Phase Transformation and Diffusion [1 ed.] 9783038132066, 9783908451617

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Phase Transformation and Diffusion

Phase Transformation and Diffusion

Special topic volume with invited papers only

Guest edited by:

G. B. Kale, M. Sundararaman, G. K. Dey, and G. P. Tiwari

TRANS TECH PUBLICATIONS LTD Switzerland • UK • USA

Copyright  2008 Trans Tech Publications Ltd, Switzerland All rights reserved. No part of the contents of this book may be reproduced or transmitted in any form or by any means without the written permission of the publisher. Trans Tech Publications Ltd Laubisrutistr. 24 CH-8712 Stafa-Zurich Switzerland http://www.scitec.net

Volume 279 of Defect and Diffusion Forum ISSN 1012-0386 (Pt. A of Diffusion and Defect Data – Solid State Data ISSN 0377-6883) Full text available online at http://www.scientific.net

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Preface The basic purpose of all research activities in materials science and technology is to tailor the properties of materials to suit specific applications. Phase transformation is the key to fine tune the structural, mechanical as well as corrosion properties of any material. A basic understanding of kinetics and mechanism of phase transformation is, therefore, of vital importance. Barring a few cases involving martensitic transformations, all phase transformations are mediated through diffusion. A proper control and understanding of the process of diffusion during nucleation, growth, oxidation, sintering etc is essential for optimising the properties of material to meet specific needs. Keeping in mind the above consideration, this special topic volume on, “Phase Transformation and Diffusion” was organised. Researchers and scientists from academia and industries immensely contributed to the volume. We are grateful to the authors for providing the manuscripts in time and the referees for their ready co-operation in reviewing these papers. We are also thankful to Prof. G. E. Murch, editor-in-chief, Defect and Diffusion Forum, who wholeheartedly supported the publication of this special topic volume. Finally, thanks are due to Trans. Tech. Publications who have agreed to publish this volume. The special volume project coincided with the sixtieth year of one of the editors, namely Dr. G. B. Kale. He has spent 35 years of his distinguished career on studying diffusion phenomena in materials. His co-editors and former colleagues would like to devote this volume in his honour and wish him a long, healthy and prosperous active life. G. B. Kale Bhabha Atomic Research Centre Mumabi 400085, India August 5, 2008

M. Sundararaman G. K. Dey G. P. Tiwari

Table of Contents Preface Diffusion in the Lattice and Interfaces of Real Engineering Materials: A Unified Approach D. Gupta The Lattice Monte Carlo Method for Solving Phenomenological Mass and Thermal Diffusion Problems I.V. Belova, G.E. Murch, T. Fiedler and A. Öchsner Diffusion and Melting G.P. Tiwari and R.S. Mehrotra Thermodynamic Diffusion Coefficients G.B. Kale Novel Method of Evaluation of Diffusion Coefficients in Ti-Zr System K. Bhanumurthy, A. Laik and G.B. Kale Diffusion in Cu(Al) Solid Solution A. Laik, K. Bhanumurthy and G.B. Kale A Probabilistic Approach to Analyze Austenite to Ferrite Transformation in Fe-Ni System G. Mohapatra and S.S. Sahay Recrystallization Kinetics in 17Cr 1Mo Ferritic Steel M.N. Mungole, P.C. Trivedi, S. Sharma and R.C. Sharma Effect of Thermal Aging on the Transformation Temperatures and Specific Heat Characteristics of 9Cr-1Mo Ferritic Steel B. Jeya Ganesh, S. Raju, E. Mohandas and M. Vijayalakshmi Gibbs Free Energy Difference in Bulk Metallic Glass Forming Alloys H. Dhurandhar, K.N. Lad, A. Pratap and G.K. Dey Effect of Cerium Addition on the Microstructure and Mechanical Properties of Al-Zn-MgCu Alloy A.K. Chaubey, S. Mohapatra, B. Bhoi, J.L. Gumaste, B.K. Mishra and P.S. Mukherjee δ-Hydride Habit Plane Determination in α-Zirconium at 298 K by Strain Energy Minimization Technique R.N. Singh, P. Ståhle, L. Banks-Sills, M. Ristinmaa and S. Banerjee Calorimetric Studies of Dissolution Kinetics of Ni2(Cr,Mo) Phase in Ni-Cr-Mo Alloys Using Non-Isothermal Approach H.C. Pai, B.C. Maji, A. Biswas, M. Krishnan and M. Sundararaman Kinetics and Mechanism of Growth of β-Solid Solution during Reaction Diffusion in Binary Titanium and Zirconium Alloy Systems G.P. Tiwari, O. Taguchi, Y. Iijima and G.B. Kale Effect of Cu Addition on Nanocrystallization Behavior in a Co-Based Soft Magnetic Metallic Glass A.P. Srivastava, D. Srivastava, K.G. Suresh and G.K. Dey Alpha to Omega Transition in Shock Compressed Zirconium: Crystallographic Aspects G. Jyoti, R. Tewari, K.D. Joshi, D. Srivastava, G.K. Dey, S.C. Gupta, S.K. Sikka and S. Banerjee Selection of Lattice Invariant Shear in Dilute Zr-Nb Alloy for bcc-hcp Martensitic Transformation D. Srivastava, G.K. Dey and S. Banerjee Development of a Thermodynamic Criterion to Predict the Alloy Compositions for Amorphous and Nanocrystalline Phase Formation during Mechanical Alloying N. Das, S.K. Pabi, U.D. Kulkarni, B.S. Murty and G.K. Dey

1 13 23 39 53 63 71 79 85 91 97 105 111 117 125 133 139 147

Defect and Diffusion Forum Vol. 279 (2008) pp 1-12 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.1

Diffusion in The Lattice and Interfaces of Real Engineering Materials: A Unified Approach D. Gupta*, Emeritus Research Staff Member, *IBM Research Division, Thomas J. Watson Research Center, P. O. Box 218, Yorktown Heights, NY 10598 Email: [email protected]; [email protected]

Abstract: Engineering materials are commonly polycrystalline in nature and chemically inhomogeneous containing hetero-phases and interfaces. Because diffusion is ubiquitous in dissimilar phases and defects, it plays a vital role in the performance and reliability. During the last several decades, a synergism of the microstructural properties and chemical inhomogeniety with diffusion has been attempted to unify their apparently diverse behavior. We discuss the methodology and the thermodynamical analyses of the diffusion data needed to obtain this synergism quantitatively and illustrate it by the results obtained in a wide variety of materials, both metallic and non-metallic. Investigations carried out in pure polycrystalline metals have yielded grain boundary energies comparable to those directly measured. Furthermore, we discuss the role of solute segregation at grain boundaries and interfaces in alloys in altering diffusion. From the perturbations caused, the solute segregation parameters - the enthalpy and the entropy of binding have been extracted and levels of solute concentrations estimated. Finally, it is shown that similar analyses when applied to complex materials, e.g. the Pb-Sn eutectic alloy, the Ni3Al intermetallic compound, and an Ag-ceramic system, also result in acceptable values of interface energies and segregation factors. Introduction Interfaces are ubiquitous in all engineering materials and determine their performance and reliability. Microstructure, diffusion, energy of the interface and microchemistry (interface solute segregation effects) are some important properties that play vital roles in the performance of materials as they are subjected to various stress fields and hostile environments in fabrication and service conditions. Over hundred years ago, a pioneer of the modern physical metallurgy, RobertAusten [1], had remarked: “One thousand part of antimony converts first-rate copper into the worst conceivable (metal)”. Some sixty years ago, while an undergraduate the author himself was inexplicably perplexed by the failure of Pt wire used to hold a borax bead in the flame test to identify metals in mineral powders. Every thing went well with the test until the powder happened to contain Pb, which quickly diffused into the Pt wire even at low temperature snapping it from its mount and loss of the expensive contraption. We know now that even a few parts per million impurities like antimony and lead, in solid or liquid form, in Cu, Au and Pt metals result in intergranular failures. The change in the interfacial energy due to the presence of impurities manifests itself in reduction of grain boundary cohesion and in failures such as the temper embrittleness, fatigue, stress corrosion, Coble creep etc. It also changes many interfacial kinetic processes such as the recrystallization, formation of textures. However, presence of solutes at interfaces is not always detrimental and has many times beneficial effects. Small amounts of Y2O3 doping to Al2O3 have been found to help its sintering. The improvement of ductility of Ni3Al by B residing at the grain boundaries is now well known and understood. Then, there are hundreds of examples of improved electromigration and diffusion barriers performances of thin film metallizations in the microelectronic industry [2]. Interfacial microchemistry, microstructure, fields are in fact all interrelated and affecting each other and result in development of many reliability problems. The

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Phase Transformation and Diffusion

synergism is best depicted in a tetrahedron type of diagram as shown schematically in Fig. 1. where all the variables are seen to be interdependent in their entirety.

Fig. 1 Interdependence of important interface properties on diffusion. In this paper, we first provide the theoretical framework, which enables extraction of interface energies and solute segregation parameters from diffusion measurements in the lattice and interfaces. Interface data thus obtained in (a) several polycrystalline FCC metals and alloys, (b) the Pb-Sn eutectic alloy displaying unstable lamellae microstructure, (c) several polycrystalline intermetallic compounds, notably the Ni3Al and B doped Ni3Al and, finally, (d) in a metal-ceramic system have been described. It is shown that the procedures appear to be universally applicable encompassing diverse materials ranging from the ductile metallic alloys to hard and brittle ceramics. We note that the interface and grain boundary terms will be used interchangeably. The former is more general and the latter specific to the polycrystalline pure metals. Procedures to obtain interface energy (γi) and solute segregation parameters from diffusion data Extraction of interface energies from diffusion data. Grain boundaries and interfaces are mesocrystalline characterized by lower density and presence of defects such as grain boundary dislocation, vacancies and interstitials. In recent years, these defects have been computer simulated by Mishin [3] and the findings, more or less, are consistent with the experimental findings. The lower density of the interface material intuitively implies and easier diffusion process with lower activation energy at least for motion. Several investigators have addressed the problem in the last four decades, notably, Borisov et al. [4] and Gibbs and Harris [5]. The basic postulate in obtaining interface energy (γi) from self diffusion data is that it is the difference between the Gibbs free energies (∆Gl and ∆Gi) for vacancy diffusion in the lattice and the interface respectively [4], as shown schematically in Fig. 2. The difference is positive because diffusion in interfaces is many orders of magnitudes faster compared to that in the adjoining lattice. Hence the interface energy can be written as γi = 1/2(∆Gl - ∆Gi).

(1)

The factor of 1/2 enters because the energies are shared between the two adjoining crystals. In diffusion studies the Arrhenius nature of the coefficient, D, is defined as: D = Γ α2 f ν exp – ( ∆H – T ∆ S ) / RT

( 2)

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where Γ is a geometrical factor, α the jump distance, ƒ a correlation factor specific to the tracer studies, ν the jump frequency and ∆H and ∆S are the associated enthalpy and entropy terms. D for lattice and interfaces can be easily described by using the respective subscripts l and i on the ∆H and ∆S terms.

Fig 2. Interface energy and free energy for diffusion in lattice and interface. ∆G ' is the change in the former upon a solute addition. Assuming that the pre-exponential terms don't differ much for lattice and interface diffusion, the difference between ∆Gl and ∆Gi can be written as:

γ i ( J / mole

or

) =

1 1 (∆ G l − ∆ Gi ) = RT { l n ( D i / D l ) } . 2 2

γ i ( J / m 2 ) = ρ {( Q l − Q i ) + RT l n ( D

o i

/D

o l

)},

( 3)

(4)

where Dol and Doi (m2/sec) are the pre-exponential terms for lattice and interface diffusion respectively and the conversion parameter ρ = 1/2 α 2No with α the mean distance between atoms at the interface and No is Avogadro’s number. Ql and Qi, and R within the brackets are to be taken in J/mole. Grain boundary or interface diffusion measurements are generally carried out in Harrison's B regime [6], in the temperature range of 0.3 - 0.5Tm where Tm is the melting temperature of the host in degrees Kelvin. In this range, diffusion in the interface is coupled with that in the adjoining lattice in a systematic manner. From the knowledge of the amount of lattice diffusion, the product (δDi) of the width of the interface (δ) and the interface diffusion coefficient (Di) respectively is obtained in the pure host. In some special situations, it is also possible to measure Di itself independent of δ at low enough temperatures < 0.3Tm because the lattice diffusion vanishes [7, 8]. This is known as Harrison's regime C. For all practical purpose, δ is approximated to 0.5nm i.e. of one lattice parameter width. In a following section, the values of the interface energies extracted from diffusion data would be compared with those obtained directly in several metals. We emphasize that only inputs of the self diffusion parameters for lattice and interfaces, obtained by employing either a tracer of the host or a good surrogate in close proximity in the Periodic Table, result in a satisfactory agreement with those obtained by direct measurements.

Interface Solute Segregation Effects from Diffusion Data. Equations 3 and 4 are also applicable to alloys as well. In diffusion measurements in alloys in the Harrison's regime B, the product sδDi is measured where s (T) is the solute segregation factor. The segregation factor has its own Arrhenius

4

Phase Transformation and Diffusion

temperature dependence. The solute addition results in reduction of the interface energy denoted by γio and increase in the free energy for diffusion by ∆G' as shown in Fig. 2. Here also, self diffusion measurements of the solvent species are required in both the lattice and the interface. From the difference of the interface energies between the pure host and the alloy, (γi -γio), it has been possible to compute the enthalpy and entropy for the solute segregation using Gibb's adsorption isotherm [9] combined with the McLean statistics [10]. The free energy for solute grain boundary segregation, ∆G', is given by: (γi -γio) =∆γi = R T ln [ 1+Co exp (∆G' / RT)].

(5)

From the free energy ∆G' and the solute concentration Co, the segregation coefficient s = Cb/Co can be readily obtained as function of temperature according to McLean's statistics:

Cb = Co exp (∆G'/ RT) /[1+ Co (∆G' / RT)].

(6)

Furthermore,

∆G' = (∆H' - T∆S').

(7)

It may be noted that mono-layer coverage (Cb) is implicit in Eq. 6 beyond which it is not valid. The limiting value of s depends on the input of Co and it also determines the magnitude of the entropy ∆S' in Eq. 7 but the enthalpy ∆H' remains unaffected. It is also possible to obtain s in the Eq. 6 directly by measuring self diffusion in the Harrison's B and C regimes in which sδDi and Di are measured respectively. A comparison of the two quantities and assuming a value for δ, leads to the values of s as function of temperature. Such an approach has been used recently by Herzig and co-workers [7, 8] for Ag and Au segregation in Cu grain boundaries and will be discussed in a following section.

Interface energies in several diverse materials Interface energies in pure metals. To test that grain boundary energies computed from diffusion data are indeed meaningful, diffusion parameters and the resulting interface energies obtained from Eq. 4 for several pure metals are listed in Table 1. Also listed therein are the interface energies obtained by direct measurements employing non-diffusion techniques such as the zero creep and transmission electron microscopy (TEM). Fig. 3 shows a plot of the two kinds of energies. It is seen that, despite the semi-empirical nature of the basic postulate in Eq. 1, the two kinds of energies agree well within a few percent. Furthermore, the temperature coefficient of the interface energies is small and negative implying a positive entropy. These are in agreement with the predictions of Mclean [10] made long time ago for thermal behavior of grain boundaries. Physically also, a negative temperature coefficient is meaningful as grain boundaries are known to be less cohesive at higher temperatures. In Fig. 3, the grain boundary energies seem to be proportional to the melting temperature of the host metal, for instance, Pb shows the smallest value of 200mJ/m2 whereas Ni has the highest value of 700mJ/m2. Fig. 3 Comparison of diffusion related interface energies with those directly measured.

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Other metals have intermediate values as their melting temperatures fall between these two extremes. However, the dependence does not appear to be linear at this stage and this effect needs to be further explored. Table1. Diffusion in lattice ( Dl ), interfaces ( δ Di ) and interfacial energies in some pure metals. The directly measured values are marked by (*). Host (Trace r) Au (195Au) Au*

Dl = Dlo exp( −Ql / RT )

Ql (kJ/ mol) 169.6 -

Dlo (10-4 m2/s) 0.04

-

δ Di = δ Dio exp( −Qi / RT ) Qi (kJ/ mole) 84.8

δ Dio (1016 m3/s) 3.1

-

-

Ag (110Ag) Ag 110 ( Ag) Ag*

169.6

0.04

77.1

0.13

169.6

0.04

74.4

0.31

-

-

Cu ( Ag) Cu 64 ( Cu) Cu 67 ( Cu) Cu*

199.4

0.16

75.2

23

196.8

0.1

84.75

11.6

199.4

0.16

91.5

29

-

-

170.5

222

-

-

110

-

-

Ni (63Ni) Ni*

277.5

Pb (203Pb) Pb (119Sn) Pb*

100

0.16

44.3

61

99.4

0.41

39.5

73

-

-

-

-

0.92 -

-

-

Ref.

γi 2

(mJ/m )

3960.016T 400 -(1173K) 382 (1173K) 392 (1173K) 400 (1173K) 776 -0.123T 7190.134T 692 -0.0054T 590 (1123K) 1126 -0.214T 1373 -0.4T 1930.0047T 206 -0.03T 200 (588K)

[11] [12] [13] [14] [15] [16] [17] [18] [12] [19] [20] [21] [22] [23]

Interfacial diffusion, energies and solute segregation in alloys. As mentioned in section 2.2, grain boundary energies are affected significantly due to the solute segregation effects in alloys. This effect is schematically shown in Fig. 2 where the interface energy drops and the activation energy for interface diffusion increases upon alloying. For a detailed discussion of this phenomenon, the reader is referred to the excellent article of Hondros and Seah [6]. Using, Eqs. 5-7, we obtained quantitative information on interface solute segregation and their physical parameters in materials in general. In Table 2, we have listed results in a large number of polycrystalline materials. For our discussion, we divide the various materials into 4 categories: (a) dilute alloys, (b) a eutectic Alloy, (c) an intermetallic compound and (d) a metal-ceramic system.

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Phase Transformation and Diffusion

(a) Solute segregation in dilute alloys: For Pb, Au and Cu alloys listed in Table 2, we have used Gibbs-McLean approach to calculate the enthalpy and entropy of solute segregations. Although all these systems behave similarly, the Pb alloys have a larger data base and hence are amenable to a detailed discussion. In Fig. 4, the grain boundary energies in pure Pb obtained using 203Pb and 119Sn tracers and in several Pb-Sn-X alloys are shown where X = In /Au. It is seen that in pure Pb (see Table 1); the grain boundary energies have similar values for the 203Pb and 119Sn tracers. The former is, of course, truly self diffusion measurement but the latter is close to self diffusion since Sn is in close proximity to the Pb host in the Periodic Table and carrier-free 119Sn tracer was used in infinite dilution. In both cases, the temperature dependence of the interface energy is small and negative as expected in pure metals. In the Pb alloys, only the 203Pb tracer is employed in the self-diffusion measurements. A strong solute segregation effect is seen on the grain boundary energies and their temperature dependence becomes positive in all cases indicating negative entropy of segregation. Furthermore, in the ternary alloys the segregation effect appear to be additive.

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Fig. 4 Grain boundary energies in Pb-In, Pb-Sn and Pb-Sn-X alloys

Fig. 5 Grain boundary binding energy parameters and elastic strain energies in some alloys Using Eq. 5, the solute parameters ∆H' and ∆S' have been computed for various alloys listed in Table 2. The ∆H' enthalpy term has elastic and electronic components. McLean [10] and later Wynblatt and Ku [30] have proposed the following relationship for the elastic strain component ∆Hε:

∆Hε = [24πQkGr1r0(r1 – r0)2 / (4 G0 r0 + 3 kr1)]

(8)

where κ is the bulk modulus of the solute, G the shear modulus of the solvent, ro and ri are the effective radii of the solvent and the solute atoms respectively. These values were obtained from the Smithell's Ref. Book [31]. The elastic strain enthalpies, ∆Hε, for various alloys are listed in Table 2

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Phase Transformation and Diffusion

and displayed by a solid curve in Fig. 5 as function of the relative atomic size difference. The data points are for the total ∆H' enthalpy values obtained from diffusion data according to the Eqs. 6 and 7. It is seen that the elastic strain energies are invariably lower due to the simple fact that they contain binding effects only partial. Other contributions such as the valency difference and the range of solid solubility at the measurement temperatures should also be taken into account. Hondros and Seah [9, 26] have provided a relatively complete treatment of the problem; its discussion is beyond the scope of this paper. We can, however, draw several valid conclusions from Fig. 5. Firstly, the binding energies in alloys having similar atomic size and valency, such as the PbSn, Pb-In, Cu-Ag, Cu-Au etc., are small and close to the elastic strain values. In the Cu-Bi and CuSb alloys, the valence and atomic size difference is large and consequently their binding energies deviate from the elastic strain energies. Secondly, contributions arising from the valency difference and the limited solid solubility can be very large as may be seen in the case of the Cu-Cr alloy. The atomic size difference is negligible in the Cu-Cr alloy but the binding energy is extremely large owing to the transition nature of Cr and it is almost totally immiscible in Cu. (b) Interface energies in Pb-62%Sn eutectic alloy: The eutectic Pb-Sn alloy is an important soldering material for microelectronic industry. Its reliability is determined by its diffusion behavior from melting temperature down to the room temperature. Diffusion and interfacial energies, however, may be expected to change in this alloy as the unstable interfaces undergo transformation upon annealing.

Fig. 6. Microstructure of Pb-62Sn eutectic alloy: (A) Pb and Sn lamellae in the as grown condition and (B) polygonized interface boundaries between the Pb and Sn phases are formed upon annealing at 424K for 5 hours

In Fig. 6 (A and B), the microstructures of the Pb-62%Sn eutectic alloy in the as grown condition and upon annealing at 424K for 5 hours are shown. Upon annealing the Pb-Sn lamellae progressively curl up, spheroidize and finally decompose into equiaxed grains of Pb and Sn metals. The preferred crystallographic relationships of the interface in directionally solidified Pb-Sn eutectic specimens have been studied by Hopkins and Kraft [32] and Takahashi and Ashinumal [33] and showed: interface || ( 111 ) Pb || (0 11 ) Sn and growth direction || [211] Pb || [211] Sn. It is certain that the interfaces in the Pb-Sn eutectic alloy have a special atomic arrangement akin to the

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coincident grain boundaries in the early stages of annealing. After prolonged annealing, the interface between the Pb and Sn grains behave more or less like a high grain boundary as shown in Fig. 6(B). As seen in Fig. 7, the low temperature interface diffusion in the eutectic Pb-62%Sn is much slower and the activation energy much higher, about 77- 85 kJ/mole. The diffusion data in the decomposed Pb-Sn eutectic alloy fall in the band comprising of data in the polycrystalline Pb and low Pb-Sn alloys (see Tables 1 and 2) with lower activation energies in the range of 40 - 50 kJ/mole. The activation energy for diffusion in the Pb and Sn lattice is about 100 kJ/mole (see Table 1). Thus, the activation energy for the PbSn interface diffusion at low temperatures is about 80% that of the lattice diffusion which is of the same order as has been reported in coincident boundaries. The corresponding interfacial energies in the two cases are 150 and 233mJ/m2 respectively (see Table 2) and reflect the degree of coherency discussed above. Fig. 7 Interface diffusion in Pb-62%Sn alloy The instability of the interface in the Pb-Sn eutectic alloy manifests itself in its applications as well. In its deformation characteristics, there is a transition of the attendant activation energy from a high value of 84.4 kJ/ mole at low temperatures (331 – 386 K) to a low value (44 – 57 kJ/ mole) at higher temperatures. The former is, in fact, diffusion in the Pb–Sn coherent interfaces assisted by conservative motion of coherency dislocations [34] and the latter in high angle interfaces between the Pb and Sn grains akin to high angle grain boundaries. In the reliability studies of the eutectic solder as well such as the electromigration, two activation energies have been reported [35]. (c) Interface energies in the intermetallic compounds: Grain boundary self diffusion coefficients in the stoichiometric undoped and B doped Ni3Al compound have been made by Frank et al. [28] and are listed in Table 2. Applying the Borisov's et al. analysis discussed in Sec. 2, the interface energies γi have been determined in both cases and are shown in Fig. 8. Values for γi in undoped Ni3Al were found to be higher than in the B doped material at 930 and 900mJ/m2 at 1400K respectively, the difference gets larger at lower temperatures. Both show positive but unequal temperature coefficients. The observations are reasonable as grain boundaries in the Ni3Al compound should have alloy characteristics and segregation of B atoms at the grain boundaries should further decrease their energy. The grain boundary energies for pure Ni are also shown in Fig.8 and as expected for pure metals, display negative temperature dependence. Very recently Divinski et al. [36] have measured 44Ti self diffusion in the interfaces of a two phase TiAl alloy consisting of alternating layers of α2-Ti3Al and twinned γ-TiAl intermetallic compounds. The value of the interface energy obtained from the self diffusion data is 350mJ/m2 at 1100K. Considering that the α2 and γ phases are stable up to at least 1440K, a much higher value of the order of 500mJ/ m2 would have been expected. The observation of such a lower value of the interface energy in this layered intermetallic compound reflects high atomic registry at the interfaces.

10

Phase Transformation and Diffusion

Fig. 8. Interface energies in pure and boron doped Ni3Al (Frank et al. 28) and in pure Ni (Lange et al Ref. 17)

(a)

(b)

Fig. 9. (a) Interface energy, (b) Ag segregation and their parameters at the PNN-PT-PZ ceramic interface. Interface energies in a metal-ceramic system: Diffusion measurements of the radioactive 110mAg tracer in the polycrystalline 50Pb (Ni1/3Nb2/3) O3 - 35PbTiO3 - 15PbZrO3 piezoelectric ceramics have been made recently [29] and the resulting parameters are listed in Table 2. At the chosen composition, the PNN-PT-PZ ceramic is a single phase material and fast diffusion of Ag is attributed to grain boundaries. Using the approach discussed in Sec.2, it was possible to compute metal (Ag)-ceramic interface energies as function of temperature and are shown in Figure 9(a) and (b) A comparison of the interface energies in PNN-PT-PZ can be made with the recent measurements of interface energies by Saiz et al.[37] for liquid metals (Ni, Al)-Al2O3 system using atomic force microscope. Their values are 1.3 and 1.2 J/ m2 respectively at 1500oC. An extrapolated value of 1.2 J/ m2 for Ag-PNN-PT-PZ interface energy at 1500oC seems to agree well, even though the two ceramics are different. From γi (T) values, Ag solute binding energy of 90 kJ/mole and

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entropy of -0.9R were obtained and the grain boundary solute enrichment of Ag atoms estimated as shown in Fig. 10(b). Segregation of Ag atoms at the PNN-PT-PZ grain boundaries is very strong and its solubility in the lattice almost negligible and follow the inverse concentration rule discussed by Hondros et al. ([9] earlier.

SUMMARIZI#G REMARKS We have examined a large number of diverse materials to establish that the absolute interface/grain boundary free energy is the difference between the free energies for self- diffusion in the lattice and the interfaces. This hypothesis is borne out well by the excellent agreement between the interface energies computed from the self diffusion data with those directly measured in several pure polycrystalline metals e. g. Ag, Au, Cu, Ni and Pb. In further analyses, the decrease in the interface energies in alloys and other complex materials due to solute segregation has been combined with the Gibb's adsorption and Mclean's statistics. The approach has yielded acceptable values for the interface solute segregation parameters, the enthalpy and entropy, which have been considered elusive to measurements in ductile materials particularly. It is noteworthy that no adjustable parameters have been used in these analyses.

REFERE#CES [ 1] W. C. Robert-Austen, Phil. Trans. Soc., Vol. A179 (1888), p. 339 [ 2] Chao-Kun Hu, Lynne M. Gignac and Robert Rosenberg, in Diffusion Processes in Advanced Technol. Mater., Devendra Gupta Ed. (William Andrew Pub. Norwich, New York, 2004), p.405 [ 3] Y. Mishin, in Diffusion Processes in Advanced Technol. Mater., Devendra Gupta Ed. (William Andrew Pub. Norwich, New York, 2004), p.113 [ 4] V. T. Borisov, V. M. Golikov and G. V. Scherbedinskiy, Phys. Metals Metallogr., Vol. 17 (1964), p.80 [ 5] G. B. Gibbs and J. E. Harris, in Interfaces, edited by R. C. Gifkin (Butterworths, London, 1969), 53 [ 6] L. G. Harrison, Trans. Faraday Soc., Vol. 57 (1961), p.1191 [ 7] T. Surholt, Yu. M. Mishin and Chr. Herzig, Phys. Rev., Vol, B 50 (1994), p.3577 [ 8] S. Divinski, M. Lohmann and Chr. Herzig, Acta. Mater. Vol. 49 (2001), p.249 [ 9] E. D. Hondros, M. P. Seah, S. Hoffmann and P. Lecek, in Physical Metall., R. W. Cahn and P. Haasen (Eds.), North-Holland Phys. Pub. (1996), p 1201-1288 [10] D. McLean, Grain Boundaries in Metals, Oxford Uni. Press London,(1957), 116 [11] D. Gupta and R. Rosenberg, Thin Solid Films, Vol. 25 (1975), p.171 [12] J. E. Hilliard, M. Cohen and B. L. Averbach, Acta. Met., Vol. 8 (1959), p.26 [13] D. Turnbull and R. E. Hoffman. Acta. Met., Vol. 2 (1954), p. 419 [14] J. T. Robinson and N. L. Peterson, Act. Met., Vol. 21 (1973), p.1181 [15] A. P. Greenough and R. King, J. Inst. Met., Vol. 79 (1951), p. 415 [16] G. Barreau, G. Brunel, G. Cizeron and P. Lacombe, Mem. Sci. Rev. Metall., Vol. 68 (1971), p. 357 [17] T. Surholt and Chr. Herzig, Acta. Mater., Vol. 45 (1997), p. 3817 [18] D. Gupta, Defect and Diffusion Forum, Vol. 156 (1998), p. 43 [19] W. Lange, A. Hassner and G. Mischer, Phys. Stat. Sol., Vol. 5 (1964), p. 63 [20] L. E. Murr, R. J. Horilev and W. N. Lin, Philos. Mag., Vol. 22 (1970), p. 52 [21] D. Gupta and K. K. Kim, J. Appl. Phys., Vol. 51 (1980), p. 2066 [22] K. K. Kim, D. Gupta and P. S. Ho, J. Appl. Phys., Vol. 53 (1982), p. 3620 [23] K. T. Aust and B. Chalmers, Proc. Roy. Soc., Vol. A 204 (1950), p. 359 [24] D. Gupta and J. Oberschmidt, in Diffusion in Solids: Recent Developments, M. A. Dayananda and G. E. Murch (Eds.), The Metall. Soc of AIME, Warrendale, 1985, 121

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Phase Transformation and Diffusion

[25] D. Gupta, K. Vieregge and W. Gust, Acta Mater., Vol. 47 (1999), p. 5 [26] E. D. Hondros: in Proc. Interfaces Conf., Australian Inst. Of Metals, Butterworths, London. 1969, 77 [27] D. Gupta, C.-K. Hu and K. L. Lee, in proc. DIMAT-96 Conf. held in Germany in Aug, 1996, Defect and Diffusion Forum, 143 - 147, (1997), 1397 [28] S. Frank, J. Rüsing and Chr. Herzig, Intermetallics, Vol. 4 (1996), p. 601 [29] D. J. Lewis, D. Gupta, M. R. Notis and Y. Imanaka, J. Amer. Ceram. Soc., Vol. 84 (2001), 1777 [30] P. Wynblatt and R. C. Ku, in Proc. ASM Sci. Seminar Interfacial Segregation, W. C. Johnson and J. M. Blakely (Eds.), ASM Metals Park, Ohio, (19790, 115 [31] J. Smithells and E. A. Brandes (Eds.), Metals Reference Book, Butterworths, London, (1976), 15, 2-4 [32] R. H. Hopkins and R. W. Kraft, Trans. Metall. Soc., A. I. M. E., Vol. 242 (1968), p.1627 [33] N. Takahashi and K. Ashinuma, J. Inst. Metals, Vol. 87 (1958), p. 19 [34] T. G. Langdon and F. A. Mohamed, Philos. Mag., Vol. A. 32 (1975), p. 697 [35] K. N. Tu, and A. M. Gusak, J. Appl. Phys., Vol. 93 (2003), p. 1335 [36] S. V. Divinski, F. Hisker, A. Bartels and Chr. Herzig, Scripta Mater.,Vol. 45 (2001), p. 161 [37] E. Saiz, R. M. Canon and A. P. Tomsia, Acta Mater., Vol. 47 (1999), p. 4209

Defect and Diffusion Forum Vol. 279 (2008) pp 13-22 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.13

The Lattice Monte Carlo Method for Solving Phenomenological Mass and Thermal Diffusion Problems Irina V. Belova#, Graeme E. Murch#, Thomas Fiedler# and Andreas Öchsner† #

Centre for Mass and Thermal Transport in Engineering Materials, School of Engineering, The University of Newcastle, Callaghan, New South Wales 2308, Australia

†

Department of Applied Mechanics, Faculty of Mechanical Engineering, Technical University of Malaysia, 81310 UTM Skudai, Johor, Malaysia Email: [email protected]

ABSTRACT. In this overview, we introduce the recently developed Lattice Monte Carlo method for addressing and solving phenomenologically-based mass and thermal diffusion problems especially for composite and porous materials. With examples, we describe the application of this numerical method to calculate effective mass diffusivities and concentration profiles. Next, we describe the application of this method to the calculation of effective thermal conductivities/thermal diffusivities and temperature profiles. I TRODUCTIO

The solving of phenomenological mass or thermal diffusion problems is often required in developing many technological processes. When exact solutions are unavailable, such problems are generally solved with finite element or finite-difference numerical methods. This is generally facilitated by the widespread availability of commercial software dedicated to the purpose. However, there are still many occasions when such numerical methods are almost impossible to implement for these problems. Recently, a Lattice Monte Carlo (LMC) method has been intensively developed for addressing phenomenological mass and thermal diffusion problems[1-4]. Although the Monte Carlo method has a reputation for being a computationally very demanding one, this is now much less of a problem as computers have become faster, less expensive and more accessible. The special advantage of the LMC method over the other methods above is that it can be used to address virtually any phenomenological diffusion problem. The LMC method makes use of a very fine-grained lattice that is overlaid on the phenomenological problem. This lattice is then explored by virtual mass or thermal particles to represent mass or thermal diffusion phenomena respectively. In this sense, the LMC method is a form of finite-difference method that is embedded in a quasisimulation of the diffusion process. In this review paper, we trace the history of applications of the LMC method with reference to phenomenological mass and thermal diffusion problems. As examples of application to mass diffusion problems we discuss the calculation of effective diffusion coefficients in grain boundary diffusion (Harrison Type-A kinetics regime), the effective diffusivity in two-phase systems and the effective ionic conductivity of composite electrolytes. Next, we discuss the determination of concentration profiles for grain boundary diffusion and the segregation of oxygen at interfaces during in-diffusion and out-diffusion of oxygen in cer-mets. For phenomenologically-based thermal diffusion problems, we discuss the calculation of the effective thermal conductivity/diffusivity in some model composites and porous metals and transient temperature profiles in composites.

14

Phase Transformation and Diffusion

MASS DIFFUSIO

The effective mass diffusivity represents the long time limit mass diffusivity in a ‘composite material’. This effective diffusivity is probably best known as the diffusivity measured in a tracer diffusion experiment made on a polycrystalline material in the Harrison Type-A kinetics regime[5]. In this regime, the diffusion length of the lattice diffusivity is of the order of the grain size and the effective diffusivity is some weighted average of the lattice diffusivity of the diffusant and the grain boundary diffusivity. Other examples are the effective ionic conductivity of a composite solid electrolyte[6] and the effective diffusivity of a diffusant in a two-phase material[7]. The LMC Method for Determining the Effective Mass Diffusivity. Mass diffusion is a random process that can be represented by random walks of particles. The century-old EinsteinSmoluchowski (ES) Equation[9,10] (often referred to simply as the Einstein Equation) describes the self-diffusivity D of randomly walking particles in d dimensions (d = 1,2,3): D=

< R2 > 2dt

(1)

(where R is the vector displacement of a given particle after some long time t and the Dirac brackets < > refer to an average over a very large number () of particles). The ES Equation refers to a system already at equilibrium. In a mass diffusion context, the ES Equation refers then to the diffusivity of individual particles that can be followed or traced in a system that is already at chemical equilibrium, i.e. with no concentration gradient acting or external field(s) acting. Each particle is followed for some long time t in order to determine its displacement R from its original position. For complete i.e. uncorrelated random walks on a simple cubic lattice the diffusivity can be partitioned from the ES Equation (eq. (1)) as; see, for example, ref [10]: D=

Γr 2 6

(2)

where Γ is the particle jump rate, r is the jump distance (the distance between sites). It is important to recognize that eq. (1) remains valid for long times even when the material has different diffusivities in different regions of the material (for example in a composite), provided that the material remains isotropic in its diffusion properties overall. The implication also is that each particle ‘explores’ a sufficiently large portion of the composite structure to be representative of the structure. The diffusivity represented in the ES equation is then the effective diffusivity Deff of the structure. By mapping a very fine grained lattice over the composite structure, and keeping the jump distance r the same everywhere (in principle this can be varied however), the diffusivities D in different regions of the structure can be simply represented by correspondingly different jump rates Г. It should be noted that since each particle is released independently and diffuses independently there are no correlation effects to be concerned with. Thus these LMC calculations are rather different in concept from atomistic Monte Carlo simulations; see, for example, ref [11]. Because the size of the jump distance r can be varied LMC calculations can be considered as spatially multiscale calculations. In many mass diffusion problems apart from spatially different diffusivities there may also be segregation of the diffusant between regions of the structure. This is readily represented by different rates in and out of the different regions. Thus if the segregation factor s between two regions 1 and 2 is defined as: s = C2/C1

(3)

Defect and Diffusion Forum Vol. 279

15

where C1 and C2 are the concentrations of the diffusant in regions 1 and 2 respectively, then the jump rate (Г12) into region 1 from region 2 and vice versa (Г21) are related to s by: s = Г12 / Г21

(4)

The simulation of actual time is not realizable in any Monte Carlo kinetics calculation. A discrete quantity that is proportional to actual time is generally used. This quantity is the number of jump attempts per particle during the calculation. Using the ES Equation, with this quantity acting as ‘time’, one can then calculate a relative diffusivity, i.e. a diffusivity that is relative to one of the specified diffusivities (usually the highest) in the system. Further details on how these multi-scale modelling LMC calculations are performed can be found in refs 1 and 2. Here, we describe several examples of applications to mass diffusion. Examples of the Effective Mass Diffusivity Calculated by the LMC method. The effective diffusivity of a diffusant in a material with grain boundaries is very important, especially for describing the rate of penetration of solute in (and out) of the material. As contributions to the effective diffusivity in a microcrystalline or nanocrystalline material, we can identify the diffusivity of the diffusant in the grains: this is the bulk or lattice diffusivity Dl and the diffusivity Dgb of the diffusant in the grain boundaries. In the well-known tracer diffusion experiment, a very thin layer of tracer (the diffusant) is deposited at the surface at time t = 0, the tracer is then permitted to diffuse into the material for some diffusion anneal time t. In this scenario it has been traditional to represent grain boundaries as parallel ‘slabs’ that are normal to the surface5. In the Harrison Type-A kinetics regime, the diffusivity that is determined experimentally from the (Gaussian) tracer concentration depth profile is the effective diffusivity of the material. (In this regime, the tracer diffusion length is of the order of the grain size.) The effective diffusivity is given exactly (for parallel slabs) by the (corrected) Hart-Mortlock Equation[5]: D eff = ( sgD gb + (1 − g ) Dl ) /(1 − g + sg )

(5)

where g is the grain boundary volume fraction. For self diffusion s = 1. However, eq. (5) had never been tested for more realistic grain structures. In Fig. 1 the effective diffusivity as calculated by LMC is shown as a function of grain boundary fraction g for a cubic grain model[12]. It is clear that the Hart-Mortlock Equation does not in fact provide a good description of the effective diffusivity in this situation. Perhaps surprisingly, the century-old Maxwell-Garnett Equation with the recent correction for possible segregation[13]:

Deff Dgb

=

2(1 − g ) Dgb + (1 + 2 g ) sDl (1 − g + sg )((2 + g ) Dgb + (1 − g ) sDl )

(6)

provides very good agreement with the LMC data, as can be seen in Fig. 1. For other grain models, a similar finding is obtained. For microcrystalline materials the Hart-Mortlock Equation overestimates the actual effective diffusivity by between 10 and 50% depending on the volume fraction of material in the grain boundaries and the degree of solute segregation. On the other hand, for nanocrystalline materials the use of the Hart-Mortlock equation would overestimate the effective diffusivity by about 45% and this is largely independent of the degree of solute segregation[14]. The Maxwell-Garnett Equation always does very well in all of the cases that have been examined so far and should therefore routinely replace the Hart-Mortlock Equation.

16

Phase Transformation and Diffusion

Hart Maxwell

Figure 1. Dependence of the effective diffusivity Deff/Dl on the volume fraction of grain boundaries g for the grain model for a solute segregation factor s = 10 and Dgb /Dl = 103 . Data points are LMC estimates.

