THEORETICAL DETERMINATION OF SOME THERMODYNAMIC PROPERTIES OF SELECTED BINARY LIQUID ALLOYS Yisau Adelaja ODUSOTE July 2013 1 UNIVERSITY OF IBADAN LIBRARY THEORETICAL DETERMINATION OF SOME THERMODYNAMIC PROPERTIES OF SELECTED BINARY LIQUID ALLOYS BY Yisau Adelaja ODUSOTE B. Sc Physics(Ogun), M. Sc Physics(Ibadan) Matric. No. 91366 A Thesis in the Department of Physics Submitted to the Faculty of Science in partial fulfilment of the requirements for the award of the degree of DOCTOR OF PHILOSOPHY University of Ibadan, Ibadan. July 2013 2 UNIVERSITY OF IBADAN LIBRARY CERTIFICATION This is to certify that the work described in this thesis was carried out under our supervision by Mr. Yisau Adelaja Odusote in the Department of Physics, University of Ibadan, Ibadan. L. A. Hussain B. Sc (Ibadan), Ph.D (Manc.) Professor of Physics Department of Physics University of Ibadan, Ibadan, Nigeria. O. E. Awe B. Sc, M.Sc, Ph.D (Ibadan) Senior Lecturer Department of Physics University of Ibadan, Ibadan, Nigeria. i UNIVERSITY OF IBADAN LIBRARY DEDICATION This work is dedicated to Madam (Mrs) D. O. Ogunsanya (Grandma) and her children. ii UNIVERSITY OF IBADAN LIBRARY ACKNOWLEDGEMENTS I give unprecedented thanks to the Ever-Living, Self-Subsisting God for His Mercy and for making this research study achievable. This thesis could not have become a reality if my supervisor Professor L. A. Hussain had not offered me the oppor- tunity to come back to the University of Ibadan. Professor Hussain’s erudition and insights, enthusiasm and optimism provided me with such an endless source of knowledge and energy. His real scholarly generosity made it possible for me to complete this research work. He is the very first person at University of Ibadan who deserves my full gratitude. Secondly, I express my gratitude to my co-supervisor, Dr. O. E. Awe who initiated the topic and provided insights into a better understanding of the many analytical aspects of the work. I am grateful for many constructive criticisms and discussions that have allowed me to take a step back, refocus and see the big picture on more than one occasion. This has been invaluable to me. My appreciation goes to many faculty and staff members of Department of Physics, University of Ibadan under the headship of Prof. I. P. Farai for their encouragement and moral supports in many ways. I specially thank Drs F. O. Ogundare, O. I. Popoola, N. N. Jibiri, J. A. Adegoke, A. A. Adetoyinbo, Dr. (Mrs) J. A. Ademola and O. O. Popoola . I am greatly indebted to Prof. Rada Novakovic of National Research Council- Institute for Energetics and Interphases, Italy for fruitful discussions and provision of some of the literatures used; Prof. Iwao Katayama of Department of Material Science and Processing, Osaka University, Japan and Prof. Ana Kostov of Copper Institute, University of Belgrade, Serbia for providing the experimental data for Ga-Tl and Al-Si alloys, respectively, used in the study. I also thank Drs U. E. Vincent, S. O. Lawal, Miss Yetunde Awosanya and Mr. Kayode Odusote for assisting me to get relevant reference materials and textbooks from UK for this research work. I am highly grateful to all academic and non-academic staff of the Department iii UNIVERSITY OF IBADAN LIBRARY of Physics, Olabisi Onabanjo University, Ago-Iwoye and the Federal University of Technology, Akure for their moral support, encouragement and assistance through- out the study. I should also thank Profs. Tunde Ogunsanwo, A. O. Olorunnisola, Alhaji Oladunjoye, Alh. Lateef Osinaike and Dr. G. A. Adebayo (FUNAAB) for their unquantifiable roles towards the success of this work. A big thank you goes to all my friends Messrs. N. O. Bakare, Kunle Babalola, Kayode Alao and others for moral and financial support. May Allah bless you all. Finally, I wish to appreciate my wife, children and siblings for their patience and understanding during the many months I spent working on this thesis. iv UNIVERSITY OF IBADAN LIBRARY Table of Contents Title page i Certification i Dedication ii Acknowledgements iii Table of Contents v List of Tables viii List of Figures x List of Symbols xv List of Abbreviations xvii Abstract xviii 1 Introduction 1 1.1 Background to the research . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Aim and Objectives of this work . . . . . . . . . . . . . . . . . . . . 5 1.3 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Literature Review 8 2.1 Study techniques for binary liquid alloys . . . . . . . . . . . . . . . 8 v UNIVERSITY OF IBADAN LIBRARY 2.2 Observable thermodynamic indicators . . . . . . . . . . . . . . . . . 13 2.2.1 Structure factors S(q) of Liquid . . . . . . . . . . . . . . . . 13 2.2.2 Small-angle scattering experiment . . . . . . . . . . . . . . . 15 2.2.3 Segregating liquid alloys . . . . . . . . . . . . . . . . . . . . 16 2.3 Short-range order parameter . . . . . . . . . . . . . . . . . . . . . . 17 3 Methodology 19 3.1 Quasi Lattice Theory of Liquid Mixtures . . . . . . . . . . . . . . . 19 3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.2 Basic Theory of Quasi-Lattice Theory . . . . . . . . . . . . . 21 3.2 Quasi-Chemical Approximation Model . . . . . . . . . . . . . . . . 24 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.2 Theory of QCAM . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Four Atom Cluster Model . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.2 Theory of FACM . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4 Statistical Thermodynamic Theory . . . . . . . . . . . . . . . . . . 37 3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4.2 Theoretical formulation . . . . . . . . . . . . . . . . . . . . . 41 4 Results and Discussion 44 4.1 Quasi-lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1.1 Concentration Fluctuations in the long-wavelength limit, the Free Energy of Mixing and the Chemical Short-range order parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1.2 Dynamic properties: diffusion and Viscosity . . . . . . . . . 49 4.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 Quasi Chemical Approximation Model . . . . . . . . . . . . . . . . 54 4.2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3 Four Atom Cluster Model . . . . . . . . . . . . . . . . . . . . . . . 61 vi UNIVERSITY OF IBADAN LIBRARY 4.3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4 Statistical thermodynamic theory . . . . . . . . . . . . . . . . . . . 73 4.4.1 Sb-Sn liquid alloys at 905K . . . . . . . . . . . . . . . . . . 73 4.4.2 In-Pb liquid alloys at 673K . . . . . . . . . . . . . . . . . . . 77 4.4.3 Al-Si liquid alloys at 2000K, 2400K and 2473K, respectively. 83 4.4.4 Ga-Tl liquid alloys at 973K, 1073K and 1173K, respectively. 93 4.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5 Conclusions and Recommendations 104 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2 Limitations to this study . . . . . . . . . . . . . . . . . . . . . . . . 108 5.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 References 111 Appendix A 121 vii UNIVERSITY OF IBADAN LIBRARY List of Tables 1.1 Table showing melting points, boiling points and atomic volume of some elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.1 Values of the parameters for Al-Zn and Bi-In liquid alloys . . . . . . 46 4.2 Computed chemical short range order parameter (α1) for Al-Zn and Bi-In alloys at T = 1000K and 900K, respectively. . . . . . . . . . . 50 4.3 Concentration dependence of DM for Al-Zn and Bi-In alloys at T = Did 1000K and 900K, respectively. . . . . . . . . . . . . . . . . . . . . . 51 4.4 Fitted interaction parameters for the systems. . . . . . . . . . . . . 55 4.5 Interchange energy ω(eV ) for the systems. . . . . . . . . . . . . . . 65 4.6 Computed and experimental activity ratio for Bi-Pb alloy. . . . . . 66 4.7 Computed and experimental activity ratio for Sb-Sn alloy. . . . . . 67 4.8 Parameters used in the model calculations for Sb-Sn and In-Pb. . . 74 4.9 Parameters used in the model calculations for Ga-Tl and Al-Si. . . 75 4.10 Activity of Sb in Sb-Sn liquid binary alloys at 905K. . . . . . . . . 76 4.11 Activity of In in In-Pb liquid binary alloy at 673K . . . . . . . . . . 81 4.12 Computed activities of Al in Al-Si liquid binary alloy at 2000K, 2400K and 2473K, respectively . . . . . . . . . . . . . . . . . . . . 86 4.13 Computed entropy of mixing for Al-Si liquid binary alloy at 2000K, 2400K and 2473K, respectively . . . . . . . . . . . . . . . . . . . . 92 4.14 Activity of Ga in Ga-Tl liquid binary alloy at 973K. . . . . . . . . 94 4.15 Activity of Ga in Ga-Tl liquid binary alloy at 1073K. . . . . . . . . 95 4.16 Activity of Ga in Ga-Tl liquid binary alloy at 1173K. . . . . . . . . 96 viii UNIVERSITY OF IBADAN LIBRARY 4.17 Computed entropy of mixing of Ga-Tl liquid alloy at 973K, 1073K and 1173K, respectively . . . . . . . . . . . . . . . . . . . . . . . . 101 ix UNIVERSITY OF IBADAN LIBRARY List of Figures 4.1 Concentration fluctuations in the long-wavelength limit (Scc(0) and Sidcc(0)) vs. concentration for Al-Zn and Bi-In liquid alloys at 1000K and 900K respectively. The solid line denotes theoretical values while the triangle and the cross denote experimental values for Al-Zn and Bi-In respectively. The dot denotes the ideal values Sidcc(0). cAl and cBi are the Al and Bi concentrations in the alloy. . . . . . . . . . . . 47 4.2 Concentration dependence of GM for Al-Zn and Bi-In liquid alloys RT at 1000K and 900K, respectively. The solid line denotes theoretical values while the triangles and the crosses denote experimental values for Al-Zn and Bi-In, respectively. cAl and cBi are the Al and Bi concentrations in the alloy. The experimental data were taken from Hultgren et al., (1973). . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Concentration dependence of viscosity ∆η eqns.(3.11) and (3.12) for ηo Al-Zn and Bi-In at 1000K and 900K respectively. The solid line are for calculated ∆η . The triangles and the crosses denote experimental ηo values for −HM for Al-Zn and Bi-In respectively. c and c are the RT Al Bi Al and Bi concentrations in the alloy. The experimental data were taken from Hultgren et al., (1973). . . . . . . . . . . . . . . . . . . 52 x UNIVERSITY OF IBADAN LIBRARY 4.4 Free energy of mixing, GM versus concentration for Ga-Zn, Ga-Mg RT and Al-Ga liquid alloys at 700K, 923K and 1073K, respectively. The solid line denotes theoretical values while the crosses, triangles and the stars denote experimental values for Ga-Zn, Ga-Mg and Al-Ga respectively. cGa and cAl are the Ga and Al concentrations in the alloy. The experimental data were taken from Hultgren et al., (1973). 56 4.5 Concentration-concentration fluctuations Scc(0) versus concentra- tion for Ga-Zn, Ga-Mg and Al-Ga liquid alloys at 700K, 923K and 1073K, respectively. The solid line denotes theoretical values while the crosses, triangles and the stars denote experimental values for Ga-Zn, Ga-Mg and Al-Ga respectively. The dot denotes the ideal values. cGa and cAl are the Ga and Al concentrations in the alloy. . 58 4.6 Calculated Warren-Cowley short range order parameter α1 versus concentration for Ga-Zn, Ga-Mg and Al-Ga liquid alloys at 700K, 923K and 1073K, respectively. cGa and cAl are the Ga and Al con- centrations in the alloy. . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.7 Enthalpy of mixing, HM versus concentration for Ga-Zn, Ga-Mg and RT Al-Ga liquid alloys at 700K, 923K and 1073K, respectively. The solid line denotes theoretical values while the crosses, triangles and the stars denote experimental values for Ga-Zn, Ga-Mg and Al-Ga respectively. cGa and cAl are the Ga and Al concentrations in the alloy. The experimental data were taken from Hultgren et al.,(1973). 62 4.8 Entropy of mixing, SM versus concentration for Ga-Zn, Ga-Mg and R Al-Ga liquid alloys at 700K, 923K and 1073K, respectively. The solid line denotes theoretical values while the crosses, triangles and the stars denote experimental values for Ga-Zn, Ga-Mg and Al-Ga, respectively. cGa and cAl are the Ga and Al concentrations in the alloy. The experimental data were taken from Hultgren et al.,(1973). 63 xi UNIVERSITY OF IBADAN LIBRARY 4.9 Free energy of mixing, GM versus concentration for Bi-Pb and Sb-Sn RT liquid alloys at 700K and 905K respectively. The solid line represents theoretical values using Eqn. (3.74) while the triangle and the cross represent experimental values for Bi-Pb and Sb-Sn respectively. cBi and cSb are the Bi and Sb concentrations in their respective alloys. The experimental data were taken from Hultgren et al., (1973). . . 68 4.10 Concentration-concentration fluctuations Scc(0) versus concentra- tion for Bi-Pb and Sb-Sn liquid alloys at 700K and 905K respec- tively. The solid line represents theoretical values obtained from Eqn. (3.82) while the triangle and the cross represent experimental values for Bi-Pb and Sb-Sn respectively. cBi and cSb are the Bi and Sb concentrations in their respective alloys. . . . . . . . . . . . . . . 69 4.11 Calculated Cowley-Warren short range order parameter, α1 versus concentration, using Eqn. (3.71) for Bi-Pb and Sb-Sn liquid alloys at 700K and 905K, respectively. CBi and CSb are the Bi and Sb concentrations in the alloys. . . . . . . . . . . . . . . . . . . . . . . 70 4.12 Calculated ratio of mutual diffusion and self-diffusion coefficients, DM/Did versus concentration, using Eqn. (3.83) for Bi-Pb and Sb- Sn liquid alloys at 700K and 905K, respectively. cBi and cSb are the Bi and Sb concentrations in the alloys. . . . . . . . . . . . . . . . . 72 4.13 Concentration dependence of GM for Sb-Sn liquid binary alloy at RT 905K. The solid line denotes theoretical values while the times de- note experimental values respectively. cSb is Sb concentrations in the alloy. The experimental data were taken from Hultgren et al., (1973). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.14 Concentration dependence of HM for Sb-Sn liquid binary alloy at RT 905K. The solid line denotes theoretical values while the times de- note experimental values respectively. cSb is Sb concentrations in the alloy. The experimental data were taken from Hultgren et al.,(1973). 79 xii UNIVERSITY OF IBADAN LIBRARY 4.15 Concentration dependence of SM for Sb-Sn liquid binary alloy at R 905K. The solid line denotes theoretical values while the times de- note experimental values respectively. cSb is Sb concentrations in the alloy. The experimental data were taken from Hultgren et al.,(1973). 80 4.16 Concentration dependence of GM for In-Pb liquid binary alloy at RT 673K. The solid line denotes theoretical values while the times de- note experimental values respectively. cIn is In concentrations in the alloy. The experimental data were taken from Hultgren et al.,(1973). 82 4.17 Concentration dependence of HM for In-Pb liquid binary alloy at RT 673K. The solid line denotes theoretical values while the times de- note experimental values respectively. cIn is In concentrations in the alloy. The experimental data were taken from Hultgren et al.,(1973). 84 4.18 Concentration dependence of SM for In-Pb liquid binary alloy at R 673K. The solid line denotes theoretical values while the times de- note experimental values respectively. cIn is In concentrations in the alloy. The experimental data were taken from Hultgren et al.,(1973). 