Structural relaxation in some metallic glasses

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ITRUCTURAL RELAXATI

SOME METALLIC GLAS.

I SOME METALLIC GLASS!

PROEFSCHRIFT

Ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus, prof. dr. J.M. Dirken, in het openbaar te

verdedigen ten overstaan van een commissie aangewezen door het College van Decanen op dinsdag 19 mei 1987 te 16.00 uur

door Erik Huizer geboren te Rotterdam, Metaalkundig ingenieur.

Dit proefschrift is goedgekeurd door de promotor prof. dr. ir. A. van den Beukei.

1. De weerstandsverandering tijdens isotherm gloeien van as-quenched materiaal gemeten na een konstante tijd, kan niet zonder meer vergeleken worden met een isochrone weerstandsverandering. Dit temeer als de isotherme metingen bij de gloeitemperatuur zijn verricht en de isochrone metingen op 4 K.

E. Woldt andJA. Leake, in proc. of6th Int. Conf. on liquid and Amorphous metals, Garmisch-Partenkirchen, (1986), verschijnt binnenkort.

E. Kokmeijer, afstudeerverslag, Lab. v. Metaalkunde, T.U. Delft. (1986).

2. De "threshold stress" bij kruipproeven aan amorfe metalen, zoals voorgesteld door Taub, is een overbodige parameter als men rekening houdt met strukturele relaxatie.

Al. Taub, Acta Metall., 20. (1982) 2129. Dit proefschrift hoofdstuk 6.

3. Hoewel in de literatuur gepresenteerde metingen op het gebied van strukturele relaxatie in amorfe metalen meestal isochrone metingen zijn, verdient het de voorkeur om isotherm te meten, daar isotherme metingen makkelijker te interpreteren en te vergelijken zijn.

4. De leidende marktpositie van IBM op het gebied van personal computers is niet te danken aan de superioriteit van het produkt, maar voornamelijk aan de naam die IBM al had op het gebied van kantoormachines.

5. Bij het bepalen van het beleid voor de aanschaffing van personal computers aan de T.U. Delft, is men vergeten waar de T in T.U. voor staat.

Delta Jrg. 18 No 20 3-6-1986 blz. 6.

6. De moderne Franse speelfilm wordt in Nederland ondergewaardeerd. 7. Indien deelneming van personeelsleden aan de brandweerploegen van de

faculteiten en diensten van de T.U. Delft van hogerhand gestimuleerd zou worden, zodat deze brandweerploegen weer op behoorlijke sterkte zouden komen, zou de veiligheid in de gebouwen aanzienlijk toenemen.

8. Het is merkwaardig dat iemand die militaire dienst weigert psychologisch getest wordt, terwijl iemand die zonder bezwaar in dienst gaat, niet psychologisch getest wordt.

voor Opa

Omslag: Schematische weergave van de ontwikkeling van een blokspektrum van aktiveringsenergiëen voor het R3 effekt en voor CSRO bij isochroon gloeien.

Surveys and Conference proceedings on Amorphous Metals.

B1 - F.E. Luborsky, ed., Amorphous Metallic Alloys, Butterworths, London, 1983. B2 - H.-J. Guntherodt and H. Beck, eds., Glassy Metals I, Springer, Berlin, 1981. B3 - Metallic Glasses, Proc. Sem. on Metallic Glasses, 1976, eds. H.J. Leamy and J.J.

Gilman, ASM, Ohio, 1978.

B4- Proc. 3rd Int. Conf. on Rapidly Quenched Metals, Brighton, 1978, 2 vols., ed. B. Cantor, Metals Society, London, 1978.

B5 - Proc. Int. Conf. on Metallic Glasses: Science and Technology, Budapest, 1980, 2 vols., Kultura, Budapest, 1981.

B6 - Proc. 4th Int. Conf. on Rapidly Quenched Metals, Sendai, 1981, 2 vols., eds. T. Masumoto and K. Suzuki, Japan Inst. Met., Sendai, 1982.

B7 - Proc. 5th Int. Conf on Liquid and Amorphous Metals, Los Angeles, 1983, 2 vols., eds. C.N.J. Wagner and W.L Johnson, J. Non-Cryst. Sol., S1M£ (1984).

B8 - Proc. 5th Int. Conf. on Rapidly Quenched Metals, Wurzburg, 1984, 2 vols., eds. S. Steeb and H. Warlimont, North-Holland Physics Publishing, Amsterdam, 1985. B9 - Proc. 6th Int. Conf. on Liquid and Amorphous Metals, Garmisch-Partenkirchen,

The work described in this thesis was partly published as Chapter 4:

A . L Mulder, S. van der Zwaag, E. Huizer and A. van den Beukel, Accurate contraction and creep measurements during structural relaxation of amorphous Fe40Ni40B2n,; Scripta Metall., I S (1984) 515.

Chapter 5:

A. van den Beukel and E. Huizer, On the analysis of structural relaxation in Metallic Glasses in terms of different models., Scripta Metall., 12 (1985) 1327.

E. Huizer and A. van den Beukel, Change of Young's modulus during structural relaxation in amorphous FeB, FeNiB, FeNiP and FeNiPB., Proc. 3rd Int. Conf. on the Structure of Non-Crystalline Materials, Grenoble, 1985, eds. Ch. Janot and A.F. Wright, J. de'Phys. Coll. C8, (1985) C8-561.

E. Huizer, A.L. Mulder and A. van den Beukel, The influence of structural relaxation on the Curie-temperature of amorphous FeMnBSi., B8, (1985) 639.

E. Huizer, J. Melissant and A. van den Beukel, Resistivity measurements during structural relaxation of Fe4QNi4QB2Q-, B9, in print.

Chapter 6:

A. van den Beukel, E. Huizer, A.L. Mulder and S. van der Zwaag, Change of viscosity during structural relaxation of amorphous Fe4QNi4QB2n.., Acta Metall.,

24(1986)483.

E. Huizer, A.L. Mulder and A. van den Beukel, On the stress dependence of viscosity changes during structural relaxation of Fe4QNi4QB2Q-, Acta Metall., 34

(1986)493.

Chapter 7:

E. Huizer and A. van den Beukel, Reversible and Irreversible length changes in amorphous Fe4QNi4uB2Q during structural relaxation., Submitted for publication.

BIBLIOGRAPHY

CONTENTS

1 GENERAL INTRODUCTION

1.1 Metallic Glasses 12 1.2 Structure of Metallic Glasses 13

1.3 Scope of this Thesis 16

Literature 18 2 DEFECTS AND ATOMIC TRANSPORT IN THE

AMORPHOUS STRUCTURE

2.1 The Ideal Amorphous Structure 22

2.2 Defects 23 2.2.1 Fluctuations in stress 24

2.2.2 Fluctuations in configurational entropy 25

2.2.3 Fluctuations in density 26

2.3 Atomic Transport 27 2.3.1 Homogeneous plastic flow (creep) 29

2.3.2 Diffusion 35 2.3.3 Scaling of diffusivity and viscosity 36

Literature 39 3 STRUCTURAL RELAXATION IN AMORPHOUS METALS.

