High-Temperature Levitated Materials

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HIGH-TEMPERATURE LEVITATED MATERIALS

One of the major experimental difficulties in studying materials at extreme temperatures is unwanted contamination of the sample through contact with the container. This can be avoided by suspending the sample through levitation. This technique also makes metastable states of matter accessible, opening up new avenues of scientific enquiry, as well as possible new materials for technological applications. This book describes several methods of levitation, the most important being aerodynamic, electromagnetic and electrostatic. It summarizes the state of the art of the measurement of structural, dynamic and physical properties with levitation techniques, the considerable progress made in this field in the past two decades and prospects for the future. It also explores the concepts behind the experiments and associated theoretical ideas. Aimed at researchers in physics, physical chemistry and materials science, the book is also of interest to professionals working in high-temperature materials processing and the aerospace industry. David L. Price is a scientist at the CNRS, Orle´ans, France. His research interests include order and disorder in solids and liquids, dynamics of disordered systems, the glass transition and melting, neutron diffraction with isotope substitution, deep inelastic and quasielastic neutron scattering, and anomalous, high-energy and inelastic X-ray scattering.

HIGH-TEMPERATURE LEVITATED MATERIALS

DAVID L. PRICE CNRS, Orle´ans, France

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521880527 © D. L. Price 2010 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2010 ISBN-13

978-0-511-72779-5

eBook (EBL)

ISBN-13

978-0-521-88052-7

Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

For Gordon Squires

Contents

page ix

Preface 1 2

Scientific and technological context Levitation methods 2.1 Aerodynamic levitation 2.2 Electromagnetic levitation 2.3 Magnetic levitation 2.4 Superconducting levitation 2.5 Electrostatic levitation 2.6 Acoustic levitation 2.7 Optical levitation 2.8 Gas-film levitation 2.9 Free-fall experiments 2.10 Quantum levitation 3 Heating methods 3.1 Laser heating 3.2 Electromagnetic heating 3.3 Resistive pulse heating 3.4 Temperature measurement 4 Experimental techniques 4.1 Electromagnetic and optical properties 4.2 Thermophysical properties 4.3 Diffraction 4.4 Small-angle scattering 4.5 X-ray absorption spectroscopy 4.6 Inelastic scattering 4.7 Nuclear magnetic resonance 4.8 Numerical simulation vii

1 3 4 7 11 13 16 18 19 19 20 21 23 23 23 24 25 28 28 37 42 48 51 53 59 63

viii

5

6

7

8

9

Contents

Levitation in materials research 5.1 Advantages of levitation methods 5.2 Cooling and metastable states Liquid metals and alloys 6.1 Early transition metals 6.2 Late transition metals 6.3 Zirconium nickel and Ti Zr Ni alloys 6.4 Aluminium transition metal alloys 6.5 Cobalt palladium alloys Molten semiconductors 7.1 Silicon 7.2 Germanium and Ge Si alloys 7.3 Boron and boron compounds Molten oxides 8.1 Pure trivalent oxides 8.2 Silica 8.3 Mixed trivalent oxides 8.4 Divalent trivalent oxide mixtures 8.5 Silicates Conclusions and prospects

66 66 67 71 71 86 93 105 120 131 131 152 155 162 162 179 181 193 201 204

References Index

206 224

Preface

The origins of this monograph can be found in a game of bridge played one evening at the 1994 Gordon Conference on High-Temperature Chemistry in Meriden, USA, the participants being Jimmie Edwards of the University of Toledo, Shankar Krishnan, then at Containerless Research Inc. (CRI), and Marie-Louise Saboungi and myself, then at Argonne National Laboratory (ANL). The outcome of the game is best left unrecorded, but a more fortunate consequence of the evening’s proceedings was the inauguration of a CRI ANL collaboration on structural studies of aerodynamically levitated liquids, first with neutrons at the Intense Pulsed Neutron Source at Argonne and subsequently with X-rays at the National Synchrotron Light Source at Brookhaven, supported by a Small Business Innovative Research Grant from the US Department of Energy. Many interesting experiments ensued, some of which are described in this work. A few years later, Marie-Louise Saboungi and I were invited by Jean-Pierre Coutures, Director of the Center for Research on Materials at High-Temperature (CRMHT), Orle´ans, France, for a three-month visit. This led to an eventual move to Orle´ans, with occasional breaks at places like Trinity College, Cambridge, where the idea of writing a book for Cambridge University Press came up. The monograph that resulted aims to summarize the state of the art of the measurement of structural, dynamic and physical properties with levitation techniques, the considerable progress made in the past two decades and the prospects for the future. In addition to exploring for the benefit of scientists in other fields the various levitation and heating techniques currently in use, I have tried to explain the concepts behind the experiments and the associated theoretical ideas, so as to familiarize a student reader with a considerable section of modern condensed matter physics and materials science. The book is aimed primarily at research students, scientists and faculty in physics, physical chemistry and materials science. While the overall style is ix

x

Preface

appropriate for an academic audience, I hope that it will also be of interest in the industrial community especially regarding high-temperature materials processing and the aerospace industry. I should like to take the opportunity to express my appreciation to my Ph.D. supervisor Gordon Squires, who played a key role in the conception of the work, collaborators at ANL, CRI especially Shankar Krishnan, already mentioned and CRMHT especially Louis Hennet. Finally, I am most indebted to Marie-Louise Saboungi who came up with most of the creative ideas in this and many other projects in which we have worked together. I am grateful to the Master, Lord Rees, and the Fellows of Trinity College for the Visiting Fellow Commoner award during which this work was initiated. I also wish to thank the anonymous reviewers for excellent suggestions.

1 Scientific and technological context

The present time appears appropriate for a monograph summarizing the current state of the art of investigation of high-temperature materials with levitation techniques. Although methods for levitating solid and liquid samples in a containerless environment have existed for the best part of the century the patent for electromagnetic levitation dates back to 1923 it is only in the past 20 years that their potential has been fully exploited by combining the levitation and heating aspects with new capabilities for structural and dynamic studies at synchrotron X-ray and high-flux neutron sources and refined techniques for thermophysical and transport property studies such as digital imaging, noncontact modulation calorimetry and electrodeless conductivity measurements. There has also been a rapid diversification in the types of levitation methods aerodynamic, electromagnetic, electrostatic, and others each of which have special advantages and disadvantages. The 2006 American Physical Society meeting in Baltimore, USA, featured a symposium of invited talks focusing on just one of these methods, electrostatic levitation combined with synchrotron X-ray studies. Measurements of the structural, dynamical, thermophysical and transport properties of materials at high temperature are important in advancing condensed matter theory, in developing predictive models, and in establishing structure property process. Major experimental difficulties are encountered in obtaining reliable data on contained materials at temperatures above 1000 K owing to reactions of the samples with container walls and to the influence of the containers on scattering measurements. These problems are compounded when dealing with high-melting, corrosive liquids. A number of research groups around the world have overcome these difficulties by employing levitation techniques, eliminating container interactions and containerderived impurities and providing rapid access to high temperatures under controlled gaseous environments. Their results have not only advanced our 1

2

Scientific and technological context

understanding of high-temperature materials and phenomena but also provided important technological information, especially in the aerospace and semiconductor industries. A special advantage of levitation techniques for high-temperature experiments is the reduction and, in favourable cases, virtual elimination of heterogeneous nucleation, so that normally inaccessible metastable states can be realized. These include not only substantially supercooled liquids opening up new possibilities for exploring liquid liquid phase transitions but also metastable crystalline and glassy solid phases obtained on cooling from these liquids that were not previously available. The arrangement of the book is quite transparent, starting with a brief description of some of levitation and heating techniques currently in use and proceeding to outline the concepts of the transport, thermophysical, structural and dynamic properties measurement and simulation techniques performed on high-temperature materials, particularly in the liquid state. The aim is to give the non-specialist reader sufficient background to appreciate some of the recent results discussed in the following chapters, devoted to liquid metals and alloys, molten semiconductors and molten oxides. The final chapter presents some subjective ideas on where this field may be heading. The philosophy is to be illustrative rather than comprehensive, with some arbitrary choices of materials that seem interesting to the author. This applies especially to the chapter on metals and alloys, where an enormous body of research has been carried out with several levitation techniques. At the same time, earlier work on contained samples is included whenever it enhances our understanding.

2 Levitation methods

A variety of levitation techniques are available to the researcher to study high-temperature materials in the normal and supercooled states. The most widely practised techniques at the present time are aerodynamic levitation, in particular conical nozzle levitation (CNL), various kinds of magnetic levitation including electromagnetic levitation (EML), and electrostatic levitation (ESL). Other methods developed for specific applications such as acoustic levitation and gas-film levitation are less widely used and will be discussed only briefly. In order for levitation to be useful in a scientific experiment, it is important not only to supply a force that can counteract the gravitational field but also to maintain the sample in a configuration that is sufficiently stable to allow the measurements to be performed. The issue of stability may be quite complex. A prime example is the Levitron®, a popular toy in which a spinning magnetic top is suspended above a flat surface of a magnetic material (http:// www.levitron.com/). The levitating force is obvious, but the stability requires a complex physical analysis (Berry, 1996; Berry & Geim, 1997). The variety of methods in current use suggests that each one has particular advantages and disadvantages, depending on the application in hand. CNL is a relatively simple and versatile technique and can be readily incorporated into different kinds of experimental apparatus. EML is restricted to conducting samples, generally metals, in which case relatively large samples (up to 1 2 cm diameter if desired) can be levitated. ESL has the advantage that samples can be held under vacuum, removing the possibility of contamination by a surrounding gas, or alternatively under controlled gas pressures up to a few atmospheres; on the other hand, it involves a relatively complex setup that makes it harder to use with certain spectroscopies. The principles of these techniques are described in the following sections. 3

4

Levitation methods

2.1 Aerodynamic levitation Like most of the techniques described in this chapter, aerodynamic levitation has a long history, with, for example, micrometre- to millimetre-sized water drops being studied in a wind tunnel in the late 1960s (Beard & Pruppacher, 1969). The most widely used technique of this kind employed today is CNL, based on the early work of Winborne et al. (1976) and Coutures et al. (1990) and on subsequent developments by Weber & Nordine (1995), in which a sample is levitated by gas flow in a convergent-divergent nozzle in which Bernouilli forces push it back to the axis of the nozzle. Stable levitated samples can then be heated to temperatures in excess of 2500 K with a laser or RF heating system. A typical CNL system, developed at Containerless Research Inc. (CRI) in Evanston, Illinois, is schematically illustrated in Fig. 2.1. In the setup shown here the levitation system is combined with laser heating and integrated with an X-ray goniometer for diffraction experiments at a synchtrotron source, to be described in Chapters 6 8. The figure shows all of the key components including two pyrometers, two video cameras and video microscope, curved beryllium window, six-circle goniometer and X-ray detector. The levitation chamber is located at the centre of the goniometer. The heating laser is inclined at 15 with respect to the normal to avoid physical interference with the X-ray detector. The nozzle and plenum chamber assembly are supported on three tubes connected to three flexible bellow feed-throughs. Two of these tubes circulate water for nozzle cooling, while the third supplies the levitation gas that feeds into the nozzle’s plenum chamber. The CNL system is enclosed in an

Fig. 2.1. Schematic view of CNL apparatus used by the CRI–Argonne team for measurements of liquid structure, showing two pyrometers (P1, P2), two video cameras (V1, V2) and video microscope (VM3), curved beryllium window, six-circle goniometer and X-ray detector (Krishnan & Price, 2000).

2.1 Aerodynamic levitation

5

environmentally controlled chamber with suitable ports for laser heating, pyrometry, sample injection and retrieval, and pressure measurements. The upper part of the chamber is a 23-cm diameter sphere. Two beryllium windows are included in this section: a small window for the incident X-ray beam and a larger curved window, approximately 8 mm wide and 0.127 mm thick, subtending an angle of approximately 120 around the nominal sample position at the centre of the chamber. This window extends to about 5 below the normal height of the levitated sample, so that the transmitted part of the direct X-ray beam can pass through. The gas flow to the CNL system is regulated by electronically controlled mass flow controllers. The chamber is connected to a vacuum system that permits evacuation to about 10 5 bar followed by a back-fill with a purified inert gas or other special environment when required. A servo-controlled exhaust throttle valve, placed between the pump and the chamber, enables control of the chamber pressure in the range 0.1 0.8 bar during levitation. The samples are heated with the aid of a 270-W CO2 laser and the sample temperature is measured with two pyrometers operating at wavelengths of 0.65 and 1 2.5 mm, respectively. Precise sample positioning is achieved by moving the nozzle assembly with a three-axis motorized translator and using a phosphor screen to observe the shadow of the sample in the X-ray beam. Samples can be levitated, heated, melted and positioned stably for up to 3 h with this system. The condition for levitation is derived from the law of momentum conservation applied to a control volume that contains the sample:  ð 1 2 ð2:1Þ
u þ p dA ¼ Mg; 2 where r, u and p are the gas density, vertical gas flow velocity and gas pressure, respectively, and Mg is the sample weight. The integral is performed over the surface A of the control volume. Free-jet and conical nozzle levitation differ in the magnitude of the two components on the left side of Eq. (2.1). Stable free-jet levitation occurs when the gas flow is sufficient to form a free jet with a momentum flow rate equal to about twice the sample weight. Stable levitation in a conical nozzle occurs when the momentum flow is less than the sample weight, and the levitation results mainly from pressure differences over the sample surface. The pressure differences are a small fraction of the total pressure: for example, a 3-mm diameter sphere of liquid aluminium oxide is levitated when the pressure difference across the sample is approximately 0.001 bar, or about 0.1% of the total pressure at one atmosphere.

6

Levitation methods

Fig. 2.2. Left: schematic view of the CNL arrangement used by the CRMHT group for measurements of liquid structure, showing the laser head (a), first mirror (b), goniometer (c), high-temperature chamber (d), and curved X-ray detector (e). Right: detailed view of the high-temperature chamber showing the laser beam (a), focusing mirror (b), flat mirror (c), levitator (d), pyrometer (e), X-ray beam (f), X-ray collimator (g) and direct beam stop (h) (Hennet et al., 2002).

For most experiments the chamber is initially pumped down to low pressure and then purged for short durations by flowing through an inert gas such as argon. The chamber is then filled with the gas and maintained at the desired pressure by controlling the throttle valve. Typical flow rates used for metals at 0.4 bar vary from 150 to 600 STP cm3 min 1. For experiments on metals, UHP argon is generally employed as the levitation gas, and for experiments on ceramics, air, oxygen, nitrogen or a nitrogen/hydrogen mixture can be used. A CNL system developed independently at the Centre de Recherche sur les Mate´riaux a` Hautes Tempe´ratures (CRMHT, now CEMHTI) in Orle´ans, France, is shown in Fig. 2.2. The basic components are similar to the CRI setup shown in Fig. 2.1, but the geometry is somewhat different. This type of arrangement was first used for NMR experiments and subsequently for EXAFS measurements on molten ceramics, to be described in Chapter 8. One of the drawbacks of the CNL method is that samples have temperature gradients between the laser-beam-heated top of the sample and the bottom of the sample where the cold levitation gas first impinges. Based on observations with the optical pyrometer and the observed maximum undercooling for well-known materials, this difference was estimated to be about 25 K for liquid metals and on the order of 50 75 K for liquid oxides due to

2.2 Electromagnetic levitation

7

their reduced thermal conductivities. The advantage of X-ray diffraction in this context is that the scattering takes place from the region of the sample whose temperature is measured by pyrometry. Temperature gradients become more serious, however, for neutron diffraction and NMR experiments and for X-ray measurements when the X-ray energy becomes sufficiently high to penetrate the interior of the sample. A setup recently developed by the CRMHT group for neutron diffraction experiments (Hennet et al., 2006) incorporates a second laser heating the sample from below through a small hole in the conical nozzle in order to reduce temperature gradients. While the CNL technique appears to have been first used for scientific experiments in the 1970s, the basic aerodynamic principle the Bernouilli effect was of course already well known. Paradis et al. (1996), who made a detailed study of nozzle behaviour in both terrestrial and microgravity environments, point out that the corresponding hydrodynamic effect was employed in fire hoses early in the twentieth century. Waltham et al. (2003) describe a system in which air is blown vertically downwards through a hose that exits in a flat horizontal sheet. Another sheet brought up to the orifice is held in place despite the fact that the air is pushing downwards. The acceleration of the air in the gap causes a drop in pressure that more than compensates for the high pressure in the hose. Weights of 2 kg can be suspended in this way. The past 30 years have seen a continual improvement in design of the nozzle for CNL, principally by the CRI and CRMHT groups whose systems have just been described. Because of the simplicity of the method, it can be readily combined with other types of apparatus, for example a secondary levitation apparatus, different types of heating or rapid cooling, and probes of structural or physical properties. These cases will be discussed in the chapters that relate to these specific aspects. 2.2 Electromagnetic levitation ‘Magnetic levitation’ is a general term that can comprise several distinct techniques, in which either: (a) an inhomogeneous electromagnetic field is generated in a radio-frequency (RF) coil and induces eddy currents in the sample; these interact with the applied magnetic field via a Lorentz force that counteracts gravity; this clearly depends on a significant electrical conductivity in the sample; (b) a large inhomogeneous magnetic field is generated in a magnet, either conventional or superconducting, and induces a magnetic moment in the sample that

8

Levitation methods

interacts with the applied field; either diamagnetic or weakly paramagnetic samples can be levitated; or (c) a magnetic field is generated in a magnet, generally a permanent magnet, and induces an electric current in a superconducting sample; this interacts with the applied field via a Lorentz force as in (a).

For convenience we will refer to these three methods as (a) electromagnetic levitation, (b) magnetic levitation and (c) superconducting levitation, although other terms are frequently used in the literature. The three methods are discussed in turn in the present and following two sections. At this point it is helpful to introduce expressions based on the quasistatic approximation of electrodynamics, valid when the wavelength of the electromagnetic radiation is much greater than the sample size and its frequency is smaller than the relaxation time of the current carriers. Following Enderby et al. (1997), we consider an isotropic conducting nonmagnetic sphere of radius a and conductivity s suspended in a uniform periodic external magnetic field of angular frequency o. The magnetic response is mainly due to the conduction currents set up in the sphere and can be characterized by a complex susceptibility given by a ¼ a0

ia00 :

ð2:2Þ

The real part is given by a0 ¼

a 2G ; d

the parameter d is the skin depth given by s 2 ; ¼ o0 

ð2:3Þ

ð2:4Þ

where m0 is the magnetic permeability of free space and m is the relative permeability of the material, and the function G is given by   3 3 sinhð2qÞ sinð2qÞ : ð2:5Þ GðqÞ ¼ 1 4 2q coshð2qÞ cosð2qÞ For typical values of s ¼ 104 O 1 cm 1, m0 ¼ 1.25mO.s.m 1 and frequency o/2p ¼ 300 kHz, the skin depth d is 0.92 mm. The imaginary term in the susceptibility is given by a ; ð2:6Þ a00 ¼ H 

9

2.2 Electromagnetic levitation 1.0 0.9

Efficiency G: Force H: Power

0.8 0.7 Efficiency

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

1

2

3 4 5 6 7 Sample Radius / Skin Depth

8

9

10

Fig. 2.3. Efficiency of levitation force and inductive heating power as a function of the ratio of sample radius to skin depth (Mathiak et al., 2005).

where the function H is given by  9 sinhð2qÞ þ sinð2qÞ HðqÞ ¼ 2 q 4q coshð2qÞ cosð2qÞ

 1 :

ð2:7Þ

The dimensionless functions G(q) and H(q) are plotted in Fig. 2.3. We now proceed to discuss EML, which is one of the oldest techniques used for containerless experiments. The method was patented by Muck (1923) and developed further in the 1950s (Okress et al., 1952). The Lorentz force that can act to counteract gravity can be expressed, to lowest order in a multipole expansion, as (Jacobs et al., 1996) F¼

rB2 4 3 a aG : 20 3 

ð2:8Þ

The sample to be levitated is positioned in a potential well generated by the electromagnetic (e-m) field B. Accordingly, the sample will perform oscillations about its equilibrium position with a frequency that is determined by the spring constant of the field and its mass. In addition, liquid samples will display free surface oscillations with a restoring force due to the surface tension. In particular applications it may be important to design the setup to minimize both types of motion (Jacobs et al., 1996). On the other hand, it may be useful to study these oscillations to derive information about the

10

Levitation methods

Fig. 2.4. Schematic view of EML apparatus used by the DLR (Germany) group for (a) optical measurements to determine sample density and (b) fluorescence radiation measurements in an X-ray beam to obtain EXAFS spectra (Jacobs et al., 1998).

physical properties of the levitated sample. Investigations of this kind will be discussed in Chapter 6. A typical EML apparatus, developed by the group at the Institut fu¨r Raumsimulation, Deutsche Forschungsanstalt fu¨r Luft- und Raumfahrt (DLR) in Cologne, Germany, is shown in Fig. 2.4. Two setups are shown, (a) for optical measurements to determine the sample density and (b) for fluorescence radiation measurements in an X-ray beam to obtain EXAFS spectra. The levitation coil, fed by an RF generator with 260-kHz frequency and 6-kW power, is built into a vacuum chamber that is equipped with quartz windows in setup (a) and Kapton® windows in setup (b). The temperature is measured by a pyrometer viewing the sample from above through a quartz window from the top of the chamber and is controlled by a variable flow of cooling gas, making it possible to maintain a constant sample temperature for several hours. EML apparatus was also developed in the group of the late J. L. Margrave at Rice University and subsequently at CRI for measurements of the optical properties of liquid metals, to be described in Chapter 6 (Krishnan et al., 1993; Krishnan & Nordine, 1993).

2.3 Magnetic levitation

11

Special apparatus for EML measurements with neutron diffraction and energy-resolved X-ray diffraction has been developed at the DLR (HollandMoritz et al., 2005), and similar apparatus has been developed in Japan for measurements on liquid silicon (see, for example, Higuchi et al., 2005). Liquid silicon is metallic, as will be discussed in Chapter 7, and has considerable technological interest for growth of large silicon crystals for the electronics industry. Normally a single split RF coil is used for sample positioning and heating, but if necessary the two functions can be separated with different frequencies. Loho¨fer (1994; 2005) used a second coil for electrical conductivity measurements, while Wunderlich & Fecht (2005) use a second, sinusoidally modulated RF coil for measurements of thermal properties with modulated induction calorimetry. These techniques will be discussed in Chapter 4. The shape of a levitated liquid sample is determined by Lorentz forces, hydrostatic forces, surface tension and, in many applications, aerodynamic forces. The shape of a sample in a vertical axis levitation coil looks like a pear with the tip at the bottom (Gagnoud et al., 1986).

2.3 Magnetic levitation In the previous section we have seen how the application of an inhomogeneous magnetic field to a conducting material induces eddy currents that interact with the applied field through a Lorentz force that can be used to position or levitate the sample. In an insulating material, the strength of such forces will be negligible, but an applied inhomogeneous field will induce a magnetization that interacts with the applied field through a magnetic force F¼

wg dB B 0 dz

g;

ð2:9Þ

per unit mass, that can made equal to zero with sufficiently strong field gradients. In Eq. (2.9), wg is the mass magnetic susceptibility and B(z) is the flux density at a distance z from the centre of the magnet. With this technique it is possible to levitate rather large objects, for example a living frog (Berry & Geim, 1997). (Berry and Geim won the 2000 IgNobel prize in physics for this accomplishment: Berry & Geim, 2000). At the Institute for Materials Research (IMR) at Tohoku University in Sendai, Japan, a large hybrid magnet, consisting of a large-bore superconducting magnet with a water-cooled conventional magnet inside, has been used to levitate diamagnetic materials (Fig. 2.5).

12

Levitation methods

3

4 2

5

7

8

9

6 1

Fig. 2.5. Schematic view of magnetic levitation apparatus at IMR, Japan, showing the hybrid magnet (1), heating laser (2), alignment laser (3), focusing mirror (4), cryostat (5), sample cell (6), illumination power supply (7), video system (8) and gas flow tube (9) (Kitamura et al., 2000).

Values of BdB/dz over 4000 T2 m 1 are obtained with this facility, making it possible to levitate materials with absolute values of the susceptibility wg greater than 3.10 9 m3 kg 1 that include, for example, a wide range of oxide ceramics and glasses. By heating the levitated samples with a CO2 laser beam, spherical glass samples are obtained. Some advantages of containerless conditions for materials processing will become apparent in subsequent chapters. As already mentioned, stability issues are often paramount in levitation experiments and these are especially interesting in the case of magnetic levitation. The classic theorem of Earnshaw (1842) states that no stationary object made up of charges, magnets and masses can be held in stable equilibrium by any combination of e-m and gravitational forces. This directly follows from Laplace’s equation, whose solutions can have no isolated minima or maxima. Nevertheless, a diamagnetic object can be levitated stably, because its magnetism originates in the orbital motion of the electrons and hence is dynamical rather than stationary. The Levitron® is stable

2.4 Superconducting levitation

13

because of an adiabatic coupling between the magnetic moment and the field that derives from the fast precession of the top (Berry & Geim, 1997). It is generally impossible to levitate non-conducting paramagnetic objects stably, essentially because the magnitude B(r) of a magnetic field in free space can possess a minimum but not a maximum. Ikezoe et al. (1998) found an ingenious way out of this restriction by pointing out that the relevant property is the difference in volume susceptibilities of the sample and atmosphere, (ws wa). Since air has a small positive susceptibility, due to the oxygen, which can be increased with pressure or increased oxygen content, the difference can be made negative for weakly paramagnetic substances such as ionic solutions. Jones (1979) considers the stability of a levitated sample as a function of electrical conductivity and magnetic permeability. Paramagnetic samples with sufficiently high conductivity, or applied field frequency, can be stably levitated, as discussed in the previous section. 2.4 Superconducting levitation In the case of a superconducting sample, an applied magnetic field induces currents that interact with the applied field to produce levitation through the Lorentz force. The principle is similar to that used in EML of normal metallic samples except that the currents involved can obviously be much higher, enabling the use of permanent magnets, and a high degree of stability can be readily achieved due to the particular properties of superconductors. This concept is of great importance from both scientific and technological points of view. The character of superconducting levitation depends both on material and geometry. In Type-I superconductors, the magnetic field is excluded and the material behaves as a perfect diamagnet, leading to a special case of the magnetic levitation described above. In the practically useful Type-II materials, which include the high-Tc oxide superconductors as an extreme case, and in the large-field region, Hc1 < H < Hc2, the magnetic field penetrates in the form of magnetic flux lines that are pinned at various types of inhomogeneity. The field distribution depends on the previous history of the sample, and flux may be trapped even at zero field. Flux trapping leads to free levitation of the superconductor above a permanent magnet or even suspension below it. When the superconductor is moved through the inhomogeneous field of the permanent magnet, the flux lines become unpinned and jump to new positions, dissipating part of their elastic energy in the process. As a consequence, the levitated superconductor behaves as if it were ‘stuck in sand’, and can be pushed to a wide range of positions and orientations where it will remain fixed, without swinging or rotating (Brandt, 1989).

14

Levitation methods

In practical devices based on superconductors, two main types of symmetry are used: cylindrical (Ruiz-Alonso et al., 2005) and translational (Del Valle et al., 2007). The latter is especially found in applications such as linear bearings and novel transportation systems including the superconducting magnetic levitation (‘maglev’) trains being developed in China, Brazil and Germany. Del Valle et al. give specific results for a model in which the superconductor obeys the critical-state model with constant critical-current density Jc and is uniformly divided into n elements with rectangular cross-sectional area
S. The vertical magnetic force per unit length Fz/L on the superconductor arising from the interaction of the currents induced in it with the field created by the permanent magnets is given by ð n X Fz Hey; j Jj
S; ¼ 0 Jx Hey dS  0 L j 1

ð2:10Þ

S

where dS is the differential of area in the cross-sectional surface of the superconductor, Hey, j is the y component of the field from the permanent magnets at the desired position and Jj is the x component of current density in element j. The authors give numerical results for a particular geometry in which the permanent magnets have a square cross section with lateral dimension aPM equal to the vertical dimension of the superconductor b ¼ 5 cm and uniform vertical magnetization equal to 1T/m0, and the superconducting elements have a width 3b and a critical-current density Jc ¼ 2Hz0/3b where Hz0 is the field at the centre of the top surface of the permanent magnet. The calculated force per unit length is shown in Fig. 2.6 for five different geometries. It can be seen that forces on the order of 103 N m 1 can be obtained with this kind of arrangement. As mentioned above, a system optimization has to include not only levitation force but stability as well. It is found that all configurations are stable vertically and stable horizontally at low separation values dPM but become unstable at higher values. The exceptions are the configurations PM3N, which is stable along the range of dPM, and PM2, which is horizontally stable for separations dPM/aPM > 0.7. In an alternative configuration, the same principles can be used to levitate a permanent magnet above a slab of hard superconductor (Davis et al., 1988). Perhaps the most widely known example of superconducting levitation is the popular demonstration of high-temperature (high-Tc) superconductivity after it was discovered in 1986, an example of which is shown in Fig. 2.7. Hull (2004) has given a detailed description of the theory of superconducting

2.4 Superconducting levitation

15

Fig. 2.6. Levitation force per unit length for a superconducting levitation system for five configurations shown in the legend, at a distance d ¼ 2.5 cm, as a function of separation between magnets dPM; aPM is the lateral dimension of the permanent magnets, assumed to have a square cross section. The straight line corresponds to the case of a single magnet (Del Valle et al., 2007).

Fig. 2.7. A magnet levitating above a high-temperature superconductor, cooled with liquid nitrogen.

levitation with particular reference to high-Tc materials, together with an account of applications to cryogenic structures, maglev transportation and flywheels. As an intriguing aside, Podkletnov & Nieminen (1992) reported a reduction in the weight of an object held above a rotating disc of the high-Tc

16

Levitation methods

compound YBCO levitated by a magnetic field. This first scientific report of anti-gravity has not yet been confirmed (Perkowitz, 2009).

2.5 Electrostatic levitation In ESL, the positioning of a charged sample is achieved through the application of feedback-controlled electrostatic fields that are generated by a set of appropriately positioned electrodes around the sample. Since a threedimensional electrostatic potential minimum does not exist, as we have already seen, according to Earnshaw’s theorem, electrostatic sample positioning is only possible with an actively controlled applied electric field. ESL systems therefore use a feedback control to correct any deviation in sample position from a preset position. For an experiment on the ground, the forcebalance equation for levitation when the sample is positioned at the centre of a pair of infinite parallel electrodes a distance L apart and a voltage difference V is given approximately by mg ¼ Qs

V ; L

ð2:11Þ

where m is the mass of the sample carrying a charge Qs. For Qs ¼ 0.69  10 9 C, m ¼ 140 mg, and L ¼ 10 mm, V is approximately 10 kV. Like the other levitation methods we have described, suspension of charged particles by electrostatic forces has a long history, going back at least as far as the 1970s. For example, Wyatt & Phillips (1972) studied aerosol particles levitated in time-varying electrostatic fields. The application to high-temperature materials processing and measurements of the properties of large samples followed later, carried out at several laboratories including the Jet Propulsion Laboratory (JPL) in Pasadena, California, the Tsukuba Space Center (TSC) in Japan, and the NASA Marshall Space Flight Center (MSFC) in Huntsville, Alabama. Despite the apparent simplicity of Eq. (2.11), a practical ESL system must address two problems that make the final setup more complicated: stable positioning, and charging of the sample. In their earlier work, Rhim et al. (1985) at JPL used a ‘dish’ configuration in which a potential minimum in the horizontal plane was achieved by curving the electrodes, so that positive control was needed only in the vertical direction. However, the amplitudes of the horizontal oscillations were unacceptably large, so that three-dimensional control was required. One way of achieving this was with a quadrupole cell (Bolsaitis et al., 1989). The geometry chosen by Rhim et al. (1993), which has been adopted in most

2.5 Electrostatic levitation

17

Fig. 2.8. Schematic view of the electrode assembly for ESL (side and top views), showing the sample (1), top electrode (2), bottom electrode (3), side electrodes (4), and the hole that allows access to the sample storage system (5) (Rhim et al., 1993).

subsequent ESL setups, consists of an upper and lower electrode with the addition of two pairs of side electrodes surrounding the bottom electrode (Fig. 2.8). Two orthogonal HeNe lasers together with two position detectors provided positional information that was used to generate appropriate damping voltages on these side electrodes to prevent sample oscillation in the lateral directions. For metallic materials, charging actually involved three processes: capacitive, photoelectric and thermionic charging. Capacitive charging was employed for sample launching and levitation at the beginning of each experiment. The surface charge on the sample increased as the top-electrode potential was raised until electrical contact with the bottom electrode was broken and the sample levitated. A 1-kW UV-rich high-pressure xenon arc lamp was then used to heat the sample. To compensate the discharging effects of ions produced by the sample and the surrounding electrodes under strong UV irradiation, photoelectric charging induced by UV radiation collected from the xenon lamp was used to maintain the charge on the sample. Finally, when the sample temperature reached about 1200  C, the more powerful thermionic charging mechanism took over. The dynamics of these three mechanisms is discussed in detail by Rhim et al.

18

Levitation methods

Subsequent developments at the TSC include replacement of the xenon lamp by a UV laser for sample charging by the photoelectric effect and a CO2 laser for heating (Paradis et al., 2001b), preheating the sample on the pedestal with part of the CO2 laser beam, and use of UV instead of visible radiation for sample positioning (Ishikawa et al., 2001). A similar setup was installed at the MSFC (Rogers et al., 2001). The initial two charging mechanisms discussed above work, of course, only for metallic materials. For refractory liquid oxides, the apparatus described above has the additional disadvantage that the high vacuum required will promote vaporization that, moreover, may be selective for certain elements and thus change the stoichiometry of the sample. To retain some of the advantages of ESL in the investigation of liquid oxides, Paradis et al. (2001b) developed a hybrid system in which aerodynamic levitation was used to levitate the sample so that it could be heated to the point where thermionic emission could maintain a sufficient charge for levitation. The environment consisted of a gas with the correct composition to maintain stoichiometry at pressures of a few atmospheres, sufficient to avoid breakdown from the high voltage applied to the electrodes. An additional benefit of the gaseous environment was that it provided sufficient damping of the horizontal oscillations that only one-dimensional (vertical) position control was required. 2.6 Acoustic levitation Acoustic levitation takes advantage of the impedance difference between the suspension medium normally a gas and a solid or liquid sample. Again, the basic concepts were developed quite early: Allen & Rudnik (1947) describe experiments with a 2-kW siren in which pennies placed on a stretched silk screen do somersaults with ‘Rockette’-like precision (1947 readers would have understood this allusion right away); or in which a penny could be made to rise slowly to a vertical position, appearing to support another penny that finally assumed a horizontal position about the first, touching rim to rim. A motivating factor behind the development of this technique for scientific applications (Whymark, 1975; Oran et al., 1980) was the need for containerless processing of insulating materials in space experiments, in contrast to EML, which was the prevalent levitation technique at that time. In terrestrial applications, the acoustic intensities required to levitate high-density samples appear to result in large surface oscillations causing fragmentation of the liquid drop. This seems to have limited the use of purely acoustic levitation techniques. Weber et al. (1994) developed a hybrid aerodynamic-acoustic levitator in which the levitating force is provided by a gas jet and the acoustic

2.8 Gas-film levitation

19

forces control the sample’s position, rotation and shape. They concluded that high-energy efficiency could be achieved by this method in rapid containerless melt processing of materials. Subsequent refinements have been made by Xie et al. at the Northwestern Polytechnical University in Xi’an, PRC, making use of a result derived by Gor’kov (1962) for the time-averaged potential U of an acoustic radiation force: ! 2
f in p2in 3 ; ð2:12Þ U ¼ 2Rs 3
f c2 2 2 are the mean square fluctuations of the incident pressure and where p2in and in velocity at the point where the sample with radius Rs is located, rf is the density of the medium and c is the sound velocity. With these improvements they have been able to levitate 3-mm diameter tungsten spheres (Xie & Wei, 2001), solid iridium and liquid mercury (Xie et al., 2002) and living insects and small fishes (Xie et al., 2006).

2.7 Optical levitation With the optical analogue of the acoustic levitation described above, charged and neutral liquid drops with diameters between 1 and 40 mm can be stably levitated and manipulated with laser beams (Ashkin & Dziedzic, 1975). This technique has particular importance in cloud physics, aerosol science, fluid dynamics and optics, but relatively limited applicability in materials science. The interactions of the drops with light, the electric field, the surrounding gas, and one another can be observed with high precision.

2.8 Gas-film levitation This technique was developed by Granier & Potard (1987). A typical setup is shown in Fig. 2.9. A flow of inert gas is forced through a porous membrane placed below the sample to be levitated, and the resulting gas flow acts as a supporting cushion with a typical thickness of 10 100 mm. The pressure difference between the bottom and top of the porous membrane is of the order of 1 bar. The porous membrane is slightly concave to guarantee the lateral stability of the sample (Haumesser et al., 2002). Advantages of this technique are that, in principle, there is no limitation on the type of material, and samples of fairly large mass (up to 200 g in the case of oxide glasses) can be routinely levitated. The method has been used to

20

Levitation methods LIQUID liquid

gas

Fig. 2.9. Schematic view of a liquid drop levitated by a gas film (Haumesser et al., 2002).

study thermophysical properties such as viscosity, surface tension and mass density in ceramic oxides (Piluso et al., 2002).

2.9 Free-fall experiments We will also encounter experiments in which metastable solid phases are prepared, or materials properties measured, in samples subjected to free fall in a drop tube. These share with levitation experiments the containerless environment and a motivating factor in the original work to prepare for microgravity and space experiments. The drop tubes in operation in the 1990s were reviewed by Greer (1994). They fall into two basic types. The first type is ‘long’ drop tubes, with heights on the order of 50 m, generally used to prepare metastable phases of refractory materials. Normally they are evacuated, providing a type of microgravity environment. A liquid drop is formed at the top of the tube either by EML or by the pendant drop technique, and the tube is equipped to measure the temperature of the drop as it falls down the tube. One of the earliest drop tubes, the 32-m tube at the MSFC, providing 2.6 s of free-fall time, was used to prepare amorphous metallic glasses (Steinberg et al., 1981). A 105-m tube was added subsequently. Figure 2.10 shows a schematic diagram of the 50-m tube at the CEA laboratory in Grenoble, France, used for studies of metastable phases in transition metal alloys. All these facilities have been since shut down. The other type, ‘short’ drop tubes, a few metres high, are generally filled with an exchange gas and produce an array of drops either by ejection or by melting a powder during the fall. Temperature measurement is impossible, but a wider range of materials can be produced.

2.10 Quantum levitation

21

wire drop filament

50 m

photodiode

Collection: Sn

in Sn granules or on a pivoting shock-absorber

final collector

Fig. 2.10. Schematic view of the 50-m drop tube at the CEA Laboratory, Grenoble, France (Berne, 2000).

The effectively zero-gravity environment in a drop tube, along with the microgravity environment in space flights, simplifies the free oscillations of liquid drops that are important for measurements of surface tension and viscosity (Matsumoto et al., 2005). These aspects will be discussed in Chapter 4.

2.10 Quantum levitation Although its application will be necessarily limited to devices at the nanometre scale, it is worth pointing out another kind of levitation made possible by quantum mechanics. An attraction between two neutral, conducting surfaces in a vacuum, a quantum effect involving the zero-point oscillations of the electromagnetic field surrounding the surfaces, was first described by Casimir (1948). These fluctuations exert a radiation pressure that tends to be slightly greater outside the plates than between the plates and so the plates are attracted to one another. Subsequent work by Dzyaloshinskii et al. (1961) showed that the vacuum could be replaced by a liquid and that, if the material of the plates and the liquid had particular dielectric permittivities, the force between the plates would be repulsive.

22

Levitation methods (b)

(a)

1

Net repulsive force

100 10

3

e 2

1

Gold Bromobenzene Silica 1015 iξ (rad s–1)

1016

Fig. 2.11. (a) The quantum mechanical interaction between material 1 and material 2 immersed in a fluid (material 3) is repulsive when e1(i o) > e3(i o) > e2(i o) where e(i o) is the dielectric function at imaginary frequency i o; (b) the optical properties of gold, bromobenzene and silica are such that this condition is satisfied (Munday et al., 2009).

Munday et al. (2009; see also Miller, 2009) have demonstrated this effect using gold and silica separated by liquid bromobenzene. The liquid was placed in a cell between a gold plate and a 40-mm diameter polystyrene sphere coated with a thin gold film and suspended from an atomic force microscope cantilever. The bending of the cantilever caused by interactions between the sphere and the plate could be determined by bouncing a laser beam off the cantilever. The attractive force between the plate and the sphere became repulsive when the gold plate was replaced by one made of silica (Fig. 2.11).

3 Heating methods

3.1 Laser heating Carbon dioxide (CO2) infrared lasers provide a natural choice of heating system in conjunction with aerodynamic levitation. For the CNL setup shown in Fig. 2.1, a Synrad Model 60 2 270-W cw CO2 laser in the infrared range is used to heat and melt the samples. The laser beam is tilted with respect to the vertical plane by approximately 15 to avoid interference with the motion of the X-ray detector. It is directed at the sample by means of two mirrors and a ZnSe lens placed between them. Two controllers, one located inside the X-ray hutch and the other outside it, were used to control the laser power independently. Heating with a single laser leads to significant temperature gradients, especially with insulating samples. More recent setups (Krishnan et al., 2005; Hennet et al., 2006) incorporate a second laser heating the sample from below through a small hole in the conical nozzle in order to reduce these gradients. In the CRMHT apparatus, there are actually two lasers heating the sample from above. At the lowest specimen temperatures, the power delivered to the specimen from below is roughly equal to the power delivered from above. With these modifications, temperature gradients from top to bottom of the sample are expected to be reduced below 25  C, even for oxide samples. Laser heating, generally with one or two CO2 infrared lasers, is also used in recent high-temperature experiments with electrostatic levitation. 3.2 Electromagnetic heating The power P absorbed by a sphere of radius a in an electromagnetic field of frequency o due to ohmic losses of the induced eddy currents is given by P¼

B2 o 4p 3 hai aH ; 2m0 3 d 23

ð3:1Þ

24

Heating methods

Fig. 3.1. Sketch of the configuration for X-ray diffraction measurements with a system combining CNL with e-m heating. The sample diameter was 3 mm and the cylindrical RF coil had an air gap of 8 mm and inner diameter of 12 mm. The diffraction cone was 20.5 (Mathiak et al., 2006).

where the skin depth d and the function H(q) have been defined in Eqs. (2.4) and (2.7). It is seen that the expressions for the levitation force F (Eq. (2.8)) and power absorbed P are similar, but F is proportional to ▽B2a3 while P is proportional to B2oa3. When the RF field is used for both levitation and heating, ▽B and B, and thus F and P, can be brought within the desired range with an appropriate coil design. However, the heating rate depends strongly on the equilibrium position of the sample and so the temperature of the sample cannot be controlled independently by adjusting the coil current. This has motivated the design of a hybrid system combining aerodynamic levitation with e-m heating, carried out under a collaboration between CRMHT, Orle´ans, and DLR, Cologne (Mathiak et al., 2005). In principle, such a system should provide high sample stability, homogeneous melting, and good temperature control, enabling relatively easy access to the deeply undercooled state in conducting samples. To achieve high temperature with a relatively small EML effect, it is clear from the discussion above that a relatively small magnetic field gradient and high frequency are required, together with as large a sample as possible to minimize heat losses by radiation or convection. Figure 3.1 shows the configuration adopted by Mathiak et al. (2006) for X-ray diffraction measurements at a synchrotron source. A Hu¨ttinger TIG 5/300 high-frequency generator with maximum power of 5 kW and maximum frequency of 300 kHz was used for power supply. Metallic samples could be stably levitated in the X-ray beam and melted. They were observed to be slightly flattened at the bottom. 3.3 Resistive pulse heating Pulse-heating (PH) techniques are also used extensively for thermophysical and electronic properties measurements at high temperature and, being somewhat complementary to levitation techniques, should be mentioned briefly.

3.4 Temperature measurement

25

In the setup at the Graz University of Technology (GUT), a wire-shaped specimen is assembled into a discharge circuit (Hu¨pf et al., 2008a,b). A capacitor bank of 500 mF is charged with a high-voltage power supply and then discharged through the wire, leading to a rapid increase in temperature. Heating rates of 108 K/s eliminate chemical reactions between the wire and the surrounding atmosphere and the cylindrical sample maintains its shape until the end of the liquid phase. This leads to an experimental duration of about 50 ms. The current is measured with an induction coil, and the voltage drop is measured with two knife edges placed on the wire. Temperatures are detected pyrometrically as in levitation experiments. Because thermal expansion leads to smaller densities, which affects the calculation of quantities such as electrical resistivity, the volume expansion has to be taken into account. This is accomplished by measuring the change in the diameter of the wire with a fast CCD camera as a shadowgraph when the wire is backlit with a photoflash. A collaboration has been set up between the DLR and GUT to measure the electrical resistivities of liquid metals with both PH and EML, involving both terrestrial and microgravity experiments. Some of their results are discussed in Chapter 6.