Engineering materials are frequently used in multiphase form. Their microstructural stability and resulting longevity at high temperatures are determined by mass transport associated with the interphase boundaries (where the diffusivity is the highest). Addressing this problem requires in the first instance the calculation of the effective diffusivity in a three ‘phase’ system (the two different grains and the interphase boundary). An equation analogous to the Hart-Mortlock equation can be readily derived for this situation[15]:

DeffHart Db

g D  g D = k ′ A A + B B + g b   s A Db s B Db 

(7)

where

 g g  k ′ =  g b + A + B  s A sB  

−1

and where DA and DB are the bulk diffusivities, gA and gB are the volume fractions in the two types of grain A and B and sA and sB are the corresponding two (in general, different) segregation factors. An equation directly in the spirit of the Maxwell-Garnett Equation for the effective diffusivity has also been derived[15]:

s A Db − D A s D − DB − 2g B B b Deff 2 s A Db + D A 2 s B Db + DB = k′ Db  s D − DA s D − DB  1 + g A A b  + gB B b 2 s A Db + D A 2 s B Db + DB   1 − 2g A

(8)

Defect and Diffusion Forum Vol. 279

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and a very useful version of this:

Deff DbA

1 − 2g B = k′

s B DbA − DB 2 s B DbA + DB

 s D − DB 1 + g B B bA 2 s B DbA + DB 

  

(9)

Where 2 g A ( Db s A − D A ) DbA (1 − g B )(2 Db s A + D A ) = g A ( Db s A − D A ) Db 1+ (1 − g B )(2 Db s A + D A ) 1−

(10)

For a model of cubic alternating A and B grains, the effective diffusivity has been calculated by the LMC method. eq. (7) was found to be in poor agreement with the results but on the other hand eq. (8) and (9), (10) were shown to describe the effective diffusivity very well[15]. It is well-known that very considerable enhancement of the ionic conductivity of a material is possible by the addition of small, insulating and insoluble particles[6]. In these ‘composite electrolytes,’ it is believed a highly conducting space-charge layer develops at the interface between the matrix and the insulating particles and this is responsible for the conductivity enhancement. A phenomenological treatment of the problem can shed light on the role of the shape, distribution and size of the insulating particles on the effective conductivity of the material. The ionic conductivity of each of the three ‘phases’ (matrix, insulating particles, highly conducting layer) is related to the diffusivity of that phase by the Nernst-Einstein Equation: σi= Ciq2Di /kT

(11)

where q is the charge on the ion, C is the concentration (of charge carriers), k is the Boltzmann constant and T is the absolute temperature. A Maxwell-Garnett type equation can be derived to describe the effective ionic conductivity[16]:

σ eff =

σ 2 ((3 − 2 g 2 )σ 10 + 2 g 2σ 2 ) g 2σ 10 + (3 − g 2 )σ 2

(12)

where

σ 10 =

σ 1 ((3g 2 + g1 )σ 0 + 2 g1σ 1 ) g1σ 0 + (3 g 0 + 2 g1 )σ 1

and {σ0, g0}, {σ1, g1} and {σ2, g2} are the ionic conductivities and the fractions of the whole occupied by the insulating core (usually σ0 → 0), highly conducting layer (coating phase) and the matrix respectively. LMC calculations[16] of the effective ionic conductivity show that this equation well describes the enhancement of the effective ionic conductivity for both cubic and spherical insulating particles at relatively low volume fractions of the insulating phase. At higher volume fractions of the insulating particles, the LMC calculations clearly show that the highly conducting coatings of adjacent particles start to touch. This percolation-type behaviour then results

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Phase Transformation and Diffusion

in a very rapid increase of the ionic conductivity as the percolation threshold is approached. Then there are substantial deviations from the predictions of eq. (12).

The LMC Method for Determining Concentration Profiles. Obtaining the concentration profile using the LMC method is equivalent to solving the Diffusion Equation (Fick’s Second Law) for the diffusion problem once initial and boundary conditions have been specified for the problem. The basic LMC procedure to generate concentration profiles has a many similarities to that described already for determining the effective diffusivity. We illustrate the procedure with the well-known tracer diffusion situation mentioned above wherein a very thin layer of tracer (the diffusant) is deposited at the surface at time t = 0 and then permitted to diffuse into the material for some diffusion anneal time t. A source of particles is established from which  particles are released sequentially. Each particle is permitted to diffuse for the (anneal) time t, at which time the final position of is recorded. This process is repeated. The final positions of all of the particles are then simply assembled to form a concentration penetration profile. The example presented in Fig. 2 shows the familiar Gaussian profile for tracer diffusion into a semi-infinite solid from a very thin source at the surface (x = 0). Here we plot the concentration ratio C/C0 as function of the distance from the source, C0 is the initial concentration at the source. In order to simulate a constant source at the surface, all particles are released simultaneously at t = 0 but are still allowed to diffuse completely independently. As each particle leaves the source it is immediately replaced. On the other hand, if a particle returns to the source it is immediately annihilated.

0.15 time = t1 0.10 C/C0

0.05 time = 10t1 0

0

20

40

60

x Fig. 2. Typical Gaussian concentration profiles obtained by LMC for diffusion into a semi-infinite solid from a thin-film source at x = 0 for two different diffusion anneal times.

Examples of Concentration Profiles Calculated by the LMC Method. The determination of concentration profiles in the presence of grain boundaries for diffusion from the thin-film tracer source is of interest to determine the transition between the Harrison Type-A kinetics regime, where the (Gaussian) concentration depth profile provides the effective diffusivity as discussed above, and the Harrison Type-B kinetics regime, where the concentration depth profile has two sections: the

Defect and Diffusion Forum Vol. 279

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first section providing Dl and the second or tail-section providing sδDgb where δ is the grain boundary ‘width’. Analysis of profiles computed by the LMC method enabled the transition point between the two regimes to be estimated to occur at Λ = L/(Dl t )1/2 = 0.4 (L is the average spacing between the grain boundaries) for the parallel slabs model, which is much higher then had been estimated previously[17]. Other grain models have also been analyzed with LMC and have shown transition points at Λ = 0.4 for the square grains model and Λ = 1.5 for the cubic grains model[18]. The oxygen that diffuses into metal/metal-oxide cer-mets in-service or during the initial cermet synthesis can strongly segregate to the interfaces between the metal and the metal oxide inclusions. As a consequence of this there can be serious degradation of mechanical properties of the composite. High temperature annealing can remove the oxygen from the cer-met by the process of diffusion-limited evaporation (out-diffusion). Both processes have been modelled by the LMC method[19]. A typical example profile is shown in Fig. 3 for oxygen diffusion into a Ag/MgO cermet from a constant oxygen source at the surface.

Fig. 3. A typical distribution map of oxygen in a model Ag/MgO cer-met with randomly distributed inclusions of MgO with allowance made for segregation of oxygen at the metal-ceramic interface (s = 1000). The (constant) source of oxygen is at plane x = 1 here. Time is 544.2 Monte Carlo time units. Np is the number of particles which is linearly related to the oxygen concentration.

THERMAL DIFFUSIO

The LMC Method for Determining the Effective Thermal Diffusivity and Conductivity. Thermal diffusion, like mass diffusion, is a random process that can be represented by random walks of particles. In the case of thermal diffusion, the particles are virtual thermal ‘particles’. The ES Equation also describes the thermal diffusivity Κ in d dimensions (d = 1,2,3):

Κ =

< R2 > 2dt

(13)

It should be noted that the thermal conductivity λi in a phase i is directly related to the thermal diffusivity Ki in that phase by the well-known expression Ki = λi / ρi Cp,i where ρi is the density of phase i and Cp,i is its specific heat. It has been shown[19] that for the purposes of a LMC calculation the following procedure of calculating effective thermal conductivity indeed works very well. In a

20

Phase Transformation and Diffusion

model composite, the conductivities λi are treated in exactly the same way as if they were thermal diffusivities (λi ⇒ K'i) and calculate effective thermal conductivity λeff which is then simply equal to the calculated effective thermal diffusivity K'eff in the same model. In the general case, the actual effective thermal diffusivity in a composite is related to the effective thermal conductivity using the following equation that can be derived by inspection: K eff =

λeff

(14)

( ρ C p )eff

where ( ρ C p ) eff =

∑ρ

i all phases i

C p,i g i

Examples of the Effective Thermal Diffusivity/Conductivity Calculated by the LMC method. In a model of a composite the LMC method for the calculation of the effective mass diffusivity (but without segregation) is thus exactly the same as the calculation of the effective thermal conductivity. Several recent LMC calculations of the effective thermal conductivity include calculations of the effective thermal conductivity of models of syntactic metallic hollow sphere structures (MHSS materials)[20] and compact heat sinks based on cellular metals[21,22]. An example of the results of a LMC calculation of the effective thermal conductivity is shown in Fig. 4 for the case of circular inclusions in a matrix[4]. In the same figure are also the results for the effective thermal conductivity using finite element analysis. The agreement is seen to be excellent.

Area fraction of inclusions, g Fig. 4. Comparison of results from LMC and Finite Element calculations of the relative effective thermal conductivity of a composite with circular inclusions (subscript 0) in a matrix (subscript 1) in a square planar arrangement as a function of area fraction of inclusions for several values of the matrix and dispersed phase thermal conductivities.

Defect and Diffusion Forum Vol. 279

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The LMC Method for Determining Temperature Profiles. Obtaining the temperature profile using the LMC method is equivalent to solving the Heat Equation for the problem once initial and boundary conditions have been set. Because of the special role of the density and specific heat in transient thermal transport problems, the determination of temperature profiles requires a rather more involved procedure from that described for concentration profiles given above. For the common situation where the surface temperature is held constant at T0, the number of virtual thermal particles at the source is held at n. The results of the LMC analyses after some time t are virtual thermal particle distributions. In order to obtain temperature profiles, the thermal particles are translated into site temperatures T according to: T=

n ⋅ T0 ρ ⋅ C p  n ⋅ ( ρ i ⋅ C p,i ) min

=

Ctherm ⋅ T0 ρ ⋅ C p ( ρ i ⋅ C p,i ) min

,

(15)

where n is the number of virtual thermal particles currently located at the site in a phase with density ρ and thermal capacity Cp, (n relates to the concentration of the virtual thermal particles as n = Ctherm ·n); and (ρi · Cp,i )min is the minimum value of that product over all phases. Further details can be found in ref [2]. Fig. 5 shows an example temperature profile obtained by LMC in a layered composite of aluminium and paraffin with the layers arranged normal to the heat flow. The thermal parameters of the two phases here are of course very different. In the same figure are the results of a determination of the temperature profile using finite element analysis. It can be seen that there is excellent agreement between these two methods.

Fig. 5 The temperature profile determined by LMC and the finite element method for a layered aluminium – paraffin composite for a constant surface temperature.

CO CLUSIO S In this overview, we introduced the recently developed Lattice Monte Carlo method for addressing and solving phenomenologically-based mass and thermal diffusion problems especially for composite materials. With examples, we describe the application of this numerical method to calculate effective mass diffusivities and concentration profiles. Next, we describe the application

22

Phase Transformation and Diffusion

of this method to the calculation of effective thermal conductivities/thermal diffusivities and temperature profiles.

ACK OWLEDGEME TS We wish to thank the Australian Research Council for its support of this work. One of us (T.F.) wishes to acknowledge the Portuguese Foundation of Science and Technology (FCT) for financial support.

REFERE CES [ 1] I. V. Belova and G. E. Murch , in Cellular and Porous Materials: Thermal Properties, Simulation and Prediction, Wiley-VCH, Weinheim, (Eds.) Öchsner A, Murch G E and de Lemos J S, in press. [ 2] I. V. Belova and G. E. Murch, Fiedler T and Öchsner A, Diffusion Fundamentals, in press. [ 3] I. V. Belova and G. E. Murch, Solid St. Phenom., Vol. 129 (2007), p.1 [ 4] I. V. Belova, G. E. Murch, N. Muthubandara and A. Öchsner , Material Science Forum, Vol. 553 (2007), P. 51 [ 5] I. Kaur , Y. Mishin and W. Gust , in Fundamentals of Grain and Interphase Boundary Diffusion, Wiley, Chichester, (1995). [ 6] P. Heitjans and S. Indris , J. Phys. Condens. Matter, Vol, 15 (2003), R1257 [ 7] I. Stloukal and J. Čermak , Defect Diffusion Forum, Vol. 263 (2007), p.189 [ 8] A. Einstein , Ann. Phys. (Leipzig), Vol. 17 (1905), p.549 [ 9] M. von Smolukowski , Ann. Phys. (Leipzig), Vol. 21 (1906), p.756 [10] H. Mehrer , in Diffusion in Solids, Springer, (2007). [11] G. E. Murch , in Diffusion in Crystalline Solids, Academic Press, Orlando, (Eds) A. S. Nowick and G. E. Murch , (1984), p 379. [12] I. V. Belova and G. E. Murch, in Mass and Charge Transport in Inorganic Materials, Techna, Fienza, (Eds.) Vincenzini and Buscaglia V, (2003) p 225. [13] J. C. Maxwell-Garnett , Phil. Trans. Royal Soc., Vol. 203 (1904), p.386 [14] I. V. Belova and G. E. Murch, J. Meta. Nanocryst. Mat., Vol. 19 (2004), p. 23 [15] I. V. Belova and G. E. Murch, Phil. Mag., Vol. 84 (2004), p.17 [16] I. V. Belova and G. E. Murch, J. Phys. Chem. Solids, Vol. 66 (2005), p. 722 [17] I. V. Belova and G. E. Murch, Phil. Mag. A, Vol. 81 (2001), p. 2447 [18] I. V. Belova and G. E. Murch, in Defect and Diffusion Forum, in press. [19] I. V. Belova , N. Muthubandara , G. E. Murch , M. Stasiek and A. Oechsner , Solid State Phenomena, Solid. St. Phenom, Vol. 129 (2007), p. 111 [20] I. V. Belova and G. E. Murch, Journal of Materials Processing Technology, Vol. 153-154 (2004), p.741 [21] U. Ramamurty , A. Paul , Acta Mater., Vol. 52 (2004), p.869 [22] A. G. Evans , J. W. Hutchinson and M. F. Ashby , Prog. Mater. Sci., Vol. 43 (1999), p.171 [23] Nemat-Nasser , W. J. Kang , J. D. McGee , W. G. Guo and J. B. Isaacs , Int. J. Impact Eng., Vol. 34 (2007), p.1119

Defect and Diffusion Forum Vol. 279 (2008) pp 23-37 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.23

Diffusion and Melting G.P. Tiwari1,a and R.S. Mehrotra2,b 1

Post Irradiation Examination Division, 2Radiometallurgy Division Bhabha Atomic Research Centre, Mumbai 400 085, India E-mail: [email protected], [email protected]

Keywords: Diffusion, melting, self-diffusion coefficient, entropy of fusion, melting temperature

Abstract. The paper reviews the correlation between the processes of diffusion and melting. It is shown that the entropy of fusion and the melting temperature have a governing influence on the self-diffusion rates in solids. The relationship between self-diffusion coefficient (D) in solids and the melting parameters can be expressed as follows: D = fa2ν exp (κSm / R) exp (– κSmTm / RT) , where f is the correlation factor, a the lattice parameter, ν the vibration frequency, Sm the entropy of fusion, Tm the melting temperature in degree K, κ a constant and R, T have their usual meaning. The above equation has been derived on the basis that the free energy of activation for diffusion is directly proportional to the free energy of liquid phase. The well known relationships of the activation energy for self-diffusion with the melting point and enthalpy of fusion can be derived on the basis of this assumption. The constant κ is a group constant for any class or group of solids having identical physical and chemical properties. The validity of the above equation is demonstrated by the fact that when the self-diffusion coefficients are plotted as a function of homologous temperature, they scale inversely with the magnitude of the entropy of fusion. The hierarchy of self-diffusion rates within any group of solids is governed by the magnitude of the entropy of fusion and the melting temperature. The paper also discusses some interesting fall out of the close relationship between the diffusion and the melting parameters concerning (a) the diffusion in elemental anisotropic lattices, (b) anomalous diffusion behavior in bcc transition metals, lanthanides and actinides and (c) congruently melting compounds. Introduction The first suggestion regarding an empirical correlation between the activation energy for selfdiffusion and the melting point was made by van Leimpt [1], Fig. 1. The next important step in establishing the kinship between diffusion and melting parameters was the demonstration of a correlation between the activation energy for self-diffusion and enthalpy of fusion by Nachtireb and Handler [2], Fig. 2. The most creditable thing about these two hypotheses by van Leimpt and Nachtrieb et al. is that more precise extensive measurements of the diffusion parameters in subsequent years have strongly validated them [3-6]. The third significant development in this field was the discovery of the role of entropy of fusion in influencing self-diffusion behavior within a group of solids having identical physical and chemical properties. It was pointed out by Tiwari [7] that when the self-diffusion rates are expressed as a function of homologous temperature (the ratio of melting temperature and the ambient temperature in absolute scale, Tm/T), their magnitudes scale inversely with entropy of fusion. This is shown in the Fig. 3 where data on self-diffusion in lanthanides are plotted as a function of the homologous temperature. The values of the entropy of fusion are shown in each case in the parentheses. Lanthanide elements are so very similar in their physical and chemical characteristics that they constitute an ideal group to exhibit the role of

24

Phase Transformation and Diffusion

melting parameters on diffusion characteristics. The significance of Fig. 3 lies in the fact that it connects the

Fig. 1 Correlation between the activation energy for self-diffusion and the melting point for metals; Slope : 146 J mol-1K-1.

1.0

1.1

1.2 1.3 Tm / T

1.4

Fig. 2 Correlation between the activation energy for self-diffusion and enthalpy of fusion for metals; Slope : 14.8.

1.5

Fig. 3 Self-diffusion coefficient in lanthanides as a function of homologous temperature. Figures in parentheses are the values of entropy of fusion in J mol-1K-1.

1.0

1.2

1.4

1.6

1.8 2.0 Tm / T

2.2

2.4

Fig. 4 Self-diffusion coefficient in metals as a function of homologous temperature. Figures in parentheses are the values of entropy of fusion in J mol-1K-1.

rate of self-diffusion in solids, i.e. the activation energy with two melting parameters and provides the basis for the correlation between the processes of diffusion and melting. As shown later, the relationship exhibited in the Fig. 3 is universal and is followed by metals, alkali halides, organic compounds and even the inert gas solids. This article attempts to put the relationship between

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diffusion and melting parameters on more firm basis than has been done before, in the following way: 1. Discuss a thermodynamic relationship between the processes of diffusion and melting. 2. Show that the correlations proposed by van Leimpt and Nachthrieb et al. are not independent but complementary to each other. 3. Discuss the implications of the universality of relationship between diffusion and melting parameters on our understanding of the self-diffusion behavior in solids. Derivation of the Basic Equations The phenomenon of melting is characterized by three parameters, viz, the melting point temperature, enthalpy of fusion and the entropy of fusion. Out of these, only two are independent owing to the following relation between these quantities:

Sm =

Hm , Tm

(1)

where S m , H m and Tm are respectively the entropy of fusion, enthalpy of fusion and the melting temperature. A proportionality between the free energy of activation for diffusion (∆Gd) and the free energy of liquid state (∆Gl) was first proposed by Dienes [8]. This can be expressed as [7,9]:

∆Gd = κ ∆Gl ,

(2)

κ being the constant of proportionality. Left hand side and right hand side of the Eq. (2) can be expanded as follows: ∆Gd = Q – TSd, and

κ ∆Gl = κ (Hm – TSm),

(3) (4)

where Q and Sd are respectively the activation enthalpy and activation entropy for self-diffusion. From the comparison between temperature-dependent and temperature-independent terms in the Eq. (3) and Eq. (4), we have: Q = κ Hm , and

Sd = κ Sm .

(5) (6)

Equation (5) represents the correlation proposed by Nachtrieb and Handler [2]. The correlation between Q and Tm proposed by van Leimpt [1] is represented as: Q = KTm .

(7)

In view of the Eq. (1), it is evident that κ and K are related as: K = κS m .

(8)

Equation (8) shows that the correlations proposed by van Leimpt and Nachtrieb et al. are interrelated via the entropy of fusion. Figures 1 and 2 validate the assumption contained in Eq. (2). The general expression for self-diffusion coefficient in solids, after Zener [10], is as follows:

26

Phase Transformation and Diffusion

D = fa2ν exp [– ∆Gd / RT] = fa2ν exp [– (Q – TSd) / RT] , = fa2ν exp (Sd / R) exp (– Q / RT) .

(9)

Here f is the correlation factor, a the lattice parameter and ν the vibration frequency of the diffusing atom. Substituting for Q and Sd from Eq. (5) and Eq. (6) in Eq. (9), we get: D = fa2ν exp (κSm / R) exp (– κHm / RT) .

(10)

And using Eq. (1), it can be written as: D = fa2ν exp (κSm / R) exp (– κSmTm / RT) .

(11)

κS Tm will have slope equal to − m . This equation T R also shows that for any group of solids having identical values of κ , the self–diffusion rates will scale inversely with the magnitude of the entropy of fusion. Self-diffusion plots based on Eq. (11) for lanthanides, metals, halides, organic solids and inert gas solids are shown in Figs. 3-11 respectively. Data on self-diffusion in metals, lanthanides and actinides from the compilation in Smithell’s Metals Reference Handbook [11]. Data on ionic conductivities of halides have been taken from Uvarov et. al [12] and other references given there in. Data on self-diffusion in organic solids and inert gas solids have been taken solids have been from references [13] and [14] respectively. Melting points and the entropies of fusion in all cases have been obtained from Barin [15].

The Eq. (11) shows that a plot of lnD against

Fig. 5 : Logarthmic plot of product of ionic conductivity (σ) and the temperature (T) against homologous temperature (Tm/T) for silver and Lithium halides. σ.T is directly proportional to the ionic diffusivity. Figures in parentheses represent the entropy of fusion.

Fig. 6 : Same as Fig. 5 for potassium halides.

Defect and Diffusion Forum Vol. 279

-2

LOG σΤ (OHM -1CM -1)K

120

-1

-1

LOG σT (OHM CM )K

-3

-4

-5 Nal (25.27)

NaBr (25.60)

80 RbCl (23.94)

20

0

RbBr (24.13)

1.0

1.2

Rbl (23.97) 1.4

1.6

1.8

T /T

-6

-7

27

NaCl (26.06)

1.0

1.2

1.4 Tm / T

1.6

1.8

Fig.7 : Same as Fig. 5 for sodium halides.

Fig. 9 Self-diffusion coefficient in organic solids methane and succinonitrile as a function and of homologous temperature. Figures in parentheses are the values of entropy of fusion in J mol-1K-1.

Fig. 8 : Same as Fig. 5 for Rubidium halides.

Fig. 10 Self-diffusion coefficient in organic solids benzene, phenanthrene, naphthalene biphenyl as a function of homologous temperature. Figures in parentheses are the values of entropy of fusion in J mol-1K-1.

28

Phase Transformation and Diffusion

0 -20

1.0

2.0

3.0

4.0

5.0

6.0

7.0

Ln D (sq.m./s)

-40 -60 -80 -100 -120 -140

Neon (13.64) Krypton (14.16) Argon (14.08) Xenon (14.06)

-160 -180

Tm / T

Fig. 11 Self-diffusion coefficient in inert gas solids neon, krypton, argon and xenon as a function of homologous temperature. Figures in parentheses are the values of entropy of fusion in J mol-1K-1. The standard form of diffusion equation is D = Do exp(−

Q ) , RT

(12)

Where Do is called the frequency factor and all the terms have same meaning as before. From the Eq. (11), Do is given by-

Do = fa 2ν exp(

κS m R

)

(13)

For each class of solids Do varies within a narrow range. In addition to near constancy of the product fa 2ν , the variation of Do is also dependent on κ and S m . These two parameters behave as group properties and show only a small variation within any group. Since the plots based on the Eq. (11) shown in the Figs.3-11 are linear, we may write-

κSm / R = constant , say λ .

(14)

With the help of Eq. (8) and Eq. (14), we may write Eq. (7) as: Q/N = λ. k Tm ,

(15)

where N is Avogadro’s number and k is Boltzmann’s constant. The Eq. (15) has an interesting interpretation. The LHS is equal to the average energy of the atoms forming the saddle-point configuration which governs the rate of diffusion. On the other hand, the RHS represents a product of λ and the average thermal energy of the atoms at the melting point. Hence, for diffusion to occur, the diffusing atom and its neighbors forming the saddle-point configuration must collectively acquire an energy, which is lambda times the average thermal energy of the atoms at the melting point. This observation provides a physical basis for the

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interpretation of Eq. (15). Lambda is nearly 71 for metals and its value is different for different types of solids. Discussion Vacancy Mechanism and the Correlation between Diffusion and Melting Parameters : Extensive experimentations [16-19] as well as numerous theoretical investigations [20-22] have established that self-diffusion in metals takes place through the exchange of atomic sites between an atom and a vacancy. This process is termed as the vacancy mechanism. On the basis of analogy between the diffusion parameters and the melting parameters, similar inferences have been drawn for alloys [23-25], alkali halides [26] and organic solids [13]. Theoretical calculations as well as the correlations of the activation energy for self-diffusion with the entropy of fusion and the melting temperature have again established atom-vacancy exchange as the predominant mode of diffusion in the inert gas solids [14]. There is thus overwhelming evidence that in a majority of cases, the self-diffusion in solids occurs predominantly through vacancy mechanism. Historically, the correlation of diffusion data with the melting parameters first started with the metals and subsequently extended to cover other types of solids. Hence, we may conclude that the correlation between the diffusion and the melting parameters is an evidence for the operation of the vacancy mechanism. These correlations manifest as the invariance in the magnitude of κ and K – Eq. (5) and Eq. (7) - for each group of solids having identical physical and chemical properties. The correlations between diffusion and melting parameters are almost universally obeyed so that any deviation in the diffusion data from the Eq. (5) and Eq. (7) is viewed with suspicion [3]. Most commonly observed deviations arise when the diffusion takes place mainly via dislocations and the grain boundaries [27-29]. The basis of correlation between the diffusion and melting parameters is the suggestion [1, 2, 8-10] that during the process of diffusion the matrix structure of a solid in the region surrounding the vacancy is similar to that obtained in its liquid state. As the vacancy is an equilibrium defect, its concentration in the lattice is maintained through thermal fluctuations. The exchange of atoms, which is the unit process for diffusion, occurs in this liquid-like region. Such liquid-like regions are being continuously created and annihilated through thermal fluctuations in such a way that a definite fraction of atoms, which is governed by the magnitude of ∆Gd, is present in liquid-like regions. The proportionality expressed in the Eq. (2) owes its origin to these considerations. Nachtrieb and Handler [2] found the value of constant κ to be approximately 14. These authors suggested that nearly 14 atoms from among the first and second nearest neighbors formed a liquidlike region around the diffusing atom. Apart from the Eq. (5) and Eq. (7), the following equation relating activation volume for self-diffusion with the rate of change of melting point as a function of pressure also provides a firm support to the close interrelationship between the phenomena of diffusion and melting [30]:

dT Q ∆V = ( m )( ) . dP Tm

(16)

A critical test of the validity of the Eq. (16) was made by Zanghis and Calais [31] who experimentally determined the activation volume in a series of zirconium and plutonium alloys. The authors found that the activation volume for self-diffusion turned negative for compositions for which the dTm/dP is negative. At the same time, the activation volume is positive whenever the dTm/dP is positive. Somewhat later, Brown and Ashby [32] compared the experimental values of the activation volumes for self-diffusion with those given by the Eq. (16) and a satisfactory agreement was noted. The equations (5), (7), (11) and (16) taken together constitute incontrovertible evidence for the existence of a close and intrinsic relationship between the phenomena of diffusion and melting. In a similar vein, Ablitzer [33] has shown that in case of solute diffusion in niobium, the rapidly diffusing elements are those which, when added in solution to pure Nb, lower the

30

Phase Transformation and Diffusion

melting temperature. In the same way, the slowly diffusing elements are those which raise the melting temperature. Mehl et al [34] and Gorecki [35] have also been successful in establishing a correlation between the melting parameters and volume change associated with the formation of vacancies in metals. Anomalous Diffusion in BCC Transition Metals, Lanthanides and Actinides : The most important conclusion to be drawn from Eq. (11) and Figs. 3-11 is that the hierarchy of the selfdiffusion rates within any group of solids is governed by the magnitude of the entropy of fusion and the melting temperature. In other words, in the two dimensional space bounded by the self-diffusion coefficients and the homologous temperature, the location of each element with respect to one another is governed by the magnitude of the entropy of fusion. This statement holds for all classes of solids. Universal validity of this observation accords to it the status of an inviolable rule. In Fig.12, self-diffusion data for a number of bcc metals are plotted as a function of homologous temperature. β-Zr and β-Ti are located in between β-Pr and ε-Pu on the upper side and Nb, V and Cr on the lower side. The self-diffusion rates scale inversely with the magnitude of the entropy of fusion. This behavior is similar to metals and other solids whose diffusion data are featured in Figs. 3-11. Hence, we can conclude that the anomalies ascribed to these elements disappear when the diffusion behavior is rationalized on the basis of the Eq. (11). In common fcc, bcc and hcp metals, satisfactory and reliable data on vacancy formation and migration energies are available. To a very great extent, this has helped in establishing the mechanism of self-diffusion in an unequivocal manner. This is not so in the case of elements like β-Zr, β-Ti, lanthanides and the actinides. In this context, the self-diffusion mechanism in β-Zr has received lot of attention [36-37]. Basic information on vacancy parameters is needed to establish, in a firm manner, the mode of diffusion in metals like β--Zr, ε-Pu. etc. Presently, we shall review theories regarding anomalous diffusion in bcc zirconium. Kidson [36] postulated that the anomaly in the diffusion behavior of bcc transition metals arises from the extrinsic contribution to the over all diffusion by the impurity-vacancy complexes similar to the role of divalent and trivalent elements in alkali halides [38]. The main effect of impurity-vacancy association would be to create an excess vacancy concentration at low temperatures which would result in appreciable enhancement of diffusion. Kidson’s model requires an impurity-vacancy energy of 125 kJ/mole to fit the experimental results, which is unrealistically high. Hence, this model is not favored. The curvature seems to be an intrinsic characteristic of diffusion in these systems. It persists for solute diffusion as well as diffusion in alloys. In latter cases, impurities will have a negligible role to play in diffusion. Sanchez and de Fontaine [37] proposed another model for anomalous self-diffusion in group IV B transition metals Ti, Zr and Hf. According to these authors, the complex formed by the jumping atom at the saddle point has a structure identical to that of hexagonal ω-phase and the low temperature diffusion in these systems is enhanced by structural fluctuations which lead to the formation of omega-embryo. However, omega-embryos have not been observed in pure matrices although their presence has been experimentally verified in low beta region for zirconium alloys [39-40]. Application of pressure favors the formation of omega phase [41]. However, the pressure reduces the self-diffusion rates in β-Ti [42]. A similar behavior is expected for β-Zr. This is contrary to the prediction based on the theory of Sanchez and de Fontaine [37]. Thus the model proposed by Sanchez and de Fontaine is at odds with the experimental data for the effect of pressure on self-diffusion. Anomaly in the diffusion characteristics of some bcc metals has two features; first, the curvature in the lnD vs 1/T plots and second, the low activation energies (and correspondingly higher diffusion rates) despite the high melting points. Careful measurements by several authors show that the curvature, albeit slight, exists in many common fcc metals, such as Cu, Ag, Au, Ni and also Al [43-44]. Curvature is also found in several bcc metals such as Ta [45], Nb [46], V [47] and W [48] as well. These elements, however, do not exhibit anomalously low activation energies

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for self-diffusion as found in the case of β-Zr [49], β-Ti [50], β-Pr [51], δ–Ce [52] and ε-Pu [53] etc.. A review of experimental data on self-diffusion in metals shows that whenever the available temperature range of measurement exceeds 500 K, a non-linearity is likely to occur in the lnD vs 1/T plots. Theoretically a non-linearity in the diffusivity plots is admissible, the only condition being that the variation in the entropy and enthalpy with temperature should be related to each other by the following equation [54]:

∂ (∆S ) 1 ∂ (∆H ) . = ⋅ ∂T T ∂T

(17)

In fact, Federer and Lundy [51] described the measurements on self-diffusion in β-Zr as a polynomial in temperature. Subsequently it was shown [55] that self-diffusion in β-Zr indeed obeys equation (17). The type of diffusion characteristics exhibited by β-Zr and β-Ti is generic to a class of elements which undergo phase transformation upon going from low temperature to high temperature. High temperature phase is characterized by low activation energies and high diffusivities. Typically, the diffusion rates near the melting point are in the range of 10-11 m2/sec. Fig.12 shows the diffusion data in bcc transition metals, lanthanides and actinides. In conformity with the diffusion behavior shown in Figs. 3-11, the diffusion rates scale inversely with the magnitude of the entropy of fusion. In other words, as a group, these elements follow same pattern as observed in other solids where the vacancy mechanism is well established as a mode of selfdiffusion. Further, the elements which lie on the upper side of the figure (Fig. 12) and exhibit relatively higher diffusion rates are all allotropic and also have lower values of the entropy of fusion [9]. An explanation for higher diffusion rates in the high temperature phases on account of phase transformation has been given in an earlier paper [56]. A smaller value of the entropy of fusion is thus an indication of allotropy and relatively higher diffusion rates in every single class of solids. Among lanthanides and actinides, δ–Ce [57] and ε–Pu [58] constitute a special pair. The activation volume for self-diffusion is negative in these cases. As a result, the diffusion rates are enhanced upon the application of pressure. In conventional sense, this will rule out vacancy mechanism because, in general and barring these two cases, the activation volume for self-diffusion is always positive. In order to account for the negative volume of activation, Cornet [58] proposed that the self-diffusion in ε-Pu takes place via an activated interstitialcy mechanism. According to this model, the migrating atom at the saddle point configuration is in electronically excited state having given up to two electrons to the conduction band. Using calculations based on the elastic theory, Cornet was able to obtain a satisfactory agreement with the high pressure diffusion data as well as the negative activation volume for self-diffusion. As against this, Brown and Ashby [29] applied Eq. (16) to δ-Ce and ε-Pu and found satisfactory agreement with experimental data on self-diffusion in these cases. In particular, it should be noted that equation (16) predicts that the activation volume for self-diffusion should be negative on account of the fact that pressure reduces the melting point in these cases. This is contrary to the general behavior in systems where the activation volume for self-diffusion is positive. It can, therefore, be concluded that the relationship between the diffusion and melting parameters remains valid even for δ-Ce and ε-Pu. However, the saddle point configuration formed

32

Phase Transformation and Diffusion

-10

10

γ-L -11

2

D (m / sec)

10

a (5

δ - Ce .19)

(5.04)

ε - Pu β

γ-Y

b (6

(3.12)

- Pr (5

.72)

.98)

β-Z

γ-U

-12

10

Nb (

r (7

.96)

(6.6

0)

9.68

) V (9

-13

Cr

10

(9.6

.68)

8)

-14

10

1.05

1.10

1.15

1.20

Tm / T Fig. 12 : Logarithmic Plot of self diffusion coefficient against homologous temperature (Tm/T) for BCC metals. The figures in parentheses represent the entropy of fusion in J mol-1K-1.

during diffusion experiences a contraction because of the volumetric contraction observed upon melting in these systems. Independent measurement of vacancy formation energies and the volume changes associated with formation and migration of vacancies can resolve this issue. Congruently Melting Solids : The preceding discussion has concentrated on the manner in which the correlation between the diffusion and melting parameters controls the diffusion within any one group of solids. The pattern of behavior remains unchanged in going from one group to another, the difference being only in the magnitude of κ. In this section we discuss how the correlation between diffusion and melting parameters influences the self-diffusion characteristic in any congruently melting system. Equation (5) gives the activation energy for self-diffusion as a product of κ and Hm. Both these parameters are the characteristics of the matrix. Hence, the Eq. (5) as well as Eq. (7) require that the activation energy for self-diffusion remains invariant along different crystallographic directions and for different components of a congruently melting compound. Equations (5) and (7) are based on absolute reaction rate theory which implies the formation of an activated state. If the activated state is analogous to the liquid state, then there can be only one kind of activated state in the congruently melting solids and hence the need for the equivalence of activation energy for self-diffusion along different directions in an anisotropic matrix as well as for different constituents of a congruently melting compound. As a test of this hypothesis, we have compiled in the tables 1 and 2 the self-diffusion data for anisotropic elements and for different types of compounds. Before discussing the results listed in the tables 1 and 2, it should be pointed out that the invariance dictated by the equations (5) and (7) is restricted only to the activation energy and

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not to the absolute diffusion rates. In addition, the invariance of activation energy will be further restricted to situations where the diffusion occurs solely via mono-vacancy migration. As a corollary, it may also be inferred that (a) the conformance of the diffusion data to equations (5) and (7), and (b) the invariance of the activation energy for self-diffusion along different directions in an anisotropic matrix and for different constituents of a congruently melting solid may be considered as an indication of the diffusion via migration of single vacancies. The rate of diffusion at any temperature is controlled, in addition to the activation energy, by the frequency factor which is defined by the equation (13). Vibration frequency of the diffusing specie is the most important parameter in this regard. In an anisotropic lattice, the changes in the correlation factor and jump distance of the diffusing atom will also affect the diffusion rates to a different extent along different crystallographic directions. For compounds with cubic lattices, vibration frequency may be the most important factor responsible for the differences in the diffusion rates of the different components of a congruently melting compound. Application of equation (5) to individual systems is discussed below. Anisotropic Metals : Self-diffusion data for anisotropic metals and other lattices [45] are listed in Table 1. According to the data listed in this table, the activation energies are nearly the same for two principal crystallographic directions in magnesium, thallium and tin. For the rest, the differences are less than 10% and can be considered to lie within the limits of experimental uncertainty. The agreement of data in the Table 1 with the Eq. (5) is satisfactory, the sole exception being antimony for which the activation energies in two directions differ by nearly 35%. Such a difference can be either due to the presence of impurities or the change in diffusion mechanism itself. A redetermination of the data on self-diffusion in antimony is called for. Table 1 Self-diffusion data for anisotropic metals [43] S.No.

Metal

Structure

1 2 3 4 5 6 7 8

Mg Tl Sn Cd Zn α-Hf Be Sb

hcp hcp bct hcp hcp hcp hcp hex

D0|| 104 m2/s 1.5 0.4 10.7 0.18 0.18 0.28 0.52 0.10

D0⊥ 104 m2/s 1.0 0.40 7.7 0.12 0.13 0.86 0.68 56

∆H||

kJ/mol 136 95.5 105 82.0 96.4 349 157 149

∆H⊥

kJ/mol 135 95.8 107 78.1 91.6 370 171 201

%Diff. ∆H 0.74 0.31 1.9 5.0 5.24 6.02 8.92 34.9

Intermetallic Compounds: Table 2 lists the activation energies for self-diffusion in some compounds, which include intermetallics. The compounds considered here include only those for which diffusivities for both components of the matrix have been experimentally determined. Among the five intermetallics listed in the table, the activation energies of the two components differ by 0.34% to 6.94% and can be said to conform to the Eq. (5). Six-jumps-cycle, triple defect mechanism and antistructural bridge are some of the mechanisms proposed for diffusion in intermetallic compounds [59]. However, the invariance of the activation energies for either component in their parent compound favors the operation of nextnearest neighbor jump mechanism for self-diffusion in these systems.

34

Phase Transformation and Diffusion

Table 2. Diffusion data for Intermetallic compounds, Alkali halides and Polymers S.No.

Matrix

Diffusing Species

1

AuCd

2

AuZn

3

AgMg

4

NbCo2

5

Ni3Al

Au Cd Au Zn Ag Mg Nb Co Ni Al

6

KCl

7

NaI

8

14

Intermetallics 0.17 0.23 0.33 1.93 0.17 0.051 9.24x10-3 0.284 150.0 80.0 Alkali halides 4.72 5.19 3.50 1.95 Organic Solid

∆H kJ/mol 116.6 117.0 138.4 148.0 188.5 198.6 292.0 294.7 347.0 340.0 208.39* 217.05* 156.97 168.52

C 1.43x109 96.7 3 H 1.53x109 96.9 Values estimated from the data given in Ref. [72] Benzene

•

K+ Cl− Na+ I−

D0 10 m2/s 4

%Diff. ∆H

Ref.