3.1 Introduction 42 3.2 The Kinetics of Structural Relaxation 45

3.2.1 The free volume model 46 3.2.2 The Activation Energy Spectrum model 48

3.2.3 Other models 51 3.3 Recent Developments 51

4 EXPERIMENTAL DETAILS

4.1 Materials 70 4.2 Young's Modulus Measurements 71

4.3 Curie Temperature Measurements 75 4.4 Resistance Measurements 77

4.4.1 General principle 77 4.4.2 Resistance measurements at the annealing temperature 79

4.4.3 Resistance measurements at77K 81

4.5 Creep Experiments 81 4.6 Length Measurements 84

Literature 85

5 THE SEPARATION OF TSRO AND CSRO

5.1 Introduction 88 5.2 The Influence of Composition on CSRO and TSRO 88

5.2.1 Introduction 88 5.2.2 Results and discussion 89

5.3 Changes in Young's Modulus during Structural Relaxation 91

5.3.1 Introduction 91 5.3.2 Comparison with experiment 93

5.4 Changes in the Curie Temperature during Structural Relaxation 97

5.4.1 Introduction 97 5.4.2 Results and discussion 99

5.5 Changes in the Resistivity during Structural Relaxation 101

5.5.1 Introduction 101 5.5.2 Results and discussion 101

5.6 Conclusions 105 Literature 106

6 VISCOSITY CHANGES AND THE FREE VOLUME MODEL

6.1 Introduction 108 6.2 Theory and Experimental Program 108

6.3 Experimental Results and Analysis 110

6.3.1 Specimen A 111 6.3.2 Specimen B 115 6.4 Discussion 121

6.4.4 High temperature limit: Crystallization 127 6.4.5 Dependence of physical properties on free volume 127

6.4.6 Strain rate corrections due to structural relaxation 127

6.4.7 Comparison with other investigations 128 6.5 The Stress Dependence of Viscosity Changes

during Structural Relaxation 129

6.5.1 Introduction 129 6.5.2 Results 131 6.5.3 Discussion 133 6.6 Conclusions 136 Literature 137

7 CSRO AND REVERSIBLE STRUCTURAL RELAXATION

7.1 Introduction 141

7.2 Reversible and Irreversible Length Changes 141

7.2.1 Introduction 141 7.2.2 The reversible length effect 143

7.2.3 The thermal expansion coefficient 153

7.2A The master curve 155

7.2.5 Conclusions 157 7.3 Resistivity Measurements 157

7.3.1 introduction 157 7.3.2 Experimental results and discussion 157

Literature 162 FINAL CONCLUSIONS 163

SUMMARY 164 SAMENVATTING 166

Chapter 1

1.1. Metallic Glasses.

This chapter is meant as a short introduction to metallic glasses. Some aspects of metallic glasses, which will not be treated any further in this thesis, will be briefly reviewed.

Metallic glasses or amorphous metals are metals and metal alloys in a solid state without (translational) long range atomic order. They can be obtained by rapid quenching from the liquid or gas phase. The solidification occurs so rapidly that the atoms are frozen in a 'liquid-like' structure. A diffraction pattern from a metal alloy that contains only a series of broad maxima indicates that the alloy is amorphous. Local or nearest-neighbour order exists in most amorphous alloys, but no long range translational periodicity. The present interest in metallic glasses begun with the invention, by Duwez [1] in 1959 of the first practical way to produce a metallic glass by quenching at a rate of more then 105 K/s, though historically the first metallic glass

was probably produced by use of vapour deposition by Wurtz as early as 1845. Various preparation techniques have been developed since [2]:

- quenching techniques like splat quenching, spray deposition and melt spinning. - deposition techniques like sputtering, evaporation, chemical deposition, electrode

deposition.

- irradiation techniques like ion implantation. - diffusion techniques

For producing structures which are highly disordered, compositionally as well as topologically, the deposition and irradiation methods appear to have far greater flexibility than rapid melt quenching. Especially the range of possible alloy compositions is limited when using the quenching techniques, while with the other techniques a broad range of compositions can be achieved. However, melt spinning has been much superior to the other methods in producing amorphous continuous filaments, ribbons or tapes [3]. The first three forementioned techniques have in common that the size of the product in one dimension has to be very small. Thicknesses hardly ever exceed 100 u.m, which restricts the applications for these materials. Recently, however, it has been found that amorphous alloys can also be produced by a solid state reaction, e.g. diffusion in a compositionally modulated alloy [4]. The solid state reaction creates, in principle, the possibility to produce amorphous alloys in bulk form. Even more recently Blatter and Von Allmen [5] reported the production of an amorphous Cr-Ti alloy by spontaneous vitrification of a crystalline phase. Paradoxally, the amorphous alloy is made in this process by heating. The bulk

Introduction.

dimensions that can be obtained in this way are still limited however and the method only works for very few alloys.

With respect to composition metallic glasses can be devided into three classes: The metal-metalloid glasses, the metal-metal glasses and the glasses based on the group II elements. In this thesis we will only deal with glasses from the first class. These glasses consist mostly of 75% to 85% transition metal (Fe,Ni,Co) and 25% to 15% metalloid (B,P,Si).

Metallic glasses have interesting magnetic, mechanical, electrical and corrosion properties, directly related to their structure:

- They behave as very soft magnetic materials [6,7];

- They are exceptionally hard, } - have an extremely high tensile strength, } [8-11]

- are wear resistant, } - and posses a large fracture toughness;- }

- The electrical resistivity of iron-nickel based amorphous alloys is three to four times higher than those of conventional iron or iron-nickel alloys [12,13];

- Some metallic glasses are exceptionally corrosion resistant [14].

On the negative side of the balance is the relatively low resistance to fatigue under tension [15].

Based on these properties several applications have been reported in literature [16], but, probably due to the lack of bulk amorphous materials on commercial scale and the low fatigue resistance, only a few have appeared on the market as yet. Most of the applications already commercially available are related to the remarkable magnetic properties of amorphous metals, e.g. recording heads, magnetic transducers and transformer cores.

Apart from the commercial interest, metallic glasses provide a fascinating subject for scientific research. They present the possibility to study unique systems of thermodynamic non-equilibrium. The fields covered range from new preparation techniques, the stability of glasses, structure and structural changes to the special properties and their relation to the structure.

1.2. Structure of Metallic Glasses.

Excellent reviews concerning the experimental data and modelling of structure of glasses have been given by Cargill [17], Gaskell [ 18] and Finney [19].

In a (metallic) glass the structure can only be described statistically. There are no unit

6 0 0 3 400 3 0 0 S 2 0 0

J

j

1

N

Figure 1.1 - The evolution of X-ray scattering data into structural information. For explanation see text, (after Cahn [20]).

Introduction.

cells, which define a crystalline structure, there is no translational symmetry and the environments of different, chemically identical, atoms will vary. Hence the difficulty in determining and describing the structure of any glass.

Experiments, with either neutrons or X-rays, result in scattering curves from which a Radial Distribution Function (RDF) can be derived. This RDF gives the average number of atoms in a spherical shell at distance r from a chosen central atom, averaged over all the atoms. The RDF yields only a statistical average projection of the structure on to one dimension and thus does not establish a unique description of the atomic configuration in three dimensional coordinates.

To illustrate the transformation of raw experimental data into structural information we show figure 1.1 from a review by Cahn [20]. In figure 1.1a the observed scattered X-ray intensity of an amorphous NisiPjg sample is shown as a function of the scattering angle 28. The interference function l(K)as a function of K = 4jtsin0/X, with X the wavelength, is obtained (figure 1.1b) by a normalizing procedure. Fourier transformation yields the RDF as a function of the radius, r,(figure 1.1c). In figure 1.1d local deviations from random coordination are shown in the reduced radial distribution function G(r). The use of RDF's and the experimental difficulties in obtaining them is extensively discussed by Sietsma [21].

The theoretical approach consists of modelling in the laboratory or on the computer. Modelling can be static (packing models) or dynamic (molecular dynamics simulations). Once a model has been constructed its structural properties, in particular the RDF and density, can be calculated and compared with experiment.