85 4.19 Concentration dependence of GM for Al-Si liquid binary alloy at RT 2000K. The solid line denotes theoretical values while the times de- note experimental values respectively. cAl is Al concentrations in the alloy. The experimental data were taken from Kostov et al.,(2007) . 88 4.20 Concentration dependence of GM for Al-Si liquid binary alloy at RT 2400K. The solid line denotes theoretical values while the times de- note experimental values respectively. cAl is Al concentrations in the alloy. The experimental data were taken from Kostov et al.,(2007). 89 4.21 Concentration dependence of GM for Al-Si liquid binary alloy at RT 2473K. The solid line denotes theoretical values while the times de- note experimental values respectively. cAl is Al concentrations in the alloy. The experimental data were taken from Kostov et al.,(2007). 90 xiii UNIVERSITY OF IBADAN LIBRARY 4.22 Computed enthalpy of mixing, HM , of Al-Si liquid binary alloy at RT 2000K, 2400K and 2473K, respectively. The solid line denotes the- oretical values. cAl is Al concentrations in the alloy. . . . . . . . . . 91 4.23 Concentration dependence of GM for Ga-Tl liquid binary alloy at RT 973K. The solid line denotes theoretical values while the times de- note experimental values respectively. cGa is Ga concentrations in the alloy. The experimental data were taken from Katayama et al.,(2003). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.24 Concentration dependence of GM for Ga-Tl liquid binary alloy at RT 1073K. The solid line denotes theoretical values while the times de- note experimental values respectively. cGa is Ga concentrations in the alloy. The experimental data were taken from Katayama et al.,(2003). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.25 Concentration dependence of GM for Ga-Tl liquid binary alloy at RT 1173K. The solid line denotes theoretical values while the times de- note experimental values respectively. cGa is Ga concentrations in the alloy. The experimental data were taken from Katayama et al.,(2003). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.26 Computed enthalpy of mixing, HM , of Ga-Tl liquid alloy at 973K, RT 1073K and 1173K, respectively. The solid line denotes theoretical values. cGa is Ga concentrations in the alloy. . . . . . . . . . . . . . 100 xiv UNIVERSITY OF IBADAN LIBRARY LIST OF SYMBOLS Except otherwise specified the following symbols used in this thesis are as define below: Symbols Quantity A,B Components of a binary A-B alloys ai(i = A,B) Activity of component i Ci(i = A,B) Composition of component i (where cA + cB = 1) DM Inter-diffusion coefficient of any binary alloy Did Intrinsic diffusion coefficient for an ideal mixture Di Self-diffusion coefficient of component i E Configurational energy of the bulk GM Gibbs free energy of mixing GXSM Excess Gibbs free energy of mixing HM Enthalpy of mixing kB Boltzmann’s conctant NA,NB Number of atoms of components A and B N Total number of atoms NA Avogagro’s number Pij(i, j = A,B) Probability of finding one of the components of the i-j bond as a part of the complex AµBν qNii (T )(i = A,B) Atomic partition function of component i for the bulk R Gas constant SM Entropy of mixing Scc(0) Concentration-concentration fluctuations in long-wavelength limits Sidcc(0) Concentration fluctuations for the ideal mixing condition Scc(q) Concentration-concentration structure factor SNN(q) Number-Number structure factor T Absolute temperature xv UNIVERSITY OF IBADAN LIBRARY Vi(i = A,B) Atomic volume of the component i Z Coordination number α1 Short-range order parameter γ Ratio of activity coefficients of A and B components γi(i = A,B) Activity coefficient of component i η Viscosity ϵij Energy of i-j bond for regular solution ∆ϵij Difference in the energy of an i-j bond when one of the bond forming component is present in the compound AµBν µ, ν Stoichiometric coefficients µi(i = A,B) Chemical potential of component i Ωi(i = A,B) Atomic volume of the component i ω Order or interchange energy parameter Ξ Grand partition function for the bulk xvi UNIVERSITY OF IBADAN LIBRARY LIST OF ABBREVIATIONS CFM Complex Formation Model CSRO Chemical Short Range Order FACM Four Atom Cluster Model FT Fourier Transform HOCP Higher Order Conditional Probability MIVM Molecular Interaction Volume Model RASM Regular Associated Solution Model SAM Self Association Model TACM Two Atom Cluster Model QCM Quasi Chemical Model QCAM Quasi Chemical Approximation Model QLM Quasi Lattice Model QLT Quasi Lattice Theory WIAM Weak Interaction Approximation Model xvii UNIVERSITY OF IBADAN LIBRARY ABSTRACT Binary liquid alloys have found wide applications in electronics, communications, automotive and aerospace technologies. The development of new alloys and im- provement of existing ones require accurate knowledge of thermodynamic variables of the constituent systems which may not be obtainable experimentally. Hence, there is always the need for theoretical investigation to complement experiment. This study was aimed at theoretical determination of the thermodynamic proper- ties of nine binary liquid alloys. Nine liquid alloys (Al-Zn, Bi-In, Ga-Zn, Ga-Mg, Al-Ga, Sb-Sn, In-Pb, Ga-Tl and Al-Si) with insufficient theoretical investigation were selected for investiga- tion using three different models. Quasi-lattice theory which connects thermo- dynamic and dynamical properties was employed to determine the concentration- concentration fluctuation in the long-wavelength limit [Scc(0)], chemical short range order parameter (α1), free energy of mixing (GM), the concentration dependence of diffusion and viscosity of Al-Zn and Bi-In liquid alloys. Quasi-chemical approxi- mation model for strongly interacting systems was used to investigate ordering and glass formation tendencies in Ga-Zn, Ga-Mg and Al-Ga binary alloys. A statisti- cal thermodynamic theory was used to determine the mixing properties of Sb-Sn, In-Pb, Ga-Tl and Al-Si liquid alloys. The choice of model was influenced by the type of available experimental data. The Scc(0) and α1 showed that a reasonable degree of chemical order existed in Bi-In system, while Al-Zn liquid alloys exhibited a liquid miscibility gap or phase separation at all concentrations. The values of free energy of mixing at the equiatomic composition were -0.4986RT and -0.9344RT for Al-Zn and Bi-In alloys respectively. The free energy of mixing of Ga-Zn, Al-Ga and Ga-Mg alloys are almost symmetrical around the equiatomic composition with Ga-Mg being the most interacting, while Al-Ga is more interacting than Ga-Zn alloy. Apart from Ga-Mg alloys with negative enthalpy of mixing, both the enthalpy of mixing and entropy of mixing of the three Ga-based systems are positive and symmetrical xviii UNIVERSITY OF IBADAN LIBRARY about the equiatomic composition. In the concentration range 0 ≤ cAl ≤ 0.3 and 0.7 ≤ cAl ≤ 1, Al-Ga systems exhibited a glass-forming potential. The free energy of mixing, enthalpy of mixing and entropy of mixing of Sb-Sn and Al-Si liquid alloys exhibited negative deviations from ideality and were symmetric around the equiatomic composition, while In-Pb and Ga-Tl systems exhibited positive deviations with Ga-Tl alloys showing asymmetry behaviour. The compound formation in liquid phase was weaker in Al-Zn than in Bi-In alloys. The Sb-Sn and Al-Si liquid alloys had tendency for heterocoordination, while In-Pb and Ga-Tl exhibited homocoodination. The Ga-Zn and Al-Ga alloys exhibited homocoordination, whereas Ga-Mg alloys had tendency towards hetero- coodination. Keywords: Binary liquid alloys, Phase separation, Concentration-concentration fluctuations, Short range order Word Counts: 415 xix UNIVERSITY OF IBADAN LIBRARY Chapter 1 Introduction 1.1 Background to the research Understanding the properties of liquid alloys is really a matter of interest in the field of liquid science mainly because most of the binary solid alloys are formed by cooling from their respective liquid state. The properties of liquid alloys provide some unique information that can improve the processing and qualities of materials in the solid state. Recent advancement in the use of liquid alloys and their composites in high temperature applications in many diverse fields call for better understanding of the thermodynamic and dynamical properties of alloys in the liquid state to aid the development of new alloys, improving the existing ones and characterizing the glassy solids (Anusionwu and Adebayo, 2001; Müller, 2003; Novakovic et al., 2003, 2004) and (Novakovic and Takana, 2006). Though our daily experience testifies to the classification of three different classi- cal phases of matter: solid, liquid and gas or vapour (other phases of matter include plasma and superfluid). Solids are rigid, and when studied by diffraction experi- ment give rise to sharp Bragg reflections. This is the hall-mark of crystallinity: an ordered array of the building blocks, be they atoms or groups of atoms. Diffraction experiments have shown that liquids in general and metallic liquids in particular exhibit some kind of short-range order (SRO), that is, absence of long-range order, demonstrating that there is no long-range structural order among the atoms or 1 UNIVERSITY OF IBADAN LIBRARY molecules (Singh and March, 1995; Müller, 2003). However, a very characteristic property of liquids and gases is that they have low viscosity and are thereby endowed with fluidity. In spite of our understanding of solid state of matter, the qualitative distinction drawn from diffraction exper- iments between solid and liquid does not carry through to distinguish liquid and vapour. The physical principles underlying the solid-liquid or liquid-vapour phase transformations are not fully understood. Due to the non-existence of exact liq- uid state theory, the understanding of the solid phases have served as guidelines to characterize the structural behaviour of respective liquids and vapour. Their characteristic behaviour is usually ascribed to an outcome of the interplay of the energetic and structural readjustment of the constituent elemental atoms that hold the system together, yet our present knowledge provides no clear-cut message as to the ways the structure and forces are coupled (Singh and March, 1995). Modelling the thermodynamic, structural and surface properties of liquid bi- nary alloys and liquid metals require the data regarding their structures and the relevant forces that quantify the interatomic interactions (i.e. the energetics of sys- tems). This is why understanding the mixing behaviour of two elemental metals forming a binary alloy has engaged the attention of many physicists, chemists, and metallurgists (Bhatia and March, 1975; Singh, 1987; Alblas et al., 1982; McGreevy and Pusztai, 1988; Singh and March, 1995; Singh and Sommer, 1997; Kaban et al., 2003; Dubinin, 2003; Novakovic et al., 2004; Anusionwu, 2006; Awe et al., 2005). A number of different theoretical approaches have been proposed and used to study the thermodynamic and structural properties of these systems. Some of these models include the Quasi-lattice approximation (Guggenheim, 1952), the electron theory (Ashcroft and Stroud, 1978), the Bhatia and Hargrove formalism (1974) which was later reformulated in Bhatia and Singh (1984) and the ab-initio methods (Jank and Hafner, 1988; Dalgic et al., 1998) which involve the calculations of thermodynamic quantities using the pseudopotential formalism. The first one is based on simple theoretical model that makes it possible to express the energetics 2 UNIVERSITY OF IBADAN LIBRARY of a system in terms of the interaction parameters reproducing its thermodynamic properties as well as to explain the ordering and phase separation phenomena in liquid binary alloys. Consequently, the thermophysical properties, as surface tension, viscosity and chemical diffusion (Bhatia and Hargrove, 1974; Prasad et al., 1998; Jha and Mishra, 2001; Novakovic et al., 2002 and Awe et al., 2006) can then be described. The mixing behaviour of two metals forming binary alloys is the result of the interplay of energetic and structural readjustment of the constituent atoms. On mixing, A and B atoms can prefer to remain self-coordinated forming A-A or B-B pairs, or can exhibit a strong interactive tendency between unlike atoms forming heterocoordinated A-B pairs. Based on this, all liquid binary alloys can be classified according to deviations of their thermodynamic and thermophysical properties from the additive rule of mixing (Raoult’s law) into two main categories: segregation (positive deviation) or short-range ordered (negative deviation) alloys (Bhatia and Hargrove, 1974; Sommer, 1982; Bhatia and Singh, 1984; Singh and Singh,1995; Singh and Sommer, 1997; Prasad et al., 1998). However, there are some liquid alloys which do not belong exclusively to any of the above two categories. For example, the excess Gibbs energy of mixing GXSM for Ag-Ge and Cd-Na is negative at certain compositions, while positive at other compositions (Singh and Sommer, 1997). Also in liquid alloys such as Au-Bi, Bi-Cd and Bi-Sb, the enthalpy of mixing (HM) is a positive quantity but G XS M is negative (Singh and Sommer, 1997). Bi-Pb has positive H and GXSM M in the solid phase as against the negative values of H XSM and GM in the liquid phase. In addition, systems such as Au-Ni and Cr-Mo exhibit immiscibility in the solid phase which is not evidently visible in the corresponding liquid phase. While alloys such as Ag-Te show intermetallic phases and large negative HM values in the liquid phase together with a liquid miscibility gap (Singh and Sommer, 1997). It is of interest to note that many binary liquid alloys such as Al-Bi, Al-In, Al-Pb, Ga-Pb, Ga-Hg, Pb-Zn and Cu-Pb exhibit characteristic features as func- 3 UNIVERSITY OF IBADAN LIBRARY tion of concentration based on their thermodynamic properties like free energy of mixing, enthalpy of mixing, entropy of mixing (Hultgren et al, 1973) and many other electrical properties (Heine, 1970) and (Busch and Guntnerodt, 1974). They are characterized by liquid miscibility gaps and exhibit large positive HM (Singh and Sommer, 1997). Their properties in the liquid phase tend to change markedly as a function of composition (c), temperature (T) and pressure (P). The liquidus lines for these systems are complicated and usually S-shaped while their enthalpy of mixing and excess free energy of mixing are large negative quantities at one or more concentrations (Anusionwu, 2006). The understanding of such anomalous behaviour of liquid alloys as functions of concentration is still a great challenge and demands extensive theoretical investigation. In addition, the development of new alloys and improvement of existing ones require accurate knowledge of thermodynamic variables of the constituent systems due to their scientific and technological applications in electronics, communications, medical science, automotive and aerospace technologies. For instance, for many years the traditional silver solders have been used in joining ceramics and glasses with metallic materials (Eustathopoulos et al., 1999). Other examples are silver based alloys containing the additions of the group IV transition metals, Ti, Zr and Hf. Ti-based alloys are principally used in aircraft, spacecraft, naval ships and medicine as reported by Ratner et al.,(1996) and Novakovic and Zivkovic (2005). Hf and its alloys are constituents or subsystems of complex metallic glasses (Saida and Inoue, 2003), superconducting alloys (Novakovic and Zivkovic, 2005) or are used in nuclear propulsion (Frost, 1981). Many properties of Cu-Zr and Cu-Si liquid alloys have been studied by many researchers (Akinlade et al., 1998; Anusionwu and Adebayo, 2001) due to their manifest behaviours and their potential usefulness in technological applications. It may therefore be necessary that a detail theoretical and thorough under- standing of the structural readjustment and energetic preferences of atoms in binary liquid metallic systems go beyond the knowledge of the empirical metallurgical con- 4 UNIVERSITY OF IBADAN LIBRARY structs to explain the characteristic behaviours. Advances in these areas have been made possible due to the combination of a number of simple theoretical approaches or models to discuss the deviations and the anomalies in terms of heterocoordina- tion or self-coordination species that might exist in liquid and vapour. The foremost task in this work is to put together the various experimental and theoretical information with a view to establishing a respectable understanding between experimental results, theoretical approaches and empirical models via a theoretical determination of some thermodynamic properties of nine binary liquid alloys. Binary liquid Ga-Tl and Al-Si alloys whose experimental data were recently published Ga-Tl (Katayama et al.,(2003) and Al-Si (Kostov et al.,2007) and seven others (Al-Zn, Bi-In, Ga-Zn, Ga-Ma, Al-Ga, Sb-Sn and In-Pb) from Hultgren et al,(1973) with insufficient theoretical investigation, thus forming the basis to eval- uate the energetics of the systems and the driving forces behind the anomalous behaviour in binary liquid alloys. 