0.34

67

6.94

68

5.35

69

0.92%

70

2.06

71

4.15

72

7.36

73

0.21

74

Alkali Halides: The majority of information on the point defects in alkali halides has been obtained through the measurement of ionic conductivities as a function of temperature. For the present purpose, we have relied upon experimental values of the diffusivities of anions and cations measured with the help of radioactive tracers. Such data are available only for potassium chloride and sodium iodide. These are listed in the Table 2 and clearly show that activation energies for the negatively and positively charged components are nearly same in both the compounds. Despite significant differences in the sizes, cations and anions have the same activation energy for diffusion. This is highly significant. This observation underscores the fact that in diffusion the size and charge of the diffusing species may have a relatively small role to play and the crucial factor governing the rate of diffusion is the composition and the energy of the activated complex. Organic Solids : Diffusion studies on polymeric materials have been reviewed by Chadwick [13]. However, in a majority of cases, only the diffusion of carbon has been measured. To authors’ knowledge, reliable values for the diffusion of carbon as well as tritium have been determined only in the case of benzene; Table 2 .The results show that the activation energies for diffusion of carbon and tritium are practically the same. McGhie et al [60] studied the self-diffusion in imidazole single crystal, which crystallizes in fcc structure, using tritium and C-14 isotopes. Diffusion experiments were performed only perpendicular to (100) cleavage plane due to difficulty in cutting and polishing (001) plane without introducing internal cleavage on (100). In this system, diffusion occurs via molecular movement

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vacancy mechanism. Activation energies as well as overall diffusion rates were identical for both species. Self-diffusion of tritium in natural as well as synthetic ice single crystal was studied by Ramseier [61]. Diffusion experiments were performed along c-direction and perpendicular to cdirection. An activation energy of 0.62 eV was observed for both directions. Anisotropy of diffusion rates also persists in both cases - natural (7%) and synthetic (16%). The authors analyzed the results on the basis of Zener’s theory of D0 and concluded that diffusion occurs through molecular movements via vacancy mechanism. Absolute Reaction Rate Theory versus Dynamical Theory : There are two different approaches to model the diffusive motion of atoms in crystalline lattices. The first approach is based on the absolute reaction rate theory [62-63] and postulates the formation of an activated complex at the saddle point, which lies at a point between the initial and the final positions of the diffusing atom. The saddle-point configuration is visualized as a kind of activated complex formed between the diffusing atom, a vacancy and the next nearest neighbors. The energy associated with the process of diffusion is equivalent to the difference between the potential energy of the atom in the two states. This assumption is often criticized on the grounds that the activated complex cannot be assigned thermodynamic properties because of its transitory nature. It will be difficult for the configuration of atoms forming a short lived activated complex to interact and exchange energy with the surrounding medium. In the dynamical theory of diffusion, Rice [64-65] proposed that diffusion occurs when the diffusing atom has acquired a large properly oriented amplitude of vibration and its surrounding atoms execute an out of phase motion. The energy associated with the diffusion in this model is the sum of (a) energy required to form a vacancy, (b) kinetic energy associated with the central diffusing atom and its neighbors, and (c) the potential energy of the diffusing configuration. The final form of the diffusion equation given by both approaches is exactly the same [66]. Therefore the determination of diffusion parameters cannot be employed to choose between these two models. However, the data listed in the tables 1 & 2 favor activated complex model due to the fact that in the dynamical theory of diffusion the activation energy is expected to vary with the size, atomic weight and charge of the diffusing specie. Conclusion The relationship between the diffusion and melting parameters in solids is as old as the subject itself. This paper discusses a formal basis for relationship between the process of melting and selfdiffusion characteristics of solids. Entropy of fusion and melting temperature turn out to be the basic physical parameters linking the two processes. The most important aspect of this relationship is that when the diffusion rates are plotted as a function of the homologous temperature, the selfdiffusion rates scale inversely with the magnitude of the entropy of fusion. The reach of this relationship between the entropy of fusion and the self-diffusion rates is truly global. It includes all the solids for which reliable data are available, including metals, alkali halides, inert gas solids and the organic solids. It has also been observed that the elements which have smaller values of the entropy of fusion and relatively higher self-diffusion rates undergo allotropic phase transformations. These observations provide a basis for the assumption that the process of diffusion and phase transformation behavior are interrelated. Next to alloying, diffusion and phase transformation are the two important physical processes utilized to strengthen the metallic matrices and the entropy of fusion is one property which governs either of them. Hence, the knowledge of the true value of the entropy of fusion is an imperative necessity from technological considerations. In this context, an unfortunate fact pertaining to several technologically important metals such as Ti, Zr, Nb, Ta, Cr etc. is that experimentally determined values of the entropy of fusion are not available. In such cases, the values of this parameter are derived from the extrapolated values of the specific heats of the solid

36

Phase Transformation and Diffusion

and liquid phases. Hence, it is suggested that efforts should be directed to obtain the experimental values of the entropy of fusion all the technologically important metals and compounds since this parameter is an important aid in the interpretation of diffusion data for elements as well as the compounds. Acknowledgement: The autors express grateful thanks to Dr D. Gupta for his valuable comments on the manuscript. References [ 1] J.A.M. van Leimpt , Z. Physik, Vol. 96 (1935), p. 534 [ 2] N.H. Nachtrieb and G. S. Handler, Acta Metall., Vol. 2 (1954), p. 791 [ 3] A.D. LeClaire, in Diffusion in Body Centered Cubic Metals (J. A. Wheeler, Jr., and F. R. Winslow, eds.), ASM, Metals Park, Ohio (1965) p.3 [ 4] J. Philibert, in Atom Movements Diffusion and Mass Transport in Solids, Les Editions de Physique, les Ulis Cedex A, France (1991), Chap. IV, p. 112 [ 5] P.G. Shewmon, in Diffusion in Solids, Second Edition, TMS AIME, Warrandale, Pa.(1989), Chap.2, p.86 [ 6] G.P. Tiwari, R.S Mehrotra and Y.Iijima” in D. Gupta (Ed.) Diffusion Processes in Advanced Technological Materials, William Andrew Publishing , New York (2005), Chap.2, p.69. [ 7] G.P. Tiwari, Trans. Jap. Inst. Metals, Vol. 19 (1978), p. 125 [ 8] G.J. Dienes, J. Appl. Phys., Vol. 21 (1950), p. 1189 [ 9] G.P. Tiwari, Z. Metallkunde, Vol. 72 (1981), p. 211 [10] C. Zener in W. Shockley et. al (Ed.) Imperfection in Nearly perfect Crystals, Wiley, New York (1952) p.289 [11] Smithell’s Metals Reference Handbook, Sixth Edition, Vol. I , Eric Brandes (Edi.), 1983, Chapter 13. [12] N. F. Uvarov, E. F. Hairetdinov and V. V. Boldyrev, Jour. Solid State Chemistry, Vol. 51 (1984), p. 59 [13] A. V. Chadwick in Point Defects and Diffusion in Solids J. N. Sherwood (Ed.). John Wiley, New York , 1985, P.285. [14] A. V. Chadwick and H. R. Glyde in M.L. Klein and J.A. Venables (Ed.) Rare Gas Solids Vol. II (1977), p’1151. [15] Ihsan Barin , Thermochemical Data of Pure Substances .Vol. I and II , 3rd. ed. ,VCH Weinheim, FRG (1955) [16] R. O. Simmons and R.W. Balluffi, Phys. Rev., Vol. 117 (1960), p. 52 [17] R.O. Simmons and R.W. Balluffi, Phys. Rev., 119 (1960), p.600 [18] R.O. Simmons and R.W. Balluffi, Phys. Rev., 125 (1962), p. 862 [19] R.O. Simmons and R.W. Balluffi, Phys. Rev., 129 (1963), p. 1533 [20] H.B. Huntington , Physical Review, 61 (1942), p. 325 [21] H.B. Huntington and F. Seitz , Physical Review, 76 (1949), p. 1728 [22] F. Seitz, Acta Crystallographica, 3 (1950), p. 355 [23] A.Vignes and C.E. Birchenall, Acta Met., 16 (1968), p. 1117 [24] V.S. Raghunathan, G.P.Tiwari and B.D. Sharma, Met. Trans. Vol. 3 (1972), p. 783 [25] L.E.Toth and A.W. Searcy , Trans. TMS-AIME, Vol. 230 (1964), p. 690 [26] L.W. Barr and A.D. Lidiard, in Physical Chemistry, Vol. 10, Academic Press, NY, 1972. [27] E.W. Hart, Acta Metall., Vol. 5 (1957), p. 597 [28] I. Kaur and W. Gust. In Fundamentals of Grain and Interface Boundary Diffusion, Zieggler Press, Stuttgart, (1988) [29] L.G. Harrison, Trans. Faraday Society, Vol. 57 (1961), p. 1191 [30] N.H. Nachtireb, H. Resing and S.A. Rice, J. Phy. Chem. Solids, Vol. 31 (1959), p. 135 [31] J.P. Zanghi amd D. Calais, J. Nucl. Mater., Vol. 60 (1976), p. 145

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[32] A.M. Borwn and M.F. Ashby, Acta Metall , Vol. 28 (1980), p. 1085 [33] D. Ablitzer, Philos. Mag., Vol. 35 (1977), p. 1239 [34] R.F. Mehl, M. Swanson and G.M. Pound , Acta Metall., Vol. 9 (1961), p. 256 [35] T. Gorecki, Z. Metallkunde, Vol. 67 (1980), p. 269 [36] G.V. Kidson, in Diffusion in Body-Centered Cubic Metals, A.S.M., Metals Park, Cleveland, 329-347 (1965) [37] J.M. Sanchez and D. de Fontain, Acta Metallurgica, Vol. 26 (1978), p. 1083 [38] S. Chandra and J. Rolfe, Can. Jour. Phys. , Vol. 48 (1970), p. 397 [39] S. K. Sikka, Y. K. Vohra and R. Chidambram: Prog. Mater. Sci. Vol. 27 (1982), P. 242 [40] S. Baneerjee, R. Tewari and G. K. Dey : Inter. J. Mater. Research, Vol. 97 (2006), p. 963 [41] J.C. Jamieson, Science, Vol. 140 (1963), p. 72 [42] R. N. Jeffrey, Phy. Rev., B3, (1971), p.4044 [43] N. L. Peterson, J. Nucl. Mater., Vol. 69 & 70 (1978), p. 3 [44] H. Hehrer, J. Nucl. Mater., Vol. 69 & 70 (1978), p. 38 [45] R. F. Peart and T. S. Lundy, J. Phy. Chem. Solids, Vol. 26 (1965), p. 937 [46] R.E. Einziger, J.N. Mundy, J. Nucl. Mater., Vol. 69 & 70 ( !978), p. 523 [47] R.F. Peart, J. Phy. Chem. Solids, Vol. 26 (1965), p. 1853 [48] J.N. Mundy, S.J. Rothman, N.Q. Lam, L.J. Nowicki and H. A. Hoff, J. Nucl. Mater. Vol. 69 & 70 ( !978), p. 525 [49] J.I. Federer and T.S. Lundy, Trans. AIME, Vol. 227 (1963), p. 592 [50] N. E. Walsoe de Recca and C. M. Libanti, Acta Met , Vol. 16 (1968), p. 1297 [51] F. H.Spedding and K. Shiba , J. Phys. Chem. Solids , Vol. 57 (1962), p. 612 [52] M. P. Dariel, G. Erez and G. M. J. Smidt , Phil. Mag. Vol. 19 (1969), p. 1045 [53] M. Dupuy and D. Calais , Trans. Met. Soc. AIME, Vol. 242 (1968), p. 1679 [54] J. Philibert,, in Atom Movement Diffusion and Mass Transport in Solids, Les Editions de Physique, les Ulis CedexA, France (1991), Chap. 4, p.105 [55] G.P. Tiwari and B.D. Sharma, Acta Metall, Vol. 15 (1967), p. 155 [56] G.P. Tiwari and K. Hirano, Trans. Jap. Inst. Metals, Vol. 21 (1980), p. 667 [57] A. languille, D. Calais and M. Formont , J. Phys. Chem. Solids, Vol. 35 (1974), p. 1373 [58] J.A. Cornet J. Phy. Chem. Solids, Vol. 32 (1971), p. 1489. [59] H. Mehrer, “ Diffusion in Solids”, Springer Series in Solid State Sciences 155, SoringerVerlag, Berlin ( 2007), Chapter 20, p341. [60] A. R. McGhie, H. Blum and M. M. Labes, Jour. Chem . Physics, Vol. 92 (1970), p.6141 [61] R. O. Ramiseier, J. Appl. Physics, Vol. 38 (1967), p. 2533 [62] C. Wert and C. Zener, Phys. Rev., Vol. 76 (1949), p. 1169 [63] S. Glastone, K.J. Laidler and H. Eyring, The Theory of Rate Processes, McGraw-Hill Book Company, NewYork, pp. 184-191 (1941) [64] S.A. Rice, Physical Review, Vol. 112 (1958), p. 804 [65] S.A. Rice and N.H. Nachtrieb, J. Chem. Physics, Vol. 31 (1959), p. 139 [66] O.P. Manely, J.Phy. Chem Solids, Vol. 13 (1960), p. 244 [67] D. Gupta, D. Lazarus and D.S. Lieberman, Phys. Rev., Vol. 153 (1967), p. 863 [68] D. Gupta and D.S. Lieberman, Phys. Rev. B,Vol. 4 (1970), p. 1070 [69] H.A. Domien and H.I. Aaronson in Diffusion in Body Centered Cubic Metals, ASM, Cleveland, p.209 (1965) [70] M. Denkinger and H. Mehrer, Philos. Mag. A, Vol. 80 (2000), p. 1245 [71] C. Herzig and S. Divinsky in D. Gupta (Ed.) Diffusion Processes in Advanced Technological Materials William Andrew Publishing , New York (2005), Chap.2,p.69. [72] M. Beniere and M. Chemla J. Phys. Chem. Solids, Vol. 37 (1976), p. 525 [73] F. Beniere, D. Kostopolous and K. V. Reddy , J. Phys. Chem. Solids, Vol. 41 (1990), p. 727 [74] R. Fox and J. N. Sherwood, Transactions of Faraday Society, Vol. 67 (1971), p. 3364

Defect and Diffusion Forum Vol. 279 (2008) pp 39-52 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.39

Thermodynamic Diffusion Coefficients G. B. Kale Materials Science Division, Bhabha atomic Research Centre, Mumbai 400 085, INDIA e-mail: [email protected] Keywords : Thermodynamic, binary diffusion, ternary diffusion, intermetallic compound

Abstract: A new form of diffusion coefficient termed as thermodynamic diffusion coefficient is introduced in this paper. Conventionally, diffusion coefficients are evaluated using concentration gradient as driving force. But truly, chemical potential gradient is the actual driving force that determines the material flow in any part of the system. Thermodynamic diffusion coefficients are based on chemical potential gradient as driving force. The relation between thermodynamic diffusion coefficients and phenomenological coefficients has been established. The advantages of thermodynamic diffusion coefficients have been underlined, especially, in the cases of line compounds where concentration difference across the phase is zero or in case of intermetallic compounds with narrow homogeneity range. The intrinsic thermodynamic diffusion coefficients are equal to tracer diffusion coefficients. This helps in estimating tracer diffusivities in cases where tracers are not easily available. The advantages of thermodynamic diffusion coefficients are shown in binary and ternary systems by illustrating them in Ni-Al and Fe-Ni-Cr systems. Introduction The conventional methods of estimation of diffusion coefficient in a system employ concentration gradient considering it to be the driving force for diffusion. But in reality, it is the chemical potential gradient that determines the material flow in any part of a system. The use of the concentration gradient as the driving force gives unrealistic values of diffusion coefficients, especially in cases where concentration gradient and chemical potential gradient differ across the phase. The main reason for not employing chemical potential gradient was the non-availability of thermodynamic data. In the last two decades thermodynamic data from experimental measurements and theoretical estimations for several systems have been reported. It has now become possible to evaluate the chemical potential gradients in various systems. With this in mind, new form of diffusion coefficient, termed as thermodynamic diffusion coefficient, D(µ) has been introduced. It is related to conventional chemical diffusion coefficient, D() through the thermodynamic properties of the system. The basic relationship between thermodynamic diffusion coefficient and phenomenological transport coefficients has been shown here. The advantages of this diffusion coefficient are described, especially in case of the diffusion coefficient of line intermetallic compounds where the concentration gradient across the compound is nil, but the chemical potential gradient is finite. To illustrate this, the thermodynamic diffusion coefficients have been evaluated for Al-Ni binary system. The concept of thermodynamic diffusion coefficients has been extended to ternary system also. The effect could be more pronounced in case of ternary systems where chemical potentials may not vary systematically with concentrations as in case of binary systems. The thermodynamic diffusion coefficients have been applied to Fe-Ni-Cr ternary system.

40

Phase Transformation and Diffusion

Thermodynamic Diffusion Coefficients General flux equations as given by Flick’s first law and can be written as J =- D(∂/∂x)

(1)

where J is the flux of component and D is the conventional diffusion coefficient of component under the influence of concentration gradient (∂/∂x). Garg et al [1] have described the thermodynamic diffusion coefficients using chemical potential gradient as the driving force. A more general form of flux equation using the chemical potential gradient as driving force can be written as J = -M(∂µ/∂x)

(2)

where M is another form of diffusion coefficient of the component. The chemical potential µi of a component i in an alloy is given as

µ = µ0 + RTlna

(3)

where a is the activity of component and µ0 is the chemical potential of component in the standard state. Substituting for µ, eq. (2) can be written as J = -MRT (∂lna/∂x)

(4)

In order to solve this differential equation, it is essential that the driving force (∂lna/∂x) should be complementary with respect to other components in an n component system. Hence, eq. 4 can be rewritten as J = -(MRT /N)(∂lna/∂x)

(5)

The term MRT/ is defined as the thermodynamic interdiffusion coefficient D(µ). Conventional diffusion coefficients are denoted as D() and thermodynamic diffusion coefficients as D(µ) in order to differentiate between them. On introducing the thermodynamic interdiffusion coefficient, D(µ), the flux equation takes the form J = - D(µ)(∂lna/∂x)

(6)

The chemical diffusion coefficient, D() and thermodynamic diffusion coefficients as D(µ) are related by D() = (∂lna/∂) D(µ)

(7)

Interrelation between D(µ) and Phenomenological Coefficients The phenomenological approach to diffusion in solids is a very general formalism, which accounts for the flux of each component in a system due to the influence of nearly all kinds of driving forces, be it chemical, electrical or thermal in origin[2,3]. Since phenomenological coefficients are more fundamental parameters in describing diffusion processes [2], a relationship with the thermodynamic diffusion coefficient would, therefore, be beneficial in understanding the diffusion processes better. Laik and Kale [4] have shown such interrelation between thermodynamic diffusion coefficient and phenomenological coefficients.

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According to the linear Onsager [5] phenomenological theory of diffusion, the flux of the component i in an n component system can be given by Ji = ∑LijXj (i = 1,2,3,…n)

(8)

Lij are phenomenological coefficients and are independent of driving force, Xj. We assume that Xj is chemical driving force and all other forms of forces such as electrical, thermal are essentially zero. For one dimensional, isothermal binary system A-B, the intrinsic fluxes, JA and JB of A and B and that of vacancies Jv can be written as JA = LAAXA + LABXB + LAvXv

(9)

JB = LBAXA + LBBXB + LBvXv

(10)

Jv = LvAXA + LvBXB + LvvXv

(11)

Assuming that lattice sites in the diffusion zone are conserved, we can write Jv = - (JA + JB )

(12)

Following the condition in eqn (12), eqns. 9, 10, 11 lead to LAA + LBA + LvA = 0

(13)

LAB + LBB + LvB = 0

(14)

LAv + LBv + Lvv = 0

(15)

On the basis of microscopic reversibility principle, Onsager reciprocity relations[5] can be stated as Lij = Lji

(16)

Following this, the flux equations take the form JA = LAA(XA - Xv) + LAB (XB - Xv)

(17)

JB = LAB(XA - Xv) + LBB(XB - Xv)

(18)

If the concentration of vacancies is at local equilibrium at all points, then the driving force Xv = 0. The driving force in terms of chemical potential or activity gradient can be written as XA = -(∂µA/∂x) = - RT (∂lnaA/∂x)

(19)

XB = -(∂µB/∂x) = - RT (∂lnaB/∂x)

(20)

Substituting the driving forces in eqns. (17) and (18) and using Xv = 0, we get JA = LAA(- RT (∂lnaA/∂x)) + LAB (- RT (∂lnaB/∂x))

(21)

JB = LAB(- RT (∂lnaA/∂x)) + LBB(- RT (∂lnaB/∂x))

(22)

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Phase Transformation and Diffusion

Equating the intrinsic flux of the component expressed in terms of thermodynamic diffusion coefficient in eqn. (6) and that expressed in terms of phenomenological coefficients in eqns. (21) and (22) , we can write DA(µ)A = LAART + LABRT(∂lnaB/∂lnaA)

(23)

Gibbs-Duhemn equation gives A dlnaA + B dlnaB = 0

(24)

Substituting eqn. (24) in eqn. (23) and simplifying equations we can write DA(µ) = RT[(LAA/A ) – (LAB/B)] for component A

(25)

DB(µ) = RT[(LBB/B ) – (LAB/A)] for component B

(26)

These relationships given by eqns. (25) and (26) provide interrelationship between thermodynamic intrinsic diffusion coefficient and phenomenological coefficients. It is noteworthy that a similar relation exists between tracer diffusion coefficient and phenomenological coefficients. The relation between tracer diffusion coefficient and phenomenological coefficient is given as AD*A/kT = LAA – ALAB/B

(27)

BD*B/kT = LBB – BLAB/A

(28)

Comparing eqn. (25) with eqn. (27) and eqn. (26) with eqn. (28) it can be concluded that the thermodynamic intrinsic diffusion coefficient is same as the tracer diffusion coefficient. The present relationship with phenomenological coefficients, which are more fundamental in nature, gives strong support for thermodynamic diffusion coefficients. Thermodynamic Diffusion Coefficients in Binary Al-i System The Al-Ni binary system has been selected for illustration because it shows five different types of intermediate phases[6]. The intermediate phase ε is a line compound. The intermediate phases α’, δ and β’ have varying homogeneity ranging from very narrow to very large homogeneity. The detailed experimental diffusion data [7] and thermodynamic data[8] for all the four phases (ε, α’, δ and β’) have been reported. The thermodynamic data for Al3Ni5 is not available. Incidentally, this phase does not appear in the diffusion zone in the temperature range of investigation. Table 1 Types of couples and phases formed in diffusion zone in Al-Ni system Annealing Type of Annealing Couple Temperature(K) Time(h) 883 50 Al-δ 883 16 Al-α’ Al-Ni 883 16 883 16 Al-β’ 869 36 Al-β’ 792 66 Al-β’ 701 67 Al-β’ 1273 32 δ-Ni 1073 102 δ-Ni 928 271 δ-Ni

Phase ε Width(µm) 103.0 10.5 14.0 12.5 17.0 41.0 15.0 -

Phase δ Width(µm) 144.0 124.0 155.0 139.0 39.0 6.0 -

Phase β’ Width(µm) 113.0 30.0 10.0

Phase α’ Width(µm) 6.0 10.0 3.0

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Janssen and Rieck [7] have studied diffusion couples between various phases to form various intermetallics in the diffusion zone. The types of couples studied and intermetallics formed in diffusion zone with their widths are given in Table 1. Evaluation of D(µ) : Boltzmann-Matano equation generally used for evaluation of chemical interdiffusion coefficient D() is given as D() = - 1/2t ∫xd (dx/d)

(29)

The equation can be modified to evaluate thermodynamic diffusion coefficient D(µ) and can be expressed as D(µ) = - 1/2t ∫xd (dx/dlna)

(30)

The ∫xd is the flux of material and is taken as area under the concentration profile. Estimation of dlna : The intermediate phases form in layers in diffusion zone. The values of chemical potential are required at the interface compositions. The chemical potential values at the interface of the two intermediate phases may differ from the equilibrium values and hence were calculated by adopting the following procedure. A steady state equilibrium between the two intermediate phases, Al1-xix and Al1-yiy in the diffusion zone can be represented by the following general reaction ((1-y)/(y-x))Al1-xix + i = ((1-x)/(y-x))Al1-yiy

(31)

The chemical potential at the interface of these two intermediate phases is given as

µii/RT = lnaii = ∆GR/RT

(32)

where ∆GR is the free energy change of reaction in eqn. (31). ∆GR has been calculated from the ∆formGi values of the phases involved in the reaction eqn. (31). Barin[8] has reprted ∆formGi values for the four intermediate phases in the Al-Ni system for their average compositions at different temperatures. These values can be expressed as a function of temperature using least square analysis as

∆formGi = a + bT + cTlog T

(33)

The calculated values of the coefficients a, b and c for the four intermediate phases are given in Table 2. Table 2 Free energies of formation of the phases as a function of temperature expressed as a + bT + cTlogT(J/g atom) Intermediate Phase ε δ β’ α’

Average Composition i 0.25 0.385 0.500 0.750

a

B

C

-13622.6 -41787.0 -46715.2 -32449.5

-242.336 -160.301 -139.944 -70.5632

73.6795 49.8258 43.9741 22.4901

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Phase Transformation and Diffusion

The values of lnai for the different two phase interfaces in the Al-Ni system were calculated by substituting ∆GR in eqn. (32). The calculated values of Ν∆lnai across the intermediate phases at the temperatures studied and the experimentally observed values ∆ across these phases are given in Table3. Table 3 Calculated values of Ν∆lna I and experimental values of ∆ across the phases in Al-Ni system at different temperatures. Temperature(K) 701 792 869 883 928 1073 1273

Across ε

Across β’

Αcross δ

Across α’

∆Ν

Ν∆lna

∆Ν

Ν∆lna

∆Ν

Ν∆lna

∆Ν

Ν∆lna

0 0 0 0 0 0 −

0.360 0.358 0.320 0.310 0.270 0.105 −

0.045 0.045 0.045 0.045 0.045 0.045 0.045

4.67 4.08 3.70 3.64 3.46 3.00 2.57

0.080 0.080 0.080 0.080 0.080 0.080 0.110

4.31 3.76 3.39 3.32 3.13 2.63 2.11

0.04 0.04 0.04 0.04 0.04 0.04 0.04

2.19 1.91 1.73 1.70 1.61 1.36 1.11

The Values of thermodynamic diffusion coefficients D(µ) have been calculated using the estimated values of ∆lna in eqn. (30). The values of ∫xdC have been estimated from the experimental concentration profiles. These experimental profiles are taken from Janssen and Rieck’s work[7] and are plotted in Fig.1. The corresponding calculated profiles of lna are also plotted in the same figure. The estimated values of D(µ) for various phases at different temperatures are tabulated in Table 4. Table 4 The values of thermodynamic diffusion coefficients D(µ) for various phases in Al-Ni system at different temperatures D(µ) x1016 m2/s

Τemperature(K) 701 792 869 883 928 1073 1273

ε 1.917 19.53 20.67 62.87 − − −

δ 0.0371 1.28 19.26 80.74 − − −

β’ − − − − 0.1436 4.146 205.7

α’ − − − − 0.07 2.588 12.14

It is interesting to note that the values thermodynamic diffusion coefficients in ε and δ phase are of the same order at 883 K (Table 4). But the growth δ phase is much larger than that of ε phase in the diffusion zone (Table 1 and Fig. 1). The large growth is due to large driving force (activity gradient) across δ phase as compared to that across ε phase (Table 3). This clearly shows the real estimates of diffusivities using true driving force in case of thermodynamic diffusion coefficients.

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Fig. 1 Concentration and activity profile for the typical diffusion couples in Al-Ni System Estimation of the thermodynamic intrinsic diffusion coefficients : The velocity, f , of the Kirkendall markers is given[9] as f = (DB() – DA())( ∂B/∂x)

(34)

The equation for marker velocity in terms of thermodynamic diffusion coefficients can be written as f = (DB(µ) – DA(µ))B(∂lna/∂x)

(35)

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Phase Transformation and Diffusion

Darken’s relation can be extended to thermodynamic diffusion coefficients and relation between thermodynamic interdiffusion coefficient and thermodynamic intrinsic diffusion coefficients can be written as D(µ) = (BDA(µ) + ADB(µ))

(36)

Janssen and Rieck [7] have carried out experiments with Kirkendall markers in Al-Ni and δNi diffusion couples in Al-Ni system. The marker position after diffusion anneal remained at the Al-ε interface in Al-Ni diffusion couple and at Ni-α’ interface in δ-Ni diffusion couple. The position of marker at Al-ε interface suggests that the value of Di(µ) in ε phase can be taken as zero. Using this fact and Eq. (36) the value of DAl(µ) in ε phase has been evaluated. Similarly, the value of Di(µ) in α’ has been evaluated. The values of these diffusivities are tabulated in Table 5. As discussed earlier, thermodynamic intrinsic diffusion coefficient is equal to tracer diffusion coefficient. The tracer diffusion coefficient of Ni in α’ phase is available in the literature[10]. The reported value is also tabulated in Table 5. The values of thermodynamic intrinsic diffusion coefficient and tracer diffusivity values are comparable. The values tracer diffusivity of Al is not available for comparison. Table 5 Thermodynamic intrinsic diffusion coefficients and tracer diffusion coefficients in Al-Ni system Type of couple

Temperature(K)

Al-Ni

883

δ-Ni

1273

Thermodynamic intrinsic diffusion coefficient(m2/s) DεAl(µ) = 2.96x10-14 Dα’i(µ) = 4.86x10-16

Tracer diffusion coefficient(m2/s)

Ref.

−

−

D*α’i = 1.35x10-16

[10]

Ternary thermodynamic diffusion coefficients in Fe-i-Cr system The general flux equation written in terms of thermodynamic diffusion coefficients (eqn. (6)) can be extended to multicomponent system and can be written as Ji = -∑Dij(µ)j(∂lnaj/∂x)

(37)

where Ji is the flux of component i and Dij(µ) is the thermodynamic diffusion coefficient of component i under the influence of activity gradient of component j, (∂lnaj/∂x). In general, for i = j, Dij(µ) is called direct or major thermodynamic diffusion coefficient and for i ≠ j, Dij(µ) is known as cross or minor thermodynamic diffusion coefficient. For a ternary system 1-2-3, the flux equations for the three components can individually can be expressed as J1 = -D11(µ)1(∂lna1/∂x) -D12(µ)2(∂lna2/∂x) -D13(µ)3(∂lna3/∂x)

(38)

J2 = -D21(µ)1(∂lna1/∂x) –D22(µ)2(∂lna2/∂x) –D23(µ)3(∂lna3/∂x)

(39)

J3 = -D31(µ)1(∂lna1/∂x) –D32(µ)2(∂lna2/∂x) –D33(µ)3(∂lna3/∂x)

(40)

Since the sum ∑i(∂lnai) of all the components in a ternary system is zero following Gibbs Duhemn equation and the summation of the interdiffusion fluxes, ∑Ji is zero, the eqs. (38), (39) and

Defect and Diffusion Forum Vol. 279

47

(40) can be reduced to two independent equations. Assuming component 3 as dependent, these two equations can be represented as J1 = -D311(µ)1(∂lna1/∂x) –D312(µ)2(∂lna2/∂x)

(41)

J2 = -D321(µ)1(∂lna1/∂x) –D322(µ)2(∂lna2/∂x)

(42)

Here D311(µ) and D322(µ) are the direct thermodynamic interdiffusion coefficients and D312(µ) and D321(µ) are the cross thermodynamic interdiffusion coefficients. For a ternary system, Fick’s second law can be written as ∂1/∂t = -∂J1/∂x and ∂2/∂t =- ∂J2/∂x

(43)

~ ~ Substituting J 1 and J 2 from eqs. (41) and (42), eq. (43) can be written as ∂1/∂t = ∂[ D311(µ)1(∂lna1/∂x) + D312(µ)2(∂lna2/∂x)]/∂x

(44)

∂2/∂t=∂[D321(µ)1(∂lna1/∂x)+D322(µ)2(∂lna2/∂x)]/∂x

(45)

and

Boltzmann solution for the binary system can be extended to ternary system for solving eqs. (44) and (45). The initial boundary conditions of experiment are

(+x,0) = (+∞,0) =+∞ and (-x,0) = (-∞,0) = -∞ The Boltzmann solution can be employed to evaluate thermodynamic interdiffusion coefficients at any composition ′ (-∞ 0

(49)

D311(µ)D322(µ) - D312(µ)D321(µ) > 0

(50)

Defect and Diffusion Forum Vol. 279

49

(D311(µ) + D322(µ) )2 > 4(D311(µ)D322(µ) – D312(µ)D321(µ))

(51)

D311(µ) > o

(52)

D322(µ) >0

(53)

D312(µ)D321(µ) > 0

(54)

D311(µ)D322(µ) - D312(µ)D321(µ) > 0

(55) Table 6

Thermodynamic Interdiffusion Coefficients At Various Compositions Composition (at%) Ni Cr 70.3

17.2

Thermodynamic interdiffusion coefficients (X 1016 m2/s) Fe Fe D ii(µ) D iCr(µ) DFeCri(µ) DFeCrCr(µ) 9.410 -6.956 -3.159 8.804

71.3

19.6

12.188

-3.545

-5.166

37.245

62.5

6.0

16.244

-0.947

-0.882

12.056

66.6

11.6

25.973

-15.680

-6.727

12.130

75.9

23.6

8.691

-9.724

-10.054

14.424

42.0

23.0

18.271

0.911

-2.930

84.081

54.0

23.0

17.361

1.685

1.383

23.652

83.6

13.2

15.628

-3.990

-2.770

43.480

91.9

1.8

9.466

-7.359

-3.729

41.465

58.5

23.2

12.415

-1.360

0.635

20.650

63.7

23.4

13.340

-3.262

-0.129

16.026

67.4

23.5

12.580

-9.419

0.780

21.757

75.4

23.9

8.584

-9.213

-1.649

14.516

53.9

25.9

18.522

1.562

0.648

21.621

46.5

9.0

28.331

-8.611

-1.412

28.360

49.2

15.3

22.110

-6.166

0.765

37.577

46.0

29.3

15.694

-9.704

-3.823

22.149

55.5

30

16.517

5.621

0.749

11.798

58.0

30.3

15.284

-0.287

-0.579

1.032

41.46

31.1

9.905

-1.370

16.436

42.346

46.0

8.4

28.544

-18.573

8.108

60.090

43.5

1778

20.492

-3.839

12.775

108.953

50

Phase Transformation and Diffusion

All the thermodynamic interdiffusion coefficients evaluated in the present study (Table 6) comply with the above relations at most of the compositions. All the major thermodynamic diffusion coefficients are positive and cross thermodynamic diffusion coefficients are either positive or negative as indicated by above relationships. In some cases the cross diffusion coefficients are of opposite sign. This may be due to the non-equilibrium vacancy concentrations available during the diffusion process[16]. The relation between thermodynamic interdiffusion coefficients and thermodynamic intrinsic diffusion coefficients [11] can be given as D311(µ) = D11(µ) + 1(D33(µ) – D11(µ))

(56)

D312(µ) = 1(D33(µ) –D22(µ))

(57)

D321(µ) = 2(D33(µ) – D11(µ))

(58)

D322(µ) = D22(µ) + 2(D33(µ) – D22(µ))

(59)

The values of thermodynamic intrinsic diffusion coefficients Dii(µ), DCrCr(µ) and DFeFe(µ) have been evaluated at various compositions from thermodynamic interdiffusion coefficients using eqs. (56)-(59) and are tabulated in Table 7. Table 7 Thermodynamic Intrinsic Diffusion Coefficients At Various Compositions Composition (at %) Ni Cr 62.5 6.0

Dii(µ) 25.4285

Intrinsic Diffusivity (X1016m2/s) DCrCr(µ) DFeFe(µ) 12.1474 16.7333

63.7

23.4

13.6921

17.2238

13.1400

67.4

23.5

10.3429

25.0412

13.6620

75.4

23.9

13.7850

17.4358

6.8865

53.9

25.9

17.1738

20.8705

23.7679

46.5

9.0

35.6265

30.0269

19.9368

83.6

13.2

33.1713

44.1100

12.1865

As discussed in one of our earlier papers[1] , the thermodynamic intrinsic diffusivities are equal to tracer diffusivities. Accordingly, the values of Dii(µ), DCrCr(µ) and DFeFe(µ) should be equal to tracer diffusivities of Ni, Cr and Fe respectively. Rothman et al[18] and Million et al[19] have reported tracer diffusivity values of Fe, Cr and Ni in some Fe-Ni-Cr alloys. The tracer diffusivity values at the exact composition and temperature of this study are not available. The values of the tracer diffusivities at similar composition within ± 5 % and at temperature of 1286 K have been compared with our thermodynamic intrinsic diffusivity values in Table 8. The intrinsic diffusivity values in these alloy compositions are comparable to values reported from experimental investigations.

Defect and Diffusion Forum Vol. 279

51

Table 8 Comparison Of Thermodynamic Intrinsic Diffusivities, D( µ ) And Experimental Tracer Diffusivity D ∗ Values For Various Compositions In Fe-Ni-Cr System At 1223k Diffusivities (X 1016 m2/s )

Composition (at%) Ni

Cr

∗ DCrCr(µ) DCr

DFeFe(µ)

∗ D Fe

Ref.

62.5

6.0

12.1

11.1

16.7

10.1

[18]

63.7

23.4

17.2

9.5

13.1

7.2

[18]

67.4

23.5

25.0

---

13.7

4.6

[18]

75.4

23.9

17.4

----

6.9

3.8

[18]

46.0

29.3

28.3

1.5

7.2

1.3

[19]

55.5

30.0

8.8

1.9

18.9

7.2

[18]

54.0

23.0

22.9

----

20.1

3.5

[18]

83.6

13.2

44.1

----

12.2

5.3

[18]

Summary and Conclusion A better alternative approach to diffusion analysis, based on the chemical potential gradient has been suggested. The salient features are summarised below: 1) With approach better estimates of diffusivities in the intermediate phases can be obtained where homogeneity range is very narrow or nil. 2) Thermodynamic intrinsic diffusivities are equal to tracer diffusivities. Hence, tracer diffusivities in intermetallics can be estimated in the absence of availability of proper radioactive tracer. 3) The thermodynamic diffusivity analysis can be extended to ternary system.

References [ 1] S. P. Garg, G. B. Kale, R. V. Patil and T. Kundu : Intermetallics Vol. 7 (1999) p. 901 [ 2] R. E. Howard and A. B. Lidiard : Rep. Prog. Phys. Vol. 27 (1964) p.161 [ 3] P. G. Shewmon : Diffusion in solids, The Minerals, Metals and Materials Society, Pennsylvania (1989). [ 4] A. Laik and G. B. Kale: Trans. Indian Inst. Met. Vol. 58 (2005), p. 133 [ 5] L. Onsager : Phys. Rev. Vol. 32 (1932) p. 2265 [ 6]T. B. Massalski, editor, Binary Phase Diagrams, Vol. 1 2nd ed. Materials Park, Ohio: ASM International, (1992) p. 183 [ 7] M. M. P. Janssen and G. D. Riecks : Trans. Met. Soc. AIME Vol. 239 (1967) p. 1372 [ 8] I. Barin : Thermochemical data of pure substances Vol. 1 3rd ed. Weinheim (Germany) VCH (1995) [ 9]J. Philibert, Atom movement – diffusion and mass transport in solids, Les Ulis Cedex A. France: les Editions de Physique (1991) [10] K. Hoshino, S. J. Rothman and R. S. Averback, Acta Metall. Vol. 36 (1988) p. 1271

52

Phase Transformation and Diffusion

[11] G. B. Kale, K. Bhanumurthy, A. Laik and S. P. Garg : Trans. Indian Inst. Met., Vol. 157 (2004), p. 35 [12] G. B. Kale, K. Bhanumurthy , S. K. Khera and M. K. Asundi : Mater. Trans. JIM, Vol.32 (1991) p.1034 [13] B. Million , Met. Mater., Vol. 18 (1980) p.154 [14] G. J. Hooyman , Physica, Vol. 22 (1956) p.751 [15] J. S. Kirkaldy , D. Weichert and Zia Ul Haq, Can. J. Phys., Vol. 40 (1963) p.2166 [16] G. W. Roper and D. P. Whitle , Metal Sci, Vol. 5 (1980) p.41 [17] S. J. Rothman , L. J. Nowick and G. E. Murch , J. Phys. F Met, Phys, Vol. 10 (1980) p.383 [18] B. Million , J. Ruzickova and J. Verstal , Mater. Sci. and Engg, Vol. 72 (1985) p.85

Defect and Diffusion Forum Vol. 279 (2008) pp 53-62 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.53

Novel Method of Evaluation of Diffusion Coefficients in Ti-Zr System K. Bhanumurthy*a, A. Laikb and G. B. Kalec Materials Science Division Bhabha Atomic Research Centre Mumbai 400 085. India E-mail:a: [email protected] b: [email protected]

c: [email protected]

Keywords: Interdiffusion coefficient, Impurity diffusion coefficient, junctions, EPMA and Microstructure

Diffusion couple,

Triple

Abstract: The incremental diffusion couples are used for evaluating interdiffusion couples in a narrow composition range and these results are extrapolated to get an estimate of impurity diffusion coefficients. In fact, several incremental couples are needed to get impurity diffusion coefficients at different compositions. This process is generally tedious. The present method describes a relatively simple method for evaluating the diffusion coefficients using “step diffusion couples”. A simple experimental method is described to prepare a step diffusion couple. This method involves preparation of diffusion couples in two stages. In the first stage, diffusion couple is made between the two materials in a conventional way and annealed for extended period of time to have a large diffusion zone typically of the order of 2-3 mm. In the second stage, the starting materials are placed on the diffusion couple in a direction perpendicular to the diffusion zone and annealed at a suitable temperature for diffusion to occur between the diffusion zone and the starting materials. This method is applied to study the interdiffusion behavior in the β phase of the Ti-Zr system. Boltzmann-Matano and Hall’s methods were used to determine the interdiffusion coefficients and their composition dependence. Kirkendall shift is observed towards Ti side and the intrinsic diffusion coefficients of Ti is approximately three times that of Zr. The width of the diffusion zone is strongly dependent on the composition of the step diffusion couple. It is observed that the interdiffusion coefficients evaluated at the terminal compositions matched well those published values in the Ti-Zr system. This experimental technique offers an easy and elegant method to determine the diffusion parameters without the tedious preparation of incremental diffusion couples. Introduction Interdiffusion coefficient essentially describes the rate of mixing of two species in a given system and it generally refers to the rate of smoothening of the original concentration gradient. Estimation of interdiffusion coefficient is of great practical importance in delineating several processes such as diffusion bonding, cladding, aluminizing, oxidation, development of ohmic contacts, diffusion barriers in semiconductors and of metal-fiber interactions composite materials [1-5]. Because of the technological importance, extensive work has been reported on the evaluation of interdiffusion coefficients in many binary and ternary systems. Boltzmann-Matano method [6-7] is generally ~ used for evaluating the interdiffusion coefficients ( D ) as a function of composition. However, for systems, which involve large variation of molar volume (Vm), Den Broeder method [8] is adopted ~ for estimating D from the concentration profiles. The estimation of interdiffusion coefficient by Boltzmann-Matano method involves the exact calculation of slope (dC/dx) and the area ( ∫ xdC ) at a given concentration from the concentration profile and this estimation at the end compositions is rather difficult even with the best possible numerical computations [9]. Inherent typical error in the diffusivity values obtained by this method is around 10% in the middle range of composition and is around 20-50% near the end compositions. Diffusion couples involving two materials are formed by holding the interfaces in contact at high temperature for diffusion to take place [10]. Recently, diffusion couples consisting of three or more metal blocks are discussed [11-12] and these studies involved joining of these blocks at high

54

Phase Transformation and Diffusion

pressure and temperature for intimate contact at the contact surface. This leads to interdiffusion at the surface possibly resulting in the formation of solid solution phases or/and intermetallic compounds. This data is used for evaluation of phase diagrams and diffusion coefficients in the solid state. In addition, incremental diffusion couples are used for evaluating interdiffusion couples in a narrow composition range and these results are extrapolated to get an estimate of impurity diffusion coefficients. In fact, several incremental couples are needed to get impurity diffusion coefficients at different compositions [13]. This process is generally tedious. The present method describes a relatively simple method for evaluating the diffusion coefficients using “step diffusion couples” and this method is unique for evaluating the diffusion coefficients in a given binary or ternary system. Step type diffusion couples involve the preparation of diffusion couples in two stages. In the first stage, diffusion couple is made between the two materials A and B in a conventional way and annealed for extended periods to have a large diffusion zone typically of the order of 2-3 mm. In the second stage, the starting materials A and B are placed on the diffusion couple in a direction perpendicular to the diffusion zone and annealed at a suitable temperature for diffusion to occur between the diffusion zone and the materials A and B. This method is applied to a Ti-Zr system to demonstrate the usefulness of this technique for evaluating the diffusion coefficients at large number of compositions. A brief summary of the diffusion studies carried out in literature for Ti-Zr system is described below. Titanium in its hcp and bcc structures form complete solid solution with zirconium and interdiffusion studies in the Zr/Zr-80.0% Ti alloy have been studied in the temperature range of 901-1168 0C [14]. Based on these studies, interdiffusion coefficients were measured as a function of Ti and also the impurity diffusivity values of Zr in β-Ti and Ti in β-Zr were estimated. These experiments did not show the presence of Kirkendall shift in the diffusion zone. The impurity diffusion coefficient of Zr in β-Ti has been studied in the temperatures range 900 to 15000C under pressure of 0.1 MPa and also at 1-3 GPa using Ti/Ti-3.0Zr diffusion couples [15]. The authors have reported the impurity diffusion coefficient of Zr in β-Ti to be 2.0x10-13 m2/sec at 9000C at a pressure of 0.1 MPa. This value is about an order of magnitude more than that determined by Raghunathan et al [14] under similar conditions. Recently, Duruka et al [16] have studied ~ ~ Kirkendall shift in Ti-Zr system and reported the ratio of ( D Ti/ D Zr) = 0.7 at 950 0C. There seems to be some inconsistency in the reported impurity diffusion coefficient of Zr in β-Ti. In addition, the interdiffusion data at the composition higher than 75 at.% of Ti are not reported in the literature. The objective of the present investigation is to prepare step diffusion couples using pure Ti and Zr and determine the interdiffusion coefficients over the whole range of compositions in the β (bcc) Ti-Zr solid solution. An attempt has also been made in this investigation to evaluate the interdiffusion coefficient more accurately at the end composition of Ti and Zr. The present work also attempts to compare the diffusion coefficients evaluated using this method with those reported in the literature.