One of the first static models for an amorphous structure was the Dense Random Packing model of Hard Spheres (DRPHS) proposed by Bernal [22]. In this model the atoms are considered as hard spheres all of the same diameter. Although there is a rather good agreement between the RDF computed with this model and those measured on some metallic glasses, there are some deficiencies. The biggest problem is that a two component real alloy was described by a single component model with infinitely hard atoms. Because this original model is static and of high density, it should perhaps be thought of as a reference structure ( an ideal glass of spherical atoms at T0) [19]. Boudreaux and Gregor [23] adjusted the model to overcome some of the

deficiencies, which resulted in improved agreement between modelled RDF and observed RDF. This was also done by Polk [24] who suggested that the relatively small metalloid atoms, like the phosphorous atoms, could be build into the DRPHS model by placing them in the largest polyhedral holes available. He argued that the number of larger holes is large enough to accomodate =20 at% of metalloid atoms.

It was soon recognised that these models, based on complete randomness, are most appropriate for structures having many alternative configurations with little or no bias imposed by chemical bonding, energy, space filling etc. For other structures, however, more ordered models may be preferable. To ensure the right Chemical Short Range Order (CSRO), Gaskell [25] proposed a model based on trigonal prismatic units. The topological constraints are brought in by (computer-)relaxing the model [21]. Another more recent approach is that of bond orientational order and quasi-crystals[26,27]. Already in 1974 Penrose [28] showed that it was possible to tile the infinite plane with two different unit cells, showing a fivefold symmetry. It was, however, not until 1983, when Schechtmann et al. [29] observed a tenfold symmetry in patterns of Bragg peaks in rapidly quenched AlgMn, that Penrose tiling was recognised as a possible way of describing apparently disordered structures. It leads to a long range order in the orientation of the bond (bond orientational order) and a quasi-translational order. Indeed Nelson et al. [30] and Yonezawa et al. [31] have shown by molecular dynamics simulation of a Lennard-Jones fluid that an icosahedral orientational correlation grows with undercooling and dissappears again near crystallization upon annealing. Some systems show a continuous phase transition amorphous --> quasi-crystal --> crystal.

The problem with most of these models is that they are statistical descriptions, not descriptions of interpretable structural properties (like density, symmetry, energies, defects etc.) which can be linked to macroscopic properties. For example annealing an amorphous metal causes changes in various macroscopic properties, but it hardly influences the RDF.

1.3. Scope of this thesis.

Although there may be no phase transitions in amorphous metals, except of course for crystallization, more subtle changes do occur at temperatures well below the crystallization temperature. These changes, which are visible in a number of physical properties, are attributed to atomic rearrangements in the amorphous state and the term structural relaxation is generally used to indicate this process. To optimise the properties and ensure their stability throughout an application lifetime therefore requires suitable heat treatments. However, to design such treatments it is necessary to gain knowledge of the changes caused by this phenomenon of structural relaxation.

To improve the understanding of the underlying processes is one of the aims of this thesis. Most of the investigations on structural relaxation so far have been concerned with measurments of the change of only one property. Therefore another aim of this thesis is to present a more systematic investigation on various property changes and to

Introduction.

show that all these changes can be described with the same model.

First, in chapter 2 the theory of defects and atomic transport in the amorphous structure are briefly reviewed. Subsequently in chapter 3 several models for the kinetics of structural relaxation are introduced. Chapter 4 deals with the experimental details. In chapter 5 the applicability of two models for the kinetics of structural relaxation, the AES model by Gibbs et al. [32] and the free volume model [33,34] as interpreted by Van den Beukel et al. [35], are tested. In chapter 6 the latter model is used to analyse the behaviour of the viscosity in amorphous metals upon structural relaxation, while in chapter 7 the reversible structural relaxation and Chemical Short Range Order are investigated on basis of this same model.

Literature.

A citation beginning with the capital B followed by a number, such as "B4", refers to an entry in the Bibliography on page 6.

1 - P. Duwez, R.H. Willens and W. Klement Jr., J. Appl. Phys., 3 1 (1960) 1136. 2 - H.H. Liebermann, B1, (1983) 26.

3-H.A.Davies, B4,1(1978) 1.

4 - R.B. Schwarz and W.L Johnson, Phys. Rev. Lett. 5 1 (1983) 415. 5 - A. Blatter and M. von Allmen, Phys. Rev. Lett. 54 (1985) 2103.

6 - C D . Graham and T. Egami, Ann. Rev. Mater. Sci., eds. R.A. Huggins, R.H. Bube and R.W. Roberts, Annual Reviews Inc., Palo Alto, CA, S (1978) 423.

7 - T. Masumoto, B5,1(1980) 121.

8 - C.A. Pampillo, J. Mater. Sci., 1Q (1975) 1194. 9 - L.A. Davis, B3, (1978)190.

10 - T. Masumoto, Sci. Rep. RITU, A26_(1977) 246.

11 - J.C.M. Li, Treatise on Material Science and Technology, ed. H. Herman, Academic Press, N.Y..2Q (1981) 326.

1 2 - K.V. Rao, B1, (1983) 401.

13 - P.J. Cote and L V . Meisel, B2, (1981) 144. 14 - T. Masumoto and K. Hashimoto, B4, (1978)435. 15 - L.A. Davis, J. Mater. Sci., H (1976) 711. 16 - D. Raskin and C.H. Smith, B1, (1983) 381. 17 - G.S. Cargill (III), Solid State Phys., 3Q (1975) 227. 18 - P.H. Gaskell, J. Phys., £ 1 2 (1979) 4337.

1 9 - J.L Finney, B1, (1983) 42.

20 - R.W. Cahn, Contemp. Phys., 21 (1980) 43.

21 - J. Sietsma, thesis, Delft University of Technology, Delft, (1987) 22 - J.D. Bernal, Proc. R. Soc, A2&4 (1964) 299.

23 - D.S. Boudreaux and J.M. Gregor, J. Appl. Phys., 42 (1977) 152,5057. 24 - D.E. Polk, A d a Metall., 2Q (1972) 485.

25 - P.H. Gaskell, J. Non-Cryst. Sol., 32 (1979) 207. 26 - S. Sachdev and D.R. Nelson, Phys. Rev. B, in print.

27 - D.R. Nelson and S. Sachdev, Proc. workshop Amorph. Met. and Semicond., may 1985, eds. P. Haasen and R. Jaffe, Acta Metal, in print.

Introduction.

29 - D. Schechtmann, I. Blech, D. Gratias and J.W. Cahn, Phys. Rev. Lett., 53 (1984) 1951.

30 - D.R. Nelson, in Topological order in Condensed Matter, eds. F. Yonezawa and Ninomya, Berlin (1983).

31 - F. Yonezawa, S. Nose and S. Sakamoto, B9,1986, in press

32 - M.R.J. Gibbs, J.E. Evetts and J.A. Leake, J. Mater. Sci., 13. (1983) 278. 33 - F. Spaepen, Acta Metall., 25 (1977) 407.

34 - A. van den Beukel and S. Radelaar, Acta Metall., 31 (1983) 419.

35 - A. van den Beukel, S. van der Zwaag and A.L. Mulder, Acta Metall., 32 (1984) 1895.

Chapter 2

D©f®cts and Atomic Transport in the Amorphous Sïaacïur®.

AS i

non -equilibrium

/ _

Figure 2.1 - The difference in entropy between the liquid (equilibrium) or the glass (non-equilibrium) and the crystalline phase as a function of temperature [1]..

Defects and atomic transport.

2.1. The ideal amorphous structure.

The absence of sharp diffraction peaks in amorphous metals is caused by the lack of translational symmetry. In most amorphous systems however, a high degree of short range order, i.e. a well defined coordination number and nearest neighbour distance, is usually established. Spaepen [1] suggested that it is possible to conceive an amorphous structure which has long range order. This simply means that the structure has a low configurational entropy, i.e. that the position of all the atoms can be known from a small number of construction rules. This does not necessarily mean translational symmetry (e.g. Penrose tiling).