3.4 Temperature measurement Pyrometry or radiometry must be used for temperature measurements in containerless experiments. In the CNL setup shown in Fig. 2.1, two automatic optical pyrometers are simultaneously employed to measure the sample temperature: a Mikron Model M90 V pyrometer operating at 0.65 mm in the visible range and a Heimann Model KT19.99 operating in the 1 2.5-mm band in the infrared; the latter is used primarily for low-temperature measurements and diagnostic purposes. The pyrometers are oriented at 45 on either side of the vertical plane and directed at a region on the surface of the sample where it is hit by the X-ray beam. Minimum spot sizes are about 1 mm for the visible-range pyrometer and about 1.9 mm for the infrared range. Sample temperatures were found to stable to within about 20 K over the duration of a diffraction scan. A major problem with the used of pyrometry for temperature measurement is the need for spectral emissivity data in order to derive true temperatures from the radiance measurements. If these are not available, the emissivity has to be derived from the radiance temperature observed at the melting point combined with the known melting or liquidus temperature, and assumed to be independent of temperature. The radiance (apparent) temperature Ta is related to the true temperature T through Wien’s law

26

Heating methods

1 T

1 l ¼ lnð"l Þ; Ta C2

ð3:2Þ

where el is the spectral emissivity of the material at the operating wavelength l of the pyrometer and C2 ¼ 1.4388 cm.K is Planck’s second radiation constant. A 92% transmission correction was applied to take account of the pair of close-focusing lenses used with the visible-range pyrometer, and window corrections 92% and 91%, respectively, were applied for the visible-range and infrared-range pyrometers, experimentally determined in separate experiments. The absolute accuracy of radiance temperature measurements is estimated as 4 K at 2000 K, taking into account the uncertainties, including window corrections, pyrometer calibrations and measurement errors. In the work of Krishnan & Price (2000) the accuracy was checked at the melting point of liquid nickel, reported as 1729 K, where a temperature of 1726  4 K was obtained with the visible-range pyrometer applying the corrections just described. Rapid measurement of relative temperatures may also be of interest. An example is the measurement of the velocity of a crystallization front during the rapid cooling of a liquid sample. Willnecker et al. (1989) developed an experimental setup for such measurements using fast-responding silicon photodiodes, shown in Fig. 3.2. The image of a molten drop levitated by

sample

T

upper diode lower diode

T

t (s1)

^

ΔT = 245 K

P

^

(a)

upper diode

upper diode

lower diode

T

s2

s1

20 μs

lower diode T

^

(b)

t (s2)

^

ΔT = 245 K P

(c) t

Fig. 3.2. Schematic illustration of apparatus for measuring solidification velocities in levitated undercooled drops: (a) images of the sample are focused onto two fast photodiodes; (b) solidification is initiated at the point P, proceeds radially and is recorded in the two photodiodes; (c) typical output traces of the photodiodes (Willnecker et al., 1989).

3.4 Temperature measurement

27

EML is focused onto the two photodiodes indicated in Fig. 3.2(a) that provide relative temperature measurements with a time resolution of 1 ms. The melt is undercooled to a predetermined temperature and crystallization is initiated by a solid needle at a point P in a plane normal to the viewing direction. Solidification proceeds rapidly from P, as indicated schematically in Fig. 3.2(b). The recalescence signals (to be described in Section 5.2) recorded in the two photodiodes are shown in Fig. 3.2(c). The velocity of the crystallization front is then determined from the time difference between the two signals and the difference in paths from P to the focal points of the two diodes.

4 Experimental techniques

We now discuss the techniques used to derive information about the properties of levitated materials.

4.1 Electromagnetic and optical properties Information about the electrical and magnetic properties of a levitated sample can be obtained by placing it inside an RF coil and observing changes in the impedance and quality factor of the coil. A schematic of apparatus developed by Enderby et al. (1997) for use with the aerodynamic levitation setup of Fig. 2.1 is shown in Fig. 4.1. If L is the inductance of the empty coil carrying the RF current and Q is its quality factor, the changes in these quantities induced by the insertion of a conducting sphere are given by L Q ¼  0 ; ¼ L Q

Q 00 ;

ð4:1Þ

where expressions for the real and imaginary parts of the susceptibility, 0 and 00 , have been given earlier in Eqs. (2.3 2.7) and the filling factor  is a geometrical quantity that scales, for a given coil, as a3 and can be determined by calibration measurements with spheres of known conductivity. For fixed values of a and s, 0 and 00 depend only on the frequency o. In analogy with Cole Cole plots used in dielectric measurements, the parametric representation of 0 and 00 can be explored by varying o, providing a consistency check on the reliability of the data. The results of four test experiments carried out at room temperature are plotted in Fig. 4.2. The changes in Q and L were measured with an HP4285A high-precision LCR meter. A sample of pure copper, two of pure lead with different sphere diameters and one of tin telluride, an example of a moderate conductor, were studied. The values derived for the conductivities were 5.4  0.04.107 O 1.cm 1 for Cu, 28

29

4.1 Electromagnetic and optical properties

Fig. 4.1. Schematic view of apparatus developed for conductivity and permeability measurements in conjunction with conical nozzle levitation (Enderby et al., 1997).

4.85  0.03.106 O 1.cm 1 for Pb and 5.08  0.05.103 O 1.cm 1 for SnTe, all in agreement with accepted values. This technique was subsequently extended to include a simultaneous determination of the magnetic permeability and the conductivity (Schnyders et al., 1999). For a uniform sphere with a radius a, conductivity s and relative permeability m placed in an oscillating magnetic field in the z direction, Beiot, the vector potential is given by   D~ 1 ð4:2Þ A~ ¼ B r þ 2 sin : r 2 In Eq. (4.2) D~ is a complex quality defined as

  ð2 þ 1Þ~  coth ~ 1 þ ~2 þ 2 3 ~ a; D¼ ð 1Þ~  coth ~ þ ð1 þ ~2 Þ

ð4:3Þ

where ~ ¼ ð1 þ iÞa=. The vector potential in Eq. (4.2) behaves as a dipole with ~ 0 so that the complex susceptibility becomes magnetic moment 3pD= ~ 3 Þ. ¼ 0 i 00 , where 0 and 00 are the real and imaginary parts of ð 3pD=2a

30

Experimental techniques 0.05 SnTe Cu Pb (a) Pb (b)

0.04

a″

0.03

0.02 theoretical 0.01

–0.12

–0.1

–0.08

–0.06 a′

–0.04

–0.02

0

0

Fig. 4.2. The 0  00 plot for the susceptibility measured for four spherical samples at room temperature on the apparatus shown in Fig. 4.1, compared with the theoretical curves (Enderby et al., 1997).

For the nonmagnetic case, m ¼ 1, these expressions reduce to the simpler ones given in Eqs. (2.2 2.7). As defined above, 00 remains positive even in the presence of a magnetizable sphere and achieves a maximum value of 0.93, substantially greater than that for the nonmagnetic case. Thus, the effective resistance and quality factor of the coil are enhanced by the permeability. On the other hand, the bounds for 0 change from 3/2  0  0 for the nonmagnetic case to 3=2  0  3ð 1Þ=ð þ 2Þ. As expected, the positive values of 0 can be accessed in the low-frequency limit where d > a. Parametric representations of 0 vs. 00 are shown in Fig. 4.3 for fixed conductivity and different values of m. It is clear that very low permeabilities are detectable, given the large changes obtained for small increases in m. Experimental data yield directly 0 and 00 , as shown in Eq. (4.1) above. At low frequencies, 0 tends to a limiting value determined directly from experiment: lim o!0 0 ¼ 3

 1 ; þ2

ð4:4Þ

which provides a first approximation for m. Numerical studies have shown that at the angular frequency oc for which 0 ¼ 0; a=   provided  > 1:2, a result anticipated by Jones (1979) in his studies of magnetic levitation referred to in Section 2.3. This gives an estimate for the value of d at the

31

4.1 Electromagnetic and optical properties 1 w 0.8

a″

0.6

0.4

0.2

m=2

m = 10

m=1 0 –2

–1

0

m = 25

m=5 a′

1

2

3

Fig. 4.3. The 0  00 plot for the susceptibility calculated for spherical samples with fixed conductivity and different values of the permeability m (Schnyders et al., 1999).

frequency oc, providing an approximate value of s. It is then possible to determine numerically the values of s and m that give the best fit to the data. Results of a measurement for the case of a solid spherical sample of Ni are shown in Fig. 4.4. A least-squares fit of s and m, made according to the technique just described, converged rapidly to the values m ¼ 12 and s ¼ 1.4.105 O 1cm 1: the authors were not aware of a value for initial permeability at the low magnetic fields present in their coil where the current was typically 20 mA, but the conductivity agreed well with the published value. A similar technique was developed at the DLR for contactless measurements of conductivity in their EML apparatus, both ground-based and microgravity-based. Loho¨fer (1994) derived expressions for spherical samples, with a correction for asphericity given in Loho¨fer (2005). This situation is complicated by the fact that the eddy currents induced in the conducting sample by the primary coil, which give rise to the levitating force, generate an additional magnetic field, and the sum of this field and that generated by the primary coil produce a voltage in the secondary, measurement coil (Fig. 4.5) given by jU2 j expði Þ ¼ ðZ~coil þ
Z~sample Þ jI1 j;

ð4:5Þ

where
Z~sample and hence the magnitude and phase of U2 depend on the conductivity s and radius a of the sample and the deviation e of its shape from spherical symmetry. Measurement of |U2|, f and |I1| determines the

32

Experimental techniques 2 Ni sphere

1 a

a″

0

a′ –1

0

200

400

600

800

1000

Frequency, w (kHz)

Fig. 4.4. Real and imaginary parts of the susceptibility measured for an Ni sample. The points represent the data and the lines least-squares fits (Schnyders et al., 1999).

pyrometer

sample

UHV chamber

rf - generator f ~ 200 kHz

levitation coil

U2

~

I1

transformer windings

Fig. 4.5. Schematic diagram of the primary and secondary measurement coils integrated with the EML in an ultrahigh vacuum chamber (Loho¨fer, 2005).

total impedance Z~coil þ
Z~sample , from which Z~coil can be subtracted by an independent measurement without the sample. If the measurement coils are designed such that the magnetic fields they produce are nearly homogeneous around the sample,
Z~sample is independent of the exact sample position and can be written as

4.1 Electromagnetic and optical properties

a 
Z~sample ¼ iko0 a3 F~ ; e ; 

33

ð4:6Þ

where k is a constant that depends on the coil geometry. The complex function F~ is defined by    ~ ~ ~ ð  Þ ð  Þ ð  Þ 1 j j j 2 1 2 ~ eÞ ¼ Fðq; þ e~  1þ ; ð4:7Þ j0 ð~Þ j0 ð~Þ 3 j0 ð~Þ where ~ ¼ ð1 þ iÞa= as in Eq. (4.3) and jn denotes the nth-order spherical Bessel function. Weak deviations of the sample from the spherical shape are expressed by the parameter e given by the lowest-order coefficient of the expansion of the sample surface in spherical harmonics. Equation (4.7) describes the influence of the sample on the coil impedance via the skin depth d. For spherical samples, e ¼ 0 and Eq. (4.7) reduces to ~ ¼ FðqÞ

2 ½2GðqÞ þ iHðqÞ ; 9

ð4:8Þ

where the functions G and H have been defined in Eqs. (2.5, 2.7). The geometrical constant k is determined by a calibration measurement with a spherical solid sample with known values of s and a. For the sample under investigation,
Z~sample is measured for a large number of frequencies ~ eÞ is fitted to the data by variation of the unknown and the function Fðq; parameters s, a and e. A Faraday balance was developed by Reske et al. (1995) to measure the magnetization of an electromagnetically levitated sample. A CoSm permanent magnet was fixed at one end of a torsion balance and placed in the vicinity of the levitating coil. A small compensation coil was mounted close to the permanent magnet, and the electrical current IM passing through the coil was adjusted so that the permanent magnet stayed at the same position, as measured with a laser optical system, when the sample temperature was altered. IM is directly proportional to the change in magnetization, with the proportionality constant obtained from a measurement with a known sample such as solid Co. Rhim & Ishikawa (1998) developed an ingenious contactless technique to measure the electrical conductivity of liquid metal drops suspended in their ESL apparatus. This technique utilizes the principle of the asynchronous induction motor and measures the relative changes in torque as a function of temperature by applying a rotating magnetic field to the sample. Changes in electrical resistivity are related to the changes in measured torque using the formula developed for the induction motor.

34

Experimental techniques

Four coils positioned around the top electrode of the ESL produce a horizontal magnetic field that rotates at an appropriate frequency in order to induce sample rotation around a vertical axis. The four-coil assembly works as the stator while the levitated sample acts as the rotor. According to the principle of the induction motor, if an ac voltage E1 at a frequency os is applied to a stator having resistance R1 and inductance L1, the torque T experienced by the rotor having its own resistance R2 and inductance L2 and rotating at an instantaneous frequency o, is given by   E21 R21 sR2

; ð4:9Þ T¼ 2 os R1 þ o2s L21 R22 þ o2s L22 where s ¼ (os o)/os. It is clear from Eq. (4.9) that in order to measure the resistivity of the rotor it is important to keep E1 and os constant throughout the measurement process. This requirement is satisfied by maintaining the stator currents constant. Under the condition that R22  s2L2, which was the case for the measurements on liquid aluminium described by the authors, Eq. (4.9) reduces to the simpler form     os E21 1 o : ð4:10Þ 1 T os R21 þ o2s L21 R2 This equation shows that when the left parenthesis the stator term is kept constant, the measured torque decreases linearly as a function of the rotor frequency, and its gradient is inversely proportional to the resistance of the rotor. The torque is measured by timing the rotational acceleration over a given range in o and deriving the moment of inertia from the measured radius and mass. In practice, to reduce systematic errors, the average of equivalent acceleration and deceleration processes is taken. Taking a value in the literature to calibrate the first term in Eq. (4.10), the authors obtained a temperature dependence about 10% smaller than the literature value, while the change on melting agreed within 1%. Another electronic property that can be measured in an ESL setup is the work function f: the amount of energy needed to release an electron from the attraction of positive ions in a material (Paradis et al., 2005b). This can be evaluated using the Richardson Dushman equation:   J ; ð4:11Þ f ¼ kT ln AT 2 where J is the current density resulting from the thermionic emission from a hot sample and A is a constant ¼ 120 A.cm 2K 2. The current density can be obtained from the relation

35

4.1 Electromagnetic and optical properties

J¼

Q ; St

ð4:12Þ

where S is the surface area of the sample, t the charging time prior to levitation and Q, the charge on the levitated sample, is given by Q¼

mg : @E=@z

ð4:13Þ

In the above equation, @E/@z is the electric field between the two parallel electrodes in the ESL apparatus. For measurements in the optical wavelength range, Krishnan & Nordine (1993; 1996) developed an ellipsometric technique for measuring the complex index of refraction of levitated liquid metal drops. In this approach, laser radiation of a known incident polarization is allowed to reflect from the sample. The optical properties of the material can then be determined from measurements of the state of polarization of the reflected radiation at a known angle of incidence. The system operated with a horizontal plane of incidence such that the detected laser radiation was reflected from an equatorial position on the sample, which was a sphere of 4 5 mm diameter suspended by EML. The angle between the incoming source and the ellipsometer axis was fixed, and the angle of incidence on the sample was chosen to be f ¼ 67.38  0.16 , corresponding to the positions of the entrance and exit windows of the EML chamber (Fig. 4.6). D1 Rotating Analyser

D2

Dye Laser

Nitrogen Laser Trigger Generator

Grating Tuning Scan Controller Stepper Motor Controller

E.M. Levitator Linear Polarizer

Two-axis Beam Steering

Box Car Averager

Time Delay Computer

Fig. 4.6. Schematic of the electromagnetic levitation, pulsed-dye laser ellipsometry and data acquisition and control system used for optical property measurements on liquid metals (Krishnan & Nordine, 1996).

36

Experimental techniques

The incident beam was provided by a dye laser pumped by a nitrogen laser providing radiation in the wavelength range 360 990 nm (photon energy range of 3.44 1.25 eV). The incident source optics included a two-axis beam steering device followed by a 2 mm aperture and two linear polarizers. The second polarizer was fixed at 145 with respect to the plane of incidence and the first could be rotated to adjust the intensity of radiation incident on the sample. A beam-splitting prism was used as the analyser to provide orthogonal components of the polarization for simultaneous detection, with a deviation of 45 between the two components, independent of wavelength. Measurements of the intensities could be recorded simultaneously for individual azimuth pairs of (0 , 90 ), and (45 , 45 ) or (135 , 45 ), by rotating the analyser to three fixed orientations. The analyser optics consisted of a strain-free fused silica lens that focused the image of the sample onto a 1-mm diameter aperture, with a second lens placed behind this aperture to collimate the beam through the beamsplitting prism. A pair of lenses refocused the radiation onto the active surface of the two photodetectors. The analyser prism, detector assembly, and secondary optics were all rigidly mounted on a motorized rotation stage. A set of 15 narrow-band interference filters was used to reduce the incandescent radiation incident on the detectors to a negligible value. Intensity measurements were made using a pair of high-speed Si photodiodes, coupled to a boxcar averager with two gated integrators. The intensity ratios I2/I1 ¼ I(0 )/I(90 ), I4/I3 ¼ I(135 )/I(45 ) and/or I6/I5 ¼ I(45 )/I( 45 ) were measured by rotating the analyser to fixed positions. The ellipsometric parameters c and two values for D were obtained from these ratios (Krishnan & Nordine, 1993). The complex index of refraction is then obtained with the relation  1=2 4~ r 2 ; ð4:14Þ sin

N~ ¼ n ik ¼ tan 1 ð1 þ r~Þ where r~ is the complex reflectance defined by r~ ¼ tan cei
:

ð4:15Þ

The normal incidence reflectivity at the wavelength l is easily calculated from Rl ¼

ðn

1Þ2 þ k2

ðn þ 1Þ2 þ k2

ð4:16Þ

and, assuming that the material is opaque, Kirchhoff’s law gives el ¼

4n ðn þ 1Þ2 þ k2

;

ð4:17Þ

37

4.2 Thermophysical properties

where el is the normal spectral emissivity. Other quantities of interest derived from these measurements include the real part of the dielectric function, e1 ¼ n2 k2 , and the optical conductivity ð Þ ¼ 4pe0 nk.

4.2 Thermophysical properties Thermophysical properties include density, surface tension, viscosity, specific heat and thermal conductivity. In a levitated sample, the first three quantities can be determined, in increasing order of difficulty, by videographic observations of the profile of the sample and its time dependence. The density can be determined from the volume and the mass: the first is determined from a fit to the visible cross section, assuming rotational symmetry about an axis Ox normal to the line of sight, and the second by weighing the sample before and after the measurement, in case there are evaporation losses. The profile is fitted using a series expansion in Legendre polynomials Pl, yielding the radius r as a function of angle y with respect to Ox: X al Pl ðuÞ; u ¼ cos : ð4:18Þ rðuÞ ¼ l

The volume is then given as ð1 2p 3 r ðuÞdu: V¼ 3

ð4:19Þ

1

The major experimental difficulties lie in the high spatial resolution required to resolve volume changes of the order of DV/V  10 4 and in the elimination of the non-rotationally symmetric surface oscillations. Measurements of surface tension and viscosity depend on the damped oscillational modes undergone by a freely suspended drop, which can be described in terms of spherical harmonics Yl,m by rðtÞ ¼ al;m Yl;m ð; Þ cos ol;m t e

l;m t

;

ð4:20Þ

where al,m, ol,m and Gl,m are the amplitude, frequency and damping of the mode (l, m) (Egry et al., 1995). For an inviscid drop of mass M and surface tension g, free of external forces, the mode frequencies were derived by Rayleigh (1879): ol;m 2 ¼

4p lðl 3

1Þðl þ 2Þ

g ; M

ð4:21Þ

38

Experimental techniques

independent of m, where M is the mass of the sphere. The frequency of the fundamental mode, l ¼ 2, is oR 2 ¼

32p g : 3 M

ð4:22Þ

In terrestrial levitation experiments, corrections to this simple formula have to be made to account for the Lorentz force and the gravitational force, which result in a distortion of the otherwise spherical sample, leading to a splitting of the fundamental frequency according to the different values of m and to a shift of all frequencies. Cummings & Blackburn (1991) derived an approximate correction formula: oR 2 ¼ 22

40 2 otr 21

3 g2 ; 10otr 2 a

where g is the gravitational acceleration,  1 2 22 ¼ o2;0 þ 2o22;1 þ 2o22;2 5

ð4:23Þ

ð4:24Þ

is the mean square of the split frequencies of the surface oscillations, and otr is the mean frequency of the translational oscillations of the centre of mass, determined by the strength of the applied levitation field and the mass of the sphere. It should be noted that this derivation assumed a perfect conductor with the gravitational force counteracted by a uniform magnetic field, and it is not obvious that it should apply to insulating liquids with CNL. Bratz & Egry (1995) generalized this derivation by including viscosity, taking the limiting case of high Reynolds number Re ¼ a2doR/, d being the density and  the shear viscosity, which is valid in the case of normal liquids above their melting point. They found that Eq. (4.24) still held under these conditions. Egry et al. (1995) applied this theory by measuring the oscillations of a liquid metal drop suspended by EML with a high-speed video camera and analysing the images by digital processing. A typical frequency spectrum is shown in Fig. 4.7. The low-lying peaks correspond to the translational oscillations and the high-frequency ones to the surface oscillation modes, which split into three resolved peaks. Measurements with samples of different masses led to consistent results for the surface tension derived from Eq. (4.22) when the correction of Eq. (4.23) was applied. A conventional method for measuring the surface tension of a liquid is the sessile drop technique, in which the surface tension is derived from the measured contour of a liquid drop on a solid substrate. A schematic diagram

39

4.2 Thermophysical properties

Fig. 4.7. Frequency spectrum of the oscillations of a liquid metal drop suspended by EML (Egry et al., 1995).

of a setup used for liquid silicon is shown in Fig. 4.8. The values obtained by Egry et al. as a function of temperature were in good agreement with those obtained previously with this technique. Fujii et al. (2006) point out that the oscillating levitated drop technique has several advantages: there is no possibility of contamination from a solid surface, the supercooled state is much more accessible, and the mass of the sample is required as opposed to the density that enters in the sessile drop technique, a more difficult quantity to measure. In ESL experiments, a correction for the non-uniform surface charge distribution must be applied. This depends on the applied electric field and the charge on the drop (Rhim et al., 1999). In the absence of external forces, the macroscopic shear viscosity is related to the damping constants in Eq. (4.20) (Chandrasekhar, 1961): l;m ¼

4pa ðl 3M

1Þð2l þ 1Þ;

ð4:25Þ

which equals 20pa/3M for the l ¼ 2 mode. In terrestrial measurements, as we have seen, the modes for different m become split. Bratz & Egry (1995) found that the average damping constant for the five components of the l ¼ 2 mode still obeyed Eq. (4.25), but those for the individual components had correction terms involving (otr2/oR2) and (g2/a2otr2oR2) as in Eq. (4.23).

40

Experimental techniques

Fig. 4.8. Schematic of a sessile drop apparatus used to measure the surface tension of liquid silicon. The silicon sample is first placed in a glass tube outside the chamber. After reaching the measurement temperature, the sample is inserted into the bottom of the alumina dropping tube and molten silicon is then forced from a small hole at the bottom of the tube onto the boron nitride substrate below (Fujii et al., 2006).

A relation between specific heat and total hemispherical emissivity can be obtained by studying the temperature T of a levitated sample as a function of time t after the heating source is removed. Assuming that radiation is the only significant form of heat loss, the resulting energy equation governing the cooling process is given as:



d Cp T 4 ; ð4:26Þ ¼ eT ASB T 4 Tamb m dt where m is the sample mass, Cp the specific heat at constant pressure, eT the total hemispherical emissivity, A the sample surface area, sSB the Stefan Boltzmann constant, and T and Tamb the sample and the ambient temperatures, respectively. Unfortunately, the total hemispherical emissivity is not easy to measure, since it involves an integral over all directions and wavelengths. If it can be assumed to be independent of temperature, and the specific heat is known at a certain temperature, Eq. (4.26) can be used to derive its temperature dependence (Paradis et al., 2001a). Alternatively, if the heat of fusion Hf is known by other means, an average value of Cp for the undercooled liquid can be determined as hCpi ¼ Hf / (Tm Thyp), where Tm is the melting point and Thyp the temperature

4.2 Thermophysical properties

41

corresponding to the hypercooling limit, at which the heat released on cooling is exactly equal to the heat of fusion (Rulison & Rhim, 1994). The concept of hypercooling is discussed in more detail in Section 5.2. In a more fundamental approach to the specific heat, Fecht & Johnson (1991) developed a noncontact modulation calorimetry technique to determine the specific heat in levitation experiments. The heater power input into a heating coil is modulated sinusoidally with a frequency o and amplitude Po, resulting in a modulated temperature response with amplitude To in the sample. If heat loss is caused by radiation only and if the modulation frequency is chosen appropriately so that various corrections become negligible, a simple relation for the temperature variation can be derived: Cp ¼

Po : oTo

ð4:27Þ

A detailed description of all the measurements and corrections involved is given by Wunderlich & Fecht (2005). Kobatake et al. (2007) have developed a contactless modulated laser calorimetry technique for measurement of thermal conductivity of hightemperature liquid silicon in a static magnetic field. Both surface oscillations and convection, which is a problem with the a.c. calorimetry technique described above when used for thermal conductivity studies, are suppressed because of the Lorentz force resulting from interaction between the fluid flow and the magnetic field. Figure 4.9 shows a schematic illustration of the apparatus. A static magnetic field is imposed along the vertical direction in a vacuum chamber using a superconducting magnet. The top surface of the liquid metal is heated sinusoidally with a semiconductor laser modulated with a frequency o. The power absorbed by the liquid is given by the product of the normal spectral emissivity el and the laser power P(t). The temperature response at the bottom surface is measured with a two-colour pyrometer. In a dynamic equilibrium state of modulation heating, a phase shift f between the measured temperature and the laser power is observed. The relation between f and o is obtained by solving the time-dependent heat conduction equation in spherical polar coordinates:      @T 1 @ 1 @T 2 @T sin  ¼k 2 r þ 2 þ Q; ð4:28Þ dCp @t r @r @r r sin  @ where k is the thermal conductivity and Q is the heat generation rate per volume by the RF induction heating, subject to boundary conditions for (a) the laser irradiated area, (b) the non-irradiated area and (c) the centreline of the liquid drop; these depend on the total hemispherical emissivity eT.

42

Experimental techniques static magnetic field

semiconductor laser (wavelength: 808 nm)

rf coil

liquid silicon (R : 4 mm)

mirror

pyrometer

Fig. 4.9. Schematic illustration of the noncontact laser AC calorimetry apparatus for thermal conductivity measurements (Kobatake et al., 2007).

The specific heat Cp is separately determined, for example by the a.c. calorimetry technique described above. Numerical solution of Eq. (4.28) by finite element analysis subject to the boundary conditions (a) (c) gives the spatial distribution of f with parameters k and eT. At lower values of o, f is governed predominantly by eT and at higher values of o mainly by k. A curve fit of the numerically obtained f o relation to the data obtained over the entire frequency range gives the values of both k and eT.

4.3 Diffraction We now pass from techniques designed to measure macroscopic bulk or surface properties to experiments in which detailed information is obtained about the atomic and electronic structure. We start with diffraction, which is generally taken to mean the measurements of atomic or magnetic structure by scattering experiments. In principle, any particle can be used, but for most investigations of atomic structure, neutrons, X-rays and electrons are most

43

4.3 Diffraction

common while neutrons and, in certain cases, X-rays can be also used to provide information about magnetic structure. Sometimes the terms wideangle neutron or X-ray scattering (WANS, WAXS) are used to distinguish such measurements from the small-angle scattering measurements described in the next section. The structural information obtained in a diffraction experiment is described by the variation of the intensity of the scattering as a function of scattering vector Q: Q ¼ k0

k1 ;

ð4:29Þ

where k0 and k1 are the wave vectors of the incident and scattered particles. Levitation experiments are normally made on samples that are directionally isotropic polycrystalline solids, glasses and liquids in which case the scattering depends only on the magnitude of the scattering vector, the scalar quantity Q ¼ |Q|. It is usual to fix the directions of k0 and k1 by means of appropriate collimators, detector placement, etc., and to fix the magnitude of one of these, generally k0, or sometimes a combination of k0 and k1, for example one that corresponds to the total time-of-flight from sample to detector in the case of neutron scattering. The total intensity of the scattered particles measured in the detector is normally recorded, irrespective of any energy transfer that may take place, and Q is evaluated from Eq. (4.29) under the assumption that the scattering is elastic, i.e. there is no energy exchange between the particle and the sample and so jk0 j ¼ jk1 j. In practice, of course, inelastic scattering is always present and in the case of neutron diffraction this can affect the structural interpretation. However, the experiments are usually designed to minimize the errors that result from these approximations, which can usually be taken care of by straightforward corrections. After taking into account appropriate factors such as beam intensity, number of atoms in the sample, detector efficiency and absorption and multiple scattering, the intensity measured in the detector is reduced to a fundamental quantity, the differential cross section per atom ds/dO (Price & Sko¨ld, 1986). For convenience, comparison between experiment and theory is usually done in terms of a dimensionless quantity, the structure factor S(Q). In the case of neutron diffraction, this is related to the differential cross section by the relation  2  n d X  ca ba  ðSðQÞ ¼  d  a 1

1Þ þ

n X a 1

ca ba 2 ;

ð4:30Þ

44

Experimental techniques

where ca, ba and ba 2 are, respectively, the atomic concentration, average (over isotopes and spin states) of the neutron nucleus scattering length, and mean square scattering length of element a present in the sample. This can be rewritten 2   2 2 3     n n n X X X d     ð4:31Þ ca ba  SðQÞ þ 4 ca ba 2  ca ba  5; ¼     d a 1 a 1 a 1 where the leading term contains the structural information that is being sought here. The second term arises from random distributions of different elements as well as in the case of neutron scattering isotopes and spin states over all the atoms belonging to a given element: these two contributions are often referred to as Laue diffuse scattering and incoherent scattering. By definition, S(Q) tends to unity at large Q, a property that is often used to normalize the intensities measured in a diffraction experiment. Its low-Q limit is related to the isothermal compressibility wT:  
0 @V Sð0Þ ¼ kB T ¼
0 wT kB T; ð4:32Þ V @P T where r0 is the number of atoms per unit volume. In between, S(Q) exhibits a complex behaviour that reflects the detailed atomic structure. In a crystalline sample either single crystal or polycrystalline there are sharp peaks called Bragg peaks that arise from diffraction from parallel crystallographic planes at Q values corresponding to 2pn/d, where n is an integer and d the plane spacing. There is also a continuous component, called diffuse scattering, arising from static and/or dynamic disorder. In fully disordered materials like liquids and glasses, all the scattering is diffuse, with a generally oscillatory pattern that reflects the short- and intermediate-range order in the sample. A well-defined distance of closest approach between atoms that can be characterized by an equivalent hard-sphere diameter sHS will be reflected in oscillations in S(Q) with a period 2p/sHS. A typical form for S(Q) in a simple classical liquid is shown in Fig. 4.10. In the case of X-ray diffraction, ba in the above equations and those that follow is replaced by fa(Q), the atomic form factor for species a. This results from the fact that the electrons in the atom from which the X-rays are scattered have a spatial distribution while the nucleus from which the neutrons are scattered can be treated, for the present purposes, as a point object. Since the form factors are generally well tabulated this is not a major problem, but it can complicate the interpretation of the scattering from multi-component systems.

45

4.3 Diffraction Qmax ≈ 2p /sHS

S(Q )

3

2 S(Q → ∞ ) = 1 1

S(Q = 0) = cr kT 0

0

1

2

3

4

5

Q, A –1

Fig. 4.10. A typical form for S(Q) in a simple classical liquid or glass.

A pair correlation function g(r) that contains the structural information about the sample in real space is then calculated from S(Q) via the Fourier transform 1 gðrÞ ¼ 1 þ 2 2p
0

Qð max

QðSðQÞ

1Þ

sin Qr MðQÞdQ; r

ð4:33Þ

0

where M(Q) is a modification function that is often used to force the integrand to go smoothly to zero at Qmax and reduce the ripples that result from the finite limit of the integration. For systems with more than one type of atom different elements, and sometimes different isotopes of the same element S(Q) is a weighted sum of partial structure factors Sab(Q). Unfortunately there are a number of alternative definitions of these in the literature: the S(Q) appearing in Eqs. (4.30 4.32) is called the Faber Ziman definition after its originators (Faber & Ziman, 1965). Another definition of partial structure factors for binary systems was proposed by Bhatia & Thornton (1970), where SNN(Q) describes the fluctuations in total particle density, SCC(Q) those in the relative concentrations, and SNC(Q) the cross-correlation of the two. For a two-component system Eq. (4.32) applies to SNN(Q). The various definitions are linear combinations of each other are given in the textbooks, e.g. that of March and Tosi (1992). For a multi-component system, g(r) is correspondingly a weighted sum of partial pair correlation functions gab(r), where the indices a, b refer to atom types. In the neutron case, this is given by

46

Experimental techniques

gðrÞ ¼

X a;b

Wab gab ðrÞ ¼

X ca cb ba bb gab ðrÞ: P ca ba j2 a;b j

ð4:34Þ

The partial pair distribution function gab(r) can be considered as the relative probability of finding a b atom at a distance r from an a atom at the origin. In a one-component system, the indices a, b disappear and only a single S(Q) and a single g(r) exist. In a system with n components, a full structural analysis requires n(n þ 1)/2 different measurements with different coefficients in Eq. (4.34) (Price & Pasquarello, 1999): in favourable cases, this may be accomplished with the use of isotope substitution in the case of neutron diffraction (Enderby et al., 1966), by anomalous X-ray scattering (AXS) near an absorption edge, where the form factor has an additional component that varies rapidly with X-ray energy (Raoux, 1993; Price & Saboungi, 1998), or by a combination of the neutron and X-ray scattering (Price et al., 1998). With a single measurement, only the average structure factor S(Q) can be determined; nevertheless, this may still contain useful information. For example, if a particular peak n in g(r) can be associated uniquely with a coordination shell for a pair of atom types a,b, the coordination number of b atoms about an average a atom for that shell is given by ð cb n rTðrÞdr; ð4:35Þ Ca ðbÞ ¼ Wab n

where T(r) ¼ 4pr0rg(r) and the integral is taken over the peak n, while the n . centroid of T(r) over the same peak gives the average coordination distance rab In a magnetic system, neutrons can also scatter from the magnetic moments associated with the unpaired electrons. In simple cases where the unpaired electrons can be associated with a particular atom, the magnetic scattering can be described by making the substitution 1 ba ! gn r0 fma ðQÞ ga Sa 2

ð4:36Þ

in the formalism given above, where gn ¼ 1.9132 is the g-factor for the neutron, r0 ¼ 2.8179 fm is the classical radius of the electron, and fma(Q), ga and Sa are the magnetic form factor, g-factor and spin operator of the unpaired electrons on the atoms of element a. It is clear that if there are correlations between the orientations of the magnetic moments in the system with their positions, i.e. some kind of magnetic ordering, there will be a structuredependent term in the magnetic scattering analogous to the first term in the expression for the nuclear scattering, Eq. (4.30). If, on the other hand, the orientations of the magnetic moments are completely random, as in a

47

4.3 Diffraction

Fig. 4.11. Schematic view of the experimental arrangement for neutron diffraction measurements with CNL, showing laser heads (a,b), spherical mirrors (c), NaCl windows (d), video camera (e) and levitation device (f). Incident and scattered neutron beams are in a horizontal plane through the sample, positioned above the nozzle of the levitation device (Hennet et al., 2006).

paramagnetic system, the magnetic scattering is independent of the structure and can be described by a term 2 2  gn r0 fm;a ðQÞ a 2 ; 3

ð4:37Þ

analogous to the second term in Eq. (4.30), ma ¼ 1/2 ga Sa being the magnetic moment on element a. We return to these ideas in the discussion of possible magnetic ordering in liquid alloys in Chapter 6. Details of the apparatus and data analysis required for X-ray diffraction experiments on levitated materials are given by Krishnan & Price (2000) and Hennet et al. (2002), and schematic views of the two setups have been shown in Figs. 2.1 and 2.2. Corresponding information for neutron diffraction is given by Hennet et al. (2006), and a schematic view of the apparatus used at the D4C diffractometer at the Institut Laue-Langevin (ILL) is shown in Fig. 4.11. It is clear from the Fourier transform in Eq. (4.33) that short-range structural information will tend to dominate the scattering at high Q and long-range at low Q. Thus, the need to get accurate information about nearest-neighbour correlations, such as bond distances and coordination numbers, has driven the development of diffractometers with a large Q range, exploiting epithermal neutrons from pulsed spallation sources and highenergy X-rays from third-generation synchrotron sources.

48

Experimental techniques

4.4 Small-angle scattering Information about long-range correlations requires extending the range of the measurements to low Q in what has come to be known as small-angle neutron scattering (SANS) and small-angle X-ray scattering (SAXS). This can be carried out with both neutrons and X-rays but, since these measurements are normally made in a transmission geometry, the weaker scattering power of neutrons can be an advantage, since the samples are usually of manageable size 1 2 mm thick. In the following discussion we will give the formalism for the neutron case (Price & Sko¨ld, 1986). It is useful for this purpose to write the differential cross section in terms of a continuous scattering-length density (SLD) X bi ½r Ri : ð4:38Þ
b ðrÞ ¼ i

Similarly, it is convenient to deal with a differential cross section per unit volume of sample, dS/dO, instead of the differential cross section per atom, ds/dO introduced in the previous section. The quantity dS/dO is sometimes called a macroscopic differential cross section since it gives the probability of a scattering event per unit distance travelled in the sample. Since a uniform SLD gives rise to scattering at the forward angle, Q ¼ 0, only, it is appropriate to write
b ðrÞ ¼
b þ 
b ðrÞ;

ð4:39Þ

where the first term is the average value over the sample and the second term the fluctuations around it. The macroscopic differential cross section then becomes  2 ð    d 1  iQ r  ð4:40Þ ¼  
b ðrÞe dr d V   V

for a sample of volume V. A simple case that produces small-angle scattering is that of Np identical particles with volume Vp and uniform scattering length density rp embedded randomly in a matrix of uniform scattering length density rm. Then Eq. (4.40) becomes d ¼ np jFðQÞj2 ; d

ð4:41Þ

where np ¼ Np/V is the number density of particles and we define a singleparticle scattering amplitude

4.4 Small-angle scattering

ð FðQÞ ¼ 
p eiQ r dr:

49

ð4:42Þ

V

We note that the scattering probability is proportional to the particle number density, as expected, but also to the square of the particle volume. Thus, SANS scattering intensities can be much higher than those encountered in normal diffraction experiments. In this section we will treat Q as a vector since the samples of interest may have a preferred orientation, which will show up as a dependence on azimuthal angle in a position-sensitive detector measuring scattering events at small angles. As Q ! 0, the differential cross section reduces to d ð4:43Þ ¼ np Vp2 
2p : d Further useful information can be obtained from the integrated intensity per unit volume ð   d ~ ð4:44Þ ðQÞdQ ¼ ð2pÞ3 
2 ðrÞ av ; Q¼ d where the average is taken over the whole sample. For the simple case discussed above this reduces to

ð4:45Þ Q~ ¼ ð2pÞ3 cp 1 cp 
2p ; where cp ¼ npVp is the volume fraction of the particles. Equation (4.45) shows that this quantity can be obtained from an absolute measurement of the total scattered intensity if the scattering length densities of the particle and matrix materials are known. Under the same conditions, the Q ¼ 0 limit, Eq. (4.43), gives npVp2 ¼ cpVp, from which the average particle volume Vp can also be obtained or, alternatively, the particle molecular weight Mw ¼ dVp NA, where d is the mass density in the particle and NA Avogadro’s number. The precise form of F(Q) depends on the shape and orientation of the particle. In the limit of small Q values such that Ql 1, where l is a characteristic particle size, the Guinier approximation generally holds: ! 2 2 Q R

2 g ; ð4:46Þ jFðQÞj2 ¼ 
p Vp exp 3 where Rg is the radius of gyration given by ð 1 2 Rg ¼ r2 dr: Vp Vp

ð4:47Þ

50

Experimental techniques

Equation (4.46) shows that Rg can be directly obtained from the experimental data by plotting the logarithm of the intensity vs. Q2 at low Q. So far we have assumed that the particles are arranged randomly so that there is no interference in the scattering from neighbouring particles. This is generally a good approximation at low concentrations. In many cases, however, the arrangement of large structural units is important, for example in colloid solutions at high concentrations and biological systems where the internal arrangement of the subunits is often of vital importance for the biological function. To treat this more general case, we divide the system into Np cells such that each cell contains exactly one particle. Denoting the centre of the Ith cell by lI and the position of the particle in the cell by di, the differential cross section can be written d 1 ¼ d V

2 + * N p X    iQ ‘I e FI ðQÞ ;  I 1 

ð4:48Þ

where FI(Q) is the scattering amplitude for cell I: FI ½Q ¼

NI X

bi eiQ di :

ð4:49Þ

i 1

For a system of identical particles that are either isotropic or anisotropic with the same orientation, the form factors are the same for all particles and Eq. (4.48) can be written as d ¼ np jFðQÞj2 SðQÞ; d

ð4:50Þ

where F(Q) is the form factor of each particle and S(Q) is the interparticle structure factor *N + p 1 X eiQ ð‘i ‘j Þ : ð4:51Þ SðQÞ ¼ Np i;j 1 As Q ! 1, the terms for i 6¼ j in Eq. (4.51) average out to zero, and the intensity gives information about the surface area of the individual particles. Equation (4.50) leads to the Porod approximation d 2p 
p 2 S ; ! Q4 d V

ð4:52Þ

4.5 X-ray absorption spectroscopy

51

where S is the total surface area of the particles. Thus, the high-Q limit of the differential cross section decreases as 1/Q4, with a proportionality constant that depends on the specific surface area of the particles. Magnetic inhomogeneities may give rise to magnetic SANS, in which case the substitution of Eq. (4.36) can be made in the formalism just given.

4.5 X-ray absorption spectroscopy X-ray absorption spectroscopy is a general term referring to experiments in which the absorption of an X-ray beam by a sample is measured as a function of incoming energy. Sometimes it is more convenient to measure the fluorescence produced following the absorption rather than the attenuation of the beam. The absorption increases when the energy is raised through and above an absorption edge of one of the elements in the sample. In materials science, the important energy regions are near the edge (X-ray absorption near-edge spectroscopy, XANES) and extending for some range above the edge (X-ray absorption fine-structure spectroscopy, EXAFS). Since the absorption edge is associated with the transition of an electron in the sample from a core level to a free state, the detailed energy dependence of the XANES spectrum gives information about the electronic structure of the valence and conduction electrons. EXAFS, on the other hand, is essentially a diffraction phenomenon in which the photoelectron is scattered back from the neighbouring atoms: the back-scattered wave interferes with that of the primary photoelectron to produce a change in the absorption probability. The higher the X-ray energy E above the energy EA of the absorption edge, the larger the wave vector k of the photoelectron and hence the scattering vector 2k characterizing the diffraction process: E

E0 ¼

2 k2 h : 2me

ð4:53Þ

The spectrum of absorption vs. 2k then shows oscillations similar to that of S(Q) in diffraction experiments. This has the advantage that the structural information in the spectrum is element-specific, i.e. relates only to the environment of the absorbing atoms. A disadvantage is that in liquids and glasses it is difficult to get structural information beyond the nearest neighbours of the absorbing atoms. With the powerful X-ray synchrotron sources now available, it is often preferable to use the anomalous X-ray scattering technique referred to above. However, in that case there is a correlation between the energy of the absorption edge and the maximum magnitude of the

52

Experimental techniques

scattering vector Q, which determines the spatial resolution of the measurement, so with light elements, e.g. those lighter than germanium (EK ¼ 11.1 keV, Qmax ¼ 11.25 A˚ 1), other methods must be used: EXAFS, neutron diffraction with isotopic substitution, if suitable isotopes are available, or a combination of two or three techniques. Other advantages of EXAFS are that it is possible to obtain information about three-body correlations (Di Cicco et al., 2003) and, since the measurements are relatively rapid, structural changes can be studied during rapid variation of temperature or pressure. The EXAFS signal w(k) is defined as the normalized deviation of the absorption coefficient m(k) of the sample from its value for an isolated atom m0(k): wðkÞ ¼

ðkÞ 0 ðkÞ : 0 ðkÞ

ð4:54Þ

From scattering theory, jaðkÞj wðkÞ ¼ k

1 ð

pðr Þ e r2

2r lðkÞ

sin½2kr þ ðkÞ dr;

ð4:55Þ

0

where |a(k)| and f(k) are the characteristic back-scattering amplitude and phase shift due to scattering from the neighbouring atoms, l(k) is the mean free path of the photoelectron and p(r) is the bond length probability density, proportional to g(r). In the case of small disorder, a Gaussian probability density with variance s is normally assumed, and Eq. (4.55) becomes       X Cn ð jÞ  2  2r n 2 1   wðkÞ ¼  2 fj ðkÞ exp 2 1j k exp lðkÞ n n r1j h i n þ j ðkÞ dr; ð4:56Þ sin 2kr1j where, in the notation introduced in Section 4.3, the absorbing atom is taken as type 1 and the nth coordination shell is occupied by atoms of type j. For disordered systems like liquids, glasses and crystalline materials at high temperatures, the assumption of symmetric peaks in g(r) is no longer valid, since the backscattering atoms feel the anharmonicity of the pair potential. Filipponi and co-workers (Filipponi & Di Cicco, 1995; D’Angelo et al., 1994) have developed a more realistic, asymmetric model function:



    2 r r1j 2 r r1j 2 exp qj þ ; ð4:57Þ qj þ pðrÞ ¼   1j j 1j j 1j j ðqÞ

53

4.6 Inelastic scattering pyrometer quartz glass window fluorescence radiation ionization chamber

1500 °C

UHV chamber

photodiode

sample in levitation coil RF generator 260 kHz 6 kW

focused, monochromatic synchrotron radiation Beryllium window photodiode He/H2 cooling gas

pumping unit

Fig. 4.12. Experimental setup used for EXAFS measurements with EML. The monochromatic synchrotron radiation reaches the levitating sample after passing an ionization chamber. The secondary fluorescence radiation is detected by four photodiodes, which are placed concentrically around the incident beam, two vertical and two horizontal (not shown) (Jacobs & Egry, 1999).

where G is the Euler gamma function, b is an asymmetry parameter and q ¼ b 2. The parameters C1n( j), r1j, s1j and bj are determined for each shell by inserting Eq. (4.57) into Eq. (4.55) and fitting to the measured w(k). In addition, the procedure takes into account the many-body correlations mentioned above (Filipponi et al., 1995). Figure 4.12 shows the setup of EXAFS measurements on liquid metal samples suspended by EML, developed by Jacobs and Egry (1999). The thickness of the sample made it impossible to measure the absorption directly in transmission, so the fluorescence radiation arising from the transition of an electron from the L to the K state was recorded. To minimize the influence of the sample movement on the spectra, a horizontal and a vertical pair of photodiodes was used, placed concentrically in a backward direction (about 16 ) around the incident beam.