1.2 Aim and Objectives of this work The aim of this study is to theoretically determine some thermodynamic properties of selected liquid binary alloys while the objectives of the thesis are summarized as follows: 1. determine the thermodynamics and microscopic structural properties of bi- nary liquid Al-Zn, Bi-In, Ga-Zn, Ga-Mg, Al-Ga, Sb-Sn and In-Pb alloys at different temperatures. 2. evaluate the phase separation and compound forming tendencies in the liquid alloys for a proper understanding of the anomalous behaviour in liquid alloys. 3. extract useful microscopic information on the energetics of formation in bi- nary liquid alloys. 4. carry out a theoretical determination of mixing properties and activities of 5 UNIVERSITY OF IBADAN LIBRARY binary liquid Sb-Sn, In-Pb, Ga-Tl and Al-Si alloys using a statistical ther- modynamic theory. 5. evaluate existing thermodynamic data related to the peculiarities of the atomic size. 6. show that thermodynamic data obtained could serve as a basis for comparison with some future critical experimental works. The properties of some relevant parameters of constituents pure metals which will be used in the present work are listed in table 1.1. 1.3 Outline of Thesis The thesis is composed of five chapters. The present chapter introduces the main theme of the research. Chapter 2 reviews the relevant literatures on thermodynam- ics and its consequences. Methodology in chapter 3 along with introductions and the theoretical basis of each of the models used in this study. The Quasi-Lattice Theory of liquid mixtures relating two dynamic properties such as diffusion coef- ficient and viscosity with thermodynamic properties used in the study of Al-Zn and Bi-In liquid alloys is presented in section 3.1. The theoretical basis of the Quasi-Chemical Approximation Model (QCAM) in determining the probable exis- tence of chemical ordering and glass formation tendencies in binary liquid Ga-Zn, Ga-Mg and Al-Ga alloys is given in section 3.2. The Four Atom Cluster Model (FACM) used to study liquid Sb-Sn and Bi-Pb alloys follows in section 3.3. The theoretical formulation of an approach for determination of the mixing properties and activities of liquid binary Sb-Sn, In-Pb, Ga-Tl and Al-Si alloys using a statis- tical thermodynamic theory is given section 3.4. While, results and discussion for various thermodynamic models used are given in chapter 4. And finally in chap- ter 5, conclusions and recommendations on the possible future research prospects realized from this study along with contributions to body of knowledge are given. 6 UNIVERSITY OF IBADAN LIBRARY Table 1.1: Table showing melting points, boiling points and atomic volume of some elements. Melting point∗ Boiling point∗ Atomic Volume+ Elements Symbols Tm(K) Tb(K) Vm(10 −6m3g.atom−1) Aluminium Al 933.35 2333.15 10.00 Antimony Sb 903.00 1913.15 18.19 Bismuth Bi 544.05 1833.15 21.31 Cobalt Co 1768.05 3458.15 6.67 Cadmium Cd 594.05 1039.15 13.00 Casium Ca 1124715 1513.15 26.20 Copper Cu 1356.15 2903.15 7.11 Gallium Ga 302.93 2573.15 11.80 Iron Fe 1808.15 3003.15 7.09 Lead Pb 600.55 2033.15 18.26 Magnesium Mg 923.15 1380.15 14.00 Mercury Hg 234.28 629.73 14.09 Nickel Ni 1728.15 3448.15 6.59 Silicon Si 1687.15 2608.15 9.80 Sodium Na 378.05 1150.65 23.78 Tin Sn 504.79 2548.15 16.29 Thallium Tl 575.65 1730.15 18.62 Zinc Zn 692.62 1203.15 9.16 For instance, Liquid range = Tb - Tm (Iida and Guthrie, 1993) e.g. the liquid range for Al = 1400K, Ga = 2270K ∗ Iida and Guthrie, (1993); + Singman, (1984) 7 UNIVERSITY OF IBADAN LIBRARY Chapter 2 Literature Review 2.1 Study techniques for binary liquid alloys The alloying behaviour of binary liquid alloys can be studied using two distinct theories. The electron theory of mixing and statistical mechanical theory of mix- ing. According to the first theory a liquid alloy is assumed to consist of a system of ions and electrons. The problem, usually, in this approach is tackled through the use of pseudo-potential theory (Harrison, 1966; Heine, 1970; Busch and Gunt- nerodt, 1974; Thakur et al., 2005; Kanth and Chakrabarti, 2009) and hard sphere model which had been used by Faber (1972) and Shimoji (1977). The major draw- back of the approach is that it cannot be used to obtain information regarding the concentration fluctuations in the long wavelength limit Scc(0), an important thermodynamic function which determines the stability of alloys. The approach is also computationally demanding and has several limitations in situations where there is formation of chemical complexes (Akinlade, 1994). The second theory also known as thermodynamic model approach is a very useful tool to study the behaviour of liquids and liquid mixtures. It involves the use of analytical solutions of liquid state equations based on some approximations. It has been used to obtain analytical expressions for various thermodynamic functions that are not possible otherwise and proved useful for qualitative description of thermodynamic and microscopic structural properties of binary liquid alloys. The 8 UNIVERSITY OF IBADAN LIBRARY conformal solution model has been used to study Scc(0) of different binary alloys (Bhatia and Hargrove, 1974; Alonso and March, 1990; Thakur et al., 2005). But the model cannot be used to study the short-range order parameters. The Quasi- chemical theory of Guggenheim (1952) is successful in studying the chemical short- range order parameters of binary liquid alloys, but cannot be used to explain the observed asymmetry in the properties of mixing as a function of concentration. In order to overcome the identified shortcomings of the models, a number of theoretical models have been established by theoreticians within the framework of statistical mechanical theory of mixing to explain the energetics of liquid alloys and the various concentration dependence thermodynamic properties based on the deviations from ideality in terms of heterocoordination or homocoordination species that might exist in liquid and vapour phases. Some of the established thermodynamic models include: the Quasi Lattice Model(QLM), the Quasi-Chemical Approximation Model(QACM); the Self- As- sociation Model(SAM) in demixing liquid alloys and vapour species; the Complex Formation Model (CFM) in liquid binary alloys; the Four-Atom-Cluster Model FACM), the Regular Associated-Solution Model(RASM) and the Weak Interac- tion Approximation Model (WIAM)(Singh and March, 1995). Thermodynamic and thermophysical data of alloys play an important role in the present production processes and design of reliable materials for high temperature applications but due to experimental difficulties they have not been successfully measured well above and well below the melting temperature (Novakovic et al., 2005). Experimental investigation of the thermodynamic properties of binary liquid alloys and higher component systems is a rather difficult, time consuming and very expensive endeavour. The main reason for the experimental difficulties are: 1. high investigating temperature required, 2. high melting points and their strong chemical affinity for oxygen, in pure states or alloyed form, and 3. difficulties of making experimental measurements over a wide range of 9 UNIVERSITY OF IBADAN LIBRARY temperature well above and below the melting temperatures with high precision. It is anticipated that most of the thermodynamic data on binary and multi- component systems will come from theoretical calculations, rather than from direct experimentation. Therefore, theoretical calculation is an indispensable and effec- tive approach to obtaining thermodynamic data for alloy of interest from some other data available on that system. In this regards, researchers have been moti- vated to develop different models to solve the difficulties and complexities of ob- taining thermodynamic quantities such as activity, free energy of mixing, enthalpy of mixing, entropy of mixing as well as thermophysical quantities like diffusion and viscosity of many binary liquid alloys by experiments. However, these mod- els are rarely valid (Chen et al., 2002) over the whole concentration ranges for real liquid and solid metallic solutions. For instance, extensive description of the hard sphere model to calculate the entropy of mixing of binary liquid alloys have been reported by Hoshino and Young (1981); Visser et al.,(1980) and Alblas et al., (1982), but the model could not explain thermodynamic properties of systems with chemical short-range order (Pettifor, 1993) and (Dalgic et al., 1998). The Melting point depression method (Chou and Wang, 1987) was used for simple eutectic sys- tems based on two approximations: the regular solution model of Hildebrand and Scott, (1950) and the melting enthalpy of component is assumed to be tempera- ture independent. The approximation dealing with regular behaviour was found to cause great errors in calculated activity values. Arising from this, in 1990, Chou developed a new method based on Rao-Belton method (Rao and Belton, 1981) and Richardson assumption (Richardson, 1974) to calculate thermodynamic properties of binary liquid alloys based on their known phase diagram. Sommer and Lee used two different models, an association model(Sommer et al., 1983) and the Miedema model(Godbole et al., 2004) to study thermodynamic properties of aluminum-calcium, aluminum-Strontium, aluminum-nickel and calcium-nickel as well as calculated the liquid phase enthalpy of mixing of liquid copper-magnesium 10 UNIVERSITY OF IBADAN LIBRARY alloys. Takana and Gokcen (1995) derived a statistical thermodynamic solution model based on the free volume theory that each atom moves in a liquid metal within a restricted region in a cell made by its nearest-neighbour atoms. Empirical relation for estimation of entropy of mixing of liquid alloys was proposed by Wi- tusiewicz and Sommer (2000) on the basis of experimental enthalpy of mixing data at constant temperature. The molecular interaction volume model (MIVM) which is a two-parameter model proposed by Tao (2000; 2003) was able to predict some component activities of liquid alloys and solid solution by using only the partial molar mixing enthalpies and the coordination numbers of the constituent elements in liquid alloys. This model has been used (Tao et al., 2002; Yang et al., 2008, 2009) for prediction of the mixing enthalpies of 23 binary liquid alloys. Whereas, Iwata et al.,(2003) predicted thermodynamic properties, including activity coefficients and the interaction parameters of the solute elements in infinite dilute Si solutions, by the use of first-principles calculations based on density functional theory. In addition, Chen and co-workers (2002) derived a statistical thermodynamic solution model based on the free volume theory. Their main aim was to calculate the excess entropy and excess Gibbs free energy in solid binary alloys through the experimental results of enthalpy of mixing. However, for successful application of the model the experimental results of mixing enthalpy should be known for different concentration ranges, otherwise the calculation cannot be carried out. Broadly speaking, the main problems associated with all the present models on liquid alloys are as follows: In a general way, the excess entropy was not accounted for or taken into consideration in the models formulations (Chen et al., 2002). The regular solution model proposed by Hildebrand and Scott (1950) and the sub- regular solution model proposed by Hardy (1953), also had not taken the problem of excess entropy SEij ̸= 0 into consideration. The SEij does not equal to zero in some molten solutions. It can be either positive or negative, mainly because the mixing of two components can not be completely random. In order to overcome this problem, Lupis and Elliott (1967) proposed the sub-regular solution model in 11 UNIVERSITY OF IBADAN LIBRARY which SEij is directly proportional to ∆H. And, Guggenheim (1976) proposed the solution model of standard chemical theory and provided the expression of excess free energy for binary systems. Though these models have physical background in some degree, they still however depend on the experimental data. It is thus obvious, that each of the proposed thermodynamic model requires inputs from thermodynamic measurements such as order energy or activity coef- ficients for the determination of the various thermodynamic properties and the models fail to make accurate prediction for liquid alloys over a wide range of con- centration and temperature with high precision. Thus underlining the importance of investigations of thermodynamic studies for binary liquid alloys in the frame- work of theoretical investigation. Therefore, a good, economic and effective method for obtaining thermodynamic data from the theories or from thermodynamic mod- els for binary liquid alloys without experimental values as input becomes a great necessity. In this study, using a statistical thermodynamic theory within the framework of Flory model (1942), an improved calculation method for obtaining the mixing properties and activities of binary liquid alloys based only on analytical expressions have been introduced (in chapter 3.4) without recourse to experimental data as input. The approach is capable of determining and evaluating the mixing properties and activities of components in binary liquid alloys. It is also a very useful approach to cut down the cost and time for the development of new alloys because the conventional alloy development has been primarily based on experimental approach and is frequently performed by trial and error methods (Liu and Fan, 2002). Such experimental approach to alloy design is cost intensive and time consuming. The thermodynamic data obtained using the analytical expressions are compared with the experimental data and the results obtained are found to be in satisfactory agreement with experimental values. Thus suggesting that the approach is reliable, convenient and economic and has a good physical basis. 