In easy glass forming materials (organic compounds.polymers), the specific heat of the liquid can be measured at all undercooled temperatures. For these materials, the difference in entropy between the liquid and crystalline phase, AS|C, can be

determined. Figure 2.1 shows AS|C as a function of temperature, it decreases with

increasing undercooling (drawn line). For amorphous metals no measurements can be done in the temperature region between the glass transition temperature Tg and the

melting point TM, because crystallization will intervene. However, a reasonable

interpolation of the data results in a temperature dependence of AS|C similar to the one

presented in figure 2.1 for easy glass forming materials.

We must realise however, that, strictly speaking, AS|C can only be determined at

temperatures where the liquid is still in internal (metastable) equilibrium. Below the glass transition temperature Tg the liquid (now a glass) cannot be considered to be in

internal equilibrium, since the time necessary for molecular rearrangements is long on the time scale of thermodynamic measurements. While the difference in specific heat between the glassy and the crystalline phase is small, the apparent (i.e. non-equilibrium) value of AS|C will not change much below Tg (thin dashed line in

figure 2.1). This results in a finite residual entropy difference ASr at absolute zero. The

equilibrium value of AS|C below Tg can only be estimated by extrapolation (heavy

dashed line in figure 2.1). As pointed out by Kauzmann [2], this entropy difference seems to vanish at T0 where the amorphous phase is fully configurationally ordered. It

is this hypothetical structure that will be referred to as the ideal amorphous structure. The Dense Random Packing model of Hard Spheres (DRPHS) is a good approximation of the equilibrium amorphous state at T0 [3].

2.2. Defects.

Defects in amorphous solids can be described as deviations from the ideal ordered structure. These deviations can be approached in two different ways [1]:

- by (computer) introducing defects similar to crystalline point or line defects in highly ordered structures and to observe their evolution and stability, or

- by observing local fluctuations of a property (such as the density, in the free volume approach).

Although several analogies between the classical dislocation theory and the plastic behaviour of amorphous metals have been reported in the literature, defects in amorphous metals still differ in many ways from those in crystals. Defects are more often described as local fluctuations in the Short Range Order (SRO), that is in the order of the immediate environment of a particular atom, than as generalized versions of crystalline defects. Depending on the property used to characterise the SRO, these fluctuations can be described in several ways.

2.2.1. Fluctuations in stress.

Egami et al. [5] created a monatomic DRPHS model, which was computer relaxed, using a gradient method of energy minimization and assuming the Johnson potential for iron to describe the atomic interaction. Based on this model they defined structural defects in amorphous materials in terms of the distribution of internal stresses on atomic scale and symmetry in the surroundings of individual atoms. According to the authors this definition does not require an ideal reference structure. This concept of internal stresses on atomic scale has been previously applied to describe the core structure of crystalline dislocations. Egami et al. [5] found that there is a significant variation in the magnitude and direction of internal stresses, and that there are regions of 10 to 20 atoms over which the stresses remain either high or low. They also showed that there are significant correlations between the internal stresses and the local symmetry: Low stress, high symmetry regions resemble microcrystalline clusters, while high stress low symmetry regions are comparable to crystalline defects.

Based on these two distinct regions observed in the computer model, Srolovitz et al. [6,7] define two types of defects. First they define the defects based on fluctuations of local pressure (and consequently fluctuations in local density). This first type of defects can, in turn, be sub-devided into two types:

- a vacancy-like defect, which is not a well defined point vacancy like in crystalline materials, but a collection of split vacancies [4] which determine a region of lower density under a large tensile stress. This is called a n-type defect, implying the negative local density fluctuations.

- Similarly an interstitial-like defect simply means a crowded region under a large compression. This is called a p-rype defect.

Defects and atomic transport.

Although these types of defects are comparable to crystalline defects, there remain significant differences in the nature of defects as defined here and those in crystalline solids. The defects in amorphous solids appear more diffuse than those in crystalline alloys and the total amount of the mass deviation in one defect is not an integral multiple of the atomic weight. Furthermore they frequently occur in pairs, although isolated p- and n- defects have been shown to exist.

The n- type defects resemble the defects suggested by applications of the free volume theory to amorphous solids (see section 2.2.3). However, in that framework no analogue of the p- type defects, which would have to be called anti-free volume, has been considered. According to Sorolovitz et al. [6,7] the n- type defects, resembling free volume, account for only the half of the structural defects in amorphous solids, and there are almost an equal number of p- type defects.

According to Vitek [8] the separation between p - and n - type defects in the model has been made arbitrarily in the number of nearest neighbours. The n - type defect has more than 12.5 nearest neighbours and the p - type defect has less than 12.5 neighbours. The identification of the different regions as defects remains problematic however, since it is not a priori clear which of these stressed regions would not be present in the (unknown) ideal reference structure.

The second important class of defects, according to Srolovitz et al. [6,7], are those formed by concentrations of shear stresses. The high shear stress regions with a large deviation from spherical symmetry are called T - defects. These are likely to initiate a local slip and thus could play a role in the plastic deformation similar to that of (screw-) dislocations in crystals. The T - type defects are not entirely independent from the p-type and n- p-type defects. A pair of neighbouring n- and p- p-type defects must indeed lead to a shear in between them. It is therefore possible that the pair of n- and p- type defects associated with a r-type defect in between them represents in fact one defect which may be analogous to an edge dislocation in a crystal.

The volume increase due to n- type defects and the volume decrease due to the p- type defects nearly cancel each other. The slight net change in density caused by the recombination of n- and p- type defects during structural relaxation (annealing) is primarily due to the non-linear relationship between pressure and volume [5]. Therefore, according to Egami, the macroscopic excess volume does not directly represent the total free volume.

2.2.2. Fluctuations in Configurational Entropy.

Chen [9] adapted the description of fluctuations in configurational entropy, originally

devised by Adam and Gibbs [10], for amorphous metals. In this concept transport properties in an amorphous system are considered as a thermally activated cooperative rearrangement of the atoms over an energy barrier, which is defined by the configurational entropy. A defect is defined as a sub-system with a configurational entropy larger than some critical value Sc*. necessary to allow rearrangements. If the

total configurational entropy is known from thermodynamic measurements, the defect concentration can be calculated from the fluctuation theory. This approach, which is inherently phenomenological, has the advantage that atomic transport properties can be calculated from measurable thermal quantities. The disadvantage is that it does not reveal the details of the defect motion on an atomic scale.

2.2.3. Fluctuations in density (Free Volume).

As early as 1913 Batschinski [11] introduced the concept that the resistance to flow in a liquid should depend upon the relative free volume per molecule. However, it was not until 1951 that Doolittle [12] provided satisfactory experimental evidence for this proposal. Doolittle's measurements on liquid normal parrafins disclosed a logarithmic relation between the viscosity and the free volume.

Tl = Aexp(B'vm/vf) (2.1)

where A and B' are constants, Vf is the average free volume and vm is the molecular

volume at 0 K.

If we consider the DRP to be the ideal amorphous structure, then we can assign the average atomic volume v° in the ideal structure. The average atomic free volume, vj, of an amorphous system can now be defined as the difference between its average atomic volume, £2, and v°: Vf = £2-v°. In an amorphous system the free volume is supposed to be statistically distributed over all atoms. It is further assumed that the free volume can be redistributed among all atoms without changing the energy of the system.

Turnbull and Cohen [13,14] used this free volume concept in assuming that a diffusive jump can only take place if there is locally a critical free volume v* available, i.e. it becomes part of a defect. They arrive at a temperature dependence of the diffusion coefficient, D, related to v* as given by:

Defects and atomic transport.

where 7 is a geometric overlap factor of order unity. In metallic amorphous systems v* is roughly equal to the volume of the metal ion core.