4.6 Inelastic scattering The dynamics of a system can be measured in an inelastic scattering experiment. For the past 50 years, this has been principally the province of neutron scattering, taking advantage of the fact that neutron beams emerging from

54

Experimental techniques

moderators at reactors or spallation sources have typical energies on the order of 0.025 eV, corresponding to a temperature of about 300 K and comparable with typical energies of collective excitations in solids and liquids (Squires, 1978; Price & Sko¨ld, 1986). With cooled moderators and developments in neutron spectroscopic techniques such as neutron spin-echo spectrometry (Mezei, 1972), the usable energy range has been pushed down to 10 6 eV and below, which provides a powerful probe of relaxation processes in complex materials. X-rays start with the disadvantage that the energies must be on the order of 104 eV to access the Q values  10 A˚ 1 of interest for investigations of the structure and dynamics of materials, so that an energy resolution of 10 7 is required to get useful information about collective excitations in solids and liquids. Remarkably, this has been achieved with sophisticated design of energy monochromators and analysers at the third-generation neutron sources such as the European Synchrotron Radiation Facility (ESRF) in Grenoble, France, the Advanced Proton Source (APS) in Argonne, Illinois, and SPRing-8 in Hyogo Prefecture, Japan. High-resolution inelastic X-ray scattering (IXS) techniques have the advantage of overcoming the kinematic limitations affecting many neutron scattering studies, and make it possible to study collective excitations in liquids and glasses at low Q (Burkel, 1991). This follows from the requirement that the velocity of the probe in such measurements must be appreciably higher than that of the collective excitation under study. A schematic view of the spectrometer developed for high-resolution inelastic scattering at the APS (Sinn, 2001) is shown in Fig. 4.13. In inelastic scattering experiments, the energy transfer E or equivalently the excitation frequency o ¼ E/ h is measured in addition to the scattering vector Q. In the neutron case this is given by 6 m horizontal arm ~11 cm

10 cm

D

J 1m

m

C

3 mm H G F

E

B

A

Fig. 4.13. A schematic view of the inelastic scattering spectrometer developed at beam line 3-ID at the Advanced Photon Source. The X-ray beam comes from an undulator (A) and pre-monochromator (C), and then passes through the high-resolution monochromator (D) and focusing mirror (E) before it illuminates the sample (G). The scattered intensity is energy analysed and focused back by crystal analyser (I) into detector (J). The ionization chamber to monitor the incident flux onto the sample is (F), and the slit systems that determine the source size are (B) and (H) (Alatas et al., 2005).

55

4.6 Inelastic scattering

2 k02 h 2mn

2 k12 h ; 2mn

ð4:58Þ

E¼ hck0

hck1 : 

ð4:59Þ

E¼ and in the X-ray case by

To accomplish this measurement with either probe, both the magnitude and direction of both k0 and k1 must be defined in the design of the scattering apparatus. The intensity of this scattering process is reduced to a double differential cross section, which for neutron scattering is given by:  " # ( ) n n    X d2  k1 X 2 2 inc 2 S ðQ; EÞ ; ð4:60Þ ca ba  SðQ; EÞ þ ca ba ba ¼   ddE k0  a 1 a 1 where the SðQ; EÞ and Sinc ðQ; EÞ are the coherent and incoherent partial scattering functions (sometimes called dynamical structure factors). Their physical significance can be understood if we make Fourier transforms to (Q,t) space: 1 SðQ; EÞ ¼ 2p h Sinc ðQ; EÞ ¼

1 2p h

1 ð

IðQ; tÞe

1 ð

iEt= h

dt

1

Is ðQ; tÞe

ð4:61Þ iEt= h

dt;

1

where I(Q,t) and I ðQ; tÞ are called, respectively, the total and self intermediate scattering functions. Their values at zero time, or alternatively the integrals of S(Q, E) and Sinc ðQ; EÞ over the entire energy region, are: 1 ð IðQ; t ¼ 0Þ ¼ SðQ; EÞdE ¼ SðQÞ s

1 1 ð

Is ðQ; t ¼ 0Þ ¼

ð4:62Þ Sinc ðQ; EÞdE ¼ 1:

1

Thus, I(Q, t) represents the time development of the instantaneous partial structure factor S(Q) introduced in Section 4.3. On the other hand, Is ðQ; tÞ is related to the distribution in space that a single particle of type a is likely to occupy after time t. The time-dependent quantity thus contains useful information about the trajectories of individual particles, whereas its value at time zero is trivially equal to one: each particle has not had time to move.

56

Experimental techniques

The astute reader will notice that the averages are defined slightly differently in Eqs. (4.31) and (4.60). The latter in fact corresponds to the Ashcroft & Langreth (1967) definition of partial structure factors, which translates more naturally into the inelastic regime. For multi-component systems, SðQ; EÞ and correspondingly I(Q, t) are weighted averages over the different pairs of atom types, as in the diffraction case, while Sinc ðQ; EÞand I s ðQ; tÞ are averages over the different individual atom types. This complication must be taken into account when interpreting inelastic scattering data. In neutron scattering, the relative contributions of coherent and incoherent scattering will depend on the various elements and isotopes in the samples. Most natural elements, as well as 2H and 6Li, are mostly coherent scatterers, whereas naturally abundant hydrogen is mostly incoherent and natural lithium and silver, for example, are a mixture of both. These facts must also be taken into consideration in the interpretation of the scattering data. In inelastic X-ray scattering, ba in the above equations and those that follow is replaced by fa(Q), the atomic form factor for species a. In this case, every atom of a given element scatters identically so the incoherent term does not appear. (In the X-ray field the term incoherent is often used instead to denote the Compton scattering, in which an X-ray scatters inelastically from an individual electron, providing information about the momentum distribution of the electrons in the ground state.) In the case of a solid sample it is convenient to distinguish four dynamical regimes of neutron scattering, illustrated schematically in Fig. 4.14 for the case of a disordered system like a liquid or a glass. For convenience we frame the discussion in terms of the weighted average structure factor S(Q) introduced in Section 4.3 and the corresponding weighted average scattering function S(Q, E). Figure 4.14(a) shows a typical structure factor S(Q) in which we pick out a particular value of Q, say Q1, and discuss the time development I(Q1, t) of the correlations contributing to S(Q1). Figure 4.14(b) shows various time regimes that may appear in I(Q1, t), and Fig. 4.14(c) the corresponding features in S(Q1, E) obtained by the Fourier transform, Eq. (4.61). (1) The conceptually simplest scattering event is one that takes place as if the target nucleus is independent of its neighbours. This is in fact what happens at short times, where I(Q1, t) falls off from its value at t ¼ 0, generally with an approximately Gaussian behaviour. In the limit of large Q, this recoil scattering is the dominant contribution to S(Q1, E), consisting of a peak on the neutron energyloss side (E > 0) centred at the recoil energy ER ¼ h2Q12/2M with a shape that reflects the momentum distribution of the system in its ground state. In particular, the variance in energy is related to the mean kinetic energy K: sE2 ¼ Kh2Q12/M. (2) If there are collective excitations with a frequency op, I(Q1, t) has an oscillatory part and S(Q1, E) has a peak centred at hop, generally referred to as one-phonon

57

4.6 Inelastic scattering (a) S (Q )

Q

Q1 (b) I (Q1,t ) (2)

S (Q1)

(1)

(3) (4) t

(c)

S (Q1E)

(4) ΔE (3) Ea

(1) (2)

បwp

o

បwp

sE ER

E

Fig. 4.14. Schematic illustration of dynamical regimes probed by inelastic neutron scattering: (a) structure factor S(Q), highlighting a specific scattering vector Q1; (b) intermediate-scattering function I(Q1, t); (c) scattering function S(Q1, E). The numbers denote the (1) recoil, (2) one-phonon, (3) quasielastic and (4) elastic scattering regimes (Price et al., 2003). scattering. In single crystals the phonons can be labelled by a wave vector Q and a branch index j: if the vibrational motion is harmonic, S(Q1, E) has a delta-function form S1(Q1) d(E   hop). In a polycrystalline sample S(Q1, E) is an orientational average over all directions of Q. In a glass, phonons still exist although they can no longer be labelled by a single value of Q. (3) If there are relaxation processes in which the correlations decay at some characteristic rate a(Q) at a given value of Q, S(Q1, E) has a broadened component centred at E ¼ 0, called quasielastic scattering. Because of the higher energy resolution, quasielastic neutron scattering (QENS) is more commonly used. In the case of an exponential time decay, S(Q1, E) has a Lorentzian form Sqe(Q1) La(Q, E), where

58

Experimental techniques L ðQ; EÞ ¼

h ðQÞp E2 þ ½h ðQÞ 2

:

ð4:63Þ

(4) If there are structural correlations that last for long times (more precisely, times t  h/DE, where DE is the energy resolution of the experiment), which is the case in a  solid where the atoms execute thermal motions about fixed equilibrium positions, I(Q1,t) contains a non-zero time-independent term. This will give a delta-function Sel(Q1)d(E) term in the scattering function, generally referred to as elastic scattering.

In a liquid, purely elastic scattering (regime (4)) does not exist, and both the quasielastic scattering (regime (3)) and collective excitations (regime (2)) will generally be heavily damped and show up as recognizable peaks only at sufficiently low Q, sometimes called the generalised hydrodynamic regime. In the purely hydrodynamic regime that can be probed by light scattering, the collective excitations represent density fluctuations and are often referred to as Brillouin peaks, while the quasielastic scattering is the result of entropy fluctuations and is called the Rayleigh peak. At higher Q the scattering reflects the dynamics of single particles, which evolves continuously into the recoil scattering at very high Q (regime 1). As shown in Eq. (4.62), the integral of S(Q, E) over the entire range of E is equal to S(Q). There is also a simple expression for the second energy moment of S(Q, E): 1 ð

SðQ; EÞE2 dE ¼ 1

h2 Q2 kT M

ð4:64Þ

which, taken together with Eq. (4.62), implies that the standard deviation of the width is given by s kT ð EÞ ¼ hQ: ð4:65Þ MSðQÞ This shows that the distribution in E narrows at values of Q where S(Q) has a maximum, a behaviour first pointed out by de Gennes (1959). A typical behaviour of S(Q, E) for a liquid, illustrating these various regimes, is shown in Fig. 4.15. The expressions given above strictly refer to the coherent scattering function. In regime (1), the incoherent scattering function will have the same form at high Q, while in regime (2) the delta function is replaced by a continuous function of Q describing a density of states. In regimes (3) and (4) the general form will be the same but with a different, generally more slowly varying, Q dependence. In an actual condensed system, especially as the complexity

59

4.7 Nuclear magnetic resonance

single particle regime hydrodynamic regime

w Q 0

De Gennes narrowing Qmax

Fig. 4.15. A typical form of S(Q, E) for a classical liquid, as a function of scattering vector Q and frequency o ¼ E/ h.

increases, there will be coupling between the different types of motion and the simple forms given above must be replaced by more complicated functions of Q and E. Nevertheless, the distinction between the four dynamical regimes will generally be meaningful. Neutrons from a moderator in a reactor or placed adjacent to a target in a pulsed spallation source have typical energies that correspond to ambient temperature 25 meV and wavelengths on the order of 1.8 A˚, both of which are well matched to the energy transfers and scattering vectors of interest in the dynamics of condensed matter. Measurement of these requires a definition of the energy, and hence the wave vector, of both incident and scattered beams. For measurements at high E as in regime (1), the higher epithermal neutron intensities at spallation sources are advantageous. Conversely, QENS measurements at high E resolution are generally carried out with neutron beams produced by cold moderators.

4.7 Nuclear magnetic resonance Nuclear magnetic resonance (NMR) is a powerful technique for obtaining short- and intermediate-range structural information and studying relaxation processes in materials. Here we will give a short description, and for a deeper discussion the reader is referred to textbooks such as that of Slichter (1980). Nuclei with spin angular momentum m ¼ hI exhibit a permanent magnetic moment m ¼ g hI;

ð4:66Þ

60

Experimental techniques

where g ¼ (e/2mpc)gN is called the gyromagnetic ratio, gN is the nuclear g-factor (equal to 2.793 for a proton), mp the proton mass and I the spin operator whose eigenvalues are positive integers or half-odd integers. Because of the large ratio of proton and electron masses, mp =me ¼ 1836, the nuclear magnetic moments will be some 10 3 10 4 times those characteristic of electronic magnetism. Classical electrodynamics dictates that such a magnetic moment under the action of a magnetic field B experiences a torque B that satisfies dm 1 dm ¼ ¼ m  B: dt g dt

ð4:67Þ

If the field is applied along the z direction with constant intensity B0, the Cartesian components of m evolve in time as dy dx ¼ gy B0 ; ¼ dt dt

gx B0 ;

dz ¼ 0; dt

ð4:68Þ

and therefore d2 x ¼ dt2

g2 B20 x ;

d 2 y ¼ dt2

g 2 B0 2 y :

ð4:69Þ

With a suitably chosen time origin the solution of the linear equations can be written x ¼ ? cos o0 t; y ¼

? sin o0 t; z ¼ jj ;

ð4:70Þ

where ? ; jj are the components of the nuclear dipole normal and parallel to the applied field. These equations tell us that the dipole will precess around B0 with the Larmor frequency o0 ¼ gB0 . The energy of a nucleus in such a field is ho0 ; ðI 1Þho0 ; . . . ; ðI 1Þho0 ; Iho0 . These thus z B0 with energy levels I levels are perturbed by fields arising from neighbouring magnetic moments. In the simplest case, which usually holds for normal liquids, the level scheme just described still holds because these fields average out to zero. Under such circumstances, irradiation by radio waves with frequency o0 will lead to resonance absorption. Since we are dealing with macroscopically large numbers N of particles under steady-state conditions, Boltzmann statistics usually hold. The relevant macroscopic quantity is then the magnetization per unit volume M that results from averaging the magnetic moments over the energy levels: M¼

Ng2  h2 IðI þ 1Þ B0 : 3kB T

ð4:71Þ

In the absence of an external field, the energy levels have equal populations but such degeneracy is broken by the applied field, resulting in an increase in

61

4.7 Nuclear magnetic resonance

energy per unit volume of the spin system given by ðM B0 Þ. This energy is dissipated throughout the ensemble of particles that for historical reasons is usually called the lattice. Under thermal equilibrium the nuclear spins obey Boltzmann statistics and preferentially occupy states where ð B0 Þ has large values. The macroscopic magnetization follows the spins and lies in the direction of B0. If we now apply a perturbation that drives the field to another value B00 along the same direction, the magnetization will have components parallel and normal to the field, the latter vanishing when thermal equilibrium is finally attained. Equilibrium is then approached through magnetic relaxation with different rates for the normal and parallel components of the macroscopic magnetization. The Cartesian components of their time dependence become My dMz dMx Mx dMy ¼ ¼ ¼ ; ; dt T2 dt T2 dt

Mz

M0 T1

;

ð4:72Þ

where M0 is the equilibrium value and T1 and T2 are the longitudinal (or spinlattice) and transverse (or spin-spin) relaxation times. The solutions of these equations are exponential functions with relaxation rates equal to the inverse of the relaxation times. The detailed behaviour of the relaxation processes depends on the detailed structure and dynamics of the sample under study. The magnetic moments on the nuclei can interact with each other and with the lattice. At least three types of interactions are involved: (a) magnetic dipole interactions; (b) interactions with the perturbations in the electron cloud caused by the external field, known as the chemical shift interaction; (c) interactions with electric quadrupole moments in the case of nuclei with I > 1/2.

These interactions are anisotropic and in the case of glasses and polycrystalline systems lead to a broadening of the resonance lines. In liquids the fast atomic motions have the effect of averaging the anisotropic part and one observes narrow resonance lines centred on the isotropic part, corresponding to the trace of the interaction tensor. This is known as motional narrowing. An apparatus designed for NMR studies of liquids in conjunction with CNL is shown in Fig. 4.16. Some results from this setup will be discussed in Chapter 8. The magnetic dipole interactions (a) can in turn be of two kinds. Two nuclei with magnetic moments m separated by a vector r will have an interaction that can be described by a variation in the local magnetic field:

62

Experimental techniques CO2 (120 W) Laser

Video

Pyrometer

T~2000 ˚C RF Coil Levitator

Sample

Laser Beam ~ ~

~ ~

Probe Magnet

90

80

70

ppm

Gas CO2 (120 W) Laser

Fig. 4.16. High-temperature CNL NMR setup at CEMHTI and a characteristic one-shot spectrum (Massiot et al., 2008).

30 3 cos2 
B ¼  4p r3

1

;

ð4:73Þ

where y represents the angle between r and the principal field B0. In the case of liquids the terms in parentheses vanishes due to motional narrowing. In addition, there can be an indirect coupling through the electrons involved in the chemical bonds between two atoms. If one atom is coordinated to C chemically equivalent atoms, the resonance line will be split into (Cþ1) components. The separation between the lines is described by a coupling constant J. The chemical shift interaction leads to a field variation
B ¼

s B0

ð4:74Þ

where s is a second-rank tensor. It is customary to describe the isotropic and anisotropic terms of this tensor by chemical shifts diso, dij:

ref 6   ¼ 10 ; ð4:75Þ ref where n refers to the frequency of the resonance line in the sample and nref to a chosen reference material. In liquids, only the isotropic part remains due to motional narrowing. This can give information about the chemical environment of the atom corresponding to the resonant nucleus. In solids, the direct

4.8 Numerical simulation

63

dipole coupling the anisotropic part of the chemical shift interaction can be eliminated by magic angle spinning about an angle ym ¼ 54.74 , which reduces Eq. (4.73) to zero. Muon spin spectroscopy (mSR) is a similar experimental technique based on the implantation of spin-polarized muons in matter and the detection of the influence of the atomic or molecular surroundings on their spin motion. The motion of the muon spin is caused by the magnetic field experienced by the particle and may provide information on its local environment in a similar way to other magnetic resonance techniques such as NMR. Facilities for mSR are located at several proton accelerators, often co-located with a spallation neutron source. 4.8 Numerical simulation The use of molecular dynamics (MD) computer simulations, in which the trajectories of the particles are followed by solving Newton’s equations in a sequence of time steps, in conjunction with realistic interatomic potentials was pioneered by Rahman in the 1960s (Rahman, 1967). The use of such simulations to delve into details of the microscopic structure and dynamics of disordered matter is now widespread. This is the result, in part, of the tremendous increase in computer power over the past decades and in part to the development of sophisticated computational tools. Today, most simulations are carried out either to get some insight into properties not directly amenable to, or difficult to access by, experiment, or to facilitate analysis of data that are incomplete or ambiguous. Thus, while not strictly an experimental technique, a brief discussion in the present chapter is appropriate. Among the most remarkable advances registered within the past two decades, the advent of methods based on electron Density Functional Theory (DFT) coupled to Newtonian MD constitutes a real milestone. The basics of electron-structure calculations by means of DFT were laid down in the mid 1960s (Hohenberg & Kohn, 1964; Kohn & Sham, 1965), establishing the existence of a unique functional of the electron density that defines the total wave function of the ground state, which attains a minimum energy when the charge density reaches a correct value on variational grounds. Its practical implementation to study complex solid structures had to wait, however, until several difficulties were circumvented. In particular, the development of Local Density Approximations (LDA) able to model the exchange and correlation energies, as well as the employment of flexible basis sets for expanding the atomic orbitals, such as Linearized Augmented Plane Waves, now make it possible to simulate systems of a few hundred particles over times of a few

64

Experimental techniques

tens of picoseconds. The use of pseudopotentials provides a further simplification, taking only the valence and conduction electrons into account, reducing the number of wave functions in the basis set. Ab initio MD (AIMD) techniques, in which the electronic distribution is recalculated at each motional step of the atomic positions, are now commonly used to investigate the structure of matter devoid of long-range order, such as liquids and glasses (Car & Parrinello, 1985). Their use is restricted for the time being, however, to the study of structural details and the high-frequency dynamics (on the time scales of a few picoseconds) because of the increased computational burden. Nevertheless, computer simulations of this type have had a significant impact in this field. In fact, reconstruction of a threedimensional structure from one-dimensional projections encompassed in the static structure factor is hampered by the strong directionality of the bonding in many compounds of interest. Computer simulations are also essential for elucidating the structure of multi-component systems for which the determination of the partial structure factors Sij(Q) is experimentally impossible. Furthermore, AIMD simulations provide a powerful complement to inelastic scattering experiments in probing the dynamics of liquids, for example the liquid alkali metals (Scopigno et al., 2005). To carry out an AIMD simulation, an initial configuration of the atoms in the system is set up in which the average density is fixed by the volume of the simulation cell chosen and the numbers and types of atom. The system is extended to infinity in the three spatial dimensions by means of periodic boundary conditions. The atoms are initially placed in userdefined starting positions {RI, I ¼ 1, . . . N} within the cell and initial values taken for the valence electron wave functions {ci(r), i ¼ 1, . . . n}. The charge density
ðrÞ ¼ 2

n=2 X

c i ðrÞci ðrÞ

ð4:76Þ

i 1

(assuming a non-spin-polarized system) is then calculated, and hence the Kohn Sham energy

EKS

ð ð n=2 1X 1 1 2 ¼ c ðrÞr ci ðrÞdr þ
ðrÞVH ðrÞdr þ
ðrÞVex ðrÞdr 2i 1 i 2 2

ð4:77Þ

þ Exc ½
ðrÞ ; where VH(r) is the Hartree potential representing the Coulomb interactions of this set of particles:

4.8 Numerical simulation

r2 VH ðrÞ ¼

4p
ðrÞ:

65

ð4:78Þ

Vex(r) represents the interaction with the atom cores (nuclei þ non-valence electrons in the pseudopotential scheme) and Exc is the exchange-correlation energy that in the Kohn Sham scheme is also a functional of the charge density. The exact form of Exc is obtained from the LDA or equivalent approximations. A new set of wave functions is then calculated that minimizes the value of EKS, and the process is reiterated until a self-consistent set of {ci(r)} and r(r) is obtained. The forces on each atom FI ¼

dEKS drI

ð4:79Þ

are then calculated, and Newton’s equations solved over a certain time step Dt to give a new atomic configuration {RI}. When equilibrium is established and the total energy no longer varies, further atomic trajectories are used to calculate the statistical properties of the system such as the time-averaged partial pair correlation functions gab(r) and their time-dependent equivalents Gab(r, t). From these the quantities measured in scattering experiments, such as the partial structure factors Sab(Q) and partial scattering functions Sab(Q, o), can be obtained by the appropriate Fourier transformations.

5 Levitation in materials research

5.1 Advantages of levitation methods Investigations of contained materials at high temperature have to contend with two distinct problems: (a) interactions between the sample and container and the resulting possibility of contamination, and (b) the effects of the container walls on the measurements, causing, for example, additional background and beam attenuation in scattering experiments. Both types of problem are removed by the use of levitation. Figure 5.1 shows the melting points and sound velocities (in the solid phase) of some representative materials. Levitation methods (with laser or other kinds of heating) are generally advantageous for measurements at temperatures above the line marked ‘levitation’. This is of course especially true for liquids, but levitation may also be beneficial for studying solid phases at high temperature. For corrosive materials, including many oxide melts, levitation methods may also be advisable at lower temperatures. The regions marked ‘n’ and ‘x’ in Fig. 5.1 indicate roughly the regions where inelastic neutron and X-ray scattering, respectively, will be advantageous for studying collective excitations, taking into account the higher energy resolution of neutron spectrometers as opposed to the greater ability of X-ray experiments to sample larger regions of Q and E, given the requirement that the velocity of the probe must be considerably larger than that of the excitation being studied (Price & Sko¨ld, 1986). In the following three chapters we will give some examples of such investigations in different kinds of high-temperature solids and liquids. The results for a given class of material are collected in one section, presented in roughly the same order as that of Chapter 4, namely, electromagnetic, optical and thermophysical properties, followed by microscopic structural and dynamic information provided by scattering and resonance techniques and numerical 66

5.2 Cooling and metastable states

67

Fig. 5.1. Melting points and sound velocities (in the solid phase) of some representative materials. The horizontal line connected to the arrow marked ‘levitation’ indicates an approximate temperature above which levitation methods are advantageous. The vertical line connected to the arrows marked ‘n’ and ‘x’ indicates approximately the sound velocities for which neutron and X-ray inelastic scattering techniques, respectively, are advantageous.

simulation. When appropriate, these will be followed by descriptions of the phenomena observed in cooling the levitated liquids, including nucleation, crystallization, formation of metastable and glassy solid phases, and coexisting liquid phases. To place these in a general context, we give below a brief discussion of the various types of behaviour that may be observed in cooling experiments. 5.2 Cooling and metastable states The absence of container walls has the effect of greatly reducing heterogeneous nucleation and thus making it possible to achieve deeply supercooled liquid and other metastable states on cooling. A typical cooling curve is that obtained by Rulison & Rhim (1994) in their study of the thermophysical properties of supercooled Zr using ESL, shown in Fig. 5.2. The variation of the temperature measured with a pyrometer is shown as function of time. After the instant that the heater is blocked off (1), the sample cools to and past the equilibrium melting point Tm (2) and remains in a supercooled liquid state until solidification starts (3). Solidification progresses rapidly and the latent heat of melting raises the temperature back to Tm (4), a phenomena known as recalescence that can be observed by a flash of emitted

68

Levitation in materials research 2300 1 2200

Zr #1 4

2

5

Tm

2100 T (K)

6 2000 1900 1800 3 1700 0.0

0.5

1.0

1.5

2.0

2.5

3.0

t (s )

Fig. 5.2. A typical temperature vs. time curve for a Zr sphere undergoing radiative cooling. At point 1, a shutter positioned in front of the heating lamp was closed to block the beam completely. The section of the curve before point 1 is unusable because light from the lamp reflected from the sample surface into the pyrometer. The section after point 6 was obtained by blocking the pyrometer and is used for calibration purposes. The significance of points 2–5 is discussed in the text (Rulison & Rhim, 1994).

light. This temperature is approximately maintained until the temperature begins again to fall (5). At point (6) the pyrometer is blocked off for calibration purposes. The state reached just before point (3) is known as hypercooling, in which the energy gained by the sample on cooling is greater than the latent heat of fusion, so that the significant time interval between (4) and (5) is needed to lose the excess heat by radiation. Reduced hypercooling results in a shortening of this time, and it approaches zero in the hypercooling limit when the energy gained by the sample on cooling is exactly equal to the latent heat of fusion:
Hf
Thyp ¼ ; Cp

ð5:1Þ

where DHf is the enthalpy of fusion and hCpi the average specific heat. The large degrees of undercooling achievable under containerless conditions can make high-temperature metastable phases accessible. In such cases additional recalescences may be observed. Figure 5.3 shows the cooling curve observed in drop tube experiments on Ta by Cortella et al. (1993). The first temperature plateau at 2930  10 K, 340 K below the equilibrium melting

5.2 Cooling and metastable states

69

Fig. 5.3. Brightness trace during solidification of a tantalum droplet falling though a drop tube. The pyometric measurements were performed with radiation from the X point of Ta (800 nm) and the melting temperature of bcc Ta was taken as a reference. The fall in signal between the two recalescences is due to increasing distance between the measuring diode and the droplet (Cortella et al., 1993).

point of the bcc phase, indicates the crystallization of a metastable phase, assigned to an A15 structure as discussed in Section 6.1. In the case of glass-forming liquids, the crystallization and recalescence step may be avoided, and the liquid cools through a liquid glass transition with no discontinuity in the physical properties. The liquid glass transition has been characterized as an ‘apparent, diffuse, second-order transition’ (Zallen, 1983). Physical properties such as entropy and volume do not exhibit discontinuities at the transition, although their temperature derivatives, such as specific heat, may show abrupt changes. In cooling from the liquid state above the equilibrium melting point Tm through the supercooled liquid state to the liquid glass transition, changes in dynamic properties are more striking than changes in the structure: the shear viscosity and the associated relaxation time can change by 15 orders of magnitude over this temperature range. The liquid glass transition temperature Tg is conventionally taken as the point where the shear viscosity reaches 1012 Pa.s, corresponding to a relaxation time around 100 s, a time long compared with most laboratory measurements (Angell, 1991). Cooling in a levitated environment may enhance glass formation, for example extending the composition range in a binary system where glasses

70

Levitation in materials research

Supercooled liquid

Tm (1878 K)

1900

1467 K

1500

Tg (1173 K)

1100

Glass

Temperature (K)

Liquid

2300

700 0

1

2

3

4

5

Time (s)

Fig. 5.4. Time dependence of the temperature during cooling of a levitated liquid (CaO)0.5(Al2O3)0.5 drop from 2173 K down to room temperature. The melting Tm and glass transition Tg temperatures are shown by solid lines. The significance of the line at 1467 K is discussed in Section 8.5 (Hennet et al., 2007a).

can be formed beyond what is possible with contained samples. An example of such behaviour is liquid (CaO)0.5(Al2O3)0.5, for which a cooling curve measured with a levitated sample (Hennet et al., 2007a) is shown in Fig. 5.4. In addition to the clearly apparent change of slope at 1467 K, discussed in Section 8.4, subtler changes can be observed at Tm and Tg. More complex temperature dependences can be found in tetrahedrally coordinated systems in which high- and low-density liquids may coexist in certain regions of temperature and pressure. The two liquid phases may undergo distinct transitions into high- and low-density glasses on cooling. Possible examples of such behaviour are discussed in connection with liquid silicon in Section 7.1 and with molten (Y2O3)x(Al2O3)1 x in Section 8.3.

6 Liquid metals and alloys

Since all the levitation techniques described in Chapter 2 work with conducting materials, it is not surprising that they form the subject of the majority of investigations carried out to date. In this chapter we discuss some representative systems, chosen either because several of their properties have been investigated, especially by levitation techniques, or because they exhibit especially interesting phenomena. 6.1 Early transition metals It is convenient to divide the results obtained on levitated samples of transition metals into early (Groups IVB VIIB) and late (Group VIIIB) transition metals. The earlier group contains the refractory metals, conventionally taken as Nb, Ta, Mo, W and Re, although relatively few measurements have been made in these even with levitation techniques, presumably because of their very high melting points. A notable exception are the early optical property measurements made on Nb by Krishnan et al. (1991a), using EML in conjunction with laser heating. Values obtained for the high-temperature solid and liquid are shown in Figs. 6.1 and 6.2. It can be seen that the values exhibit rather small discontinuities at the melting point but their temperature dependences change considerably, in the case of n and k going from small negative to large positive temperature coefficients. According to Eq. (4.26), the temperature dependence of the emissivity required a 16% correction to the previous value for the specific heat of liquid Nb at the melting point, with a revised value of 35  1 J mol 1 at Tm. In the same work Krishnan et al. also made measurements on liquid Ti. The value for the emissivity was similar but those for n and k somewhat lower. The temperature dependence of the emissivity was close to zero in liquid Ti, indicating that it was approaching free-electron behaviour at 633 nm. 71

72

Liquid metals and alloys

Normal emissvity at 633 nm

0.5

0.4

0.3

Tm 0.2 1800

2000

2200

2400

2600

2800

3000

Temperature (K)

Fig. 6.1. Normal spectral emissivity el at a wavelength l ¼ 633 nm for solid ( ) and liquid (■) Nb, together with an earlier value from the literature (þ). The solid lines represent least-squares fits to the data (Krishnan et al., 1991a).

▴

7

Refractive indices at 633 nm

6

k

5

4 n 3

2 Tm 1 1800

2000

2200

2400

2600

2800

3000

Temperature (K)

▴

Fig. 6.2. Refractive index n and extinction coefficient k at a wavelength of 633 nm for solid ( ) and liquid (■) Nb. The solid lines represent leastsquares fits to the data (Krishnan et al., 1991a).

6.1 Early transition metals

73

Agreement with literature values was less good than with Nb, which the authors ascribed to oxide and nitride contamination in the earlier work. Later, spectral emissivity measurements were made on liquid Zr by Krishnan & Nordine (1996), using EML and RF heating. They found a very small temperature dependence as in Ti but a slightly higher absolute value, around 0.35. Rulison & Rhim (1994), following the procedure described in Section 4.2 and using a literature value for Hf, obtained a value for eTl, the total hemispherical emissivity of undercooled liquid Zr, of 0.28  0.01. The physical properties of many liquid transition metals have been measured at the TSC in Japan by ESL and at the DLR in Germany by EML. A recent compilation of the TSC results has been given by Paradis et al. (2005c). For the five refractory metals listed above together with Ti, Zr, Hf and V, and the densities range from 4.1 to 18.4 g.cm 3, although the spread is largely the result of atomic mass differences: the number densities r0 have a more compact range, from 0.039 A˚ 3 in Zr to 0.057 A˚ 3 in W. Temperature coefficients range from 0.4 to 1.0 (10 3 g.cm 3.K 1). Their values for the surface tension of these metals, excluding V but including the Group VIIIB metals Ru, Rh, Ir and Ni are shown in Fig. 6.3. Much of the spread, as with the viscosity results shown in Fig. 6.4, can again be ascribed to atomic mass differences. It can be seen that the temperature coefficients are again all negative, which must be the case for pure elements but not necessarily for alloys (Egry, 2005). The temperature coefficients of the viscosities in Fig. 6.4 are also negative. Perhaps because of the difficulties of levitating and melting samples at the highest temperatures, there has been rather little work on the microscopic properties of refractory metals, although Nb has recently been measured by the CEMHTI group (L. Hennet, private communication, 2008), and Jakse et al. (2004a) have made an MD simulation of equilibrium and supercooled liquid Ta. The wealth of data becomes richer when we pass to the intermediatemelting-temperature Group IVB transition metals Ti and Zr. The structure factors of both metals in the equilibrium and supercooled liquid states have been measured with levitation techniques, in addition to earlier measurements on contained samples. Results from an X-ray structural measurement on equilibrium and supercooled liquid Ti with ESL by Lee et al. (2004) are shown in Fig. 6.5, along with the earlier experimental data in the literature and results derived from a theory based on a central interatomic pair potential derived from an embedded atom model. The general form of S(Q) is typical of a liquid with hard-core interactions, similar to the schematic illustration in Fig. 4.10. The maximum of the first peak is close to the value of 7.725/r1 expected if g(r) is dominated

74

Liquid metals and alloys

Fig. 6.3. Surface tension of several normal and supercooled liquid transition metals versus temperature (Paradis et al., 2005c).

Fig. 6.4. Viscosity of several normal and supercooled liquid transition metals versus temperature (Paradis et al., 2005c).

75

6.1 Early transition metals Tl – 282K

Tl + 33K 2 S(Q)

4

5

6

7

8

Tl – 282K 1 Tl + 33K

0 1

2

3

4

5

6

7

8

–1

Q (Å )

Fig. 6.5. Experimental X-ray structure factors for equilibrium and supercooled liquid Ti (connected open circles); Tl ¼ 1940 K. The two curves are displaced vertically for clarity. The dashed line is derived from a theory based on an interatomic pair potential. X-ray data from earlier work are shown by the dotted line. The inset shows a fit of an icosahedral model discussed in the text (solid line) to the experimental S(Q) in the region around the second peak (Lee et al., 2004).

by a sharp nearest-neighbour peak at r1 ¼ 2.87 A˚. The peak positions in the ESL study are significantly different from those found in the earlier experimental study but in better agreement with the theoretical results. The relative intensity and shapes of the peaks do not change significantly with the supercooling. The inset shows a fit to an icosahedral model that we will discuss later. Neutron diffraction measurements on equilibrium and supercooled liquid Zr with EML by Schenk et al. (2002) are shown in Fig. 6.6. Again, the general form of S(Q) is typical of a liquid with hard-core interactions. The maximum of the first peak is close to the value of 7.725/r1 with r1 ¼ 3.12 A˚. Figure 6.6 reveals one advantage of the neutron technique: the ease of extending the measurements to high Q, owing to the constancy of the neutron scattering length b compared with the fall-off of the X-ray form factor f(Q) (with the lost intensity going into the Compton scattering which then has to be subtracted as an unwanted background). The availability of high-Q data helps in two ways: it makes easier to normalize the measured S(Q), which must go asymptotically to unity as Q ! 1, and it leads to improved spatial resolution in the g(r) obtained by the Fourier transformation. Analysis of the g(r) thus obtained led to values at Tl of the nearest-neighbour distance r1 ¼ 3.12  0.01 A˚,

76

Liquid metals and alloys

S (Q )

4 3

T = 1830 K T = 1890 K T = 2135 K T = 2290 K

2 1 0 2

3

4

5

6

7

8

9

10

11

12

Q [ Å–1 ]

Fig. 6.6. Experimental neutron structure factors for equilibrium and supercooled liquid Zr; Tl ¼ 2125 K (Schenk et al., 2002).

the second-neighbour distance r2 ¼ 5.73  0.03 A˚, and nearest neighbour coordination number C1 ¼ 12.0  0.3. The value of r1 remained essentially unchanged as the temperature was decreased into the supercooled state, while that of r2 decreased slightly and that of C1 increased slightly. The values of 12 obtained for C1 are characteristic of several types of densely packed short-range order (SRO): icosahedral, fcc and hcp. Over 50 years ago, Frank (1952) postulated that icosahedral short-range order (ISRO) in the melt might explain the high degree of supercooling possible in pure liquid metals, since substantial rearrangements would be needed to transform into crystals that had close-packed structures. Support for this conjecture came 30 years later from theoretical work of Sachdev & Nelson (1984) based on a Landau description and MD simulations on systems interacting via a Lennard Jones potential, a relatively hard central potential (Steinhardt et al., 1983). Interest in the idea was further stimulated by the experimental observation of quasicrystals in late 1984 (Shechtman et al., 1984). An experimental signature of ISRO is provided by a feature on the right side of the second peak in S(Q), both in the theoretical work of Sachdev & Nelson and in an experimental measurement on vapour-deposited amorphous cobalt (Leung & Wright, 1974), shown in Fig. 6.7. This has stimulated workers in this field to look for such features in diffraction experiments on supercooled liquids. The region in question for the liquid Ti measurement of Lee et al. is shown in the inset to Fig. 6.5. The expected position of the feature just mentioned is highlighted by the vertical arrow, and the solid curve represents the fit of a 13-atom perfect icosahedron to the data in this region. If the position of the second peak in a liquid governed by central forces is taken as 5.0 A˚, based on an embedded atom calculation in the literature, the feature deriving from ISRO at 5.1 A˚ 1 can be regarded as a shoulder on the

77

6.1 Early transition metals 4.0

THEORY

S (Q )

EXPERIMENT

2.0

0 0

7.0

14.0

21.0

28.0

35.0

Q /k

Fig. 6.7. Theoretical structure factor derived from a Landau description of icosahedral short-range order compared with experimental results for amorphous Co (Leung & Wright, 1974). The peak heights and widths are determined by parameters in the theory but the peak positions are fixed by the value of k (the inverse radius of the tangent sphere to the icosahedron) (Sachdev & Nelson, 1984).

high-Q side of this peak and is, in fact, slightly higher than the peak itself, in the equilibrium as well as the supercooled liquid. The fit of the perfect icosahedron model does not reproduce this behaviour although, as pointed out by the authors, introduction of distortions of the regular icosahedron can change the relative heights of the two features. A similar feature can be seen in the liquid Zr measurements of Schenk et al., already shown in Fig. 6.6. In this case it becomes more pronounced with supercooling, a behaviour consistent with Frank’s conjecture and also with the MD simulations mentioned above. Jakse & Pasturel (2003) have carried out ab initio MD simulations on equilibrium and supercooled liquid Zr. Their results for the pair correlation function g(r) are compared with those of Schenk et al. in Fig. 6.8 and are seen to be in excellent agreement; the MD peaks are slightly sharper, which could be at least partly the result of the broadening produced by the finite cut-off at high Q in the experimental data.

78

Liquid metals and alloys 7

6

5

4 g (r )

T = 2500 K 3 T = 2200 K 2 T = 2000 K 1 T = 1850 K 0 0

1

2

3

4

5 r (Å)

6

7

8

9

10

Fig. 6.8. Pair-correlation function for liquid Zr. The open circles correspond to experimental neutron diffraction data of Schenk et al. (2002) and the solid lines to MD simulations. The curves are shifted upwards by 1, 2, and 3 units for 2000, 2200, and 2500 K, respectively (Jakse & Pasturel, 2003).

Jakse and Pasturel explored the issue of ISRO by performing a commonneighbour analysis of their MD results. This method (Honeycutt & Andersen, 1987) characterizes the local environment of each pair in terms of the number and properties of their common nearest neighbours. A set of four indices ijkl is associated with each pair: the first index i denotes to what peak of g(r) the pair under consideration belongs; the second index j represents the number of nearest neighbours shared by the root pair; the third index k is the number of nearest-neighbour bonds among the shared neighbours; and a fourth index l is used to distinguish configurations with the same ink but a different topology. Some typical configurations for i ¼ 1 are shown in Fig. 6.9. To remove the complicating effects of thermal disorder, the analysis is performed on inherent structures in which the atoms are brought to local minima of the potential energy surface by applying a technique similar to the steepest descent minimization proposed by Stillinger & Weber (1982). The abundances of some of the pairs classified

79

6.1 Early transition metals Table 6.1. Some bonded pairs identified in an MD simulation of equilibrium and supercooled liquid Zr. Indices

Characteristic environment

Zr (2500 K)

Zr (2000 K)

1551 1541 1421 1422 1431 1321 1661 1441 2211 2101 2441 2331 2321

I, FK I0 F, H H I0 I B, FK B, FK F, H, B, FK F, H, B, FK F, H, B H, I, FK I0

0.32 0.16 0.01 0.01 0.11 0.01 0.16 0.10 0.96 0.89 0.31 0.90 0.04

0.27 0.10 0.00 0.01 0.14 0.01 0.23 0.14 1.03 0.72 0.41 0.82 0.15

Note: The indices in the first column refer to a common neighbour analysis of a given pair (see text, Honeycutt & Andersen, 1987). The notation in the second column indicates whether the pair is a dominant feature in local structures with fcc (F), hcp (H), bcc (B), icosahedral (I), distorted icosahedra (I0 ) and Frank Kaspar phase (FK) environments. The third and fourth columns show the relative abundances of each pair in equilibrium and supercooled liquid Zr, respectively; the errors in each value are estimated as 0.01 (after Jakse & Pasturel, 2003).

Fig. 6.9. Local configurations identified from MD simulations of liquid transition metals for pairs contributing to the first peak in g(r). The indices 1jkl refer to a common-neighbour analysis of a given pair (Jakse et al., 2004a).

in this way are given in Table 6.1 for the simulated equilibrium and supercooled liquids, along with a notation indicating whether the pair is a dominant feature in various local structures.