12 UNIVERSITY OF IBADAN LIBRARY 2.2 Observable thermodynamic indicators 2.2.1 Structure factors S(q) of Liquid The structure factor S(q) of liquid is one of the most important properties to study the various electronic, magnetic, static and dynamic properties of a material in liquid states (Thakor et al., 2002). It is a measure of particle correlations in the reciprocal space. The liquid structure factor S(q) is related to I(q) as I(q) S(q) = (2.1) Nf 2 where I(q) is the intensity of the scattered radiation (say neutron or X-ray) is measured as a function of scattering angle, N is the number of scattering points and f is the atomic scattering factor, q = 4πsinθ/λ, 2θ is the scattering angle and λ is the wavelength of the radiation. The physical significance of S(q) is defined in terms of the pair distribution function g(r) of a liquid with density ρ (Ganesh and Widom, 2006), which measures the probability of finding an atom at a distance r from the origin (r=0) ∫ ∞ − sin(qr)S(q) = 1 + 4πρ [g(r) 1] r2dr (2.2) 0 qr Evidently, one needs a knowledge of the radial distribution function up to large values of r to get a good S(q). g(r) and S(q) both oscillate between 0 and 1. Besides providing the information regarding the distribution of atoms, g(r) can be utilized to determine the coordination number, z, for the first coordination shell, ∫ z = 4πr2ρg(r)dr (2.3) The limit of integration can be chosen to vary from zero to the distance where g(r) exhibits the first minimum. According to Bhatia and Thornton (1970), the concentration-concentration fluc- tuations Scc(q) for a binary mixture at q = 0 (long wavelength) may be expressed as Scc(0) = N⟨(∆c)2⟩ (2.4) 13 UNIVERSITY OF IBADAN LIBRARY where ⟨(∆c)2⟩ represents the mean square fluctuations in the concentration and are readily derived from statistical mechanics in terms of the Gibbs free energy G, i.e. NKBT Scc(0) = (2.5) (∂2G/∂c2A)T,P,N For a binary mixture consisting of NA = NcA and NB = NcB)g moles of A and B atoms where cA + cB = 1, G may be expressed as G = N [c G0A A + cBG 0 B] +GM (2.6) with G0A and G 0 B being the Gibbs free energies per atom of the species A and B, respectively. GM is the free energy of mixing. In view of equation (2.6), Scc(0) becomes NKBT Scc(0) = (∂2GM/∂c2A)T,P,N aAcB = (∂aA/∂cA)T,P,N cAaB = (2.7) ∂aB/∂cB T,P,N where aA and aB are the thermodynamic activities of components A and B in the mixture. It should be pointed out the (∂2G/∂c2A) is also known as the stability of the solution, which was originally introduced by Darken (1967) (Singh and March, 1995). Thus, the reciprocal of Scc(0), i.e. NKBT/Scc(0), is a measure of the sta- bility of the mixture. The Scc(0) is directly related to the excess stability function by the equation ( 2 xs) ( ) Exs ∂ G 1 1 = M = RT − (2.8) ∂c 2∑A) Scc(0) cAcB For ideal solutions, GM = RT i ci ln ci, and Scc(0) and E xs become Sidcc(0) = cAcB; E XS = 0 (2.9) It is obvious from equation (2.7) that Scc(0) can be determined directly from the measured activity data or from the Gibbs free energy of mixing. In the literature, this is usually referred to as an experimental Scc(0). It should pointed out that such 14 UNIVERSITY OF IBADAN LIBRARY a numerical differentiation is very sensitive, in particular around the critical region when activity and GM start to flatten (Singh and Sommer, 1997). However, a direct measurement of ∂aA/∂cA, for instance, with the help of electromotive force (emf) measurements, can be a suitable experimental basis for the calculation of Scc(0) as reported by Singh and Sommer (1997). 2.2.2 Small-angle scattering experiment In principle, the long wavelength limit of the structural factors Scc(0) can be deter- mined from the small-angle scattering experiments either by using neutron or x-ray as sources of radiation for extracting structural information from liquid metals and liquid alloys, but it poses a much more experimental problem that has never been successfully solved. As such, there is no direct information on Scc(0) from the diffraction experiments, although it can be determined directly from the measured activity data or measured free energy of mixing data as earlier mentioned. Owning to experimental difficulties, theoretical determination of Scc(0) is of great importance to anyone trying to visualize the nature of atomic interactions in the mixture, particularly if the region of interest is of the order of atomic dimen- sions. The mixing behaviour of liquid binary alloys can be deduced from the devia- tion of Scc(0) from the ideal value S id cc(0). If, at a given composition, S id cc(0)>Scc(0), then there is a tendency for segregation (preference for like atoms to pair as nearest neighbours); while Scc(0)S id cc(0). 16 UNIVERSITY OF IBADAN LIBRARY 4. The Warren-Cowley short-range order parameter, α1 > 0. 5. In the framework of regular solution theory, the interchange energy, ω > 0. For instance, the following liquid alloys Bi-Zn, Cu-Pb, Bi-Cu, Cd-Ga, Ag-Te, Na-Cs, Na-K (Singh and Sommer, 1997), which exhibit liquid immiscibility and comparatively large positive HM values are typical examples of segregating liquid alloys. On the other hand, short-range ordered alloys exhibit negative deviations from Raoultian behaviours in contrast to the above highlighted empirical criteria. Ex- amples of short-range ordered alloys are Bi-K, Al-Au, Bi-Mg and Hg-Na (Singh and March, 1995), Bi-Pb (Novakovic et al., 2002) and Cu-Zr (Novakovic et al., 2004). These alloys are usually strongly interacting and their mixing properties such as free energy of mixing, GM , heat of mixing, HM and concentration fluctuations Scc(0), etc. are often symmetrical about the equiatomic composition cc = 0.5. 2.3 Short-range order parameter The Warren-Cowley (Warren, 1969; Cowley, 1950) chemical short-range order (CSRO) parameter α1 which is frequently used to quantify the degree of order and segregation in the melt, is directly related to the conditional probability, [A/B], [A/B] = c(1− α1) (2.10) where [A/B] defines the probability of finding A atom as a nearest neighbour of B atom. The sign of α1 indicates whether atoms in a given mixture prefer AB ordering (α1 < 0) or segregating (α1 > 0 (Singh,1987). The SRO parameters are normalized such that the limiting values of α1 lie in the range − c ≤ α1 ≤ 1 1, c ≤ (2.11) (1− c) 2 −1− c ≤ 1α1 ≤ 1, c ≤ (2.12) (c) 2 17 UNIVERSITY OF IBADAN LIBRARY For a perfect random distribution of atoms i.e. an alloy without atomic correlations, [A/B] is simply c and then α1 = 0. If α1 < 0, then A-B pairs of atoms are preferred over A-A or B-B pairs as nearest neighbours, and for the converse case α1 > 0. For c = 1/2, one has −1 ≤ α1 ≤ 1. The minimum possible value, αmin1 = −1, means complete ordering of A-B pairs in the melt, whereas the maximum value, αman1 = 1, suggests that the A-A and B-B pairs in the melt are totally segregated. However, since α1 can be determined from diffuse x-ray and neutron diffraction experiments (Müller, 2003), a quantitative comparison between calculation and measurement is possible. Experimentally, α1 is determined from the concentration- concentration structure factors Scc(q) and the number-number structure factors SNN(q). For mixtures in which the A and B atoms are nearly the same size, (Singh and Sommer, 1997) showed that ∫ r1+ϵ 4πr2ρcc(r)dr = α1z (2.13) r1−ϵ where r1 is the mean distance of the first neighbour shell from a given atom. The actual distance is, however, taken between r1−ϵ and r1+ϵ. z is the number of atoms in the first shell, usually known as the coordination number, and is determined from SNN(q). ρcc(r) is called the radial concentration-correlation function, which is obtained from the Fourier transform (FT) of concentration-concentration structure factor, Scc(q), i.e., [ ] 2 Scc(q)4πr ρcc(r) = 4πrFT − 1 (2.14) c(1− c) However, because of practical difficulties involved in the measurement of Scc(q), experimental data on α1 are very scarce. 18 UNIVERSITY OF IBADAN LIBRARY Chapter 3 Methodology 3.1 Quasi Lattice Theory of Liquid Mixtures Bulk and Dynamic properties in Al-Zn and Bi-In Liquid Alloys Using a Theoretical Model Keywords: Al-Zn, Bi-In, Entropic, Enthalpic, Dynamic properties, Segregation 3.1.1 Introduction Different kinds of theoretical models proposed to explain the concentration depen- dence of the thermodynamic properties of liquid binary alloys and by so doing, extracting useful microscopic information on them has attracted the attention of Physicists, Chemists and Metallurgists for a long time (Bhatia and Hargrove, 1974; Singh, 1987; Akinlade, 1994 and 1995; Singh and Singh, 1995). In this regard, a substantial effort has been directed towards the understanding of the mechanism for the phenomenon of compound formation in liquid binary alloys (Singh, 1987; Singh and March, 1995). In addition, their energetics are reasonably well understood theoretically. On the contrary, relatively little is known and written on the other class of liquid alloys that exhibit segregation (i.e. preference of like atoms as nearest neighbours) or on the liquid alloys endowed with a miscibility gap. Singh and Sommer (1997) have 19 UNIVERSITY OF IBADAN LIBRARY documented extensively on thermodynamic information (both experimentally and theoretically) on phase separating systems. They have equally mentioned different kinds of theoretical methods that could be used to explain the mechanism for the energetics of liquid alloys. Singh and Sommer (1992a, 1992b) established an empirical model which pro- vides a possible way of relating two of the dynamics properties, as diffusion coeffi- cient (D) and viscosity (η) with thermodynamic properties like the concentration- concentration fluctuations in the long wavelength limit, Scc(0), and the free energy of mixing, GM , for liquid binary alloys. These quantities can then be connected in RT terms of entropic and enthalpic contributions to the free energy of mixing. How- ever, no practical application was made to a particular liquid alloy in their work so as to know the limit of suitability of the model. In this section, an attempt was made in this direction to use the theory by Singh and Sommer (1992a, 1992b) and show that it could be used to model the structure of liquid binary alloys and to facilitate our understanding of the energetics of the liquid alloys. The model was applied to Al-Zn and Bi-In liquid alloys. The choice of Al-Zn and Bi-In was however influenced by the availability of all the required thermody- namic data for both systems. Moreso, due to the difficulties in performing high temperature surface and viscosity measurements, in the literature only a few ref- erence data for metals (Lucas, 1984) are available. For binary systems they are scarce, and in the case of complex alloys a nearly complete lack of data is evident. Therefore, it is of interest to estimate the missing values of the viscosity data by theoretical models. The dynamic properties of Al-Zn and Bi-In liquid alloys are calculated in the framework of quasi-lattice theory (QLT) with the aim to analyze existing thermodynamic data and use these data as the input for the fitted param- eter calculations, and to explain the ordering and phase separating phenomena in liquid binary alloys. In addition, available information on phase diagrams and other thermodynamic properties indicate the existence of liquid miscibility gap in Al-Zn 20 UNIVERSITY OF IBADAN LIBRARY system. Its Scc(0) exhibit a tendency for phase separation over the concentration range, whereas the Scc(0) for Bi-In alloy exhibits compound formation tendency. The above mentioned facts motivated the study of these two systems to widen the theoretical understanding of the energetics in liquid binary alloys. 3.1.2 Basic Theory of Quasi-Lattice Theory The quasi-lattice theory (QLT) of liquid mixtures is based on the Guggenheim theory (Guggenheim, 1952) which assumes the existence of a binary mixtures con- sisting of NA = NcA and NB = NcB atoms of elements A and B, respectively. The theory also assume that a small gradient of composition cA is maintained in an equilibrium condition along the x-direction by the application of a force FA, such that for dilute alloy, it is possible to write (Akinlade et al., 1998) ( − c ) A FA = kBT d ln (3.1) dx where T is the temperature and kB, the Boltzmann’s constant. Using equation (3.1), Singh and Sommer (1992a, 1992b) have shown that Scc(0) is related to the diffusion coefficient, DM for liquid alloys (Singh and March, 1995; Singh and Som- mer, 1997) by ( ) kBT cA cB cAcB η = + (3.2) DM λ2 λ1 Scc(0) where λ1 and λ2 are functions of the size and shape of the constituent particles. Thus, the quantity ϕ is defined as cAcB ϕ = (3.3) Scc(0) and the ideal value of Sidcc(0) as Sidcc(0) = cAcB (3.4) Equation (3.3) can be rewritten as η = ηoϕ (3.5) 21 UNIVERSITY OF IBADAN LIBRARY with ( ) kBT cA cB ηo = + (3.6) DM λ2 λ1 Thermodynamic investigation of ϕ have been performed by Osman and Singh (1995) for the more general case in which one introduces the entropic contribution (i.e. the size ratio γ= ΩA , ΩB > ΩA), where Ω is the atomic volume and the en-ΩB thalpic contributions (via the interchange energy, ω). They obtained an expression for ϕ given by ϕ = 1− cAcBf(γ,W ) (3.7) where 2γ2W − (γ − 1)2(cA + γcB) f(γ,W ) = (3.8) ((c 3A + γ)cB)ω W = ΩA (3.9) KBT with ( ) w = Z ϵAB − (ϵAA + ϵBB)/2 (3.10) Equation (3.10) is known as the interchange energy and it should be stated that ϵAB, ϵAA, and ϵBB are the energies for AB, AA and BB pairs of atoms, respectively and Z is the coordination number of the liquid alloys. Obviously, if ω < 0, there is tendency to form unlike atom pairs, and if ω > 0, like atoms tend to pair together. ω=0, however, shows that atoms in the mixture are perfectly disordered(Singh, 1987). By taking together equations (3.5), (3.7), (3.8) and (3.9), one expresses ∆η ηo as ∆η = −cAcBf(γ,W ) (3.11) ηo The factor f(γ,W ) which introduces both the entropic and enthalpic effects is responsible for the characteristic behaviour of ∆η for a given binary alloy. In the light of the result obtained by Singh and March (1992a, 1992b), it is readily shown that ∆η −HM= (3.12) ηo RT R is the universal gas constant. It is important to note that in using the formulae given above, one