Experimental Procedure The present studies involved several steps like the preparation of “sandwich” type of diffusion couples, “step type diffusion couples”, diffusion annealing, establishing the true concentration – penetration profiles across the diffusion zone using an electron microprobe analyser (EPMA), microstructure by scanning electron microscopy (SEM) and transmission electron microscopy (TEM) and evaluation of diffusion coefficients using Boltzmann-Matano analysis [6-7] and Hall’s Method [17]. The details of various steps are mentioned below.

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Preparation of diffusion couple: Electron beam melted iodide grade pure titanium (99.9%) and nuclear grade zirconium (99.9% purity) have been employed for these studied. The impurities present in these alloys are mentioned in Table 1. Titanium and zirconium pieces were rolled, cut into rectangular shape, annealed at 900oC for 12 hours. This long time of annealing was an essential step to increase the grain size up to 3 mm and thereby minimize the contribution of the grain boundary diffusion. The pieces of Ti and Zr were subsequently polished to 1µm diamond finish. The couples of size 10 X 10 X 5 mm consisting of titanium and zirconium were made by placing the polished faces in contact with each other in a specially made fixture and then heated in a vacuum (better than 10-5 torr) at 900oC at a pressure of 0.1 MPa for 15 min. The schematic of this diffusion couple is shown in Fig. 1 (a). The electron probe microanalysis of the diffusion zone for the couple revealed that the width of the diffusion zone was 5-10 µm and was free from porosity. The width of the diffusion zone before annealing was Figure.1 Schematic of the preparation and found to be very small compared to that of the width the annealing schedule of the of the diffusion zone obtained after diffusion conventional and step diffusion couple annealing and hence was neglected for the analysis. (Ti/Zr). (a) conventional diffusion couple The diffusion couples thus prepared were sealed in annealed at 900oC for 0.15 minutes. (b) the quartz tube in Ar atmosphere and annealed in a conventional diffusion couple annealed at preheated furnace at 900oC for 72 hrs. The o 900oC 72 hrs, (c) step diffusion couple temperature of the furnace was controlled within ± 1 annealed at 900oC for 0.15 minutes. and C throughout the diffusion annealing experiments. (d) step diffusion couple annealed at The diffusion couples were subsequently quenched 900oC for 72 hrs. in air and metallographically polished across the cross section. The top and bottom side of the diffusion couples were metallographically prepared to a 1 µm finish. The schematic of the conventional diffusion couple is shown in Fig.1 (b). The step type diffusion couple of size 20 X 10 X 5 mm was prepared by placing polished pieces of Ti on the top and Zr at the bottom side of the diffusion couple in a specially designed die and annealed at 900o C for 15 min. for getting intimate contact. Schematic of this step type diffusion couple is shown in Fig. 1(c). The diffusion couple consisting of the conventional and the step type diffusion couple was sealed in quartz tube in Ar atmosphere and subsequently annealed in a preheated furnace at 900oC for 72 hrs. The schematic of this step type diffusion couple is shown in Fig.1 (d). Table 1. Chemical analysis of the Ti and Zr Impurities O

N

C

Al

B

Ca

Cr

Cu

Fe

Mn

Mg

Ni

Mo

Hf

(ppm) Zr

780 22

40

10

50

50

48

20

500 10

10

10

-

100

Ti

600 40

80

40

-

-

25

50

240 50

-

60

10

-

56

Phase Transformation and Diffusion

Determination of concentration profiles across the diffusion zones : The diffusion annealed couple was sectioned perpendicular to the bond interface using a slow-speed diamond saw. The sample was mounted, ground on various grades of emery papers and then polished on a lapping wheel with 0.25 µm diamond paste. The polished, unetched diffusion couples were analysed using a CAMECA SX 100 Electron Probe Micro-Analyser (EPMA) equipped with three wavelength dispersive spectrometers (WDS). LiF crystal was employed for diffracting the Ti-Kα and PET for diffracting Zr-Lα lines. The operating voltage and beam current were kept at 20 kV and 20 nA respectively. Pure Ti and pure Zr were used as standards for the analysis. The standard PAP correction program was used for atomic number (Z), absorption (A) and fluorescence (F) corrections [18]. Quantitative analysis on point to point basis was done at a regular interval of 1-2 µm by scanning the sample across the bonding interface to determine the concentration profile. For each sample, at least three scans were taken at different locations to confirm the consistency of the concentration profiles. Several line scan profiles were taken across the diffusion zone (DZ) and the locations of these scans are marked as regions- 1 to 5 and these s are shown in Fig. 1 (d). The regions- 1, 4 and 5 correspond to conventional diffusion couple and regions- 2 and 3 correspond to step type diffusion couple. The concentration profiles have been smoothened by using the cubic spline equation with the help of a computer programme. Specimen preparation for microstructural characterization by (SEM) and (TEM): For SEM studies, the etching was carried out using a solution consisting of 8% HF + 45% HNO3 +47% H2O (by volume). TEM examination was carried out in a JEOL-200 FX microscope. Disc samples of 3mm diameter were cut using electo-discharge machining (EDM) close to the Ti side of the diffusion zone (region-1 in Fig.1 (d)). These specimens were subsequently jet thinned in 20% perchloric acid + 30% n-butanol + 50% methonal solution at –40 0C and 20 V for TEM examination. Data analysis and evaluation of interdiffusion coefficient from concentration profiles : The most widely used method to determine the concentration dependent interdiffusion coefficient is the Boltzmann Matano method [6,7]. This method involves numerical solution of the Fick’s second law and estimation of slope of the concentration profile at the point concerned and the major drawback of the method is that it incorporates a lot of error in the calculation of the slope at the extreme ends of the profile. Since the Hall’s method is analytical, one can yield sufficiently accurate diffusion data in such a low concentration range [17]. Therefore to eliminate such erroneous results, Hall’s analytical method [15,17] was used instead for the composition range 0 ≤CTi,, Zr ≤2 at%, and Boltzmann-Matano method for CTi, Zr≥10 at%. In a binary system A-B, the interdiffusion coefficient at composition C* is calculated by the relation: ~ D AB (C * ) =

1 dx ( ) * 2t dC C=C

C*

(x ∫ ∞

x 0 )dC

(1)

C=C

with the initial boundary condition as at t=0 and x ≤xo, C=C-∞ and at t=0 and x ≥xo, C= C+∞ where t is the time of annealing, (

dx * -∞ ) C=C* is the inverse of the concentration gradient at C , C is the dC

concentration at the extreme left end, C+∞ is the concentration at the extreme right end of the diffusion couple and x0 is the position of the Matano Interface (MI). In Hall’s method, a variable λ is defined in such a way that, C* 1 = (1 +erf ( λ )) C0 2

(2)

Defect and Diffusion Forum Vol. 279

57

where C* is the solute concentration at time t and at position x with respect to the Matano Interface , C0 = C+∞ -C-∞ and erf(λ) is the error function of λ.. λ is found to bear a linear relationship with the variable η(=x/√t),as:

λ=hη+k

(3)

~ The concentration dependent interdiffusion coefficient D (C) can be determined from the values of h and k by the following relation [17,19]: 1 k π ~ D(C * ) = 2 + 2 (1 +erf ( λ )) exp( λ 2 ) 4h 4h

(4)

The intrinsic diffusivities of the constituent elements A and B, i.e. DA and DB, at the marker positions were calculated by using Darken’s equation [20,21] and the velocity of the markers, v: ~ D AB = (D A NB +DBN A )

v=

∆x dC = (D A - D B )( ) 2t dx

(5) (6)

where, ∆x is the displacement of the markers from the original interface in the annealing time t, A, dC B mole fraction of A and B respectively and ( ) is the concentration gradient at the marker dx position. The significance of DA and DB at terminal compositions relevant to Ti and Zr system has been described below. The interdiffusion coefficient at Zr → 0 is symbolized as D0Zr, can be regarded as the impurity diffusion coefficient of Zr in Ti. Similarly, at Ti → 0, DoTi can be regarded as impurity diffusion coefficient of Ti in Zr.

Results and discussions Microstructure of the diffusion zone : Backscattered electron (BSE) image corresponding to the region-1 (Fig.1(d)) is shown in Fig.2. This micrograph clearly exhibits five different regions (a) αTi parent phase, (b) transformed α-Ti needles (c) retained β Ti-Zr solid solution (d) transformed αZr needles and (e) α-Zr-parent phase. In addition, the presence of the Kirkendall interface close to the Ti side of the diffusion couple can be noticed in this micrograph. The microstructures corresponding to the region-2 and -3 (Fig 1(d)) are nearly same. The microstructure corresponding to the titanium side (region-2) showed (a) α-Ti parent phase, (b) transformed α-Ti needles (c) retained β Ti-Zr solid solution and that of the Zr side (regions-3), showed (a) β Ti-Zr solid solution (b) transformed α-Zr needles and (c) α-Zr parent phase. TEM micrograph (bright field image) taken in the diffusion zone close to the Ti rich side (region-1, Fig 1(d)) is shown in Fig.3. The micrograph clearly shows plate type martensite.

58

Phase Transformation and Diffusion

Figure. 2 Back-scattered electron (BSE) image for the diffusion couple annealed at 900oC for 144 hrs. for region-1 (Fig 1(d)) taken at lower magnification (100X).

Figure.3. Bright field electron image for the conventional diffusion couple annealed at 900oC for 144 hrs, corresponding to Ti rich side of the diffusion couple .

Concentration profiles and diffusion widths: The concentration variation of Ti and Zr across the diffusion couple annealed at 900 0C for 144 hrs, corresponding to region -1 (Fig.1 d) is represented in Fig.4. It can be noted that Ti and Zr variation is continuous across the diffusion zone. The position of the Matano Interface (MI) is marked on this profile. The profile is typical of the solid solution as is expected on the basis of the Ti-Zr phase diagram, which shows complete solid solubility in β-region [22]. The occurrence of the Kirkendall interface (Fig.2) close to the Figure. 4.Concentration penetration titanium clearly suggests, higher mobility of the Ti. Higher profile for the conventional mobility of the Ti is also reported in the Ti-Ta [23] and Ti-Hf diffusion couple annealed at 900oC [24] system. However, Raghunathan et al [14] did not report for 144 hrs corresponding to the presence of the Kirkendall interface. The concentration location 1 as shown in Fig.1 (d). variation of Ti and Zr across the step diffusion couple 0 annealed at 900 C for 72 hrs. corresponding to the region- 2 (Fig 1d) is shown in Fig.5 and the nature of these profiles are similar to that of conventional diffusion couple. Several concentration profiles of Ti and Zr were taken at regular intervals at regions- 1 to 5 (Fig.1 d) for calculating the diffusion widths. It needs to be mentioned that the cumulative annealing time for the region- 1 is 144 hrs and for all other regions, namely 2 to 5 the annealing time is 72 hrs. It can be seen from Fig.4, that the diffusion width corresponding to region- 1 is 1780 µm and for other regions- 4 and 5 are 1420 and 1428µm respectively. Assuming parabolic growth at regions- 4 and 5, the diffusion width at the region- 1 should be 2010 µm. Though the region- 1 has seen 144 hrs of annealing at 900 0C relatively smaller diffusion width could be due to the imparting of a quenching step after 72 hrs.

Defect and Diffusion Forum Vol. 279

Figure 5.Concentration penetration profile for the step diffusion couple annealed at 900oC for 72 hrs corresponding to region-2 as shown in Fig. 1 (d). The starting composition of the step diffusion couple is Ti/Zr-98.44 Ti.

59

Figure 6 . Diffusion widths corresponding to regions-2 and 3 in Fig. 1 (d). The starting composition of the step diffusion couple is clearly marked in this figure.

The end compositions of β(Ti,Zr) solid solution of the step diffusion couple and the diffusion width estimated from the concentration profiles are plotted on a three dimensional plot in Fig.6. The step diffusion couple corresponding to region-2 is essentially between Ti and β(Ti,Zr) solid solution and that corresponding to region-3 is between Zr and β(Ti,Zr). For the Zr rich side of the step diffusion couple corresponding to the region-3 (Fig.1d), (Zr/Ti-92.8 Zr), the width of the diffusion zone is around 460 µm and for the Ti rich side of the step diffusion couple corresponding to region-2, (Ti/Ti-1.55 Zr), the width of the diffusion zone is 1250 µm. This result confirms that the growth of the diffusion zone is strongly dependent on the start and end composition of β(Ti,Zr) solid solution. It is important to mention that the dependency of the diffusion width on the composition is very specific to the configuration of the step diffusion couple. Configurations of this type involving the triple junctions were made to measure the local chemistry at the interface [11]. This localized property measurements were used to determine the ternary phase diagrams, diffusion coefficients and these results also reflected on the development of multiple joints by hot isostatic pressing [12,25]. These studies involved interdiffusion measurements between different starting materials and this geometry is not suitable for measuring the impurity diffusion data.. The present study describes the dependency of the growth of the diffusion zone as a function of composition in a given binary system. Thus based on the single step type diffusion couple experiment, it is possible to obtain the dependence of diffusion width with composition and this information will be useful to understand the joining of multiple specimens under isothermal conditions. ~ Interdiffusion coefficients and their temperature dependence : The values of the D for the regions- 1, 4, 5 were calculated using Eqn. (1) in the β region for the Ti composition ranging from ~ 10 to 90 at.% . These values for regions- 4 and 5 were nearly identical and hence D the have been ~ plotted for regions- 1 and 4 in Fig.7. The values of D for region- 1 are small compared to those of ~ the region- 4. As discussed, the marginal difference in the D values at the same temperature could be due to the nature of the annealing scheme used in these studies. It needs to be mentioned

60

Phase Transformation and Diffusion

~ that the D values evaluated for the region- 4 are more realistic compared to the region- 1. The ~ curve shows an increase in the value of the D with increase in the Ti composition and reaches a maximum diffusivity at around 55-65 at.% of Ti. These results can be understood using the Vignes~ Birchenall correlation of interdiffusivity D with ∆TSAB [26]. Here ∆TsAB is the difference between the solidus temperature of the alloy and the temperature, which could be computed if the solidus temperature varied linearly. The phase diagram of Ti-Zr system shows complete solid solubility in the β-region and the solidus temperature is lowest at 65 at.% Ti [22]. In addition, the value of ~ ∆TsAB is maximum around 200oC for this composition and hence maximum D is expected at this composition. The values of the diffusion coefficients evaluated by Raghunathan et. al [14] at 901 ~ 0 C for the Ti composition 10 to 75 at.% are also plotted in Fig.8. The values of the D in the present investigation are generally higher than that of the values reported by Raghunathan et al [14] and this could be due to the composition of the starting materials used for making the diffusion couples.

~ ~ Figure 7. Interdiffusion coefficients ( D ) for the Figure 8 .Interdiffusion coefficients ( D ) conventional diffusion couple (Ti/Zr) annealed evaluated at terminal compositions for at 900oC for 144 hrs. corresponding to the the step diffusion couples annealed at region-1 in Fig. 1 (d) and also for the step 900oC for 72 hrs corresponding to o diffusion couple (Ti/Zr) annealed at 900 C for regions-2 and 3 as shown in Fig.1 (d). 72 hrs, corresponding to the region- 4 as shown in Fig. 1 (d). ~ In order to evaluate the D for step diffusion couples, several location close to region-2 and region-3 (Fig.1.d) are identified in a narrow range of composition (< 10 at.%). The geometry of the step diffusion couple corresponding to region-2 and region-3 clearly suggest the presence of diffusion triples and determination of diffusion coefficients and mapping of Young’s modulus based on available local equilibrium at the interface is reported in the literature [27-29]. It should be ~ mentioned that several scans were taken at regions-2, 3 and D values were obtained for compositions up to 5 at.% by Boltzmann-Matano method [6-7] and these values are plotted in Fig.8. ~ In addition, the D values were also evaluated for compositions < 2 at. % by Halls method ~ from the region –4 are these values are plotted in Fig.8. It can be seen that the D values increases with increase in Ti composition on the Zr rich side and decreases with the increase in the Ti composition on the Ti rich side. The values of the diffusion coefficients evaluated by the Halls method are generally larger than those evaluated by step diffusion couples. In addition, the usage of Halls method is generally restricted up to 2 at.%. Based on the present method it is possible to ~ evaluate D for all compositions more particularly at the terminal compositions up to 5 at.%.

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Intrinsic(DTi ,DZr ) and impurity diffusion coefficients (D0Ti,, D0Zr) : Based on the position of the Kirkendall interface and the extent of its displacement from the original interface, the values of intrinsic diffusivities were evaluated for region- 1 at Zr-55 at.%Ti using Eqs. 5 and 6 and these values are DTi =3.9 X 10 –13 m2/sec and DZr=1.24 X 10 –13 m2/sec and the DTi value is 3 times larger ~ than DZr. In addition, the impurity diffusion coefficients were determined by extrapolating the D values determined by Boltzmann-Matano method [6,7] at the terminal compositions for region -2 and 3 and also those determined by Halls method for region-4 and these values are listed in Table 2. Table 2 Impurity diffusion coefficients at 900oC.

Sr.

Diffusion of Zr in Ti

Diffusion of Ti in Zr

No.

D0Zr (m2/sec)

D0Ti (m2/sec)

1.

1.41 x 10 -13

1.78 x 10-14

Remark

Present work, for region-2, 3, Step diffusion couple, Boltzmann-Matano method [6-7]

2.

4.58 x 10-13

1.07 x 10-14

Present work, for region-4, Conventional diffusion couple, Hall method [17]

3.

3.63 x 10 -14

2.63 x 10-14

4.

2.00 x 10-13

-

Raghunathan et al [12] Araki et al [13]

The extrapolated values of impurity diffusion coefficients obtained using the Zr/Zr-80.0% Ti diffusion couple by Raghunathan et al [14] and also those determined by incremental couple Ti/Ti3.06 Zr by Araki et al [15] are also listed in Table 2. It can be seen from the Table 2 that the D0Zr values based on the step diffusion couple is close to the value determined by Araki et al[15]. On the contrary the D0Zr value reported by Raghunathan et al [12] is an order of magnitude smaller compared to that reported by Araki et al [13]. Similar comparison could not be made for D0Ti, value as there are no reported values. In odder to evaluate the impurity diffusion data, it is important to make diffusion couples close to the terminal compositions. Araki et al [15]. Araki et al [15] have made diffusion couple between pure Ti and Ti-3.06Zr and extrapolated the interdiffusion data to estimate the D0Z . On the other hand Raghunathan et al [14] had made diffusion couples from Pure Zr and Zr-80%Ti and had extrapolated the interdiffusion data at terminal compositions to arrive at the impurity diffusion data (D0Ti,, D0Zr). It can be seen from Table 2 that there is an order of magnitude difference in the measured impurity diffusion data by these two studies [14, 15]. In the case of the present method, the geometry provides the necessary compositions close to the terminal compositions and hence extrapolations provide a better estimate of the impurity diffusion data.

Conclusions ~ A novel method is described for evaluating the D values at terminal compositions and thereby estimating the impurity diffusion coefficients by extrapolation. This method is applied to relatively well-studied Ti-Zr system. Based on the present studies the following conclusions are inferred.

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Phase Transformation and Diffusion

~ Interdiffusion coefficients D in the β phase of Ti-Zr system strongly depend on the ~ composition. D values show a maximum around 65 at.%. Ti and closely follows the solidus curve of the Ti-Zr phase diagram. Kirkendall shift is observed towards Ti side. The intrinsic diffusion coefficient of Ti (DTi) is approximately three times greater than that of Zr (DZr) for composition corresponding to the Kirkandall interface. The diffusion width is a strong function of the starting composition of the step diffusion couple. In fact, based on a single diffusion couple it is possible to determine complete composition dependent diffusion width and this information may be useful for joining of multiple specimens involving plane geometries. The present method offers an easy way of estimation of interdiffusion coefficients at the terminal compositions without the tedious preparation of incremental diffusion couples. In addition, this method may offers an attractive way of estimation of interdiffusion and impurity diffusion coefficients in those systems, which form brittle compounds in the diffusion zone.

REFERE)CES [1] D. Ansel, I. Thibon , M. Bolivean and I. Debligine : Acta Metall Vol 46(2) (1998) p. 423. [2] M. Maeda, R. Oomoto, M. Naka and Shibayanagi: Metall. Mater.Trans. A Vol 34A (2003) p.1647. [3] A. Guillon , D. Ansel and J. Debugne : Scripta Mater. Vol 34 (6) (1998) p. 981. [4] D. Gupta : Defect and Diffusion Data Vol 59 (1988) p. 137. [5] C. R. Kao, J. Woodford and Y. A. Chang : J. Mater. Res Vol 11(4) (1996) p. 850. [6] C. Matano: Jap. J. Phy Vol 8 (1933) p. 109. [7] L. Boltzmann: Annal. Phys Vol 53 (1984) p. 960. [8] F. J. A. Der Broeder: Scripta Mater, Vol 3 (1969) p.321. [9] K. Bhanumurthy , Ph.D Thesis, University of Bombay, Bombay, 1992. [10] K.Bhanumurthy, G.B.Kale, S.K.Khera and M.K.Asundi.:Metallurgical Transactions, Vol 21A (1990) p. 2897. [11] J. –C. Zhao : Adv.Eng.Mater Vol 3 (2001) p. 143. [12] J. –C. Zhao : Annu. Rev. Mater. Res Vol 35 (2005) p. 51. [13] Y. Iijima, K. Hoshino and K. I. Hirano : Metall. Trans A. Vol 8A (1977) p. 997. [14] V. S. Raghunathan, G. P. Tiwari and B. P. Sharma Metall. Trans. Vol 3 (1972) p. 783. [15] H. Araki, Y. Minamini, T. Yamane , T. Nakatsuka and Y. Miyamoto: Metall. Trans. A Vol 27A., (1996) p. 1807. [16] I. Daruka, I. A. Szabo, D. L. Beke, C. S. Cserhati, A. Kodentsov and Van Loo FJJ: Acta Mater. Vol 44(12) (1966) p. 4981. [17] L. D. Hall : J. Chem Phys. Vol 21 (1953) p. 87. [18] J. L.Panchou and F. Pichoir: Microbeam Analysis, 1985; San Fansisco Press, California [19] J. Crank: Mathematics of Diffusion, Oxford University, Oxford, 1956; p.219. [20] L. S. Darken : Trans AIME; Vol 175 (1948) p. 184. [21] P. G. Shewon : Diffusion in Solids, 2nd Edn., The Minerals, Metals and Materials Society, Pennsylvania, PA, (1989) p.135. [22] T. B. Massalski: Alloy Binary Alloy Phase Diagram, 2nd Edn., Oh: ASM International, (1990) p.241. [23] D. Ansel, I. Thibon, M. Boliveau J. Debuigne: Acta Metallurgica Vol 35 (1998) p.423. [24] G. Le Gale, D. Absel and J. Debuigne: Acta Metallurgica Vol 35 (1998) p. 2297. [25] Jin Z. Scand: J. Metall; Vol 10 (1981) p. 279. [26] A. Vignes and C. E. Birchanall : Acta Metall; Vol 16 (1968) p.117. [27] J. –C. Zhao, X. Zheng and D. Cahill : Materials TodayVol 10 (2005) p. 28. [28] Y. –T. Cheng and C.-M. Cheng: Mater Sci.Engg R Vol 44 (2004) p. 91. [29] H.Xu, Z.Jin and R. Wang : Scripta Materialia Vol 37 (1997) p. 1469.

Defect and Diffusion Forum Vol. 279 (2008) pp 63-69 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.63

Diffusion in Cu(Al) Solid Solution A. Laik1, K. Bhanumurthy2, G. B. Kale3 Materials Science Division, Bhabha Atomic Research Centre, Mumbai 400 085, India Email: [email protected]; [email protected]; [email protected] Keywords : diffusion coefficient; impurity diffusion; activation energy; copper; aluminium

Abstract. The solid state diffusion characteristics in the Cu(Al) solid solution phase, was investigated in the temperature range of 1023–1223 K using single phase bulk diffusion couples between pure Cu/Cu- 10 at.% Al. The interdiffusion coefficients, D, were calculated using Boltzmann–Matano method and Hall’s method from the concentration profiles of the couples that were determined using EPMA. The interdiffusion coefficients (D) calculated ranges between 1.39 X 10−14 and 3.97 X 10−13 m2/s in the temperature range of 1023 to 1223 K. The composition and temperature dependence of D were established. The activation energy for interdiffusion varies from 123.1 to 134.2 kJ/mol in the concentration range 1 at. % ≤ CAl ≤ 9 at. %. The impurity diffusion coefficient of Al in Cu is determined by extrapolating the interdiffusion coeffficient values to infinite dilution of the alloy i.e CAl →0 and its temperature dependence was also established. The activation energy for impurity diffusion of Al in Cu was found to be 137.1 kJ/mol. Introduction The Cu-rich side of the Cu-Al system finds commercial application in form of aluminium bronzes which are Cu-based alloys containing up to 16% Al. These alloys are considered as potential materials for a wide variety of applications due to their corrosion resistance and exceptionally high strength [1]. The Cu-side terminal solid solution is single phase Cu(Al) (α-phase) and can dissolve up to 19.7 at. % of Al and can be strengthened by cold working. Additions of certain elements, such as cobalt and nickel render these single phase binary alloys age hardenable. The Cu(Al) solid solution takes part in two solid state reactions in the system i.e. a eutectoid reaction β ↔ γ1 +Cu(Al) at 567°C and a peritectoid reaction Cu(Al) + γ1 ↔ α2 [2]. These reactions being solid state in nature, their kinetics are essentially controlled by the diffusion behaviour of the participating phases. It is worth noting that Sprengel et al. [3] showed that the interdiffusion coefficients obtained from diffusion couple consisting of multiple phases differ from those consisting a single-phase. Kim and Chang [4] have reported a large difference in the activation energy for interdiffusion determined by multi-phase diffusion couples and single-phase diffusion couples in the NiAl phase. The large difference in the activation energy could be due to the fact that phase boundaries formed in multi-phase diffusion couples influence the diffusion flux to a large extent. The phase boundaries act both as source and sink for point defects, and the grain boundaries in an intermediate phase, formed during annealing, may alter the overall diffusion rate in the system. Therefore the diffusion coefficients calculated from such experiments are not truly representative. Besides, no data for tracer diffusivity of aluminium (DAl) in Cu exist in the literature to the best of knowledge of the authors. It is worth mentioning here that this is due to the experimental limitations in determining the impurity diffusion coefficient of Al in Cu primarily due to nonavailability of suitable radioactive isotope of Al which could be used as tracer. The only isotope which is suitable for such experiments is 26Al (t1/2 = 0.75 X 106 yr), which has low specific activity and is extremely expensive. The production and isolation of the 26Al isotope, required for the radiotracer diffusion experiments is extremely difficult [5] and can be achieved through a (p,pn) nuclear reaction with naturally occurring 27Al in a cyclotron [6]. However, the impurity diffusion

64

Phase Transformation and Diffusion

coefficient of Al can also be determined from concentration dependent interdiffuson coefficients using diffusion couples experiments. Recently such studies on binary systems Ni-Al [7] and Zr-Al [8] have been reported. The present study reports a detail investigation of the diffusion behaviour in the fcc Cu(Al) solid solution in the concentration range 0 ≤ CAl ≤10 at.% Al in the temperature range of 1023 to 1223 K. The impurity diffusion coefficient of Al in Cu is determined by extrapolating the interdiffusivity values to infinite dilution of the alloy i.e. CAl → 0 and its temperature dependence was also established. Experimental procedure Diffusion couple preparation. A dilute alloy of nominal composition Cu-10 at. %Al was prepared by vacuum induction melting in yttria coated graphite crucible using proportionate mixture of pure (99.9%) Cu and pure (99.95%) Al. The chemistry and homogeneity of the alloy was determined by analysing in a CAMECA SX100 electron probe micro analyser (EPMA) and was confirmed to be having homogenous composition of Cu-10 at. % Al. Rectangular pieces of size 10 X 8 X 3 X mm3 were cut from the rolled Cu-Al alloy and pure Cu. These pieces were encapsulated in quartz tubes in He atmosphere at a residual pressure of about 16 kPa and annealed at 1223 K for 72 h to homogenise the materials and to form stable coarse grained structure, which reduces the contribution of grain boundary diffusion in the diffusion annealing stage. The 10 X 8 mm2 surfaces of the samples were prepared by grinding on successive grades of emery paper from 80 grit through 2400 grit and then subsequently polishing on a lapping wheel with 0.25 µm diamond paste. The diffusion couples were made by keeping the polished surfaces of the pure Cu and the Cu-10 at. %Al alloy pieces in contact with each other under a pressure of about 5 MPa in an Inconel die and then heating in vacuum better than 10−3 Pa at 1223 K for 1 h. Chemical analysis using EPMA across the interface of the as-bonded couples showed a negligible diffusion width. The couples were then sealed in quartz capsules in He atmosphere and diffusion annealed isothermally for different time intervals in a preheated resistance heating furnace in the temperature range 1023–1223 K for 24–72 h. The details of the heat treatment schedule of the isothermal diffusion annealing are given in Table 1. The temperature of the furnace was controlled within ±1 K by a proportional temperature controller. Table 1. Heat treatment schedule for the diffusion couples. Couple No. 1 2 3 4 5

Temperature [K] 1023 1073 1123 1173 1223

Time [hrs] 72 62 48 36 24

Metallographic preparation. The diffusion annealed couples were sectioned perpendicular to the bond interface using a low-speed diamond saw. The cross-sections of the couples were prepared to 0.25 µm finish using standard metallographic techniques as described earlier. The polished surfaces of the couples were etched with an etchant with the composition: 5 parts water, 5 parts ammonium hydroxide and 2 parts hydrogen peroxide to reveal the microstructure. Characterisation. The cross-sections of the polished couples samples were characterised using an optical microscope and an electron probe micro analyser (CAMECA SX100) equipped with three wavelength dispersive spectrometers. The operating voltage and beam current were kept at 20 kV

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and 20 nA repectively. Pure Cu and pure Al were used as standards for the analysis. Lithium fluoride and thallium acid pthalate(TAP) crystals were used for diffraction of Cu-Kα and Al-Lα lines respectively. The standard PAP correction program was used for atomic number (Z), absorption (A), and fluorescence (F) corrections. Quantitative analysis on point-to-point basis was done at a regular interval of 1-2 µm by scanning the sample across the bonding interface to determine the concentration profile. For each sample, at least three scans were taken at different locations to confirm the consistency of the concentration profiles. Results and discussions Interdiffusion in the Cu(Al) phase. Figure 1 shows a typical concentration profile for the couple annealed at 1223K. The plot shows typically solid solution type of concentration penetration curve across the interface. The concentration profiles for all the couples are of the same nature and is represented by Fig. 1. During the process of annealing, the initial compositions of the individual pieces of the couples are preserved at the ends, thereby fulfilling the requirement of the infinite geometry. The interdiffusion coefficients for various compositions within the diffusion zone for all the diffusion couples were calculated by using the Boltzmann-Matano method [9,10] and Hall’s method [11]. The interdiffusion coefficients (D) calculated ranges between 1.39 X 10−14 and 3.97 X 10−13 m2/s in the present temperature range of annealing.

Figure 1: Concentration profile across the interface of Cu/Cu-10 at. % Al couple annealed at 1223K for 24 h. The interdiffusion coefficients were calculated at compositions alloy ranging from 0.5 at. % to 9.5 at. % at intervals of 0.5 at. %. The concentration dependence of interdiffusion coefficients at various temperatures by Boltzmann-Matano method is presented in Fig. 2. The interdiffusion coefficients at various temperatures are fitted in a cubic relation of the type: D = a + b CAl + c CAl2 + dCAl3 The values of the parameters a, b, c and d for various temperatures are given in Table 2.

66

Phase Transformation and Diffusion

Table 2 Values of the parameters a, b, c and d in the relation D = a + b CAl + c CAl2 + d CAl3 (CAl is in at. %). Temperature [K] 1023 1073 1123 1173 1223

a -29.54 -30.48 -30.74 -31.75 -31.93

b 0.24 0.04 0.01 0.14 0.09

c -0.03 0.01 0.01 -0.01 -0.01

d X10-4 16.21 -15.02 -10.63 7.02 5.04

Such enhancement in the interdiffusion coefficient, D, of the terminal solid solution with the addition of Al was reported in other system as well. For example, increase in the interdiffusion coefficient in the β-Zr(Al) solid solution phase was observed in single phase [8] as well as in multiphase [12] diffusion couple experiments. Similar behaviour was reported in the β-Ti(Al) solid solution phase by Hirano and Iijima [13]. Addition of Al to Cu decreases the solidus temperature line in the Cu–Al phase diagram [2]. The activation energy of diffusion is directly related to the solidus temperature [14]. Therefore with increase in Al concentration solidus temperature and hence activation energy decreases and the diffusivity increases.

Figure 2: Concentration dependence of interdiffusion coefficient in Cu(Al) solid solution at various temperatures as indicated. In order to establish the temperature dependence of the interdiffusion coefficient D, ln D is plotted against the reciprocal of absolute temperature of diffusion annealing for various compositions in Fig. 3. A linear relationship in this plot shows that D follows an Arrhenius type of relationship, D = D0 exp(−Q/RT). D0 is the pre–exponential factor, Q is the activation energy and T is the absolute temperature. The activation energy for interdiffusion, Q, varies between 123.1 ± 3.9 to 134.2 ± 3.9 kJ/mol in the concentration range 1 at. % ≤ CAl ≤ 9 at. %. The values of the activation energy (Q) and the pre–exponential factor (D0) at different compositions of the Cu-Al system are given in Table 3.

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Figure 3: Temperature dependence of interdiffusion coefficient, D, at various concentrations of Al. Table 3 Pre-exponential factor, D0, and activation energy, Q, for interdiffusion in the Cu(Al) solid solution phase at various compositions of Cu and Al. Composition Al [at. %] 1 2 3 4 5 6 7 8 9

Pre-exponential factor (D0 X 108) [m2/s] 4.88 7.01 8.22 9.60 10.83 9.68 9.97 8.58 3.82

Activation energy [kJ/mol] 129.5 131.9 132.8 133.6 134.2 132.5 132.3 130.4 123.1

Impurity diffusion of Al in Cu. According to the Darken’s relation [15], the interdiffusion coefficient in an infinitely dilute solid solution corresponds to the impurity diffusion coefficient of the solute in the solvent matrix. The impurity diffusion coefficient of Al in Cu phase was determined by extrapolating the D values in the range 0 ≤ CAl ≤ 1 at. %, calculated by Hall’s method, to infinite dilution, i.e., CAl → 0. In the narrow composition range 0 ≤ CAl ≤ 1 at. %, D bears a linear relationship of the type D = a + b CAl. Here the impurity diffusion of Al in Cu is denoted as D AlCu (CAl = 0). Table 4 shows the values of D AlCu at various temperatures of investigation. The logarithm of D AlCu (CAl = 0) is plotted against the inverse of absolute temperature in Fig. 4. The frequency factor D0 and the activation energy QAl have been estimated by fitting a straight line to the data points in Fig. 4 by linear regression method. The activation energy of impurity diffusion is 137.1±4.0 kJ/mol. The temperature dependence can be expressed by an Arrhenius relation:  − 137.1 ± 4.0  2 DAlCu = 1.49 X 10 −7 exp  m /s. RT  

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Phase Transformation and Diffusion

Table 4 Impurity diffusion coefficients of Al in Cu. Temperature Impurity diffusivity D AlCu [K] [m2/s] 1023 1073 1123 1173 1223

-29.54 -30.48 -30.74 -31.75 -31.93

Figure 4: Temperature dependence of impurity diffusivity of Al in Cu.

Summary The interdiffusion coefficient in the Cu(Al) solid solution phase was determined in the temperature range of 1023–1123 K using single phase bulk diffusion couples between pure Cu/Cu- 10 at.% Al. The interdiffusion coefficients, D, calculated using Boltzmann–Matano method and Hall’s method, were found to range between 1.39 X 10−14 and 3.97 X 10−13 m2/s in the temperature range of 1023 to 1223 K. The composition and temperature dependence of the interdiffusion coefficient (D) were established. The interdiffusion coefficient (D) was found to follow a cubic relation of the type D = a + b CAl + c CAl2 + d CAl3 with the concentration of Al in the range 1 at. % ≤ CAl ≤ 9 at. %. The interdiffusion coefficient (D) shows an Arrhenius type of temperature dependence and the activation energy for interdiffusion varies from 123.1 to 134.2 kJ/mol in this composition range. The impurity diffusion coefficient of Al in Cu is determined by extrapolating the interdiffusion coefficient values to infinite dilution of the alloy i.e CAl →0. The temperature dependence of the impurity diffusion coefficient of Al in Cu ( D AlCu ) was established and the activation energy for impurity diffusion of Al in Cu was found to be 137.1±4 kJ/mol.

Acknowledgements The authors are grateful to Dr. G. K. Dey, Head Materials Science Division for his keen interest in the work.

References [1] Metals Handbook, Vol. 2, 10th Edn., ASM International (1990), Materials Park, Ohio, p. 216. [2] J. L. Murray, in: Binary Alloy Phase Diagrams, edited by T. B. Massalski, H. Okamoto, P. R. Subramanian and L. Kacprzak, 2nd Edn., ASM International, Materials Park, OH (1990) pp. 141-143.

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[3] W. Sprengel, N. Oikawa, H. Nakajima, Proceedings of the Workshop on Defects, Dynamics and Diffusion in Intermetallics, Vienna, Austria, 1994. [4] S. Kim, Y. A. Chang: Metall. Mater. Trans. A Vol. 31A (2000) p. 1519. [5] H. Mehrer: Mater Trans. JIM, Vol. 37 (1996) p. 1259. [6] T. S. Lundy and J. F. Murdock: J. Appl. Phys. Vol. 33 (1962) p. 1671. [7] K. Fujiwara and Z. Horita: Acta Mater. Vol. 50 (2002) p. 1571. [8] A. Laik, K. Bhnaumurthy and G. B. Kale: J. Nucl. Mater. Vol. 305 (2002) p. 124. [9] C. Matano: Jpn. J. Phys. Vol. 8 (1933) p. 109. [10] L. Boltzmann: Annal. Phys. Vol. 53 (1894) p. 960. [11] L. D. Hall: J. Chem. Phys. Vol. 21 (1953) p. 87. [12] A. Gukelberger, S. Steeb: Z. Metallkd. Vol. 69 (1978) p. 255. [13] K. Hirano, Y. Iijima, in: Conference Proceedings of TMS-AIME Fall Meeting Detroit MI, edited by M.A. Dayananda, G.E. Murch, 7 September 1984, p. 141. [14] A. Vignes and C. E. Birchenall: Acta Metall. Vol. 16 (1968) p. 1117. [15] L. S. Darken: Trans. AIME Vol. 175 (1948) p. 184.