The defect concentration, Cf, can also be calculated from the statistical distribution of the free volume among all the atoms [3]:

cf = exp(-yv7vf) (2.3)

As the viscosity, T|, is inversly proportional to Cf [1], we get:

Tl = T|0exp(7v*/vf) (2.4)

which is exactly the Doolittle equation (2.1). At T = T0 (see figure 2.1) free volume

goes to zero. Therefore a simple estimate of the temperature dependence of Vf is:

vf(T) = <xvn(T-T0) (2.5)

where the the thermal expansion coefficient oty is assumed constant. Substituting this in (2.4) gives us the Fulcher-Vogel [15,16] equation, frequently used to describe the temperature dependence of the viscosity of metallic glasses in thermal equilibrium:

il = r|0exp(B/(T-T0)) (2.6)

This equation can also be obtained by using the configurational entropy model as proposed by Adam and Gibbs [10], described in the previous section.

2.3. Atomic Transport.

In crystalline materials, the motion of defects gives rise to atomic transport. Vacancies, for example, govern diffusion and some forms of creep and dislocations govern high-stress deformation and, in combination with vacancies, some other forms of creep. Defects in amorphous metals, although different from their crystalline counterparts, must perform the same basic transport function. It is therefore important to study the atomic transport properties, like creep and diffusion, of amorphous metals.

Figure 2.2, from [1], shows schematically the temperature dependence of the shear viscosity, r|, for a typical amorphous metal. The liquid phase is in stable equilibrium above the melting temperature TM and in metastable equilibrium between TM and the

10 20 r ^ ,«15 'g 10 in Z, J 0 — 10 >- 1-Ü > 1 " T* ! - 7 - ' _'mi 10 ,-5 2.0

Figure 2.2- Shear viscosity, TI, in the various stability regimes: stable equilibrium above TM; metastable equilibrium from TM to Tg and

extrapolated below Tg; isoconfigurational (1) and (2) with

associated fictive temperatures Tj, below Tg. The direction of

viscosity change due to structural relaxation towards equilibrium is indicated [1].

Defects and atomic transport.

glass transition temperature Tg The general shape of the curve in figure 2.2 is described by the Fulcher-Vogel equation (2.6) as given in the previous section. At the higher temperatures (T>TM) the viscosity is low (= 10"3 Ns/m2) and changes only

slowly with temperature. That is because the free volume, Vf, is large and the defect concentration does not change very much with temperature (exp(Yv*/Vf) -> 1). When T approaches T0, however, vj becomes small and the defect concentration decreases

very rapidly, resulting in a sharp rise in the viscosity. On decreasing the temperature, the system must make configurational rearrangements to increase its structural order, i.e. to eliminate the defects. The rate at which these rearrangements occur decreases as the order, and therefore the viscosity, increases. When the viscosity reaches a certain value, the time necessary for the configurational rearrangements of the system will be getting too long to reach equilibrium. The system will be configurationally frozen. The temperature at which this occurs, the glass transition temperature, is dependent on the time scale of the experiment or on the quench rate of the sample and will thus be different for each annealing treatment of an amorphous sample. Therefore Tool [17] introduced the fictive temperature, Tf, to characterize the structural state of a system below Tq. Tf is defined as the temperature at which the structure of the system

would be the equilibrium structure. Figure 2.2 shows how Tf can be obtained from the isoconfigurational lines. Below Tg the system is not in internal equilibrium and it will

relax towards the metastable state at that temperature (arrows in figure 2.2). This process is called structural relaxation.

2.3.1. Homogeneous Plastic Flow. (Creep).

In this section we will discuss the theory of homogeneous plastic flow following part of an extensive overview concerning homogeneous and inhomogeneous plastic flow [18].

In order to get physically meaningfull activation energies for flow it is necessary to perform isoconfigurational viscosity measurements at a low stress level, i.e. where the plastic flow is homogeneous. To perform isoconfigurational measurements the system needs to be annealed first, to lower the relative structural relaxation rate (dln(T|)/dt). This rate, which decreases continuously during annealing, has to be slow enough to allow for viscosity measurements at different temperatures without appreciable structural changes. It seems surprising that structural changes can be neglected while measuring the viscosity, but only 10~4 jumps per atom are needed to measure the

viscosity.

In the Newtonian viscous region, i.e. the low stress region, where the stress a is

Figure 2.3 - Schematic creep curve showing the elastic, anelastic and plastic components of the strain, e, after loading and unloading [B10].

Defects and atomic transport.

proportional to the strain rate ê, the homogeneous flow is characterized by the shear viscosity, which in metallic glasses is often measured by using creep experiments:

T| ■ ! = -S- (2.7)

It 3e

using the Von Mises criterium. Figure 2.3 shows a schematic creep curve.

The first accurate creep experiments on amorphous metals were performed by Taub and Spaepen [19] on Pd82Si-|8- They found that the viscosity changes over five orders of magnitude during structural relaxation, while the rate of viscosity change drj/dt shows an Arrhenius type of temperature dependence:

d-n. ,-E\

■dT ~ 6 X P ( RT} <Z8>

with E' = 32 kJ/mole. This makes the viscosity a rather sensitive measure for the structural state of an amorphous metal.

From isoconfigurational measurements they found an Arrhenius type temperature dependence for T|, at least within the T and r\ region of their tests:

i l ~ exp(ET 1/RT) (2.9)

with an activation energie E „ = 192 kJ/mole.

Analogous to a dislocation in a crystalline material, Spaepen [1] defines a flow defect. The basic feature of a flow defect is that, upon application of a shear stress, it undergoes a local shear transformation which is transferred elastically to the specimen boundary to produce a macroscopic strain. Spaepen further argues that in an amorphous system a free volume fluctuation can be a flow defect if, after the central atom's jump out of the nearest neighbour "cage", the collapse of the new "cage" around it produces a local shear strain. He then generalizes the original formulation of the free volume model to involve the movement of several atoms contained in a local defect volume v0. When enough free volume is collected in v0, the atoms can, under

action of a stress, go through a shear transformation and produce a local shear Y0. This

is illustrated schematically for a two atom defect in figure 2.4.

For any system subjected to a shear stress i , the plastic flow rate can be formulated basically in t e r m s of these flow defects:

©

©

®

Figure 2.4 - Schematic diagram of the motion of a flow defect (volume v0)

under the action of a shear stress x, producing a local shear strain

m-Defects and atomic transport.

y = ^ ovok'f (2-1°)

where y is the shear strain rate of the system, Cf is the concentration of flow defects and kf' is the net jump frequency of a flow defect under the action of stress. This equation is a general one and it can be applied, for example, to calculate the movements of dislocations in crystalline materials, leading to the well-known Orowan equation [18].

The net jump frequency kf', under the action of stress, can be found from the stress unbiased jump frequency kf, according to Spaepen[1]:

kf' = kfp(x) (2.11) with P(T) a stress dependent factor. It is expected that kf is of the form:

kf = vf exp(-EF/RT) (2.12)

where Vf is the attempt frequency (<= Debye frequency) and Ep is the activation energy for defect motion. Since TY0V0 is the work done by the applied stress upon motion of

the defect it represents the difference in potential energy between the two positions of the defect. Using rate theory the stress biasing factor can now be calculated as the difference of two fluxes of exponential form [20]:

P(T) = s i n h ( ^ ) (2.13) If the stress is small (t < kT/y0v0), this equates to:

Ty v

P(T) = - ^ (2.14) In this regime the flow is Newtonian viscous, and the shear viscosity as defined by

(2.7) can be calculated:

r, s 1 = - J ™ _ (2.15)

y

ta)

S

This is the general equation for viscous flow in the free volume model.