does not have sufficient information on thermodynamic quantities 22 UNIVERSITY OF IBADAN LIBRARY to relate it to η. In order to do this, the general expression for the Gibb’s free energy of mixing GM is employed GM = −TSM +HM (3.13) where SM is the entropy of mixing and HM is the enthalpy of mixing. Using Guggenheim’s theory of mixtures (Guggenheim, 1952), QLT makes it possible to write an explicit expression for GM as G ( ω )M = cB lnΨ + cA ln(1−Ψ) + cAΨ ΩA (3.14) RT KBT = cB lnΨ + cA ln(1−Ψ) + cAΨW (3.15) with γcB Ψ = (3.16) cA + γcB And from GM as in equation (3.14) or (3.15), Scc(0) can easily be calculated from standard relationship in terms of free en(ergy of m)ixing, 2 −1∂ GM Scc(0) = RT (3.17) ∂c2 T,P,N or in terms of activity, aA and aB(, )−1 ( )−1 − ∂aA ∂aBScc(0) = (1 c)aA = caB (3.18) ∂c ∂(1− c) T,P,N T,P,N as cAcB Scc(0) = (3.19) 1− cAcBf(γ,W ) Once Scc(0) is fitted from equation (3.19), then all other parameters can be calcu- lated. The degree of order and segregation in the melt can be quantified by another important microscopic function, known as Warren-Cowley short range order pa- rameter or simply chemical short-range order (CSRO), α1, (Cowley, 1950) and (Warren,1969). The parameter α1 is related to the Scc(0) by Scc(0) 1 + α1 = (3.20) cAcB 1− (Z − 1)α1 23 UNIVERSITY OF IBADAN LIBRARY For equiatomic composition, the CSRO α1, is found to be in the range −1 ≤ α1 ≤ 1. The negative values of this parameter indicate the ordering in the melt, and complete ordering is shown by αmin1 =-1. On the contrary, the positive values of α1 indicate segregation, whereas the phase separation takes place if αmax1 =1. In addition,the mixing properties of binary molten metals forming alloys can as well be analyzed at the microscopic scale in terms of the quantity ϕ defined in equation (3.3) known as diffusion. The formalism relation that connects diffu- sion and Scc(0) (Singh and Sommer, 1992) combines the Darken’s thermodynamic equation for diffusion (Darken,1967) with the basic thermodynamic relation in the form (Singh and Sommer, 1992): D idM S = cc (0) (3.21) Did Scc(0) where DM is the mutual diffusion coefficient and Did is the intrinsic diffusion coef- ficient for an ideal mixture, given as Did = cADB + cBDA (3.22) with DA and DB being the self-diffusion coefficients of pure components A and B, respectively. For ideal mixing, Scc(0)→Sidcc(0), i.e. DM → Did; for ordered alloys, Scc(0) Did; and similarly for segregation, DM < Did. The highest peak of DM as a function of composition indicate the presence of maximum Did chemical order in molten alloy system as well as the composition of the most likely associates to be formed in the liquid phase (Singh and March, 1995). 3.2 Quasi-Chemical Approximation Model Thermodynamic properties of some gallium-based liquid alloys Keywords: Chemical order, Homocoordination, Interaction energy, Raoultian be- haviour 24 UNIVERSITY OF IBADAN LIBRARY 3.2.1 Introduction Many investigations have been reported in literature on liquid binary alloy systems which are of importance from both the scientific and also the technological points of view. An accurate knowledge of the thermodynamic properties and phase diagrams of the alloy systems are essential to establish a respectable understanding between the experimental results, theoretical approaches and empirical models for liquid alloys with a miscibility gap. The interatomic interactions and the related energies of the bond between the A and B component atoms of a binary alloy play an essential role in understanding the mixing behaviour of two metals. Because of this, the energetically preferred heterocoordination of A-B atoms as nearest neighbours over self-coordination A- A and B-B, or vice versa lead to the classification of all binary alloys into two distinct groups: short-range ordered (Bhatia and Hargrove, 1974; Bhatia and Singh, 1982; Singh and March, 1995; Novakovic et al., 2002; Novakovic et al., 2004) or segregating (demixing) alloys (Singh, 1993; Singh and Sommer, 1997; Prasad and Mikula, 2000a). The three binary alloys namely Ga-Zn, Ga-Mg and Al-Ga are the alloys of in- terest. The choice of these alloys arises from the fact that the three alloys have various industrial applications. Alloy systems containing semiconducting compo- nents such as gallium which is a group IIIB element like aluminium have been the subject of an increasing attention in the semiconductor production for their impor- tant application in the solid-state electronic devices and as a useful thermometric liquid (Awe et al., 2005). Also, magnesium alloys offer lightweight alternatives to conventional metallic alloys and consequently, research on Mg alloys is fuelled nowadays by the need for low-density materials that suit aerospace and automobile industries (Islam and Medraj, 2004) and as a result of which the world consump- tion of magnesium alloys in the automobile industry has been on the increase in the last decade (Awe et al., 2006). Moreover, aluminium and its alloys are used in many aspects of modern life, from soda cans and household foil to automobiles and 25 UNIVERSITY OF IBADAN LIBRARY aircraft in which we travel. Al-based alloys have also been found useful as additives in various fuel formulations for propellants, explosives, incendiaries or pyrotechnics (Awe et al., 2006). Large quantities of zinc are used to produce die castings. In addition, all the essential experimental data that are required for the calculation of the thermodynamic properties are available. It has been established in this work that both Ga-Zn and Al-Ga systems are characterized by positive interaction energies indicating segregation in the melts, while Ga-Mg alloy is characterized by a negative interchange energy indicating ordering in the melt. This characteristic behaviour is likely to be a reflection of the interplay of the energetic and structural re-adjustment of the constituent elemental atoms in the alloys. In this section, the composition dependence of thermodynamic properties of these liquid alloys has been studied using a quasi-chemical approximation model by Singh (1987) for compound forming binary alloys and that for simple regular alloys. The energetics of mixing as well as the positive deviation from Raoultian behaviour was discussed for the various thermodynamic quantities calculated. 3.2.2 Theory of QCAM The fundamental idea about the quasi chemical model is that the thermodynamic properties of a compound forming A-B alloy can be explained by treating the alloy as pseudo ternary mixture of A atoms, B atoms and AµBν (µ and ν are small integers) complexes. The grand partition function Ξ of a binary molten alloy AB, which consists of NA = Nc and NB = N(1 − c) atoms of elements A and B, respectively, where the total number of atoms, N, is equal to NA + NB,can be expressed as ∑ Ξ = qNAA (T )q NB B (T )exp[(µANA + µBNB − E)/kBT ] (3.23) E where qNi (T ) and µi are atomic partition function and chemical potential of i th components (i = A,B), kB is Boltzmann’s constant, T is the absolute temperature 26 UNIVERSITY OF IBADAN LIBRARY and E is the configurational energy of the alloy. The QCM is utilized to determine the probable chemical complexes existing in a liquid binary alloy. The solution of equation (3.23) in line with (Prasad et al., 1998) is given as the ratio of the activity coefficient γ (γ = γA ); (γ and γ are activity coefficients of γ A BB A and B atoms, respectively) for the compound (AµBν) forming alloys as Z(1− c) (β + 2c− 1) ln γ = . (3.24) 2c (β − 2c+ 1) with [ ]Z β − 1 + 2c 2 γA = (3.25) [ c(1 + β) β + 1− ]Z2c 2 γB = (3.26) (1− c)(1 + β) The excess free energy of mixing GXSM is related to the free energy of mixing GM by the expression: { } GXSM = GM −RT c ln c+ (1− c) ln(1− c) (3.27) The quasi chemical expression for the excess free energy of mixing GXSM is given as: GXS ∫ c[ ] M = Z lnσ + (2kBT ) −1(PAA∆ϵAA − PBB∆ϵBB) dx+ ϕ (3.28) RT 0 where Z is the coordination number, c is the concentration of atom A and R is the universal gas constant and 1 (1− c)(β + 2c− 1) lnσ = ln (3.29) 2 c(β − 2c+ 1) with β = {1 + 4c(1− c)(η2 − 1)}1/2 (3.30) and ( ) ( 2 2ω 2PAB∆ϵAB − PAA∆ϵAA − PBB∆ϵ )BBη = exp exp (3.31) zkBT kBT For a simple regular alloy, η is given as ( ) ω η = exp (3.32) zkBT 27 UNIVERSITY OF IBADAN LIBRARY ω is the interchange or order energy, expressed as: 1 ω = Z[∆ϵAB − (∆ϵAA +∆ϵBB)] (3.33) 2 ∆ϵAB, ∆ϵAA, ∆ϵBB are the interaction parameters and ∆ϵij is the change in the energy of the ij bond in the complex AµBν . Pij is the probability that the ij bond is a part of the complex. Pij may be expressed as P = cµ−1(1− c)ν−1[2− cµ−1(1− c)ν−1AB ] (3.34) P = cµ−2AA (1− c)ν [2− cµ−2(1− c)ν ], µ ≥ 2 (3.35) P µBB = c (1− c)ν−2[2− cµ(1− c)ν−2], µ ≥ 2 (3.36) The constant ϕ in equation (3.28) is determined from the requirement that GxsM = 0 at c=0 and c=1. The concentration-concentration fluctuations in the long-wavelength limit, Scc(0), has emerged as an important microscopic function to understand the mixing be- haviour of liquid alloys in terms of compound formation and phase segregation (Singh, 1987). The Scc(0) can be expressed by GM , or the activity, ai(i = A,B), as ( ) ( ) ∂2 −1 −1 GM ∂aA Scc(0) = RT = cBaA ∂c2A ∂cT,P,N ( A )T,P,N−1 ∂aB = cAaB (3.37) ∂(1− cA) T,P,N For ideal mixing the energy parameters, ω, given in equation (3.33) is equal to zero, and equation (3.37) becomes Sidcc(0) = cAcB (3.38) Substituting equation (3.27) for GM into equation (3.37), one obtains c(1− c) Scc(0) = ( ) (3.39) 1 + Z 1 − 1 2 β Equation (3.37) is usually utilized to obtain the experimental values of Scc(0) from the measured activity or the free energy of mixing data (Akinlade, 1998). The 28 UNIVERSITY OF IBADAN LIBRARY mixing behaviour of liquid binary alloys can be inferred from the deviation of S (0) from Sidcc cc(0). The presence of chemical order is indicated by S id cc(0)S id cc(0), the segregation and demixing in liquid alloy take place. The Warren-Cowley short-range order parameter α1 for the first nearest neigh- bours is expressed in term of β-function (Anusionwu, 2006), (equation (3.30)) as: β − 1 α1 = (3.40) β + 1 The enthalpy of mixing, HM within the QCM can be obtained from the standard thermodynamic relation (Novakovic and Zivkovic, 2005) (∂G )M HM = GM − T (3.41) ∂T P Making use of GM in equation (3.27), HM is expressed as 8RTc2(1− c)2exp( 2ω ) ( ) H = − Zk 1 dω ωBTM − (3.42) (β − 1 + 2c)(1 + β)(β + 1− 2c) kB dT T and thus the entropy of mixing, HM −GM SM = (3.43) T 3.3 Four Atom Cluster Model Energetics of Mixing in Bi-Pb and Sb-Sn Liquid Alloys Keywords: Four atom cluster model; Bi-Pb; Sb-Sn; heterocoordination; Bulk properties 3.3.1 Introduction Different theoretical models or formalisms have been established with a view to explaining or describing thermodynamic, structural and thermophysical properties of binary molten alloys. Some of these are Bhatia and Hargrove formalism (1974), electron theory (Aschroft and Stroud, 1978), four atom cluster model (FACM) 29 UNIVERSITY OF IBADAN LIBRARY (Bhatia and Singh, 1982; 1984) and the pseudopotential formalism (Dalgic et al., 1998; Thakur et al., 2005). FACM is a model established by Singh (1993) as an extension of his earlier work on a two atom cluster model (TACM) (Singh et al., 1988). This model is established to give a more realistic value of the ordering energy ω and the chemical short range order (CSRO) parameter α1 and hence, a better quantitative approach to the processes of compound formation or phase separation in liquid binary sys- tems (Akinlade, 1997). Essentially, in FACM a simple scheme is used to connect conditional probabilities enumerating the occupation of neighbouring sites by the atoms of the constituent elements in the liquid alloys. One significant advantage of the FACM, over the other available methods of calculating CSRO parameter α1 is the fact that in obtaining this parameter, the model formulation does not require the assumption of any preferred complexes (Akinlade et al., 2003). The implication of this is that we can obtain information on the properties of liquid alloys without having to necessarily look for information on such features like the phase diagrams, which normally provides information on the possible complexes (Akinlade, 1997). Furthermore, the calculation of α1 using the FACM only requires as input data, a knowledge of the activity ratio aB/aA, where A and B are the individual compo- nents of the alloy AB of interest. In addition to the aforementioned, FACM makes it possible to obtaining higher order conditional probabilities (HOCPs) encompassing atomic distribution in near- est neighbour shells. Hence, it is possible to define i/iji as the probability of finding i atoms on a given lattice site while the other three neighbours in the cluster are occupied by i, j, and i atoms. For the present study one has chosen Bi-Pb and Sb-Sn for some reasons. The choice of Bi-Pb for investigation was influenced by the recent work on the surface properties of Bi-Pb liquid alloys which suggested that Bi-Pb system is characterized by short-range ordering in the liquid phase (Novakovic et al., 2002). In addition, bismuth metal is used primarily as an alloy and metallurgical additive. One class of 30 UNIVERSITY OF IBADAN LIBRARY bismuth alloys comprises the low fusible (low-melting-point) alloys-combinations of bismuth and other metals, such as cadmium, lead, gallium, indium and tin. These alloys found applications in fuel tank safety plugs, holders for lens grinding and other articles for machining, solders, and sprinkler triggering mechanisms (Brown, 2000). Although, not quite much is known about the uses of Sb-Sn, yet one notes that since antimonial lead (Sb-Pb) is primarily used in grid metal for lead acid storage batteries and tin and lead are elements belonging to the same group of the periodic table (meaning that they have similar chemical properties), then, it is not unlikely that Sb-Sn will be relevant in the battery making industries. In addition, the availability of enough data on the two systems makes them good candidates for investigation. In this section, the FACM has been used to compute the energetics of Bi-Pb and Sb-Sn systems in terms of the ordering energy parameter reproducing their respective activity ratios and two other thermodynamic properties (free energy of mixing and the concentration-concentration fluctuations in the long wavelength limit) as well as CSRO parameter and chemical diffusion. 