80

Liquid metals and alloys

Several conclusions can immediately be drawn from Table 6.1. First, the absence of certain indices in the liquid structure such as 1421 and 1321 means that it cannot be made up entirely of simple close-packed arrangements like fcc, hcp and icosahedral arrangements, while the presence of indices like 1551 and 2331 show that bcc is not predominant either. We therefore have to include more complicated structures such as distorted icosahedra and higher coordinated polyhedra characteristic of Frank Kaspar phases (Frank & Kaspar 1958; 1959), designated by I0 and FK respectively. On the other hand, the presence of indices such as 2441 shows that bcc-type local environments are significant also, and that their role becomes more pronounced on supercooling, at the expense of the Frank Kaspar environments. This is not entirely unexpected, since the hightemperature crystalline phase of Zr has bcc symmetry. On the other hand, the coordination number remains significantly higher than the bcc value of 8, and in fact increases slightly on supercooling in both the MD and experimental results. Overall, the analysis of the simulation shows that, while there is pronounced polyhedral short-range order in the liquid structure as the experimental results suggest, its exact nature is more complicated than the one originally proposed on geometric grounds and discussed in the experimental papers. Jakse et al. (2004a) derived similar conclusions in their AIMD studies of equilibrium and supercooled liquid Ta for which, as pointed out above, no diffraction studies have yet been made. In this case they relate the FK-type short-range order to the A15 crystal structure, which is the FK phase with the lowest proportion of icosahedral sites and highest proportion of polytetrahedral ones. The dynamics of levitated liquid Ti have been studied by inelastic X-ray scattering by Said et al. (2006). The measurements were carried out at the APS with the spectrometer shown earlier in Fig. 4.13. The X-ray energy of 21.7 keV corresponded to the (18,6,0) reflection for Si, and the energy resolution was determined from a Plexiglas® sample to be 2.2 meV full-width at half-maximum. The levitation apparatus was enclosed in a bell jar designed for back-scattering geometry with a separation of 10 cm between sample and detector. The temperature was measured by a pyrometer directed at the upper part of the sample, using an emissivity value of 0.33. IXS measurements were made at 2020 K, just above the melting point. The IXS data, shown in Fig. 6.10 for several Q values, show well-defined sound excitations propagating up to a Q value of 16 nm 1, more than halfway to Qp, the position of the first peak of S(Q) at 26 nm 1. After subtracting the background due to the atmosphere of the Ar levitating gas, the corrected data were fitted by a

81

6.1 Early transition metals –10

0

10

20

0.3

30

40

50

–10

0

10

20

60

0.0 1.8

Q = 11.80 nm–1

Q = 21.70 nm–1

0.9 0 2.4

0 0.3 I (counts/s)

50

Q = 30.00 nm–1

6.0

Q = 10.15 nm–1

Q = 20.11 nm–1

1.2 0 1.8

0 0.3 Q = 8.49 nm–1

Q = 18.45 nm–1

0.9 0 0.6

0 0.3 Q = 6.80 nm–1

Q = 16.79 nm–1

0.3 0

0 0.3

0

20 E (meV)

40

Q = 15.13 nm–1

0.3

Q = 5.17 nm–1 0

40

12.0

Q = 13.47 nm–1

0 0.3

30

60

0 –10

0

10

20

30

40

50

60

E (meV)

Fig. 6.10. The IXS spectra for liquid Ti at T ¼ 2023 K for different Q values indicated (open circles). Note that 10 nm 1 ¼ 1 A˚ 1, the unit used elsewhere for Q. The solid lines are the fits of the resolution-broadened generalized hydrodynamic description, Eq. (6.1), together with the levitating gas background.The strong quasielastic intensity at low Q is due to the levitating gas (Said et al., 2006).

function used previously in connection with a generalized hydrodynamic model (de Schepper et al., 1983): " #  s þ bðo þ os Þ s  bðo  os Þ ho 1 SðQ; oÞ ¼ SðQÞe2kT þ As þ As ; ð6:1Þ A0 2 p o þ 2 ðo þ os Þ2 þ 2s ðo  os Þ2 þ 2s where G and Gs are the widths of the Rayleigh and Brillouin peaks and os is the frequency of the Brillouin peak; the fit is shown by the solid line in Fig. 6.10. Figure 6.11 shows the dispersion relation forpthe sound mode frequencies renormalized by the damping term, s ¼ o2s þ 2s (Glyde, 1994), as a function of Q. The linear dependence of the dispersion relation at low Q corresponds to a high-frequency longitudinal sound velocity c1 ¼ 4520  50 m s 1, about 2.5% higher than the reported adiabatic sound velocity shown by the dashed-dotted line in Fig. 6.11. This figure also shows the half-widths Gs of the sound excitation peaks derived from the fits.

82

Liquid metals and alloys 40

Ω Γs

Ω,Γs (meV)

30

20

10

0 0

5

10

15

20

25

30

Q (nm–1)

Fig. 6.11. Frequencies (closed symbols) and half-widths (open symbols) of the sound excitations in liquid Ti, derived from a fit of Eq. (6.1). The solid lines and dotted lines represent MCT calculations using two different experimental structure factors in the literature. The straight dashed-dotted line corresponds to the reported adiabatic sound velocity (Said et al., 2006).

The nanometre distance scale presents a challenge for the dynamical theory of liquids, being intermediate between the regions where hydrodynamics and kinetic theory may be expected to be valid (Egelstaff, 1992; Price et al., 2003; Scopigno et al., 2005). A relatively fundamental approach to a microscopic description of liquid dynamics, as well as the approach to the glass transition, is mode-coupling theory (MCT). In its original version (Go¨tze & Lu¨cke, 1975) the memory functions for the current fluctuation spectra were related to coupled modes of density and current correlations. Subsequently, a simpler version was found to successfully describe the excitation spectrum of liquid helium (Go¨tze & Lu¨cke, 1976) and the relaxation processes in the vicinity of the glass transition (Bengtzelius et al., 1984). The scattering function is written as   1 SðQÞ o20 ; ð6:2Þ Re io þ SðQ; oÞ ¼ io þ MðQ; oÞ p where o20 ¼ kB TQ2 =mSðQÞ

ð6:3Þ

is the second frequency moment of S(Q, o)/S(Q), m being the atomic mass. In Eq. (6.2) M(Q, o) is the Fourier Laplace transform of the second-order

83

6.1 Early transition metals

memory function M(Q, t), which is related to the intermediate scattering function I(Q, t) the Fourier transform of S(Q, o) by the generalized Langevin equation: ::

ðt

I ðQ; tÞ þ dt0 MðQ; t

:

t0 Þ IðQ; tÞ þ 20 ðQÞIðQ; tÞ ¼ 0:

ð6:4Þ

0

In this version of MCT the memory function is approximated as a linear combination of products of intermediate scattering functions: MðQ; tÞ ¼ 20 ðQÞ

1 X VðQ; Q1 ; Q 2V Q

Q1 ÞIðQ1 ; tÞIðQ

Q1 ; tÞ;

ð6:5Þ

1

where V is the volume of the system and the coupling coefficient VðQ; Q1 ; Q2 Þ depends only on the structure factor S(Q) and the number density r0. Equations (6.3) and (6.5) represent a closed set of equations that can :be solved numerically, subject to the initial conditions I(Q, 0) ¼ S(Q) and IðQ; 0Þ ¼ 0 (Schirmacher & Sinn, 2008). Said et al. use this theory to calculate I(Q, t) and hence S(Q, o) for liquid Ti at 2020 K and r0 ¼ 0.0514 atoms A˚ 3. They used the two measurements of S(Q) those of Lee et al. (2004) and Waseda (1980). For the MCT calculations it was necessary to extrapolate to low Q, using the limit S(0) ¼ 0.0278 obtained from Eq. (4.32) using the value of r0 just cited and the compressibility w from Tamaki & Waseda (1976). The results of the MCT calculations using the data of Lee et al. and Waseda are shown by solid and dotted lines, respectively, in Fig. 6.11. The prominent difference in the amplitudes of the two calculated dispersion curves in Fig. 6.11 originates from small differences in the two measurements of S(Q) at Q values below Qp, the position of the first peak in S(Q). Considering the large spread in the two S(Q) data sets used as input, the agreement of the MCT calculation with the IXS data for both the frequencies and widths of the Brillouin peaks is quite impressive. In the hydrodynamic region M(Q, t) reflects two physically distinct processes: density fluctuations that give rise to sound waves that correspond to the Brillouin side peaks in S(Q, o), and entropy fluctuations that give rise to non-propagating excitations that correspond to the Rayleigh central peak. The ratio of the central peak intensity to the total intensity S(Q) is given by (g 1)/g, where g ¼ Cp/Cv, the ratio of specific heats. In the Q region (nm 1) that we are dealing with here, it is found that the central peak intensity ratio is much higher than this. Furthermore even for insulating liquids like oxides, as we shall see later, the width of the central peak is several orders smaller than the hydrodynamic value DTQ2, where DT is

84

Liquid metals and alloys 8 4/3hs

hl (10–3 Pa s)

6

4

2

0

0

5

10

20

15

25

30

–1

Q (nm )

Fig. 6.12. The longitudinal viscosity of liquid Ti as a function of Q. The point marked 4/3s, where s is a value of the shear viscosity in the literature, represents a lower limit for the longitudinal viscosity at Q ¼ 0. The lines have the same notation as in Fig. 6.11 (Said et al., 2006).

the thermal diffusivity k/dCp (k ¼ thermal conductivity, d ¼ mass density). Consequently, it is usual to neglect the entropy fluctuations and treat the central peak observed as a consequence of high damping associated with the density fluctuations (Mountain, 1966). Support for this approach comes from MD simulations, for example those of liquid Ni (Ruiz-Martin et al., 2007), that show that both DT and (g 1) have a strong Q dependence and fall to values close to zero at Q values a little below Qp. In this case it is possible to extract a generalized longitudinal viscosity from the central peak as l ðQÞ ¼ pr0 kT

SðQ; 0Þ SðQÞ2

:

ð6:6Þ

Values of l obtained in this way are plotted in Fig. 6.12, and the width G0 of the central peak in Fig. 6.13. The curves again represent the MCT results with the two data sets of S(Q), based on the same neglect of entropy fluctuations. The longitudinal viscosity at Q ¼ 0 is related to the shear viscosity s and bulk viscosity b by l ¼ 43 s þ b . The value of 4s/3, where s is an experimental value from the literature, is a lower limit for l at Q ¼ 0 and is also marked in the figure. The MCT result is seen to be in satisfactory agreement with all the quantities derived from the IXS measurement, especially if the S(Q) data of

85

6.1 Early transition metals 15

Γ0 (meV)

10

5

0

0

5

10

15

20

25

30

Q (nm–1)

Fig. 6.13. The half-width of the central peak in liquid Ti as a function of Q. The lines have the same notation as in Fig. 6.11 (Said et al., 2006).

Waseda are used. Thus, the simplified MCT theory, incorporating decay of the excitations into two longitudinal modes and neglecting the coupling to transverse modes, appears capable of describing the dynamics over the Q range investigated. This implies that the cage effect that produces the structural arrest in the supercooled liquid at the glass transition also appears to dominate in the less viscous, equilibrium liquid regime. Several experiments on the cooling of transition metals and alloys were carried out in ultra-high-vacuum drop tubes in the 1990s. Vinet et al. (1991) reported undercooling of 900 K and 550 K, respectively, in W and Rh millimetre-size droplets. Cortella et al. (1993) studied solidification of deeply undercooled droplets of Ta and Rh. The nucleation temperature Tn can be estimated in two ways: by numerical integration of the cooling law, taking account of the expected temperature dependence of the material parameters, and by assuming that the droplet reaches its melting temperature at the end of recalescence. The latter procedure is valid as long as the degree of undercooling is less than the hypercooling limit defined in Eq. (5.1). Some samples showed a single recalescence, with values of Tn around 625 660 K and 860 900 K below the equilibrium melting temperatures for Ta and Rh, respectively. Others showed a double recalescence, like that shown earlier for Ta in Fig. 5.3. As discussed in Section 5.2, the first temperature plateau at Tmmet ¼ 2930  10 K indicates the equilibrium of a metastable phase.

86

Liquid metals and alloys

Although it was not possible to characterize this phase, a first-principles electronic structure calculation indicated that the A15 structure was a likely candidate, giving a value for Tmmet close to that observed, on the assumption that the entropies of melting of the bcc and A15 phases were equal. Berne et al. (1999) made first-principles electronic structure calculations for refractory metals across the 4d and 5d transition series, considering three tetrahedrally close packed (tcp) phases with characteristic polyhedra in addition to bcc and hcp. For the earlier elements in the two series Nb, Mo and Ta, W the order of stability is bcc, tcp (A15), tcp (sigma), tcp (chi) and hcp. Among the tcp phases, the A15 has the smallest number of icosahedral Z12 sites (i.e. the atom at the centre of the polyhedron is 12-fold coordinated), the rest being Z14. Thus, it is the high coordination effects that stabilize the A15 phase rather than the icosahedral units. These considerations support the conjecture that the metastable phase observed in Ta by Cortella et al. (1993) has the A15 structure. In drop tube experiments on RexW1 x and RexTa1 x systems, double recalescence was observed for x ¼ 0.18 0.35 in Re W and x ¼ 0.14 0.17 in Re Ta; in the latter case, two different kinds of morphologies were observed in the final droplets, suggesting two different nucleation paths with different transitory phases (Berne et al., 2000). A theoretical analysis on the lines described above suggested that the metastable phase was the A15 in the Re W system, while in the Re Ta system the sigma phase was competing with the A15, even though it was far from its known composition range. 6.2 Late transition metals There is a considerable body of work on liquid iron and nickel, on levitated as well as contained samples. This is due not only to the intrinsic interest in the liquid state of these ubiquitous metals but also to the fact that liquid Fe, alloyed with appreciable amounts of Ni and light elements such as sulphur (Nasch, 1996), is believed to be the main constituent of the earth’s outer core and thus responsible for terrestrial magnetism arising from the electrical currents associated with convective motions in the liquid. Cagran et al. (2007) report a study of electrical transport in liquid Cu, Ni and Cu0.53Ni0.47, comparing EML and pulse-heating (PH) techniques under the joint project between the DLR and GUT already mentioned. They report values for the density of liquid Ni at Tm ¼ 1728 K of 7.92 (EML) and 8.11 (PH) gm.cm 3, and for the electrical conductivity of 1.15.104 (EML) and 1.18.104 (PH) O 1.cm 1. The 2 3% differences in the measurements of both quantities by the different techniques, and the considerable difference

6.2 Late transition metals

87

Fig. 6.14. Real part of the dielectric constant e1 for liquid Ni at 1764 K as a function of photon energy (solid circles). Earlier results in the literature are shown by open circles, and data for the ferromagnetic solid at room temperature, also from the literature, by the dashed line. The error bars illustrate the estimated uncertainty at three photon energies (Krishnan et al., 1997).

observed in the temperature dependence of the conductivity, await further investigation. It is planned to extend these comparisons to refractory metals, for which PH measurements have already been reported (Hu¨pf et al., 2008a; 2008b). Krishnan et al. (1997) made a detailed study of the optical properties of liquid Fe and Ni as a function of wavelength in the energy range 1.2 3.5 eV, which includes the IR, visible and UV regions. Figures 6.14 6.16 show their results for liquid Ni, along with results of some earlier work. Their experiments were made with EML and were considered by the authors to be more reliable than earlier measurements on samples contained in alumina crucibles and those obtained with rapid pulse heating, both of which methods are susceptible to oxygen contamination. The data for the real part of the dielectric constant, e1, and the optical conductivity s(n) resemble qualitatively those for the room-temperature solid, with a minimum in the conductivity at about 2.2 eV, compared with 3 eV in the solid. The normal spectral emissivity shows a monotonic decrease with wavelength and a slowly decreasing slope. The corresponding data for Fe, not shown here, are similar to those of liquid Ni but quite different from solid Fe. Krishnan et al. suggest that the ferromagnetic contributions to the optical properties of Ni may, unlike in Fe, lie outside the energy range of the

88

Liquid metals and alloys

Fig. 6.15. Optical conductivity s for liquid Ni at 1764 K as a function of photon energy. The circles and line have the same notation as in Fig. 6.10 (Krishnan et al., 1997).

Fig. 6.16. Normal spectral emissivity el for liquid nickel at 1764 K as a function of wavelength l. The solid line is a multifunctional least-squares fit to the data. The open circles represent earlier results in the literature (Krishnan et al., 1997).

6.2 Late transition metals

89

measurements. Another reason may lie in the crystal structure, with iron having the less closely packed bcc structure at the temperature of the solid measurements. Krishnan & Nordine (1996) found a 4% increase in the emissivity of liquid Ni at a wavelength of 633 nm between the deeply undercooled liquid and Tm, where the value was 0.35, and a slight further increase at a temperature about 200 K above Tm. This contrasted with the almost temperature-independent behaviour observed in liquid Zr and Ti, discussed in the previous section. Rulison & Rhim (1994), following the procedure described in the previous section, obtained a value for the total hemispherical emissivity eTl of undercooled liquid Ni of 0.16  0.01. In the work cited in the last section, Paradis et al. (2005c) give values for the densities of liquid Ru, Rh, Ir, Ni and Pd. The number densities range from 0.060 A˚ 3 in Pd to 0.081 A˚ 3 in Ni, a higher range than found for the early transition metals. The value of d(Tm) for Ni agrees better with the EML value of Cagran et al. quoted above than their PH value, but the temperature coefficient is closer to the PH value. The surface tension and viscosity of the first four metals have already been shown in Figs. 6.1 6.2. Values for the surface tension of Ni obtained by Egry et al. (1995) with EML, applying the corrections discussed in Section 4.2, and by Millot et al. (2002b) with CNL, are in good agreement with those shown in Fig. 6.3. Wille et al. (2002) report values for the density and surface tension of liquid Fe using CNL. The diffraction studies discussed in the last section also included measurements in Fe and Ni. Lee et al. (2004) observed in liquid Ni a similar behaviour to that discussed above for liquid Ti, except that the feature ascribed to ISRO is less evident, but its temperature dependence more pronounced, in Ni compared with Ti. Schenk et al. (2002) found essentially the same behaviour in liquid Fe and Ni as in liquid Zr. Figure 6.17 shows their result for Ni in the region of the second, third and fourth peaks in S(Q), compared with model calculations for fcc, icosahedral and dodecahedral clusters. It is clear that the fcc cluster gives an extremely poor fit, while the icosahedral cluster gives a slightly worse fit than the dodecahedral one. Since the dodecahedral cluster is a polytetrahedral aggregate similar to the Frank Kaspar phases discussed above, these results are quite consistent with the MD simulations of Jakse & Pasturel (2005), for which a commonneighbour analysis gave results comparable with those of Zr described in the previous section. Similar diffraction results were found for liquid iron by Holland-Moritz et al. (2005) and liquid cobalt by Holland-Moritz et al. (2002a): in the latter case, energy-dispersive X-ray diffraction (i.e. scattering

90

Liquid metals and alloys 1.2 Ni

1.1

S(Q)

1.0 0.9 0.8 0.7 0.6

5

6

7

8

9

10

11

Q [Å–1]

Fig. 6.17. Measured structure factor S(Q) of an Ni melt at T ¼ 1435 K in the range 4.5–11.4 A˚ 1 (open circles), and simulated S(Q) assuming clusters with different symmetries prevailing in the melt: fcc clusters (dotted line), icosahedra (dashed line) and dodecahedra (solid line) (Schenk et al., 2002).

measurements at fixed angle with a wide incident energy spectrum) was used for the structural measurement on account of the relatively high neutron absorption cross section of Co. To the author’s knowledge there have been no measurements to date of the collective motions of late transition metals with levitated samples, but we will briefly discuss the INS measurements of Bermejo et al. (2000) on liquid nickel contained in alumina tubes. Figure 6.18 shows the excitation frequencies and damping coefficients (fwhm) derived by a fit of Eq. (6.1) to the S(Q, o) data. A comparison of the two quantities note that GQ corresponds to 2Gs in the notation of Eq. (6.1) shows that propagating excitations exist at Q values up to 1.5 A˚ 1 halfway to Qp. This result was somewhat surprising since it was thought that the existence of well-defined excitations out to these large Q values, as found in Rb and Pb, for example, was a consequence of the harmonic character of the force field. Liquid Ni, on the other hand, is generally considered a rather anharmonic system, with a specific heat Cp ¼ 4.6R compared with the harmonic value of 3R, and a specific heat ratio g ¼1.88, compared with, e.g., 1.2 in Rb. The survival of sound-like excitations out to relatively large Q can be explained by a consideration of the parameter gG quantifying the departure of the liquid from ideal behaviour (Lovesey, 1986):  r ]3 V=dr3  ; ð6:7Þ gG ¼ ]2 V=dr 2 r0

91

6.2 Late transition metals 40 (a)

ΩQ (meV)

30

20

10

0 (b)

40

ΓQ (meV)

30

20

10

0

0

1

2

3

4

Q (Å−1)

Fig. 6.18. Spectral parameters characterizing the collective excitations in liquid Ni. (a) Excitation frequencies; the experimental INS data are depicted by filled symbols, open symbols stand for simulation data in the literature and the line going through the data points is drawn as a guide to the eye; the straight line represents a hydrodynamic dispersion Ohyd ¼ cTQ with cT ¼19.82 meV.A˚ (b) Damping coefficient as given by the fwhm of the inelastic peak; the line shows a fit to a hydrodynamic damping Ghyd ¼ Dhyd Q2 with Dhyd ¼ 15 meV.A˚2 (Bermejo et al., 2000).

where V(r) is the pair potential and r0 the position of its minimum. This parameter has values of 1.45 1.8 for the heavier alkali metals and 2.1 for Ni, but a considerably greater value of 3.5 for Ar, for example, where the excitations do not extend much beyond the hydrodynamic region (van Well et al., 1985). In other words, it is the curvature of V(r) at the first minimum that drives the microscopic dynamics, as opposed to other regions of V(r) that are important for the thermodynamic properties. More recently, dynamical studies of liquid Ni have been extended to lower Q with IXS (Cazzato et al., 2008) and QENS (Ruiz-Martin et al., 2007; Meyer et al., 2008). In the latter case, the scattering is dominated by the incoherent scattering, yielding estimates for the diffusion constant and the vibrational density of states. The measurements of

92

Liquid metals and alloys 80 Ni

V [m/s]

60

40

20

0

ΔT ∗

0

100

300

200 ΔT [K]

Fig. 6.19. Growth velocity as a function of undercooling for primary solidification of liquid Ni; dots: measured values; solid line: prediction from a theory that takes account of the dendrite tip curvature in addition to dissipation of the enthalpy of crystallization into the undercooled melt ahead of the solidification front; dashed line: prediction including a term that describes atomic attachment kinetics at the interface. Below a critical undercooling DT* the data follow a power-law dependence, V  DT3, and above it a linear dependence (Willnecker et al., 1989).

Meyer et al. were carried out with EML over an extended temperature range from 214 K above to 212 K below the equilibrium melting point and showed an Arrhenius temperature dependence over this entire range, with an activation energy of 0.47  0.03 eV. Willnecker et al. (1989) measured the solidification velocity v in undercooled Ni and Cu0.7Ni0.3 alloy using EML in conjunction with the technique described in Section 3.3. Their results for pure Ni are shown by the points in Fig. 6.19. The solid line represents a theory that takes account of the dendrite tip curvature in addition to dissipation of the enthalpy of crystallization into the undercooled melt ahead of the solidification front (Lipton et al., 1987; Trivedi et al., 1987). The total undercooling is expressed by DT ¼ DTt þ DTr ¼ ðTi

T Þ þ ðTm

Ti Þ;

ð6:8Þ

where the thermal undercooling DTt is the difference between the temperature Ti of the dendrite tip and the temperature T of the melt far from the dendrite, and DTr is the depression of the melting point below the equilibrium value Tm due to the Gibbs Thomson effect arising from the curvature of the dendrite tip. This theory, which involves only known thermophysical properties and has no adjustable parameters, is seen to over estimate the growth velocity.

6.3 Zirconium nickel and Ti Zr Ni alloys

93

If an empirical kinetic term DTk ¼ v/m is added to the right-hand side of Eq. (6.8), where m is a coefficient describing the atomic attachment kinetics at the interface (dashed line), good agreement is found for undercooling less than a critical value DT*, where the dependence changes from power-law to linear. The authors do not provide a theory for the behaviour above DT*, but note that DT* also represents a point at which the solidified product changes from coarse- to fine-grained morphology. Herlach et al. (1991) investigated the solidification of Fe Ni alloys in samples heated in alumina crucibles lined with a glass flux. A two-step solidification was observed, with primary crystallization into a metastable bcc phase followed by equilibrium solidification into the fcc phase. 6.3 Zirconium–nickel and Ti–Zr–Ni alloys Of the considerable body of work on liquid metal alloys carried out by both levitation and conventional techniques, we highlight in the rest of this chapter three examples of particular interest. Zr Ni alloys can form glasses over a wide range of compositions by cooling from the melt and also by mixing of the two pure metals in a ball mill, properties that distinguish them from the general class of metal alloys in which the amorphous state is reached only by thin-film deposition or by rapid cooling techniques such as splat cooling and ribbon spinning. The reason for the glass-forming ability of these alloys is not completely clear, but is probably connected with the viscosity of the liquid alloy, the mobility of the atoms in the liquid phase playing an important role in the glass-forming process, as discussed by Ohsaka et al. (1998). Some of the results discussed below support this conjecture. The electrical conductivity of the eutectic composition Zr64Ni36 was measured by Egry et al. (1996). The electrical resistivity measured in microgravity is shown in Fig. 6.20: the small increase at the melting point is typical of transition metals and alloys. The value of the conductivity of the liquid just above the melting point, 6620  50 O 1.cm 1, is intermediate between the literature values of the end members, 6500 and 11 800 O 1.cm 1 for Zr and Ni, respectively, just above their own melting points. The spectral emissivity el of liquid Zr75Ni25 measured by Krishnan & Nordine (1996) over a range of l, shown in Fig. 6.21, exhibits a gradual increase with decreasing wavelength without the sharp rise below 600 nm exhibited by liquid Ni. The values are somewhat higher than those for pure Zr measured in the same work. On the other hand, Wunderlich et al. (1997) found a value of 0.32 for the total hemispherical emissivity eT of Zr64Ni36 just

94

Liquid metals and alloys 170

Resistivity (mΩcm)

160 150 140 130 120 110 100 90 80 600

800

1000

1200

1400

1600

Temperature (K)

Fig. 6.20. The electrical resistivity of Zr64Ni36 measured in microgravity (Egry et al., 1996).

Fig. 6.21. Wavelength dependence of the spectral emissivity for liquid Zr75Ni25 at 1300 K, Ni at 1800 K and Zr at 2350 K (Krishnan & Nordine, 1996).

above the melting point, similar to that for pure Zr but higher than that for pure Ni. They also obtained a value for Cp, 45.5  1 J.mol 1K 1, at the same temperature. The specific volumes (1/d) and viscosities of liquid Ni Zr alloys were measured by Ohsaka et al. in the work just cited, using ESL and heating by a high-intensity xenon arc lamp. Their results are shown in Figs. 6.22 and 6.23, respectively. In the first figure the specific volume of the NiZr2 alloy

95

6.3 Zirconium nickel and Ti Zr Ni alloys 0.18

Specific Volume (cm3/g)

1393 K

0.16

Zr

Ni36Zr64

0.14

Ni24Zr76

NiZr

0.12

NiZr2

Ni

0.10 0.0

0.2

0.4

0.6

0.8

1.0

Atomic Fraction of Zr

Fig. 6.22. Specific volumes of liquid Ni–Zr alloys at 1393 K (Ohsaka et al., 1998).

Fig. 6.23. Viscosities of liquid Ni–Zr alloys as a function of temperature. Arrhenius fits are shown by solid lines; the equilibrium melting point of each alloy is indicated by the arrow (Ohsaka et al., 1998).

is seen to be significantly smaller than that of the ideal mixture, while in the second figure the viscosities of both the NiZr and NiZr2 alloys are seen to be much higher than for the other compositions. At 1500 K they are also higher than the values that the pure elements would take if supercooled to this temperature, as seen from Fig. 6.4. These results show that associated species

96

Liquid metals and alloys Surface Tension of Zr(64)-Ni(36) 1.9 1.8

γ [N/m] = 1.54 + 1.07 ∗ 10–4 (T[⬚C]–1010) Teutec.

γ [N/m]

1.7 1.6 1.5

1g-results (corrected)

μg-results (un-corrected)

1.4 1.3 1.2 900

1000

1100

1200

1300

1400

1500

1600

1700

Temperature [⬚C]

Fig. 6.24. The surface tension of Zr64Ni36. Microgravity data points (triangles) are compared with terrestrial data (diamonds) corrected according to the procedure discussed in Section 4.2 (Egry et al., 1996).

in the form of NiZr and NiZr2 exist in the liquid alloys at temperatures well above the melting point. The reduced mobility of these species compared with single atoms can be expected to reduce the solid phase growth rate and give the glass the chance to form. The surface tension of liquid Zr64Ni36 measured by Egry et al. in the work cited above is shown in Fig. 6.24. The terrestrial data agree well with the microgravity results when corrected with the procedure of Cummings & Blackburn (1991) discussed in Section 4.2. The value at 1650 K, the highest temperature measured, is intermediate between those for pure Zr (1.52 at 1800 K) and pure Ni (1.78 at 1550 K), shown earlier in Fig. 6.3. Turning now to microscopic measurements, Hennet et al. (2006) have briefly reported neutron diffraction measurements on the liquid with equimolar composition ZrNi, made on samples levitated by CNL at the D4C diffractometer at ILL with 0.7 A˚-wavelength neutrons. Figure 6.25 shows the total structure factor obtained in the liquid close to the melting point (1533 K). The curve shows a principal peak at 2.8 A˚ 1 and a small prepeak around 1.8 A˚ 1. The former reflects the usual topological SRO, while the second results from a considerable but not complete cancellation of pronounced peaks in the like and unlike partial structure factors resulting from chemical SRO. The structure in the liquid closely resembles that obtained for

97

6.3 Zirconium nickel and Ti Zr Ni alloys

S(Q)–PF(Q)

2.0

1.5

0.1 0.0 1.51 Å–1

S(Q)

–0.1 0 1.0

1

Q (Å–1)

2

3

Prepeak

0.5 ZrNi 1653 K Parabolic function PF(Q) 0.0 0

2

4

6

8

10

12

14

Q (Å–1)

Fig. 6.25. Total structure factor for liquid ZrNi at 1653 K. The inset shows the prepeak region after removing the parabolic base line (Hennet et al., 2006).

the glass by neutron diffraction with isotope substitution (Fukunaga et al., 1985) and theoretical work based on a tight-binding approach (Hausleitner & Hafner, 1992). Figure 6.26 shows the corresponding average pair correlation function obtained by a Fourier transformation of S(Q) with a high-Q cut-off of 12 A˚ 1, using a number density r0 ¼ 0.0568 A˚ 3 calculated from the average of the liquid nickel and liquid zirconium densities. This average pair correlation function is a weighted average of the three partial functions: gðrÞ ¼ 0:35gNiNi ðrÞ þ 0:48gNiZr ðrÞ þ 0:17gZrZr ðrÞ:

ð6:9Þ

From the metallic radii for the elements, the shortest Ni Ni, Zr Ni and Zr Zr interatomic distances are expected at 2.5, 2.8 and 3.20 A˚, respectively. Based on these assignments, the authors performed a Gaussian fit of the function T(r), also shown in figure, with the first three Gaussians attributed to these correlations and a fourth around 4.3 A˚ representing an average of the second-neighbour correlations. The bond distances and correlation numbers derived in this way are compared in Table 6.2 with those obtained for the glass by the experimental and theoretical work referred to above and to the corresponding values for the ZrNi crystal. The latter is made up of trigonal

98

Liquid metals and alloys

Table 6.2. Nearest-neighbour distances r1 and coordination numbers C.N. in NiZr (Hennet et al., 2006). Phase

Crystal

Ni–Ni Ni–Zr Zr–Zr

r1(A˚) 2.62 2.68–2.78 3.27–2.44

C.N. 2 7 8

Glass (theorya)

Glass (exptb)

Liquidc

r1(A˚) 2.68 2.75 3.50

r1(A˚) 2.63 2.73 3.32

r1(A˚) 2.46 2.7 3.15

C.N. 2.2 6.1 7.8

C.N. 3.3 6.7 7.8

C.N 1.6 6.5 8

Notes: a Hausleitner & Hafner (1992). b Fukunaga et al. (1985). c Hennet et al. (2006).

2.0 ZrNi 1653 K Ni-Zr

G (r )

1.5

1.0

Ni-Ni

Zr-Zr

Gaussian fit

0.5

0.0 1

2

3

4

5

6

7

8

9

10

r (Å)

Fig. 6.26. Total pair correlation function for liquid ZrNi at 1653 K (Hennet et al., 2006).

prisms arranged in layers with the Ni atoms at the prism centres arranged in zigzag chains, and is illustrated in Fig. 6.27. It can be seen from the table that the SRO in both liquid and glass closely resembles that of the crystalline phase. Voigtmann et al. (2008) have recently made a comprehensive study of the structure and diffusion at the eutectic composition Zr64Ni36, carried out on samples prepared by arc-melting in a high-purity He atmosphere suspended by EML; neutron diffraction experiments with isotopic substitution of the

99

6.3 Zirconium nickel and Ti Zr Ni alloys

Fig. 6.27. Crystal structure of NiZr. The small and large spheres represent the Ni and Zr atoms, respectively.

2.0 Zr–Zr Ni–Ni Zr–Ni

Sab (Q)

1.5

1.0

0.5

0.0

–0.5 0

2

4

6

8

10

12

–1

Q [Å ]

Fig. 6.28. Partial static structure factors Sab(Q) of liquid Zr64Ni36 at 1350 K from neutron scatterring (symbols) and a calculation based on a mixture of hard spheres with literature values of covalent radii (solid lines) (Voigtmann et al., 2008).

Ni atoms (60Ni, 58Ni and naturally abundant Ni) and QENS measurements were also carried out in an EML environment. In Fig. 6.28 the partial structure factors are compared with those obtained from a simple model of hard spheres using literature values for the covalent radii of Zr and Ni. It is

100

Liquid metals and alloys 101 Zr64 Ni36 D [10–9 m2/s]

ZrTiNiCuBe 100

MCT Zr

10–1

MCT Ni MCT interdiff. MCT interdiff. (kinetic only)

10–2

0.6

0.7

0.8

0.9

1

–1

1000/T [K ]

Fig. 6.29. Diffusion and interdiffusion coefficients for liquid Zr–Ni alloys as a function of inverse temperature. Closed diamonds and circles represent QENS results for Zr64Ni36 and Zr41.2Ti13.8Cu12.5Ni10Be22.5, respectively. MCT results at 1350 K are also shown for DZr (open square), DNi (open circle) and DZrNi (filled triangle: including thermodynamic factor F(cNi), see text, Eq. (6.11); open triangle: without it) (Voigtmann et al., 2008).

seen that while the hard-sphere model gives a generally adequate representation, it does not account for the peak in SNiNi(Q) around 1.9 A˚ 1: this peak represents a degree of chemical SRO not found in the hard-sphere model and will result in a prepeak in the average S(Q) measured in a simple neutron diffraction experiment, similar to that observed by Hennet et al. at the equiatomic composition. Results for the diffusion constant extracted from the QENS measurements are shown in Fig. 6.29. As in pure Ni, the values for DNi follow an Arrhenius temperature dependence (a straight line in the figure) but with a high value for the activation energy, 0.64  0.02, compared with 0.47 in pure Ni, for example. Figure 6.29 also shows the predictions of an MCT calculation, similar to that made for Ta in Section 6.1, for DNi, DZr and the interdiffusion constant DNiZr (discussed in the next section in the context of Al Ni alloys) at 1350 K. When the thermodynamic factor F(cNi) is included in the expression for DNiZr (see Eq. (6.11)), the predicted values are all close to each other, being slightly higher than the value for DNi obtained by the QENS measurement. In connection with the discussion in the previous two sections on the structure of liquid Zr and Ni, it is interesting to note the observation of

6.3 Zirconium nickel and Ti Zr Ni alloys

101

Hausleitner & Hafner that there is a consistent structural trend in going from the Zr-rich to the Ni-rich compositions in Zr Ni glasses. They found that the local environments in Zr65Ni35 and, as we have seen, NiZr resemble the trigonal prisms of the ZrNi crystal, while in Zr35Ni65 the Ni environment has an icosahedral character and the Zr environment resembles the 14-, 15and 16-coordinated Frank Kaspar phases. On the other hand, an analysis of neutron diffraction data of Voigtmann et al. by Holland-Moritz et al. (2009) did not find a satisfactory fit on the basis of trigonal prismatic structural units unless values were chosen for interatomic distances that were in conflict with the real-space diffraction results. Other types of SRO such as icosahedral, dodecahedral, fcc, hcp, or bcc did not produce reasonable fits either. The authors speculate that the anomalous structure of liquid Zr64Ni36, characterized by an unusually large average local coordination number of 13.9, may result from the comparatively large difference of the atomic radii of Ni and Zr, RZr/RNi ¼ 1.29. Teichler (1999) made a simulation of melting in ZrNi using the Hausleitner Hafner interatomic potentials. Within the constraints of the MD model (960 atoms), the crystal can be superheated to about 500 K before melting on the ns timescale of the simulation takes place. Below this temperature, thermal vibrations and Frenkel pair production by themselves do not produce enough atomic displacements to induce melting, which apparently requires additional defects in the Zr sublattice, perhaps through antisite defect production. Such degrees of superheating are not of course possible during heating of bulk samples, but the phenomena predicted may be essential for solid-state amorphization (Devanathan et al., 1993). Adding a third element to Zr Ni alloys brings in the possibility of quasicrystals, a class of materials exhibiting long-range orientational order without long-range translational order. Although mathematical models of such entities were known since the 1960s, the first demonstration of a material with such a structure was made by Shechtman et al. in 1984. A large class of quasicrystals is formed from aluminium alloys, for example Al86Mn14, the original alloy studied by Shechtman et al., and Al63Cu25Fe12 (Cornier-Quiquandon et al., 1991, prepared by levitation melting). These will be discussed in the next section. Another large class is found in titanium alloys. A typical diffraction result obtained with a TEM instrument is shown in Fig. 6.30 for a Ti45Zr38Ni17 alloy (Kelton et al., 1997), which like most, but not all, quasicrystalline materials has icosahedral symmetry. In their further research on Ti Zr Ni quasicrystals, Kelton et al. (2002) carried out levitation experiments on the properties of the molten state

102

Liquid metals and alloys

Fig. 6.30. X-ray Laue diffraction images for a Ti45Zr38Ni17 alloy, showing (a) five-fold (b) three-fold and (c, d) two-fold symmetry axes. The actual compositon of the quasicrystalline phase was measured to be approximately Ti41.5Zr41.5Ni17 (Kelton et al., 1997).

and subsequent solidification of the melt. Using electrostatic levitation they measured the cooling curves of a sample with relatively low Ni concentration Ti37Zr50Ni13 and one with a higher Ni concentration in which the icosahedral quasicrystalline phase is anticipated Ti37Zr42Ni21. The two alloys exhibited quite different behaviour: in Fig. 6.31(a) Ti37Zr50Ni13 exhibits a recalescence at 941  C due to the heat evolved during transformation to a mixture of crystalline b-Ti Zr and the liquid. The second recalescence at 766  C is due to the solidification of the remaining liquid to a Ti Zr Ni Laves phase. The change in slope just after, at 800  C, is due to an emissivity change with surface crystallization. In Fig. 6.31(b) Ti37Zr42Ni21 exhibits a recalescence at 700  C corresponding to the nucleation and growth of the icosahedral phase. Because the equilibrium phase field for this phase does not extend to the liquidus temperature, however, it is a metastable solidification product; the plateau at 790  C corresponds to the metastable solidus temperature. The second rise to 810  C corresponds to the formation of the Ti Zr Ni Laves phase. These results suggest that the local order of the liquid is more similar to that of the icosahedral quasicrystal than to the polytetrahedral Laves phase. The reduced undercooling is defined as DT 0 ¼

Tm

Tc Tm

;

ð6:10Þ

6.3 Zirconium nickel and Ti Zr Ni alloys 1200

(a)

103

Ti37Zr50Ni13

1002 ˚C 941 ˚C

900 800 ˚C

β to α transformation

Temperature (˚C)

766 ˚C 600 1200

900

(b)

Ti37Zr42Ni21

810 ˚C 790 ˚C

701 ˚C 600 Time (s)

Fig. 6.31. Cooling curves for electrostatically levitated liquid 3 mm droplets of (a) Ti37Zr50Ni13 and (b) Ti37Zr42Ni21; the various recalescence temperatures are indicated (Kelton et al., 2002).

where Tc is the observed temperature of crystallization and Tm is the equilibrium melting temperature. For a known driving free energy, DT0 provides a measure of the interfacial free energy. The values obtained for the three phases studied are: 0.11  0.02 for the icosahedral phase, 0.15  0.01 for the Laves phase, and 0.19  0.01 for the a-Ti Zr phase, where the error bars reflect the uncertainty in the measured liquidus temperatures. These results indicate that the reduced undercooling increases as the structure of the resulting crystal becomes less tetrahedral, signalling an increasing interfacial free energy with the liquid phase. Similar conclusions have been reached for Al-based quasicrystalline alloys (Holland-Moritz, 1998). Kelton et al. (2003) went on to make XRD measurements on a Ti39.5Zr39.5Ni21 liquid alloy levitated by ESL. Figure 6.32 shows the X-ray structure factors as a function of undercooling. The data exhibit an enhancement of a shoulder on the high-Q side of the second peak in S(Q) with increasing undercooling, in the same temperature range where nucleation of the icosahedral phase becomes favourable. As we have seen, this shoulder is

104

Liquid metals and alloys

S (Q)

1473 K 1173 K 1073 K 1029 K

1

2

3

4

5

6

7

8

Q (Å–1)

Fig. 6.32. Structure factor for the Ti39.5Zr39.5Ni21 liquid as a function of temperature. An increase in the intensity of the shoulder on the second peak (indicated by the arrow) is observed as the temperature is lowered below the liquidus temperature (1083 K) (Kelton et al., 2003).

consistent with local icosahedral order. The relative locations of the first two peaks in S(Q), Q2/Q1 ¼ 1.72, and the location of the shoulder on the second peak, Qshoulder/Q1 ¼ 1.97, are in good agreement with those expected for a perfect icosahedron (Sachdev & Nelson, 1984), indicating little distortion in the icosahedral order. Figure 6.33 shows the fit to a 13-atom icosahedral cluster with Ni at the centre and Ti atoms on the 12 vertices, consistent with a structural model for the Ti Zr Ni quasicrystal at two temperatures. The shoulder in the experimental data is reproduced in the S(Q) calculated from the cluster if no thermal effect is considered. When the Debye Waller factors are included to account for the temperature dependence, the shoulder becomes less prominent, particularly at lower temperatures, contrary to the experimental results, indicating that the Debye Waller factors are overestimated and are possibly anisotropic. The locations and the intensities of the second and third peaks in S(Q) fit well, however. These results demonstrate an enhanced icosahedral short-range order with undercooling in Ti Zr Ni liquids that form icosahedral quasicrystals, decreasing the barrier for the nucleation of the metastable icosahedral phase, favoured over the formation of stable polytetrahedral crystal phases. Kim et al. (2007) report a common-neighbour analysis of a numerical simulation fitted to their XRD results and find that symmetric and

105

6.4 Aluminium transition metal alloys 1.2 (a)

1473 K

S (Q)

1.0

1.2 (b) 1029 K 1.0

4

5

6

7

8

Q (Å–1)

Fig. 6.33. Comparison between the experimental data (circles) for the structure factor for a Ti39.5Zr39.5Ni21 liquid at (a) 1473 K and (b) 1029 K, and the fit to a 13-atom icosahedral cluster decorated by a Ni atom at the center and Ti atoms at the 12 vertices (solid line) (Kelton et al., 2003).

distorted icosahedra and polyhedra (denoted by I, I0 , and FK in Table 6.1) are the most significant components and become increasingly so on supercooling. 6.4 Aluminium–transition metal alloys We now pass to the other major class of systems that form quasicrystals, aluminium transition metal alloys. In fact, interest in these alloys originated much earlier owing to a remarkable increase in viscosity and a decrease in the coefficient of thermal expansion upon the addition of a few atomic per cent of the transition metals to pure molten aluminium. Turnbull (1990) ascribed this behaviour to the formation of large Al clusters around the transition metal atom. The phase diagram of these binary systems is generally quite complex, displaying eutectic as well as intermetallic compositions. At a eutectic composition the solid phase is destabilized with respect to the liquid phase, leading to a melting point lower than that of the constituent elements. Upon directional solidification, the microstructure of a eutectic shows a characteristic lamellar structure, indicating a tendency for phase separation of unlike atoms. On the other hand, the solid phase is stabilized at intermetallic compositions, giving rise to much higher melting points. The variable stability

106

Liquid metals and alloys 2.5 2.0

S (Q)

1.5 684 ˚C

1.0

766 ˚C 0.5

832 ˚C 1019 ˚C

0.0 1165 ˚C –0.5 –1.0 2

4

6

8 Q,

10

12

14

Å–1

Fig. 6.34. Average X-ray structure factor for liquid Al99Ti1 at different temperatures. Spectra are shifted vertically for clarity (Pozdnyakova et al., 2006).

of the solid phase as a function of composition indicates a subtle interplay between energetic and entropic contributions to the corresponding thermodynamic potentials of the individual phases. Based on these findings in the solid state, it is natural to ask whether a similar behaviour can be found in the liquid state, i.e. whether the structure of the liquid phase shows a distinct concentration dependence and, in particular, whether traces of the clusters of intermetallic composition suggested by Turnbull exist in the liquid. The structures of dilute alloys of Ti and Fe with Al have been studied by Pozdnyakova et al. (2006), using the hybrid aerodynamic levitation electromagnetic heating facility described in Section 3.2. According to their phase diagrams, the liquidus temperatures of the Al Fe alloys are higher than those of the Al Ti alloys, which are closer to the melting temperature of Al (660  C), so that the former have a larger region of coexistence between liquid Al and the intermetallic compositions. In the structure factors for liquid Al99Ti1 and Al95.8Fe4.2, shown in Figs. 6.34 and 6.35, respectively, for various temperatures in the stable liquid, it can be seen that in Al99Ti1 crystallization is not setting in before cooling to 684  C, but is appearing at 779  C in Al95.8Fe4.2. Similar behaviour was observed at the other concentrations measured, Al99.5Ti0.5 and Al92.5Fe7.5. The crystalline phase was identified by

107

6.4 Aluminium transition metal alloys 4.0 3.5 3.0 2.5

S(Q )

2.0 1.5 679 ˚C 779 ˚C

1.0 0.5

941 ˚C 1071 ˚C

0.0

1152 ˚C –0.5 –1.0 2

4

6 Q, Å–1

8

10

12

Fig. 6.35. Average X-ray structure factor for liquid Al95.8Fe4.2 at different temperatures. Spectra are shifted vertically for clarity. At 779  C and 679  C, the sample was partly solid (Pozdnyakova et al., 2006).