1 08 -10 10 > lö12 E ~~" -14 >-io H > , A « w ' 0 =3 M- -18 ÜL 10 Q -20 10 -22 10 -c ■ — . -\ " _ 1 in f - F e ^ ^ C in a \ ^ C in y-Fe V \ Fe in y - Fe \ \ Fe in a-Fe i i -Fe C in 1 -~ m D Fe3C % " + + i i i 0.6 0.8 1.0 1.2 1.4 1.6 1.8 1 03/ T (K_ 1)

Figure 2.5 - Comparison of diffusivities in Fe - based systems: FeinFe4 0Ni4 0P1 4B6 (+)

B in Fe4 0Ni4 0B2 0 (•)

Si in Fes2Bi2Si6 (°) After Greer [26].

Defects and atomic transport.

2.3.2. Diffusion.

Direct measurement of the diffusion rate in amorphous metals is difficult, while it has to be done below the crystallization temperature. The measurement temperature is usualy chosen around Tg, which still implies diffusivities of only 10"20 m2s"1- For

these measurements special techniques have to be used. A recent survey of the measured diffusion coefficients of amorphous metals is given by Cantor and Cahn [21].

The statement that physically interpretable activation energies can only be measured isoconfigurationally is also valid for diffusion. In diffusion measurements on materials with a profiled composition the time needed for a measurement is about 200 jumps per atom. Here relaxation during the experiment seems inevitable. However, measuring the chemical interdiffusion coefficient D~ in a compositionally modulated material means that only 0.2 jumps per atom are needed per measurement. This is of course much more than needed for viscosity measurements, but still small enough to restrict structural relaxation. This technique, devised by Hilliard et al. [22] for crystalline materials, is not only very sensitive, but it also allows for a continuous check of the diffusivity.

Greer et al. [23] observed a linear increase of 1/D~ with time on annealing for a Pd85Sii5/Feg5Bi5 film. Chen [24] showed that the relative changes in diffusivity (dln(1/D~)/dt) and viscosity (dln{T|)/dt) were of the same order of magnitude, measuring Au in Pd77 sCusSi-ig 5. Greer et al. [23] also did some isoconfigurational experiments, finding an Arrhenius type temperature dependence for D~, with an activation energy of 195 kJ/mole close to the 192 kJ/mole found for viscous flow in Pdg2Siig [19]. Kijek et al. [25] suggested that the diffusivities fall into two groups: fast diffusion of small atoms (generally metalloid) by an interstitial mechanism; and slow diffusion of large atoms (generally metal) by a substitutional mechanism. Greer [26] showed that this is probably not true. Figure 2.5 shows that the metal and metalloid diffusivities in Fe-based glasses are remarkably close, compared to substitutional and interstitial diffusivities in crystalline iron. It appears that the metalloid atoms in metal-metalloid glasses do not diffuse interstitially, but that their movement is linked to that of the metal atoms. According to Greer [26] this is not surprising in view of the strong metal-metalloid bonding in these systems and the high degree of chemical short range order.

Spaepen suggests, analogous to flow, the existence of diffusion defects. Their basic feature is a change of nearest neighbours upon atomic rearrangement. An example of a diffusion defect in an amorphous metal is the free volume fluctuation described in

section 2.2.3.

For an extensive review of diffusion results the reader is referred to Cantor and Cahn [21].

2.3.3. Scaling of Diffusivity and Viscosity.

Essential to the understanding of atomic transport mechanisms in amorphous metals is the relation between diffusivity and viscosity. The similarities between the defects governing diffusion and those governing flow are brought out by a number of observations:

- The similarity between their isoconfigurational activation energies (195 kJ/mole and 192 kJ/mole respectively in PdSi).

- The similar relative relaxation rates for both properties (dln(1/D~)/dt) and (dln(ii)/dt) respectively.

- D and r\ scale according to the Stokes-Einstein relation for TsTg [28].

In the simple free volume theory diffusion and flow occur by the same mechanism (cf. fig. 2.4). It is therefore to be expected that for amorphous systems the Stokes-Einstein relation will also apply below Tg:

kT

, D = - (2.16) where r is a characteristic particle radius. Chen et al. [24] have compared diffusivity

data on Au in Pdjj^CugSiig 5 around Tg with viscosity data on the same glass in the

same temperature region. They found a value for r close to the ionic radius of Pd. Greer compared the isoconfigurational data on TI in Pdg2Si-)8 a nd o n 1/D~ in a

Pds5Sii5/Feg5B-|5 modulated film. He found that good fits with (2.16) were obtained using radii 160 to 590 times smaller than the ionic radius. He therefore concludes that the Stokes-Einstein relation probably is not valid for metal-metalloid glasses well below Tg. The measured diffusivities are significantly greater than would be expected from measured viscosities. This means that there are diffusive jumps which do not contribute to flow. Greer further suggests that the movement of vacant sites on the metal-metalloid network are involved, which could occur without substantial alteration of the topology of the network.

Spaepen [1] calculated the relation between TI and D by using the free volume model. First he calculates the diffusivity from the 'random walk' theory:

D = aDk 'D^ (2.17)

Defects and atomic transport.

dimensionality of the system and the correlation of the jumps; kD' is the diffusive jump

frequency per atom and XQ is the distance of of a diffusive jump (expected to be of the order of an interatomic distance X). If the concentration of diffusion defects is cD and

their jump frequency kD, then:

kD' = cDkD (2.18)

Combining (2.17) and (2.18) with the general equation for viscous flow (2.15) gives: cn kn Xnii

nD = aD( ^ ) ( ^ ) - 2 -Tk T (2.19)

C' k' (Yovo)

If there exists some relation between the flow and diffusion defects (for example if one defect performed both functions, or one type is a subset of the other), r\ and D are expected to scale with a characteristic length:

(Y v )

L = -^f- (2.20) Assuming now the simple free volume picture, in which the same defect performs

both functions we get cD = Cf, kD = kf, XD = X. Furthermore we take y0 = 1 and

v0 = Q = X3. This means L = X and

r|D = aDkT/X. (2.21)

which, for aD = n/3, is the Stokes-Einstein relation.

In a crystal, viscous flow can occur through the movement of vacancies . This is the Nabarro-Herring creep, in which viscosity is given by [29]:

kTd

Ti = (2.22) 4QD

with d the grain size. The grain boundaries act as sinks and sources for vacancies. In metallic glasses there could be a link between D and TI analoguous to that in Nabarro-Herring creep, but it seems likely that, in addition atom movements of the type shown in figure 2.4 occur and contribute to both diffusion and flow.

In the absence of a stress the atomic rearrangement in a flow/diffusion-defect will not have any preferential direction: the defects will not contribute to macroscopic flow but only to diffusion and in some cases free volume annihilation. At higher temperatures

the mobility of the atoms is higher and therefore the chance that a rearrangement leads to free volume annihilation.

When a stress is applied, the atoms in a flow/diffusion defect will, according to Spaepen, have a preferential direction for jumping, causing macroscopic flow. The number of rearrangements in the direction of the applied stress will increase with increasing stress. Because of this the number of diffusion jumps associated with flow will increase. In this interpretation the rate of free volume annihilation is independent of the applied stress, which will be confirmed later on.

At higher temperatures thermal equilibrium is quickly established and the Stokes-Einstein relation is valid. This is consistent with the observation that for most amorphous metals the activation energies for diffusion controlled processes like crystallization and phase separation, are close to those measured for the viscosity in that temperature region.

While the Stokes-Einstein relation is not valid at lower temperatures one can question the validity of equations (2.1) and / or (2.2). Therefore it is also not certain wether equation (2.4) should follow from (2.3) [8]. However, these equations form the basis of the free volume model, as discussed in section 2.2.3. Results obtained so far with this model are in good agreement with experiments, so the use of these equations seems justified.

Defects and atomic transport.

Literature.

A citation beginning with the capital B followed by a number, such as "B4", refers to an entry in the Bibliography on page 6.

1 - F. Spaepen, in Les Houches: Lectures on Physics of defects, eds J.P. Poirier and M. Kleman, France, 1980.