3.3.2 Theory of FACM The general expression for the grand partition function Ξ of a binary alloy AB consisting of N atoms, of which NA = Nc are A atoms and NB = N(1-c) are B atoms can be expressed as (Cartier and Barriol, 1976; Singh, 1993) ∑ Ξ = qNA(T )qNBA B (T )e β(µANA+µBNB)e−βE (3.44) E where qi(T ) are the particle functions of atoms i (A or B) associated with inner and vibrational degrees of freedom, µA and µB are the chemical potentials and E is the configurational energy. In solving equation (3.44), some simplifying assumptions are essential (Singh, 1993). The first assumption is that the interactions between the atoms should be of short range and effective only between nearest neighbours. The second assumption is that the atoms are located on lattice sites such that each 31 UNIVERSITY OF IBADAN LIBRARY site has Z nearest neighbours. Also, the lattice sites are further subdivided into smaller cluster of just a few lattice sites in domain 1 and the remainder in domain 2. In view of the above assumptions, one could then write the grand partition function as the product of the partition function of the two domains (i.e Ξ = Ξ1·Ξ2). The grand partition function for the cluster can be expressed as (Singh, 1993; Cartier and Barriol, 1976) ∑ Ξ = ξN1AξN1BϕZAϕZBe−βE11 A B A B (3.45) E1 where ϕA and ϕB are constants to be eventually eliminated NA = N1A +N2A ; NB = N1B +N2B ; E = E1 + E2 + E12 (3.46) In equation (3.46), E1 denotes the configurational energies of domain 1 and E12 takes into account the interactions between atoms in cluster and the remainder. The ξA and ξB in equation (3.45) can be expressed as ξ = q (T )eβµAA A ; ξB = qB(T )e βµB (3.47) For a cluster of one lattice site, E1 = 0 because there are no AA, AB, or BB atom in the bond. Hence, the single lattice site can be occupied by either an A-atom or a B-atom and ZA = ZB = Z (the coordination number), and equation (3.45) becomes (1) Ξ1 = ξ ϕ Z A A + ξ Z BϕB (3.48) The superscripts (1) in equation (3.48) is necessary to show that the expression for Ξ1 is for a cluster of one atom. For a cluster of four lattice sites in domain 1, taking into consideration, the possible arrangements of atoms that can be obtained statistically, one can then write equation (3.45) as (4) Ξ = ξ4ϕ4ZLP 6 + 4ξ3 ξ ϕ3ZLϕZLP 3 P 3 + ξ4 ϕ4ZLP 61 A A AA A B A B AA AB B B BB +6ξ2 ξ2 ϕ2ZLϕ2ZL 4 3 ZL 3ZL 3 3A B A B PAAPBBPAB + 4ξAξBϕA ϕB PBBPAB (3.49) 32 UNIVERSITY OF IBADAN LIBRARY Where ZL = Z − 3, P βEijij = e , (i,j = A,B) and the Eij’s are bond energies for ij nearest neighbour bond. Further simplification of equation (3.49) leads to an expression of the form σ12 −B σ91 −B σ62 −B σ33 −B4 = 0 (3.50) where ( ) ϕB PAA σ = (3.51) ϕA PBB B4 = x 3 (3.52) B = 3x2 (1− x/3) 3 (3.53) η3 (1− x) B2 = 3x (3.54) η4 1− 3x B1 = η3 ( ) (3.55) 1− c βω x = and η = exp (3.56) c ( Z) w = Z ϵAB − (ϵAA + ϵBB)/2 (3.57) Equation (3.57) is the interchan(ge energy or o)rder energy for the alloy. With the activity ratio earlier defined i.e a = aB/aA , one recalls the thermodynamic relation for the chemical potentials µA and µB for the two components elements A and B in the alloy, µ A BA = µo + β ln aA and µB = µo + β ln aB (3.58) where µAo and µ B o are the chemical potentials of the pure species A and B. Using the grand partition function and the above equation, one obtains 1 µoA = −β ln qA(T ) + ZϵAA (3.59)2 µo 1 B = −β ln qB(T ) + ZϵBB (3.60)2 Equations (3.47), (3.59) and (3.60) combine(d to y)ieldZ/2 aB ξB PBB a = = (3.61) aA ξ A PAA 33 UNIVERSITY OF IBADAN LIBRARY Where aA and aB are the activities of component A and B in the alloy. An expres- sion connecting a and σ can be found in (Singh, 1993) as cf1(a, σ) = (1− c)f2(a, σ) (3.62) where f1(a, σ) and f2(a, σ) are defined as 4 4ZL 3a 3σ3ZL 3a2σ2ZL aσZL f1(a, σ) = a σ + + + (3.63) η3 η4 η3 and a3σ3ZL 3a2σ2ZL 3aσZL f2(a, σ) = + + + 1 (3.64) η3 η4 η3 where ZL in both equations (3.63) and (3.64) has the same meaning as in equation (3.49). The activity ratio for a given binary alloy can be determined by solving equation (3.64) numerically based on the knowledge of σ obtained from the numer- ical solution of equation (3.50). The value of σ needed in the calculations should be optimized in such a way that it gives a good overall representation of activity at all concentrations as reported by (Akinlade, 1997). The major idea behind the FACM is to express the degree of CSRO in terms of probabilities. Even in the framework of this model, the probability of finding an A atom or B atom on any lattice site still depends on the nature of atoms already existing in the neighbouring sites (Akinlade, 1995). One begins by stating the probability that all four lattice sites occupied by atoms A as (A, A, A, A) and similar probabilities (i, j, k, l) can readily be reduced to HOCPs such that i/iji (the probability of finding i atom on a given lattice site while the other three sites in the cluster are occupied by i, j, and i atoms) and similar others. Having this in mind, one can express the pairwise conditional probability PAB as (A/BB) PAB = (A/B) = (3.65) (B/AB) + (A/BB) In terms of HOCP, one writes for the terms in equation (3.65) (A/BBB) (A/BB) = (3.66) (B/ABB) + (A/BBB) (B/AAB) (B/AB) = (3.67) (B/AAB) + (A/ABB) 34 UNIVERSITY OF IBADAN LIBRARY and one further notes that (B/ABB)=1-(A/ABB). However, equations (3.65)- (3.67) are only useful in this context if they can be expressed in terms of a and σ. Hence, the results are 1 (A/BBB) = (3.68) 1 + aσZL exp(3βω/Z) 1 (A/ABB) = (3.69) 1 + aσZL exp(βω/Z) aσZL exp(−βω/Z) (B/AAB) = (3.70) 1 + aσZL exp(−βω/Z) The value of PAB defined by equation (3.65) can then be obtained by solving equations (3.68)-(3.70). A useful quantity which can be obtained from knowledge of PAB is the Cowley-Warren short range order parameter α1 (Cowley, 1950). For nearest neighbour sites, α1 can be defined as: PAB α1 = 1− (3.71) c From a simple probabilistic approach the limiting values of α1 lie in the range − c ≤ α1 ≤ 1 c ≤ 1 (3.72) (1− c) 2 −1− c ≤ 1α1 ≤ 1 c ≥ (3.73) (c) 2 For c = 1 , one has −1 ≤ α1 ≤ 1. The minimum possible value, αmin1 , means2 complete ordering of A-B pairs in the melts, whereas the maximum value αmax1 = +1, suggests that the A-A and B-B pairs in the melts are totally segregated (Singh, 1987). A situation in which α1 = 0, implies a random alloy. On the basis of the discussion on the equations written so far, one can say that we have obtained all the essential equations required for FACM. In order to calculate the bulk thermodynamic properties, the values of ω ob- tained above have been used; hence, one has computed the excess Gibb’s free en- ergy of mixing, GXSM and the concentration-concentration fluctuations in the long wavelength limit, Scc(0). In obtaining these, one has used G = Gid XSM M +GM (3.74) 35 UNIVERSITY OF IBADAN LIBRARY where { } GidM = RT c+ (1− c) ln(1− c) (3.75) and { } GXSM = RT c ln γA + (1− c) ln γB (3.76) Here γA and γB are activity coefficients and are related to the activity a by the standard relations: aA = cγA ( ; aB =)(1− c)γB (3.77) − Z/2β 1 + 2c γA = (3.78) ( c(β + 1) ) − Z/2β + 1 2c γB = (3.79) (1− c)(β + 1) with β = {1 + 4c(1− c)(η2 − 1)}1/2 (3.80) From equation (3.74), Scc(0) can be determined using thermodynamic rela- tions: ( )−1 ( )2 −1∂ GM − ∂aAScc(0) = RT = (1 c)a ∂c2 ( A ∂cT,P,N )T,P,N−1 ∂aB = caB (3.81) ∂(1− c) T,P,N Substituting equation (3.74) in equation (3.81), one obtains c(1− c) Scc(0) = ( ) (3.82) 1 + Z 1 − 1 2 β Hence, from equations (3.74) and (3.81), it means that once a suitable optimized value of ω is obtained, it can be used to compute GM and Scc(0) which can thenRT be compared with experiments and deductions made therefrom. One observes that β in equation (3.82) depends on the interchange energy ω through equation (3.56) and brings about the deviation from ideality. As ω → 0, β → 1 and Scc(0) = c(1- c) = Sidcc(0), the ideal values. Equation (3.81) is usually utilized to obtain the experimental values of Scc(0) from the measured activity or the free energy of mixing data (Bhatia and Har- groove, 1974). 36 UNIVERSITY OF IBADAN LIBRARY The knowledge of Scc(0) has been further used to investigate the nature of chemical diffusion in the two liquid alloys, which is likely to play an important role in many technological and corrosion phenomena. Using the Darken (1948) thermo- dynamic equation for diffusion, an expression that relates diffusion and Scc(0) can be established (Singh and Sommer, 1992a; Prasad et al., 1998); thus one can write D SidM = cc (0) (3.83) Did Scc(0) where DM is the mutual diffusion coefficient and Did is the intrinsic diffusion coef- ficient for an ideal mixture, given as Did = xADB + xBDA (3.84) with DA and DB being the self-diffusion coefficients of pure components A and B, respectively. For ideal mixing, S idcc(0) →Scc(0), i.e. DM→Did; for ordered alloys,Scc(0) Did; and similarly for segregation, DM < Did. The maximum peak of DM/Did as a function of composition indicate the presence of maximum chemical order in molten alloy system as well as the composition of the most probable associates to be found in the liquid phase (Prasad et al., 1998; Singh and March, 1995). 3.4 Statistical Thermodynamic Theory Determination of Mixing Properties and Activities of Binary Liquid Alloys Keywords: Statistical thermodynamic, size factor, activity, enthalpy of mixing, entropy of mixing 3.4.1 Introduction A great deal of thermodynamic data on binary liquid alloys have been compiled in monographs such as (Hultgren et al., 1973), but multi-component data is scattered over some journals and quite scarce because the determination of thermodynamic data of these liquid alloys needs not only the consummate skill of a researcher 37 UNIVERSITY OF IBADAN LIBRARY and excellent instruments but also the continuous financial support (Tao, 2000). Although, the development of new and reliable materials requires sometimes the knowledge of thermodynamic and thermophysical data which due to the experimen- tal difficulties, especially high investigation temperatures required have not been successfully measured (Novakovic and Takana, 2006; Kostov et al., 2007; 2008) and coupled with the growing numbers of liquid alloys and numerous multi-component systems in industrial processes. The main reasons for this high scientific attention is certainly due to increasing applications of liquid alloys, especially in aerospace industry, due to their high oxidation resistance, low density and high melting point. It was obviously unrealistic to measure all the data experimentally. At present, sev- eral experimental methods, such as calorimetric method, electromotive force (emf) method and Knudsen effusion method, are in use to measure thermodynamic data for liquid binary alloys and for ternary compounds. Although, the experimental method is regarded to be the reliable one, but it is usually costly, laborious and time-consuming. Studies in both theoretical and computational modelling of liquid alloys have led to rapid increase in the development of advanced high performance materials in industrial world. These high performance liquid and metallic alloys find usage in various segments of materials and chemical industry as catalysts, low weight and high strength structural materials (Liu and Fan, 2002; Tao 2000; 2003; Tao et al., 2002). In this regards, focus is on research areas having direct impact on innovative development of such materials. Thus, the need for timely, cost efficient and envi- ronmentally compliant methods for development of such advanced materials. This aroused a genuine interest to carry out a theoretical determination of the mixing properties and activity of the liquid Sb-Sn, In-Pb, Al-Si and Ga-Tl alloy systems based on the recently published experimental data on Al-Si (Kostov et al., 2007) and Ga-Tl (Katayama et al., 2003). Although, different computational methods have been developed in literature to obtain thermodynamic properties such as free energy of mixing, heat of mixing, 38 UNIVERSITY OF IBADAN LIBRARY etc. Witusiewicz and Sommer, (2000); Islam and Medraj, (2004); Kanth and Chakrabarti, (2009) as well as thermophysical properties such as viscosity and diffusion (Akinlade et al., 1998; Novakovic et al., 2008; Anusionwu et al., 2009; Yong and Chen, 2011) of many binary liquid alloys. But these methods require inputs from thermodynamic measurements such as ordered or interchange energy, activity or activity coefficients for the determination of various thermodynamic properties, otherwise the calculations cannot be done. In this section, an effective and cost efficient approach to determine the mixing properties like the free energy of mixing, GM , enthalpy of mixing, HM , entropy of mixing, SM and activities of binary liquid alloys at different temperatures within the framework of a statistical thermodynamic theory without the need for input from thermodynamic measurements has been proposed. With this approach lim- ited experiments can then be focused on the most promising alloy compositions identified by thermodynamic calculations, while avoiding the experiments with the less promising ones. This approach is a very useful method to cut down the cost and time for the development of new alloys. Analytical expressions have been obtained for the various thermodynamic quan- tities such as GM , HM , SM and ai using standard thermodynamic relations by incorporating the atomic size mismatch volume. These expressions can be used for determination of the mixing properties and activities of binary liquid alloys at any temperature of interest and, the results obtained can then be compared with the experimental data. The analytical expressions were applied to liquid Sb-Sn at 905K, In-Pb at 673K, Ga-Tl at 973K, 1073K and 1173K and Al-Si at 2000K, 2400K and 2473K, respectively. The free energy of mixing GM and activity of liquid alloys play important role in predicting the glass-forming ability of the multi-components metallic alloys to the extent that many multi-component glass forming alloys like Ca- Senkov et al., (2004), Fe- Liu et al.,(2004), Cu- Cai Anhui et al.,(2008), and Mg-based alloys- Jha and Mishra, (2001) and Cai Anhui et al.,(2006) possess excellent glass-forming 39 UNIVERSITY OF IBADAN LIBRARY ability. Mg-Zn is similar in structure to Ca-Mg. Ca-Mg is a good glass former while Mg-Zn glass was reported by Jha and Mishra,(2001) as the first transition metal-free metallic glass. The two quantities GM and ai are among the fortunate thermodynamic functions which are obtained directly from experiment. However, there is no universal applicable expression forGM , though attempts have been made to derive analytical expressions for free energy of mixing of different materials (Cai et al., 2008; Lad Kint et al., 2004). Besides the factors mentioned above, the alloys or the individual constituent elements of the chosen candidates are known for their industrial relevance and the availability of the required experimental data for the liquid alloys is another impor- tant factor. Some of the important uses of these materials are as follows: Aluminum alloyed with tin, indium and zinc is used as sacrificial anodes and corrosion-proof anodic coating for steel (Popescu and Taloi, 2007), while Al-Sn-Zn alloys are used as soldering materials (Prasad and Mikula, 2006). Also, Mg-based alloys with dif- ferent additives such as Pb, Ga, Al, Zn, Hg and Tl have been widely used as anode materials due to their rapid activation, low density, low electrode potential and high current capacity (Islam and Medraj, 2004; Aljarrah et al., 2007). In addition, various properties of Mg liquid alloys show anomalous behaviour as a function of concentration, with Mg being the lightest structural metal, with a density of 1.741g/cm3, in comparison with the densities of aluminium (2.70g/cm3) and iron (7.86g/cm3). This makes magnesium alloys particularly attractive for weight re- duction and higher fuel efficiency transportation applications (Islam and Medraj, 2004; Aljarrah and Medraj, 2008). As a result, thermodynamics of liquid alloys are currently engaging attentions of numerous researchers. Moreover, no systematic thermodynamical study of Ga-Tl and Al-Si liquid al- loys has been done or completely reported in literature until very recently, when Katayama et al., (2003) carried out an activity measurement of Ga in liquid Ga-Tl alloys by electromotive force (emf) method and measured the free energy of mixing, GM and aA of binary Ga-Tl liquid alloys at 973K, 1073K and 1173K, respectively. 