Defect and Diffusion Forum Vol. 279 (2008) pp 71-77 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.71

A Probabilistic Approach to Analyze Austenite to Ferrite Transformation In Fe-Ni System G. Mohapatra and Satyam S. Sahay Tata Research Development and Design Centre A division of Tata Consultancy Services Ltd. 54, Hadapsar Industrial Estate, Pune - 411 013, India Email : [email protected], [email protected] Keywords: Dilatometer; induction heating; temperature gradient; austenite ferrite phase transformation, probabilistic

Abstract. Dilatometer is often used for in situ measurement of phase transformation by monitoring the length change during heating or cooling cycle. However, the inevitable temperature gradient across the specimen length during inductive heating, introduces uncertainty in temperature measurements and hence the associated phase transformation kinetics. Due to this uncertainty, it is more meaningful to interpret the transformation kinetics from dilatometry in terms transformation ranges instead of unique values of fraction transformed. In the present work, a probabilistic approach has been used to predict the fraction transformed ranges, arising due to the temperature gradient during dilatometry. The approach has been validated for Fe-5.93 at.% Ni undergoing austenite to ferrite phase transformation at various constant cooling rates. Introduction Precise determination and modelling of austenite to ferrite phase change kinetics in iron based alloys is important for optimization and control of commercially significant thermo-mechanical operations (e.g. hot rolling). Moreover, this also consequently determines the final microstructure and product quality. In a recent work, a differential dilatometer was used to precisely examine the austenite to ferrite phase change kinetics in binary iron based system [1]. This novel dilatometer has the capability to monitor phase change under uniaxial stress (compressive as well as tensile) and under unstressed conditions. Precise calibration methods (both temperature and length change) [2] and a temperature correction procedure [3] were developed to improve the accuracy of this technique. The temperature correction procedure [3] is highly sensitive to the fluctuation in the experimental length change data, making it unsuitable for very noisy data. As was reported earlier [3], there is a distinct temperature gradient across the solid cylindrical specimen length during the induction heating/cooling experiment. Although the total dilation during cooling is precisely measured, due to the thermal gradient, different location in the specimen will have different transformed fraction at any specific time. In prevalent methodology, this uncertainty in temperature measurement is circumvented by assuming an equivalent temperature, for which unique values of fraction transformed are determined. However, due to the uncertainty in temperature measurements, it may be more meaningful to interpret the dilation results in terms of range of transformations. In the present work, dilation studies are carried out on Fe-5.93% Ni system to examine the influence of cooling rates on γ→α transformation. Furthermore, a probabilistic approach is proposed, to estimate the upper and lower bounds of uncertainty in the fraction transformed due to the temperature gradient within the specimen during γ→α transformation. The method is validated with the experimental data obtained for γ→α transformation, for Fe-5.93% Ni system, under various applied cooling rates in the range of 20 to 220 Kmin-1.

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Phase Transformation and Diffusion

Experimental Fe-Ni alloy was prepared by melting appropriate amount of high purity Fe (99.98%) and Ni (99.99%) in a vacuum-melting furnace, and casting the molten alloy in a copper mould. The as-cast ingots of 7 mm in diameter were hammered down to rods of 6 mm diameter and homogenized in argon filled quartz capsule at 1423 K for 100 h and subsequently furnace cooled to room temperature. The composition of the Fe–Ni rods as determined by Inductive Coupled Plasma–Optical Emission Spectrometry (ICP-OES) was Fe-5.93 at. % Ni. Finally, the rods were machined into solid cylindrical specimens with 5 mm diameter and 10 mm length. A dilatometer DIL-805 A/D (BaehrThermoanalysis GmbH), employing inductive heating/cooling in normal mode [2] was used to measure the thermal dilation of the specimen as a function of temperature and time. In all the dilatometer specimens two thermocouples were spot welded at the specimen centre (Tcentre) and specimen end (Tend) along the longitudinal direction [3]. With this instrument, the specimen length change with a resolution of ±0.05 µm can be measured along with the specimen temperature at the centre and end of specimen. The measured length change and the temperature were calibrated according to the procedure described in Ref. [2]. The temperature calibration was done with cooling rate independent Curie temperature of pure Fe and the length change calibration was done with a Pt reference specimen. Five homogenized cylindrical specimens of Fe-5.93at%Ni alloy with similar initial grain size were heated from room temperature up to 1273 K with a rate of 20 Kmin−1 and soaked for 30 min to get rid of the build up-of misfit deformation energy associated with α→γ transformation during heating so that during cooling the specimens would be free from any residual stress. Subsequently, all the specimens were cooled down to room temperature at various constant applied cooling rates (20-220 Kmin-1), during which length change was monitored as a function of temperature. After the dilatometry measurements, the specimens were sectioned along the longitudinal direction, polished, etched (2.5% Nital) and analyzed under optical microscope. The line intercept method [4] was employed to measure grain size and size distribution. From this 2D measurement, the true average ferrite grain diameter was obtained by multiplying the intercept length by a factor of 1.5 [4]. Results and Discussion 90

Dilation,µ m

Applied The measured length changes as a function of Cooling Rate -1 temperature (Tcentre) for various applied cooling 100 Kmin 70 20 rates are shown in Fig. 1. The specimen 140 thermocouple temperatures (Tcentre and Tend), as 180 well as the length change as a function of time 50 220 during γ→α transformation for an applied cooling rate of 100 Kmin-1 is shown in Fig. 2. 30 The difference between Tcentre and Tend confirms the temperature gradient in the longitudinal 10 direction of the specimen. The temperature 700 800 900 1000 gradient exists due to the conductive heat loss Temperature, K from the specimen to the pushrods holding the specimen. It is interesting to note that although Fig.1. Measured length change as a the applied cooling rate remains isochronal function of temperature (Tcentre) during throughout the cooling cycle, due to the temperature recalescence associated with γ→α transformation, the specimen exhibits nonisochronal cooling (Fig. 2). This temperature recalescence effect was found to increase with increase in cooling rate, probably due to the associated faster transformation kinetics. It was also observed that with increase in cooling rate, the length change pertaining to γ→α transformation decreases from 49.3 µm to 39.7 µm, except for the applied cooling rate of 100 Kmin-1 (see Fig. 3). This could

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be attributed to a number of factors, including enhancement in temperature gradient, enhancement in transformation kinetics and possibility of increase in retained austenite fraction. The transformed ferrite fraction, fα, was determined from the length change data employing lever rule (see Fig. 4 for an applied cooling rate of 20 Kmin-1). In this analysis, it is assumed that the entire specimen is undergoing uniform transformation. The unique values of transformed fraction were modelled for different cooling rates using an average temperature ((Tcentre+Tend)/2), with the Johnson-Mehl-Avrami (JMA) expression [5-7].

(

n

f = 1 − exp − (k (T )t )

)

(1)

where, f is the transformed fraction, k(T) is the rate constant, t is the time and n is the Avrami exponent. Further, the k(T) is be given by an Arrhenius like temperature dependence.

 k (T (t )) = k 0 exp  − Q RT ( t )  

(2)

with ko is the pre-exponential factor, Q is the overall, effective activation energy and R is the gas constant. The obtained kinetic parameters for transformed fractions obtained for various cooling rates experiments are given in Table 1. As shown in Fig. 4 (applied cooling rate of 20 Kmin-1), the predicted transformed ferrite fraction values using the average temperature approach fit closely with the measured values. Instead of the average temperature, one could also determine the equivalent temperature [3] which is expected to closely fit with the measured transformed ferrite fraction data. Although both these approaches predict a close fit to the experimental measurement, they are incapable of predicting the range of transformed fraction arising due to temperature gradient in the specimen. This can be achieved by using a probabilistic approach outlined below. 930

80

Applied cooling rate:

65

-1

890

60

Tcentre Tend

40

870

D ila tio n ,µ m

T em p era tu re, K

910

20

L e n g th c h a n g e , µ m

100 Kmin

55

45

35 850 5010

5020

5030

0 5040

Time, s Fig. 2. Measured length change and temperatures (Tcentre and Tend) as a function of time.

0

50

100

150

200

250

-1

Cooling rate, Kmin

Fig. 3. The length change due to γ→α transformation for various cooling rates.

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Phase Transformation and Diffusion

T ra n sfo r m ed fr a c tio n ( f α )

1

Cooling rate:

0.8

20 Kmin-1

0.6 0.4 0.2

Measurement Fit

0 5860

5900

5940

5980

6020

Time, s Fig. 4. The experimental and the JMA model fit of the transformed fraction as a function of time Table 2 The kinetic parameters (n, K0 and Q) obtained after fitting JMA expression to the measured experimental transformed fraction as a function of time for various applied cooling rates. Cooling rate (Kmin-1) 20 100 140 180 220

 3.28 1.53 2.10 1.89 2.28

K0 1.7e+006 6.0e-012 1.5e-007 1.4e-007 1.1e-006

Q (Jmol-1) 1.4e+005 1.8e+005 9.8e+004 9.8e+004 8.9e+004

In the probabilistic approach, it is assumed that the temperature within the entire specimen length lie between the two extreme temperatures (Tcentre and Tend). During the time march, at any given time, the temperature is probabilistically selected (shown as dots in the Fig. 2) between the two limits. Subsequently, a temperature profile is obtained by connecting these probabilistically temperature points. In the next step, the transformed fraction is simulated using the JMA expression (Eq. 1), taking the obtained probabilistic temperature profile along with the kinetic parameters (Table 1). The transformed fraction for applied cooling rate of 20 K.min-1 obtained by this probabilistic approach is shown in Fig. 5a. It must be noted that although, the average values are close to the experimental result, the probabilistic approach shows a fluctuations in transformed ferrite fraction in comparison to the measured value (adopting lever rule) due to the scatter in the probability temperatures selection (see Fig. 2) between Tcentre and Tend. In order to mitigate such fluctuations which are characteristic to probabilistic approach, the above procedure was repeated several time to obtain the upper and lower bounds of fraction transformed. It was noted that from 15 such simulations, smooth bounds of transformed fractions can be obtained (Fig. 5b for 20 Kmin-1). It was observed that the uncertainty bounds were low for the initial period (0 < fα < 0.2), which can be attributed to the lower thermal gradient in the initial stages. Furthermore, it was noted that the

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uncertainty range achieved by these simulations increased with the cooling rate, which can also be attributed to the enhancement in thermal gradient with the cooling rate.

1

Cooling rate

Cooling rate

-1

0.8

20 Kmin

0.6

Measured Simulation

0.4 0.2

(a) 0

Transformed fraction ( f α )

Transformed fraction (fα )

1

20 Kmin-1 0.8 Measured Upper bound Lower bound

0.6

0.4

0.2

(b) 0

5860

5900

5940

Time, s

5980

6020

5840

5880

5920

5960

6000

6040

Time, s

Fig. 5.The measured and the probabilistically simulated transformed fraction as a function of transformation time for the cooling rate 20 Kmin-1 (a) an individual simulation, (b) lower and upper bounds from multiple simulations.

The transformation rates as a function of transformed fraction were determined for various applied cooling rates. The results (Fig. 6) show a single peak in transformation rate as a function of transformed ferrite fraction for all the cooling rates experiments, thus confirming normal transformation behavior [8]. The absolute value of the transformation rate increases significantly with increasing applied cooling rate from 20-220 K.min-1 (within the limit of lower and upper bound after incorporating the results from Fig. 5(b)), suggesting transformation rate is cooling rate dependent. The scatter in the transformation rate increases with increase in cooling rate. This large scatter in transformation rate observed for the applied cooling rate of 100 and 220 Kmin-1 is not an instrument effect. The accuracy of the measured length change for the applied dilatometer is about ±50 nm, which causes a relative error of ±5×10-4 in the value determined for the ferrite fraction. This error in transformed ferrite fraction introduces the same relative error in transformation rate i.e. ±5×10-4, which is significantly smaller than the observed scatter (±0.02-0.05) in 100 and 220 Kmin1 applied cooling rates, respectively. This phenomenon was also observed for Fe-2.26 at.% Mn, Fe1.79 at.% Co [8], pure Fe [9] and Fe-3.1%Ni [1] systems. The observed fluctuation in transformed ferrite fraction (see Fig. 6) might correspond to a succession of periods of acceleration and deceleration (stop and go) in the interface migration process, in correspondence with the observation by in situ transmission electron microscopy analysis [10]. This stop-and-go mechanism is due to a time and temperature dependent build up-of misfit deformation energy (because of difference in specific volume of the product (α) and the parent (γ) phase) and subsequent relaxation. In other words, this stop-and-go behavior can be attributed to the interplay of misfit deformation energy and Gibbs free energy. At smaller cooling rate, transformation rate is smaller at a particular transformed ferrite fraction (see Fig. 6) so the build up-of misfit deformation energy relaxes quickly, but as the rate of transformation increases (220 Kmin-1 cooling rate in Fig. 6) the misfit deformation energy can not be relaxed significantly because of the lack of sufficient time, thus giving rise to larger fluctuations in the transformation rate at a particular transformed ferrite fraction. Therefore, it can be concluded that the observed larger fluctuations at larger cooling rates is primarily due to lower relaxation of misfit deformation energy, which can not be attributed to the instrumental error.

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Phase Transformation and Diffusion

Transformation rate (df /dt )

0.24

-1

220 Kmin - upper bound - lower bound

0.18

0.12

-1

100 Kmin

0.06

-1

20 Kmin

0 0

0.2

0.4

0.6

0.8

1

Transformed fraction (f α )

Fig. 6. The transformation rate as a function of transformed ferrite fraction,r for different applied cooling rates. The upper and the lower curves for each applied cooling rate corresponds to the upper and lower bounds due to temperature uncertainty. The ferrite grain size distributions of the dilatometer specimens transformed under various cooling rate were determined. The normalized grain size distributions plotted in Fig. 7(a), show an invariant behavior for various cooling rates. Such an invariant nature of normalized grain size distributions has also been experimentally observed [11, 12] as well as found to be in agreement with Feltham’s model [13]. The normalized grain size distributions follow a log-normal distribution (see solid line; Fig. 7(a)). The average grain size of the fully transformed ferrite phase (at room temperature) subjected to various applied cooling rates is shown in Fig. 7(b). The average grain size shows a logarithmic dependence on cooling rate, and it decreases from about 54 to 37 µm when the cooling rate was increased from 20 to 220 Kmin-1. The decrease in average grain size with the increasing cooling rate can be attributed to the reduction in γ→α start temperature and the resultant higher nucleation rate, if site saturation for nucleation is assumed [1, 8]. (a)

60 y = -6.68Ln(x) + 74.73

80

2

R = 0.94

Grain size (µm)

Cumulative frequency .

100

60 20 K/min 100 K/min 140 K/min 180 K/min 220 K/min Log-normal fit

40

20

0 0

2

4

6

-ormalised grain size (d/dmedian)

50

40

(b) 8

30 0

50

100

150

200

250

-1

Cooling rate, Kmin

Fig. 7(a). The cumulative frequency as a function of normalised grain size along with the lognormal fit of ferrite grain-size distribution at various indicated cooling rates for Fe-5.93at%Ni, (b) the average grain size of ferrite at room temperature as a function of various applied cooling rates.

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Summary Dilatometry is becoming increasingly popular for in situ monitoring of phase transformation. However, a distinct temperature gradient occurs along the longitudinal direction of an inductively heated dilatometer specimen during heating and/or cooling, which increases with increasing heating/cooling rate. Hence, the obtained kinetic information obtained from length change measurements and corresponding transformed fraction can not be directly interpreted. In the present work, a probabilistic temperature selection approach has been proposed, which can provide the lower and upper bounds of transformed fractions. In this approach, temperature is probabilistically selected from the measured temperature limits and transformed fraction is determined using Johnson-Mehl-Avrami model. Several such simulations are aggregated to obtain the lower and upper bounds of transformed fraction. The probability approach developed in this work was applied successfully to determine the kinetics of γ→α phase transformation for Fe–5.93 at.%Ni subjected to various cooling rates. The γ→α transformation in Fe-5.93at.%Ni shows normal transformation behavior (single peak in the rate of transformation as a function of transformed fraction) for all the imposed cooling rates. Increase in cooling rate results in increasing γ→α transformation rate. The rate of transformation shows fluctuations (stop-and-go mechanism), which is ascribed to the buildup of misfit deformation energy and subsequent time and temperature dependent relaxation. The fluctuations in the rate transformations increase with increasing cooling rate. This is ascribed to the lack of sufficient time for the relaxation of the misfit deformation energy with increasing transformation rate resulting from increasing cooling rate. The normalized grain size distributions of the transformed specimens exhibited invariant nature, with reduction in mean grain size with increase in cooling rate. Acknowledgements The experimental dilatometry work was carried out at MPI-MF in Stuttgart, Germany. The first author would also like to thank Prof. E.J. Mittemeijer and Prof. F. Sommer for allowance to use these experimental data which were performed in Stuttgart under their guidance. The authors are grateful to the management of Tata Research Development and Design Centre, Pune for approving and supporting this work. References [1] G. Mohapatra, F. Sommer and E.J. Mittemeijer: Acta Mater Vol. 55 (2007), p. 4359 [2] G. Mohapatra, F. Sommer, E.J. Mittemeijer: Thermochim Acta Vol. 31 (2007), p. 453 [3] G. Mohapatra, F. Sommer, E.J. Mittemeijer: Thermochim Acta Vol. 57 (2007), p. 453 [4] ASTME112, Annual Book of ASTM Standards (1988), 03.01:297. [5] W. A. Johnson and R.F. Mehl: Trans. m. Inst. Min. (Metall.) Engs., Vol. 135 (1939), p. 416 [6] M. Avrami: J. Chem. Phys. Vol. 7 (1939), p. 1103 [7] M. Avrami: J. Chem. Phys. Vol. 8 (1940), p. 212 [8] Y.C. Liu, F. Sommer, E.J. Mittemeijer: Acta Mater Vol. 51 (2003), p. 507: [9] Y.C. Liu, F. Sommer, E.J. Mittemeijer: Philos Mag Vol. 84 (2004), p. 1853 [10] M Onink, F.D. Tichelaar, C.M. Brakman, E.J. Mittemeijer, S. Ven der Zwaag: J Mater Sci Vol. 209 (1995), p. 209 [11] S.S. Sahay, K. Joshi: J. Mater. Eng. Perform. Vol. 12 (2003), p. 157 [12] S.S. Sahay, C.P. Malhotra, A.M. Kolkhede: Acta Mater. Vol. 51 (2003), p. 339 [13] P. Feltham: Acta Metall. Vol. 5 (1957), p. 97 [14] D. A. Porter, K. E. Easterling: Phase Transformations in Metals and Alloys, 2nd ed., CRC Press, 1992

Defect and Diffusion Forum Vol. 279 (2008) pp 79-83 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.79

Recrystallization Kinetics in 17Cr 1Mo Ferritic Steel M.N.Mungole1 , P.C.Trivedi2, Satyam Sharma2, and R.C.Sharma1 1

Department of Materials and Metallurgical Engineering, IIT Kanpur 208016 2

Metallurgical Engineering Department IT-BHU, Varanasi 221005

E-mail: [email protected], [email protected] Keywords: Ferritic steel, Cold rolling, Microstructures, Recrystallization kinetics.

Abstract: The recrystallization kinetic of 17Cr 1Mo ferritic steel was studied using 60% cold rolled samples. The recrystallization was carried out at 750 , 800 and 9000C for 3, 5, 15, 30 and 60 minuets. The volume fraction of the recrystallized grains was used as a kinetic parameter. The magnitude of time exponent “n” was much less than the ideal value because of the heterogeneous recrystallization. The estimated activation energy for recrystallization “Q” was more as compared to those for pure metals. Introduction Ferritic steels containing Cr of various level with 1-2 wt% Mo are known for their improved mechanical properties and superior corrosion resistance comparable to that of 316 austenitic stainless steel [1]. They are also resistant to stress corrosion cracking. Similarly, the low nickel content in these steels provides good resistance in sulphur-containing environment at high temperature [2]. Hence, they are recommended for chemical, petrochemical, power generation and nuclear technology applications. In this family “SR” trademark steel, containing higher Cr (6-24 wt%) with 18SR “Scale Resistance” series is found capable of resisting 100 hour exposure up to 12000C [3]. Moreover, 17Cr 1Mo composition which represents a very narrow range in Schaeffer diagram is widely recommended for automobile components like exhaust systems by virtue of its excellent corrosion resistance [3]. Hence, very good formability of these steels like rolling, deepdrawing are important for above applications. Since, these cold working processes introduce large defect concentration (highly strained) and make the steel susceptible to quick failure either by mechanical or chemical/corrosion process, it becomes necessary to recrystallize-anneal. This is a process of polymorphic change in which composition and crystal structure remain the same as that of parent material and it is more stable state. The information on the recrystallization kinetics of this specific composition is not available in the literature. The present paper reports the results of an experimental study to determine kinetic parameters i.e. exponent “n” in Johnson-Mehl-Avrami equation and the activation energy “Q” in this alloy. Experimental Procedure 17 Cr 1 Mo ferritic sample in the form of 8 mm thick slab in annealed condition was obtained from Oak Ridge National Laboratory, Tennessee, USA. The chemical composition in wt % determined by spectrometer is C:0.015, Si:0.09, Mn:0.016, S: 0.5 Tm) for 3, 5, 15, 30 and 60 minutes in Argon gas atmosphere followed by air cooling. The sample coupons after the above treatment were hotmounted in Buehler USA make transonic resin and polished successively with P400, P600, P1000 and P1200 grit (FEPA) emery polishing papers.. The final polishing was carried out with velvet polishing cloth fitted on rotating wheel using 5 micron and finally with 1 micron alumina slurry.

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Phase Transformation and Diffusion

Fully polished samples were chemically etched with ferric chloride solution (anhydrous FeCl3 : 10 gms, HCl :30 ml, water: 120 ml) at room temperature using chamois leather. The microstructures were observed in the optical microscope: Zeiss: Axioskop (2 MAT) using “Axiovision AC” image analysis software. The volume fraction of the recrystallized grains in each sample coupon was estimated by a systematic point counting technique (an average of nearly 25 observations for each sample) through a graticule (square grid) fitted in the eyspiece. Results and Discussion 60 % cold rolled sample exhibited a microstructure with elongated grains along the rolling direction shown in Fig. 1. Recrystallize-annealed samples at 750, 800 and 9000C for different time have shown new stress free equiaxed grains nucleated in cold rolled region whose volume fraction increased with time at each temperature. Similarly, the recrystallized volume fraction increased with temperature after a fixed time interval. The representative micrographs of this treatment carried out at 8000C are shown in Fig. 2 (a) for 15 minutes, partially recrystallized (b) for 30 minutes, partially recrystallized and (c) for 60 minutes, fully recrystallized.

Fig. 1: Micrograph of cold rolled 17Cr 1 Mo ferritic steel.

(a)

(b)

( (c)

Fig. 2: Micrograph of recrystallized 17Cr 1Mo ferritic steel at 8000C (a) for 15 minutes, (b) for 30 minutes and (c) fully recrystallized. It can be observed in these micrographs that the volume fraction of recrystallized grains (equiaxed grains) increased with ageing time and finally after about 60 minutes the process of recrystallization is nearly completed which led to 100 % recrystallized grains structure. The complete data obtained for the samples recrystallized at 750, 800 and 900 0C is presented in Fig. 3. These plots exhibited an expected sigmoidal form of the transformation. Since, the magnitudes of these parameters (n and Q) help in understanding the mechanism of the recrystallization process the estimation of these in the present system is summarized in the following sections:

Fraction recrystallized (X)

Defect and Diffusion Forum Vol. 279

1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

81

750°C 800°C 900°C

0

10

20

30

40

50

60

Time (t) minutes Fig. 3: Variation of recrystallized volume fraction with time during isothermal transformation. (i) Exponent “n”. In a solid state transformation large number of atomic processes occurs and the rate of each atomic process is exponentially related to the activation energy/enthalpy and reciprocal of the absolute temperature [4]. During heavy cold working of the metals/alloys which introduces inhomogeneous strain, various kind of defects (vacancy, dislocations, stacking faults, shear bands, high-angle boundaries within the grains) are introduced and the free energy of the system is increased [4]. These defects act as nucleation sites during the solid state transformation e.g. recrystallization and hence the system has a tendency to go to its low energy state by making it defect free. This defect-free state is achieved by the nucleation of new defect-free lattice and the subsequent growth of these nuclei at the expense of strained parent lattice. Therefore, the rate of transformation in this process also depends on the amount of untransformed fraction. An empirical rate equation derived on this basic which follows a sigmoidal curve is represented by: Vv = 1-exp (-Ktn )

(1)

Where “Vv“ is the transformed/recrystallized volume fraction, “t” is time, “K” and “n” are constants. This is a well known Johnson-Mehl-Avrami equation [5,6,7]. To obtain the exponent “n” equation (1) can be rewritten as follow: ln ln (1/1- Vv) = ln K + n ln t

(2)

Thus plotting ln ln (1/1- Vv) against ln t for each temperature as shown in Figure 4, the slopes of the linear fit of these plots give the value of exponent “n” and the intercept “K”. The data is presented in Table 1. It is reported that the ideal value of “n” for homogeneous nucleation is 4 [4]. However, in the present case these values are much less than the ideal. The reason for this difference is obvious that the nucleation process in the present case is heterogeneous and this is because of the stored energy due to cold working process which is not uniform in the system [4]. The low value of “n” could also be due to decreasing nucleation and growth rate with time [4]. The micrographs in figure 2 (a and b) confirmed this where the recrystallized grains are not uniformly distributed. Belyakov et al. has observed such heterogeneous nucleation in the recrystallization process of 22Cr 3 Ni ferritic stainless steel [8]. It is observed (Table 1) that the exponent n decreased and the constant K increased with increase in temperature

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Phase Transformation and Diffusion

3

o

750 C o 800 C o 900 C

2

ln [ln(1/(1-x)]

1 0 -1 -2 -3 o

n(750 C)=1.50 o n(800 C)=1.46 o n(900 C)=1.23

-4 -5 3

4

5

6

7

8

9

ln t

Fig. 4: Linear variation of ln ln (1/1-Vv) with ln t for recrystallization. Table1 The exponent “n” and constant “K” obtained at various temperatures. 750oC N K

800oC

1.50 3.92x10

900oC

1.46 -6

3.14x10

1.23 -5

6.17x10-4

(ii) Activation energy “Q”. The process of recrystallization in the cold worked metal/alloy is an activated process where a transformation involves creation of new strain-free lattice at the expense of its strained lattice. The activation energy of recrystallization does not refer to any unique atomic process, rather, it is related to many different atomic processes and hence, Q is considered as an empirical constant [9]. It is determined experimentally using recrystallized volume fraction and the corresponding time period. It is assumed that the mechanism of recrystallization remained the same over the temperature range. This yields the temperature dependence of the transformation rate and the process is known as strain-induced boundary migration. In the present case the activation energy is estimated by “the rate constant technique” where rate constant k is related to effective activation energy Q in the equation [5]: k = ko exp(-Q/RT)

(3)

where ko is a constant, R and T have their usual meanings. Knowing K and n of the equation (1) above the rate constant k is obtained using the relation: k = (K)1/n

(4)

Based on the eq. (3), a logarithmic form known as Vent’s Hoff plot between ln k against reciprocal of the absolute temperature (1/T) is obtained and is shown in Fig. 5. The slope of the linear fit to data is equal to Q/R and “Q” is calculated using the known value of gas constant R (8.34 J/mol). The estimated value of “Q” is 150 kJ/mol. This is well comparable to that of stainless steel

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(114kJ/mol) reported by K.Kato et al.[10] .The presence of the alloying elements Cr, Mo, V etc in alloys make the process of recrystallization more complex which influence the atomic movement in the lattice / the migration of the grain boundaries resulting its activation energy to be more than that of pure metals.

-6.0

-6.5

ln k

-7.0

-7.5

-8.0

-8.5 8.4

8.6

8.8

9.0

9.2

9.4

9.6

9.8

-4

(1/T) X 10

Fig. 5: Plot of logarithm of the rate constant ln k versus reciprocal of absolute temperature 1/T for recrystallization. Conclusions Recrystallization kinetics in cold rolled 17 Cr 1 Mo ferritic steel exhibited the magnitude of exponent n much lower than 4, indicating heterogeneous nucleation dominating in the recrystallization process. The activation energy Q is found to be more than those of pure metals because of the presence of the alloying elements which make the process of recrystallization more complex. References [ 1] H.F.G. de Abreu, A.D.S. Bruno, S.S.M. Tavares, R.P. Santos and S.S. Carvalho: Materioals Characterization, vol. 57 (2006), p. 342 [ 2] P. J. Grobner: Metall Trans, vol. 4 (1973) p.251 [ 3] Joseph A.Douthett: Proc. of an Int. Conf. on Production, Fabrication, Properties and Application of Ferritic Steels for High Temperature Applications, ed. by A. K. Khare, American Society of Metals, Metal Park, Ohio, (1982), 296. [ 4] A.K.Jena and M.C.Chaturvedi, Phase Transformation in Materisls, Prentice Hall, Englewood Cliffs, New Jersey, 1992, p.240. [ 5] Romesh C.Sharma, Phase Transformations in Materials, CBS Publishers, New Delhi, 2002, p.163. [ 6] W.A.Johnson and R.F.Mehl: Trans. AIME, Vol.135 (1939) p.416 [ 7] M.Avrami: J.Chem. Phy., Vol.7 (1939) p.1103; Vol.8 (1939) p.416 [ 8] A.Belyakov, Y.Kimura and K. Tsuzaki: Materials Science and Engineering A, Vol. 403 (2005), p. 249 [ 9] R.E.Reed-Hill, Physical Metallurgy Principal, East-West Edition, New Delhi 1964, p. 267. [10] K. Kato, Y. Saito and T. Sekai : Transactions of the Iron and Steel Institute of Japan, Vol. 24 (1984), p. 1050

Defect and Diffusion Forum Vol. 279 (2008) pp 85-90 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.85

Effect of Thermal Aging on the Transformation Temperatures and Specific Heat Characteristics of 9Cr-1Mo Ferritic Steel B. Jeya Ganesh, S. Raju*, E. Mohandas, and M. Vijayalakshmi .

Physical Metallurgy Division Indira Gandhi Centre for Atomic Research, Kalpakkam, 603 102, India E-mail: *[email protected] Key Words: Ferritic Steel; Specific Heat, Phase Transformation, Calorimetry

Abstract: The effect of thermal ageing on the heat capacity and transformation behaviour of behaviour of 9Cr-1Mo-0.1C (wt.%) ferritic / martensitic steel has been studied using differential scanning calorimetry (DSC) in the temperature range 473 to 1273 K. It is found that α-ferrite + carbide → γ-austenite phase transformation temperature is only mildly sensitive to microstructural details; but the enthalpy change associated with this phase transformation and especially, the change in specific heat around the transformation regime are found to be dependent on the starting microstructure generated by thermal ageing treatment. Prolonged ageing for about 500 to 5000 hours in the temperature range 823 to 923 K contributed to a decrease in heat capacity, as compared to the normalised and tempered sample. The martensite microstructure is found to possess the lowest room temperature CP among different microstructures. Introduction Although, the physical metallurgy of standard 9Cr-Mo steels is replete with many experimental and theoretical investigations on phase stability and microstructure evolution during heat treatments [1-14], there seems to be a general paucity of experimental data on various thermodynamic quantities [13]. It is clear that a consistent thermodynamic data set is essential from the point of view of understanding as well as predicting the long-term phase stability of alloys [3, 9-11]. The microstructure of ferritic steels being very sensitive to heat treatments [8, 12], it is probable that the thermodynamic quantities measured there of exhibit certain microstructural dependence, in direct relation to the phase fractions of different microstructural constituents present in the alloy. A preliminary survey of literature on this account has revealed that the effect of different starting microstructures on the heat capacity of high chromium steels, especially the enthalpy changes associated with phase transformations has not been investigated so far. In view of this fact, it is planned to study the effect of typical ageing treatments on the transformation temperatures and heat capacity of a nuclear grade 9Cr-1Mo steel using high temperature differential scanning calorimetry. The results of this study are reported in this paper. Experimental details The chemical composition (wt. %) of the steel used in the present study is as follows: 8.44 Cr, 0.94 Mo, 0.49 Si, 0.46 Mn, 0.17 i, 0.11 Cu, 0.1 C, 0.008 P, 0.011 Al, 0.002 S, 0.001V, 0.008 ; balance being Fe. The alloy has been supplied in plate form in normalised and tempered (N&T) condition (austenitisation at 1323 K for 15 minutes, followed by air cooling to room temperature). Subsequent tempering is carried out at 1053 K for 2 hours, and the tempered sample is air-cooled to room temperature. In the present study, a set of samples is further subjected to the following thermal ageing schedules: (i). 823 K for 500 hours, (ii). 823 K for 5000 hours, and (iii). 923 K for 500 hours.

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Phase Transformation and Diffusion

In order to study the thermal effects that accompany the tempering reaction of a fully martensitic microstructure, a few samples are directly quenched from 1323 K /1050 0C, to obtain fully martensitic microstructure. The samples were prepared in the standard manner for metallographic characterisation. In Fig.1, the scanning electron micrographs of various aged samples are illustrated. The average value of the lattice parameter of the α-ferrite phase is found to be 0. 287 ± 0.003 nm. It may be mentioned that within the limits of experimental uncertainty, the lattice parameter of ferrite phase did not exhibit any noticeable microstructural dependence. The samples for DSC experiments were sliced from the heat-treated plate using diamond coated wire saw. These were further cleaned and polished to regular and nearly identical shapes of mass varying from 50 to 100 ± 0.1 mg. The DSC experiments were performed with Setaram Setsys 16® heat-flux type high-temperature differential scanning calorimeter, employing recrystallised alumina crucibles of about 100µ L volume. The description of the equipment and the calibration procedure has been discussed in our previous publication [15]. The temperature calibration is performed using recommended high pure melting point standards, namely, Sn, Al, Pb, In and Au. In addition, the measurement of enthalpy of α (bcc) → γ (fcc) allotropic transformation in pure iron (Aldrich, impurities ≤ 80 ppm) under identical experimental conditions is also employed for the heat flow calibration. The measured transformation temperatures are accurate to ± 2 K; the transformation enthalpies are accurate to ± 5 % at 10 K min-1. 20 µ

20 µ

b)

a) 20 µ

d)

20 µ

c) 20 µ

e)

f)

g) a) N & T b) 823 K/ 500 h aged c) 823 K / 5000 h aged d) 923 K / 500 h aged e) Quenched (Martensite) f) Quenched from 898K g) Quenched from 973K

Fig.1. SEM (a-e), optical (f&g) micrographs of 9Cr-1Mo samples A typical heat capacity measurement using DSC in the continuous heating mode involves recording of at least three consecutive experimental runs under identical heating, holding and cooling schedule [16, 17]. In the present study, the experimental schedule consisted of heating the system from room temperature to an initial temperature of 473 K at the rate of 10 K min-1 and holding at this temperature for about 15 minutes. This is followed by the actual programmed heating, holding and cooling schedules. In the present study, the sample is heated at a steady rate of 10 K min-1 to 1273 K, followed by an isothermal hold of about 15 minutes at this temperature. Subsequently, the sample is cooled to 473 K at 10 K min-1 and is allowed a resident time of 15 minutes at this temperature, before it is finally cooled again to room temperature. The three basic DSC runs that constitute a CP measurement are [16, 17]: The three basic DSC runs that constitute a CP measurement are [16, 17]: (i). the baseline run employing only identical empty crucibles on both sides of the DSC plate, (ii). the reference or calibration run with the sample crucible containing pure iron of known mass, and (iii). the sample run with a known mass of ferritic steel loaded onto the sample crucible. The following formula, based on the method of ratios has been used to calculate CP, from the three run DSC data [17]. CPS = CPR × (mR/mS) × {(µS - µb)/(µR - µb)}.

(1)

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In the above expression, CPS and CPR represent the specific heat of the sample and the reference whose masses are given by mS and mR respectively. µS is the microvolt DSC signal obtained with the sample, µR is the corresponding signal obtained with the reference or standard material and µb is the baseline signal obtained with empty crucibles. Results

4

TP 2

3

1

endo

DSC output (a.u.)

(i) Principal transformation characteristics. In Fig 2, typical DSC profiles obtained during the heating cycle with the steel sample, pure iron and with empty crucibles (base line) are presented. In order to illustrate the role of starting microstructure on the nature of the DSC thermogram, the profile of the normalised and tempered (N&T) sample is compared with that of the near 100 % martensite microstructure. It is clear from Fig. 2, that in the temperature range of interest, two wellresolved endothermic peaks characterise the DSC profiles of pure iron and N&T steel sample.

TC

TC

1 : normalised and tempered 2 : quenched from 1273 K : martensite 3 : Pure Iron 4 : Baseline

α-γ α-γ

500

600

700

800

900

1000 1100 1200

Temperature (K) Fig. 2. On-heating DSC profiles for 9Cr-1Mo steel The first peak marked as TC comes from the ferromagnetic to paramagnetic transformation, while the second one is due to the α-ferrite + carbides → γ-austenite, phase change. In literature, it is conventional to represent the onset temperature of the α-ferrite+carbide→γ-austenite transformation as Ac1 and the offset point as Ac3. It is also noteworthy that in the case of steel sample, the Curie point and the onset (Ac1), peak (Acp) and finish (Ac3) temperatures for the α → γ structural transformations are lower than that of pure iron values. In case of martensite, an extra exothermic peak is observed at about 920 K (Fig. 2). This peak corresponding to the precipitation of M23C6 type carbide from virgin martensite [20] is however absent in the DSC profiles of tempered and aged samples. This is because of the fact that the carbide precipitation had already been realised to full measure in these samples during prior tempering and subsequent long ageing treatments. The enthalpy changes associated with the magnetic transition, α→ γ structural change and M23C6 carbide precipitation reactions are directly obtained from the DSC profile as the total or integrated area of the respective peak. It is reasonable to assume in the case of pure iron that for a heating rate of about 10 K per minute, the α→ γ allotropic transformation goes to near completion; but in the case of high chromium steels, the presence of carbides and its sluggish dissolution in austenite, results in an incomplete realisation of the α→ γ transformation at the transformation offset Ac3 temperature [14,19,20]. Because of this fact, there is a mild under estimation of the true

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Phase Transformation and Diffusion

enthalpy change that is associated with austenite formation reaction under equilibrium conditions. In Table 1, the α→γ transformation onset (Ac1) and offset (Ac3) temperatures and the associated transformation enthalpy (∆Htr) values measured in the present study are listed. Table 1 α→γ transformation onset (Ac1) and offset (Ac3) temperatures and the associated transformation enthalpy (∆Htr) values

Sample Designation:

TC (K)

Ac1 (K)

Acp α→γ peak temperature

Ac3 (K)

∆Hα →γ

(Jg-1)

(K) Pure Iron

1040

1156

1184

1200

16

9Cr 1Mo : N&T 9Cr 1Mo : 823 K (550 0C) / 500 h 9Cr 1Mo : 823 K (550 0C) /5000 h 9Cr 1Mo: 923 K (650 0C) / 500 h

1013 1014 1015 1015

1094 1095 1099 1100

1115 1115 1116 1117

1143 1146 1150 1148

15 12 13 13

9Cr 1Mo : Martensite

1011

1092

1112

1145

15

Tps = 881

Tpeak = 920

Tpf = 947

- 5.6

Carbide precipitation peak data in martensite

The Curie (TC) temperature turns out to be relatively insensitive to the effects of ageing (Table 1). However, Ac1 and Ac3 values show slight microstructure sensitivity. In general terms, ageing serves to increase the austenite start (Ac1) temperature. It is also instructive to note that in the case of martensite microstructure, the width of the α → γ transformation zone as measured by Ac3-Ac1, is the highest recorded in the present study. This suggests that for a given heating rate, the presence of considerable strain in the starting martensitic microstructure serves to retard the overall kinetics of reaustenitisation reaction upon heating. The measured enthalpy change ∆Htr associated with αferrite + M23C6 → γ -austenite transformation in 9Cr 1Mo steel is slightly smaller than the corresponding α → γ value for pure iron (see, Table 1). ( ii) ageing induced changes in CP. In Fig. 3, the measured CP data for different aged samples of 9Cr 1Mo steel are collated together. Although, the qualitative nature of the CP curve remains much the same for all samples other than the one that had martensite as the starting microstructure, the following specific points deserve special mention. Not withstanding the maximum uncertainty of about ± 5 %, the CP of the normalised and tempered (N&T) sample is distinctly higher than that of the aged ones. Further, among the aged samples, the 923 K - 500 h aged ones have higher heat capacity than the 823 K aged one. An appeal to their respective microstructures (Fig. 1) suggests that the 923 K ageing treatment has resulted in a refinement or coarsening of the ferrite grain size, besides a marginal increase in carbide density. In a similar vein, an increase of ageing time from 500 to 5000 hours at 823 K has served to decrease CP by a small amount. Thus, it may be inferred in general terms that prolonged ageing, accompanied by a concomitant increase in carbide volume fraction induces a decrease in CP. The heat capacity of martensite on the other hand, starts with a lower value, but quickly rises upon approaching Tps, the M23C6 precipitation start temperature. At Tps, the carbide precipitation causes a drop in the specific heat, and on reaching Tpf, the apparent

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precipitation finish temperature, the CP increases once again to catch up with the 923 K aged sample.