2- W. Kauzmann, Chem. Rev. 42 (1948) 219.

3 - M.H. Cohen and D. Turnbull, J. Chem. Phys. 21 (1959) 1164. 4 - F. Spaepen, J. Non-Cryst. Sol., 21 (1978) 207.

5 - T. Egami, K. Maeda and V. Vitek, Phil. Mag. A, 41 (1980) 883.

6 - D. Srolovitz, K. Maeda, V. Vitek and T. Egami, Phil. Mag. A, 44 (1981) 847. 7 - D. Srolovitz, V. Vitek and T. Egami, Acta Metall., 21 (1983) 335.

8 - V. Vitek, Invited talk at Delft University of Technology, (1986). 9 - H.S. Chen, J. Non-Cryst. Sol., 22 (1976) 135.

10 - G. Adam and J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. 11 - A. Batschinski, Z. Phys. Chem., 24(1913) 644.

12 - A. Doolittle, J. Appl. Phys. ££ (1951) 1471.

13 - D.Turnbull and M.H. Cohen, J. Chem. Phys., 29 (1958) 1049. 14 - D.Turnbull and M.H. Cohen, J. Chem. Phys., 34 (1961) 120. 15 - H. Vogel, Z. Phys., 22 (1921) 645.

16 - G. Fulcher, J. Amer. Ceram. Soc, 6 (1925) 339. 17 - A.Q. Tool, J. Amer. Ceram. Soc, 22 (1946) 240.

18 - G. Leusink, Internal Report, lab. of Metallurgy, Delft University of Technology, (1986).

19 - A.I. Taub and F. Spaepen, Acta Metall., 22 (1980) 1781.

20 - S. Gladstone, K.J. Laidler and H. Eyring, The theory of Rate Processes, McGraw-Hill, NY (1941) 480.

21 - B. Cantor and R.W. Cahn, B1, (1983) 487.

22 - H.E. Cook and J.E. Hilliard, J. Appl. Phys., 40.(1969) 2191. 23 - A.L. Greer, C.J. Lin and F. Spaepen, B6, (1982) 567.

24 - H.S. Chen, L.C. Kimerling, J.M. Poate and W.L. Brown, Appl. Phys. Lett., 22 (1978) 461.

25 - M. Kijek, M. Ahmadzadeh, B. Cantor and R.W. Cahn, Scripta Metall., 14 (1980) 1337.

26 - A.L. Greer, B7, (1984) 737.

27 - A.I. Taub, Acta Metall., 22 (1980) 663.

28 - H.S. Chen, B6, (1982) 495.

Chapter 3

a)

T2 Tg

quilibrium

Figure 3.1 - Schematic behaviour of

a) the specific volume V or the internal energy H as a function of temperature.

Structural relaxation.

3.1. Introduction.

Structural relaxation is the process or collection of processes by which an amorphous system relaxes towards a more stable configuration. The coordination numbers of the atoms will rise and the free volume will decrease during structural relaxation, influencing a lot of physical properties [1]. It is therefore often difficult to discuss the properties of an amorphous metal, without knowing its relaxation condition. Figure 3.1a sketches the behaviour upon structural relaxation of the specific volume V of a glass forming material as a function of the temperature T. This behaviour is very general and applies to polymer- and oxide glasses as well as to metallic glasses. Instead of the specific volume V as a parameter one can use the figure equally well to describe the internall enthalpy H as a function of temperature. Structural relaxation decreases the free energy of a glass.

It has been shown several times in the literature, that some properties of amorphous alloys change irreversibly upon relaxation, while others appear to have a reversible component i.e. in a temperature region with sufficient atomic mobility properties can be changed reversibly with temperature. In the simple picture of figure 3.1a it is shown that the irreversible property changes are coupled to the lowering of the free volume to the extrapolated metastable undercooling curve. The arrow on the left in figure 3.1a indicates the change of the free volume upon isothermal annealing well below the glass temperature. At these temperatures the metastable equilibrium line is never reached, and thus the change is irreversible. The reversible property changes are a result of the temperature dependence of the metastable equilibrium value of the free volume in the vicinity of Tg. This is demonstrated by the arrows on the right in figure 3.1a: Upon isothermal annealing at temperature T1 the free volume is lowered until the metastable equilibrium line is reached. If the temperature is now instantaneously changed to T2 the free volume has to increase to reach the metastable equilibrium line, thus giving a reversible change.

A reversible effect is, however, also observed at temperatures well below Tg. The relationship between this reversible relaxation observed below Tg and the irreversible relaxation has been extensively discussed by Scott and Kursumovic [3]. Egami [4] compared these reversible changes to order-disorder phenomena in crystalline materials and the irreversible changes to annealing out of excess defects. Consequently he attributed the irreversible behaviour to what he termed Topological Short Range Order (TSRO), i.e. the more efficient packing of the constituent atoms irrespective of their chemical identity. The reversible (below Tg) behaviour he ascribed to Compositional or Chemical Short Range Order (CSRO), which describes the process by which the relative atomic positions itself are unaltered but by which the

TABLE 3.1

Changes of physical properties during structural relaxation. (After [2], slightly modified.)

Property irrev. rev. (T=Tg) rev. (T<Tg)

Density Enthalpy Young's modulus Diffusivity Viscosity Curie temperature Electrical resistivity Internal friction Ductility Hardness Magnetic after-effect Peak heights in RDF

Superconducting transition temp.

I D I D I I.D l,D D D I D I D + + + + +

irrev.= irreversible change; rev. (T=Tg) = reversible change near Tg; rev. (T<Tg)= reversible change below Tg

l,D indicate an increase or decrease.

chemical species of the atoms at these positions is changed. Although little direct structural information (X-ray, EXAFS , Mössbauer etc.) on atomic scale is at present available, this suggestion, of two processes playing a major role in structural relaxation of metallic glasses, has been generally accepted. Scott [5] presented strong , though indirect, evidence from isothermal micro-calorimetry on thermally cycled Fe4oNi4gB2o glass that the degree of CSRO varies reversibly between temperature dependent equilibrium values. Similar calorimetric evidence for other glasses has been obtained by Drijver et al.[6], while Balanzat [7] has obtained similar evidence for thermally cycled Fe4nNi4gB2o glass by monitoring electrical resistivity.

The behaviour of CSRO is sketched in figure 3.1b. Equilibrium is easily obtained well below the glass temperature making a reversible change possible. This opposed to

Structural relaxation.

TSRO, figure 3.1a, where the equilibrium would only be obtained very close to Tq. E.g. the reversible effect observed in the vicinity of Tg on viscosity by Tsao and Spaepen [8,9], is due to TSRO (i.e the annealing out of free volume) which reaches the (metastable-) equilibrium line. This effect is only noticeable in those amorphous metals that have a crystallization temperature, Tx, far enough from Tg so that crystallization will not intervene. As this is not the case for any of the materials discussed in this thesis, TSRO will further be considered as an irreversible process.

Chemical Short Range Ordering is a process that involves nearest neighbours and is therefore a relatively fast process which cannot be explained by the same mechanism as TSRO. Egami [10] suggests that CSRO takes place by minor cooperative atomic rearrangements. Greer [2] suggests that that the ordering takes place due to diffusive jumps, governed by the movements of defects analogous to the movement of vacancies in crystals. As pointed out in a previous section there are diffusive jumps which do not contribute to viscous flow. Greer also shows that the measured inter-diffusion coefficients are consistent with CSRO kinetics. Kursumovic et al. [11] showed by means of the so-called "cross - over" experiment, a special variant of temperature cycling, that the kinetics of the reversible part, i.e. CSRO, can only be interpreted on the basis of at least two activation energies.

