (b) atomic jump fre- quency, and (c) activation enthalpy (from ref.9).

i" A : // ~I,. . . . . . . - / ~ , , .

Temp~atme K i=

Fig. 4.Scanning trace from a sputtered A183Mn17 f i lm, heated at 40K/min. (from ref .7) .

"111.58 1.6 1 , ~ 1 . ~ 1 . ~

Fig.5. Kissinger plots of the f i rst transfor- mations of Fig.4. (from ref.8).

I

~ K 0,6

~ = 1 ~

~o., ~

O / I ~ . . . .

0.8 573K

. 17.4A 0.6

O.4

O.2

5

5~K 1.2

1o.5 O4 ~ _

O ..... .-_,_ - i 10 15 20 ~me 5 rain

Fig. 6. Isothermal calorimetry traces at three temperatures from A183Mn17 sputtered f i lms. The dashed l ines are simulations according to

equation (6) (from ref.9).

• o ~K

573K

n=l.~

3.5 4 4.6 5 5,5 In H

Fig~ 7.Plot of the isothermal data according to eq.(7), to determine the grain growth exponent. n=1.9±0.2 from ref. 8) .

6 . 5

5.5 o o

!l- 170 1 ~ 174 176 176 150

1 ~ 10 -S K'I

Fig. 8. Plot of the isothermal decay time to dete~ine the activation enthalpy. Q=212kd. (from ref. 8).