X-ray diffraction and chemical analysis at both compositions: SEM measurements showed the crystalline phases to be needle-like inclusions in the Al matrix. These results are in agreement with a previous study where the first intermetallic compound formed in the Al matrix was identified as Al13Fe4. The quantity of crystalline phase was found to be enhanced by cooling rapidly, which may explain why the phase was not observed in other previous work. The microstructure formed in dilute Al Fe alloys can play a crucial role in the high specific strength exhibited by these alloys by impeding dislocation motion. In all the alloys measured, and also in pure liquid aluminium measured as a reference, S(Q) exhibits two well-defined peaks at 2.6 2.8 A˚ 1 and 4.8 5.1 A˚ 1. At temperatures well above the liquidus, the ratio between the second and first peak positions, Q2/Q1, is temperature-independent at values of about 1.86 for pure Al and the Al Ti alloys and 1.8 for the Al Fe alloys. This behaviour, together with the symmetrical shapes of the first peak in S(Q), are characteristic of homogeneous liquids interacting via hard-sphere interactions. The shapes of the second peak in S(Q) do not show any of the structural features associated with icosahedral ordering discussed in the

108

Liquid metals and alloys 1152 1071

1034

Q1ds

9.0

8.8

999 1138 1165 1058 965 872

1019 800 738

832

8.6

941 916

dense-packing limit Al99.5Ti0.5

779

766 684

Al99Ti1 Al95.8Fe4.2 Al92.5Fe7.5 Al

7.2

7.4

7.6 Q1r1

7.8

8.0

Fig. 6.36. Scattering vectors of the first peaks in the structure factors of liquid Al–Ti and Al–Fe alloys, scaled by the nearest-neighbour distance r1 and the mean interatomic spacing ds. The values near the data points indicate the temperature ( C) (Pozdnyakova et al., 2006).

previous sections although, as we discuss below, they are observed at higher Fe concentrations in Al Fe liquid alloys. There is no temperature dependence of the first- and second-neighbour peak positions in the g(r)s for pure Al and Al Ti alloys, but in the Al Fe alloys the position of the first peak decreases slightly with increasing temperature. It should be noted that the weighting factors for the contribution of Al Al interactions to the measured average S(Q) and g(r) are 0.97 and 0.85 for Al99Ti1 and Al95.8Fe4.2, respectively, which explain the more distinct behaviour in the Al Fe alloys. The nearest-neighbour coordination numbers are in the same range, 10 11, in all the liquids measured, and decrease with increasing temperature, as is typical for liquid metals. This behaviour can be clearly seen in Fig. 6.36, which plots Q1ds against Q1r 1 for all the liquids and temperatures measured, where ds is the mean interatomic distance (6/pr0)1/3 (Moss & Price, 1985; Price et al., 1989). At low temperatures the points are close to the dense random packing limit for hard spheres, 0.637, but move away with increasing temperature with Q1ds increasing while Q1r1 remains approximately constant.

6.4 Aluminium transition metal alloys

109

Passing now to more concentrated alloys, we review a series of measurements on liquid binary and ternary alloys containing Al and Mn that, as already mentioned, form one of the major classes of quasicrystals in the solid. Since the melting temperatures are not extremely high, these measurements were made on samples contained in single-crystal sapphire cells: it is appropriate to include them here in view of our interest in the relation between the structure of the liquid and that of the corresponding quasicrystal. Maret et al. (1989; 1991) obtained partial structure information on Al Mn alloys by substituting FeCr (equiatomic mixture of the two elements bracketing Mn in the periodic table) for Mn in neutron diffraction experiments. Under the assumption that the substitution is isomorphous, i.e. that the structure of the liquid alloy is not appreciably affected by the substitution, Faber Ziman and Bhatia Thornton partial structure factors were obtained for liquid Al1 xMnx at the x ¼ 20% and 40% compositions; the former is within, and the latter outside, the range of compositions where icosahedral quasicrystalline phases are formed. For comparison, Maret et al. (1990) obtained partial structure factors for Al80Ni20, which also does not form an icosahedral phase, by neutron diffraction with isotopic substitution (NDIS), taking advantage of the very different mean neutron scattering lengths of naturally abundant Ni and the isotopes 58Ni and 60Ni. Magnetization and neutron scattering measurements of Hippert et al. (1996) on liquid Al Pd Mn alloys (Fig. 6.37) showed that a localized moment appears on Mn atoms in the liquid state and disappears in the solid state. Both measurements are consistent with average values of the spin in the range 1.3 1.5, compared with the value of 5/2 expected for a free Mn atom. This implies that at a given temperature only a fraction of the Mn atoms in the liquid bears a localized moment; there is an equilibrium between two different Mn atom species in the liquid: magnetic and nonmagnetic. The presence of paramagnetic scattering was previously detected in an Al80Mn20 liquid in the work of Maret et al. (1989) already cited. The obtained spin value, S 1 (assuming that all the Mn atoms in the liquid are equivalent), was close to the values obtained for liquid Al Pd Mn, showing that the presence of Pd is not a necessary ingredient to get magnetic Mn in the liquid state. Simonet et al. (2001) made a neutron scattering study of liquid Al88.5Mn11.5, obtaining partial structure factors by substituting Cr for Mn. This substitution is simpler than that of FeCr and had been shown to be truly isomorphous in the mAl4Mn and mAl4Cr approximants (crystalline phases with large unit cells and SRO similar to the quasicrystalline ones). The Faber Ziman and Bhatia Thornton structure factors and pair correlation functions are shown in Fig. 6.38. The positions of the first peak of the gAlM and gAlAl

110

Liquid metals and alloys (a)

106 c (en)

40

Al.765Pd.20Mn.035

20

TE

0

TL

–20 –40 –60 300

500

700

900

1100

1300

T (K) (b) 0.4

s 0 (barns)

0.3 0.2 0.1 0.0 850

950

1050

1150

1250

T (K)

Fig. 6.37. (a) Temperature dependence of the susceptibility of Al72.1 Pd20.7Mn7.2; the dashed line is a fit to w(T) ¼ w(T¼ 0) þ AT2 in the solid in the temperature range (300 K, TE), where TE is the melting point of the ternary eutectic. The differential thermal analysis at decreasing temperature is also shown; TL is the liquidus temperature. (b) Temperature dependence of s0, the total neutron scattering cross section as Q ! 0 (Hippert et al., 1996).

partial pair correlation functions are at 2.65 and 2.78 A˚, respectively, slightly higher than in the mAl4Mn approximant. Their ratio, 1.048, is very close to that of a regular icosahedron of 12 Al atoms surrounding an Mn atom (1.0515), and explains the shoulder clearly visible on the right side of the second peak in SNN(Q). A model fit of an Al12Mn icosahedron and an Al12Mn cuboctahedron to the neutron-weighted average S(Q) for the Al92.3Mn7.7 composition, shown in Fig. 6.39, is a convincing demonstration of the important role that icosahedra play in the structure of these liquid alloys. In order to understand the role that Mn plays in the formation of quasicrystalline phases, Jakse et al. (2004b) carried out ab initio molecular dynamics simulations of liquid Al80Mn20 and Al80Ni20. Their results are compared with the experimental Bhatia Thornton structure factors of Maret et al. in

111

6.4 Aluminium transition metal alloys 2.5

SAIAI

gAIAI

2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0 2.5

gAIM

SAIM

0.0 2.5

2.0

2.0

1.5

1.5

1.0

1.0

0.5

g(r )

S (Q )

2.5

0.5

0.0 2.5

S NN

2.0

S NC

gNN

0.0 2.5 2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0 0

2

4

6

8

Q (Å–1)

10 12

2

4

6

8

0.0 10 12 14

r (Å)

Fig. 6.38. Faber–Ziman and Bhatia–Thornton partial structure factors (left) and pair correlation functions (right) extracted from Al 88.5(MnyCr100 y)11.5 liquid spectra for y ¼ 74.5, 49.4, 24.6, and 0 at 1343 K (Simonet et al., 2001).

Fig. 6.40. The agreement is remarkably good, the most significant discrepancy being that the simulations slightly underestimate the shift in the peaks of SCC(Q) for Al80Ni20 to higher Q compared with Al80Mn20. The increased width and shift to the right of the main peak in SNN(Q) for Al80Ni20 compared with Al80Mn20 are well reproduced. Figure 6.41 shows the partial pair distribution function GMnMn(r) calculated by Jakse et al. with and without explicit treatment of spin on the Mn atoms. The two curves are found to be very different both in the location and the height of the first and second peaks, spin effects leading to a correct description. Jakse et al. went on to make a common-neighbour analysis, described above in Section 6.1, of their results for Al80Mn20 and Al80Ni20. The local

112

Liquid metals and alloys 1.2 A192.3Mn7.7 T = 1223 K icosahedron 1.1 S (Q )

cuboctahedron

1.0

0.9

4

6

8

10

12

14

Q (Å–1)

Fig. 6.39. Fit of the neutron-weighted average structure factor based for Al92.3Mn7.7 to models of an Al12Mn icosahedron (thin line) and an Al12Mn cuboctahedron (thick line) (Simonet et al., 2001).

environment of Mn atoms is dominated by icosahedral and distorted icosahedral inherent structures since the 1551 and 1541 bonded pairs are preponderant. However, although the 2331 pairs are relatively numerous, the absence of the 1321 pairs and the presence of 1661 pairs is a strong indication that the local order is more complex than the one found in the 13-atom icosahedron (Jakse & Pasturel, 2003). In Al80Ni20, on the other hand, the 1441, 1431, 1421 and 1422 pairs become preponderant. The sum of these four types of pairs amounts to more than 50% of the total number of all bond types, indicating a strong tetrahedral local order. On the contrary, the 1551 and 1541 bond types, related to the icosahedral configuration, contributes only 34% to the total number of all bond types. Such behaviour suggests that the icosahedral packing is still present in Al80Ni20 but strongly hidden by the tetrahedral packing. These results indicate that the fivefold symmetry is preponderant in the liquid alloys from which quasicrystalline phases may be formed by quenching techniques. In subsequent work, Jakse & Pasturel (2007a) investigated the concentration dependence of liquid Al1 xMnx alloys, contrasting two compositions (x ¼ 0.14 and 0.2) in the quasicrystal-forming range with one (x ¼ 0.4) outside it, making comparison with the measurements of Maret et al. (1991) at these compositions. They found that at the x ¼ 0.4 composition the mean Mn Mn distance shortened considerably, from 2.9 A˚ to 2.75 A˚, a value closer to the Ni Ni distance (2.65 A˚) in Al80Ni20, and the common-neighbour analysis found a much more complex polytetrahedral

113

6.4 Aluminium transition metal alloys 2.5 (a)

Al80Mn20

(b)

Al80Ni20

2.0 1.5 1.0

SBT (Q)

0.5 0.0 2.5 2.0 1.5 1.0 0.5 0.0 0

2

4

6 Q

8

10

(Å–1)

Fig. 6.40. Bhatia–Thornton partial structure factors for (a) Al80Mn20 and (b) Al80Ni20 liquids from AIMD simulations (solid lines), compared with the neutron diffraction results of Maret et al. (1990; 1991). The triangles, circles, and squares represent the experimental SNN(Q), SNC(Q) and SCC(Q), respectively (Jakse et al., 2004b).

symmetry. Thus, a strong icosahedral ordering in the liquid phase is a preponderant feature of the formation of quasicrystalline phases in Al Mn alloys with fast quenching. Al13Fe4 and Al13Co4 form polytetrahedral phases closely related to quasicrystals: Al13Fe4 is a monoclinic approximant phase, while one of the Al13Co4 phases is isostructural to the Al13Fe4 monoclinic phase. Holland-Moritz et al. (2002b) investigated the short-range order of deeply undercooled liquids of Al76.5Fe23.5 and Al74Co26 alloys by combining electromagnetic levitation with neutron diffraction. The experimentally determined structure factor S(Q) was fitted with a model that assumed that clusters with different structures exist in the liquid. The best fit of the experimental data was obtained under the assumption of an icosahedral short-range order. The similarity between the two systems suggests the possibility of Fe Co isomorphous substitution in the liquid state. With this in mind,

114

Liquid metals and alloys 7 6 5 4 G (r )

3 2 1 0 –1 –2 –3 –4 0

1

2

3

4

5

6

7

8

9

r (Å)

Fig. 6.41. The partial pair correlation function GMnMn(r) in liquid Al80Mn20. The symbols represent the experimental result of Maret et al. and the solid and dashed line the ab initio molecular dynamics result with and without, respectively, magnetic interactions (Jakse et al., 2004b).

Schenk et al. (2004) carried out neutron diffraction experiments with electromagnetic levitation on liquid Al13M4 alloys where M ¼ CoxFe1 x with x ¼ 0, 0.25, 0.5, 0.75 and 1. The measured structure factors at 150 K below and above the liquidus, shown in Fig. 6.42, clearly exhibit the shoulder on the right side of the second peak that, as we have seen, is the signature of ISRO. This feature becomes more visible as the liquid is supercooled and as Fe is substituted for Co. The latter changes are relatively subtle and support the notion of an isomorphous substitution. Under that approximation, three partial structure factors can be derived from the five data sets at each temperature. The results for TL 150 K are shown in Fig. 6.43 in two ways: (a) Bhatia Thornton partial structure factors, in which the NN partial function corresponds to the structure factor of a hypothetical monatomic liquid with the same topological structure as the one being studied, and (b) Faber Ziman partial structure factors. The shoulder on the second peak in SNN(Q) still visible here. In the inset, the SNN(Q) at TL þ 150 K is compared with results of simulations for a cuboctahedron (fcc) cluster and dodecahedron (33-atom icosahedral) cluster, the latter showing good agreement with the experimental data. The idea of an ISRO prevailing in the liquid is thus supported by the diffraction data as well as by the rather low undercooling observed in Al13(CoxFe1 x)4 liquids, which implies a similar SRO in the solid and in the corresponding liquid phase (Holland-Moritz, 1998). The strong chemical SRO characterized by the deep minimum in the Co Co partial structure factor is also consistent with that observed in the solid phases.

115

6.4 Aluminium transition metal alloys 3.0

Al13(CoxFe1–x )4

2.5 2.0

S (Q )

1.5 1.0 TL – 150 K

0.5 0.0 1.0

TL + 150 K

0.5 0.0 4

2

6

8

10

12

Q [Å–1]

Fig. 6.42. Structure factors at TL  150 K (top) and TL þ 150 K (bottom) of Al13(CoxFe1 x)4 liquids. TL denotes the liquidus temperature, which ranges from 1420 K (Al13Fe4) to 1440 K (Al13Co4). The curves are displaced upwards with decreasing x ¼ 1, 0.75, 0.5, 0.25 and 0, respectively (Schenk et al., 2004). (a)

(b) 3

1.1

2.5 SNN 2.0

2

1.0

SAITM 1

0.9 1.5 4

6

8

10

12

STMTM

1.0

0 1

SCC

0

0.5 S AIAI

SNC

1

0.0 0 2

4

6 8 Q [Å–1]

10

12

2

4

6 8 Q [Å–1]

10

12

Fig. 6.43. (a) Bhatia–Thornton and (b) Faber–Ziman partial structure factors at TL  150 K extracted from the total structure factors shown in Fig. 6.42. Inset: experimentally derived SNN(Q) at TL þ 150 K (dots) compared to calculated structure factors based on models with cuboctahedron (dashed line) and dodecahedron (solid line) clusters (Schenk et al., 2004).

116

Liquid metals and alloys 2

Al8058Ni20

S (Q )

1

Al80natNi20 1

Al8060Ni20 1

0

2

4

6

8

10

12

14

Q (Å–1)

Fig. 6.44. Neutron-weighted average structure factors for liquid Al80Ni20 at 1330 K for three different Ni isotopic compositions (Maret et al., 1990).

Liquid Al Ni alloys, already mentioned in comparison with Al Mn, have been the subject of several investigations with both contained and levitated samples. The Bhatia Thornton structure factors for the 20%Ni alloy obtained by Maret et al. (1990) from NDIS measurements on samples contained in sapphire cells have already been shown in Fig. 6.40. It is also instructive to view the neutron-weighted average structure factors obtained directly from the three measurements, shown in Fig. 6.44. A prepeak is clearly visible at Q ¼ 1.8 A˚ 1 in the data with 58Ni and natNi, where the Ni scattering length is much larger than that of Al, but not with 60Ni where the two have similar magnitudes, confirming that the feature arises from chemical ordering of the two metals in the liquid. Brillo et al. (2006) have made X-ray diffraction measurements of the 2.7% and 25% Ni alloys, corresponding to the eutectic and Al3Ni compositions, respectively, with the hybrid aerodynamic levitation electromagnetic heating facility used for the Al Ti and Al Fe measurements described above. The X-ray weighted average structure factors exhibit a prepeak at the Al3Ni composition but not at the eutectic. EXAFS measurements by Egry et al. (2008) with a similar setup also indicated a considerable degree of chemical ordering between 10 and 30 at.% Ni. Similar diffraction results were obtained

117

6.4 Aluminium transition metal alloys 2.5 Al33Cu@750 °C Al17Cu@750 °C

2.0

S (Q )

1.5

1.0

0.5

0.0

–0.5 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Q (Å–1)

Fig. 6.45. X-ray-weighted average structure factors for liquid Al–Cu alloys at 1023 K for two different compositions (Brillo et al., 2006).

for Al Cu alloys at 17% and 33.3% Cu, corresponding to the eutectic and Al2Cu compositions, respectively, shown in Fig. 6.45. The absence of a prepeak at the eutectic is consistent with the thermodynamic evidence that homoatomic Al Al pairs are preferred at this composition. A similar result was obtained by Qin et al. (1998) who studied the complete range of liquid Al Fe alloys with X-ray diffraction from the liquid vapour interface in y y geometry: the prepeak intensity reaches a maximum at the concentration corresponding to the Fe2Al5 intermetallic compound. In the Al67.7Cu33.3 measurement of Brillo et al., a shoulder is observed on the right side of the second peak of S(Q) reminiscent of that observed by Holland-Moritz et al. (2002b) in Al76.5Fe23.5 and Al74Co26 alloys. The ternary systems Al65Cu25Co10 and Al65Cu34Fe6 form decagonal and icosahedral quasicrystalline phases, respectively. Holland-Moritz et al. (1993) investigated the undercooling of the liquids at these compositions with electromagnetic levitation. The temperature time profile obtained for Al65Cu25Co10 is shown in Fig. 6.46. Onset of solidification is observed at a temperature 205 K below the liquidus TL for the quasicrystalline phase, inferred from the DTA trace shown in the left inset in the figure: this corresponds to a relative undercooling DT/TL ¼ 0.16, considerably less than the values 0.20 normally reached with EML. The TEM pattern in the right inset shows that the decagonal quasicrystalline phase is formed in the solidification. From the duration of the recalescence and the diameter of the

118

Liquid metals and alloys 1800 1600 1400 TL=1274 K Temperature [K]

1200

TLQC=1259 K TN =1055 K

1000 Δt = 0.5 s

800

1.5 1.4

600 Q

400

1.3 1.2

200

1.1 1000

5s

1100 1200 1300 1400 Temperature [K]

Time [s]

Fig. 6.46. Temperature–time profile for cooling of an electromagnetically levitated Al65Cu25Co10 melt. The left inset shows a DTA trace and the right inset shows a TEM diffraction pattern of the solidified sample (Holland-Moritz et al., 1993).

pyrometer, the solidification velocity is estimated as 0.7 cm s 1, also considerably less than typical values for pure metals and simple alloys (Willnecker et al., 1989). Holland-Moritz et al. applied classical nucleation theory to estimate the activation threshold for nucleation of the quasicrystalline phase and find it also to be appreciably smaller than for crystalline phases. Egry et al. (1999) made EXAFS measurements at the Co K edge on the melt above TL with EML. They found that more than 90% of the neighbours of Co atoms were Al, compared with the 65% expected for a random alloy, consistent with an icosahedral SRO. Self-diffusion and interdiffusion in Al80Ni20 melts have been studied by Horbach et al. (2007) using tracer diffusion, QENS and numerical simulations with embedded-atom potentials. The tracer diffusion measurements were carried out with the long-capillary technique, in which samples of Al85Ni15 and Al875Ni25 prepared with naturally occurring Ni sandwich a short length of Al80Ni20 enriched in 62Ni isotope, all contained and heated inside thin quartz capillaries. The QENS measurements were carried out on a sample contained in a thin-walled Al2O3 cell. In fact, two types of diffusion constant can be obtained from the tracer measurement: the self-diffusion constants DAl and DNi describing the random walk motion of Al and Ni

119

Da, DAB (m2/s)

6.4 Aluminium transition metal alloys

10–8

sim. DAB sim. DAI sim. DNi QNS D Ni

Al80Ni20

LC DAB LC exp. DNi 10–9 0.5

0.6

0.7

0.8

1000/T (K–1)

Fig. 6.47. Arrhenius plot of interdiffusion DAB (A ¼ Al, B ¼ Ni) and selfdiffusion (DAl and DNi) constants for liquid Al80Ni20 obtained from experiment and simulation (sim.). The experimental results are measured by quasielastic neutron scattering (QNS) and by the long-capillary (LC) tracer diffusion technique. The liquidus temperature, 1280 K, is just off-scale to the right (Horbach et al., 2007).

atoms, respectively, through the liquid; and the interdiffusion (or chemical diffusion) constant DAlNi describing the collective mass transport driven by a concentration gradient. The two are related by the expression DAlNi ¼ ðcNi ÞHcc ðcAl DNi þ cNi DAl Þ;

ð6:11Þ

where F(cNi) is a thermodynamic factor related to the second derivative of the Gibbs free energy with respect to cAl and cNi, and Hcc is a dimensionless factor, normally close to unity, expressing the contribution of cross correlations to DAlNi. The two constants can also be determined from scattering experiments. The self-diffusion constants can be obtained from the QENS profile for the incoherent scattering at low Q. Since Ni is the dominant incoherent scatterer in this system, its self-diffusion constant is given by the low-Q limit of ðQÞ ¼ DNi Q2 ;

ð6:12Þ

where ha(Q) is the half-width of the Lorentzian defined in Eq. (4.63). In principle the interdiffusion constant can be determined from the coherent scattering at small Q (Faux & Ross, 1987), but this is more difficult because its intensity is proportional to S(Q), which becomes very small at low Q. Results obtained by Horbach et al. are shown in Fig. 6.47. It is seen that the

120

Liquid metals and alloys

simulation and QENS results for DNi are in reasonably good agreement and that the simulation results for DAlNi are about twice as large, principally because of the factor F(cNi). The tracer results are consistent with the simulations within the experimental error and confirm that DAlNi is considerably higher than DNi.

6.5 Cobalt–palladium alloys Magnetic moments in liquids containing magnetic ions such as Fe and Ni and even, as we saw in the previous section, in some liquid alloys such as Al80Mn20 that are not magnetic in the solid state, can be readily detected by neutron diffraction measurements at low Q. It is natural therefore to enquire if such moments can order spatially in the liquid state. Such ordering is theoretically possible under appropriate conditions (Groh & Dietrich, 1997). It is well known that magnetic ordering can occur readily in solid amorphous systems (Zallen, 1983), so an ordered crystalline structure is not a prerequisite for such behaviour. The Curie temperatures TC of all known ferromagnetic and antiferromagnetic metallic materials are lower than their liquidus temperatures, so magnetic ordering has not been observed in stable metallic melts. However, it might well be observable in the strongly undercooled state accessible with levitation techniques. A promising candidate for such behaviour is Co80Pd20. An experiments on a melt undercooled by EML to 20 K below the Curie point of the solid, TC(s), showed an attractive interaction between the melt and an external magnet which disappeared when the sample was heated above TC(s) (Platzek et al., 1994). Measurement of the temperature dependence of the magnetization with the Faraday balance technique described in Section 4.1 on levitated undercooled drops (Reske et al., 1995), shown in Fig. 6.48, revealed a Curie Weiss behaviour with a Curie temperature TC(l) ¼ 1253  8 K, some 20 K below that of the solid, TC(s) ¼ 1271  8 K. The slopes in the Curie Weiss plots of inverse susceptibility vs. temperature were the same for the solid and liquid, showing that the local magnetic moments had similar values in the two phases. Theoretical support for this result is provided by calculations with a three-dimensional Ising spin-1/2 system in which the TC(s) of a system with bond disorder is below that of the equivalent ordered system (Falk & Gehring, 1975). In a subsequent study, Bu¨hrer et al. (2000) found that deviations from the Curie Weiss law occurred on coming closer to TC. Their results for the susceptibility w shown in Fig. 6.49 shows that w 1(T) diverges from linear

121

6.5 Cobalt palladium alloys 0.35

IM/sample mass [mA/g]

0.30 0.25 0.20 0.15 0.10 0.05 0.00 500

1000 Temperature [K]

1500

Fig. 6.48. Current in the compensating coil (proportional to the change in magnetization, DM) as a function of temperature in the undercooled liquid regime for Co80Pd20. The solid line represents a fit according to the Curie– Weiss law. The dashed line indicates the liquidus temperature and the arrow the Curie temperature of the solid (Reske et al., 1995).

behaviour at about 30 K above TC. This is an indication of the approach to a critical regime in which the zero-field susceptibility w0 is expected to follow a power-law behaviour  w0 ðT Þ /

T

TC TC



g

;

ð6:13Þ

where g(T) is a critical exponent (Kouvel & Fisher, 1964). For a threedimensional Heisenberg magnet close to the critical point, g takes the value 1.387, and far from it, in the Curie Weiss regime, g is identically 1.0. The determination of g(T) requires the knowledge of the susceptibility at zero field, which can only be accessed by making measurements as a function of external field and extrapolating to zero field. Bu¨hrer et al. added two coils in a Helmholtz geometry producing magnetic fields up to 65 mT in conjunction with a small flux-gate sensor that allowed the direct measurement of the magnetic dipole moment of the sample in the external field of the Helmholtz coils. Figure 6.50 shows a plot of the effective exponent versus relative temperature for solid Co and solid and liquid Co80Pd20. The open and closed symbols result from different procedures for extrapolating the susceptibility data to zero field, discussed in detail in the original paper. For pure Co, g(T) drops monotonically from 1.23 down to 1.0 with increasing temperature.

122

Liquid metals and alloys

Fig. 6.49. Measurement of the susceptibility of liquid Co80Pd20 versus temperature in an external field 15.6 mT of the modified Faraday balance. With decreasing temperature there is a steep rise in susceptibility below 1300 K. Below 1250 K the onset of spontaneous magnetization in the liquid metal is clearly visible. The kink is caused by the limited external field of the permanent magnet in conjunction with the minimization of the stray field energy. The inset displays the inverse susceptibility versus temperature. The deviation from an ideal Curie–Weiss law (dashed line) is clearly visible (Bu¨hrer et al., 2000).

The exponent for the chemically disordered crystalline Co80Pd20 shows a sharp maximum, while that for liquid Co80Pd20 stays approximately constant at a value around 1.4 and then rises to a broad maximum with increasing temperature. The value of g extrapolated to TC ð‘Þ was 1.42  0.05, consistent with that for a three-dimensional Heisenberg magnet. As can be seen in the inset, these three types of behaviour are found in other ordered, sitedisordered (i.e. chemically disordered but topologically ordered) and structurally disordered ferromagnetic materials, respectively. Bu¨hrer et al. obtained values of TC ð‘Þ ¼ 1251:2  0:5 K and TC(s) ¼ 1271.5  0.5 K, consistent with those of the earlier work but more precise. Herlach et al. (1998) report muon-spin-rotation results in which the relaxation rate G in the liquid supercooled sample increases by a factor of five when the temperature approaches TC ð‘Þ, indicating large spin fluctuations caused by the onset of spontaneous magnetization.

123

6.5 Cobalt palladium alloys 2.0 Co80Pd20 solid

1.8 Co80Pd20 liquid

1.8

1.6

1.7

g (T )

1.4

site disordered, crystalline alloy

1.6 amorphous alloy g (T )

1.2 Co solid

1.5

1.0 1.4

0.8

1.3

0.6

1.2

crystalline metal ordered, crystalline alloy

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 T / Tc–1

Fig. 6.50. Plot of the effective exponent versus relative temperature for solid Co and solid and liquid Co80Pd20. The open and closed symbols result from different procedures for extrapolating the susceptibility data to zero field. The inset shows that these three types of behaviour are found in other ordered, site-disordered and structurally disordered ferromagnetic materials, respectively (Bu¨hrer et al., 2000).

A definitive demonstration of magnetic ordering in the liquid could be provided by neutron scattering experiments. Fischer et al. (2007) have carried out SANS measurements on aerodynamically levitated solid Co80Pd20 above TC(s). Under the condition that the incident energy is much larger than the line width of the critical scattering, the structure factor for a metallic ferromagnet is given by SðQÞ ¼ S0

T TC x

2

1 ; þ Q2

ð6:14Þ

where x is the transverse spin spin correlation length of the magnetic fluctuations. Approaching TC from above, the correlation length at zero magnetic field should diverge as   T TC n : ð6:15Þ xðT Þ ¼ x0 TC

124

Liquid metals and alloys T calibrated using 2-colour pyrometer fit for Tc = 1293.37 K+/–2 K,n = 0.76 +/–0.05 2.2

120

T calibrated using 2-colour pyrometer fit with line of slope = –0.76

2

100

1.8 1.6 log(x)

x = magnetic correlation length (Å)

140

80

1.4 1.2

60

1 0.8

40 0.6 –2.4

0 1250

–2.2

–2

–1.8

–1.6

–1.4

–1.2

–1

–0.8

–0.6

–0.4

log[(T–TC)/TC]

20

1300

1350

1400

1450 1500 T (K)

1550

1600

1650

1700

Fig. 6.51. Divergence of the magnetic correlation length x in solid Co80Pd20 above the ferromagnetic transition temperature TC, together with a fit to x(t) ¼ x0t n, where t ¼ (TTC)/TC is the reduced temperature. The sample was in the solid phase except for the highest temperature data point at 1681 K (Tl ¼ 1606 K). The log–log plot of the inset confirms that the exponential divergence of x(t) holds true across almost two orders of magnitude in reduced temperature, with the possible exception of the liquid-phase data point at the far right (Fischer et al., 2007).

The results for x(T) together with the fit of Eq. (6.15), shown in Fig. 6.51, gave the values n ¼ 0.76  0.05 and x0 ¼ 2.05  0.25 A˚. The value of n is again consistent with that for a three-dimensional Heisenberg magnet, 0.707. At the time of writing it has not been possible to supercool the samples in these experiments down to TC ð‘Þ. Magnetic ordering in a melt should also be revealed by changes in its thermophysical properties. Loho¨fer et al. (2001) measured the temperature dependence of the surface tension and viscosity of liquid Co80Pd20 with the oscillating drop method described in Section 4.2, and that of the electrical conductivity with the applied inductive technique, described in Section 4.1. The measurements were made under microgravity with electromagnetic positioning and heating. All three quantities showed a gradual increase as the temperature was lowered in the normal and supercooled liquid states. The behaviour of the electrical resistivity, shown in Fig. 6.52, is interesting for the present discussion, since it shows a rapid rise on cooling below about 1320 K and 1300 K for the supercooled liquid and solid, respectively. The

125

6.5 Cobalt palladium alloys 160 Onset of magnetic ordering

r l (T )fit = 146 + 0.050 (T –1613) [μΩ.cm]

Resistivity, μΩ.cm

150

140

130

linear fit

120

110

r s(T )fit = 122 + 0.042 (T –1563) [μΩ.cm]

linear fit

100 1200

Tsol 1300

1400

1500

Tliq

1600

1700

Temperature, K

Fig. 6.52. Electrical resistivity of Co80Pd20 in the solid (lower curve) and liquid (upper curve) states, together with the corresponding linear fits as function of the temperature. Tsol and Tliq indicate the solidus and liquidus temperatures, respectively. The sharp rise in both curves below 1300 K is ascribed to the onset of magnetic ordering (Loho¨fer et al., 2001).

authors associate this behaviour with the onset of spin flip scattering, which is not taken into account in the derivation of the resistivity with the applied inductive technique. Additional evidence for magnetic ordering is provided by specific heat measurements by Wilde et al. (1996a) over an extended range of supercooling. The samples were cooled slowly in a differential heat-flow calorimeter. To avoid premature crystallization due to heterogeneous nucleation on container walls and thus achieve maximum undercooling, the ingots of Co80Pd20 were fluxed with Duran glass. An undercooling of 347 K was, achieved by this method, comparable to the levitation experiments already referred to. The measured curve of Cp(l), shown in Fig. 6.53, is almost constant in the temperature interval of the stable liquid and in the adjacent undercooling range. In the range of deepest undercooling, however, the specific heat shows a cusp-like rise starting at about 1270 K when the undercooled liquid approaches the temperature regime of the Curie point of the crystalline state. Similar behaviour was found for a crystalline sample, whereas the specific heats of Ni75Pd25, Fe80Pd20 and Co50Pd50 showed essentially constant behaviour in both states over comparable temperature ranges. Several attempts have been made to search for structural changes at the apparent magnetic ordering transition. Jacobs & Egry (1999) made EXAFS

126

Liquid metals and alloys 1.0

Specific Heat [J/gK]

0.8

liquid

0.6

0.4

crystalline Tl

Tc Ts

0.2

0 1150

Co80Pd20 ⌬Hf = 177 J/g

1250

1350

1450

1550

1650

Temperature [K]

Fig. 6.53. The specific heat in the liquid and crystalline states of Co80Pd20. DHf is the heat of fusion of the alloys. Both curves show a significant cusplike rise around the Curie temperature of the crystalline phase (Wilde et al., 1996a).

measurements at the cobalt edge on both liquid and solid Co80Pd20. It turned out to be impossible to make measurements in the supercooled liquid below 1293 K because of the tendency to solidify in the EXAFS setup. However, they were able to obtain structural data over a wide temperature range in both liquid and solid phases. Results for the mean Co Co and Co Pd distances are shown in Fig. 6.54. Interestingly, the Co Co distance decreases on melting while the Co Pd increases slightly. In the liquid, the two distances are 2.69  0.02 A˚ and 2.52  0.01 A˚ and neither shows a significant change with temperature, although the variances in the two distances associated with thermal motions decrease steadily with decreasing temperature. Krishnan et al. (1999) made X-ray diffraction measurements down to 1480 K. The first peak in the pair correlation function represents a weighted average of the Co Co and Co Pd distances and was found at 2.58  0.03 A˚ just above the liquidus, consistent with the EXAFS results. Assuming that Co and Pd atoms are randomly distributed about each Co or Pd, an assumption also made in the EXAFS analysis, the Co80Pd20 weighted average pair correlation function can be predicted from those for the pure liquid metals. This is shown in Fig. 6.55 and seen to agree quite well with the X-ray measurement. Co Pd melts also present an appropriate vehicle for the study of hypercooling, i.e. supercooling beyond the hypercooling limit DThyp, defined in Eq. (5.1). Miscible alloys with a large difference in atomic radii produce crystalline lattices with large strain lattices, so that the enthalpy of formation

127

6.5 Cobalt palladium alloys 2.9 Co–Co, liquid Co–Pd, liquid Co–Co, solid Co–Pd, solid

Neighbour distance R [Å]

2.8

2.7

2.6

2.5

2.4 Tl

2.3 0

500

1000 Temperature [°C]

1500

Fig. 6.54. Mean Co–Co and Co–Pd interatomic distances measured by EXAFS in liquid and solid Co80Pd20. T‘ marks the liquidus temperature (Jacobs & Egry, 1999).

Pair distribution function, G (r )

2.5 Liq. Co–Pd Weighted G (r )

2

1.5

1

0.5

0

–0.5

0

2

4

6

8

10

Radial distance, r (Å)

Fig. 6.55. Measured X-ray weighted average pair distribution function for liquid 80% Co – 20% Pd at 1480 K compared with the weighted average of the pair distribution functions of pure Co and Pd (Krishnan et al., 1999).

of the solid solution exceeds the enthalpy of melting in the liquid (Wilde et al., 1996b). Thus, the values of DHf for such alloys, and hence those of DThyp, are considerably depressed compared with those of the ideal solutions. The left and right sides of Fig. 6.56 show the values obtained by Wilde et al. (1996b)

128

Liquid metals and alloys 1600

17

Co(1–x)Pdx

16

Temperature [°C]

Melting Enthalpy [kJ/g-atom]

18

Co(1–x) Pd x

15 14 13 12

1400 Tf = 125°C

1200 1000

11 10

0

0.6 0.4 Pd - concentration x

0.2

0.8

800

1

0

0.2

0.4

0.6

0.8

1

Pd-concentration x

Fig. 6.56. (Left) Melting enthalpy of Co–Pd alloys as a function of composition: the points joined by the dashed line indicate the measured values whereas the solid line represents the calculated values assuming ideal solution behaviour; (right) phase diagram of the Co–Pd system: the points joined by the dotted line represent the calculated hypercooling while the points joined by the dashed line indicate the lowest crystallization temperatures obtained experimentally. The solid lines represent measured solidus and liquidus temperatures (Wilde et al., 1996b). 1700 Co50Pd50

Temperature [K]

1600 1500 1400

residual solidification

Tm ⌬T

Tpl < Tm

recalescence

TN ⌬T

1300 Thyp 1200

TN

10 s Time

Fig. 6.57. Temperature–time profiles of undercooling experiments on levitated drops of Co50Pd50 for an undercooling less than the hypercooling limit (left-hand side) and greater than the hypercooling limit (right-hand side); Tm: melting temperature; TN: onset of nucleation, leading to solidification; Thyp: hypercooling limit; Tp1: post-recalescence temperature (Volkmann et al., 1998).

for DHf and DThyp, respectively, for Co Pd alloys fluxed with Duran glass. It is clear that hypercooling is achieved over the range of 15 50% Pd. Volkmann et al. (1998) studied hypercooling of Co Pd alloys with EML. Figure 6.57 shows a schematical demonstration of the different solidification behaviours of Co50Pd50 samples that crystallize above and below the

129

6.5 Cobalt palladium alloys 45

Growth velocity [m/s]

40 35

Co80Pd20 collision-limited dendritic growth

30 25 20

⌬Thyp

15 10 5 50

100

150 200 250 Undercooling [K]

300

350

Fig. 6.58. Dendrite growth velocity v as a function of the undercooling DT for Co80Pd20 alloy. Experimental results are represented by the dots, while the solid line gives the prediction of dendrite growth theory (Volkmann et al., 1998).

hypercooling limit. In the first case, the primary crystallization takes place under non-equilibrium conditions during the recalescence period, in which the rapid release of the heat of fusion DHf leads to a rapid rise in temperature up to the melting temperature. In the second step, characterized by the temperature plateau in the temperature time curve, the remaining liquid solidifies under equilibrium conditions at the melting temperature. The situation is quite different in the second case when the supercooling is extended beyond the hypercooling limit. Volkmann et al. were able to accomplish this by placing a nozzle just below the levitated sample and forcing a helium gas stream through it. This gas stream removes heat and also provides an additional levitation force, reducing the RF power needed for levitation and thus the RF heating of the sample. If the sample can be cooled in this way below the hypercooling limit given by Eq. (5.1), the heat of fusion DHf is not sufficient to heat the supercooled melt back up to the melting temperature and so the re-melting processes under equilibrium conditions are avoided. As discussed earlier, the crystal growth velocity can also be measured in levitation experiments. Volkmann et al. used an observation system that imaged a part of the sample surface onto the surface areas of a rapidly responding photodiode. The growth velocity is then given by v ¼ Ds/Dt, where Ds is the path taken by the solidification front in time Dt. Results for Co80Pd20 are shown in Fig. 6.58. It can be seen that the values of v are on the

130

Liquid metals and alloys

order of 3 30 m s 1, three orders of magnitude faster than those found in the Al Cu Co alloys discussed in the previous section. The measured values of v are in agreement with dendritic growth theory (Trivedi & Kurz, 1994) at lower supercooling but not when the hypercooling limit is approached. There appears to be some additional mechanism setting in that slows down the interface kinetics.

7 Molten semiconductors

In this chapter we discuss some elements that are semiconducting in the solid state and that, because of their relatively high melting points and interest in their supercooled liquid states, have been the subject of investigation with levitation techniques. The first materials to be discussed silicon, germanium and their alloys in fact melt into metals, albeit, as we shall see, metallic liquids with quite unusual properties. 7.1 Silicon Silicon crystal growth and crystal properties are important in the semiconductor industry. Silicon is the host material for the majority of semiconductor applications, and the properties of its crystalline, amorphous and liquid phases are of substantial interest. The atomic structure, electrical, optical and thermophysical properties of the liquid phase are key factors that determine the quality of crystals grown from the melt. Mito et al. (2005) have made numerical simulations of Czochralski growth of silicon crystals and found that the Peclet number and deflection of the melt crystal interface, two important parameters in the growth process, are highly sensitive to the emissivity, thermal conductivity, temperature coefficient of surface tension in the liquid as well as the emissivity and thermal conductivity of the crystal. A graphic representation of the importance of thermophysical properties of the liquid for numerical modelling of crystal growth is given in Fig. 7.1. The conductivity and other electrical transport properties of liquid Si and Ge have been measured by several authors in contained samples. Relatively recent results obtained by Schnyders & Van Zytveld (1996) for Si contained in high-density graphite and Ge contained in high-density alumina are given in Table 7.1. The values for the conductivity are in line with previous determinations, while those for its temperature dependence and thermopower are 131

132

Molten semiconductors Viscosity Thermal diffusivity

5

Density

4

Electrical conductivity

Expansion coefficient

3 Contact angle (crystal)

Surface tension

2 1

Contact angle (crucible)

Thermal conductivity

0 Vapour pressure

Heat capacity

Melting point

SiO gas−liquid equilibrium

Spectral emissivity

Hemispherical total emissivity

Diffusivity (B,P)

Heat of fusion Diffusivity (O)

Fig. 7.1. Importance of thermophysical properties of liquid silicon required for numerical modelling of silicon crystal growth processes (Kawamura et al., 2005). Table 7.1. Experimental values for the electrical conductivity  and thermopower S, and their temperature dependences, for liquid Si and Ge. s(Tm) ds/dT S(Tm) dS/dT Sample [Ref.] Environment Tm (K) (104 O 1cm 1) (O 1cm 1K 1) (mV.K 1) (mV.K 2) Sia Sib Gea Geb

Graphite CNL Alumina CNL

1683 1211

1.33  0.01 1.79  0.06 1.50  0.005 1.53  0.07

5.4  0.4

0.3  0.5 0  5

3.4  2

2.1  0.6 9  5

Notes: a Schnyders & Van Zytveld (1996); b Enderby et al. (1997).

probably superior to earlier work in the literature. Enderby et al. (1997) measured the conductivity of liquid Si and Ge with CNL using the technique described in Section 4.1. Figure 7.2 shows the behaviour of the inductance L and quality factor Q of the resonant coil after the laser beam was switched off. The results for the conductivity obtained from these data, also shown in Table 7.1, are in reasonable agreement with those on contained samples.

133

7.1 Silicon 0.12

(dL /L)%

0.09

0.06

0.03

liquid solid solid/liquid

(dQ /Q)%

2

1

0

–1

–2

0

5

10

15 20 Time (s)

25

30

35

Fig. 7.2. Relative changes in the inductance L and quality factor Q induced by insertion of a silicon sphere into the resonant coil. The radius a of the sphere was 1.62 mm, the filling factor  ¼ 0.80, the frequency o/2p ¼ 500 kHz and Q ¼ 71.9 (Saboungi et al., 2002).