Table 3.1 illustrates the variety of properties that are affected by structural relaxation. The first column names the physical property, the second column indicates the direction of the irreversible change upon structural relaxation (due to TSRO), as reported in the literature. The third column indicates wether a reversible change in the vicinity of Tg (due to TSRO) has been reported in the literature or not and the last column indicates wether a reversible change below Tg (due to CSRO) has been observed. New in this table is the reversible density due to CSRO, which will be shown to exist in chapter 7 of this thesis.

3.2. The Kinetics of Structural Relaxation.

The description of the relaxation kinetics has been the subject of much theoretical debate. The approach by Egami [12] based on local fluctuations in stress and symmetry has already been described in a previous section. The link between this model and more'phenomenological models, which wil be described next, has not yet been made [13]. We will first describe the free volume model as it is interpreted by Van den Beukel and Radelaar[14], which is based on the previously described free volume model by Spaepen.

3.2.1. The Free Volume Model.

Taub and Spaepen [16] find that the activation energy for free volume annihilation (160 kJ/mole, in PdSi) is lower than that for isoconfigurational flow (192 kJ/mole, in PdSi). They suggest that the annealing out of free volume can be described by the annihilation of excess flow defects at special 'relaxation sites'. The annihilation rate of flow defects is then:

cf = -krctcr (3.1)

where cr is the concentration of relaxation sites; Cf is the concentration of flow defects

and kr is the jump frequency of the annihilation process. Van den Beukei et al. [18]

argue that there is no a priori reason why this frequency should differ from the frequency of flow defects:

kr = kf (3.2)

where kf is given by equation (2.12). In chapter 6 it will be shown that this is in fact a plausible assumption. It is, however, not in agreement with the interpretation of Spaepen and coworkers who argue that the rearrangements associated with diffusion and flow are different from those associated with structural relaxation, i.e. k^kf.

It was first assumed by Spaepen [17] and later on experimentally verified by Tsao and Spaepen [8], that the annihilation of flow defects is an essentially bimolecular process i. e.:

cr = p'cf (3.3)

where p' is a constant. Combining (3.1), (3.2) and (3.3) Van den Beukei et al. [18] get:

Cf^-Cfo'1 = C0t exp(-EF/RT) (3.4)

with C0 = pVf. Substituting equation (2.3) leads to the isothermal equation for free

volume annihilation in the free volume model:

exp(x_1) - exp(x0_1) = C0t exp(-EF/RT) (3.5)

with x = Vf / yv* the reduced free volume and x0 the reduced free volume at t=0. Van

den Beukel and Radelaar further showed that the annealing out of free volume under arbitrary conditions is described by the equation:

Structural relaxation.

d vf ( VVf9) EF TV*

Where Vfe is the equilibrium value of Vf at the annealing temperature given by equation

(2.5).

This formalism for the kinetics of TSRO has been used succesfully by Van den Beukei et al. [19] in describing the TSRO part of the change of length and Young's modulus in

Fe4oNi4gB2o-Van den Beukel and Radelaar [14] further introduced a kinetic model for the CSRO part of structural relaxation based on two assumptions:

- The ordering is limited to two kinds of atoms, and can be described by a Warren-Cowley parameter a^,.

- The kinetics of CSRO can be described by a single relaxation time TS, assuming first

order kinetics:

\ = To extfy") exP(pf) <37)

Van den Beukel and Radelaar arrive at the following equation for the ordering kinetics:

f " = - « V « w e > * o e XP ( w > e XP ^ ) <3-8> where

V1 = vf ' 0-9)

and o ^ g is the equilibrium value of a^. Together equations (3.6) and (3.8) form a set of differential equations describing the kinetics of structural relaxation.

Some remarks on this model: One of the important aspects of this model is that the kinetics of CSRO are dependent on the state of the free volume decay. This is what one would expect intuitively, as it seems logical that a decrease of free volume makes atomic rearrangements more difficult. In such a complex system as an amorphous metal one would expect that different atomic rearrangements require different activation energies, therefore it is surprising when the annealing out of free volume can be described by one single activation energy and CSRO by one single relaxation time. This last assumption by Van den Beuke! and Radelaar, of a single relaxation time for CSRO, will be shown in the next section to be too simple.

3.2.2. The Activation Energy Spectrum Model.

It was realized in the early stages of research on metallic glasses that the temperature and time range of structural relaxation is much too wide to be accounted for by a single thermally activated process. Structural relaxation seems to consist of two or more relaxation processes, or a series of relaxation processes with a continuous distribution of activation energies. This last suggestion forms the basic idea used by Gibbs, Evetts and Leake in the development of their Activation Energy Spectrum (AES) model [21]. Their theory is based on the work of Primak [22] . They assume that the activation energies of the processes that contribute to a change of a property P, during CSRO, are distributed over a continuous activation energy spectrum. In this case process means every possible thermally activated rearrangement of atoms or atomclusters, leading to a change in the chemical short range order. The structure is characterized by one (or several) short range order parameter, the value of which varies locally throughout the specimen. Through thermal activation the order parameter can reach equilibrium. As the associated activation energy also varies throughout the specimen there is a distribution (a spectrum) of relaxation short range order parameter values. Gibbs now defines the total density of ordering parameters with activation energy E as Q(E). The density of the parameters with an equilibrium value at a temperature T is qs(E,T). Gibbs then defines q(E,T) as the total density

available for relaxation:

Q(E,T) = q(E,T)+qs(E,T) ' (3.10)

Assuming first order kinetics for q(E,T) we get:

^=- q(E , T ) voe x p ( - J L ) (3.11)

Where v0 is a frequency factor. Integrating (3.11) and introducing q0(E,T) as q(E,T) at

t=0 and q^(E,T)=q0(E,T)-q(E,T) as the density of ordering parameters that have reached

the equilibrium value after a time t, yields (for isothermal annealing):

qt(E,T) = q0(EIT){1-exp[-v0texp(-^)]} (3.12)

Unfortunately the ordering parameters themselves cannot be measured, only their contribution to a change in a property P:

Structural relaxation.

where c(E) is the proportionality between the property P and the ordering parameter with activation energy E. p(E,T)dE is the change in P after a time t caused by the parameters with an activation energy between E and E+dE. The total change in the measured property, AP, is given by:

A P = [ p ( E , T ) d E (3.14) o

Combining (3.12) and (3.13) yields:

p (E,T)dE = po(E,T){1-exp[-v0texp(^)]}dE (3.15)

with p0(E,T) is the total available property change in the range E to E+dE. Following Primak [22] equation (3.15) can be written as:

p(E,T) = p0(E,T)6(E,T,t) (3.16)

where 9(E,T,t) is defined as the characteristic annealing function. This function is shown by Primak to change in a small energy range (width =eRT) from .99 to .01. Therefore, for an isothermal anneal, 6(E,T,t) can be approximated by a 'down' step function at E=E0 where

E0 = R T I n ( v0t ) (3.17)

In this approximation 9(E,T,t) expresses the fact that, at time t, all ordering parameters with E<E0 have contributed to the relaxation and have reached the equilibrium value at that temperature, while all parameters with E>E0 have yet to contribute. This means that p(E,T)dE = p0(E,T)dE for E < E0 and p(E,T)dE = 0 for E>E0. The measured change in a property P, AP, is then given by:

AP = Jp0(E,T)dE (3.18)

If p0(E,T) is constant with respect to E for E<E0, in other words if the activation energy spectrum is a simple box distribution (figure 3.2), it follows:

AP = p0(T) RT ln(v0t) (3.19)

From a change in a measured property the spectrum can be found by differentiating (3.18) with respect to ln(t):

a(E

Figure 3.2 - Box distribution [27].

10*

Figure 3.3 - Isothermal change of v2 of as quenched specimans at the

temperatures indicated. Calculated TSRO and CSRO contributions are indicated by T and C respectively [19].