(α + carbides)

1750 9 Cr-1Mo steel

γ

specific heat scenario

-1

-1

CP (J kg K )

1500 1250 1000

f a - Normalized & Tempered 0 b - 550 C / 500 h 0 c - 550 C / 5000 h 0 d - 650 C / 500 h e - Martensite structure f - Pure Iron

TC

a

d

750

Tpf

e

b c

500

Tps Ac1

250 600

750

900

Ac3

1050

Temperature (K) Fig. 3: CP - T for pure iron and aged 9Cr-1Mo samples In order to assess the nature of microstructure that existed just around the carbide precipitation temperature, a martensite sample was heated at the rate of 10 K min-1 up to 898 K in DSC, kept at this temperature for 1 second and then fast cooled to room temperature at 99 K min-1. It must be mentioned that the carbide precipitation start temperature (Tps) is about 881 K, and therefore it is clear that at 898 K, the carbide precipitation would have just begun. In a similar fashion, another martensite sample was heated up to 973 K, a temperature slightly ahead of the carbide precipitation finish temperature (Tpf = 947 K) and is rapidly cooled at 99 K min-1 to room temperature. The optical micrographs of these two quenched-in samples are illustrated in figures 1(f) and 1(g) respectively. While the sample quenched from 898 K is somewhat featureless and revealed barely any sign of precipitation, the sample quenched from 973 K contained copious amount of carbides. However, the ferrite matrix grain structure has not been very well refined yet. Discussion In general terms, the observed behaviour is in-line with the expected microstructural changes that take place in these materials upon continuous ageing. Thus taking the case of martensite as an example, the initial response is one of relaxation of highly dislocated and hard lath martensitic microstructure resulting in the gradual development of strain free ferrite matrix characterised by the development of low energy grain and subgrain structures. This is a continuous process and is characterised by a relatively broad exothermic smearing of the DSC profile. No sharp peak as is characteristic of a precipitation reaction will be evident for martensite relaxation. This is followed by carbide precipitation at higher temperatures. Phase stability calculations as well as experimental results on long term aged samples in generic 9Cr 1Mo steels reveal that its is mostly the M23C6 type carbide that comes out under equilibrium conditions at about 650 0C upon heating and the corresponding equilibrium fraction of the carbide precipitate is rather small [11]; thus the net CP of

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Phase Transformation and Diffusion

a low carbon tempered martensitic-ferritic steel is derived predominantly from the matrix ferrite phase only. Nevertheless, the presence of carbides and their morphology do play a significant role in deciding the kinetics of austenite formation and hence in determining the rate of CP change, especially around the phase transformation region. The carbide precipitation being a diffusion-controlled process is somewhat sluggish at low temperatures and requires about 923 K to get initiated. The rate of heating plays a crucial role in the complete realisation of the carbide formation and its subsequent dissolution at high temperatures, that is, upon reaching Ac3. The slower the rate of heating, better are the chances for the system to simulate the equilibrium transformation behaviour. In this sense, the prolonged isothermal ageing at 823 or 923 K, temperatures which are lower than Ac1, does indeed help in establishing the equilibrium volume fraction of the minor carbide phases, with of course the equilibrium composition. This probably is the reason for the comparatively lower values of CP recorded for 823 K/5000 h aged samples, because in this case the carbide particle density should have attained the maximum possible value. Another noteworthy outcome of this study is the estimation of the enthalpy of carbide precipitation from martensitic microstructure. In the present study, a value of about 5.6 J g-1, which roughly translates into 302 J mol-1 The enthalpy change associated with the α→ γ transformation in aged samples is slightly less than the corresponding value recorded for pure iron and this is certainly due to the presence of the carbide phase that serves to retard the kinetics of α→ γ structural change [20]. In case of martensite, the carbide volume fraction formed during continuous heating being relatively less, the α→ γ structure change is facilitated to a little more extent with the concomitant effect of recording a slightly higher value for enthalpy change (Table 1). References [1] J. Orr, F. R. Beckitt, G. D. Fawkes, in : Ferritic steels for fast reactor steam generators, (Eds.) S. F. Pugh and E. A. Little, BNES (1978), London, 91-109. [2] S. J. Sanderson, in : Ferritic steels for fast reactor steam generators, (Eds.) S. F. Pugh and E. A. Little, BNES (1978), London, 120-127. [3] A. Bjarbo and M. Hatterstrand: Mater. Trans., Vol, 32A (2001), p. 19. [4] D. V. Shtansky and G. Inden: Acta Metall., Vol. 45 (1997), p. 2862 [5] M. Taneike, K. Sawada, F. Abe: Metall. Met. Trans., Vol. 35A (2004), p. 1255 [6] N. Fujita and H. K. D. H. Bhadeshia: ISIJ Int., Vol. 41 (2001), p. 626 [7] H. K. D. H. Bhadeshia, ISIJ Int., Vol. 41 (2001), p. 626 [8] J. M. Vitek and R. L. Klueh: Metall. Trans., Vol. 14 A (1983), p.1047 [9] A. Kroupa, J. Havrankova, M. Coufalava, M. Svoboda and J. Vrestal: J. Phase Eq., Vol. 22 (2001), p. 312 [10] J. Hald and L. Korcakova: ISIJ Int., Vol. 43 (2003), p. 420 [11] A. Schneidner and G. Inden: Acta Mater., Vol. 53 (2005), p. 519 [12] V. Vodaraek and A. Strang: Mater. Sci. Tech., Vol. 16 (2000), p. 1207 [13] S. Raju, B. Jeyaganesh, A. Banerjee and E. Mohandas: Mater. Sci. Engg., Vol. A465 (2007), p. 29 [14] D. V. Shtansky, K. Nakai and Y. Ohmori: Z. Metallkd., Vol. 90 (1999), p. 25 [15] S. Raju, N. S. Arun Kumar, B. Jeyaganesh, E. Mohandas and U. Kamachi Mudali: J. Alloys Compd., Vol. 440 (2007), p. 173 [16] Teresa M. V. R. de Barros, Rui. C. Santos, Anabela C. Fernandes and M. E. Minas da Piedabe: Thermochim. Acta, Vol. 269/270 (1995), p. 51 [17] M. J. Richardson, in : K. D. Maglic, A. Cezairliyan, V. E. Peletsky (Eds.), Compendium of Thermophysical Property Measurement Techniques, vol. 2, Plenum Press, New York, 1992, p.519. [18] B. Nath, E. Metcalfe, J. Hald, in: A. Strang, D. J. Gooch (Eds.), Microstructural Development and Stability in high Chromium Ferritic Steels, The institute of Materials, London, 1997, P. 123 [19] U. R. Lenel: Scripta Metall., Vol. 47 (1983), p. 471 [20] U. R. Lenel and R. W. K. Honeycombe: Metal. Sci., Vol. 18 (1984), p. 201

Defect and Diffusion Forum Vol. 279 (2008) pp 91-96 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.91

Gibbs Free Energy Difference in Bulk Metallic Glass Forming Alloys Heena Dhurandhar1,a, Kirit N. Lad1, Arun Pratap1,b and G.K.Dey2 1

Condensed Matter Physics Laboratory, Applied Physics Department, Faculty of Technology & Engineering, M S University of Baroda, Vadodara - 390 001, India. 2

Materials Science Division, Bhabha Atomic Research Centre, Mumbai – 400 085, India. a

email: [email protected], bemail: [email protected]

Keywords: Gibbs free energy, bulk metallic glass, specific heat difference, under cooled region

Abstract. The Gibbs Free Energy Difference between the solid and liquid phases (∆G) is related to nucleation frequency and has played an important role in predicting the glass forming ability (GFA) of multicomponent metallic alloys. This is due to the fact that the maximum energy for nucleus formation i.e. the activation barrier for nucleation has an inverse square relation with ∆G. The Gibbs Free Energy Difference of three multi-component bulk metallic glasses namely Mg65Cu25Y10, Zr57Cu15.4Ni12.6Al10Nb5 and Zr52.5Cu17.9Ni14.6Al10Ti5 have been evaluated using two new expressions. The results show that the ∆G values calculated assuming ∆Cp to be constant lie closer to the experimental values for the Mg based system while in the case of two Zr based systems, ∆G computed using the hyperbolic variation of ∆Cp show improved agreement with the experimental data. Introduction The Gibbs Free Energy Difference between the solid and liquid phases (∆G) gives a qualitative measure of the stability of the glass phase. In fact, ∆G on crystallization of undercooled systems is an important parameter in nucleation process. The nucleation frequency has an exponential dependence on ∆G, and hence the estimation of ∆G is often critically important in the investigation of the thermal stability of amorphous phase through the study of crystallization kinetics. Bulk metallic glasses (BMG’s) are novel class of engineering materials, which possess excellent corrosion resistance, extremely high mechanical strength and have fairly good thermal stability. Gibbs free energy difference (∆G) between under-cooled liquid and the corresponding crystalline solid acts as the driving force of crystallization and it is a vital parameter in nucleation and crystal growth. In an alloy system, less ∆G indicates less driving force of crystallization, enhanced stability of super-cooled liquid and better GFA. So, estimation of ∆G is an important parameter in governing the relative thermal stability of metallic glass as well as their kinetics of crystallization. ∆G for three multi-component bulk metallic glasses namely Mg65Cu25Y10 [1], Zr57Cu15.4Ni12.6Al10Nb5 and Zr52.5Cu17.9Ni14.6Al10Ti5 [2] have been evaluated using two expressions with one assuming constant ∆Cp (heat capacity difference of under cooled liquid and solid) and other assuming a hyperbolic variation (∆CpmT/Tm) of ∆Cp. Formulation The difference in Gibbs free energy between the liquid and crystalline phases is given by ∆G = ∆H − T ∆S

(1)

92

Phase Transformation and Diffusion

where Tm

∆H = ∆H m − ∫ ∆C p dT

(2)

T

and Tm

∆S = ∆S m − ∫ ∆C p T

dT T

(3)

with ∆S m =

∆H m Tm

(4)

∆Cp is defined as C pl − C px and it is the difference between the specific heats of undercooled and corresponding crystalline phases of a material. Experimental ∆G can be calculated using equations (1)-(3). In case of non-availability of ∆Cp data, one may choose an expression which satisfactorily explains the temperature dependence of ∆Cp. Most of the expressions available in the literature [310] for estimating the thermodynamic parameters assume ∆Cp to be constant in the entire undercooled range. Hence, substituting ∆H and ∆S from Eq. 2 and 3 in Eq. 1 provides ∆H m T   ∆T + ∆C p T ln m − ∆T  Tm T  

∆G =

(5)

Treating ∆Cp to be constant in the undercooled region and taking ∆S = 0, one can easily get at T = T K, ∆C p = α

with α = ln

∆H m Tm

1 Tm

(6)

TK

substituting Eq. 6 in Eq. 5 it simplifies to

∆G =

∆H m Tm

Tm   α T ln T + ∆T (1 − α )   

(7)

The value of α can be evaluated by taking the constant value of ∆Cp = ∆Cpm at melting point in Eq. 6 as

α=

∆C pmTm ∆H m

(8)

However, the constant ∆Cp assumption shows large deviations in the ∆G values for the systems in which the specific heat difference varies appreciably with temperature. Hence, to compute ∆G accurately one must know the variation of ∆Cp with temperature. In most of the glass forming systems the specific heat difference increases with undercooling, so for such kind of systems ∆Cp at any temperature in the undercooled region can be expressed as

Defect and Diffusion Forum Vol. 279

∆C p =

∆C pmTm T

93

(9)

Substituting ∆Cp from Eq. 9 in Eqs. 2 & 3 and using Eq. 1, the ∆G values for such systems can be represented by the following equation

∆G =

 T ∆H m ∆T ∆T  − ∆C pmTm ln m −  Tm  T Tm 

(10)

The general formula derived by Thompson and Spaepen (TS) [6] is given by

∆G =

 (1 − α )Tm + (1 + α )T  ∆H m ∆T   Tm Tm + T  

(11)

This expression is derived by using the following approximation for logarithmic term in Eq.(5)

 T  2∆T ln  m  ≅  T  Tm + T

This expression used is strictly valid only for small ∆T. When α = 1, the TS expression reduces to

∆G =

 2T  ∆H m ∆T   Tm  Tm + T 

(12)

Due to the approximation of the logarithmic term only up to the first term, both the TS expressions show large deviation from experimental values at large undercooling. For multicomponent metallic glasses, which exhibit larger undercooling, the deviation in ∆G is quite appreciable. Hence, Lad et al deduced a new expression for ∆G by expanding the logarithmic function upto the second term and thus getting

ln

Tm 4T ∆T = T (Tm + T ) 2

The corresponding expression given by Lad et al [10] is

 ∆H m ∆T   4T 2  ∆G =   2  Tm   (T + Tm ) 

(13)

Dubey and Ramachandrarao [11] have derived the following expression for ∆G, using the hole theory of liquids

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Phase Transformation and Diffusion

∆C pm (∆T ) 2  ∆T  ∆G = ∆S m ∆T − 1 − 6T  2T

(14)

It may be noted that all the expressions given by previous workers consider relatively small undercooled region (∆T=Tm-T) and hence approximate the ln(Tm/T) using Taylor series expansion either upto first or second term. This limits the applicability of their expressions to small undercooled region and hence shows deviation in theoretically calculated values from experimental points in the large undercooled region particularly possessed by bulk metallic glasses.

Results and Discussion Gibbs free energy difference of experimental data and the calculated values with Eqs. 7 and 10 for amorphous alloys of Mg65Cu25Y10, Zr57Cu15.4Ni12.6Al10Nb5 and Zr52.5Cu17.9Ni14.6Al10Ti5 are shown in Figs. 1-3. For comparison, ∆G values are also calculated using equations given by Thompson and Spaepen [6], Lad et al [10] and Dubey and Ramchandrarao [11] for all the three systems.

Fig.1: ∆G as a function of T for Mg65Cu25Y10 As evident from the Fig.1 the results obtained using the present expression given by Eq. 7 seems to be more consistent with the experimental data for Mg based system in almost entire undercooled region. It can be observed that our results lie closest to the experiment as compared to the other plots of Thompson & Spaepen, Lad et al and Dubey and Ramchandrarao. The present approach not only considers the assumption of constancy of ∆Cp but also avoids approximating the logarithmic term ln(Tm/T) as done by Lad et al [10]. The difference made by the approximation of the logarithmic term is quite appreciable, which can be seen from the amount of deviation of the results through the Eq. 13. Assuming ∆Cp constant provides ∆G values close to the experiment and hence it appears that ∆Cp does not increase appreciably for Mg based system. This is further corroborated by

Defect and Diffusion Forum Vol. 279

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∆G values obtained using Eq. 10, with hyperbolic variation of ∆Cp in the entire undercooled region which show large deviation from the experimental curve for this system.

Fig.2: ∆G as a function of T for Zr57Cu15.4Ni12.6Al10Nb5

Fig.3: ∆G as a function of T for Zr52.5Cu17.9Ni14.6Al10Ti5

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Phase Transformation and Diffusion

However, for the Zr57Cu15.4Ni12.6Al10Nb5 system the ∆G values obtained using the present Eq. 10, which assumes hyperbolic variation of ∆Cp, are in excellent agreement with the experimental data, Fig. 2. The values from the equation of Lad et al also lie nearer to the experimental curve while those obtained for both TS expressions show very large deviations. Incidentally, the expression of DR also gives consistent result for this system. In the case of third system i.e. Zr52.5Cu17.9Ni14.6Al10Ti5, the calculated ∆G values either under estimate viz. Lad et al and TS(gen.) values or overestimate as observed from present results, DR and TS(α =1) plots. This deviation of the theoretical curves from experiment may be attributed to polynomial nature of ∆Cp variation and a one parameter (∆Cpm) expression for temperature dependence of ∆Cp may not be sufficient to account for the accurate temperature variation. Efforts are on to derive the Kauzmann temperature, TK through the difference in the heating rate variation of Tg and Tx using DSC.

References [1] R. Busch, W. Liu and W. L. Johnson: J. Appl. Phys., Vol.83 (1998), p. 4134. [2] S. C. Glade, R. Busch, D. S. Lee and W. L. Johnson: J. Appl. Phys., Vol 87 (2000), p. 7242. [3] D. Turnbull: J. Appl. Phys., Vol. 21(1950), p. 1022. [4] J. D. Hoffman: J. Chem. Phys., Vol. 29(1958), p. 1992. [5] D. R. H. Jones and G. A. Chadwick: Phil. Mag., Vol. 24(1971), p. 995. [6] C. V. Thompson and F. Spaepen: Acta Metall., Vol. 27(1979), p. 1855. [7] H. B. Singh and A. Holz: Solid State Commun., Vol. 45(1983), p. 985. [8] L. Battezzati and E. Garrone: Z. Metallkunde Vol. 75(1984), p. 305. [9] K. N.Lad, Arun Pratap and K. G. Raval: J. Mater. Sci. Lett. Vol. 21(2002), p. 1419. [10] K. N. Lad, K.G.Raval and Arun Pratap: J. Non-Cryst. Solids, Vol. 334 & 335(2004), p. 259. [11] K. S. Dubey and P. Ramchandrarao: Acta Metall., Vol. 32(1984), p. 91

Defect and Diffusion Forum Vol. 279 (2008) pp 97-103 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.97

Effect of Cerium Addition on the Microstructure And Mechanical Properties of Al-Zn-Mg-Cu Alloy A.K. Chaubey, S. Mohapatra, B.Bhoi, J. L. Gumaste, B.K. Mishra & P.S. Mukherjee Institute of Minerals and Materials Technology (IMMT), Bhubaneswar-751013 Email: [email protected] Key words: Precipitation hardening, microstructure, grain size, Cerium, strength

Abstract. Al–8.6Zn–2.6Mg-2.4Cu–xCe(x = 0–0.4 wt. %) alloys were prepared by metal mould casting method, the effects of Ce on the microstructure and mechanical properties of the alloys were investigated. The results showed that the dendrite as well as grain size were refined by the addition of Ce, and the best refinement was obtained in 0.25 wt % Ce containing alloy. The main phases in the as-cast alloys were α-Al, Mg-Zn32, Mg32 (Al, Zn)49, and Al4Ce phase was found in the alloys containing more than 0.1%Ce. The addition of Ce improved the mechanical properties of the alloys. The strengthening mechanism was attributed to grain refinement and compound reinforced Introduction Over the last decade several researchers[1-5] have carried out extensive work on the production of high strength aluminum alloys by micro alloying with Ce, Zr, Sr, La, Sc etc. Aluminum alloys modified with addition of rare earth metals have shown promising mechanical and corrosion properties. These rare earth alloy additions serve dual purpose: reduces grain size and clean the melt by reducing the impurities. The mechanical properties of the alloy can be further improved by controlled cooling and appropriate heat treatment. The favorable mechanical properties of the alloy deteriorate if the size of the precipitates exceeds the range by a few nanometers. Growth of crystallites can be retarded by the choice of the alloy composition that leads to higher activation energy of crystallization. According to the recent reports, there are two effective ways to improve the elevated temperature mechanical properties of Al-Mg based alloy: (1) adding rare earth and/or alkaline earth elements to the alloy. These elemental additions can effectively refine the structure of the alloy and the high melting point, heat resistant intermetallics is formed, thereby restraining the sliding of the dislocation and grain-boundary[6-9]. (2) Adding high Zn content. The density of the melt increased with Zn increasing, and the anti-oxidizing flux will always float at the melt surface, hence the protection would really operate. Moreover, high Zn addition decreases the Mg17Al12 phase and brings into the formation of heat resistance phases, such as MgZn2, MgZn and Mg32(Al, Zn)49. Zhang et al. [10] suggested that the Mg17Al12, Mg32 (Al, Zn) 49 and Mg–Zn binary phases may exist in the Mg–Al–Zn alloys according to the different Zn/Al ratio. Zeng et al.[11] reported that the tensile strength of the Al–Zn-Mg alloy would be influenced by the microstructure and work hardening effects. Rare earth elements (RE) are successfully used to improve the elevated temperature properties of some commercial Al-Mg alloys. The present work describes the influence of cerium (Ce) on microstructure and mechanical properties of Al–8.6Zn–2.6Mg-2.4Cu (C912) alloy. Experimental The alloys were prepared by liquid metal casting technique by melting high purity aluminum ingot (99.70%), industrially pure zinc (99.9%), industrially pure magnesium (99.9%), and master alloy

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A1-6.2Ce. The chemical composition of the alloys was tested by weight chemical method and the results are shown in Table 1. Table 1 Chemical composition of alloys (mass fraction, %) S. No. Cu

Mg

Zn

Fe

Ce

1

2.4

2.14

7.75

0.12

0.00

2

2.35

2.1

8.25

0.14

0.10

3

2.34

2.4

8.18

0.13

0.20

4

2.45

2.45

8.15

0.10

0.30

5

2.25

2.18

8.3

0.12

0.40

Al Balance

Pit -type electrical resistance furnace was used to melt the alloys at 780°C. The melt was cast by a 100 mm cube steel mould. The as-cast ingots were homogenized at 4500C for 4 hour, then hot forging was carried out on 3.15 T hammer with temperature range from 410 – 3500C. The hot working ratio is approximately 9.00. The forged ingot was solution treated at 4700C for 2 hour and then double aging was carried out 1070C (6hour) and 1770C (6hour) and subsequently air cooled. The hardness was measured by Brinell Hardness Machine of 1000 kg load 10 mm ball diameter. both in homogenized & after heat treatment condition and the hardness values of the alloys are shown in Table-2. Tensile testing was carried out on 40T Universal Testing Machine and the results are shown in Fig-6.The samples for metallographic examination were polished and observed in NIKON- LV-150 upright metallographic microscope. Phase analysis of as-cast alloys was completed on DMAX3C rotary-target X-ray diffractometer. The distribution of the second-phases in as-cast alloys was observed on HITACHI S-650 scanning electron microscope. Results and discussion Microstructure: Fig.1 shows the metallurgical microstructure of all the as-cast alloys. As can be seen in Fig. 1a, the semi-continuous and granular phases form a typical cored dendrite structure of primary α (Al) solid solution surrounded by inter-dendritic secondary phases. After Ce addition, the semi-continuous phases are gradually broken into granular shapes, and the α-Al dendrites are refined which can be seen from the Fig.1(b),(c) &(d). The best refining dendrite is obtained in 0.30 % Ce containing alloy. Fig.2 shows the microstructure of the homogenized and age-harden alloys. The grain size obviously decreases with addition of Ce, for instance, when the addition of 0.30% Ce, the average grain size found minimum. SEM micrograph of the age-hardened alloy (Fig.-3) shows that the second phase particle which was segregated along the grain boundary was distributed within the grains.

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a

99

b

20 um

20 um

c

d

20 um

20 um

Fig.1 As cast structure of (a) Al-Zn-Mg-Cu-0%Ce (b) Al-Zn-Mg-Cu-0.1%Ce (c)Al-Zn-MgCu-0.2%Ce (d)Al-Zn-Mg-Cu-0.3%Ce

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a

b

20 um

c

20 um

d

20 um

20 um

Fig. 2 Microstructure of the alloy after forging and heat-treatment(a) Al-Zn-Mg-Cu-0%Ce (b) Al-Zn-Mg-Cu-0.1%Ce (c)Al-Zn-Mg-Cu-0.2%Ce (d)Al-Zn-Mg-Cu-0.3%Ce

Fig. 3 Scanning electron micrograph of the age hardened C912 Al alloy. X-ray diffraction: The X-ray diffraction patterns of the as-cast alloys are shown in Fig.4 & Fig-5. From the XRD pattern it is found that the microstructure of the as-cast Al-Zn-Mg alloy is mainly composed of the following phases: α (Al), MgZn2, Mg 32 (A1, Zn) 49 and Al4Ce. The Mg32 (Al, Zn) 49 is an unstable phase based on T phase and it is similar to T phase reported in literature[9-11].

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From the Fig.4 it can be seen that as % Ce increases the impurity level in the alloy decreases. Fig 5 shows that after homogenization unstable phase MgZn2 decreases and stable phase Al2CuMg evolves in the alloy.

Fig. 4 X-ray diffraction patterns of the as cast alloys

Fig. 5 X-ray diffraction patterns of the as cast and as homogenized alloy

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Mechanical properties: Tensile testing was carried out at room temperatures and the results are shown in Fig.-6. At room temperature, the tensile strength increased from 506 to 559 MPa when the Ce content was increased up to 0.30%. However, there is a nominal increase in tensile strength when the Ce content increased from 0.30 to 0.40%. Previous research indicates that the thermally stable Al–RE dispersions can effectively restrain the dislocation slipping and strengthen the alloy at high temperature[6]. However, the low content of Al4Ce phase in the alloy cannot effectively hinder the dislocation movement, and with increase of Ce percentage rod like Al4Ce phase increases in the matrix and attain optimum value after that no major effect was found. Table-2 Variation of hardness with % Ce in as cast and heat-treatment alloys S.No.

Alloy

1 2 3 4 5

0%Ce 0.10%Ce 0.20%Ce 0.3%Ce 0.40%Ce

As cast (BHN) 110 112 114 117 118

After age hardening (BHN) 170 171 172 174 175

580

10 9 8 7

540

6 520

5 4

500

3 Yeild Strength

480

Percentage elongation

Strength (Mpa)

560

2

Tensile strength

1

Elongation

460

0 0

0.1

0.2

0.3

0.4

0.5

Percentage Cerium (Ce)

Fig-6-Mechanical properties of the present alloys tested after forging and heat-treatment Conclusions 1. The addition of Ce to Al-Zn-Mg-Cu alloy can refine the grains remarkably due to the presence of the primary Al4Ce, which can serve as nucleus for inhomogeneous nucleation and significantly improve the ratio of nucleation.

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2. The phase compositions of the C912 alloy are mainly α-Al, MgZn32 & Mg32 (Al, Zn)49 Al4C phase. 3. The addition of Ce improves the mechanical properties of the alloy. The reason is the precipitation of secondary A14Ce particles and the remarkable refinement of the grains, namely, precipitation strengthening and fine-grain strengthening. References [ 1] PAN Fu-sheng, ZHOU Shou-ze, SHI Gong-qi, et al.: Light Alloy Fabrication Technology, Vol. 3 (1990), p.1 [ 2] TANG Ding-xiang, WANG liang-xuan, ZHAO Min-shou, et al.:. The Chinese Jounal of RearEarth, Vol. 10(1) (1992), p. 66 [ 3] YANG Yu-chun. :Rare-Earth Rare Metal Materials and Engineering, Vol. 22(4) (1993), p.1 ZHAO Min-shou, DANG Ping, [ 4] X.-M. Li, M.J. Starink: Mater. Sci. Technol. Vol. 17 (2001), P. 1324 [ 5] A.K. Mukhopadhyay: Metall. Mater. Trans. Vol. 30A (1999), p. 1693 [ 6] W.G. Yang, C.H. Koo: Mater. Trans. Vol. 44 (2003), p. 1029 [ 7] G.H. Wu, Y. Fan, H.T. Gao, C.Q. Zhai, Y.P. Zhu: Mater. Sci. Eng. A Vol. 408 (2005), p. 255 [ 8] Y.Z. LU¨ , Q.D. Wang, X.Q. Zeng, W.J. Ding, C.Q. Zhai, Y.P. Zhu: Mater.Sci. Eng. A Vol. 278 (2006), p. 66 [ 9] J. Bai, Y.S. Sun, S. Xun, F. Xue, T.B. Zhu: Mater. Sci. Eng. A Vol.419 (2006), p.181 [10] Jing Zhang, Z.X. Guo, Fusheng Pan, Zhongsheng Li and Xiaodong Luo Mater. Sci. Eng. A Vol. 456 (2007), p.43 [11] X.Q. Zeng, W.J. Ding, Z.Y. Yao, L.M. Peng, C. Lu, J. Shanghai Jiaotong Univ. 39 (2005) p. 47

Defect and Diffusion Forum Vol. 279 (2008) pp 105-110 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.105

δ-Hydride Habit Plane Determination in α-Zirconium at 298 K by Strain Energy Minimization Technique R. N. Singh a, b*; P. Ståhle b ,c, Leslie Banks-Sills c,d, Matti Ristinmaac, S. Banerjeea a

SMS, Mechanical Metallurgy Section, Materials Group, Bhabha Atomic Research Centre, Mumbai-4000085, India b

Materials Science, Technology and Society, Malmö University, SE20506, Sweden c d

Division of Solid Mechanics, Lund University/LTH, SE22100 Lund, Sweden

The Dreszer Fracture Mechanics Laboratory, Department of Solid Mechanics, Tel Aviv University, 69978 Ramat Aviv, Israel

Email: [email protected], [email protected], [email protected], [email protected] and [email protected] Keywords: α-Zirconium, δ-hydride, Finite element method, strain energy minimization, accommodation energy and habit plane.

Abstract: Hydrogen in excess of solid solubility precipitates as hydride phase of plate shaped morphology in hcp α-Zr with the broad face of the hydride plate coinciding with certain crystallographic plane of α-Zr crystal called habit plane. The objective of the present investigation is to predict the habit plane of δ-hydride precipitating in α-Zr at 298 K using strain energy minimization technique. The δ-hydride phase is modeled to undergo isotropic elasto-plastic deformation. The α-Zr phase was modeled to undergo transverse isotropic elastic deformation but isotropic plastic deformation. Accommodation strain energy of δ-hydride forming in α-Zr crystal was computed using initial strain method as a function of hydride nuclei orientation. Hydride was modeled as disk with round edge. Contrary to several habit planes reported in literature for δhydrides precipitating in α-Zr crystal, the total accommodation energy minima at 298 K suggests only basal plane i.e. (0001) as the habit plane. Introduction Zr-2.5Nb alloy is used as pressure boundary for hot coolant in CANDU, PHWR and RBMK reactors [1-2]. The pressure boundary material picks up a part of hydrogen evolved during aqueous corrosion and once solid solubility is exceeded hydride phase precipitates out. Hydride acquires plate shaped morphology and the broad face of the hydride plate coincides with certain crystallographic plane of hcp (hexagonal close packed) α-Zr crystal, which is called habit plane [3]. Hydride plate oriented normal to tensile stress significantly increases the degree of embrittlement [4]. Thus key to mitigating the damage due to hydride embrittlement is to avoid the formation of hydride plates normal to tensile stress. This is possible by imparting desirable texture [5] and grainboundary structure [6] to Zr-alloy components so that habit planes of hydride are oriented in such a way that formation of hydride plates normal to tensile stress can be avoided. Recently, Une et al. [7] reviewed the data reported for habit plane of δ-hydride formation in α-Zr. Using experimental techniques such as metallographic trace, X-ray diffraction and transmission electron microscopy [7] hydride habit plane have been reported as prism plane {10 1 0} [8-9], the twinning planes of {10 1 2}, {1121}and {1122} [10], the pyramidal plane of {10 1 1} [11], the basal plane of (0001) [11-13] and the {10 1 7} [9,14-15]. TEM investigation of irradiated zircaloy cladding tube [11] reported

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{10 1 7} plane

as the habit plane for microscopic hydride whereas (0001) basal plane as the habit plane for sub-microscopic hydride. Since experimental techniques suggest several planes as habit plane of δ-hydride forming in α-Zr, it is worth determining the same using theoretical approaches. Two different theoretical approaches are used to determine the shape and habit plane of precipitates viz., geometrical and solid mechanics [16]. For the geometrical approach invariant plane and invariant-line criteria have been applied successfully and for the solid mechanics approach strain energy minimization criteria have been used successfully [17-19]. Even though the aforementioned theoretical approaches are different, they often predict same shape and orientation for plate type inclusions [20]. Solid mechanics approach using strain energy computed by FEM technique have been applied to hydride precipitation in Zr-alloys, but the emphasis has been to understand the solvus hysteresis [21-23]. The objective of the present investigation is to predict habit plane of fcc (face centered cubic) δ-hydride precipitating in hcp α-Zr using strain energy minimization technique [16]. The δ-hydride phase is modeled to undergo isotropic elasto-plastic deformation [24]. The α-Zr phase was modeled to undergo transverse isotropic elastic deformation [25] but isotropic plastic deformation [22-23]. Both perfect plastic behavior and linear workhardening behavior were considered. Finite element computation The accommodation energy of δ-hydride in α-Zr single crystal was computed by initial strain method [26]. Transformation of the zirconium hydrogen solid solution into δ-hydride is associated with about 17 percent positive change in volume [27-28]. The close packed planes of α-Zr and δhydride bear an orientation relationship of (111)δ (0001)α [28]. The misfit strain along the hydride platelet is designated as e11 and the misfit strain normal to the hydride platelet is designated as e22 [28]. For all the computations reported in this work stress free transformation strain e11 = 0.0458 and e22 = 0.0722 were used, which is the stress free transformation strain at 298 K [27-28]. The body was partitioned into two parts viz., Hydride and matrix. The phase transformation of zirconium hydrogen solid solution to hydride was simulated by imposing a temperature rise in partitioned region, which was assigned with appropriate thermal expansion coefficients [22-24] to achieve desired volume expansion. The accommodation strain energy was computed under 2D plane strain condition for fully elastic case and elasto-plastic case. Both matrix and hydride were treated as perfectly plastic and linearly work-hardening plastic. Hydride precipitate was modeled with Young’s Modulus, E=94.465 GPa, Poisson’s ratio, ν=0.333 [24] whereas transversely isotropic elastic behaviour of hcp matrix was modeled with five independent elastic constants C11 = 143.4 GPa, C12= 65.3 GPa, C22= 164.8 GPa, C13= 72.8GPa, C44 = 32 GPa and C66 = 35.3 GPa [25]. It may be noted here that Direction 1 and 3 are in the basal plane and direction 2 is normal to basal plane of the α-Zr crystal for the present investigation. The Yield strength (YS) of hydride was taken = 501.965 MPa [24] and that of matrix = 445 MPa [29]. For both hydride and matrix Ultimate strength of 1.25 times YS corresponding to a plastic strain of 0.1 was considered. Both matrix and hydride was treated as fully elastic or elastoplastic. For elasto-plastic case hydride was assumed to undergo isotropic elastic and plastic [24] deformation and the matrix was allowed to undergo transverse isotropic elastic deformation [25] and isotropic plastic deformation [22-23]. Both stress free transformation strain of hydride with respect to hcp α-Zr phase [27-28] and elastic constants of the latter are transversely isotropic [25] and hence the accommodation energy is expected to depend on the orientation of hydride with respect to the matrix. For determining the influence of hydride nuclei orientation on accommodation energy, full body model was considered (Fig. 1). The ratio of major dimension of hydride to that of full body was 20. The ratio of hydride plate thickness with its diameter was 0.05 [22]. Edge biased meshing with finer mesh near hydride matrix interface was considered. A series of computation with decreasing mesh size ensured the

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values reported in this work were mesh size independent. The number of elements in the model was 37375 and the number of nodes was 112326 and the computations were carried out for plane strain condition. The angle, Ψ, representing hydride nuclei orientation is defined as angle between the hydride disk normal and [0001] direction of hcp α-Zr (Fig. 1). The Ψ values in the range of 0° to 90° at an interval of 15° were considered. The orientation relationship between δ-hydride and α-Zr was preserved while rotating the hydride disk in matrix.

[0001]

Matrix Ψ Ψ

[1120]

h dHyd Hydride

Fig. 1: Illustration of flat disk shaped hydride nuclei with round edge [23] embedded inside a matrix. The angle, Ψ, between the normal to hydride disk with [0001] direction of hcp α-Zr were considered in the range of 0° to 90° at an interval of 15° for estimating the influence of hydride disk orientation on the accommodation energy. Results The plots of normalized accommodation energy with hydride nuclei orientation for the case where hydride is isotropic elastically and matrix is transversely isotropic elastically is presented in Fig. 2. The plots in Fig. 2 with circular, square and triangular symbol represent the accommodation energy stored in hydride, zirconium and whole model, respectively. The accommodation energy values were normalized with total accommodation energy value corresponding to the zero degree orientation, which corresponds to the case where broad face of hydride disk is oriented normal to the c-axis of α-Zr hcp crystal. The normalized accommodation energy for hydride, zirconium and whole model shows minima for zero degree orientation for fully elastic case. Thus strain-energy minimization for fully elastic case suggests basal plane as habit plane. However, transformation of α-Zr-H solid solution to δ-hydride is associated with about 17 percent increase in volume. Such a large increase in volume is likely to cause plastic deformation of both hydride and the matrix surrounding it. Hence, dependence of accommodation energy on hydride nuclei orientation was also determined for elasto-plastic case. The plots for elasto-plastic case are shown in Fig. 3. Here too, circular, square and triangular symbol represent the accommodation energy stored in hydride, zirconium and whole model, respectively. The corresponding solid symbols represent the values for work-hardening (WH) case. The accommodation energy values were normalized with total elastic accommodation energy value corresponding to the zero degree orientation. It is interesting to note that for elasto-plastic case, the accommodation energy in hydride and whole model shows minima whereas the same for matrix zirconium shows maxima for zero degree orientation. The dependence of hydride accommodation

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energy on plate orientation is much stronger as compared to that of the dependence of the same for matrix. Hence, total accommodation energy for the whole model shows a dependence on plate orientation similar to that of hydride. The plots with solid symbol in Fig. 3 are for work-hardening (WH) case. The values for WH case were marginally higher than that for perfect plastic case, though its dependence on plate orientation was similar to that for perfect plastic case. Thus for both perfect plastic and work-hardening plastic case, the total accommodation strain energy shows a minimum value for zero degree orientation. 0.34

1.60

0.32

1.40

0.30

Normalized strain energy

Normalized strain energy

1.20

1.00

0.80 0.20

0.28

0.26 0.20

0.18

0.18 Hyd, pl Zr,pl Total,pl Hyd, WH Zr, WH Total,WH

0.16

Hyd Zr Total

0.16

0.14

0.12 0

20

40

orientation

60

80

0

20

40

60

80

orientation

Fig. 2: Plot of accommodation energy for Fig. 3: The variation in accommodation energy hydride (hyd- open circle), zirconium stored for hydride (hyd), zirconium (Zr) (Zr- open square) and whole model and whole model (Total) with orientation (Total- open triangle) with hydride for elasto-plastic case. Pl represents nuclei orientation for fully elastic values for perfect plastic and WH case. represents corresponding values for workhardening case. Discussion For any solid state phase transformation there are three components of the transformation energy, viz., chemical, interfacial and strain energy [3]. For phase transformation involving no volume change strain energy is neglected and in such a case the shape and orientation of the precipitate is governed by the orientation dependence of interfacial energy [3]. The shape and orientation for which interfacial energy is minimum, is the shape and orientation exhibited by the precipitate [3]. However, for phase transformation involving volume change strain energy could play a major role in deciding the orientation of precipitates, especially when interfacial energy components are small compared to strain energy. Fig. 2 shows the variation in normalized total accommodation energy with orientation for fully elastic case. The minimum accommodation energy value for zero degree orientation suggests that the hydride disk oriented normal to c-axis of α-Zr crystal is the favored orientation and hence for fully elastic case suggests basal plane (0001) as the habit plane of δ-hydride forming in α-Zr at 298 K. For the elasto-plastic case (Fig. 3) the minimum in total accommodation is observed for zero degree orientation similar to the fully elastic case and hence suggests only basal plane as the possible habit plane for δ-hydride precipitation in α-Zr at 298 K. The hydride plate, which appears as a single entity under optical microscope is reported to exhibit two levels of organization called platelet and subplatelet level of organization [30]. At the

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platelet level of organization, several smaller platelets are stacked nearly parallel to each other. Subplatelet level of organization is observed within a platelet, where tiny sub-platelets are stacked end to end. Also, the hydride sub-platelets are not continuous, but leave a gap between two subplatelets (Fig. 4). As is evident from Fig. 4 the hydride-matrix interface for sub-platelet and hydride plate will bear different crystallographic relationship with the matrix. Hence, it is felt that the habit plane for hydride sub-platelet could be different from that of a bunch of sub-platelets as they are not perfectly aligned. Since the present computation was carried out using a single hydride plate, the predicted habit plane by strain energy minimization technique is applicable to sub-platelet level of organization. For α-Zr it has been recently reported that δ-hydrides shows basal plane (0001) [7] as sub-microscopic habit plane, which is suggested by the present investigation as well.

Fig. 4 Internal details of hydride as observed in secondary electron mode.

In literature hydride habit plane have been reported as prism plane {10 1 0} [8-9], the twinning planes of {10 1 2}, {1121}and {1122} [10], the pyramidal plane of {10 1 1} [11], the basal plane of (0001) [11-13] and the {10 1 7} [9,14-15]. In these work the hydriding of the zirconium sample was carried out above 673 K, whereas the present computation was carried out using the material property at 298 K. With increase in temperature elastic constant of hydride and α-Zr phase [24-25], the flow stress of hydride [24] and α-Zr [29] and the transformation strain of zirconium hydrogen solid solution to hydride [27-28] is likely to change and hence accommodation energy will also show temperature dependence. Thus it will be worth examining the effect of temperature on the habit plane predicted by strain energy minimization technique. Such a work is already under progress and will be reported latter. Conclusions Accommodation energy associated with δ-hydride formation in α-Zr crystal at 298 K has been computed using a FEM code for fully elastic and elasto-plastic cases. The orientation dependence of accommodation energy suggested a minimum corresponding to hydride nuclei disk oriented normal to c-axis of α-Zr crystal and hence basal plane (0001) as the hydride habit plane.