The optical properties of liquid silicon have been measured by several authors on both contained and levitated samples. Results of an early measurement with EML made by Krishnan et al. (1991a) are shown in Fig. 7.3. It can be seen that the normal spectral emissivity el falls with increasing temperature in the solid, drops sharply on melting and increases slowly with temperature in the liquid. The refractive index n drops sharply on melting while the extinction coefficient k jumps considerably. This behaviour is quite consistent with the transition from a semiconducting to a metallic state on melting. More recent measurements by Kawamura et al. (2005) are largely consistent with these results, except that they obtain a higher value for el: 0.23, independent of temperature to within experimental uncertainty. They made measurements with different incident wavelengths and found a gradual

134

Molten semiconductors 1

(b) 10

0.9

9 Refractive indices at 633 nm

Normal emissivity at 633 nm

(a)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Tm

8 7

k

6 5

n

4 3

n k

2

Tm

1 0 1000

0 900 1100 1300 1500 1700 1900 2100 Temperature (K)

1200

1400 1600 1800 Temperature (K)

2000

Fig. 7.3. Optical properties of high-temperature solid and liquid silicon in EML samples at a wavelength of 632.8 nm. Left: normal spectral emissivity; right: refractive index n and extinction coefficient k (Krishnan et al., 1991a).

decrease in el with increasing wavelength. They compared this behaviour with the Drude theory of absorption solely by conduction electrons, in which the real and imaginary parts of the dielectric function are given by the following equations: e1 ðoÞ ¼ n2 þ k2 ¼ 1 e2 ðoÞ ¼ 2nk ¼ 1

o2p t2 1 þ o2 t 2 o2p t

ð7:1Þ



; o 1 þ o2 t 2

where o is the angular frequency of the electric field, t is an appropriate relaxation time and op is the plasma frequency given by o2p ¼

Ne2 m e 0

ð7:2Þ

with N being the number of free electrons per unit volume, m* the effective mass, e the charge on the electron and e0 the permittivity of vacuum. For an opaque material in thermal equilibrium the emissivity is expressed in terms of the optical constants according to Eq. (4.17). Estimating t from the electrical conductivity according to the relation ¼

Ne2 t ; m

ð7:3Þ

135

Normal Spectral Emissivity, e

7.1 Silicon 0.3

0.2

0.1

0

600

800

1000

1200

1400

1600

Wavelength, l /nm

Fig. 7.4. Comparison between the data for the measured spectral emissivity averaged over the temperature range 1553–1797 K (solid line) and values calculated from the Drude free-electron model at the melting point 1687 K (·-·-·) and1797 K (· · ·) (Kawamura et al., 2005).

putting m* equal to the free electron mass and using literature values for the liquid density and conductivity, the authors obtained good agreement with their measured values, as shown in Fig. 7.4. The Drude theory also predicts a slow rise with increasing temperature, as observed by Krishnan et al. in the work cited above. There have been several investigations of the thermophysical properties of the normal and supercooled liquid phases. While some of these have suggested anomalous density changes, more recent work on levitated samples shows little or no anomalies in the temperature dependence of the density vs. behaviour in the liquid and supercooled regimes, suggesting effects of container contamination or problems with the Archimedean technique in the earlier work, as pointed out by Terashima & Kanno (2001). Zhou et al. (2003) made some of the more recent measurements of thermophysical properties of molten silicon with ESL. Their measurement for the density, shown in Fig. 7.5, showed a nonlinear behaviour, i.e. a downward curvature in addition to a downward slope, over a temperature range of 1370 1830 K. They suggested that this might imply either a structural change or clustering on heating the supercooled liquid. The value of the density at the melting point (Tm ¼ 1687 K), d(Tm), was 2.583 g.cm 3, in reasonable agreement with previous determinations with ESL. The thermal expansion coefficient, defined by ¼

1 @V ; V @T

ð7:4Þ

was equal to 0.77.10 4 K 1 at Tm. Langen et al. (1998) and Higuchi et al. (2005) made similar measurements with EML and, with a smaller number

136

Molten semiconductors 2.64 Experimental data

2.63

Fitting curve

2.62

Density (g /cm3)

2.61 2.60 2.59 2.58 2.57 Tm 2.56 2.55 2.54 1300

1400

1500

1600

1700

1800

1900

Temperature (K)

Fig. 7.5. Temperature-dependent density of molten silicon (Zhou et al., 2003).

of data points, obtained d(Tm) ¼ 2.52 and 2.579 g.cm 3, respectively, and a(Tm) ¼ 1.1·10 4 and 0.8·10 4 K 1, respectively. It is seen that the more recent results of Higuchi et al. are in better agreement with the ESL data. None of the EML data display the nonlinearity found in the ESL results, although there are fewer data points over a more limited temperature range (1500 1800 K). Furthermore, recent AIMD simulations by Morishita (2006) in the deeply supercooled region support the nonlinear behaviour and find a broad density maximum at about 1200 K. On the other hand, Chathoth et al. (2008) in microgravity experiments performed on a zero-g aircraft with samples positioned with a low magnetic field and melted by laser found d(Tm) ¼ 2.52  0.01 g.cm 3 and a(Tm) ¼ (1.37  0.02)·10 4 K 1 with a high density of data points over a significant temperature range, 1530 2000 K, in better agreement with the earlier EML results of Langen et al. Furthermore they did not observe any nonlinearity in the behaviour of d(T). Inatomi et al. (2007) made measurements with EML under a static magnetic field and obtained d(Tm) ¼ 2.51 g.cm 3 with a(Tm) ¼ 1.08·10 4 K 1 with a very small nonlinearity, 10 4 K 1. The surface tension of molten silicon has also been studied by a variety of methods. Zhou et al. report results with ESL in the work already cited. They found a linear decrease with temperature with the value

137

7.1 Silicon

g(Tm) ¼ 0.72  0.02 N.m 1 with the temperature coefficient dg=dT ¼ 0.615· 10 4 N.m 1.K 1. Przyborowski et al. (1995) had earlier carried out EML experiments and obtained g(Tm) ¼ 0.78  0.02 N.m 1 with a much higher absolute value of the temperature coefficient, 6.5·10 4 N.m 1.K 1. Chathoth et al. in the work already cited give a value about 0.75 N.m 1 but do not give the temperature dependence. Fujii et al. (2006) obtained g(Tm) ¼ 0.733 N.m 1 with dg=dT ¼ 0.62·10 4 N.m 1.K 1 with the oscillating drop technique in a microgravity environment achieved using the 710 m drop-shaft facility at the Japan Microgravity Centre (JAMIC). They also made measurements in the normal liquid with a sessile drop apparatus (shown earlier in Fig. 4.8), which were consistent with the oscillating drop result but less precise. Przyborowski et al. suggest that the presence of surface impurities may lower both the surface tension and the absolute value of the temperature coefficient and could explain why their values were higher than some previous determinations. The effect of oxygen impurities was studied in detail by Mukai et al. (2000) using the sessile drop technique, finding that g(Tm) fell from a value of 0.83 N.m 1 at zero P(O2) to 0.77 N.m 1 for P(O2) ¼ 10 14 MPa, while |dg/dT| fell from 7.5·10 4 to 0.15·10 4 N.m 1.K 1 over the same interval. Millot et al. (2008), reporting recent measurements made with CNL, point out that these objections may not apply to contactless experiments where appropriate care is taken with sample purity, and that the sessile drop technique may give high values with certain substrates. Figure 7.6 shows that

surface tension (mN/m)

900

800

700

600

500 1500

2000 Temperature (K)

2500

Fig. 7.6. Surface tension of molten silicon obtained with various contactless techniques. Symbols: circles, CNL (Millot et al., 2008); squares, drop shaft (Fujii et al., 2006); triangles, ESL (Zhou et al., 2003) (Millot et al., 2008).

138

Molten semiconductors 1

Present results Rhim et al. Sasaki et al. Linear fitting

0.9

Viscosity (mPa⋅s)

0.8 0.7 0.6 0.5 Tm 0.4 0.3 1600

1650

1700

1750

1800

1850

1900

1950

Temperature (K)

Fig. 7.7. Viscosity of molten silicon as a function of temperature measured by Zhou et al. (2003, closed circles) compared with the earlier results of Rhim & Ohsaka (2000, squares) and Sasaki et al. (1995, triangles) (Zhou et al., 2003).

three of the contactless measurements discussed do in fact give results in good agreement with each other. Viscosity measurements on liquid silicon are also prone to uncertainties. Figure 7.7 shows the results obtained by Zhou et al. compared with some earlier measurements of Rhim & Ohsaka (2000) and Sasaki et al. (1995). Both Zhou et al., as already discussed, and Rhim & Ohsaka used the oscillating drop technique in conjunction with ESL, whereas Sasaki et al. used the oscillating cup method in which a cylindrical cup containing the liquid sample is suspended by a wire; an instantaneous torque is applied to the cup so that it oscillates about the vertical axis, and the viscosity deduced from the damping of the motion. Sato et al. (2003), who also used the oscillating cup method, found that their results were dependent on the crucible used. Zhou et al. explain their discrepancy with Rhim & Ohsaka on the basis that the latter did not account for the sample mass loss during the measurement. Studying the free cooling of a levitated drop and assuming the losses to be purely radiative, Zhou et al. found the ratio of specific heat to hemispherical total emissivity, Cp/eT, to be 149  7 J.mol 1.K 1 at the melting point, with a negative slope and upward curvature with increasing temperature. Taking the literature value for Cp, 25.61 J.mol1.K 1, gives eT ¼ 0.17 at the melting point, with a temperature dependence similar to that of Cp/eT.

139

7.1 Silicon 100 0.5 T

k / Wm–1 K–1

80

1.0 T

FP1

60

FP3 LF 1, 2

2.0 – 4.0 T

LF3

TW

FP4

FP2

40

HD Melting point

20 1650 1700 1750 1800 1850 1900 1950 2000 2050 Temperature / K

Fig. 7.8. Thermal conductivity of liquid silicon as a function of the temperature in static magnetic fields of 0.5 T (diamonds), 1.0 T (crosses), 2.0 T (squares), 3.0 T (triangles, and 4.0 T (circles), together with values reported previously measured with various methods (Kobatake et al., 2007).

The thermal conductivity of liquid silicon has been measured by Kobatake et al. (2007), using a magnetic field to suppress convection currents as described in Section 4.2. Their results are shown in Fig. 7.8. The averaged results with magnetic fields of 2 4 T appear to be most reliable and give k ¼ 62  3 W.m 2.K 1. The results obtained using the Wiedenmann Franz law with reported values of the electrical conductivity, indicated FP1 4 in the figure, show that the thermal conduction is principally electronic, as expected from its metallic character. Inatomi et al. (2007) obtained 67 W.m 2.K 1 with a similar technique. The structure of normal and supercooled liquid silicon has been studied extensively using X-ray and neutron scattering. Waseda & Suzuki (1975) reported some of the first X-ray measurements on the liquid and showed that the structure of liquid silicon resembled that of white tin (A5 structure) with a well-defined shoulder on the high-Q side of the first peak in S(Q) and a first shell coordination of about 6. However, these studies were confined to temperatures above the melting point because these experiments and the later ones of Waseda et al. (1995) were performed on contained samples. A majority of the structural features observed for the liquid were satisfactorily reproduced by the AIMD simulations of Stich et al. (1989a,b; 1996) using a local density approximation. This pioneering work established the basic characteristics of the nearest-neighbour structure and also suggested a mean first-shell coordination of 6 at temperatures just above the melting point. Angell et al. (1996) performed conventional MD calculations using the

140

Molten semiconductors

Stillinger Weber (S-W) potential (Stillinger & Weber, 1985) and their structural results on liquid silicon were in satisfactory agreement with the experimental observations. Silicon can be prepared as an amorphous semiconductor that melts into the liquid metallic state. The possibility of the reverse transition has been the subject of considerable speculation. The existence of an intermediate phase, which might facilitate such a transition, was first suggested by Aptekar (1979) on thermodynamic grounds, with an additional phase transition around 1500 K. In the work just cited, Angell et al. made a similar prediction that has driven much of the recent interest in supercooled liquid silicon: their simulations showed a decrease in the first-shell coordination with supercooling and, with deep supercooling, a first-order transition to a tetrahedrally coordinated amorphous state, the first-shell coordination dropping discontinuously from 4.6 to around 4.2. This suggestion received support from the X-ray diffraction measurements of Ansell et al. (1998, 1999) that extended into the supercooled region and showed a coordination number and interatomic distance that decreased with decreasing temperature. These results were limited to a modest degree (140 K) of supercooling, considerably smaller than the 340 K that can be typically achieved in the laboratory. Subsequently, Angell & Borick (1999) argued that the phase just below the transition was not the amorphous solid but a viscous liquid that transformed into the solid amorphous phase at lower temperature. The transition was therefore a liquid liquid transition (LLT) from a high-density (HDL) to a low-density liquid (LDL) that they proposed takes place at a temperature around 1345 K, the experimentally observed supercooling limit (Fig. 7.9). This value is consistent with that of 1420 K obtained from calorimetric studies of heating amorphous silicon (Donovan et al., 1985) and 1480  50 K from transient conductance measurements with laser heating (Thompson et al., 1984). In MD simulations with the S-W potential, Luedtke & Landman (1988) observed the amorphous-liquid transition taking place at 1435 K at a very fast heating rate (1010 K.s 1) and 1082 K at a very slow rate in which the simulated sample was brought into equilibrium at successive small temperature steps. Subsequently Luedtke & Landman (1989) found the reverse transition at 1060 K. The temperature for the liquid amorphous transition obtained from the simulations of Angell & Borick was also 1060 K. The viscosities of the LDL and HDL calculated by Deb et al. (2001) are shown in Fig. 7.10. The two values of the interaction parameter W in the twofluid model discussed below both yield calculated  T relations that are consistent with the measurements of Rhim & Ohsaka discussed above. The closed squares correspond to a value of W that gives the HDL over the entire

141

7.1 Silicon

AVERAGE COORDINATION NUMBER

(a) 8 7

Liquid

6 5

340 K

Amorphous

s.-c. limit

4

exp Crystal

sim Tl -l

3 0

500

1000

exp

Tm, a-Si

1500

exp

Tm, c-Si 2000

2500

TEMPERATURE (K) (b) DENSITY (in atoms per cubic angstrom)

0.056 glass (hyperquenched)

0.054 liquid

0.052 amorphous

0.050 crystal

Tl -l

Tm

0.048 0

500

1000

1500

2000

2500

TEMPERATURE (K)

Fig. 7.9. (a) Nearest-neighbour coordination numbers of liquid Si as a function of temperature from simulations using the Stillinger–Weber potential (closed circles), compared with experimental values obtained from XRD measurements on levitated liquid samples (open circles). The experimentally observed melting temperature of a-Si and the experimental and simulated equilibrium melting points of crystalline Si, which are the same within simulation uncertainty, are indicated by arrows. Tllsim denotes the temperature of the LLT obtained from the simulations, somewhat lower than the actual one, postulated to be close to the experimental supercooling (s.c.) limit, also shown. (b) Density dependence of normal and supercooled liquid silicon simulated with the Stillinger–Weber potential, showing the density maximum at 1350 K, together with that observed in the crystal (Angell & Borick, 1999).

142

Molten semiconductors 12 W = 26 600 (LDA)

log h (Pa s)

8

calculated L-L

4

W = 13 350 (HDA)

Viscosity data

0

–4 4

5

6

7 10 000/T (K)

8

9

10

Fig. 7.10. Calculated viscosities of hypothetical HDL and LDL silicon liquids. Two values of the interaction parameter W were used to model the behaviour of silicon liquids over the entire temperature range: both yield calculated  T relations that are consistent with experiment above 1650 K. The closed squares correspond to a value of W that give the HDL over the entire temperature range, the open circles to one that gives an LLT. The top of the frame corresponds to a viscosity normally assigned to the glass transition, so all points within the frame represent a liquid state (Deb et al., 2001).

temperature range, the open circles to one that gives a first-order transition to an LDL phase on cooling. A clear demonstration of the existence of a first-order LLT in a numerical simulation with the S-W potential was given by Sastry & Angell (2003), who found a non-monotonic dependence of enthalpy on temperature resulting from the latent heat of transformation (Fig. 7.11). By studying the intermediate scattering function I(Q, t) they showed that the LLT marked a change in the dynamic character from a fragile liquid to a strong liquid, as observed in supercooled confined water (Liu et al., 2005) and other tetrahedrally coordinated liquids. Results for the diffusivity D (Fig. 7.12) found a non-Arrhenius temperature dependence above the LLT, another characteristic of a fragile liquid. It can be seen from the same figure that the value of D falls by two orders of magnitude at the transition but remains clearly in the liquid range (10 8 cm2s 1). Subsequent numerical simulations with a reversible scaling method by Miranda & Antonelli (2004) found a weak first-order LLT at 1135 K and a continuous liquid amorphous solid transition at 843 K. However, Beaucage & Mousseau (2005) found that small changes in the Stillinger Weber potential, e.g. increasing the strength of the three-body force by 5%, could suppress the existence of the low-density liquid.

143

7.1 Silicon Constant P and H Constant P and T, above transition Constant P and T, below transition

Enthalpy (kJ mol–1)

360 900

1500

2100

320

Crystal melting

370 360

400 380 1020

1120

1220

1320

1420

T(K )

Fig. 7.11. Main panel: plot of enthalpy against temperature, from constant pressure–constant enthalpy simulations as well as constant pressure–constant temperature simulations for supercooled liquid Si above and below the LLT. The non-monotonic dependence of the enthalpy indicates the presence of a first-order phase transition. Inset: crystal–liquid transition, shown for comparison. The non-monotonic loop in this case indicates the liquid– crystal first-order melting transition (Sastry & Angell, 2003).

D (cm2 s–1)

10–4 10–5 10–6 10–7

power-law fit VFT fit low-T liquid

5

7

9

104 /T(K–1)

Fig. 7.12. Diffusivity of liquid Si, above and below the LLT. In the hightemperature liquid, the diffusivity show a strongly non-Arrhenius temperature dependence characteristic of a fragile liquid. Below, the liquid displays a diffusivity of 6.4·10 8 cm2s 1 (filled square), roughly two orders of magnitude lower than in the high-temperature liquid (Sastry & Angell, 2003).

144

Molten semiconductors

Coordination number Nc

6.4 6.2 6 5.8 5.6 5.4 5.2 5 1000

1200

1400

1600

1800

Temperature (K)

Fig. 7.13. Temperature dependence of the nearest-neighbour coordination number Nc in liquid Si from the AIMD simulations of Morishita (2006) (closed circles) and Jakse et al. (2003) (closed triangles). Experimental results are also shown: Ansell et al. (1998, open triangles), Kimura et al. (2001, open diamonds) and Kim et al. (2005, open squares). The experimental results of Jakse et al. lie very close to their AIMD results (Morishita, 2006).

The use of AIMD simulations should in principle provide a more reliable prediction of the behaviour of the liquid as it undergoes large changes in density at the transition. Morishita (2006) found a significant drop in coordination with decreasing temperature, but the drop does not begin to set in until 1300 K, in contrast to the AIMD results of Jakse et al. (2003), both shown in Fig. 7.13 (the experimental results denoted by the open symbols are discussed below). Morishita suggested that the contribution of covalent bonding was exaggerated in the work of Jakse et al., while Jakse & Pasturel (2007b) ascribed the disagreement to the quenching rate that was two orders of magnitude higher in the Morishita simulations. Jakse & Pasturel studied the properties of the liquids above and below the LLT and found an enthalpy difference similar to that obtained with the S-W potential, confirming the first-order nature of the transition. Their values for the diffusivity were 9.2·10 6 and 4.8·10 7 cm2s 1 above and below the LLT, respectively, a decrease by a factor 20. Morishita (2005) found that the diffusivity increases with pressure, anomalously, below the LLT, but obtained a difference of only a factor 3 between the low-density liquid at 1100 K and the high-density liquid at 1500 K at ambient pressure. Discrepancies are also found in the predictions of electronic properties: Ganesh & Widom (2009) find that the low-density liquid is a semimetal with a pseudogap in the electronic density of

7.1 Silicon

145

states, while in the simulations of Jakse & Pasturel it remains a liquid metal. Given the uncertainties associated with the simulations, it appears important to resolve the experimental situation, discussed below, and, if at all possible, extend the measurements closer to the predicted transition. Liquid Si can undergo another LLT at high pressure that has been observed in XRD experiments by Funamori & Tsuji (2002) and investigated with AIMD simulations by Delisle et al. (2006). As the pressure is increased between 8 and 14 GPa with the temperature increased to stay on the liquid side of the melting curve, the shoulder on the first peak of S(Q) disappears and the coordination number increases to a value around 9, similar to the structure of liquid tin. This ‘very high density liquid’ (VHDL) appears similar to a highpressure solid amorphous phase of silicon found in AIMD simulations, which has a coordination number of 8.6 (Durandurdu & Drabold, 2002). All these studies suggest that the HDL VHDL transition and the corresponding HDA VHDA transition in the solid amorphous state are continuous rather than first-order. Benmore et al. (2005) have pointed out the structural similarities of the LDA, HDL and VHDL forms of Si and Ge (discussed below) with those of water and amorphous ice and suggest that these patterns may be characteristic of tetrahedrally coordinated liquids in general. The difficulty of reaching a sufficient degree of supercooling to observe the HDL LDL transition experimentally can be avoided by studying the analogous LDA HDA transition at high pressure. The relation between the two systems can be seen schematically in Fig. 7.14(a), which shows the dotted line representing the transition terminating in a critical point Tc, expected to be at a temperature close to a maximum in the melting curve (McMillan, 2004), both following from a two-fluid model with low- and high-density components (Rapoport, 1967) when the liquid develops a double minimum in its free energy of mixing between the two components. In Si, this is predicted to occur at a negative pressure, so the situation shown in Fig. 7.14(b) would apply. On thermodynamic grounds, the transition temperature is expected to decrease with increasing pressure (Poole et al., 1997). The pressure at which it reaches ambient temperature in Si is around 14 GPa, established by Raman spectroscopy and optical observations by Deb et al. (2001). The conductivity of the samples studied in these experiments increased by a factor of about 200 at the transition (McMillan et al., 2005). XRD measurements as a function of pressure (Daisenberger et al., 2007; McMillan et al., 2007) showed similar trends to those found in the liquid by Funamori & Tsuji. However, as pointed out by McMillan et al. (2007), correlating the transition in the amorphous phase with that postulated for the liquid involves taking into account the glass transition in both the LDL and HDL, which may have different

146

Molten semiconductors (a)

T

Liquid (2-state)

(b)

T

Tc

Tc HDL LDL Tg(A)

Tg(B) LDA

LDA

HDA P

–P

HDA +P

Fig. 7.14. Generalized P–T phase diagrams showing the development of liquid–liquid transitions (LLT) at constant composition. (a) A maximum in the melting curve can take place if the liquid contains two fluctuating components with different densities. If crystallization is bypassed on cooling, a critical point Tc is encountered below which the liquid develops a double minimum in its free energy of mixing of the two components, causing separation into low- and high-density liquids (LDL, HDL), separated by a first-order phase transition line. The LDL and HDL liquids are generally expected to have different glass transitions, Tg(A) and Tg(B), indicated schematically as dotted ranges. Below Tg, the supercooled liquid phases transform into the LDA and HDA amorphous solid states. (b) In cases for which a negative initial melting slope is observed, such as Si, Ge and H2O, the maximum in the melting curve and a critical point Tc are expected to occur at negative pressure, i.e. in a metastable tensile regime. In this case an LLT could take place at ambient pressure if a sufficient degree of supercooling could be achieved (McMillan, 2004).

characteristics in the two phases. A glass liquid transition has been observed in the low-density phase by Hedler et al. (2004) by bombarding a solid amorphous sample with energetic heavy ions. A liquid glass transition has been observed in the high-density phase of Ge, to be described in the next section. The prediction of a drop in coordination number with supercooling and its implications for a possible LLT stimulated further diffraction studies of supercooled liquid silicon. Kimura et al. (2001) performed energy-dispersive X-ray diffraction measurements with EML and reported structure factor and pair correlation functions that qualitatively confirmed the predominant features observed in the previous measurements. However, they obtained values of the first-shell coordination number that were less than 5 at high temperatures in the liquid and approached 6 as the liquid approached 1400 K. This trend in the temperature dependence of the coordination number was opposite that obtained by Ansell et al. and the value at elevated temperatures (4.9) was in considerable disagreement with those obtained in previous experimental studies and simulations, which are around 6 or higher.

147

7.1 Silicon 5.0 4.5

T = 1767 K

4.0 3.5 T = 1667 K S (Q )

3.0 2.5 2.0

T = 1542 K

1.5 1.0 T = 1458 K 0.5 0.0 0 1 2 3 4 5 6 7 8 9 10 11 12 Q (Å–1)

Fig. 7.15. Structure factor S(Q) for liquid silicon from AIMD simulations (solid lines) and XRD experiments (open circles). The equilibrium melting point is 1685 K. The curves for T ¼ 1542, 1667, and 1767 K are shifted upwards by 1.25, 2.5 and 3.75, respectively (Jakse et al., 2003).

Higuchi et al. (2005) also used EML and obtained coordination numbers between 5.0 and 5.2 with no systematic dependence on temperature, while Kim et al. (2005) used ESL and obtained an essentially constant value of 6.0 over the temperature range studied, 1400 1800 K. Meanwhile, Jakse et al. (2003) carried out new measurements using CNL on normal and supercooled liquid silicon over a temperature range from 1767 K in the normal liquid down to 1458 K (a supercooling of 227 K), using a new experimental setup at the Advanced Photon Source (APS) at Argonne. The results confirmed the finding of Ansell et al. (1998) of a decrease in coordination number on supercooling, and the observed temperature dependence of the coordination number also agreed well with their AIMD simulation. The structure factors measured by Jakse et al. at one temperature in the normal liquid and three in the supercooled liquid are compared with their corresponding AIMD results in Fig. 7.15. The shoulder observed on the high-Q side of the first peak is well defined and sharpens considerably with supercooling. This peak the highest one in amorphous Si can be viewed as a signature of the tetrahedral structure. As the temperature decreases in the

148

Molten semiconductors

g (q ) (arbitrary units)

11 10 9 8 7

20

40

60

80 100 120 140 160 180 q (deg)

g (r )

6 5

T = 1767 K

4 T = 1667 K

3 2

T = 1542 K

1 T = 1458 K

0 1

2

3

4

5

6

7

8

r (Å)

Fig. 7.16. Pair-correlation function g(r) for liquid silicon: notation as in Fig. 7.15. The curves for T ¼ 1542, 1667, and 1767K are shifted upwards by 1.5, 3, and 4.5, respectively. The inset represents the bond-angle distributions g(3)(y) at T ¼ 1458 K (solid line) and T ¼ 1767 K (dashed line). The thin lines represent the corresponding tetrahedral component of g(3)(y) calculated with a smaller cut-off, close to the covalent bond length (Jakse et al., 2003).

supercooled regime, the intensity of the first peak grows and the shoulder becomes resolved into a second peak, but their positions remain essentially unchanged. In the AIMD results, the main features of the experimental data are reproduced, although the height of the first peak is underestimated and the second peak is better resolved and slightly shifted towards higher Q. The corresponding results for g(r) are shown in Fig. 7.16. The first nearestneighbour distance over the full temperature range is around 2.48 A˚, with no significant dependence on temperature. As the temperature decreases, the first peak sharpens up while the second peak shows a change in profile with the suggestion of a splitting at 1458 K, also observed in the work of Ansell et al. The AIMD results of g(r) are in reasonable agreement, although the first peak is always slightly narrower and higher than the experimental one and the subsidiary peaks are shifted towards lower r. The inset to Fig. 7.16 gives the bond-angle distribution at two temperatures, showing that the peak near the tetrahedral angle of 109 becomes more pronounced on supercooling.

149

7.1 Silicon Beam stopper

(a)

(d)

Solid

1

41 K 0.175 s (b)

Melt

40 K 0.088 s

(e)

2

128 K 0.175 s (c)

152 K 0.224 s (f)

241 K 0.175 s

254 K 0.056 s

Fig. 7.17. Diffraction patterns from undercooled and partly crystallized Si just after recalescence obtained at different undercooling of (a) DT ¼ 41 K, (b) DT ¼ 128 K and (c) DT ¼ 242 K; high-speed video images for samples solidified at (d) DT ¼ 40 K, (e) DT ¼ 152 K and (f) DT ¼ 254 K (Nagashio et al., 2006a).

These trends, tighter with the decrease in coordination number discussed above, provide strong support for a reinforcement of the tetrahedral ordering as the temperature is reduced in the supercooled liquid. In practice, the eventual limit of supercooling occurs with nucleation of the crystalline phase, and this process is of course of enormous importance for the semiconductor industry. The crystallization of undercooled Si melts has been studied with both EML (Nagashio et al., 2006a; Panofen & Herlach, 2006) and CNL (Nagashio et al., 2007), the latter with a time resolution of 1 kHz. A representative result is shown in Fig. 7.17, which shows Debye Scherrer X-ray diffraction patterns on the left side, and high-speed video (HSV)

150

Molten semiconductors

camera results on the right, for samples solidified at different degrees of undercooling. For the sample solidified at DT ¼ 41 K, several spots were clearly observed with the broad pattern of the melt because of the semi-solid condition after recalescence. When the undercooling was increased to 128 K, strong spots with a long tail appeared. Finally, a well-connected ring pattern was observed at DT ¼ 241 K. Figure 7.17(d) (f) show HSV images for samples solidified at almost the same undercooling as in Fig. 7.17(a) (c). Just after recalescence, both the crystal and liquid have the temperature of the melting point and so the contrast resulted from the difference in emissivities of the two phases. It was already known (Nagashio & Kuribayashi, 2005) that <110> dendrites grow at undercooling of less than 100 K, while the <100> dendrites with a four-fold axial symmetry appear at DT > 100 K. When the undercooling level was 40 K (d), sharp <110> dendrites grew with {111} facet planes. At DT ¼ 152 K (e), the primary stems of <100> dendrites were well connected. Finally, randomly distributed particles were observed at DT ¼ 254 K. The clear spot pattern observed in the diffraction patterns at undercooling of less than 100 K indicates that the relatively few <110> dendrites were located within the X-ray beam, whereas long tails around the spot were observed for the sample solidified at DT ¼ 128 K. Detailed XRD peak profiles along the diffraction rings for tail 2 in (b) suggested that the main stems of the <100> dendrites were still connected but that the second and third arms, as shown by arrows in (e), were mainly detached from them. On the other hand, the diffraction patterns for spot 1 in (a) obtained at undercooling less than 100 K indicated that the relatively few <110> dendrites observed at DT < 100 K are covered by {111} planes with the lowest interface energy and have no high-order arms that connect with the main stem, preventing these dendrites from fragmentation. Moreover, the drastic transition from well-connected dendrite to the randomly distributed particles found in (f) is consistent with the dramatic change of the diffraction pattern from spots to rings, suggesting the complete fragmentation of the main stems as well as high-order arms in the <110> dendrites that produces a spontaneous grain refinement. Alatas et al. (2005) studied the dynamics of supercooled liquid silicon with IXS and CNL. The experimental setup was similar to that used for liquid titanium, described in Section 6.1. Energy scans for different scattering vectors are shown in Fig. 7.18 for supercooled liquid Si at 1620 K and also for hot solid Si at the same temperature. In the hot solid, the longitudinal phonons are still well defined and give rise to isolated peaks that were fitted with Lorentzian functions, while in the supercooled liquid, where the collective excitations show up as shoulders on the side of the central peak not as

151

7.1 Silicon Hot Solid Silicon T = 1620 K

Supercooled Silicon T = 1620 K

0.00024 Q = 9.62 nm–1

0.0003

Q = 9.62 nm–1

0.0002

Counts/Monitor

Q = 8.49 nm–1 0.00016 Q = 7.39 nm–1

Q = 8.49 nm–1 0.0002

Q = 7.39 nm–1

0.00012 Q = 6.29 nm–1 8e-05

Q = 6.29 nm–1 0.0001

Q = 5.20 nm–1

Q = 5.20 nm–1

Q = 4.10 nm–1

Q = 4.10 nm–1

4e-05

0

0 0

20

40

Energy (meV)

60

0

20

40

60

Energy (meV)

Fig. 7.18. IXS scans for supercooled liquid and hot solid silicon for different scattering vectors (Atalas et al., 2005).

well separated as in the Ti spectra shown in Fig. 6.10 the spectra were fitted with the expression used there, Eq. (6.1). The respective sound velocities determined from the initial slope of the excitation frequencies as a function of scattering vector were 7600 m.s 1 for the hot solid and 4560 m.s 1 for the supercooled liquid. In Fig. 7.19 these values are compared with those for the solid at lower temperatures extrapolated from low-temperature data for the elastic moduli in the literature: v u 4 u r L tB þ 3G ¼ ; ð7:5Þ vL ¼ d d where d is the density and L, B and G are the longitudinal, bulk and shear moduli, respectively, and with values for the normal liquid obtained by IXS measurements on contained samples by Hosokawa et al. (2003). The value for the hot solid is about 20% lower than that extrapolated from the elastic moduli at lower temperature, which might indicate a precursor of the semiconductor-to-metal transition on melting. The large change in sound velocity

152

Molten semiconductors

Sound velocity (m s–1)

9000 8000

melting point

RT solid hot solid

7000 6000 5000

liquid

supercooled liquid 4000 3000 300

1600

1650

1700

1750

Temperature (K)

Fig. 7.19. Longitudinal sound velocities for supercooled liquid and hot solid Si measured in ADL experiments compared with values for the solid at lower temperatures extrapolated from low-temperature data for the elastic moduli and for the normal liquid obtained by IXS measurements on contained samples. The vertical dotted line shows the melting temperature of Si and horizontal one shows the sound velocity of the normal liquid, as guides for the eye (Alatas et al., 2005).

between the supercooled liquid and hot solid at the same temperature, 1620 K, can certainly be attributed to this transition. Finally, the 5% decrease in sound velocity observed in going from the supercooled to the normal liquid can be related to the significant decrease in directional bonding discussed above. For comparison, the velocity of sound in films of amorphous Si at room temperature is 77% of the average velocity in the corresponding crystal, appreciably higher than in the supercooled liquid.

7.2 Germanium and Ge–Si alloys While Si and Ge have many features in common in the crystalline, amorphous and liquid states, Ge is less interesting for levitation experiments on account of its lower melting point (1210 K) and, to some extent, less interesting for IXS measurements of the dynamics on account of the lower sound velocity (see Fig. 5.1). Values of the conductivity and thermopower of liquid Ge measured by Schnyders & Van Zytveld (1996), shown in Table 7.1, are similar in magnitude to those of liquid Si. The density, thermal expansion coefficient and surface tension of Ge and Si Ge alloys obtained from the microgravity experiments of Chathoth et al. (2008) are given in Table 7.2. The marked

153

7.2 Germanium and Ge Si alloys

Table 7.2. Experimental values for the mass density d, number density
0, liquidus

temperature Tliq, thermal expansion coefficient b and surface tension g for Si, Ge, and Si–Ge alloys (Chathoth et al., 2008). Sample

d(Tliq) (g.cm 3)

r0(Tliq) (A˚ 3)

Tliq (K)

b(10

Si Si75Ge25 Si50Ge50 Si25G75 Ge

2.52  0.1 4.46  0.1 5.87  0.1 5.26  0.1 5.57  0.1

0.054  0.002 0.068  0.002 0.070  0.001 0.051  0.001 0.046  0.001

1687 1623 1548 1423 1211

1.37  0.02 3.06  0.02 1.79  0.02 0.73  0.02 1.06  0.02

4

K 1)

g(N.m 1)a 0.75  0.03 0.65  0.03 0.60  0.03 0.57  0.03 0.62  0.03

Note: a Approximate values read off from Fig. 5 in Chathoth et al. (2008).

1.5 1 S (Q )

0.5 0

–0.5 Si, T = 1767 K SiGe, T = 1433 K Ge, T = 1253 K

–1 –1.5 0

2

4

6 Q

8

10

(Å–1)

Fig. 7.20. X-ray weighted average structure factor for liquid Si and liquid SiGe compared with a literature result for pure Ge. Successive curves are displaced by 0.5 units for clarity (Krishnan et al., 2007).

increase in number density in the alloys indicates an attractive interaction between the two metals. Several measurements of the structure and dynamics of liquid Ge have been carried out with neutron and X-ray scattering. These have been on contained samples for the reason given above and hence have not extensively probed the supercooled state, although Filipponi & Di Cicco (1995) were able to achieve a supercooling of 260 K in EXAFS measurements on samples in which finely grained Ge powder was dispersed in an inert matrix. Krishnan et al. (2007) measured the structure of levitated equiatomic liquid SiGe with the setup used for liquid Si described above. The structure factor obtained for SiGe was found, unsurprisingly, to be intermediate between that for Si and the literature result for Ge, as shown in Fig. 7.20.

154

Molten semiconductors (b)

(a)

(c)

100 nm

Fig. 7.21. (a) Optical micrograph of a fragment of vitrified Ge quenched from the liquid at 7.9 GPa, showing globules in the matrix; (b) electron diffraction pattern obtained from all areas except the area including the globule indicated by the arrowhead; (c) electron diffraction pattern for the globule area, with Laue spots indicating crystalline character (Bhat et al., 2007).

Germanium, like silicon, has a negative slope to the melting curve and can therefore be expected to exhibit an LLT. Principi et al. (2004) and Di Cicco et al. (2008) observed the LDA HDA transition at a pressure of 8 GPa using optical micrographs, X-ray and Raman scattering and X-ray absorption spectroscopy to confirm the metallic character. A remarkable experiment was carried out by Bhat et al. (2007) by quenching the HDL under pressure, using insights obtained from a recent MD computer simulation (Molinero et al., 2006). By lowering the temperature at a nearly constant pressure above the point at which the LLT line crosses Tg (see Fig. 7.14), they obtained a uniform solid amorphous phase that was undoubtedly the HDA phase, although they were not able to make a definitive diffraction or spectroscopic measurement in situ. Ex-situ electron and optical micrographs taken after decompression showed that these samples had transformed from the HDA to a uniform LDA phase. By quenching at a lower pressure at which the LLT line crosses Tg, some of the decompressed samples showed nanocrystalline globules that presumably were formed from droplets of LDL that vitrified during the quench and then crystallized (Fig. 7.21). By inference, HDL and LDL components coexisted at some stage in the quench, providing further evidence for a first-order LLT. This result is reminiscent of an earlier finding in Al2O3 Y2O3, to be discussed in the next chapter.

7.3 Boron and boron compounds

155

Li and Herlach (1997) studied solidification of liquid Ge in an EML apparatus by purifying the liquid in repeated heating cooling cycles in vacuo. They were able to obtain an unusually high degree of undercooling, 426 K, in highly purified samples. For an undercooling of less than 300 K, lamellar twins were obtained, whereas a microstructural transition to equiaxed grains was observed after undercooling by more than 300 K. A significant reduction in grain size was achieved after further increasing the undercooling to 400 K. They conducted further experiments in which small droplets of the purified liquid were dispersed into an 8.5 m drop tube containing helium gas. A similar microstructural development was observed, with smaller grain size due to the higher cooling rate, and in addition single crystals, exhibiting the normal diamond structure, were found for some droplets less than 200 mm in diameter. The transition from lamellar twins to equiaxed grains was consistent with a model in which the growth mechanism is lateral or continuous, depending on whether the undercooling is below or above a critical value, estimated as 153 K for Ge. The occurrence of a grain-refined microstructure at very high undercooling was consistent with the dendrite break-up model of Schwarz et al. (1994). 7.3 Boron and boron compounds Relatively little is known about liquid boron, partly because of its high melting point (2360  10 K) and partly because it is extremely corrosive. Like Si and Ge, boron is a semiconductor under ambient conditions but transforms to a superconducting metal under pressure (Eremets et al., 2001) and a recently discovered ionic form at even higher pressure (Oganov et al., 2009). Glorieux et al. (2001) measured the electrical conductivity of liquid boron near the melting point with the contactless electrical conductivity technique described in Section 4.1. From the change in coil resistance as a function of frequency they obtained an enhancement of 510  40 O 1cm 1 of the conductivity of heated boron on melting. Taking into account the literature value for the conductivity of the hot solid at 2200 K of 450 O 1cm 1, they obtained the value 960  80 O 1cm 1 for the conductivity of liquid boron just above the melting point. Since this is less than the conventional minimum metallic value, they concluded that liquid boron remains a semiconductor upon melting. This value was consistent with indirect estimates of a lower limit of 103 O 1cm 1 on the basis that a 15-mm diameter bar could be r.f. heated at 450 kHz (J. K. R. Weber, private communication, 1998). Regarding the upper limit, Millot et al. (2002a) obtained the value of 0.34  0.1 for the total emissivity eT at the melting point, based on the cooling of an

156

Molten semiconductors

aerodynamically levitated liquid drop and the literature value for the specific heat: this is too high a value for the liquid to be metallic. Measurements of the density are less consistent. This is important both as a basic property and because it enters into the calculation of g(r) and hence coordination numbers from diffraction data. In their first XRD study, Krishnan et al. (1998a) took an earlier literature value that corresponded to a 4.5% expansion on melting. Recent measurements with CNL (Millot et al., 2002a) and ESL (Paradis et al., 2005a) gave instead contractions of 8% and 3%, respectively. Millot et al. state that the appearance of the boron drop after cooling implies a contraction on melting. In both experiments the samples underwent significant vaporization losses, which could lead to cleaning of the surfaces as well as mass loss (C. Chatillon, private communication, 2008). However, mass loss should lead to a lower value for the density if not properly accounted for, so that the discrepancy with the literature values, and that between the two recent measurements, are at present unexplained. On the other hand, the value obtained for the surface tension by Millot et al. by observing the l ¼ 2 mode vibrations, 1.088 N.m 1, is in good agreement with the literature value. Crystalline boron exhibits a remarkable variety of structures, composed of icosahedra and pentagonal pyramids and characterized by large unit cells. The stable form at low temperature is either the a-rhombohedral (Masago et al., 2006) or a symmetry-broken b-rhombohedral (Widom & Mihalkovicˇ, 2008) structure and at high temperature the b-rhombohedral. It melts into a stable liquid at 2360  10 K. The liquid structure was measured by Krishnan et al. (1998a) and more recently by Price et al. (2009), using a combination of CNL, laser heating and XRD. AIMD simulations were carried out as part of the later work. In the formulation employed for these, the only experimental input was the density, taken from Paradis et al. Figure 7.22 shows the XRD results for S(Q) in the normal and slightly undercooled liquids, together with the AIMD result obtained from a direct formulation in Q space. All S(Q)s show well-defined peaks at Q ¼ 2.5 and 4.4 A˚ 1 and weaker ones at 8 and 11.8 A˚ 1. Figure 7.23 shows the corresponding g(r)s. The experimental g(r) is fitted well by Gaussian peaks centred at 1.77, 3.13, 4.58, 6.08 and 7.56 A˚, and the same peaks can be distinguished in the AIMD. The inset of Fig. 7.23 shows a typical distribution of nearest-neighbour coordination numbers obtained from the AIMD, in which six-fold-coordinated atoms are the most numerous. The average coordination number calculated from the area of the first peak in both the XRD and the AIMD g(r)s was 6.0, compared with 6.5 and 6.4 in the a- and b-rhombohedral crystals, respectively. The peaks in the AIMD S(Q) and g(r) sharpened up on cooling down to and below the

157

7.3 Boron and boron compounds 3.0 2500 K (exp)

2.5

S(Q )

2.0 2400 K (MD)

1.5 1.0

2340 K (exp)

0.5 0.0 0

2

4

6

8

10

12

14

16

18

Q (Å–1)

Fig. 7.22. Structure factor for liquid boron measured by XRD at 2500 and 2340 K, together with the AIMD result for 2400 K. The experimental result at 2500 K is displaced upwards by 1.0 for clarity (Price et al., 2009). 4.0

100 80 D(N)

3.5 3.0

40 20 0

2.5 g (r )

60

4

5

6 N

7

2.0

8

2500 K (exp)

1.5

2400 K (MD)

1.0

2340 K (exp)

0.5 0.0 1

2

3

4

5

6

7

8

r (Å)

Fig. 7.23. Pair correlation function for liquid boron measured by XRD at 2500 and 2340 K, compared with the AIMD result for 2400 K, broadened by the Fourier transform of the Lorch modification function . Successive curves are displaced upwards by 0.75 for clarity. Inset: distribution of coordination numbers (defined within a radius of 2.37 A˚) from the AIMD at 2400 K, and a typical six-fold-coordinated atom showing a pentagonal pyramid configuration (Price et al., 2009).

equilibrium melting point, but there were no systematic temperature shifts in their positions or coordination numbers. An intriguing issue in boron involves the possible survival into the liquid of the icosahedron and pentagonal pyramid structural units of the crystalline

158

Molten semiconductors

phases. The possibility of their survival on melting as has been found for complex structural units in other semiconducting systems, for example NaSn (Saboungi et al., 1993) and CsPb (Price et al., 1991) might be invoked to explain the unusual properties of the liquid. Icosahedra would produce a first sharp diffraction peak (Price et al., 1989) around 0.8 A˚ 1, and indeed such a peak was observed in diffraction measurements on solid amorphous boron (Delaplane et al., 1991). There is no evidence of such a peak in either the XRD or AIMD results, and no complete icosahedral arrangements are found in the simulation. On the other hand, many atoms adopt with their first neighbours a geometry corresponding to the pentagonal pyramids of the crystalline phases, shown in the inset to Fig. 7.23. A further issue relates to the possibility of an LLT in liquid boron. The existence in the solid of a- and b-rhombohedral crystal structures, which can be regarded as high- and low-density phases (Masago et al., 2006; van Setten et al., 2007), suggests the possibility of high- and low-density phases in the liquid. Furthermore, the presence of icosahedra and pentagonal pyramids in both solid phases implies that two length scales are involved, which is believed to be a favourable condition for an LLT to occur (Franzese et al., 2001). It is also tempting to raise the comparison with silicon since, as we have seen, boron appears to undergo a contraction on melting accompanied by a considerable decrease in longitudinal sound velocity (45%, compared with 42% in Si). On the other hand, the entropy increase on melting is only 9.6 J.mol 1.K 1, a typical value for most elements, while it is three times larger in Si and Ge. As we have seen, the change in structure (decrease of 8% in coordination number compared with increase of 50% in Si) is also relatively modest, as is that in the conductivity (increase of 510 O 1cm 1 compared with 104 O 1cm 1 in Si). Furthermore, there is little evidence of a significant structural shift with temperature as found in Si that might suggest an eventual phase transition. This suggests that melting in boron is more similar to the transition between solid and supercooled tetrahedral liquid Si predicted by the computer simulations discussed in Section 7.1, and that any LLT must take place, if at all, at appreciably higher temperature. Price et al. (2009) also report IXS measurements of the dynamics of levitated liquid and hot solid boron, also complemented by AIMD simulations. Results for S(Q, o) and C(Q, o) are shown in Fig. 7.24. The solid lines represent the phenomenological model discussed below. The AIMD and IXS Q ranges overlap only at 0.5 A˚ 1 where the AIMD result is seen to be in reasonable agreement with the experimental data.