40 UNIVERSITY OF IBADAN LIBRARY And Kostov et al,(2007) for the first time used a FactSage Thermo-Chemical Soft- wares and Databases developed in 2001 to make thermodynamic predicting of the free energy of mixing, GM of Al-Si liquid alloys at different temperatures of 2000K, 2400K and 2473K, respectively. Thus, the result obtained using the analytical expressions proposed for theoret- ical calculations of free energy of mixing GM and activity aA of Ga-Tl and Al-Si at investigated temperatures for which the data exist were compared with Katayama et al.,(2003) for Ga-Tl and Kostov et al.,(2007) for Al-Si, respectively. 3.4.2 Theoretical formulation The grand partition function of a binary liquid A−B alloy consisting of NA = NcA and NB = NcB atoms of the element B, respectively, where the total number of atoms, N = NA +NB, is expressed in terms of the configurational energy, E, as: ∑ Ξ = qNAA (T )q NB B (T )exp((µANA + µBNB − E)/kBT ) (3.85) where qi(T) are the partition functions of atoms i (= A or B) associated with inner and vibrational degrees of freedom and it is assumed that the partition functions remain unchanged immaterial of whether the atom i is located in the pure state or located in the alloy; µ1 and µ2 are the chemical potentials, kB is Boltzmann’s constant and T is the absolute temperature. Equation (3.85) is solved in the framework of the quasi-lattice theory to obtain an expression for the activity. The activity of an element in a binary mixture is given by (Takana and Gokcen,1995) as: −zFEm ln a = (3.86) kBT where z is the valence of carrier ions of the element, F is the Faraday’s constant and Em is the electromotive force which can be observed directly from experiment. Using statistical thermodynamic theory, an expression for the Gibbs free energy of mixing[ is obtained as: ] GM (1− cA) = cA ln cA+(1−cA) ln(1−cA)+cA ln(1−ν)−ln(1−νcA) +ωc (3.87) RT 1− νc 41 UNIVERSITY OF IBADAN LIBRARY with 1 VB ν = 1− ; n = (3.88) n VA where VA and VB are the atomic volume of species A and B, respectively in the binary mixture A-B alloys and ω is the interchange energy. In order to obtain the expression for the activity of components ai, (i = A,B) in the binary alloy, one makes use of the standard thermodynamic relation: ( ) ( ) ∂GM ∂GM RT ln aA = = GM + (1− cA) (3.89) ∂NA T,P,NB ∂cA T,P,N Recalling that N = NA + NB and cA = NA/N , and upon differentiating equation (3.87) partially with respect to cA, one obtains [ ] [ ] ∂GM = RT ln cA−ln(1−cA)+ln(1− ν 1− 2cA ν(1− cA)cA) ν)+ +ω + (3.90) ∂c 2A νcA 1− νcA (1− νcA) By putting equations (3.87) and (3.90) in equation (3.89) and solving, the activities can be expressed as: cA(1− ν) ν(1− c ) ω (1− c 2A) ln aA = ln + + (3.91) 1− νcA 1− νcA RT (1− νcA)2 and cA ν(1− cA)cA ω (1− cA)ν ln aB = ln + + (3.92) 1− νcA 1− νcA RT (1− νc )2A Once the expression for GM is obtained, other thermodynamic quantities readily follows. Enthalpy of mixing and entropy of mixing are related to GM through standard thermodynamic relations: ( ) ∂GM HM −GM HM = GM − T ; SM = (3.93) ∂T T,P,N T Differentiating equation (3.87) with respect to T and substituting in equation (3.93), an expression for the enthalpy of mixing is given as cA(1− cA) − cA(1− cA) ∂ω cA(1− cA)HM = ω T [ +RT 2 1− νcA 1− νcA ∂T 1− ν]cA × ν − cA ω ∂ν (3.94) 1− ν 1− νcA RT ∂T 42 UNIVERSITY OF IBADAN LIBRARY and, the entropy of mixing as: SM = R[cA ln cA + (1− cA) ln(1− cA)[]+R[cA lnn− ln(1− ν]cA)] −cA(1− cA) ∂ω cA(1− cA) ν cA ω ∂ν+RT − (3.95) 1− νcA ∂T 1− νcA 1− ν 1− νcA RT ∂T Equations (3.87), (3.91), (3.92), (3.94) and (3.95) are the essential equations for the model calculations. 43 UNIVERSITY OF IBADAN LIBRARY Chapter 4 Results and Discussion In this chapter, results and discussion on each of the models used in the previous chapter are presented. 4.1 Quasi-lattice theory The QLT theory discussed in section 3.1, has been used to investigate the com- position dependence of the bulk thermodynamics and dynamic properties such as diffusion and viscosity for the two systems Al-Zn and Bi-In liquid alloys. It is evi- dent from the theory that there are some parameters that need to be fitted in order to perform the calculations. These parameters are the size ratio,γ, the coordina- tion number, Z and the interchange energy, ω. For consistency, the coordination number, Z in the liquid phase was chosen as 10 (Waseda, 1980; Anusionwu and Adebayo, 2001). It was observed in the course of these calculations that the choice of Z does not significantly affect the results obtain. In addition, the size ratio, γ should have been obtained from experimental density measurements but one has chosen to treat it as a free parameter in performing this calculations simply because there are no experimental measurements for the densities at the temperatures for which thermodynamics data are available. Once suitable values of these param- eters are chosen, the theory allows to calculate other quantities described above. The values of the fitted interaction parameters obtained are presented in table 4.1. 44 UNIVERSITY OF IBADAN LIBRARY 4.1.1 Concentration Fluctuations in the long-wavelength limit, the Free Energy of Mixing and the Chemical Short-range order parameter Due to the difficulties associated with diffraction experiments, the theoretical cal- culation of Scc(0) is of great interest when investigating the nature of interaction as well as the structure of binary liquid alloys(Novakovic et al., 2004). The mixing be- haviour of liquid binary alloys can be inferred from the deviation of Scc(0) from the ideal value, Sidcc(0). The presence of chemical order is indicated as Scc(0)S id cc(0), the segregation and demixing in liquid alloys take place. A perusal of Fig. 4.1, the plot of concentration fluctuations in the long- wavelength limit Scc(0) versus concentration for the systems studied show that the fitted parameters yield a good representation of the experimental data. The experimental values of Scc(0) for the two systems were derived from the experimen- tal free energy of mixing equation (3.17), the theoretical Scc(0) was obtained using equation (3.19) and thermodynamic data were taken from(Hultgren et al., 1973). The Scc(0) values for the Al-Zn system at T=1000K (fig. 4.1) clearly indicate that S idcc(0)>Scc(0) in the whole concentration range. This implies a tendency for homocoordination, i.e. segregation of preference for like atom Al-Al and Zn-Zn tend to pair as nearest neighbours. The Scc(0) curve exhibits the maximum value of about 0.409 at CAl = 0.50. For Bi-In system, the Scc(0) values show that Scc(0) < S id cc(0) over the concentration range. This suggests a tendency of complex formation i.e. preference for unlike atoms pairing as nearest neighbour. The plot of the concentration dependence of GM for the two alloys are given in RT Fig. 4.2. It is interesting to note that the computed values are in quite excellent agreement with experiment. From the result, the Gibbs energies of mixing GM for Al-Zn and Bi-In liquid alloys, have values of about -0.4986RT and -0.9344RT, 45 UNIVERSITY OF IBADAN LIBRARY Table 4.1: Values of the parameters for Al-Zn and Bi-In liquid alloys Alloy T(K) Z W γ RT Al-Zn 1000 10 0.800 0.950 Bi-In 900 10 -0.900 1.135 46 UNIVERSITY OF IBADAN LIBRARY 0.45 0.4 △ 0.35 △ △ △ 0.3 0.25 Scc(0) △ 0.2 ←Sidcc(0) ←Al-Zn × ×0.15 × ←Bi-In × × 0.1 × △ 0.05 0△× △× 0 0.2 0.4 0.6 0.8 1 CAl,CBi → Figure 4.1: Concentration fluctuations in the long-wavelength limit (Scc(0) and Sidcc(0)) vs. concentration for Al-Zn and Bi-In liquid alloys at 1000K and 900K respectively. The solid line denotes theoretical values while the triangle and the cross denote experimental values for Al-Zn and Bi-In respectively. The dot denotes the ideal values Sidcc(0). cAl and cBi are the Al and Bi concentrations in the alloy. 47 UNIVERSITY OF IBADAN LIBRARY 0△× △× -0.1 -0.2 ←Al-Zn △ -0.3 △ -0.4 △ △ × GM -0.5 △ △ RT -0.6 ×←Bi-In -0.7 -0.8 × × -0.9 × × -1 0 0.2 0.4 0.6 0.8 1 CAl, CBi → Figure 4.2: Concentration dependence of GM for Al-Zn and Bi-In liquid alloys at RT 1000K and 900K, respectively. The solid line denotes theoretical values while the triangles and the crosses denote experimental values for Al-Zn and Bi-In, respec- tively. cAl and cBi are the Al and Bi concentrations in the alloy. The experimental data were taken from Hultgren et al., (1973). 48 UNIVERSITY OF IBADAN LIBRARY respectively. This shows that the tendency of compound formation in the liquid phases is weaker in Al-Zn than in Bi-In alloy. The same mixing behaviour can be deduced from the values of the Warren- Cowley short-range order parameter, α1, table 4.2. The positive values of the parameter α1, for Al-Zn support a tendency towards segregation and the negative values of α1, for Bi-In system at all concentrations confirm a tendency of complex formation. 4.1.2 Dynamic properties: diffusion and Viscosity From the ordering point of view, the computed chemical diffusion ratio (equation (3.21)) seems to be a more realistic parameter than the Warren-Cowley short range order parameter, α1, (Novakovic et al., 2004) since equation (3.21) does not take into consideration the coordination number, Z, as an input and thus the inherent problems related to its estimation are avoided. The relationship between the Scc(0) and diffusion expressed by the ratio of the mutual and self-diffusion coefficient DM , indicates the mixing behaviour of alloys, i.e. phase separation tendency or Did segregation in Al-Zn, for DM < 1 over the concentration range, table 4.3. The Did values DM > 1 for Bi-In liquid phase at all concentrations suggest a tendency for Did compound formation in the melt. As to the other aspect of the calculations which is to investigate how well the theoretical formulation in equations (3.11) and (3.12) compare with experiment, Fig. 4.3 shows the results obtained for ∆η and −HM , but due to the lack of ηo RT experimental data for viscosity at temperatures of interest, it was not possible to compare the results from this work for viscosity with experimental values. However, examination of Fig. 4.3 reveals that the computed ∆η for Al-Zn show a negative ηo deviation while its −HM is characterized by liquid miscibility gaps and exhibit large RT positive heat of mixing, HM . This fact is substantiated by the earlier hypothesis of the existence of a liquid miscibility gap in Al-Zn liquid alloy. On the other hand, the computed ∆η for Bi-In exhibit positive deviation and ηo 49 UNIVERSITY OF IBADAN LIBRARY Table 4.2: Computed chemical short range order parameter (α1) for Al-Zn and Bi-In alloys at T = 1000K and 900K, respectively. CAl,Bi (α1)Al−Zn (α1)Bi−In 0.1 0.0697 -0.0696 0.2 0.0625 -0.0614 0.3 0.0551 -0.0534 0.4 0.0476 -0.0453 0.5 0.0399 -0.0375 0.6 0.0322 -0.0298 0.7 0.0244 -0.0222 0.8 0.0164 -0.0145 0.9 0.0083 -0.0073 50 UNIVERSITY OF IBADAN LIBRARY Table 4.3: Concentration dependence of DM for Al-Zn and Bi-In alloys at T = Did 1000K and 900K, respectively. C DM DMAl,Bi Did Al−Zn Did Bi−In 0.1 0.8510 1.1493 0.2 0.7393 1.2751 0.3 0.6631 1.3746 0.4 0.6209 1.4443 0.5 0.6112 1.4805 0.6 0.6324 1.4792 0.7 0.6824 1.4358 0.8 0.7623 1.3334 0.9 0.8683 1.2021 51 UNIVERSITY OF IBADAN LIBRARY 0.5 0.4 0.3 ←Bi-In △ △△ △ △ 0.2 △ △ 0.1 △ △∆η η0 0△× △× × × -0.1 × ← × × -0.2 Al-Zn × × × × -0.3 -0.4 0 0.2 0.4 0.6 0.8 1 CAl, CBi → Figure 4.3: Concentration dependence of viscosity ∆η eqns.(3.11) and (3.12) for ηo Al-Zn and Bi-In at 1000K and 900K respectively. The solid line are for calculated ∆η . The triangles and the crosses denote experimental values for −HM for Al-Zn ηo RT and Bi-In respectively. cAl and cBi are the Al and Bi concentrations in the alloy. The experimental data were taken from Hultgren et al., (1973). 52 UNIVERSITY OF IBADAN LIBRARY its −HM shows negative deviation which confirm that Bi-In liquid alloy is an ordered RT alloy. In order to discuss the disagreement between the calculated values of ∆η and ηo −HM for the two systems, it is important one looks at the parameters used to fit RT Scc(0) and hence to carry out the calculations. It is observed that the fitted value of γ for Bi-In is 1.135 and 0.950 for Al-Zn, (table 4.1). The calculated values of γ for Bi-In is closer to that determined at the melting point than that of Al-Zn. One reasonably inference from this is that the size effect plays a prominent role in the energetics of Bi-In than it plays in Al-Zn alloy and thus pulling the system from segregation to order phase. 4.1.3 Summary A Quasi-Lattice Theory has been utilized to obtain the fitted parameters, that are assumed to be invariant in all calculations. The bulk and dynamic properties in Al-Zn and Bi-In liquid alloys have been investigated using a theoretical approach with special interest on their bulk thermodynamic properties such as free energy of mixing, Concentration fluctuations in long-wavelength limit, chemical short range order parameter and the concentration dependence of diffusion and viscosity. The theoretical investigation of bulk properties of Al-Zn liquid alloy is substantiated by the experimental data, obtained at 1000K (Hultgren et al., 1973). The ordering in Al-Zn liquid phase has been analyzed in terms of the microscopic functions, Scc(0), and CSRO (α1). The calculated values of these functions indicate the existence of liquid miscibility gap or segregation in the melt. On the contrary, the presence of chemical order was observed for Bi-In at 900K. The bulk thermodynamic and dynamic properties of the two liquid alloys have been explained to a reasonable extent and the role of size effects notwithstanding the non-availability of the ex- perimental viscosity data. The results obtained in this work establish or confirm the applicability of this theoretical approach for a proper description of mixing properties of binary liquid alloys. 