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Acknowledgement Dr. Singh’s association with Malmo University as a Marie Curie Incoming International fellow was financially supported by European Commission under its FP6 programme to promote researchers mobility into European Union (Contract No. MIF1-CT-2005-006844). Prof. Banks-Sills is grateful for the support of Lund University as the Lise Meitner Professor, July, 2006 to December, 2006. Reference [ 1] C. Lemaignan, and A. T. Motta: in Zirconium alloys in nuclear applications, Materials Science & Techno. A Comprehensive treatment, eds. R. W. Cahn, P. Haasen & E. J. Kramer, Nuclear Materials, 10B chapter 7, (1994) 1. [ 2] C. E. Coleman, B. A, Cheadle, C. D. Cann, and J. R. Theaker, Zirconium in the nuclear industry ASTM STP 1295, Eds. E. R. Bradley and G. P. Sabol, (1996) 884. [ 3] W.-Z. Zhang, G.C. Weatherly, On the crystallography of precipitation, Prog. in Mater. Sci. Vol. 50 (2005), p. 181 [ 4] R. N. Singh, Lala Mikin R., G. K. Dey, D. N. Sah, I. S. Batra and P. Ståhle: J. of Nucl. Mater., Vol. 359 (2006), p. 208 [ 5] L. Moulin., S. Reschke, and E. Tencknoff, Zirconium in Nuclear Industry, ASTM STP 824 (1984) 255. [ 6] K.V. Mani Krishna, A. Sain, I. Samajdar, G.K. Dey, D. Srivastava, S. Neogy, R. Tewari, S. Banerjee: Acta Mater. Vol. 54 (2006), p. 4665 [ 7] K. Une, K. Nogita, S. Ishimoto, K. Ogata: J. of Nucl. Sci. & Techn. Vol. 41(7) (2004), p. 731 [ 8] J. E. Baily: Acta Metall., Vol. 11 (1963), p. 267 [ 9] G. C. Weatherly: Acta Metall., Vol. 29 (1981), p. 501 [10] F. W. Kunz, A. E. Bibb: Trans. AIME Vol. 218 (1960), p. 133 [11] H. M. Chung, R. S. Daum, J. M. Hiller, Zirconium in Nuclear Industry, ASTM STP 1423 (2002) 561. [12] V. Perovic, G. C. Weatherly: Acta Metall., Vol. 31 (1983), p. 1381 [13] M. Valeva, S. Arsene, M. Record, J. L. Bechade, J. Bai: Metall. Mater. Trans., Vol. 34A (2003), p. 567 [14] D. G. Westlake: J. of Nucl. Mater., Vol. 26 (1968), p. 208 [15] C. Roy, J. G. Jacques: J. of Nucl. Mater., Vol. 31 (1969), p. 233 [16] M. Kato, T. Fuji, S. Onaka: Acta Mater., Vol. 44(3) (1996), p.1263 [17] U. Dahmen, P. Furguson, K. H. Westmacott: Acta Metall., Vol. 32 (1984), p. 803 [18] U. Dahmen, K. H. Westmacott: Acta Metall., Vol. 34 (1986), p. 475 [19] D. Duly: Acta Metall., Vol. 41 (1993), p. 1559 [20] M. Kato, T. Fuji: Acta Metall., Vol. 42 (1994), p. 2929 [21] M. P. Puls: Acta Metall., Vol. 32 (1984), p. 1259 [22] B. W. Leitch and M. P. Puls: Metall. Trans., Vol. 23A (1992), p. 797 [23] B. E. Leitch, S. –Q. Shi: Moedlling Simul. Mater. Sci. Engg. Vol. 4 (1996), p. 281 [24] M.P. Puls, San-Qiang Shi, J. Rabier: J. of Nucl. Mater. Vol. 336 (2005), p. 73 [25] E. S. Fisher and C. J. Renken: Phy. Rev. A Vol. 135 (1964), p. 482 [26] S. Sen, R. Balasubramaniam, R. Sethuraman: Acta Mater., Vol. 44 (1996), p.437 [27] R. N. Singh, P. Ståhle, A. R. Massih, A. A. Shmakov: J. of Alloys & Comp. Vol. 436 (2007), p. 150 [28] G. J. C. Carpenter: J. Nucl. Mater. Vol. 48 (1973), p. 264 [29] R. N. Singh, S. Mukherjee, R. Kishore and B. P. Kashyap: J. of Nucl. Mater. Vol. 345 (2005), p. 146 [30] R. N. Singh, Lala Mikin R., G. K. Dey, D. N. Sah, I. S. Batra and P. Ståhle: J. of Nucl. Mater. Vol. 359 (2006), p. 208

Defect and Diffusion Forum Vol. 279 (2008) pp 111-116 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.111

Calorimetric Studies of Dissolution Kinetics of Ni2(Cr,Mo) Phase in Ni-Cr-Mo Alloys Using Non-Isothermal Approach H. C. Pai1a, B. C. Maji2b, A. Biswas2c, Madangopal K2d and M. Sundararaman1e 1

Structural Metallurgy Section, Mechanical Metallurgy Section Bhabha Atomic Research Centre, Trombay, Mumbai-400085, India 2

Functional Materials Section, Materials Science Division Bhabha Atomic Research Centre, Trombay, Mumbai-400085, India E-mail: a [email protected] ,b [email protected] , [email protected] , d [email protected] , [email protected] Keywords: Ni-Cr-Mo alloys, Ni2(Cr,Mo) phase, DSC, Dissolution kinetics, Activation Energy.

Abstract. The kinetics of dissolution of ordered phase with Pt2Mo structure has been studied in two nickel chromium alloys – one without molybdenum and another with molybdenum - using differential scanning calorimetry. The activation energy for dissolution, determined using three nonisothermal approaches was found to be ~ 418 kJ /mole for both the alloys. This value agreed very well with the activation energy for coarsening of γ″ precipitates in Ni-Cr-Mo matrix and is close to activation energy for mobility of chromium and molybdenum in complex nickel alloy matrix. Introduction Ni-Cr-Mo based multicomponent alloys (such as Hastelloys) have excellent corrosion resistance in chloride environments as well as more aggressive environments[1]. Due to this property, these alloys are being considered in design and manufacturing of containers for storage of high level nuclear wastes for periods up to 100,000 years[2]. During long term usage, precipitation of Ni2(Cr,Mo) phase has been observed along with topologically close-packed (TCP) phases in these alloys. Ni2(Cr,Mo) phase contains substantial amount of chromium and molybdenum whose formation results in lowering chromium and molybdenum contents in the matrix resulting in lower corrosion resistance[3-5]. The precipitation of Ni2(Cr,Mo) phase has been reported to very sluggish in the literature[6-8]. The dissolution of the ordered phase should result in improvement in properties of the alloy. In order to decide the appropriate heat treatment schedule for dissolution, it is important to have data on the dissolution kinetics of Ni2(Cr,Mo) phase. The present paper reports the results of a detailed study carried out in the author’s laboratory on precipitation kinetics in two Ni-Cr-Mo alloys. Experimental Procedure The alloys with Ni-33a%Cr and Ni-30.6a%Cr-2.7a%Mo compositions designated as Alloy 1 and Alloy 2 respectively, prepared by arc melting, were used in this investigation. The melted buttons were homogenized at 1200°C for 24 hours under argon atmosphere to break the cast structure and then hot rolled to thin sheets. Samples cut from rolled sheets were sealed in quartz capsules under helium atmosphere and then solution treated at 1100°C for 3 hours and water quenched. Ageing of solution treated samples was carried out at 500 °C for 500 hours to precipitate Ni2(Cr,Mo) phase in them. Thin foils for electron microscopy were prepared using the dual jet electropolishing technique containing one part of perchloric acid and four parts of ethanol. Electropolishing was carried out at

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25 V DC while maintaining the temperature of electrolyte at ~40°C. These thin foils were examined in a JEOL JEM 2000FX TEM. DSC (Differential Scanning Caloriemetry) was carried out in a heat flux SETARAM Labsys DSC 16. Accurately weighed samples were loaded in alumina crucible of 100 µl capacity and the empty alumina crucible is used as the reference. Experiments were carried out under inert nitrogen gas atmosphere. Samples were polished and made flat so that material should completely cover the bottom of the crucible to ensure good thermal contact. Dissolution kinetics studies were carried out on these alloys by heating them at different heating rates from 1 °C /min to 10 °C per min. Results Transmission electron microscopic examination revealed that the solution treatment has resulted in complete dissolution of all intermetallic precipitates. This is confirmed by the absence of any superlattice reflection in the selected area diffraction patterns. Planar array of dislocations and isolated stacking faults probably generated due to quenching stresses were only noticed in the matrix. Ageing has resulted in the nucleation and homogeneous distribution of fine Ni2(Cr,Mo) precipitates in the matrix. SAD patterns revealed superlattice reflections corresponding to different variants. These observations are illustrated in figure 1 in the case of alloy 1.

Figure 1. Microstructure of Alloy 1 (a) Solution treated and (b) Solution treated and aged at 500 °C for 500 hours. Differential Scanning Calorimetry results. Typical DSC plots of solution treated and aged Alloy 1 are shown in figure 2. An endothermic peak could be observed close to 600 °C. The starting of dissolution of temperature defined as the onset temperature (Ti), the peak transformation temperature (Tp) and the end of dissolution defined as endset temperature (Tf) are marked in the figure 2(b). Following the convention adopted in the literature[9], the peak dissolution rate temperature, Tp is taken as the solvus temperature. The heat effect due to dissolution of ordered Ni2(Cr,Mo) phase is obtained by subtracting the heat effect of solution treated sample(figure 2a) from that obtained in aged sample(figure 2b)and is shown in figure 2c. This methodology has been adopted in the literature to find out the heat effects corresponding to transformation [9] in cases where both specimens have identical mass and thickness and similar heat capacities as well. The plot of peak transformation temperature (Tp) versus the heating rate is shown in figure 3. The sigmoidal plot fits these data points better than any other fit. The extrapolation of this fit to 0°C/min heating rate should correspond to isothermal solvus temperature and the values obtained for Alloy 1 and Alloy 2 are marked in the plot itself. The intercept made by the curve corresponding to Alloy 1 and Alloy 2 on y-axis is 602 °C and 610 °C respectively. Ageing of Alloy 1 and Alloy 2 at 600°C for prolonged period did not reveal any precipitation of Ni2(Cr,Mo) phase. This is indicated by the absence of superlattice spots in SAD patterns (see inset figure 4) and a microstructure (figure 4)

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Figure 2. A typical DSC Spectra of Alloy 1 (a) Solution treated, (b) Aged at 500 °C for 500hours and (c) Net effect due to dissolution of Ni2(Cr,Mo) phase after subtracting (a) from (b)

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Figure 3. Plot of peak transformation temperature as a function of heating rate for Alloy 1 and Alloy 2. Transformation temperature for zero degree cooling rate is marked in the figure.

Figure 4. Microstructure of Alloy 1 (a) and Alloy 2 (b) aged for 500 hours at 575 °C and 600 °C respectively. This microstructure is similar to solution treated alloy. very similar to that of solution treated specimens (figure 1a). This is suggestive of the fact that the precipitation temperature is lower than the dissolution temperature. The DSC plots obtained for different heating rates for alloys 1 and 2 are illustrated in figure 5. It is clear from this plot that the peak transformation temperature and also the energy required to effect the disordering process given by area occupied by the peak increases with increase of heating rate. These results indicated that the dissolution kinetics is dominated by temperature dependent thermodynamic equilibrium. Similar reaction kinetic driven processes like precipitation and dissolution of GPB zones in aluminum alloys [10] and recrystallisation behavior of metallic glass has been studied by direct extension of Johnson Avarami equation for isothermal process to non isothermal process [9]. The same procedure is borrowed from literature to investigate dissolution of Ni2(Cr,Mo) phase in the present case. The three most commonly used expressions are derived by Kissinger [11,12], Augis and Bennett [13] and Gao and Wang [14] to determine the activation energy for crystallization kinetics. These methodologies use the hypothesis that at the

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maximum transformation rate at the peak temperature Tp, (d2X/dt2) = 0 where X is the volume fraction dissolved. This results in the expression at peak Tp as follows for different methods.

Figure 5. DSC scans at different heating rates for Alloy 1 and Alloy 2 aged at 500 °C for 500 hours.

(

)

d Ln β / TP2  Q =− d (1/ TP ) R

(

)

d Ln β /(TP − To )  Q =− d (1/ TP ) R

(

)

d Ln ( dX / dt )p  Q   =− d (1/ TP ) R

Kissinger method [11, 12]

…. (1)

Augis-Bennett method [13]

…. (2)

Gao-Wang method [14]

…. (3)

From the slope of the plots of Ln β / TP2  , Ln β /(TP − To )  and Ln ( dX / dt ) p  versus (1/ TP ) for   different values of heating rates, the value of Q could be estimated. The plots corresponding to equations (1), (2) and (3) are depicted in figure 6. The linear regime could be noticed for all three methods. The value of Q obtained by these methods is given in table 1. It is clear from this table that the value of Q obtained by different methods is in agreement with each other and it appears to be rather insensitive to molybdenum addition to binary Ni2Cr alloy.

Figure 6. Kissinger method, Augis –Benett method and Gao-Wang method plots for obtaining Q Values for Ni2(Cr,Mo) phase dissolution in Alloy 1 and Alloy 2 aged at 500°C for 500 hours.

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Table 1. Activation energies obtained by different methods

Method

Alloy 1

Alloy 2

Kissinger[11]

419kJ /mole

410 kJ /mole

Augis-Benett[13]

413kJ /mole

410 kJ /mole

Gao-Wang[14]

401 kJ /mole

426 kJ /mole

Discussion The variation of dissolution temperature with heating rate in DSC experiment is a clear indication that dissolution is a non isothermal kinetic process. The methodologies reported in the literature for finding out activation energy for recrystallization of metallic glass has been employed in this investigation to determine the activation energy for dissolution of Ni2(Cr,Mo) phase. This activation energy has been correlated in the literature either to migration energy [15-16], vacancy formation energy [178] or entropy [18,19] to get some physical meaning to theses values and in assigning some physical processes to it. For example, in the case of precipitation of GP zones, Jena and Chaturvedi [10] attributed the activation energy to migration of vacancies while in the case of dissolution of GP zones, the activation energy was compared with the activation energy for diffusion of copper in Al-Cu alloys. They arrived at these values based on the microstructural condition of the material. Delasi and Adler have [18, 19] explained the dissolution kinetics of precipitation in 7075 aluminum alloy to the influence of entropy than the activation energy. In the present work, the activation energy determined using different methods for Alloys 1 and 2 have a constant value of 418 kJ /mole (Whether it contains molybdenum or not). This value is much higher than the activation energy for diffusion of Cr in Ni (287kJ/mole) [20] and Mo in Ni (269 kJ /mole) [21]. It is interesting to note that in many γ″ precipitation alloys containing high Cr or Cr and Mo in the nickel matrix, the activation energy for precipitation found to be in the range of 400 - 450 kJ/mole [22-25]. In fact, the activation energy for precipitation of γ″ in Alloy 625 in the present work was determined to be 398 kJ /mole [26]. It is clear from these data that the activation energy obtained for precipitation or dissolution in Ni-Cr or Ni-Cr-Mo matrix is ~ 400 kJ /mole that is higher than the activation energy for diffusion of any of the solute species participating in the precipitate formation. It is surprising that activation energy for mobility of Cr and Mo in nickel has a value of 346 kJ /mole and 347 kJ /mole respectively which is very close to experimentally determined values [27]. From this discussion one is tempted to conclude that the mobility of different solute elements has more influence on dissolution than their activation energy for diffusion.

Conclusions The following conclusions could be drawn on the basis of results reported in this paper. 1. Solvus temperature for isothermal heating rates was determined to be 602°C for Alloy 1 and 610°C for Alloy 2. 2. Activation energy for dissolution of Ni2(Cr,Mo) phase is ~ 417 kJ /mole irrespective of whether the alloy contains molybdenum or not. This value matches very well with the activation energy for coarsening of γ″ precipitates in Ni-Cr-Mo alloy matrix.

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

G. E. Gdowski, LLNL Report No: - UCRL-ID-108330, March 1991. P. E. A. Turchi, L. Kaufman, Zi-Kui Liu, Calphad, Vol. 30 (2006), p. 70 H. M. Tawancy, Metall. Trans., Vol. 11A (1980), p. 1764 H. M. Tawancy, J. Mater. Sci., Vol. 16 (1981), p. 2883 H. M. Tawancy, R. B. Herchenroeder and A. I. Asphahani, J. Metals, Vol.35 (1983), p. 37 M. Hirabayashi, M. Koiwa, K. Tanaka, T. Tadki, T. Saburi, S. Nenno and H. Nishiyama, Trans. Japan Inst. Metals, Vol. 10 (1969), p. 365 S. Karmazin, J. Krejčí and J. Zeman, Mater. Sci. Engg., Vol. 183A (1994), p. 103 E. Gozlan, M. Bamberger, S. F. Dirnfeld, B. Prinz and J. KIodt, Mater. Sci. Engg., Vol. 141A (1991), p. 85 J.S. Blázquez, C.F. Conde, A. Conde, Acta Mater., Vol. 53 (2005), p. 2305 A.K. Jena, A. K. Gupta and M. C. Chaturvedi, Acta Metall., Vol. 37 (1989), p. 885 H. E. Kissinger, J. Res. Nat. Bur. Stand., Vol. 57 (1956), p. 217 H. E. Kissinger, Anal. Chem., Vol. 29 (1957), p. 1702 J. A. Augis and J. E. Benett, J. Thermal Anal., Vol. 13 (1978), p. 283 Y. Q. Gao and W. Wang, J. Non-Cryst Solids, Vol. 81(1986), p. 129 H. Kimura , A. Kimura and R. R. Hasguti, Acta Metall., Vol. 10 (1962), p.607 T. J. Koppennal and M. F. Fine, J. App. Phys., Vol. 32 (1961), p. 1781 R. Horiuchi and Y. Minonishi, J. Japan Inst. Light Metals, Vol. 39 (1970), p. 936 R. DeIasi and P. N. Adler, Metall. Trans., Vol. 8A (1977), p. 1177 P. N. Adler and R. DeIasi, Metall. Trans., Vol. 8A (1977), p. 1185 Björn Jönsson, Scanadinavian J. of Metall., Vol. 24 (1995), p. 21 M. S. A. Karunaratne and R. C. Reed, Acta Mater., Vol. 49 (2001), p. 861 K. Kusabiraki and S. Saji, Tetsu-to-Hagané, Vol. 84 (1998), p. 643 K. Kusabiraki, L. Wang, T. Ooka and Y. Yamada, Tetsu-to-Hagané, Vol. 76 (1990), p. 1341 K. Kusabiraki, S. Araie and T. Ooka, Tetsu-to-Hagané, Vol. 78 (1992), p. 650 K. Kusabiraki and T. Maekawa, Tetsu-to-Hagané, Vol. 85 (1999), p. 241 Hrishikesh C. Pai, Kinetics of Intermetallic Phase Precipitation in Nickel base Superalloys, Ph.D Thesis, submitted to Mumbai University, 2008 M.S.A. Karunaratne, D.C. Cox, P. Carter and R.C. Reed, Proc. Superalloys 2000 (Eds., T. M. Pollack, R. D. Kissinger, R. R. Bowman, K. A. Green, M. McLean, S. Olson and J. J. Schirra), TMS, Warrendale, PA, 2000, p.263

Defect and Diffusion Forum Vol. 279 (2008) pp 117-124 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.117

Kinetics and Mechanism of Growth of β-Solid Solution During Reaction Diffusion in Binary Titanium and Zirconium Alloy Systems. G.P. Tiwari1 , O. Taguchi2, Y. Iijima3 and G.B. Kale 4 1

2

Post Irradiation Examination Division, Bhabha Atomic Research Centre, Mumbai. Materials Science and Engineering, Miyagi National College of Technology.Natori, 981-1239, Japan 3 Dept. of Materials Science, Graduate School of Engineering, Iwate Univ. Japan. 4 Materials Science Division, Bhabha Atomic Research Centre, Mumbai. E-mail :[email protected],[email protected]

Key Words: Layer Growth Kinetics, β-Phase, Ti-based Systems, Zr-based Systems

Abstract: High temperature beta-phase in titanium and zirconium alloy systems decomposes through an eutectoid reaction into a Ti- and Zr-rich α-solid solution and an intermetallic compound. The present paper reports the layer growth kinetics of the β-solid solution phase in elemental diffusion couples of titanium and zirconium. The growth kinetics obeys a parabolic growth law. However, the temperature dependence of the growth rate constant shows a bimodal behavior. The Arrhenius plot of the growth rate constant, which is linear at the start, becomes curved at lower temperature ranges. The deviation from the Arrhenius plot of the growth rate constant is related to the curvature in the solvus line of the β-solid solution. A theoretical model for the reaction diffusion responsible for the growth of β-solid solution is presented. The growth rate of β-phase is described by the equation k2 =

Wβ2 t

= Dβ .∆C.ξ ,

where k is a growth rate constant and Wβ is the thickness of the β-phase formed over a period of time t, Dβ is the interdiffusion coefficient for the β-phase, ∆C is concentration range of β-phase and ξ is a parameter which is a function of the miscibility gaps in the phase diagram on the either side of the β-phase. The above equation provides a satisfactory description of the various aspect of the phenomenon of the growth of β-phase in Ti-and Zr-alloy systems.

Introduction Reaction diffusion is a controlled chemical reaction occurring due to diffusion of atomic species. It is governed by the alloying behavior of the diffusing elements and simulates the situations where different elements, alloys and compounds come in close contact and interact with each other at elevated temperatures. The prevailing high atomic mobility at elevated temperature ensures significant amount of mutual interaction. In materials technology, such situations are encountered during diffusion bonding, cladding, sintering, precipitation, oxidation and sulfurization etc. The distribution of alloy phases in the diffusion zone and their growth kinetics are the major considerations in any investigation on reaction diffusion[1-12]. The various aspects of the reaction diffusion have been extensively reviewed in a recent book by Dybkov[13]. In titanium and zirconium alloy systems, β−solid solution undergoes eutectoid transformation. The present authors have investigated growth kinetics of the β-solid solution phase during reaction diffusion between pure metals Ti-X and Zr-X systems (X= Cr, Cu, Ag and Au). It occupies a major section of the diffusion zone and its` growth is affected by the compositional range of the β-phase.

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The variations in the compositional range of the β-phase has profound influence on its growth kinetics and gives rise to a curved Arrhenius plot for the growth rate constant. The paper discusses the kinetics of formation of β−solid solution in the systems listed above. A characteristic of Ti- and Zr-alloys systems is that the diffusion rates in bcc β-phases are several orders of magnitudes higher than in hcp α–phases and adjoining intermetallic compounds[14]. This fact has been utilized to develop a model for the growth of β−solid solution phase during interdiffusion. The model successfully accounts for the experimentally observed activation energy for the growth of β−solid solution phase, the overall curved nature of Arrhenius plot for growth and also brings out the role of solubility range on the either side of the phase on its growth processes. The results and their analysis presented in the paper represent the first direct validation of Wagner’s hypothesis for the formation and the growth of a phase through reaction diffusion.

Theoretical background Consider the phase diagram in A-B binary system with β as an intermediate phase and α, γ as the phases existing on the two sides of β as shown in Fig.1. If the diffusion couple is made between pure A and pure B and annealed at temperature T1, reaction diffusion will form the diffusion zone showing the formation of an intermediate phase β sandwiched between α and γ. The schematic concentration profile across such a diffusion zone is shown in Fig.2.

Fig.1 Schematic Phase Diagram with Intermediate Phase β Let Xα and Xβ be the locations of α/β and β/γ interfaces, respectively. On the basis of material balance, the growth of β-phase is controlled by the difference in the flux entering at Xα and leaving at Xβ. The velocities α/β and β/γ interfaces with respect to the Matano Interface can be given as[15]

(C

αβ

− C βα )

dX α   ∂C     ∂C   = − Dα    − − D β    dt  ∂X  α / β    ∂X  β / α  

(1)

  ∂C     ∂C   =  − Dβ    − − Dγ    dt  ∂X  β / γ    ∂X γ / β  

(2)

and

(C

βγ

− Cγβ )

dX β

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Fig.2 Schematic Concentration Profile in which an Intermetallic Compound Formed in a Diffusion Zone If Wβ is width of β-phase, then Wβ = X β − X α

(3)

From equations (1), (2) & (3), the rate of change of width (Wβ) of β- phase can be written as dWβ dt

= =

d (X β − X α ) dt Dβ C βγ − Cγβ

  ∂C   Dγ  ∂C   Dβ  − − − −       ∂X  β / γ  Dβ  ∂X  γ / β  Cαβ − C βα

 D ∂C    ∂C     − α . (4) − −   Dβ ∂X α / β  ∂X  β / α   

If we assume that Dα and Dγ are much smaller than Dβ, i.e. Dα/Dβ and Dγ/ Dβ > Dα or Dγ

Results As mentioned earlier the growth kinetics of β Phase in Ti-X and Zr-X systems (X= Cr, Cu, Ag and Au) have been studied [16-20]. The inter-diffusion coefficients have been determined by the method of Sauer and Freise [21].

Growth of β−Solid Solution Phase in the Diffusion Zone: The growth rate of phase has been established by plotting the width of β-Phase, Wβ in the diffusion zone as a function of time, t. Parabolic growth behavior is observed in all cases. The assumption that the growth rate of β-Phase far exceeds that all the intermetallic compounds is largely true in the majority of cases[16-20] and the equation derived in the previous section can be applied to the results. Interdiffusion Coefficient (Dβ ): The temperature dependence, the interdiffusion coefficients follow classical Arrhenius pattern. A typical case is shown in the Fig. 3 for Ti-Cu system.. The activation energies are more or less constant for all compositions in β-phase. Hence, average value of activation energy is considered. The behavior of interdiffusion coefficient shown in the Fig. 3 is typical and representative of all the systems studied here. The activation energies for the interdiffusion in various systems are given in the table 1.

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Fig. 3 Temperature dependence of K2 and Dβ in Ti-Cu system.

Table 1 Activation energies for Layer growth, Diffusion and ∆C for various β- phases β-phase β-Ti(Cr) β-Ti(Cu) β-Ti(Ag) β-Ti(Au) β-Zr(Ag) β-Zr(Cu)

QL 380 380 810 600 560 830

Activation energies in kJ/mol. QD QC Qξ QD+QC+ Qξ 150 60 64 274 150 210 360 160 680 840 350 200 570 140 430 570 160 720 880

Reaction Rate Constant(k): The magnitude of k is derived from the slope of plot between layer thickness, Wβ and √t. The temperature dependence of k2 and D is plotted for typical system in Fig. 3. When plotted as the function of reciprocal of the absolute temperature, k2 exhibits bimodal behavior. According to eqn. (12), k2 is directly proportional to interdiffusion coefficient. However, temperature dependence of k2 and Dβ is markedly different. Whereas the temperature dependence of Dβ shows linear behaviour, the temperature dependence of k2 exhibits bimodal behaviour. It is linear at high temperature and has curvaceous shape at lower temperature.

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Concentration range of β -Phase(∆C) : ∆C represents the composition range of the β−solid solution phase. According to the equation (12), the magnitude of ∆C influences the growth rate constant such that a higher value of ∆C will tend to enhance the growth rate. Plots of ∆C as a function of the reciprocal of absolute temperature is also shown in the Fig. 3. The solute concentrations as measured in the diffusion zone and those measured from the phase diagram are different only marginally. This difference can be ignored and in the absence of actual measurements from the diffusion zone, the values obtained from the phase diagram have been be employed. The most significant aspect of the ∆C variation with respect to the temperature is the variations in slope of ∆C VS 1/T plots. Such plots are initially linear followed by a curve, which is concave downward. The interesting feature of the curvature in the ∆C VS 1/T plot is that it is similar to that observed in case of k2 VS 1/T and transition occurs nearly at the same temperature. Temperature dependence of ξ : ξ is a complex function of the interfacial compositions on the boundaries of the β-phase. It is related to the miscibility gaps on the either side of the β-phase. The temperature dependence of this parameter is shown in the Fig. 4 where the magnitude of ξ for different system is plotted against 1/T. Because of the paucity of data points, the temperature dependence of ξ can not be estimated accurately. It is observed that except for Ti-Cr system ξ is invariant with temperature. Hence temperature dependence of ξ has been evaluated for Ti-Cr system only.

Fig. 4 Temperature dependence of ξ in various systems

Discussion Activation Energy for the Growth of β−solid solution (QL) : Activation energy for the layer growth of the β−solid solution, QL, is determined from the slope of the k2 VS 1/T plot. The values of QL obtained from the linear section of the k2 VS 1/T plot are given in the table 1. Table 1 also lists the activation energy for the interdiffusion, QD and the activation energy for the formation of the β−solid solution in the diffusion zone, QL. The relation between QL, QD and QC can obtained from the equation (12). Taking logarithm and expanding each parameter in form of an Arrhenius equation, we may write

Or

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ln k2 = ln Dβ + ln ∆C + ln ξ

(13)

ln k02 – QL/R = ln (D0 +∆ C0 + ξ0 ) – (QD+QC+Qξ)/R

(14)

Therefore, QL = QD +QC +Qξ

(15)

In the above equations, k02 is the pre-exponential factor incorporating the temperature independent part of Dβ, ∆C and ξ and QD, QC and Qξ are respective activation energies. The magnitude of these activation energies are tabulated in the table 1. It is seen from the table that there is satisfactory agreement between QL and (QD+QC+Qξ). Conclusions The preceding discussion clearly demonstrates that the equation (12) indeed provides an unambiguous explanation of the qualitative as well quantitative aspects of the growth behavior of β−solid solution phase formed during reaction diffusion in Ti-X and Zr-X systems investigated here. The following conclusions can be drawn: 1. Equation 12 is derived on the basis of material balance equations 1 and 2 proposed by Wagner [15]. Hence the success of equation 12 in accounting for the growth behavior confirms the validity of Wagner’s material balance hypothesis for the growth of intermediate phase in the diffusion zone.

2. The success of equation 12 in describing the interdiffusion in the Ti-X and Zr-X hinges on two observations. The onset of curvature in the arrhenius plots for ln k2 Vs. 1/T and ln ∆C Vs. 1/T plots occurs at approximately same temperature. Secondly, a reasonably good agreement exits between the experimental and calculated activation energies for interdiffusion. 3. The interdiffsion coefficients and the miscibility gaps have a major effect on growth kinetics of any phase in the diffusion,

References [ 1] Th. Heumann: Z. Metal. Vol. 59 (1968), p. 455 [ 2] G.V. Kidson and G.D. Miller: J. Nucl. Mater., Vol. 12 (1964), p. 61 [ 3] J. Phillibert: Defect and Diffusion Forum, Vol. 95-98 (1993), p. 493 [ 4] K. Hirano and Y. Iijima: in Diffusion In Solids, Recent Developments, (Eds.) M.A. Dayananda and G.E.Murch , TMS Warrandale, PA, P. 144 [ 5] T. Nishizawa and A. Chiba: Trans. JIM, Vol. 16 (1975), p. 767 [ 6] P. Lamparter, T Krabichler and S. Steeb: Z. Metall.,Vol. 64 (1973), p. 720 [ 7] D.S. Williams, R.A. Rapp and J.P.Hirth: Metall. Trans. A , Vol. 12 (1881), p. 639 [ 8] G.-X. Li and G. W. Powell: Acta Metall., Vol. 33 (1985), p. 23 [ 9] M. Danielewski, Mat. Sci. Engg. A, Vol. 120-121 (1989), p. 69 [10] G.B.Gibbs: J. Nucl. Mater., Vol. 20 (1966), p. 303 [11] Y. Funamizu and K. Watanabe : Trans., JIM, Vol. 12 (1974), p. 46 [12] U. Goesele and K.N.Tu: J. Appl. Phys., Vol. 54 (1982), p. 3252 [13] V. I. Dybkov: in Reaction Diffusion and Solid State Reaction Kinetics , The IPMS

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Publications, Kyiv, Ukraine, 2002. [14] D. Gupta: in Diffusion processes in advanced technological materials Ed. D. Gupta, Williams Andrew Publications NY USA, (2005) p. 13 [15] C. Wagner: quoted by W. Jost Diffusion in Solids, Liquids and Gases , Academic Press, (1952), p. 72 [16] O. Taguchi, Y. Iijima and K. Hirano : J. Japan Inst. Metals, Vol. 54 (1990), p. 619 [17] O. Taguchi, Y. Iijima and K. Hirano : J. Alloys and Compounds, Vol. 215 (1994), p. 329 [18] O. Taguchi and Y.Iijima : Mater. Trans. JIM, Vol. 35 (1994), p. 673 [19] O. Taguchi and Y. Iijima : Defect and Diffusion Forum, Vol. 143-147 (1997), p. 597 [20] O. Taguchi, T. Watanobe, Y. Yamazaki and Y. Iijima : Defect and Diffusion Forum, Vol. 194199 (2001), p. 1569 [21] F. Sauer and V. Freise : Z. Electrochem. Vol. 66 (1962), p. 353

Defect and Diffusion Forum Vol. 279 (2008) pp 125-132 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.125

Effect of Cu Addition on Nanocrystallization Behavior in a Co-based Soft Magnetic Metallic Glass A. P. Srivastava, D. Srivastava, *K.G. Suresh and G. K. Dey Materials Science Division, Bhabha Atomic Research Centre, Mumbai 400 085, India *Department of physics, Indian Institute of technology, Mumbai 400 085, India Email: [email protected], [email protected], [email protected], [email protected] Keywords: Metallic Glass; Soft magnetic materials; Magnetism, Microstructure

Abstract. Microstructure and magnetic properties of a nanocrystalline soft magnetic material having composition Co64.5 Fe3.5 Si16.5 B13 Ni1.5 Cu1 has been studied. Amorphous ribbon could be produced by melt spinning unit. DSC analysis showed four distinct crystallization events. Heat treated samples were characterized using XRD and TEM techniques. Co2B, and CoB phases were found to crystallize before magnetic phase α−Co. Addition of copper was proved to have adverse effect on soft magnetic properties. Introduction. Nanocrystallization of amorphous metallic glass has shown to result in improved soft magnetic properties [1-4]. Herzer [5-7] has partially explained the decrease in coercivity with modified random anisotropy model [8]. Improved soft magnetic properties are observed when nanocrystalline particles (of generally size less than ferromagnetic exchange length) are homogeneously distributed in amorphous matrix [1]. It has been seen that the constituents of microstructure which includes morphology, structure, composition and magnetic nature of phases and nature of inter-phases between crystalline and amorphous phases dictate the overall magnetic property in these nanocrystalline materials. Therefore, a better understanding of the microstructure development during nanocrystallization process is required to control the magnetic properties. Cobalt based magnetic materials are known to exhibit larger magnetic induction and higher Curie temperature which are desirable for high temperature application [1]. Many of the Co-Fe based amorphous alloys have been explored as promising soft magnetic materials because in these alloys magnetostriction reduce to near-zero value (as iron and cobalt have opposite signs of magnetostriction), which results in increase in permeability and decrease in Coercivity [1, 9-13]. Metallic glass 2714A with composition Co65Fe4Si15B14Ni2, which is often used in high frequency electromagnetic devices, has shown very good soft magnetic properties [9-13]. In many systems it has been seen that addition of Copper leads to increase in the nucleation rate and hence results in finer nanocrystalline phases [1, 14]. Role of copper in the above mentioned alloy system has not been examined so far. Present paper is an attempt in that direction. In the present work, 1 at% of copper has been added by reducing nickel content in the Metallic glass 2714A alloy. In this study nanocrystallization of the modified composition Co64.5 Fe3.5 Si16.5 B13 Ni1.5 Cu1 alloy from the amorphous precursor has been examined. The effect of nanocrystallization on microstructure has been characterized by X-ray diffraction and transmission

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electron microscopy (TEM) techniques. Vibrating sample magnetometer (VSM) has been used to evaluate the magnetic properties. Experimental procedure. Alloy of composition Co64.5 Fe3.5 Si16.5 B13 Ni1.5Cu1 was produced by melting pure elements (> 99 % purity) in an electric arc furnace. The melting was repeated 5 to 6 times to achieve homogenous composition. The chemical composition and homogeneity of the alloy was ascertained using electron probe micro-analyzer (EPMA). The alloy was cut into small pieces so as to fit in the crucible of Melt spinning unit. Metallic ribbons (as received sample) were produced by planer flow casting technique. Linear wheel speed was set at 40 m/s and argon gas ejection pressure was 1 kg/cm3. These parameters were optimized after number of experiments to get amorphous ribbons. Mettler Toledo DSC 822e differential scanning calorimeter (DSC) was used for determining crystallization temperatures. Constant heating rates of 10°C/min between temperatures 300°C to 625°C were employed in DSC for kinetic studies of the crystallization process. Constant heating rate experiment was also performed at 10°C /min between temperature ranges 300 to 750°C. Nanocrystalline phases were produced by controlled heat treatment of the amorphous precursor. Samples sealed in quartz tube in helium atmosphere were heat treated at temperatures 480°C, 500°C, 550°C and 600°C for 30 min. The different phases present in as received and heat-treated samples were characterized by X-ray diffractometer (Philips, model: PW 1830). The Samples for Transmission electron microscope (TEM) were prepared using Gaton Duo Ion milling unit and examined in a 200 kV JEOL 2000FX TEM. Magnetic properties were evaluated for a few selected samples using Vibrating Sample Magnetometer (VSM). Results and Discussion. The composition of as received ribbon obtained from melt spinning unit was determined by EPMA and is given in Table 1. Table 1 Composition of as received ribbon obtained from EPMA Elements

B

Si

Fe

Co

Ni

Cu

Atomic%

11.90

16.42

4.11

65.13

1.57

0.86

The XRD pattern obtained from melt spun sample is shown in Fig. 1. The pattern exhibited typical humps normally associated with the amorphous phase. DSC technique was used to determine the different events associated with crytstallization. A constant heating rate (10°C/min) experiment was performed. DSC thermogram thus obtained for the temperature range 300°C to 625°C is shown in Fig 2. Four distinct crystallization events at peak temperatures 480, 500, 505, 512°C could be observed from this figure. The DSC scanning was also carried out for higher temperature (upto 750oC) but no additional peak could be seen. In order to examine the

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crystallizing behavior of the different phases, the temperature of the heat treatments were selected as 480, 500, 550 and 600oC for 30 min duration.

Co2B

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Fig 1. X-ray diffraction patterns obtained from as received and heat treated samples X-ray diffraction patterns obtained from heat treated samples are shown in Fig. 1. All the samples exhibited crystalline peaks. The sample heat treated at 500oC for 30 min, showed presence of Co2B and CoB phases. Samples heat treated at higher temperature for the same duration manifested presence of new crystalline peaks which confirmed presence of another phase α−Co, along with Co2B and CoB. The height of the crystalline peak was found to increase with temperature suggesting increase in the crystalline volume fraction and also suggesting presence of residual amorphous phase. 3.0

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Selected area diffraction pattern (SADP) obtained in a TEM of as received sample is shown in Fig.3a. The figure showed two diffuse halos which confirmed that sample was amorphous as seen in X-ray diffraction analysis. The presence of any crystalline phase could not be seen in this sample. Fig. 3(b) and 3(d) show the SADP and the dark field images obtained from sample heat treated at 480oC for 30 minutes respectively. It could be seen from TEM micrograph that nearly spherical shape particles were uniformly distributed in amorphous matrix. Particle sizes were found to be in the range of 15 to 30 nm. The ring pattern confirmed the fine size of the particles as observed in the TEM micrograph. These rings could be indexed with respect to Co2B phase which is consistent with X-ray analysis. In case of sample heat treated at 500oC or above for same duration, particles with different size range were found to be present. SADP and TEM micrograph obtained from the sample heat treated at 500oC are shown in Fig.3(c) and Fig.3(e). This SAD pattern could be indexed with respect to CoB phase. This sample also showed presence of Co2B phase. The size distribution of the nanocrystalline particles for a representative sample heat treated at 500°C/30min is shown in Fig.3(f). From the plot, it can be seen that majority of the particles were in the range of 30-90 nm and small fraction of particles was quite large in size. The SADP patterns obtained from these particles corresponded to the phases Co2B and CoB similar to those seen in the XRD pattern. As expected Co2B phase particles formed at the earlier stage were larger in size in comparison to the CoB phase particles. Fig 3(g) shows the bright field image obtained from sample heat treated at 550oC. Small particles of size less than 10 nm appeared along with the bigger particles. Dark field image taken from these smaller particles is shown in Fig.3(h). Indexing of the SADP’s indicated that in addition to Co2B and CoB phases, α-Co phase was also present and these smaller particles were corresponded to α-Co phase. Fig 3(i) shows the micrograph taken from sample heat treated at 600oC. No significant change in terms of phases present or particle size could be observed in this sample compared to sample treated at 550oC.The interesting observation was made that Co2B and CoB phase particles did not show much of growth in comparison to the sample treated at 500oC. This observation is also supported by the fact that XRD peaks of these two phases did not show any significant change. It can be seen that α−Co phase nucleated at temperature above 550oC were extremely fine (less than 10nm) and this phase also did not show any growth at 600oC. Although four crystallization peaks were observed in the DSC thermogram, but only three phases could be characterized through TEM and XRD. The sequence of the crystallization in Metglass 2714A is formation of α−Co primary crystallization phase at 550oC and followed by boron containing phases CoB and Co2B which are relatively nonmagnetic [15]. However, the TEM and XRD results of the modified Metglass 2714A in this study show the order of crystallization as Co2B, CoB and α-Co. It seems that copper addition has preferentially promoted nucleation of these two nonmagnetic phases and reduced their crystallization temperature substantially whereas, nucleation rate of the magnetic α-Co phase did not change much as crystallization temperature remained same (~550 oC) as in Metglass2724A alloy [15]. .