159

7.3 Boron and boron compounds Q = 0.5 Å–1

I (Q,w)

Q = 0.3 Å–1

C (Q,w)

Q = 0.4 Å–1

Q = 0.2 Å–1

Q = 0.1 Å–1

0

10

20 30 E (meV)

40

50

Experiment Model Simulation

0

20 40 E (meV)

60

80

Fig. 7.24. Scattering function S(Q, o) (left) and current correlation function o2S(Q, o)/Q2¼C(Q, o) (right) for liquid boron at 2370 K. The solid lines represent a fit with a phenomenological model described in the text. The solid circles show the AIMD result at 0.5 A˚ 1, the lowest Q value studied in the simulation (Price et al., 2009).

As a first approach to a theoretical analysis, the low-Q data were fitted with a phenomenological form of the memory function introduced previously in Eq. (6.2): MðQ; tÞ ¼
21 e

t=t1

þ
22 e

t=t2

;

ð7:6Þ

where the relaxation times t1, t2 and relaxation strengths D1, D2 are treated along with o02 as fit parameters. The low-Q spectra could be fitted reasonably well with the same relaxation times for all Q values: t1 ¼ 0.3  0.1 and t2 ¼ 0.025  0.02 ps, and a ratio D1/D2 ¼ 1.5  0.3. The value thus obtained for the isothermal sound velocity vt ¼ o0/Q was 5280  100 m s 1, while the high-frequency sound velocity vs ¼ Os(Q) derived by plotting the maxima Os(Q) in C(Q, o) against Q was 8580  100 m s 1. The corresponding value obtained for the hot solid at 2170 K was 14100  100 m s 1, consistent with the reported value of 14 300 m s 1 at room temperature (Gerlich & Slack, 1985). The large ratio (1.47) of high-frequency to isothermal sound velocity

160

Molten semiconductors

Viscosity (nm2/ps)

10 IXS data MD Simulation

1 mode

couplin

g

0.1 0.0

0.5

1.0

1.5

2.0

2.5

3.0

–1)

Q (Å

Fig. 7.25. Experimental values of generalized longitudinal kinematic viscosity l=r obtained from the IXS measurements (solid circles) and AIMD simulations (solid triangles) of liquid boron. The continuous lines bounding the hatched area represent the results of a mode coupling theory calculation with different parameters (Price et al., 2009).

in the liquid, compared with the values of 1.1 1.2 found in other monatomic liquids (Scopigno et al., 2005), indicates an unusually high viscoelastic stiffening. The generalized longitudinal viscosity in the low Q-region was extracted from the relation l ðQÞ ¼

pdv20 SðQ; o ¼ 0Þ SðQÞ2

¼

dð
21 t1 þ
22 t2 Þ ; Q2

ð7:7Þ

where d is the mass density and v02 ¼ kBT/m, using values obtained from the fit of Eq. (7.6); for the higher Q values, where the central peak dominates, a single Lorentzian function was fitted to the data and S(Q, 0)/S(Q) determined from its width. The result is plotted as l/d in Fig. 7.25, together with the values obtained from the AIMD results for S(Q, 0)/S(Q). The agreement is good at low Q but, at Qs around the peak of S(Q), the AIMD predicts a higher value, indicating a narrower shape to S(Q, o). The value extrapolated to low Q, l ¼ 15 mPa.s, is appreciably higher than in alkali metals but only slightly higher than other Group IIIA elements: Al and Ga (Scopigno et al., 2005). It remains to address the applicability of the density fluctuation version of MCT, previously employed successfully in the case of liquid Ti, discussed in Section 6.1. The generalized longitudinal viscosity obtained from the application of the MCT to the memory function M(Q, t) is shown by the hatched area

161

7.3 Boron and boron compounds Temperature

1600

2.066

T (°C)

Laser shutdown

Phase transformation

Crystallization 2.062

1200 2.06 1000

Inductance (μH)

2.064

1400

2.058 Inductance

800 50

60

80 70 Time (s)

90

2.056 100

Fig. 7.26. Cooling curve and inductance of Fe2B as a function of temperature (Saboungi et al., 2002).

in Fig. 7.25, where the two lines bounding the hatched area represent two different ways of treating the high-Q data. It can be seen that the calculation is in reasonable agreement at higher Q but fails by an order of magnitude at low Q. The poor agreement obtained here for boron, in contrast to the satisfactory fit in the case of Ti, indicates that higher-order correlation functions arising from the directional bonding and short-lived local structures observed in the AIMD simulations are playing a crucial role in the damping of the sound excitations, whereas the interatomic potential in Ti can be approximated by a smoothed hard-sphere interaction. Finally, as an example of the way in which levitation methods can be employed to synthesis materials, Saboungi and Glorieux (2005) were able to prepare cobalt and iron boron compounds in various compositions. The composition Fe2B was confirmed by X-ray diffraction and investigated with the electrodeless technique for magnetic and transport property measurements described in Section 4.1. The inductance of the sample measured in both the normal and supercooled liquid phases as a function of temperature, shown in Fig. 7.26 along with the cooling curve, suggests that a transition occurs in the supercooled phase, possibly because of magnetic ordering (Saboungi et al., 2002).

8 Molten oxides

High-temperature melts with low electrical conductivity have largely been the province of CNL since the other main levitation techniques, EML and ESL, are inconvenient if not impossible for measurements on insulating systems. In fact, from its inception, non-conducting liquids, and in particular refractory molten oxides, have presented the heaviest application of CNL. 8.1 Pure trivalent oxides The molten trivalent oxides Al2O3 and Y2O3, together with their mixtures, by themselves account for the vast majority of measurements with CNL. This is due in part to the scientific as well as technical importance of yttrium aluminium garnet (YAG), Y3Al5O12, an important laser material, discussed in the next section. In addition, pure Al2O3 is of technological interest as the reaction product in rocket engines fuelled by aluminium metal (Parry & Brewster, 1991). As a result, molten Al2O3 has often been the material of choice when the opportunity arose for exploiting a new experimental technique with CNL. The conductivity of molten Al2O3, measured with a contactless technique by Enderby et al. (1997) and Saboungi et al. (2002), is about 6 O 1cm 1 at the melting point, indicating the presence of ionic but not electronic conduction. Later measurements (Saboungi et al., unpublished work, 2003) probed the effect of changing the gaseous environment in the levitation system and found a substantial drop in conductivity in a reducing environment (Fig. 8.1). The earlier results of Shpil’rain et al. (1976) on contained samples of molten Al2O3 under argon and in vacuo are in reasonably good agreement with the upper curve in Fig. 8.1. Several authors reported optical measurements on molten Al2O3 in the 1990s. Krishnan et al. (1991b) found a value for the refractive index measured 162

163

8.1 Pure trivalent oxides

Conductivity (ohm–1cm–1)

20

16

Alumina

Oxygen

12

Argon

8 Hydrogen+Ar 4

0 1900

2100

2300

2500 2700 Temperature (K)

2900

3100

3300

Fig. 8.1. Electrical conductivity of molten Al2O3 as a function of temperature for three different gaseous environments (Saboungi et al., 2002; private communication).

at a wavelength of 0.633 mm on laser-heated pendant drops, averaged over several measurements with sapphire and ruby initial samples in oxygen and argon gaseous environments, of n ¼ 1.744  0.16, with no significant variation with either starting sample or environment. Weber and collaborators measured the spectral absorption coefficient at the same wavelength in partially molten sapphire filaments under Ar, O2 and (10%H2 90%N2) environments (Weber et al., 1995a), and in pendant drops under Ar, O2 and N2 environments, as well as different concentrations of (CO CO2) and (H2 H2O) mixtures (Weber et al., 1995b). Absorption coefficients al in the range of 10 55 cm 1 were obtained, corresponding to extinction coefficients k in the range (5 25)·10 5. Thus, the melt is semitransparent at optical wavelengths and so, according to (Eq. 4.16), the reflectivity is governed by the real part n of the complex refractive index and is thus independent of environment. On the other hand the absorption coefficient al appears to depend strongly on environment. Figure 8.2 shows the plot of al against the partial oxygen pressure p(O2) calculated by Weber et al. (1995b) for their different ambient gases and shows a minimum around p(O2) ¼ 10 4. They suggested that this might correspond to a stoichiometric composition in the melt. Sarou-Kanian et al. (2005) measured the spectral emissivity in the IR of high-temperature solid and liquid alumina with O2, Ar, and Ar þ 5%H2 levitation gases, and also obtained values that depended markedly on both temperature and levitation gas. Figure 8.3 shows their results at a wavelength

164

Molten oxides

Absorbance at ca. 2400 K, cm–1

80 Pure gases CO–CO2 H2–H2O

60

Ar CO

N2

40

O2 20

CO2

H2O

0 –10

–8

–6

–4

–2

0

2

log p (O2)

Fig. 8.2. Spectral absorption coefficient at l ¼ 0.633 mm and T ¼ 2400 K of molten Al2O3 in different gaseous environments as a function of partial oxygen pressure (Weber et al., 1995b).

1.0 Spectral emissivity at l = 3 µm

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

T = 2327 K

0.0 1600

1800

2000

2200

2400

2600

2800

Temperature, K

Fig. 8.3. Spectral emissivity of solid and liquid alumina in O2 and Ar at 3 mm vs. temperature. In O2: () liquid, (✦) undercooled liquid, (□) liquid þ solid, (e)solid; in Ar: () liquid, (□) liquid þ solid, (□) solid (SarouKanian et al., 2005).

8.1 Pure trivalent oxides

165

of 3 mm in O2 and Ar environments. The emissivity is appreciably higher in Ar in both the stable liquid and when cooled into the solid phase. Glorieux et al. (1999) measured the density of molten Al2O3 in the range 2000 2800 K using CNL and the imaging technique described in Section 4.2. They obtained a value of 2.79  0.01 g cm 3 at 2500 K, with a temperature coefficient a ¼ (4.2  0.1)·10 5 K 1. They concluded that the density did not depend significantly on the levitation gas, although with pure Ar they experienced some difficulties in melting the droplets completely and obtaining a stable levitation. Their absolute value for a is similar to that obtained earlier by Coutures et al. (1994) but much lower than those found earlier in measurements on contained samples, probably due to chemical contamination in the latter case. Glorieux et al. (2002) proceeded to make measurements of the surface tension with the same setup and found a value of 0.65  0.03 at 2500 K with a temperature coefficient of (6  2)·10 5 K 1. Slightly higher values were found in O2 compared with an Ar H2 mixture. Again, their absolute value for the temperature coefficient was much lower than those found on contained samples, presumably for the same reason. There have been several studies of the atomic structure of molten Al2O3. The stable solid form is a-Al2O3, a hexagonal structure with six-fold Al coordination, but disordered solid phases with a mixture of four- and sixfold coordination also exist. NMR studies by Coutures et al. (1990) found a coexistence of four- and six-fold coordinated sites with the four-fold predominating. In subsequent NMR measurements made with different levitation gases, Coutures et al. (1994) found substantially higher values of both the isotropic chemical shift and FWHM of the 27Al resonance under argon than under air (Fig. 8.4). They ascribed this to either a partial reduction of Al3þ to Al2þ/Alþ or a complete reduction to Al0. The observation of a weak but significant resonance line attributable to Al metal in a sample solidified rapidly under argon suggested the second explanation. Presence of metal inclusions in molten Al2O3 could explain the reduction with increasing temperature in the activation energy for conductivity observed by Shpil’rain et al. in contained samples in molten Al2O3 under argon and in vacuo, and would also be consistent with the reduced conductivity (due to scattering processes) observed by Saboungi et al. and the increased optical absorption found by Weber et al. and Sarou-Kanian et al. under a reducing atmosphere. The structure factor was measured with X-ray diffraction by Ansell et al. (1997) and Hennet et al. (2002) and with neutron diffraction by Landron et al. (2001), who used a Monte Carlo simulation to supplement the data on account of the necessarily poor statistics. The most recent XRD measurements of Krishnan et al. (2005) were performed over wide ranges of Q and

166

Molten oxides

105

1200 1000

Argon

95

Argon

(Hz)

(ppm)

800 85 75

600 400

65 55 2000

Air

2100

2200

2300

Temperature (°C)

Air

200

2400

0 2000

2100

2200

2300

2400

Temperature (°C)

Fig. 8.4. Temperature dependence of the isotropic chemical shift (left) and full width at half-maximum (right) of the 27Al resonance in molten Al2O3 for argon (, ) and air (,♦) environments (Coutures et al., 1994).

▴

temperature and with good statistics. They used CNL with both argon, purified by a titanium-getter furnace, and pure oxygen as the levitation gas. Their apparatus included a second CO2 laser (40 W) to provide additional heating of the specimen from below through the opening in the nozzle, in order to achieve high levels of supercooling and reduced temperature difference between the top and bottom of the sample, which were estimated to be less than 25 K. A number of conventional molecular dynamics (MD) computer simulations were carried out in response to these experiments. Both San Miguel et al. (1998) and Hemmati et al. (1999) obtained adequate agreement with the Xray structure factor with appropriate weighting of the three partial structure factors obtained from the simulation. However, the individual partial structures showed considerable differences, particularly for the Al O partial which gave rise to predominantly four-fold coordination in San Miguel et al. and predominantly five-fold coordination, with appreciable four- and six-fold, in Hemmati et al. Recently a new approach to MD simulation of classical liquids was introduced by Jahn et al. (2004). This captures the dependence of the interaction energy on the instantaneous coordination environment provided by the more rigorous ab initio MD technique but allows affordable simulations of larger systems and longer timescales. The potential contains contributions from the charge charge and dispersion interactions, overlap repulsion and polarization. The first two components are pair-wise additive, as in Born Mayer-type pair potentials. The overlap repulsion terms allow for spherical breathing and shape deformations of the anions up to the

167

8.1 Pure trivalent oxides 2

1.5

S(Q )

1

0.5

Experimental, 2415 K AIMD, 2350 K

0

–0.5 0

2

4

6

8 10 Q (Å–1)

12

14

16

18

Fig. 8.5. Experimental structure factor for molten Al2O3 obtained by XRD at 2415 K (open circles) compared with the X-ray weighted average obtained from an MD simulation with an ab initio interaction potential at 2350 K (Krishnan et al., 2005).

quadrupolar level. Further the anions are considered polarizable, which allows for induction of dipoles and quadrupoles due to the electric fields and field gradients of the other ions and short-range overlap effects. The potential for molten Al2O3 was validated by comparison with ambient- and highpressure solid phases (Jahn et al., 2004) and then used to compute the X-ray and neutron weighted average S(Q) and g(r) (Jahn & Madden, 2007). In Fig. 8.5 the structure factor measured by Krishnan et al. at 2415 K under argon is compared with the X-ray weighted average result from the MD simulation described above, corresponding to 2350 K. The origins of the peaks in S(Q) are readily understood from the partial structure factors obtained in the simulations. The first peak at 2.1 A˚ 1 reflects the first sharp diffraction peak (Price et al., 1989) that appears in all three partials but is most pronounced in SAlAl(Q), indicating a small degree of intermediate-range order in the cation arrangement. A chemical short-range order peak that appears at 2.6 A˚ 1 in all three partials disappears in the X-ray weighted average because Al3þ and O2 each contain 10 electrons; it appears, however in the neutron result of Landron et al. because the two neutron scattering lengths are different. The second peak between 4 and 5 A˚ 1 results from topological short-range order reflected in all three partials over a spread of Q values due to the differences in the closest-approach distances. Some small

168

Molten oxides (a) 3 Experimental Data, 2415K

2.5

AIMD, 2350K

G (r )

2 1.5 1 0.5 0 –0.5 0

1

2

3

4 r (Å)

5

6

7

8

(b)

Fig. 8.6. (a) Comparison of the XRD X-ray weighted average pair correlation function with the X-ray weighted average obtained from the MD simulation. (b) Snapshot of an MD configuration, showing approximately equal populations of lower- (three- and four-fold) coordinated cations (open circles) and higher- (five- and six-fold) coordinated cations (closed circles joined by lines) (Krishnan et al., 2005).

differences between the XRD and MD results can be observed: the first peak is less well resolved from the second in the simulation S(Q), and the shape of the second peak is more skewed to higher Q. Figure 8.6(a) shows the corresponding comparison for g(r), obtained with a number density of 0.0847 A˚ 3, based on a mass density of 2.87 gm cm 3 at 2525 K (Glorieux et al., 1999). The first peak represents the Al O bond length

8.1 Pure trivalent oxides

169

and the broad second peak an average of the Al Al and O O correlations at somewhat different distances. The ratio of the second to first peak positions, 1.75, is close to that of an ideal tetrahedron, 1.732. The first and second peaks agree quite well in the XRD and MD results, while the third, originating from the Al O second neighbours, is shifted to slightly smaller r in the simulation. A point which should be borne in mind is that the X-ray weighted average S(Q) is calculated from the MD partial structure factors on the basis of spherical ions centred at the nuclear sites with complete charge transfer, i.e. Al3þ and O2 , and this assumption is not necessarily correct. In an X-ray diffraction study of liquid FeCl3 (Badyal et al., 1997), better agreement was found with less than complete charge transfer. The nearest-neighbour coordination number obtained from the area of the first peak was found to be 4.4  0.5; neither the coordination number or peak positions displayed any systematic temperature dependence. In the MD results, the average coordination number of the cations is 4.5, in good agreement with the experimental result; the cations are predominantly four-fold coordinated (54%) with 41% five-fold, 4% six-fold and 1% three-fold coordinated. A snapshot of an MD configuration is shown in Fig. 8.6(b). Very little difference was found between the measurements in neutral and oxidizing atmospheres. It should be noted that the penetration depths for the X-rays and optical radiation are on the order of 50 mm and 0.2 mm, respectively, while the RF cavity radiation samples the entire sample in the case of Al2O3. Thus, the different effects of the gas environment found with the three types of probe cannot be simply ascribed to bulk versus surface effects. These results show that melting of Al2O3 in its stable a-form is accompanied by drastic changes in structure and coordination. These structural changes explain the complex solidification behaviour reported on cooling the melt: when the cooling rates approach 105 K s 1, amorphous Al2O3 can be formed; at somewhat lower cooling rates, the metastable g-phase can be formed directly from the melt (Levi et al., 1988), while at cooling rates in the range of 1 100 K s 1 only a-Al2O3 forms (Weber et al., 1995c). The diffraction measurements indicate that molten Al2O3 exhibits a real-space structure similar to g-Al2O3, enabling the formation of an intermediate g-phase on cooling from the melt, prior to the formation of the stable a-phase. Molten Al2O3 was also the first material whose dynamical properties were studied with IXS combined with CNL (Sinn et al., 2003). Measurements were carried out on spheres 3 4 mm in diameter suspended in an oxygen gas jet and heated with a 270-W CO2 laser beam at the Advanced Photon Source on apparatus already described in Section 6.1. The X-ray energy of 21.657 keV corresponded to an (18,6,0) reflection for Si, and the energy resolution was

170

Molten oxides

1.8 meV FWHM. The levitation apparatus was enclosed in a bell jar with a separation of 10 cm between sample and detector in order to maintain the back-scattering condition for analysis of the scattered beam. For a two-component system, as we discussed in Section 4.6, the scattering function S(Q, o) is a weighted average of the partial functions for the three atom pairs. Because Z(Al3þ)  Z(O2 ), this average comes close to the function SNN(Q, o) that represents the frequency spectra of the number density correlations, corresponding to the Bhatia Thornton structure factor SNN(Q). The IXS spectra measured in molten Al2O3 at 2323 K at the lowest six wave vectors and fitted by the sum of three Lorentzians given in Eq. (6.1), convolved with the measured resolution function, are shown in Fig. 8.7.

1

Q = 6.09 nm–1

0 Q = 5.09 nm–1 1 0 1

Q = 4.09 nm–1

0 1

Q = 3.09 nm–1

0 1

Q = 2.09 nm–1

cts/s

0 2 0 –30 –20 –10

Q = 1.09 nm–1

0 10 20 30 energy transfer [meV]

40

50

60

Fig. 8.7. Inelastic X-ray scattering spectra for molten Al2O3 at the six lowest wave vectors measured and a temperature of 2323 K. The fits of Eq. (6.1) convoluted with the resolution function are shown as solid lines and the equivalent functions before convolution as dotted lines. The resolution function is shown hatched (Sinn et al., 2003).

171

8.1 Pure trivalent oxides T [°C] 2000

2800

2400

7000

Γs [meV]

7500 ns

35

6500 ns [m s–1]

5

30

Γs@ 3 nm–1

25

4

20 Ωs

Ωs, Γs [meV]

4.5

15 10 Γs 5 0

0

1

2

3

4

5

6

Q [nm–1]

Fig. 8.8. Dependence on wave vector of the frequency Os and half-width Gs of the Brillouin peaks in molten Al2O3 at 2323 K. The solid line represents Os ¼ vsQ with vs ¼ 7350 m s 1 and the dashed line represents Gs ¼ aQ2 with a ¼ 6.2.10 7 m2 s 1. Inset: temperature dependence of vs ¼ Os/Q and Gs. The solid line is a linear fit and the dashed line denotes the values obtained from fits with a frequency-dependent viscosity, Eq. (8.1) (Sinn et al., 2003).

The linear Q dependence p of the excitation frequencies renormalized by the damping term, s ¼ o2s þ 2s , corresponded to a longitudinal sound velocity vs ¼ Os Q ¼ 7350  40 m s 1 (Fig. 8.8). This value is considerably lower than that of solid a-alumina at room temperature, 10 500 m s 1, and reflects a change on melting consistent with those of the local coordination, density, entropy and conductivity. With increasing temperature (inset to Fig. 8.8), the value of vs fell to 6530  70 m s 1 at 3073 K. The Brillouin modes remained under-damped (Gs/os< 1/2) out to Q ¼ 6 nm 1, a surprising result for excitations of short wavelength (1 nm) in this relatively viscous liquid. The damping coefficient Gs (Fig. 8.8) exhibited a quadratic dependence on Q. In the hydrodynamic limit, the memory function appearing in Eq. (6.2) is given by MðQ; oÞ ¼

l 2 Q þ MT ðQ; oÞ; d

ð8:1Þ

172

2800 ° 2600 C °C

Molten oxides 2800 °C 2600 °C 2400 °C 2050 °C

2

0 °C 240

1

20 50

°C

Γ [meV]

1.5

0.5

0 0

10

20 Q

[nm–1

30

]

Fig. 8.9. Dependence on wave vector of the half-width G of the Rayleigh peak at different temperatures. The solid line joining the points at 2050  C (2323 K) is a guide to the eye. The dashed line represents DAlQ2 and the dotted lines show predictions based on kinetic theory (Sinn et al., 2003).

where l is the longitudinal viscosity (4/3 þ z) ( ¼ shear viscosity, z ¼ bulk viscosity), d is the mass density and MT(Q, o) represents the thermal relaxation process, while the damping coefficient is given by s ¼

l 2 Q; 2d

ð8:2Þ

which is also quadratic in Q. However, the observed damping constant would then correspond to l  3.5 mPa.s, more than a factor of ten below the known macroscopic values for the shear viscosity , even neglecting the bulk viscosity term. Furthermore, the observed Brillouin line width increases with temperature while the viscosity is expected to decrease. Thus, as discussed in Section 6.1 in connection with the measurements on Ta, the damping of the sound modes cannot be described satisfactorily by the hydrodynamic expressions. The widths of the central peak (Fig. 8.9) are considerably smaller than those of the Brillouin peak at all scattering vectors and temperatures measured. Furthermore, they exhibit a more structured Q dependence and a stronger temperature dependence. In the hydrodynamic limit, the central peak width is given by G ¼ DTQ2, where DT represents the thermal diffusivity l/Cpr. Taking literature values of l for the high-temperature solid, 7.4 W.m 1.K 1,

173

8.1 Pure trivalent oxides

hI [mPas]

100 80 60 40 20 0

t [ps]

0.6 0.4 0.2 0 2200

2400

2600

2800

3000

3200

T [K]

Fig. 8.10. Results of fits with a frequency-dependent viscosity, Eq. (8.3). In the upper panel, the squares denote values obtained for 1 and the dashed lines data for 7/3 obtained from macroscopic measurements (Sinn et al., 2003).

Cp ¼ 184 J.mol 1.K 1 and d ¼ 2.89.103 kg.m 3, DT was found approximately equal to 6.10 6 m2.s 1. The observed values of G were two to three orders of magnitude smaller than this would imply. Furthermore, because even at low Q the central peak dominates the total intensity (Fig. 8.7), the intensity ratio is very different from that predicted by the hydrodynamic theory, (g 1)/g, with g ¼ Cp/Cv, the ratio of the specific heats at constant pressure and constant volume, estimated as 1.08. In a phenomenological approach equivalent to that described for the liquid tantalum measurements in Section 6.1 and the liquid boron measurements in Section 7.3, fits to the low-Q data were made by neglecting the thermal term in Eq. (8.1) and replacing l by a frequency-dependent viscosity l ðoÞ ¼

l þ 1 ; 1 iot

ð8:3Þ

where t describes a Maxwell-type relaxation time and 1 is a non-relaxing friction coefficient. At each temperature, the spectra for all Q values up to 6.09 nm 1 could be fitted within experimental error by the same set of l, t and 1 (Fig. 8.10). The first two exhibited a gradual temperature dependence while the third did not vary significantly with temperature. The longitudinal viscosities extracted from the data were consistent with the macroscopic shear viscosities if one assumed  ¼ x (dashed lines in Fig. 8.10). An Arrhenius-like fit, ,t / exp(E/kT), resulted in activation energies of E ¼ 82  4 kJ mol 1 for

174

Molten oxides

the viscosity and 72  4 kJ mol 1 for the relaxation time, consistent with the increased damping observed with increasing temperature (Fig. 8.8, inset). At larger Q, the central peak widths were similar in magnitude to those obtained from the single-particle diffusion description, G ¼ DAlQ2 (dashed line in Fig. 8.9), taking the literature value for DAl, the single-particle diffusion constant for Al3þ ions, of 1.95·10 9 m2.s 1. This is not unexpected since interference effects in the scattering become less pronounced at higher Q and S(Q, o) approaches the incoherent scattering function Sinc(Q, o) (Section 4.6), which is dominated by diffusion phenomena. A quantitative estimate is given by kinetic theory (Cohen et al., 1987): ¼

SðQÞ½1

DQ2 ; j0 ðQÞ þ 2j2 ðQÞ

ð8:4Þ

where D is an average diffusion constant and s is the interparticle separation. The results obtained using the measured S(Q), the value given above for DAl and the Al O distance for s gave a fair representation of the Q and T dependence of G, as shown by the dotted lines in Fig. 8.9. Jahn and Madden (2007; 2008) also calculated the dynamic properties from the simulations described above and reproduced the IXS spectra extremely well. The memory function of interest in the present context should closely resemble the correlation function of the elements of the stress tensor hsab(t)sab(0)i, where sab(t) is an off-diagonal element of the stress tensor of the simulation cell. For an isotropic system, an average over different elements a, b ¼ x, y, z can be performed. The values obtained from the simulation are shown in the left side of Fig. 8.11. They could be fitted in the two distinct time domains found by two stretched exponential functions of the form A·exp [ (t/t)b]. The relaxation times t were found to be 0.013 and 0.55 ps, differing by more than one order of magnitude. This simulation result provides an additional justification for the use of Eq. (8.3) to fit the IXS data. The Fourier transform of this function defines a frequency-dependent viscosity V ðoÞ ¼ kT

1 ð



dt eiot  ðtÞ ð0Þ ;

ð8:5Þ

0

shown in the right side of Fig. 8.11, together with the Fourier transform of the two stretched exponentials. The slow component dominates the low frequency spectrum and is associated with the configurational relaxation of the fluid. The value at zero frequency, (0) ¼ 25 mPa.s, indicated by a closed circle in the figure, corresponds to the macroscopic viscosity, in excellent

175

8.1 Pure trivalent oxides 2e –10

1.5e –10

t1 = 0.013 ps

h (w) [mPa.s]

ásab (t )sab (0)〉 [a.u.]

10

1e –10

MD fit (total) fit (fast) fit (slow) MD w = 0

10 meV

1

0.1

5e –11 t2 = 0.55 ps 0 0.0001 0.001

0.01 0.1 t [ps]

1

10

0.01

0

50

100 w [meV]

150

200

Fig. 8.11. Left panel: average stress tensor autocorrelation function (solid line) calculated from simulations of molten Al2O3 fitted with two stretched exponentials (dashed line). Right panel: corresponding spectrum (frequencydependent viscosity). The zero frequency value 25 mPa.s, which corresponds to the macroscopic shear viscosity, is indicated by a full circle. At 10 meV, (o) has dropped by more than a factor of ten. Also shown are the spectra of the fitted curve and its fast and slow components (Jahn & Madden, 2008).

agreement with the experimental data. The spectrum associated with the fast component, resembling the vibrational density of states of a glass, corresponds to the region probed in the IXS experiments. The solidification behaviour of molten Al2O3 has been studied by CNL in different environments. Coutures et al. (1994) found that both the temperature from which the melt was cooled and the gaseous environment had a considerable effect on the texture of the resulting solids. Solidified droplets obtained under oxygen were porous with a dendritic structure, while those obtained under argon were more compact with larger crystals. They also found a considerably lower degree of undercooling between 50 K and 175 K under argon compared with 275 K obtainable under oxygen. They ascribed this effect to the presence of metallic aluminium inclusions, acting as nucleation agents, in the samples levitated in a reducing environment, discussed above in connection with their NMR results. Weber et al. (1995c) used aerodynamic levitation with an acoustic position stabilizer to melt and solidify samples of different initial impurities under different environments. With the purest starting material, single-crystal sapphire, they obtained supercooling of 450 K in argon and 360 K in oxygen. In terms of the microstructure of the solidified samples, they obtained results similar to those of Coutures et al.: dendritic, porous polycrystalline material under oxygen and dense, larger crystals under argon.

176

Molten oxides

We now turn to the other end member of the YAG system Y2O3. The stable form at room temperature is the C-type cubic phase characteristic of the heavier rare-earth oxides. Its considerable thermal expansion on heating is interrupted by a transition at 2326 K to the more closely packed H-type hexagonal phase: the H phase is a superionic conductor in La2O3 (Aldebert et al., 1979) and probably in Y2O3 as well. The B-type monoclinic structure found in the middle rare-earth oxides at intermediate temperatures has been obtained over a limited temperature region (Hoekstra, 1966; Coutures et al, 1972). Y O coordination numbers for the C, H, and B phases are reported as 6, 7, and intermediate between 6 and 7, respectively. At 2712 K, Y2O3 melts into a liquid phase with a remarkably small range of stoichiometry (Foex, 1977): the composition has been found to be Y2O3 under air, Y2O2.998 under a vacuum, Y2O2.997 under Ar, and Y2O2.97 under H2 at 2773 K (Schneider, 1970). Total X-ray diffraction measurements on levitated samples were first made by Krishnan et al. (1998b), but the AXS measurements of Hennet et al. (2003) had the advantage of being able to study the Y and O environments separately. The experiments were carried out just above the melting point, at 2770 K, using Ar0.95(O2)0.05 as the levitation gas in order to minimize substoichiometry. Diffraction data were collected at two incident energies, one just below the yttrium absorption edge at 17.01 keV and the other further below at 16.75 keV. Figure 8.12 shows the X-ray weighted average structure factor S(Q) and the yttrium structure factor SY(Q) (Price & Saboungi, 1998) obtained from D E DIcoh ðQÞ ¼ ½SY ðQÞ 1 Djh f ðQÞij2 þD j f ðQÞj2 ; ð8:6Þ where D denotes the difference in the values for the two energies and h. . .i denotes the average over the atoms in the sample. The sharpness of the first peak at Q ¼ 2.07 A˚ 1 in both S(Q) and SY(Q) is remarkable and implies a high degree of chemical ordering. There is no sign of intermediate-range order, which would appear in this liquid as a first sharp diffraction peak (FSDP, Moss & Price, 1985) around 1.0 1.2 A˚ 1. The corresponding average pair correlation function g(r) and yttrium pair correlation function 1 gY ðrÞ ¼ 1 þ 2 2p
0

Qð max

½SY ðQÞ

1

sin Qr QdQ; r

ð8:7Þ

0

are shown in Fig. 8.13. The number density r0 was taken as 0.060 atom A˚ 3, 10% lower than in the room-temperature crystal. The correlation function

177

8.1 Pure trivalent oxides 3.5 Y2O3 2770 K

3.0 2.5

S(Q) and SY(Q)

S(Q) (16.75 keV) 2.0 1.5 1.0 yttrium structure factor SY(Q)

0.5 0.0 –0.5 1

3

5

7 Q

9

11

13

15

(Å–1)

Fig. 8.12. Average structure factor measured at 16.75 keV and yttrium structure factor for molten Y2O3 at 2770 K. The upper curve has been shifted up by 1 for clarity (Hennet et al., 2003).

g(r) is the weighted average of the partial correlation functions for the three atomic pairs: 0.557gYY(r) þ 0.379gYO(r) þ 0.064gOO(r), whereas gY(r) involves only the Y O and Y Y pair correlation functions: 0.735gYY(r) þ 0.265gYO(r). Five peaks were observed in both g(r) and gY(r): the first peak at 2.26 A˚ in both functions is due to Y O nearest-neighbour pairs; the second peak in g(r) at 3.67 A˚ is due to a combination of O O and Y Y correlations, but in gY(r) it is centred at 3.74 A˚ and is due only to Y Y pairs. Combining these results led to an O O distance of 3.06 A˚. Gaussian fits to these peaks gave 6.2  0.5 and 11.8  1.5 for the Y O and Y Y coordination numbers, respectively. These values, slightly below 7 and 12, imply that the liquid retains the local structure of the high-temperature solid H phase, supported by the fact that the sharp main peak in the liquid S(Q) of the liquid coincided with the closely spaced [002] and [101] Bragg peaks that are the strongest diffraction features of the H phase. Cristiglio et al. (2007a) have made neutron diffraction measurements on levitated Y2O3 at almost the same temperature, 2800 K. The two sets of results are compared in Fig. 8.14. Remarkably, the very sharp main peak in the X-ray weighted average structure factor SX(Q) is scarcely visible in the

178

Molten oxides Y2O3 2770 K

2.2 O–O Y–O Y–Y

2.0 1.8 G(r ) and G Y (r )

G(r) (16.75 keV) 1.6 1.4 Y–Y 1.2 1.0 0.8

yttrium pair correlation function G Y(r)

0.6 0

2

4

6

10

8

12

14

r (Å)

Fig. 8.13. Average pair correlation function measured at 16.7 keV and yttrium pair correlation function for molten Y2O3 at 2770 K. The upper curve has been shifted up by 0.5 for clarity (Hennet et al., 2003). 2.0 2

1

S N (Q )

S X (Q )

1.5

X-rays (2770K)

1.0 0 Neutrons (2800K)

–1

0.5 0

2

4

6

8

10

12

14

16

Q (Å–1)

Fig. 8.14. X-ray and neutron weighted average structure factors for molten Y2O3 (Cristiglio et al., 2007a).

8.2 Silica

179

neutron weighted average structure factor SN(Q). This is because Y Y correlations are de-emphasized, and O O ones emphasized, in the neutron case: SX ðQÞ ¼ 0:56SYY ðQÞ þ 0:37SYO ðQÞ þ 0:07SOO ðQÞ; SN ðQÞ ¼ 0:22SYY ðQÞ þ 0:50SYO ðQÞ þ 0:28SOO ðQÞ:

ð8:8Þ

Although the Q ranges and hence the r-space resolutions are almost the same in the two measurements, the O O and Y Y peaks can be resolved in the neutron g(r) because they are of similar magnitude. From the areas of these peaks values of 6.3  0.5 and 11.8  0.5 were obtained for the Y O and Y Y coordination numbers, in excellent agreement with the AXS result. The O O coordination number was estimated as 9.0  0.5. The preservation in Y2O3 of the high-temperature solid structure on melting is consistent with the relatively small changes in volume (11%, Granier & Heurtault, 1983) and entropy (0.031 kJ mol 1, Foex, 1977). This behaviour contrasts strongly with that of Al2O3 which, as we have seen, exhibits a large density change on melting, accompanied by a coordination change from 6 to approximately 4.5. These differences are accounted for by the higher ionicity of Y2O3 relative to Al2O3 on the phenomenological chemical scale of Pettifor (1986). The very different behaviour of the two end members may be part of the reason for the tendency to polymorphism in Y2O3 Al2O3 liquids that we will discuss later in this chapter.

8.2 Silica Molten silica and silicates are of considerable geological interest and many technologically important glasses are made by quenching from these melts. While it might appear that levitation techniques are ideal for reaching temperatures above the melting point 1986 K for pure SiO2 the rapid vaporization of SiO2 has discouraged such attempts. By using large-area image-plate detectors, Mei et al. (2007) were able to make diffraction measurements, even though a 3 mm diameter sphere of liquid SiO2 lost approximately half its mass in 2 min during the measurements of the liquid phase. Figure 8.15 shows the X-ray weighted average structure factors obtained for the glass at room temperature, the hot glass at 1873 K (the liquid glass transition temperature Tg is normally taken as 1450 K), and the melt at 2373 K. The main effects of heating the glass are seen to be a lowering and broadening of the FSDP and a change in shape of the second peak (enlarged in the inset), effects seen in previous diffraction measurements on the glass.

180

Molten oxides 1.6

1.2

4

SX (Q)

4.4

4.8

5.2

Liquid 2100 °C 2 Hot glass 1600 °C

Glass 25 °C

0 0

5

10

15

Q (Å–1)

Fig. 8.15. Measured X-ray average structure factors for glass and liquid SiO2. Inset: the region of the second peak in S(Q) for the glass at room temperature (dashed curve), hot glass (dotted curve) and liquid (solid curve). The star at Q ¼ 0, S(Q) ¼ 0.0123 represents the value of the measured isothermal compressibility for the liquid at Tg (Mei et al., 2007).

The additional changes on going through the glass liquid transition on heating the sample by an additional 500 K are relatively minor. Figure 8.16 compares the X-ray average pair distribution functions of the room-temperature glass and the liquid. Again, the main differences are in the first peak, due to Si O correlations (shown in the inset), which appears to be shifted to higher r in the liquid with a slightly increased intensity, and in the small second peak due to O O correlations, also shifted slightly to higher r. The Si O distance is calculated to be 1.597  0.005 A˚ in the roomtemperature glass and 1.626  0.005 A˚ in both the hot glass and the melt; the corresponding coordination numbers are 3.90  0.20 and 3.88  0.20, respectively. The commensurate increase of the Si O and O O distances implies that the tetrahedral shape of the SiO44 units is maintained, with an expansion on heating the glass and little further change at the liquid glass transition. SiO2, as another tetrahedrally coordinated system, is predicted to undergo an LDA HDA first-order transition under pressure (Lacks, 2000).

181

8.3 Mixed trivalent oxides 10

12

8 6 1.4 1.6 1.8 Experiment

G x ( r)

8

Classical MD [3]

4

Ab initio MD [2]

0

2

4

6

r (Å)

Fig. 8.16. X-ray average pair distribution functions for room-temperature glass (dashed curve) and liquid SiO2 (solid curve). Upper curves: experiment; middle and lower curves: results from MD simulations in the literature. Inset: region of the first (Si–O) peak in the experimental pair distribution functions (Mei et al., 2007).

8.3 Mixed trivalent oxides As already mentioned, (Y2O3)x(Al2O3)1 x mixtures account for a considerable amount of the interest in trivalent oxide melts and glasses, due partly to the technological importance of doped YAG the crystalline compound at x ¼ 0.375 as the host material in solid-state lasers, phosphors, light-emitting diodes and scintillators, and partly to the scientific interest in glass-forming liquids in the range x  0.24 0.32, one of the earliest examples of a first-order liquid liquid phase transition (Aasland & McMillan, 1994). Paradis et al. (2005b) used the hybrid electrostatic aerodynamic levitation system described in Section 2.5 to measure the densities d and work functions ’ of several molten ceramics. The furnace was operated at a pressure of 450 kPa UHP nitrogen to reduce evaporation. For the YAG composition they obtained at the melting point, Tm ¼ 2240 K, the values ’ ¼ 8.0 eV and d ¼ 4.08 0.29.10 3(T Tm) gm.cm 3.

182

Molten oxides Temperature, K 14

2500

2000

1750

1500

1250

Log (viscosity, Pa . s)

SiO2 YAG

10

6

2

–2

4

5

6 1E4/T, K–1

7

8

Fig. 8.17. Viscosity vs. temperature for molten YAG compared with that for molten SiO2 (Nordine et al., 2000).

Nordine et al. (2000) plotted the temperature dependence of the viscosity of molten YAG (Fig. 8.17) using data from three different experiments. The lowest three points around   0.04 Pa·s resulted from direct measurements on contained samples at temperatures above the melting point (Fratello & Brandle, 1993). The middle point is an estimated viscosity of 30 300 Pa·s at 1600 1650 K, the temperature at which glass fibres can be pulled from the undercooled liquid (Weber et al., 1998), while the highest point at 1012 Pa·s represents the glass transition at Tg  1300 K (Aasland & McMillan, 1994). The nonlinear nature of the viscosity vs. temperature curve indicates that molten YAG is a ‘fragile’ liquid (Angell, 1991), compared with molten SiO2 which, as a ‘strong liquid’, exhibits a straight-line behaviour on an Arrhenius plot. Coutures et al. (1990), in the NMR work already cited in connection with molten Al2O3, also measured the molten YAG composition and deduced that 80% of the Al3þ ions were in four-fold coordinated sites. This contrasts with crystalline YAG where 60% of the Al3þ ions are four-fold coordinated and 40% six-fold. The first XRD study of molten YAG was made by Weber et al. (2000a), who identified peaks in the average g(r) at 1.75 1.80 A˚ and 2.24 2.28 A˚, assigned to Al O and Y-correlations, respectively, in agreement with the corresponding values for the pure oxide melts discussed in the previous section. The coordination numbers obtained were consistent with predominantly four-coordinated Al3þ and six-coordinated Y3þ ions. Hennet et al. (2006) made both ND and XRD measurements, taking advantage of the

183

8.3 Mixed trivalent oxides Neutrons 1.0 X-rays

S(Q)

0.5 1.2 S(Q)

0.0

0.8 Simulation Experiment

0.4 –0.5 0.0

YAG 2373 K

0

1

2

3 4 Q (Å–1)

5

6

–1.0 0

2

4

6

8

10

12

14

16

18

20

22

24

Q (Å–1)

Fig. 8.18. Neutron- and X-ray-weighted average structure factors for molten YAG at 2373 K. The inset shows the comparison between the neutron data and the AIMD simulation results in the low-Q region (Hennet et al., 2006).

different weighting factors for the three component atoms between neutrons and X-rays, as well as the larger Q range, and hence greater r-space resolution, accessible in the neutron measurements. In addition they carried out AIMD numerical simulations. Figure 8.18 shows the neutron- and X-ray-weighted average S(Q)s. The first peak in the X-ray S(Q), very sharp as in pure molten Y2O3, is almost absent in the neutron S(Q), showing that it results from cation correlations, while the first peak in the neutron S(Q) is totally absent from the X-ray S(Q), showing that it results from oxygen correlations. The inset shows good agreement between the neutron data and the AIMD simulation results in the low-Q region. Figure 8.19 shows the experimental and simulated neutron-weighted average g(r). The Al O nearest-neighbour distance was found at 1.81 A˚ with a coordination number of 4.3. The inset shows the simulated gYO(r) in the region of the first peak. A shoulder on the right part is clearly visible, showing the existence of two Y O sites. The area of the double peak gives a total Y O coordination number of 7.3, whereas considering only the main peak at 2.33 A˚ gives a value of about 6.3 as in the X-ray studies, showing that the coordination numbers directly calculated from the experimental X-ray and neutron g(r)s are underestimated. More detailed results from the simulation made by Cristiglio et al. (2007b) show that 69% of the Al3þ ions are in three- and four-coordinated sites, with the remainder

184

Molten oxides YAG 2373 K 1.5 Experiment

G (r )

Simulation 1.0

G Y–O(r )

Y– O partial 0.5

1

2

r (Å)

3

4

0.0 1

2

3

4

5

6

7

8

r (Å)

Fig. 8.19. Experimental and simulated neutron-weighted average pair correlation function for molten YAG at 2373 K. The inset shows the simulated Y–O partial average pair correlation function in the region of the first peak (Hennet et al., 2006).

principally in five-coordinated with a small component of six-coordinate sites. The Y3þ ions are predominantly in six- and seven-coordinated sites, with smaller components of five- and eight-coordinated sites. The distributions of O Al O and O Y O bond angles are generally similar to those of crystalline YAG. Cristiglio et al. (2007c) extended the ND experiments of Hennet et al. (2006) to the x ¼ 0.15 and 0.25 compositions and in addition carried out AIMD simulations for x ¼ 0.15, 0.20 and 0.25. A typical result is shown in Fig. 8.20 for x ¼ 0.25, where the agreement between the ND experiments and AIMD simulations is seen to be good in both real and reciprocal space. The main effect of going to lower values of x was a reduction in the Y O average coordination from about 7 to about 6. Again, these are higher than the values obtained directly from the experimental g(r) because of the shoulder arising from the second Y O distance. As already mentioned, (Y2O3)x(Al2O3)1 x mixtures have attracted considerable interest as one of the earliest found examples of a liquid liquid transition (LLT). Aasland & McMillan (1994) observed a second liquid phase that appeared spontaneously on quenching liquid mixtures in the glass-forming

185

8.3 Mixed trivalent oxides Q (Å–1) 0 3.5

4

8

12

16

1.5 AY15 S(Q)

3.0

1.0

2.5

Experiment AIMD simulation

0.5 g (r)

2.0 Experiment 1.5 AIMD Simulation 1.0 Partials:

AIO YO

OO AIAI

YAI YY

0.5 0.0 1

2

3

4

5

6

7

8

r (Å)

Fig. 8.20. Neutron-weighted average pair distribution function for liquid (Y2O3)0.15(Al2O3)0.85 at 2373 K: comparison between experiment and AIMD simulations (the experimental curve is shifted up). The lower curves depict the six partial pair distribution functions obtained from the simulations. The inset shows the experimental neutron-weighted average structure factor compared to the simulation results (Cristiglio et al., 2007c).

region, x  0.24 0.32 in contained samples. This second phase formed bubbles that grew during the quench, until both phases were frozen at Tg (Fig. 8.21). The liquid liquid separation temperature increased slightly with decreasing x. The compositions of the two phases in the quenched glasses were determined by electron probe microanalysis and found to be identical to within experimental error (2s  2 relative atom%). Backscattered electron images showed that the inclusions had lower electron density, and hence lower mass density if the compositions were the same, and optical microscopy showed that they had a lower refractive index. Although the liquid phases could not be studied quantitatively, these results showed that a first-order liquid liquid transition at constant composition was occurring, as opposed to the more common phase separation into two liquids of different composition. Weber et al. (1998) were able to pull thin optical fibres from aerodynamically levitated drops of molten (Y2O3)x(Al2O3)1 x when slightly off the YAG

186

Molten oxides

Fig. 8.21. Transmitted-light optical micrograph of a (Y2O3)0.301 (Al2O3)0.699 sample showing the glassy inclusion phase in a glassy matrix of the same composition (Aasland & McMillan, 1994).