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(b)

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(c) 111

211 000

(d)

(e)

(f)

0

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Fig. 3(a). SAD pattern obtained from as received sample, 3(b). SAD pattern obtained from Co2B phase in sample heat treated at 480oC/30 min 3(c). SAD pattern obtained from CoB phase in sample heat treated at 500oC/30 min, 3(d) Dark field micrograph obtained from sample heat treated at 480oC/30 min, 3(e). TEM micrograph obtained from sample heat treated at 500oC/30 min, 3(f). The size distribution of the nanocrystalline particles sample heat treated at 500°C/30min, 3(g). Bright field image obtained from sample heat treated at 550oC/30min, 3(h). Dark field image obtained from sample heat treated at 550oC/30min, 3(i). TEM micrograph taken from sample heat treated at 600oC

Phase Transformation and Diffusion

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Fig.4. Magnetic hysteresis loops of as-received and heat treated samples TEM micrograph showed that in all the nanocrystallized samples, particles were found to be uniformly distributed throughout the volume. This ensures better ferromagnetic coupling between ferromagnetic particles, which is the essential requirement to obtain better soft magnetic property. Hysteresis loops for as received and heat treated samples at 500, 550 and 600oC for 30 min obtained from VSM at 470C are shown in Fig.4. The typical rectangular shape of the hysteresis loops confirmed that all samples possessed good soft magnetic properties. The saturation magnetization, Ms, and coercivity, Hc, values obtained from the hysteresis loops are listed in the Table 2. From this table, it could be seen that as received sample has quite low coercivity (70 Oe) which indicated that amorphous sample itself was a good soft magnetic material. It could be noticed that the nanocrystalline samples exhibited small decrease in the Ms value after nanocrystallization as compared to amorphous sample. This decrease in heat treated samples can be attributed to the formation of boron containing relatively non magnetic Co2B and CoB phases. The volume fraction

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of these two phases is expected to increase at higher temperature which should further decrease the Ms value as can be seen in Table 2. But the extent of decrease was rather insignificant due to appearance of magnetic α−Co phase, whose volume fraction seems to increase with temperature along with those of nonmagnetic phases. It is also noticed that, the coercivity in as-received sample is lower than that in the heat treated samples. This can be explained by the fact that, in the heat treated samples the nanocrystalline particles may have a higher value of magnetostriction (λs) than the amorphous matrix with the same sign which gets added up, giving greater value of λs. This in turn may result in increase in coercivity. In case of heat treated samples, it is found that the coercivity increases from 150 Oe for 500°C/30min sample to 286 Oe for sample annealed at 600°C/30min. This increase in the value of Hc may be accounted by the fact that for small grain sizes (< 100nm), Hc increases with increases in grain size [1]. So we conclude that as the sample is annealed at higher temperatures, the nanocrystalline grain size increases giving a higher value of Hc. It has been observed that the as received sample has good soft magnetic properties, which deteriorates on annealing as the particle size increases. Table 2 The saturation magnetization, Ms, and coercivity, Hc, values obtained from the hysteresis curve of as received and heat treated samples Sample description As received 500 oC/ 30 min 550 oC/ 30 min 600 oC/ 30 min

Ms (emu/g) 61.24 60.41 55.11 53.66

Hc (Oe) 70 150 255 286

Conclusion. Amorphous ribbon of modified composition of Metglas 2714A could be produced by melt spinning unit. It showed good soft magnetic properties. DSC thermogram indicated four crystallization events in the temperature range 300 to 625°C. TEM and XRD results confirmed that phases Co2B and CoB appeared before α-Co phase. α-Co phase appeared at 550°C as found in case of unmodified Metglas 2714A. Addition of copper assisted formation of boron containing nonmagnetic phases and reduced their crystallization temperature. This resulted in reversal of sequence of the phase formation. The preferential formation of the two relatively non magnetic phases resulted in moderate deterioration of soft magnetic properties of nanocrystallized samples. References. [ 1] F. M. McHenry, M. A. Willard, D. E. Laughlin: Prog. Mater. Sci. Vol. 44 (1999), p.291 [ 2] Y. Yoshijawa, S. Oguma, K. Yamauchi: J. Appl. Phys. Vol. 64 (1988), p.6044 [ 3] K. Suzuki, A. Makino, N. Kataika, A. Inoue, T. Masumoto: Mater. Trans. JIM. Vol. 32 (1991), p.93

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[ 4] M. A. Willard, M. Q. Huang, D. E. Laughlin, M. E. McHenry, J. O. Cross, V.G. Harris, C. Franchetti: J. Appl. Phys. Vol. 85 (1999), p.4421 [ 5] G. Herzer: IEEE Trans. Magn. Mag. Vol. 26 (1990), p.1397 [ 6] G. Herzer: Hand Book of Magnetic Materials, Elsevier Science, Amsterdam, vol 10 (1997) chap. 3, p. 415 [ 7] G. Herzer: J. Mag. Mag. Mater. Vol. 112 (1992), p.258 [ 8] Alben, J. J. Becker, M. C. Chi: J. Appl. Phys. Vol. 48 (1978), p.1653 [ 9] S.H. Lim, Y.S. Choi, T.H. Noh, I.K. Kang: J. Appl. Phys. Vol. 75 (1994), p.6937. [10] G. Bordin, G. Buttino, A. Cecchetti, M. Cecchetti, M. Poppi: J. Magn. Magn. Mater. Vol. 153 (1996), p.285. [11] G. Buttino, A. Cecchetti, M. Poppi: J. Magn. Magn. Mater. Vol. 172 (1997), p.147 [12] C.S. Tsai, W.J. Yang, M.S. Leu, C.S. Lin: J. Appl. Phys. Vol. 70 (1991), p. 5846 [13] J.H. Yang, Y.B. Kim, K.S. Ryu, M.J. Kim, Y.C. Chung, T.K. Kim: J. Magn. Magn. Mater. Vol. 222 (2000), p. 65. [14] M. C. Weinberg, D. P. Birnie III, and V. A. Shneidman: J Non Cryst. Solids, Vol. 219 (1997), p.89. [15] I. Betancourt, M. Jiménez, S. Aburto, V. Marquina, R. Gómez, M.L. Marquina, R. Ridaura, M. Miki and R. Valenzuela: J. Magn. Magn. Mater. Vol. 140 (1995), p.459

Defect and Diffusion Forum Vol. 279 (2008) pp 133-138 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.133

Alpha to Omega Transition in Shock Compressed Zirconium: Crystallographic Aspects G. Jyoti1a, R. Tewari1a, K. D. Joshi2b, D. Srivastava1a, G. K. Dey1a, S. C. Gupta2b, S. K. Sikka3 and S Banerjee1a 1

Material Science Division, Bhabha Atomic Research Centre, Mumbai 400085, India

2

Applied Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India 3

Secretary,Office of Principal Scientific Advisor, Government of India, New Delhi.

E-mail: [email protected], artewari@ barc.gov.in, [email protected], [email protected], a [email protected], bkdj @ barc.gov.in, bsatish@ barc.gov.in, Keywords: Phase transformations, ω phase, high pressure, orientation relationships.

Abstract. In the present study, specimens of Zr were subjected to shock compression of 11.6 GPa. TEM examination of the recovered samples revealed that during shock compression the α phase has transformed into the ω phase. The orientation relationships (ORs) between the α and ω phase have been determined using both the stereographic projection method and the correspondence matrix method. Our ORs have been found to belong to the Variant I OR given by Usikov and Zilbershtein (UZ) for statically compressed Zr samples. Our ORs are the same as the one reported by Song and Gray (SG) on dynamically compressed samples. In the present paper it has been shown that the OR of SG is a subset of the OR of UZ and is not apart from it. The mechanism of α→ω transition with respect to occurrence of an intermediate β (bcc) structure, during the transition has also been explored. We also show in this study that the amount of the transformed ω phase decreases with increasing oxygen content in the samples that were shock loaded to the same peak pressure, as is revealed by both the TEM and XRD results. Introduction. Silcock [1] was the first to propose a model for the formation of ω (hexagonal) phase from α (hcp) lattice but the model involved large atomic movements. Sargent and Conrad [2] reported the alignment of the (0 0 0 1)ω plane parallel to the (112 0) α plane in Ti metal. In subsequent studies on pure Zr and Ti, Usikov and Zilbershtein (UZ) [3] suggested an alternative mechanism for the α→ω transformation in which β(bcc) appears as an intermediate unstable phase. They arrived at the correspondence matrix between the α and ω phases from the well established correspondence matrices between α to β and β to ω transformations [4]. The two crystallographically nonequivalent variant ORs, so obtained, are, (0 0 0 1)α  (011 1)ω ; [1 12 0]α [1 0 1 1]ω

Variant I

(0 0 0 1)α  (1 1 2 0 )ω; [1 12 0]α [0 0 0 1]ω

Variant II

Selected area electron diffraction (SAD) patterns from statically compressed Zr and Ti samples have shown that all the experimental ORs between α and ω phases could be described by the correspondence matrices associated with the variant I OR. This was taken as the verification that the

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α to ω transformation proceeds via the intermediate β structure. A direct evidence of the formation of the intermediate β-structure during the α to ω transition in alloys has been provided by transmission electron microscopy (TEM) studies on high pressure treated Ti-V alloys by Vohra et al [5]. The explanation for the occurrence of the four variants of ω phase from a parent β phase and the particle morphology of the ω phase in these alloys was explained on the (111) plane collapse model of de Fontaine [6]. In this plane collapse model the formation of the ω phase from the β phase can be considered as resulting from collapse of a pair of consecutive (111) planes of the bcc lattice to the intermediate position, leaving the next plane unchanged, collapsing the next pair again and so on. And since there are four equivalent bcc directions in the bcc lattice, one expects four equivalent orientations of ω phase. Given this mode of ω formation and also the fact that a slight change in volume at the β→ω transition introduces some elastic strain, the precipitation of ω phase in tiny particle morphology is thus expected. Subsequent transmission electron microscopy measurements on pressure treated Zr samples by Rabinkin et al [7] revealed the presence of ellipsoidal ω particles, similar to those observed in as quenched β phase alloys of Ti and Zr. All the SAD patterns conformed to Variant II OR of UZ. They proposed a different diffusionless mechanism for this transformation. OR studies have also been carried out on Zr samples shock compressed under normal plate impact conditions; Kutsar et al [8] reported the α/ω OR in the shock treated Zr to be the same as the variant II OR of UZ. Later investigations by Song and Gray (SG) [9, 10] in shock compressed Zr, have reported the following O.R between the α and ω phases, (0 0 0 1)α  (1 0 1 1)ω ; [11 0 0]α  [1 12 3]ω According to Song and Gray [9], the above OR is new and did not agree with those reported previously in static high pressure experiments. They proposed a different transformation mechanism in which the α phase transforms directly to the ω phase without involving any intermediate β structure. The present study was undertaken with the objective of comparing the orientation relationships and/or the mechanism of α→ω transformation under static and shock compression by doing an independent TEM study on shock compressed Zr. We have also examined the effect of oxygen content on the α→ω transition in Zr. Experimental. Four Zr samples with differing oxygen content of 200, 480, 1600 and 2800 ppm, respectively, were emplaced in a single recovery capsule, in a proper scheme, and a dynamic compression experiment was carried out, using gas gun [11], at a projectile velocity of 0.6 mm/µs. The pressure in the Zr samples was computed to be 11.8 GPa. Fig. 1 gives the x-ray diffraction (XRD) pattern of the shock recovered samples, showing the presence of ω phase along with the α phase for 200 ppm oxygen content sample. Further, specimen samples were prepared from all the four recovered samples i.e. the aforementioned varying oxygen content samples, for carrying out electron microscopy investigations using JEOL 2000 FX electron microscope,. The sample with 200 ppm oxygen, especially, was used for detailed crystallographic studies.

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Figure 1 XRD patterns of (a) 200 and (b)1600 ppm oxygen content samples. Results and Discussion. From the x-ray diffraction patterns of the samples (Fig.1), we find that the XRD pattern of the sample with 200 ppm oxygen content (Fig. 1(a)) shows substantial amount of the ω phase coexisting with the parent α phase whereas the sample with 480 ppm oxygen content showed just the beginning of the formation of ω phase. No ω was observed in the XRD pattern of Zr samples with 1600 ppm (Fig.1(b)) and 2800ppm oxygen content. TEM investigations on the recovered samples showed plate shape morphology of the ω phase. Fig. 2b, Fig. 3a and Fig. 3b show the microstructures of 200, 1600 and 2800ppm oxygen content samples, respectively. As can be seen from these microstructures, number density of the ω plates reduces with increase in oxygen content. The width of these plates also reduces from ~160 nm to 100nm. Also, it may be noted that even though the XRD of samples containing 1600 and 2800 ppm oxygen did not show any ω phase, its presence was detected under TEM.

Fig.2 For 200ppm oxygen content sample (a) SAD pattern with zone axis [1 0 1 0]α || [1 123]ω. (b) corresponding dark field micrograph.

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Figure 3 Micrographs of the (a)1600 and (b) 2800 ppm oxygen content samples. Sample containing 200ppm oxygen content was further investigated for the crystallographic studies. After verifying the uniformity of microstructure from a number of areas in the sample, large number of SAD patterns and corresponding micrographs were obtained. ORs of many of the patterns were then determined from the region of co-existence of the two phases. In the ORs so determined we find that one of our experimental ORs is exactly the same as the variant I OR given by UZ and also interestingly another one is the same as the one given by Song and Gray. Fig. 2a shows one of our SAD pattern, which is identical to the one reported by SG [9]. The corresponding dark field micrograph is given in Fig 2b. To further examine the crystallographic relationship between the α and the ω phases, the (0 0 0 1)α and (0 11 1)ω stereographic projections were superimposed maintaining a coincidence between the [21 1 0]α and [1 011]ω directions, i.e satisfying the variant I OR of UZ. In such a superposition of stereograms, shown in Fig.4, which displays the plot of the great circles representing directions, we find [12] all our experimentally determined ORs to be satisfied including the one that was reported by SG, implying the equivalence of their OR with the variant I class OR of UZ. The correspondence matrices for the α→ω transformation were also computed both for the direct and reciprocal lattice using the experimentally determined ORs. The matrices were found to be identical to the correspondence matrices derived by UZ for variant I class O.R. and also were found compatible with the ORs reported by SG [10] also but are different from the ones given by SG. The matrix given by SG did not satisfy any of the ORs determined by us. This again implies that the OR reported by Song and Gray is equivalent to and a subset of the Variant I OR of UZ. Further, assuming that the α→ω transformation proceeds via the β phase, three strains, ∈1, ∈2 and ∈3 were identified for causing this transition. Accordingly, the formation of intermediate β structure requires strains ∈1 and ∈2 to operate together first. The strain ∈3 then causes collapse of (111) planes to effect the β→ω transition. Hence, if the intermediate β structure is formed and if the plane collapse model for β→ω transition operates, one expects to observe reflections from more than one variant in the SAD patterns, and correspondingly, particles of ω phase in the parent matrix in the micrograph. None of these features are observed in our study. In contrast, only a single variant of ω phase and a large plate–like structure is seen in the micrographs.

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Figure 4. (0 0 0 1)α and (0 11 1)ω stereographic projections showing great circles corresponding to the directions for our experimental orientation relations between the α and the ω phases.

Summary. Summarily, the amount of formation of ω phase from the α phase decreases as the oxygen content increases. Also, it is clear that the ORs obtained from SAD measurements provide the crystallographic relations between the initial and final structures of a transformation and these can not unambiguously establish the mechanism of transition in terms of atomic movements. Regarding the mechanism of α→ω transformation in Zr; either all the strains are coupled so that ω phase forms directly from the α phase without the β structure formed as an intermediate step or if the intermediate β is formed, the β→ω transformation does not operate by collapse of (1 1 1)β planes (as observed in quenched Ti and Zr alloys) but is driven by some shear strain as reported by Dey et al [13] in Zr-20Nb alloy where plate shape ω has been observed from the initial β structure in Zr20Nb under shock compression. It is difficult to conclude as of now, which of the two possibilities is experimentally realized, the determination of the habit plane between the α and the ω phases may shed some light on this issue. References [1] J. M. Silcock, Acta Metall., Vol. 6 (1958), p. 48. [2] G.A. Sargent and H. Conrad, Mater. Sci. Engg., Vol. 7 (1971), p.220.

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[3] M. P. Usikov and V. A. Zilbershtein, Phys. Stat. Sol.(a), Vol. 19 (1973), p.53. [4] A.B Notkin, L. M. Utevskii, P.V. Terentieva and M. P. Usikov, Zav. Labor, Vol. 8 (1973), p. 970. [5] Y.K. Vohra, E. S. K. Menon, S.K. Sikka and R. Krishnan, Acta Metall, Vol. 29 (1981), p. 457. [6] D. de Fontaine, Phil. Mag.,Vol. 27 (1973), p.967. [7] A. Rabinkin, M. Talianker and O.K. Botstein, Acta. Metall., Vol. 29 (1981), p.691. [8] A. R. Kutsar, I. V. Lyasotski, A. M Podurets and A. F Sanches-Bolinches, High Pressure Research, Vol. 4 (1990), p.475. [9] S. Song and G.T. Gray III, High Pressure Science and Technology-1993, edited by S.C. Schmidt, J.W. Fowles, G.S. Samara and M. Ross, AIP Press, New York (1994), p.251. [10] S. Song and G.T. Gray III, Philosophical Magazine A, Vol. 71 (1995), p. 275. [11] S. C. Gupta, R.G. Agarwal, G. Jyoti, S. Roy, N. Suresh, S. K. Sikka, A. K. Kakodkar and R. Chidambaram, Shock waves in Condensed Matter-1991, edited by S.C. Schmidt, K.D. Dick, J. W. Fowles and D. G. Tasker, Elsevier, Amsterdam (1992), p.834. [12] G. Jyoti, K. D. Joshi, S.C. Gupta, S.K. Sikka, G. K. Dey and S. Banerjee, Phil. Mag. Lett, Vol. 75 (1997), p.291. [13] G.K. Dey, R. Tewari, S. Banerjee, G. Jyoti, S.C Gupta, K.D. Joshi and S.K. Sikka, Phil. Mag.Lett, Vol. 82 (2002), p.333.

Defect and Diffusion Forum Vol. 279 (2008) pp 139-146 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.279.139

Selection of Lattice Invariant Shear in Dilute Zr-Nb Alloy for bcc-hcp Martensitic Transformation D. Srivastavaa, G. K. Deyb and S. Banerjeec Materials Science Division, Bhabha Atomic Research Center, Mumbai 400 085, India Email: [email protected], [email protected], [email protected] Keywords: Martensite, lattice invariant shear, Zr-Nb alloy

Abstract. The morphology and substructure of martensite is considered to arise from the lattice invariant shear (LIS) associated with the transformation and this may be slip, twinning or both. Out of the several possible slip shears and twin modes only a few satisfy invariant plane strain criteria of the phenomenological theory of martensite (PTMC). On the basis of crystallographic and energetic criteria, a simple model has been proposed for determining the factors which influence the selection of the preferred LIS mode. In the present work, it is found that for β → α' martensitic transformation in Zr-2.5 wt%Nb alloy, the preferred slip system is {1101}α'α' and the preferred twin system is {1101}α'α'. Introduction. The crystallographic and microstructural features of the bcc to hcp martensitic transformation in Ti, Zr, Li, Hf and their alloys have been reviewed extensively [1-2]. The presence of lattice defects such as dislocations and twins in the martensitic microstructure can generally be correlated with the inhomogeneous lattice invariant shear (LIS) of the martensitic transformation [1-2]. In internally twinned martensite {1101}α'α' twin system [1-4], and in internally slipped martensites, dislocations with predominantly α' Burgers vectors[5] at the interfaces of the laths and within the plates of the martensite have been observed. Experimental observations made on morphology and substructure of martensitic microstructure does not confirm the presence of all the possible slip and twin modes as LIS. The primary requirement for any shear system to be a LIS is that it should meet the invariant plane strain (IPS) condition and it should yield the habit plane solution. Based on this requirement, Otte has proposed generalized theory of the martensitic transformation for cubic to orthorhombic [6] in which he has worked out restrictions that have to be imposed on the starting data (possible LIS) for the existence of the habit plane solution. Crocker and Bilby [7] have considered these restrictions for the most general case of the martensitic transformation. However, in practice, amongst all the shear systems which qualify to be the LIS modes on the basis of PTMC only a few seem to be operative, in general, in the martensitic transformation process. Several studies have suggested that there are other factors, which possibly influence the selection of a specific LIS in addition to the PTMC criterion. In the bcc to hcp martensitic transformation also only a few LIS modes amongst the possible LIS modes have been confirmed in microstructural studies [1-6]. Otte [6] has employed a crystallographic approach and suggested two LIS systems ( 2 112)α'α' and {1101}α'α' in Ti base alloys.

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In the present work, a simple model has been proposed for determining the factors, in addition to the PTMC criterion, which influence the selection of the preferred LIS mode in the bcc-hcp martensitic transformation in Zr-2.5 wt%Nb alloy.

Experimental. The Zr-Nb alloy samples were solutionised in the β phase field at about 1000οC for 15 minutes and subsequently quenched in water. Preparation of samples for TEM was carried out by jet electropolishing using a solution containing 30 parts perchloric acid, 170 parts n-butanol and 300 parts methanol at temperature around -40oC and voltage 20 V. TEM thin foils were examined in a JEOL2000FX microscope operating at 200 kV.

Results and Discussion. TEM examination of the microstructure of the dilute Zr-Nb alloys from the β phase field showed two distinct types of martensite morphology, dislocated laths and twinned plates that is typical of zirconium and titanium martensites [1-5]. Packets of lath martensite were observed in several regions as shown in Fig. 1. These laths are generally aligned parallel to one another in groups (packets). The lath boundaries (lath–lath interfaces) consisted of equally spaced parallel arrays of dislocations of the type and were lying on {1011}α' plane. The detailed analysis of lath martensitic microstructure is presented elsewhere[5].Some of these plates were internally twinned and the twinning plane was found to be a {1011}α’ type (Fig.1). The average twin thickness ratio of the two fractions was observed to lie in the range of 3.4:1 to 4.2:1. The habit plane traces of both lath as well plate martensites were observed to lie close to {334}β type poles [3-5 ] . The observation of these two types of lattice invariant shear i.e. slip system {1101}α'α' (lath martensite) and twin system {1101}α' α' (internally twinned martensite) has been rationalized in the following section.

(a)

(b)

0.5µm

0.1µm

Fig.1 Bright field TEM micrograph showing internally slipped and twinned martensite in β quenched Zr-2.5 wt% Nb alloy

Selection of LIS. The possible LIS in the bcc to hcp martensitic transformation in Zr- Nb alloy system which meet the invariant plane strain (IPS) satisfying Crocker and Bilby criterion, has been computed by Srivastava et al [8-9] and those are listed in Table 1. In this study an attempt has been made to determine the basis for the selection of experimentally observed LIS system out of these possible LIS

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systems. On the basis of crystallographic and energetic criteria, the factors which could possibly influence the preference for a particular twin or slip system as the LIS in thermally induced martensites in Zr-Nb are described below. Each of these has been analyzed independently and their overall effect with regard to the bcc-hcp transformation has been discussed.

I. Magnitude of Shear and Shape Strain Shear. The magnitudes of the shear associated with all the possible slip systems had been computed by Srivastava et. al. [8-9], and those are presented in Table 1. It could be seen that the shear magnitude for different slip systems can be as low as 0.0258 (for the { 2 112}α' α' shear system) and as high as 0.4481 (for the {1101}α' [0111] α' slip system). It could be seen that the values of shear associated with the Burgers vector (slip direction) α' are much lower in comparison with those associated with the Burgers vectors α', α' and α'. It is expected that the shear system with the lowest shear magnitude would be the preferred shear system. The three shear systems {1101}α' α', ( 2 112)α'α' and {11 2 1 }α' α' have the lowest values of the magnitude of shear, which are given by 0.0288, 0.0258 and 0.0410 respectively. Since the differences in the values of the shear for these slip systems are very small, all of them could be considered as preferred slip systems. The magnitudes of the shear associated with other slip systems are much higher. In view of this they can be ruled out to be LIS in under the normal circumstances. Table 1 The possible slip systems (in hexagonal lattice) which qualify as LIS in bcc to hcp martensitic transformation in Zr-2.5wt%Nb alloy and their corresponding shear magnitude slip systems

shear magnitude 0.0288 0.0821 0.0410 0.0258 0.1270 0.1270 0.1095

(1101) [2113] (0110) [2113] (11 2 1) [2113] ( 2 112) [2113] (0001) [1 2 10] (1011) [1 2 10] (1013) [1 2 10]

slip systems (1013) [1 2 10] (0001) [0110] (2114 ) [0110] (2112) [0110] (3034) [0110] (1101) [0111] (0112) [0111]

shear magnitude 0.2246 0.0499 0.0885 0.1363 0.2372 0.4481 0.1892

slip systems (2 3 14) [0111] (1101) [1 2 13] (1121) [1 2 13] (1011) [1123] (1 2 11) [1123]

Shear magnitude. 0.2344 0.0287 0.1420 0.0287 0.1200

Table 2 Magnitude of shear and shape strain for selected LIS modes [8-9] Type LIS

of Twin Slip

LIS mode (1101)[41 5 3] (0112)[0111] (1101)[21 13] ( 2 112)[21 13] (11 2 1) [2113]

Slip/twin shear

Shape strain magnitude

0.4119 0.7850

0.154 0.215

0.0288 0.0258 0.0410

0.192 0.191 0.197

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Similarly the preference for the {1101}α'α' twin system over the {0112}α' α' twin systems in the bcc to hcp transformation in the Zr-2.5wt% Nb alloy could be seen from Table 2. Another important point to be noted is that the values of shear associated with the slip systems are much lower than those for twin modes. This suggests that just on the basis of the magnitude of shear a slip mode will be preferred over a twin mode as the LIS in the bcc-hcp martensitic transformation. The computed values of the shape strain for the possible slip modes lead to results similar to those obtained on the basis of the magnitude of shear. The values pertinent to the three slip systems and the two twin modes selected above are presented in Table 2. It may be noted that magnitude of the total shape strain is the lowest for the {1101}α' α' twin mode and the highest for the {0112}α' α' twin mode and that the values for the two slip systems (1011)α' [2113]α' and ( 2 112)α'[2113]α' are almost the same and much lower than that for (11 2 1)α'[2113]α' slip system.

II Force Required to Move a Dislocation. In a normal deformation process the preferred slip plane is usually the plane associated with the largest interplanar distance (i. e. with the largest atomic density) and the slip direction is nearly always the one that contains the shortest Burgers vector. In the bcc-hcp martensitic transformation, it can be observed that all the three slip systems associated with the lowest values of the magnitude of shear have a common slip direction (i.e., Burgers vector). Therefore, it is the slip plane which would decide the preference for a specific slip mode in this case. The atom movements required for slip to occur in hcp metals, each of the {1101}α' and {11 2 1}α' slip planes offers two α' slip directions ( therefore twelve slip systems) as compared to only one slip direction for the { 2 112}α'. slip plane (six slip systems). In mechanical deformation, owing to their higher availability of slip systems, their higher atomic density, and their less pronounced waviness, the {1101}α' and {11 2 1}α' planes are thus favored energetically compared with the { 2 112}α' planes. Although in martensitic transformation only one slip system is operative at a given time still the other two factors (interplanar spacing and waviness) would favor the {1101}α' and the {11 2 1}α' planes as slip planes. The influence of these factors is reflected in the magnitude of the force (Peierles and Nabarro (P-N)), τp, required to move a dislocation through the crystal lattice and is expressed as:

τp = 2G/(1-ν)exp [2πa/(1-ν)b]

(1)

where G is the bulk modulus, ν is Poisson ratio, a is the interplanar spacing and b is the dislocation Burgers vector. It is obvious that to move a given dislocation the preferred slip system would be that which is associated with the minimum magnitude of the P-N force. Since for the above three slip systems apart from a all other terms are constant, expression (1) can be rewritten as

τp = k1e k2.a

(2)

where k1 and k2 are constants. For Zr-2.5 wt% Nb alloy, the inter-planar spacing of the planes {1101}>α', {11 2 1}α' and { 2 112}α' is computed to be 0.244, 0.154 and 0.136 nm respectively. The magnitude of the associated Burgers vector (b) for α' is computed to be. 0.523 nm and Poisson's

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ratio is equal to 0.33 [3]. Putting these values, the ratios of the magnitudes of the P-N force for the three slip modes are given by ; τp { 2 112}α' : τp {11 2 1}α’ : τp {1101}α’ :: 7:5.5:1 Thus, in the absence of any other external factor, the {1101}α' 99.5%) Al, i and Ti powder of particle sizes of -325 mesh (45 µm) for MA, in a high energy Fritsch P5 planetary

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Phase Transformation and Diffusion

ball mill, using cemented-carbide grinding media and in toluene atmosphere, at a mill speed of 300 rpm and ball to powder weight ratio of 10:1. The identity and phase evolution at different stages of MA were studied by X-ray diffraction (XRD) analysis. To confirm the presence of phase(s), a few samples, which are relatively important for the study, were examined using transmission electron microscope (TEM) and will be published elsewhere. Results and discussion. It is clear from Table 1 that all the alloys have significantly large negative enthalpy of mixing and hence, amorphous phases are likely to form during their MA. However, the experimental results showed mainly three different kinds of phase evolution (summarized in Table 1). Again, Alloy 1 has large driving force for amorphization however it did not yield amorphous phases (see Table 1). In contrast, it could be noted that amorphous phases were found in alloys having very small driving force for amorphization. Murty et al. [5] and Nagarajan et al. [6] have reported a similar discrepancy. Therefore, there is a need for consideration of other useful criterion, which will take into account of experimental parameter(s). Table 1 Phases observed during MA of various compositions studies in the present work and their corresponding thermodynamic parameters. Alloy Composition No. (at. %) Al-Ni-Ti

1

10-15-75

2 3 4 5 6 7 8 9 10 11 12

15-5-80 20-25-55 22-22-56 25-10-65 30-5-65 30.5-37-32.5 31-8-61 51-37-12 55.5-37-7.5 60-20-20 70-15-15

Enthalpy of Driving force of Milling Phase(s) in mixing in amorphization time as-milled amorphous (kJ/mol) (h) powder phase (kJ/mol) -28.919 -2.844 30 Amorphous + Crystalline. -19.613 +1.8709 20 Crystalline -54.586 -13.265 20 Amorphous -53.287 -12.270 20 Amorphous -38.430 -4.676 30 Amorphous -33.884 -2.874 30 Am.+ Cryst. -70.035 -22.510 60 Crystalline -39.896 -5.266 30 Amorphous -48.231 -12.758 60 Crystalline -38.625 -9.263 60 Crystalline -48.959 -11.79 10 Am.+ Cryst. -36.756 -7.049 10 Am.+ Cryst.

It is seen in this study that the AFCs are aligned in straight line when Hdiff is plotted against the chemical contribution to enthalpy of solid solution, ∆Hss(chem) in case of a number of binary systems, which have been extensively studied by many authors, reviewed by Weeber and Bakker [2]. Hdiff defines the change in enthalpy in the process of amorphization from the pure crystalline elements, calculated by subtracting the sum of the enthalpies of pure solid elements from the enthalpy of the amorphous phase (Ha) (Eqn.(1)) for the binary A-B system. It can be noticed that enthalpies of the pure elements are assumed to be zero in their standard state. Therefore, magnitude of Hdiff is same as that of Ha (i.e. Hdiff ≈ Ha). Since aim of the study is to determine the AFCs and starting material is elemental crystalline powder the Hdiff is an important term rather than enthalpy of amorphous phase or enthalpy of mixing in amorphous phase.

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(

am ∆Hdiff = XA∆HAa + XB∆HBa + HAB − XA∆HAss + XB∆HBss

)

149

(1)

a

where, Xi (=A, B), ∆H i (= A,B) and ∆H iss(= A, B ) are the mole fraction of i-th element, the enthalpy of amorphous pure i-th element and the enthalpy of crystalline, pure i-th element respectively and ∆H ijam(= AB ) is the enthalpy of mixing of i-th and j-th element in amorphous phase. The aforesaid linear relationship can also be written in the form of an Eqn. (2) shown below, where, m is the slope of the line and ‘C’ is the intercept on Hdiff axis.

(2)

∆H diff = m∆H ss(chem) + C

It can be stated from Eqn. (2) that when Hss(chem) =0, Hdiff = C. In such a case, ∆Hdiff is nothing but the sum of enthalpies of fusion of the individual pure elements at the experimental temperature when there is no mixing among the elements and thus ‘C’ can be expressed, as shown in Eqn. (3) for binary A-B system.   Texp  Texp  X + ∆Hm +  Cliq,B − Csolid,B dTX ,A solid, A  C = ∆HAm +  ∫ Cliq − C dT p p A  B B ∫ p p   TfA TfB      

(

)

(

)

(3)

Where, ∆H Am (or B) , XA (or B), TfA(or B ) are the enthalpies of fusion of pure element A (or B) at its melting point, the mole fraction of A (or B) and the melting temperature of A (or B), respectively. ( or solid ), A( or B ) C liq is the molar heat capacity of A (or B) for its corresponding phases. Texp is the p experimental temperature, which is the temperature generated in the reaction front of the trapped powder-mass due to heat of reaction and the heat effect of the collision during MA and TfA(or B ) is the melting point of pure crystalline element of A (or B). The expression for enthalpy of difference between the amorphous phase and the constituent elements (i.e. Hdiff) is the same in magnitude as Ha at room temperature and is given by Eqn. (4). This difference in enthalpy is assumed to remain the same at higher temperature [7]. Therefore, slope (m) can be equated from Eqns. (2) and (4), which is given in Eqn. (5). ∆Hdiff = (SROF) ∆H ss( chem) +∆Htopo

(

)

m = SROF + ∆H topo−C ∆Hss(chem)

(4) (5)

SROF is short range-ordering factor [8] and ∆H topo is the topological contribution to enthalpy of amorphous phase [9].The calculation for ternary systems has been easily extended from the above Eqns.1-5 by simple addition rule which will be published elsewhere.

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Phase Transformation and Diffusion

10

13.253 kJ/mol

-10

-20

Experimented alloys in this study
crystalline ⌧ amorphous + crystalline u amorphous

2




Y-axis

Difference of enthalpy (∆ H (kJ/mol)

diff

)

0

Alloys calculated based on Driving force for amorphization AFCs a Non-AFCs H =0

1⌧

-30

6

⌧

5
⌧ 8 uu
12

-40

-50

4

u 3
u

AFC line,Ni-Ti AFC line, Ti-Al Experimented alloys from the literature [6,10]  crystalline  amorphous + crystalline
amorphous




-60

 -70




7

-60

-50

AFC line, Al-Ni-Ti

10

 


⌧
9 11

-40

-30

-20

-10

Chemical contribution to enthalpy of solidsolution ( H (kJ/mol)

0

ss(chem)

)

Fig. 1 Variation of the difference of the enthalpy vs. chemical (∆Hdiff) contribution to enthalpy of solid solution (∆Hss(chem)) showing superimposed enthalpy data corresponding to different alloys of the systems: Al-i-Ti, i-Ti and Ti-Al. Lines, drawn through AFCs of the different systems show different slopes.

In Fig. 1, the enthalpy data corresponding to AFCs of the three binary systems and the ternary system have been merged. It can be easily observed that based on driving force for amorphization criterion a large number of compositions, marked as open circles in the ∆Hdiff vs. ∆Hss(chem) plots, denote the predicted AFCRs of the Al-i-Ti. Whereas, all the compositions below the dashed horizontal line denoting Ha equals to zero, in the Fig.1, constitute the AFCRs based on negative enthalpy of amorphous phase or large negative enthalpy of mixing in amorphous phase. Although, Alloys 7, 9 and 10 have large negative enthalpy of mixing/ sufficiently large driving force for amorphization, according to extended Miedema model calculation (see Table 1), did not show any signature of amorphization in any stage of milling up to 60 h of milling and lie away from the ternary line in Fig.1. MA of Alloy 11, Alloys 6 and 12 showed formation of the amorphous plus crystalline phase(s) formed within 10-20 h of milling and then, on further milling, transformed to completely crystalline phase(s) and these alloys are close to the straight line in the ∆Hdiff vs. ∆Hss(chem) plot (Fig. 1). The Alloys 3, 4, 5 and 8, were found to form amorphous phase after MA of 10 to 20 h. The reported AFCs, by Nagarajan et al. [6] and Itsukaichi et al. [10] are also found to lie on the line (Fig. 1) and non-AFCs are placed on both sides away from the line. For any composition, the extent of deviation to follow the relationship between the two enthalpies (i.e. Hdiff and Hss(chem) ) is referred to as parameter δ, expressed in terms of Hdiff. δ is defined as the difference between Hdiff of the composition and Hdiff corresponding to the AFC line (Fig. 1) as shown in Eqn. (6).

δ = (∆Hdiff)Compositio n

(

)

- ∆Hdiff AFC Line

(6))

This parameter signifies the amount of extra/deficit of energy available during MA of any composition, which will effect in temperature higher/lower than its amorphization temperature. The results from the experimented compositions of this system in the earlier studies [6, 10] fully support our observation. Conclusions. The linear relationship between enthalpy difference between amorphous and crystalline elements (∆Hdiff) and chemical contribution to enthalpy of solid solution (∆Hss(chem)) or magnitude of δ coupled with the criterion of large negative enthalpy of mixing, can be used to select the AFCs in Al-i-Ti system for MA experiments and can be used for other binary or ternary systems. Nanocrystalline phase(s) embedded in amorphous phase can be synthesized by proper selection of composition considering parameter, δ, and the milling time of MA.

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References [1] R.B. Schwarz, W.Johnson: Phys. Rev. Lett. Vol.; 51(1983), p.415. [2] A.W. Weeber, H. Bakker: Physica B Vol. 153(1998), p.93. [3] A.R. Miedema, P.F.de Chatel,F.R.de Boer:Physica B Vol.100 (1980), p.1. [4] L.J.Gallego, J.A. Somoga, J.A. Alonso: J Phys B Vol.2 (1990), p.6245. [5] B.S. Murty, S. Ranganathan, M.M. Rao: Mater Sci. Eng. A Vol. 16 (1992), p.231. [6] R.Nagarajan, S.Ranganathan: Mater Sci. Eng. A Vol. 179 (1994), p.168. [7] H. Bakker: Enthalpies in Alloys, edited by H. Bakker, Volume 1. Trans Tech publication, Netherlands, (1998): Materials Science Foundation, p.1. [8] A.W.Weeber: J Phys. F Vol. 17 (1987), p.809. [9] G. J. Van der Kolk, A.R. Miedema and A.K. Niessen: J. Less Common Metals Vol.145 (1988), p.1. [10] T. Itzukaichi, M. Umemoto, J.G. Cabanas-Moreno: Scripta Mater Vol.29 (1993), p.583.

Keywords Index A α-Zirconium Accommodation Energy Activation Energy Al-Ni-Ti System Aluminum (Al) Amorphisation Austenite Ferrite Phase Transformation

105 105 63, 111 147 63 147 71

B β-Phase Binary Diffusion Bulk Metallic Glass (BMG)

117 39 91

C Calorimetry Cerium Cold Rolling Copper (Cu)

85 97 79 63

D δ-Hydride Diffusion Diffusion Coefficient Diffusion Couple Dilatometer Dissolution Kinetics Dye Sensitized Solar Cell (DSSC)

105 23 63 53 71 111 111

E Entropy of Fusion EPMA

H Habit Plane High-Pressure

105 133

I Impurity Diffusion Impurity Diffusion Coefficient Induction Heating (IH) Interdiffusion Coefficient Intermetallic Compound (IMC)

63 53 71 53 39

L Lattice Invariant Shear Layer Growth Kinetics

139 117

M Magnetism Martensite Mechanical Alloying (MA) Melting Melting Temperature Metallic Glass Microstructure Miedema Model

125 139 147 23 23 125 53, 79, 97, 125 147

N Ni-Cr-Mo Alloy Ni2(Cr,Mo) Phase

23 53

O

79, 85 105

P

ω Phase Orientation Relationship

111 111

133 133

F Ferritic Steel Finite Element Model (FEM)

G Gibbs Free Energy Grain Size

91 97

Phase Transformation Precipitation Hardening Probabilistic

85, 133 97 71

R Recrystallization Kinetics

79

154

Phase Transformation and Diffusion

S Self-Diffusion Coefficient Soft Magnetic Material Specific Heat Specific Heat Difference Strain Energy Minimization Strength

23 125 85 91 105 97

T Temperature Gradient Ternary Diffusion Thermodynamic Thermodynamic Criterion Ti-Based Systems Triple Junctions

71 39 39 147 117 53

U Under Cooled Region

91

Z Zr-Based Systems Zr-Nb Alloy

117 139

Authors Index B Banerjee, S. Banks-Sills, L. Belova, I.V. Bhanumurthy, K. Bhoi, B. Biswas, A.

105, 133, 139 105 13 53, 63 97 111

C Chaubey, A.K.

97

D Das, N. Dey, G.K. Dhurandhar, H.

147 91, 125, 133, 139, 147 91

M Maji, B.C. Mehrotra, R.S. Mishra, B.K. Mohandas, E. Mohapatra, G. Mohapatra, S. Mukherjee, P.S. Mungole, M.N. Murch, G.E. Murty, B.S.

111 23 97 85 71 97 97 79 13 147

O Öchsner, A.

13

P F Fiedler, T.

13

G Gumaste, J.L. Gupta, D. Gupta, S.C.

97 1 133

117

J Jeya Ganesh, B. Joshi, K.D. Jyoti, G.

85 133 133

K Kale, G.B. Krishnan, M. Kulkarni, U.D.

39, 53, 63, 117 111 147

L Lad, K.N. Laik, A.

147 111 91

R Raju, S. Ristinmaa, M.

85 105

S

I Iijima, Y.

Pabi, S.K. Pai, H.C. Pratap, A.

91 53, 63

Sahay, S.S. Sharma, R.C. Sharma, S. Sikka, S.K. Singh, R.N. Srivastava, A.P. Srivastava, D. Ståhle, P. Sundararaman, M. Suresh, K.G.

71 79 79 133 105 125 125, 133, 139 105 111 125

T Taguchi, O. Tewari, R. Tiwari, G.P. Trivedi, P.C.

117 133 23, 117 79

156

Phase Transformation and Diffusion

V Vijayalakshmi, M.

85