Fig. 8.22. Photograph of glass fibres pulled from levitated molten YAG at 1600–1660 K, 600 K below the melting point of crystalline YAG. The fibres are 20 mm in diameter (Weber et al., 1998).

stoichiometry (x ¼ 0.375) or when doped with 1 mol% of Nd2O3 or Er2O3 (Fig. 8.22). The fibres were drawn by inserting and rapidly removing a 100-mm diameter tungsten wire stinger. Fibres were obtained only if stinging was initiated when the melt temperature was in the range 1600 1660 K: at higher temperatures, the stinger pulled out of the melt without forming a fibre while, at lower temperatures, the drop was pushed away by the stinger. This temperature is considerably above the glass transition temperature at the YAG composition (1135 K according to Wilding et al., 2002b). At the

187

8.3 Mixed trivalent oxides

Transmittance, %

100 80

CaAl2O4

60 40 YAG, 2-phase glass 20 0

2

3

4

5

6

Wavelength, 1000 nm

Fig. 8.23. Spectral transmission of a single-phase CaAl2O4 composition glass and two-phase Y3Al5O12 (Weber et al., 2002).

YAG stoichiometric composition, fibres were rarely obtained, and the drops always crystallized when cooled below 1400 K, whereas the doped and off-stoichiometric samples generally formed glasses. In further studies, Weber et al. (2000b) obtained two-phase glasses at the YAG composition in molten Y3Al5O12 and Er3Al5O12, using both aeroacoustic and aerodynamic levitation. However, substitution of the rare earth by La suppressed the LLT, explained by the tendency of the larger cation to favour four-fold coordination. Weber et al. (2002) fabricated glasses with a wide range of rare-earth dopants and proceeded to measure their optical properties, concluding that both the optical and physicochemical properties of these glasses were well suited for optical device applications. Representative results are shown in Fig. 8.23, where the behaviour of the CaAl2O4 melt, taken as a reference, is typical of that observed in the single-phase undoped glasses, while the transmission of the two-phase glasses is decreased by scattering in the two-phase medium. Nagashio and Kuribayashi (2002a) made similar studies of quenching aeroacoustically levitated liquids at x ¼ 0.25 0.375 and found from XRD that the inclusions were composed of crystalline YAG rather than an amorphous phase. Skinner et al. (2008) made similar studies on aerodynamically levitated liquids in the composition range 0.21 0.41. They obtained single-phase glasses in the range x ¼ 0.27 0.33. At higher values of x, the glasses had YAG inclusions. At x ¼ 0.21, two-phase samples were produced in which the inclusions consisted of a multiphase crystalline material containing yttrium aluminium perovskite (YAP, x ¼ 0.5) but not YAG. At x ¼ 0.24 the samples had a cloudy appearance with sub-micron inclusions, but it was not possible to establish from the diffraction pattern whether these were (i) a second glassy phase of

188 Heat capacity (displaced) / J/g.K

Molten oxides 60 40 20 0

AY-20 AY-24 AY-28

–20 –40

AY-32

–60 –80 800

1000

1200 Temperature/K

1400

Fig. 8.24. DSC results for (Y2O3)x(Al2O3)1 x glass samples for x ¼ 0.20, 0.24, 0.28 and 0.32, heated from room temperature to 1500 K. The exothermic peak at 1200 K represents crystallization from the HDL phase, while the higher temperature exothermic peak is interpreted as a transition from the HDL phase to a more stable LDA glass; this peak merges with the peak at 1200 K as x increases (Wilding et al., 2002b).

different density but the same composition as that arising from an LLT, (ii) a second glassy phase of different composition as that arising from spinodal decomposition or (iii) a large number of sub-micron nanocrystals giving Bragg peaks with considerable particle size broadening. They observed that this sample, like some of those of Weber et al. (2000b), had holes in the interior, which could suggest a region of internal tension, possibly relevant to the inferred presence of the LLT critical point at negative pressure. Detailed microscopic studies of the HDL and LDL phases began in earnest after the year 2000. Wilding et al. (2002b) made calorimetric studies of the various transitions involved with DSC. Results for glasses with x ¼ 0.20, 0.24, 0.28 and 0.32 heated up from room temperature are shown in Fig. 8.24. Around 1150 K there is a shallow endothermic peak (most visible in the 28% composition) representing the glass liquid transition into the high-density liquid (HDL); for x ¼ 0.20 0.28, this is followed by two peaks, one at 1200 K due to crystallization and the second, at a temperature depending on composition, to a transition of the HDL to the more stable low-density amorphous (LDA) solid. Wilding et al. carried out neutron diffraction (2002a) and XRD and 27Al NMR (2002b) measurements on glasses with the x ¼ 0.20 and 0.25 compositions; the first was characterized by a mixture of LDA and HDA phases, the latter by a single HDA phase in which the HDL LDL transition was bypassed in the quench. Weber et al. (2004) made a similar comparison with neutron and X-ray diffraction studies on glasses with x ¼ 0.27 and 0.375, identified as single-phase and two-phase samples,

8.3 Mixed trivalent oxides

189

respectively. In both sets of experiments, small differences in the Y O correlations were identified, and Wilding et al. also found significant changes in the correlations involving the metal atoms. The latter results were combined with MD simulations in the liquid phase by McMillan et al. (2003), who also measured the densities of the two glasses and found that the HDA glass at x ¼ 0.24 was 4% denser than the LDA. This would seem to support the identification of the inclusions as an amorphous phase, since the YAG crystal and a metastable eutectic at x ¼ 0.24 have densities that are higher by 22% and 7.5%, respectively. The MD simulations of the two liquids showed no significant differences (beyond the effects of different compositions) in the metal oxygen coordination but a marked difference in Y Y correlations. An interesting feature of these simulations was a density fluctuation in the x ¼ 0.20 liquid with a surprisingly long time constant, 0.1 ns, and an amplitude s.d. of 0.7%. Although the temperature of the simulations was 2400 K, well above the LLT, the authors suggested that the liquid might be sampling in recurrent mode configurations characteristic of the LDL and HDL. Analysis of the cation cation coordination environments in the two limits showed that, on average, the Y cations in the low-density limit were coordinated to a greater number of Y ions than in the high-density limit, i.e. the Y component was more clustered in the low-density limit and more uniformly distributed in the high-density regime. This interpretation would be consistent with the LDL phase having lower configurational entropy than the HDL polyamorph. While microscopic data on the glasses quenched from liquids with two coexisting phases are illuminative, there is clearly no substitute for measurements performed in situ on the liquids themselves. Greaves et al. (2008) were able to carry out both small-angle (SAXS) and wide-angle (WAXS) X-ray measurements on aerodynamically levitated samples at the x ¼ 0.20 and 0.25 compositions. Figure 8.25, which can be directly compared with the P ¼ 0 intercept in the generic phase diagram given in Fig. 7.14(b), represents the scope of the problem. In order to observe the two-phase behaviour, it was necessary to supercool the liquids by more than 300 K to get below the calculated LLT temperature of 1788 K. Figure 8.26 shows the results of SAXS measurements, exhibiting a sharp rise at small Q in a narrow temperature range around 1788 K for the x ¼ 0.20 composition that can be interpreted as direct evidence for the coexistence of LDL and HDL states with a density difference DrLL at TLL. The WAXS results also indicated an LLT at the x ¼ 0.20 composition: the height of the FSDP showed a sudden increase in cooling below 1788 K that implies a decrease in density (Moss & Price, 1985), as opposed to the density increase normally expected on cooling.

190

Molten oxides 2200 Tm 2100 K 2000 TLL 1788 K

T/K

1800

HDL

C 1804 K, –0.31 GPa

LDL

1600

Tc 1520 K in situ Tc 1423 K rapid cooling

1400

1200 Tg 1150 K

0

1 P/GPa

2

Fig. 8.25. P–T phase diagram for (Y2O3)0.2(Al2O3)0.8 showing the line separating the HDL and LDL phases surrounded by those for the spinodal limits. The dashed curves represent calculations from a two-state model, while the solid curves indicate dT/dP ¼ DVLL/DSLL, as determined from the changes in entropy DSLL and molar volume DVLL. This places the critical point C at 1804 K and 0.31 GPa. The melting point Tm and Tg for the HDA phase are also shown, together with experimentally observed crystallization temperatures Tc (Greaves et al., 2008).

A further interesting feature was observed on these experiments: depending on the precise laser alignment, regular oscillations in temperature were sometimes observed around 1800 K, as illustrated in the centre panel of Fig. 8.27. Video imaging revealed the supercooled drop revolving through 180 about a horizontal axis at the start of each cycle (left panel), the movement coinciding with a temperature spike. The authors proposed a ‘polyamorphic rotor’ model for this unusual behaviour in which the rotation of the supercooled drop is driven by the LLT occurring within the levitation nozzle adjacent to the upward gas flow. When T < TLL, the liquid within the nozzle switches abruptly to the LDL phase. Since Dr/r < 0, the drop is destabilized, resulting in the low-density zone at the bottom flipping to the top (right panel), the high viscosity of the LDL phase maintaining the rigidity of the drop. Tangeman et al. (2007) used aerodynamic levitation and laser melting techniques to synthesize transparent nanophase glass ceramics of Al2O3 Y2O3 La2O3 and Al2O3 Y2O3 La2O3 SiO2. In the first case, nucleation of

8.3 Mixed trivalent oxides (a)

191

AY25

I(Q)/arbit. units

2203K 1663K

10000

(b)

AY20 1907K 1800K 1786K 1515K

(c)

AY15

I(Q)/arbit. units

10000

1000

I(Q)/arbit. units

2150K 1964K 10000

1000

0.1 Q/Å–1

Fig. 8.26. SAXS data for supercooled (Y2O3)x(Al2O3)1 x liquids for x ¼ 0.15, 0.20 and 0.25. The intensity I(Q) is plotted against Q on a log–log scale. Note the rise at small Q for the x ¼ 0.20 liquid in the vicinity of 1788 K (Greaves et al., 2008).

LaAlO3 perovskite nanocrystals in the viscous undercooled liquid caused a volume decrease that resulted in a crinkled appearance of the glass ceramic pieces. With the addition of SiO2, single-phase glass spheroids were formed and nanophase nucleation was then achieved by heating the glass into the supercooled liquid regime.

192

Molten oxides T > TLL

F HDL

1900

HD

F E D

1880

D

L

LD

E

L

Brightness Pyrometer Model

1820

D LD

1780 Rotation

L

A

H

1800 B

HDL

C

C B A

1840

LDL

T/K

1860

L

1760 0

2

4

6

8

HDL

t/s LDL T < TLL

Fig. 8.27. Left panel: video images illustrating the horizontal rotation of the levitated drop of supercooled (Y2O3)0.2(Al2O3)0.8; centre panel: fluctuating image compared with pyrometer output for one 4-s cycle; the 180 rotation takes 600 ms, after which time the drop is virtually stationary; the horizontal dashed lines indicate the temperature limits of the two-phase region; right panel: polyamorphic rotor model: the LLT occurs repeatedly at the bottom of the sphere whenever T < TLL, the mechanical instability causing the LDL zone to rotate to the top where it transforms back to HDL in the laser beam; the dotted lines in the centre figure indicate the temperature variation calculated from this model (Greaves et al., 2008).

Nagashio & Kuribayashi (2002b) studied the solidification of undercooled rare-earth orthoferrites REFeO3 with RE ¼ La, Sm, Dy, Y, Yb and Lu. Observation using a high-speed video camera revealed that the formation of the metastable phase became pronounced, and double recalescence from the metastable phase to a stable phase occurred, as the ionic radius of the rare-earth element decreased. Nagashio et al. (2006b) followed up with fast X-ray diffraction measurements of undercooled samples with RE ¼ Y and Lu suspended by CNL. The in situ diffraction pattern of the metastable phase in YFeO3 was consistent with that of the metastable hexagonal LuFeO3 phase, which could be recovered on cooling to room temperature.

8.4 Divalent trivalent oxide mixtures

193

8.4 Divalent–trivalent oxide mixtures (CaO)x(Al2O3)1 x mixtures represent another technologically important system, forming glasses in a narrow range of composition (x ¼ 0.6 0.7). In addition to good mechanical properties, these glasses present interesting optical characteristics, in particular high transparency in the mid-IR range up to 6 µm. In the past they have been considered for technological applications such as waveguides for infrared lasers, but owing to their high melting points and the need for rapid quenching they are relatively difficult to prepare. Furthermore, a tendency to devitrify easily has limited their use as common optical materials. Some of these problems can be overcome by the addition of a small amount of silica that extends the glass-forming region and lowers the liquidus temperature. In order to study such effects, there has been considerable effort in a reliable representation of the silica-free mixtures in both the liquid and glass state. (CaO)x(Al2O3)1 x was the subject of considerable early 27Al NMR work on both levitated liquid and glass samples. Coutures et al. (1990) measured the x ¼ 0.5 melt along with liquid YAG and Al2O3 and found that the proportions of Al3þ ions in tetrahedral sites were around 95%, 80% and 60% in CaAl2O4, YAG and Al2O3, respectively, so that admixture with the larger cations favours the four-fold coordination of the aluminium. Subsequent NMR measurements were made on (CaO)0.5(Al2O3)0.5 and (MgO)0.5(Al2O3)0.5 (Poe et al., 1993), (CaO)x (Al2O3)1 x, x ¼ 0 (0.1) 0.7 (Poe et al., 1994), and (CaO)0.5(Al2O3)0.5 on cooling from the liquid through the glass transition (Massiot et al., 1995). The NMR spectra in (CaO)0.5(Al2O3)0.5 and (MgO)0.5(Al2O3)0.5 consisted of a single narrow line, indicating that the four-, five- and six-fold coordinated Al species are averaged by rapid chemical exchange at a rate faster that that defined by their chemical shift (>5 kHz). The proportion of four-coordinated Al was higher in the Ca mixture that in the Mg, again because of the larger size of the metal cation. In the (CaO)x(Al2O3)1 x measurements a steady increase in isotropic chemical shift with increasing x was observed, indicating a decrease in number of four-fold coordinated Al. In the temperature-dependent studies of (CaO)0.5(Al2O3)0.5 a steady increase in isotropic chemical shift on cooling was observed (Fig. 8.28), interpreted as a decrease by 0.2 per 1000 K of the number of four-fold coordinated Al on cooling to a value of around 4.0 at Tg ¼ 1180 K  4 K. Massiot et al. also measured the T1 and T2 relaxation times as a function of temperature. From the former a correlation time tc for the quadrupolar interaction could be deduced: 1 3 2I þ 3 C2 t c ; ¼ p2 2 T1 10 I ð2I 1Þ Q

ð8:9Þ

194

Molten oxides supercooled liquid

liquid

d (ppm)

90

80

70

Tg 60 1100 1300

mp 1500

1700

1900

2100

2300

2500

T (K)

Fig. 8.28. Evolution of the 27Al NMR chemical shift versus the temperature in liquid (CaO)0.5(Al2O3)0.5. The open squares represent the data obtained during the cooling of the levitated sample, and solid square represents the value derived from an MAS spectrum of the glass. The bars represent the line width of the NMR line (Massiot et al., 1995).

where CQ is the dynamic quadrupolar product and I the nuclear spin. Using the value of CQ ¼ 6.4 MHz obtained from MAS measurements on the glass, values of tc were computed from Eq. (8.9). To compare with these, characteristic relaxation times were obtained from the Maxwell relation ts ¼

s ; G1

ð8:10Þ

where s is the macroscopic shear viscosity, taken from values measured by Urbain (1983) on contained samples, and G1 the high-frequency shear modulus, taken as 1010 Pa (Angell, 1991). The values of tc and ts thus derived are compared in Fig. 8.29. The two quantities are in reasonably close agreement in both magnitude and temperature dependence, showing that the fluctuation mechanism inducing NMR quadrupolar relaxation of 27Al is similar to that involved in the mechanical relaxation. Capron et al. (2001) carried out similar measurements on liquid (SrO)x(Al2O3)1 x at several compositions. In contrast to (CaO)0.5(Al2O3)0.5, the temperature dependence of the isotropic chemical shift showed a significant upturn from the linear behaviour at low temperature for x  0.333. As with (CaO)x(Al2O3)1 x, however, the concentration dependence of the shifts in the linear region showed a linear increase with x, indicating a decrease in Al coordination number. The line width, which can be taken as roughly proportional to the correlation time tc (Eq. (8.9)), decreases with temperature with a variation similar to that shown in Fig. 8.29 for (CaO)0.5(Al2O3)0.5. Plotted at a

195

8.4 Divalent trivalent oxide mixtures In(t) 15

–23

10

t

1000/T (K)

–27 0.40

–11

(s)

–25 10 0.50

0.60

5

0 1600

1800

2000

2200

2400

T (K)

Fig. 8.29. Evolution with temperature of the correlation times tc temperature in liquid (CaO)0.5(Al2O3)0.5 derived from 27Al NMR relaxation times (Eq. (8.9), empty squares) and the relaxation times ts derived from measurements of the shear viscosity (Eq. (8.10), full circles). The line represents a fit to the viscosity data of Urbain (1983). The inset shows an Arrhenius plot of the same data (Massiot et al., 1995).

fixed temperature (2373 K) against x, the line width exhibits a maximum at x ¼ 0.4, close to the composition (x ¼ 0.5) where the activation energy for viscosity reaches a maximum in (CaO)x(Al2O3)1 x. liquids (Urbain, 1983). Weber et al. (2003) made neutron diffraction measurements on aerodynamically levitated liquid (SrO)0.5(Al2O3)0.5 and (CaO)0.5(Al2O3)0.5. Although the statistics were limited by the relatively modest source flux, the results showed that both liquids are composed of predominantly AlO45 units, and were consistent with eight-fold coordinated Sr and six- or higher coordinated Ca species. Hennet et al. (2007b) have carried out both X-ray and neutron diffraction measurements on liquid (CaO)0.5(Al2O3)0.5. The two structure factors are shown in Fig. 8.30. The first peak at 2.15  0.02 A˚ 1 is much stronger in the X-ray S(Q) than the neutron one, indicating that it is due to cation correlations. The insert shows the Q value of the first peak from X-ray measurements on (CaO)x(Al2O3)x(SiO2)1 x glasses (Petkov et al., 1998), which show a continuous development from the position of the FSDP in pure SiO2, supporting the assignment of the first peak in liquid (CaO)0.5(Al2O3)0.5 to such a feature. The following peak at 2.62  0.02 A˚ 1 in SN(Q), which does not appear in SX(Q), can be attributed to O O correlations whose weighting factor is higher with neutrons. The corresponding pair correlation functions gX(r) and gN(r) gave an Al3þ coordination number of 4.4  0.5 in good agreement with the NMR measurements discussed above. The second

196

Molten oxides FSDP position (Å–1)

3.0 CaAl2O4 2173 K 2.5

2.0

2.1

(CaAl2O4)x(SiO2)1–x

1.9 1.7 1.5 0.0

0.2

0.4

0.8

0.6

1.0

S(Q )

x S x(Q)

1.5

1.0 S N(Q) 0.5

0.0 0

2

4

6

8

10

12

14

16

18

20

22

Q (Å–1)

Fig. 8.30. X-ray and neutron structure factors SX(Q) and SN(Q) for liquid (CaO)0.5(Al2O3)0.5 at 2173 K. The upper curve is shifted up by 0.5 for clarity. The inset shows the evolution of the FSDP position as a function of the composition x in (CaAl2O4)x(SiO2)1 x glasses taken from Petkov et al. (1998) (Hennet et al., 2007b).

Gaussian gave a Ca O coordination number of 5.5  0.5, a value similar to that obtained for the glass in measurements of Benmore et al. (2003). Cristiglio et al. (2008) have compared the neutron results on liquid (CaO)0.5(Al2O3)0.5 with their AIMD simulations and obtained good agreement in both reciprocal and real space. Their values for the Al O and Ca O coordination numbers were 4.1  0.3 and 6.2  1.1, respectively. The Al value is lower, and the divalent cation value higher, than those obtained from their simulations of (Y2O3)x(Al2O3)1 x melts discussed above, both effects being a consequence of the larger ionic radius of Ca2þ. As in the (Y2O3)x(Al2O3)1 x case, the simulation gives a higher value for the Ca O coordination, calculated with the large cut-off radius of 3.3 A˚, due to the long tail of the partial pair correlation function. The bond angle distributions are shown in Fig. 8.31, giving a graphic illustration of the different coordinations involved. The four-fold and five-fold coordinated Al ions are seen to lie predominantly in tetrahedral and octahedral sites, respectively, while the environments of all

8.4 Divalent trivalent oxide mixtures (a)

197

O–Al–O Total

[a.u.]

CN = 4

CN = 5 CN = 6 (b)

O–Ca–O

[a.u.]

Total

CN = 6

CN = 5

CN = 7 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Angle (°)

Fig. 8.31. Bond angle distributions in liquid (CaO)0.5(Al2O3)0.5 at T ¼ 2273 K for cations with different oxygen coordination numbers CN: (a) O–Al–O; (b) O–Ca–O (Cristiglio et al., 2008).

the Ca ions have a broad spread with a maximum around the cubic value of 70 . Similar results were obtained for liquid (MgO)0.5(Al2O3)0.5 but with a higher value for the Al coordination and a lower one for the divalent cation coordination, again consistent with the smaller cation radius (Hennet et al., 2007b, c). In order to study the evolution behaviour of the structure during the cooling of liquid (CaO)0.5(Al2O3)0.5, Hennet et al. (2007a) performed measurements at intervals of 100 ms after turning off the laser power using a 60 -aperture fast X-ray detector. Cooling was complete after about 6 s and cooling from the melting point temperature Tm down to Tg took about 2.7 s, corresponding to a quench rate of 360 K s 1. Figure 8.32 shows the evolution of the width and height of the first peak of S(Q) during the supercooling process. In the figure, Tm and Tg are indicated by solid lines; the value of Tm was taken as 1878 K and Tg was taken from the last inflexion point common

198

Molten oxides Tg /T 0.6

0.7

0.8 0.9 1.0

Tm

1.5 2.2

1467 K

0.5 1.1

Tg Glass 2.0

Supercooled liquid

Liquid

1.8

Height (a.u.)

Width (Å–1)

1.0

0.9 1.6

0.8 2400

2100

1800

1500

1200

900

1.4 600

T (K)

Fig. 8.32. Dynamical evolution of the width and intensity of the first peak of S(Q) during the cooling of liquid (CaO)0.5(Al2O3)0.5 from 2173 K to room temperature. The melting Tm and glass transition Tg temperatures are shown by solid lines. The dashed line indicates the temperature 1467 K at which the structural evolution begins (Hennet et al., 2007a).

to the two curves: its value of 1173  30 K agrees with that (1180  4 K) obtained by DSC (Massiot et al., 1995). Neither width nor intensity show any significant change down to a temperature of about 1467 K (1.25Tg), indicated by a dashed line in the figure, after which the width decreases linearly down to Tg, where it stabilizes. The intensity starts to increase linearly at the same point and shows a lower slope after Tg. The Al O coordination number, derived under the approximation that the Ca O coordination number is constant, is shown in Fig. 8.33 along with the values of the position r1 of the first peak. As observed for the S(Q), changes occur after a temperature of 1467 K in the supercooled state, up to which the coordination number remains constant and after which it starts to decrease. The first peak position also starts to decrease at 1467 K down to a value of 1.76 A˚ in the glass that agrees with the results of Benmore et al. (2003). This decrease in the first peak position is consistent with the decrease in coordination number since the presence of AlO5 and AlO6 observed by Poe et al.

199

8.4 Divalent trivalent oxide mixtures Tg /T 0.6 Tm

1.5 5.2

Tg

1.8

4.8

1.7

4.4

1.6

4.0 Supercooled liquid

Liquid 1.5 2400

2100

1800

1500

Al–O coordination number

First peak position r1 (Å)

0.8 0.9 1.0

0.7

1467 K

0.5 1.9

Glass

1200

900

3.6 600

T (K)

Fig. 8.33. Dynamical evolution of the first peak position r1 of the pair correlation function T(r) and of the Al–O coordination number in liquid (CaO)0.5(Al2O3)0.5. The melting Tm and glass transition Tg temperatures are shown by solid lines. The dashed line indicates the temperature 1467 K at which the structural evolution begins (Hennet et al., 2007a).

(1993) in the liquid phase increases the mean Al3þ ionic radius. The inflexion point of the curve is also consistent with the value of 1173  10 K for Tg determined in Fig. 8.32. The onset of the increase in both intermediate-range and short-range order at 1.25Tg is occurring close to the dynamic crossover where the slope of the Arrhenius plot increases, conventionally taken as 1.2Tg for fragile liquids. From a fit of the Vogel Tammann Fulcher expression   ¼ 0 exp

DT0 T T0

 ð8:11Þ

to the macroscopic viscosity data in the literature, Poe et al. (1994) determined a coefficient D ¼ 3.2, characteristic of a very fragile liquid. From this fit, it is possible to show that the change in viscosity becomes more pronounced around 1.2Tg. Over twenty years ago, Angell (1985) pointed out the

200

Molten oxides 70 w 0 fit w 0 exp

Frequencies (meV)

60

C∞aQ

50

wl

40 30 20 10 0 0

2

4 Q (nm–1)

6

8

Fig. 8.34. Scattering vector dependence of the second frequency moment o0 for liquid (MgO)0.5(Al2O3)0.5 derived from the experimental S(Q) and from the fitting parameters of the viscoelastic model, maxima of the longitudinal current correlation ol, and the high-frequency limit for structural relaxation C1aQ. The solid lines are linear fits. The errors are calculated from the least square fits to the scattered intensity (Pozdnyakova et al., 2007).

instability of intermediate-range order as a characteristic feature of fragile liquids. In the energy landscape picture, the deeper minima assumed by the CA melt must correspond to an increase in structural order on both short and intermediate length scales. Furthermore, the cooling curve (evolution of the temperature vs. time, shown earlier in Fig. 5.4) shows a kink at this temperature, indicating a change in thermodynamic properties consistent with the proposed connection between thermodynamic and kinetic fragility (Ito et al., 1999; Martinez & Angell, 2001). Pozdnyakova et al. (2007) measured the composition dependence of the microscopic dynamics of liquid (MgO)x(Al2O3)1 x with x ¼ 0.333 and 0.5. In the IXS spectra taken at 2423  10 K, a triplet structure was observed at the lower Q values, but not as well resolved as the data from pure Al2O3 shown in Fig. 8.7. The values of o0(Q) obtained from a viscoelastic model fit and the experimental values of S(Q) (Hennet et al., 2007b, c) for x ¼ 0.5 are shown in Fig. 8.34 as closed and open squares, respectively. The closeness of the two sets of values supports the validity of the model. The same figure shows the calculated acoustic dispersion relation ol(Q) as open circles, where ol is taken as the position of the maximum of the longitudinal current correlation

8.5 Silicates

201

spectrum C(Q, o), while the values of the excitation frequencies Os(Q) calculated from q ð8:12Þ s ðQÞ ¼ o20 ðQÞ þ D2 ðQÞ are shown as closed circles. These two sets of values are also seen to be in close agreement. The linear fit gave values for the longitudinal sound velocity of 9640  220 and 9270  160 m s 1 for x ¼ 0.333 and 0.5, respectively. A similar analysis of the results for Al2O3 (Sinn et al., 2003) gave 10050  250 m s 1. Thus, the values of longitudinal sound velocity decrease with increasing x as in the solid oxides. As in the case of pure liquid Al2O3, the values of the generalized longitudinal viscosity l(Q) and relaxation time t derived from the viscoelastic model fits did not change significantly with Q. A classical MD simulation was carried out using an advanced ionic interaction potential that has been shown to reproduce well the structure factor, scattering function, self-diffusion coefficient and longitudinal viscosity of molten Al2O3 (Jahn et al., 2004), extended here to the binary system. Including the results of Sinn et al. (2003) for x ¼ 0, the values of l from both experiment and simulation showed a weak maximum around the x ¼ 0.33 composition. This is reminiscent of maximum in the activation energy for viscosity observed in (CaO)x(Al2O3)1 x liquids (Urbain, 1983), in that case at x ¼ 0.5. The self-diffusion coefficients of Al3þ and O2 ions obtained from the simulation also showed a minimum at x ¼ 0.33, while the mobility of Mg2þ ions decreased steadily with increasing Mg content, suggesting that addition of relatively small amounts of MgO to liquid Al2O3 stabilizes the [AlO4]5 network. Skinner et al. (2006) report the production of BaAl2O4 and BaAl2TiO6 glasses by CNL and laser heating, followed by structural measurements with high-energy X-ray diffraction. The interest in barium aluminotitanates and related compounds known as SYNROC derives from their possible use for the encapsulation of high-level nuclear waste. The authors suggest that it may be possible to combine the advantages of encapsulation in glass (no grain boundaries) with the advantageous properties of the aluminates (high waste product loading) to form new glasses or glass ceramics for nuclear waste storage.

8.5 Silicates As already mentioned, calcium aluminosilicate glasses are attractive materials owing to their highly refractory nature and their excellent optical and

202

Molten oxides

mechanical properties, so that there is interest in the melts that they are prepared from. We first discuss the (Al2O3)x(SiO2)1 x system, before the divalent oxide is added. Poe et al. (1992) made in situ 27Al measurements on aerodynamically levitated samples with x ranging from 0.1 to 1.0. The chemical shifts varied only slightly with composition, ascribed by the authors to a cancellation between the opposing effects of increased shielding by nextnearest neighbours and increased Al O coordination number with increasing x. The correlation times tc calculated from Eq. (8.9) were in good agreement with those estimated from viscosity data in the literature (Eq. (8.10)) except at high silica content, x < 0.2, where O2 exchange between both silicate and aluminate species may govern the shear relaxation rates, whereas spin relaxation may become the dominant NMR relaxation mechanism. Krishnan et al. (2000) made X-ray diffraction measurements in the liquid at x ¼ 0.6, corresponding to the 3:2 mullite crystalline phase. Supercooling promoted an increase in the concentration of tetrahedral Si4þ ions, manifested by a shift in the first peak in g(r), offset by an increase in octahedral Al3 ions. The authors suggested that clustering of the SiO44 tetrahedral units resulted in a rapid increase in the viscosity of the liquid. There has been considerable NMR work on calcium aluminosilicate glasses (Neuville et al., 2004, and references therein) and also XRD experiments (Petkov et al., 1998). Pozdnyakova et al. (2008) have made X-ray and neutron diffraction measurements on La and Y aluminosilicate glasses. Regarding structural investigations of liquid aluminosilicates, Cote´ et al. (1992) extended the NMR studies discussed previously to [(CaO)y(Al2O3)1 y]x(SiO2)1 x melts and glasses with y  0.53 and x ranging from 0.34 to 1.0. The measured chemical shifts indicated that Al remained four-coordinated in the glasses and that there were Al sites in the melts that were structurally distinct from those in the glasses. Benoit et al. (2001) made AIMD simulations of liquid [(CaO)0.64(Al2O3)0.36]0.33(SiO2)0.67, finding nonbridging oxygens in excess of the number obtained from a simple stoichiometric prediction, links between AlO4 tetrahedra, three-fold coordinated oxygen atoms and five-fold coordinated silicon atoms. These results could provide a guide for future diffraction experiments. In a final example of the use of levitation techniques to prepare materials not accessible with conventional techniques, Kohara et al. (2004) reported the preparation of magnesium silicate glasses at the forsterite composition, Mg2SiO4, which is not possible with melt quench techniques. Normally SiO44 tetrahedral form the three-dimensional network of silicate glasses, with added components such as MgO serving to modify the network. At the forsterite composition, with two Mg ions for every Si, there is an insufficient number of

203

8.5 Silicates 10

Mg-O

Si-O

Mg-Mg Si-Si Mg-Si

O-O crystal

5

T (r )

glass

X-ray

0 10

Mg-O Si-O

5

O-O

crystal

Mg-Mg Si-Si Mg-Si

neutron

glass

0 0

1

2

3

4

5

r (Å)

Fig. 8.35. X-ray- and neutron-weighted average correlation functions T(r) of crystalline and glassy Mg2SiO4 (Kohara et al., 2004).

SiO44 to form a network. X-ray and neutron weighted pair distribution functions T(r) are shown in Fig. 8.35 for the forsterite crystal and the glass. While the Si O coordination retains its tetrahedral character, the Mg O correlations, which form octahedra in the crystal, are shifted to shorter distances and considerably broadened. This shows that the six-fold coordination of Mg ions in the crystal is being replaced by four-, five-, and six-fold (in decreasing numbers) in the glass, providing the network continuity needed to hold the structure intact.

9 Conclusions and prospects

It is hoped that the reader who has conscientiously struggled through the previous eight chapters has acquired a sense of the achievements and potential of investigating high-temperature materials with levitation techniques. The acquisition of reliable thermophysical data on solids and liquids at high temperature must be considered among the major achievements, bearing in mind the difficulties that previous workers had in obtaining consistent and reliable data on contained samples at high temperature, especially those of a corrosive nature. One need only take the example of the density, an apparently humdrum quantity that is not only technologically important, for example in determining the ideal conditions for synthesis of crystalline silicon for the semiconductor industry, but also a vital parameter in materials research: a knowledge of the number density is needed to obtain useful realspace information from diffraction experiments, and furthermore it is the unique quantity that enters into a version of mode-coupling theory that has provided one of the most successful routes to understanding the dynamics of simple liquids, as well as an important parameter in ab initio numerical simulations. A second achievement has been the ability to access metastable solid states. We have encountered several examples of new solid phases, especially glassy phases, that are not accessible with conventional techniques. Undoubtedly such phases will prove to have important technological applications in the optical and optoelectronic industries. From the point of view of fundamental science, some striking accomplishments have resulted from the ability to access the deeply undercooled liquid state. The measurements on liquid silicon discussed in Chapter 6 provide a tantalizing suggestion of an approach to the predicted first-order liquid liquid phase transition, while the results on molten (Y2O3)x(Al2O3)1 x mixtures

204

Conclusions and prospects

205

described in Chapter 8 can be interpreted as direct evidence for the coexistence of low-density and high-density liquid phases. What of the future? From the point of view of more powerful experimental facilities, there can be little doubt that the third-generation pulsed spallation neutron sources the recently commissioned Spallation Neutron Source in Oak Ridge, USA, and the neutron source at the Japan Proton Accelerator Research Complex in Tokai, as well as the recently announced European Spallation Source in Lund, Sweden will provide new opportunities for rapid structural measurements as well as high-resolution dynamical studies. For example, the detailed structure of a magnetically ordered state in a liquid alloy may soon be directly measured. On the X-ray side, the new free-electron laser sources the Linac Coherent Light Source at Stanford, USA, and the European X-Ray Laser Project in Hamburg, Germany will surely provide new capabilities for fast measurements of the crystallization and liquid glass transition in levitated liquid samples which, as we have seen, can be taken into deeply undercooled states. Predictions of new scientific breakthroughs are notoriously unreliable. The author will venture a single suggestion, namely, in view of the enormous surge in interest in new forms of carbon, a detailed investigation of the structure, dynamics and thermophysical properties of liquid carbon would seem to be important (Johnson et al., 2005). To what extent levitation techniques, for example electrostatic levitation at high pressure, or pulse heating methods will be more productive remains to be seen.

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Index

Advanced Proton Source, 54 Al transition metal alloys solidification, 105 Al clusters, 105 Al2O3 conductivity, 162 diffusion, 174 dynamics, 169 emissivity, 163 kinematic viscosity, 173 optical properties, 162 physical properties, 165 shear viscosity, 173, 174 solidification, 169, 175 sound velocity, 171 structure, 165 Al2O3 SiO2, 202 Al2O3 Y2O3 La2O3, 190 Al2O3 Y2O3 La2O3 SiO2, 190 Al Co structure, 113 Al Co Fe structure, 114 Al Cu structure, 117 Al Cu Co solidification, 117 structure, 118 Al Cu Fe solidification, 117 Al Fe structure, 106, 113, 117 Al Mn icosahedral short range order, 110 magnetization, 109 structure, 109, 110 Al Ni diffusion, 118 icosahedral short range order, 112 structure, 110, 116 Al Pd Mn magnetization, 109 structure, 109

Al Ti stucture, 106 Argonne National Laboratory (ANL), ix Ashcroft Langreth definition, 56 B compounds, 161 conductivity, 155 dynamics, 158 emissivity, 155 liquid liquid transition, 158 longitudinal viscosity, 160 physical properties, 156 structure, 156 BaAl2O4, 201 BaO Al2O3, 201 BaO Al2O3 TiO2, 201 Bernouilli effect, 7 Bhatia Thornton definition, 45 Bragg peaks, 44 C, 205 CaO Al2O3 cooling curve, 70, 200 dynamic crossover, 199 glass transition, 197 relaxation, 193 structure, 193, 195 CaO Al2O3 SiO2, 202 Casimir force, 21 Centre de Recherche sur les Mate´riauxa` Hautes Tempe´ratures (CRMHT), 6 charging, 17 chemical scale, 179 Commission E´nergie Atomique, Grenoble, 20 compressibility, 44 conductivity electric, 28, 31 optical, 37 thermal, 41 Containerless Research Inc. (CRI), 4 coordination number, 46

224

Index Co Pd conductivity, 124 hypercooling, 126 magnetic structure, 123 magnetic susceptibility, 120 magnetization, 120 physical properties, 124 specific heat, 125 solidification, 129 structure, 126 correlation function, 45 crystalline material, 44 crystallization front, 26 de Gennes narrowing, 58 dense random packing limit, 108 density, 37 density fluctuations, 83 Density Functional Theory, 63 Deutsche Forschungsanstalt fur Luft und Raumfahrt (DLR), 10 dielectric function, 37 differential cross section, 43 double, 55 macroscopic, 48 diffraction, 42 neutron, 43 isotope substitution in the case of, 46 X ray, 44 drop tubes, 20, 68 Drude theory, 134 dynamics, 53 Earnshaw’s theorem, 12 elastic scattering, 58 electrical conductivity, 33 electrodynamics, 8 electromagnetic properties, 28 electrostatic field, 16 emissivity spectral, 25, 37 total hemispherical, 40, 41 enthalpy of fusion, 68 entropy fluctuations, 83 European Synchrotron Radiation Facility, 54 Faber Ziman definition, 45 Faraday balance, 33 Fe optical properties, 87 Fe B, 161 Fe Ni solidification, 93 form factor, atomic, 44 Frank Kaspar phases, 80, 89 free fall experiments, 20 free electron laser sources, 205 Ge amorphous amorphous transition, 154 conductivity, 152 liquid liquid transition, 154

physical properties, 152 solidification, 155 structure, 153 generalized hydrodynamics, 81 glass transition, 69 glasses, 44 Gor’kov equation, 19 Graz University of Technology, 25 Guinier approximation, 49 heat of fusion, 40 heating electromagnetic, 23 laser, 23 resistive pulse, 24 hydrodynamic regime, 58 generalized, 58 hypercooling, 68 limit, 41, 68 icosahedral short range order (ISRO), 76 induction motor, 33 inelastic scattering, 53 neutron (INS), 53 one phonon, 57 X ray (IXS), 54 Institut Laue Langevin, 47 Institute for Materials Research, Tohoku University, 11 interaction chemical shift, 62 direct dipole, 61 Japan Microgravity Centre, 137 Jet Propulsion Laboratory (JPL), 16 Kirchhoff ’s law, 36 levitation acoustic, 18 aerodynamic, 4 conical nozzle, 4 electromagnetic, 9 electrostatic, 16 gas film, 19 magnetic, 7, 11 optical, 19 quantum, 21 superconducting, 13 Levitron, 3, 12 light scattering, 58 liquid glass transition, 69 liquid liquid transition, 70 liquids structure, 44 dynamics, 58 Local Density Approximations, 63 Lorentz force, 9, 13 magnetic force, 11 magnetic inhomogeneities, 51 magnetic moment, 46

225

226

Index

Marshall Space Flight Center, 16 melting point, 66 metastable states, 67, 204 Mg2SiO4, 202 MgO Al2O3 dynamics, 200 longitudinal viscosity, 201 sound velocity, 201 structure, 197 mode frequencies, 37 mode coupling theory (MCT), 82, 160 modulation calorimetry technique, 41 molecular dynamics, 63 ab initio, 64 multi component system, 45 muon spin spectroscopy, 63 Nb optical properties, 71 Newton’s equations, 65 Ni density, 86 diffusion, 91 dynamics, 90 electrical conductivity, 86 icosahedral short range order, 89 optical properties, 87 solidification, 92 structure, 89 Northwestern Polytechnical University, Xi’an, 19 nuclear magnetic resonance, 59 numerical simulation, 63 optical properties, 35 oscillating drop technique, 37, 39 oxides divalent trivalent, 193 trivalent mixed, 162 pure, 181 patent, 9 permeability, magnetic, 29 physical properties, 204 Porod approximation, 50 pyrometry, 25 quasicrystals, 101 quasielastic scattering, 57 radius of gyration, 49 rare earth orthoferrites, 192 Rayleigh equation, 37 recalescence, 67 reduced undercooling, 102 reflectance, 36 reflectivity, 36 refractive index, 36 relaxation time spin lattice, 61 spin spin, 61 Re Ta solidification, 86

Re W solidification, 86 Reynolds number, 38 RF field, 24 Rh solidification, 85 Rice University, 10 Richardson Dushman equation, 34 scattering diffuse, 44 incoherent, 44 Laue diffuse, 44 magnetic, 46 scattering function, 55 intermediate, 55 scattering length, 44 scattering vector, 43 scattering length density, 48 sessile drop technique, 38 Si amorphous amorphous transition, 145 amorphous liquid transition, 140, 146 conductivity, 131 crystal growth, 131 dynamics, 150 liquid liquid transition, 140 optical properties, 133 physical properties, 135 solidification, 149 sound velocity, 151 specific heat, 138 structure, 139, 140, 146 thermal conductivity, 139 SiGe structure, 153 silicates, 201 SiO2 amorphous amorphous transition, 180 structure, 179 skin depth, 8 small angle scattering, 48 solidification, 67 sound velocity, 66 spallation neutron sources, 205 specific heat, 40, 41 SPRing 8, 54 SrO Al2O3, 194 stability, 3, 12, 16 structure factor, 43 superconductivity, 13 high temperature, 14 supercooling, 67 surface tension, 37, 38 susceptibility electric, 8, 28 magnetic, 11 Ta cooling curve, 68 icosahedral short range order, 80 solidification, 85 temperature gradients, 6

Index temperature measurement, 25 thermophysical properties, 37 Ti dynamics, 80 icosahedral short range order, 76 optical properties, 71 structure, 73 Ti Zr Ni icosahedral short range order, 104 solidification, 101 structure, 103 transition metals early, 71 electronic structure, 86 late, 86 physical properties, 73, 89 Tsukuba Space Center, 16 viscosity, 38 longitudinal, 84 shear, 84 Wien’s law, 25 work function, 34

X ray absorption spectroscopy fine structure (EXAFS), 51 near edge (XANES), 51 Y2O3 structure, 176 Y2O3 Al2O3 liquid liquid transition, 184, 188 optical fibres, 185 physical properties, 181 solidification, 187 structure, 182 Zr cooling curve, 67 icosahedral short range order, 73 optical properties, 73 structure, 75 Zr Ni diffusion, 100 electrical conductivity, 93 melting, 101 optical properties, 93 physical properties, 94 structure, 96, 101

227