53 UNIVERSITY OF IBADAN LIBRARY 4.2 Quasi Chemical Approximation Model The mixing properties of Ga-Zn, Ga-Mg and Al-Ga binary liquid alloys have been discussed based on the model formulation described in section 3.2. The values of the relevant parameters used to obtain various thermodynamic quantities are presented in table 4.4. Keeping these fitted parameters which gives the best representation of the ob- served Gibbs free energy of mixing GM data unchanged in all calculations, one then proceeded using these fixed values to compute properties such as the Scc(0), α1, HM and SM and thus, forming a basis for discussing the energetics of the liquid alloys. The free energy of mixing for these liquid alloys has been computed using equa- tion (3.27). The experimental data given as symbols in figure 4.4 were taken from (Hultgren et al., 1973). Examination of figure 4.4 shows that the interaction pa- rameters give a good reproduction of the experimental values of the free energy of mixing for the alloys Ga-Zn, Ga-Mg and Al-Ga at temperatures of 700K, 923K, and 1073K, respectively. The good agreement obtained for GM as shown in the figure for the three systems investigated, allow the use of the fitted energy parameters to study the nature of ordering in the liquid alloys. A comparison of the figure for the three liquid alloys reveal that the free energy of mixing are almost symmetric around the equiatomic composition (cGa,Al = 0.5), with Ga-Mg [( GM )cc = 2.08]RT exhibiting a higher tendency for compound formation and is the most interacting of the three alloys; Al-Ga [(GM )cc = 0.652] is more interacting than Ga-Zn alloyRT [(GM )cc = 0.569].RT For proper analysis of the nature of ordering in the melts, it is important to critically consider the results for the structure related quantities. From this point of view the first quantity considered is the concentration fluctuations Scc(0). The deviation of Scc(0) from ideal value S id cc(0)=cAcB is an essential parameter in or- der to visualize the nature of atomic interactions in the mixture. If, at a given composition S (0)>Sidcc cc(0), then there is a tendency for segregation and vice versa 54 UNIVERSITY OF IBADAN LIBRARY Table 4.4: Fitted interaction parameters for the systems. System T(K) Z ω(eV ) dω × 10−3 µ ν dT Ga-Zn 700 10 0.0323 -1.240 1 1 Ga-Mg 923 10 -0.3932 9.215 1 1 Al-Ga 1073 10 0.0150 -0.250 1 1 55 UNIVERSITY OF IBADAN LIBRARY 0△×⋆ △×⋆ ×⋆ ×⋆ ×←Ga-Zn -0.5 ⋆ × × × × × ×⋆ △ ⋆ ⋆← ⋆Al-G⋆a ⋆ △ -1 GM RT △ △ -1.5 ←Ga-Mg △ △ △ -2 △ △ -2.5 0 0.2 0.4 0.6 0.8 1 cGa, cAl → Figure 4.4: Free energy of mixing, GM versus concentration for Ga-Zn, Ga-Mg and RT Al-Ga liquid alloys at 700K, 923K and 1073K, respectively. The solid line denotes theoretical values while the crosses, triangles and the stars denote experimental values for Ga-Zn, Ga-Mg and Al-Ga respectively. cGa and cAl are the Ga and Al concentrations in the alloy. The experimental data were taken from Hultgren et al., (1973). 56 UNIVERSITY OF IBADAN LIBRARY for heterocoordination. It is obvious from equation (3.37) that Scc(0) can be ob- tained directly from the experimental Gibbs energy of mixing or from the activity data. This is usually referred to as an experimental Scc(0) in literature. Equation (3.39) was used to determine the computed Scc(0) for these liquid alloys, while their measured Scc(0) were obtained by numerical differentiation of the Gibbs free energy of mixing data taken from (Hultgren et al., 1973). It is important to add that equation (3.38) was used to obtain Sidcc(0). The results for the concentration fluctuations as a function of composition for the three systems are as shown in figure 4.5. From the figure, it is clear that the calculated S idcc(0)>Scc(0) for Ga-Zn and Al-Ga, while calculated Scc(0)> 1, the entropic effects become dominant over the enthalpic effect and tend to pull the system from the segregation to order phase. Thus suggesting a limited solubility in the solid state of the liquid alloys. The magnitudes of GM are found to increase with increasing temperature. An indication that Ga-Tl liquid alloy is such a weakly interacting system like Al-Ca, Al-Mg, Hg-Sn, Ag-Al, Cu-Sn and Mg-Zn liquid binary alloys (Prasad and Mikula, 2006; Jha nd Mishra, 2001) with Ga-Tl alloys at 1173K being the most interacting of the three Ga-Tl liquid alloys. The results of the HM and SM computed at three different temperatures, T = 973K, 1073K and 1173K, are presented as functions of concentration in Figure 4.26 and Tables 4.17, respectively. A comparison between calculated theoretical 93 UNIVERSITY OF IBADAN LIBRARY Table 4.14: Activity of Ga in Ga-Tl liquid binary alloy at 973K. cGa Theory Experimental* 0.050 0.200 0.232 0.079 0.298 0.299 0.098 0.356 0.366 0.151 0.493 0.517 0.203 0.597 0.586 0.276 0.704 0.649 0.397 0.805 0.751 0.502 0.846 0.779 0.588 0.861 0.824 0.697 0.873 0.865 0.800 0.890 0.910 * Katayama et al.,(2003) 94 UNIVERSITY OF IBADAN LIBRARY Table 4.15: Activity of Ga in Ga-Tl liquid binary alloy at 1073K. cGa Theory Experimental* 0.050 0.158 0.206 0.079 0.238 0.270 0.098 0.286 0.339 0.151 0.404 0.492 0.203 0.498 0.553 0.276 0.601 0.612 0.397 0.716 0.720 0.502 0.775 0.757 0.588 0.807 0.803 0.697 0.841 0.840 0.800 0.874 0.891 * Katayama et al.,(2003) 95 UNIVERSITY OF IBADAN LIBRARY Table 4.16: Activity of Ga in Ga-Tl liquid binary alloy at 1173K. cGa Theory Experimental* 0.050 0.134 0.187 0.079 0.203 0.247 0.098 0.245 0.318 0.151 0.350 0.473 0.203 0.438 0.528 0.276 0.537 0.583 0.397 0.657 0.695 0.502 0.728 0.738 0.588 0.771 0.785 0.697 0.818 0.824 0.800 0.863 0.876 * Katayama et al.,(2003) 96 UNIVERSITY OF IBADAN LIBRARY 0× × -0.05 -0.1 GM -0.15 × RT × -0.2 × × × × -0.25 × × × × -0.3 0 0.2 0.4 0.6 0.8 1 cGa → Figure 4.23: Concentration dependence of GM for Ga-Tl liquid binary alloy at 973K. RT The solid line denotes theoretical values while the times denote experimental values respectively. cGa is Ga concentrations in the alloy. The experimental data were taken from Katayama et al.,(2003). 97 UNIVERSITY OF IBADAN LIBRARY 0× × -0.05 -0.1 GM -0.15 RT × × -0.2 × × -0.25 × ×× × -0.3 × × 0 0.2 0.4 0.6 0.8 1 cGa → Figure 4.24: Concentration dependence of GM for Ga-Tl liquid binary alloy at RT 1073K. The solid line denotes theoretical values while the times denote experimen- tal values respectively. cGa is Ga concentrations in the alloy. The experimental data were taken from Katayama et al.,(2003). 98 UNIVERSITY OF IBADAN LIBRARY 0× × -0.05 -0.1 -0.15 GM × RT × -0.2 × -0.25 ×× × × -0.3 × × × -0.35 0 0.2 0.4 0.6 0.8 1 cGa → Figure 4.25: Concentration dependence of GM for Ga-Tl liquid binary alloy at RT 1173K. The solid line denotes theoretical values while the times denote experimen- tal values respectively. cGa is Ga concentrations in the alloy. The experimental data were taken from Katayama et al.,(2003). 99 UNIVERSITY OF IBADAN LIBRARY 0.5 ←973K 0.45 0.4 ←1073K 0.35 ←1173K 0.3 HM 0.25 RT 0.2 0.15 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 cGa Figure 4.26: Computed enthalpy of mixing, HM , of Ga-Tl liquid alloy at 973K, RT 1073K and 1173K, respectively. The solid line denotes theoretical values. cGa is Ga concentrations in the alloy. 100 UNIVERSITY OF IBADAN LIBRARY Table 4.17: Computed entropy of mixing of Ga-Tl liquid alloy at 973K, 1073K and 1173K, respectively cGa 973K 1073K 1173K 0.1 0.281 0.331 0.331 0.2 0.326 0.511 0.511 0.3 0.432 0.625 0.625 0.4 0.515 0.690 0.690 0.5 0.602 0.711 0.711 0.6 0.688 0.690 0.690 0.7 0.710 0.626 0.626 0.8 0.695 0.512 0.512 0.9 0.629 0.332 0.332 101 UNIVERSITY OF IBADAN LIBRARY values at the investigated temperatures and the experimental data cannot be made due to lack of experimental data. Nonetheless, the calculated enthalpy of mix- ing are positive and exhibited slight asymmetrical behaviour near the equiatomic composition for Ga-Tl liquid alloys. The magnitudes of HM are found to increase with decreasing temperature. The effect of temperature is more distinct around the stoichiometric composition and the positions of the maxima in HM shift with decreasing temperature. 4.4.5 Summary The concentration dependence of the properties of mixing and activities of binary liquid Sb-Sn, In-Pb, Al-Si and Ga-Tl systems at respective temperatures were determined using analytical expressions only. This has made it possible to extract a consistent interpretation of the temperature and concentration dependence of the mixing properties. Bearing in mind that thermodynamical study of Al-Si and Ga-Tl systems have not been completely reported in the literature. Both Sb-Sn and Al-Si alloys exhibited negative deviations from ideal behaviour, while In-Pb and Ga-Tl systems exhibited positive deviations from ideality. For segregating alloys, such as In-Pb, having the size ratio value close to 1, the effects of size on their mixing properties can be neglected. On the contrary, for the segregating systems such as Ga-Tl, with the size ratio of 1.5 or higher, the size effects on the mixing behaviour becomes significant. The anomalous behaviour observed in the mixing properties of binary alloys in molten state as observed in Ga- Tl alloy as functions of concentration may then be attributed to the size difference of the constituent species. The size effect plays a significant role in the energetics of In-Pb and Ga-Tl which are segregating systems than it plays in both Sb-Sn and Al- Si liquid alloys. The effect of temperature on the thermodynamic functions is more visible around the stoichiometric composition. The theoretical results obtained are in good agreement with the corresponding experimental data for Sb-Sn, In-Pb, Al-Si and Ga-Tl liquid alloys at the investigated temperatures. At this stage, it is 102 UNIVERSITY OF IBADAN LIBRARY still not possible to single out any particular elemental properties which might be held responsible for the observed anomalous behaviours of liquid alloys. However, the empirical investigations suggest that quantities such as atomic size, heat of vapourization and electronegativity together hold the key to the understanding of the segregation or order in a liquid alloy. Hence, the approach discussed in this chapter, represent an effective and cost efficient method for obtaining the mixing properties likeGM , HM , SM and activities of binary liquid alloys at desire temperatures without the need for input from thermodynamic measurements. The approach is a very useful method to cut down the cost and time for the development of new alloys, and obtained thermodynamic data may as well be useful for comparison with some future critical experimental results and in developing new alloys. 103 UNIVERSITY OF IBADAN LIBRARY Chapter 5 Conclusions and Recommendations 5.1 Conclusions Theoretical approaches have been used in this study to determine some thermody- namic properties of selected binary liquid alloys at different concentration temper- atures, because good understanding of thermodynamic and microscopic structural properties of liquid alloys is an important prerequisite in the design and develop- ment of reliable materials for high temperature applications and also the need to find cost effective substitutes to obtain thermodynamic data as against the present conventional alloy development which is primarily based on experimental approach motivated this study. The energetics of alloy formations in Al-Zn (at 1000K), Bi- In (at 900K), Ga-Zn (at 700K), Ga-Mg (at 923K), Al-Ga (at 1073K), Sb-Sn (at 905K), In-Pb (at 900K), Al-Si (at 2000K, 2400K and 2473K) and Ga-Tl (at 973K, 1073K and 1173K) liquid alloys have been evaluated with special focus on their bulk thermodynamic and dynamical properties. From the results obtained, the following conclusions are made: The liquid binary alloys studied have been classified into two main groups: segregating (demixing) or short-range ordered based on the deviations of their 104 UNIVERSITY OF IBADAN LIBRARY thermodynamic properties from Raoultian behaviour. The study of microscopic functions of Al-Zn at 1000K indicates that Al-Zn is a segregated alloy while its heat of mixing −HM is characterized by liquid miscibility RT gaps and exhibits large positive heat of mixing values. On the other hand, Bi-In at 900K exhibits positive deviation and its −HM shows negative deviation which RT confirmed that Bi-In liquid alloy is an ordered alloy. In addition, the concentration dependence of the dynamic properties of both systems have been analyzed through the study of diffusion coefficient and viscosity within the framework of the Quasi- Lattice Theory (QLT). The size effect plays a prominent role in the energetics of Bi-In than it plays in Al-Zn alloy and thus pulling the system from segregation to order phase. The study has shown the suitability of the formalism which relates the dynamic properties with thermodynamic properties for a proper description of the alloying behaviours of binary liquid alloy. The mixing properties of binary alloys, namely Ga-Zn at 700K, Ga-Mg at 923K and Al-Ga at 1073K have been described in terms of the GM , Scc(0), αRT 1, enthalpy of mixing, HM and entropy of mixing, SM as a function of composition using the RT RT Quasi-Chemical Approximation Model (QACM). Both Al-Ga and Ga-Zn alloys were observed to exhibit positive deviations from ideal mixture behaviour. The positive deviation supports a tendency of both systems towards segregation. The results obtained for Ga-Zn agrees with the results of Awe et al., (2005) using four- atom-cluster-model (FACM). On the contrary, Ga-Mg exhibits negative deviation, an indication that a reasonable degree of chemical order exists in Ga-Mg alloy across the entire composition range. It is noted that the enthalpy of mixing and entropy of mixing for the three alloys are positive and symmetrical about the equiatomic composition in accord with the result obtained by Novakovic et al.,(2005) and Singh et al., (1997) on Ga-based alloys, except for Ga-Mg alloy whose enthalpy of mixing is negative. Furthermore, it was established that in the concentration range ≤ cAl ≤ 0.30 and 0.7≤ cAl ≤1, the calculated Scc(0) almost equal to the ideal Sidcc(0) values for Al-Ga systems. An indication that Al-Ga liquid alloy exhibited 105 UNIVERSITY OF IBADAN LIBRARY the highest glass-forming potential among the three Ga-based alloys investigated in these concentration ranges. The FACM which was successfully used in the study of thermodynamic func- tions of Bi-Pb and Sb-Sn has been reported to have some shortcomings when applied to alloys where size effect or/and electronegativity difference are signifi- cant (Akinlade, 1997); yet one is still of the opinion that the use of the FACM which is an extension of Singh earlier work on a two atom cluster model(Singh et al., 1988) and reformulated to give a more realistic value of the ordering energy ω and the chemical short range order (CSRO) parameter α1 and hence, to explain the energetics of mixing in liquid alloys in terms of compound formation or phase separation is still justified. For both systems, in view of the fact that the size ratios are quite less than 2.0 and their electronegativity differences are ignorable, one thus infers that the characteristic behaviour of each of the alloy is dependent on the energetics of the respective system. In addition, evaluation of the atomic size mismatch in each of the alloys Al- Zn, Bi-In, Sb-Sn, In-Pb, Al-Si and Ga-Tl shows that size factor is less than 2.0 and thus, the observed anomalies in the thermodynamic properties of the liquid alloys is perhaps a function of the energetics of the respective alloys. However, in this study, the anomalous behaviour observed in the mixing properties of Ga-Tl alloys as functions of concentration has been ascribed to the size factor. Several experimental evidences clearly demonstrate that the asymmetric behaviour for a large number of liquid alloys occur at or near the stoichiometric composition where stable intermetallic compound exists in the solid phase. It is, therefore, natural to propose that the chemical complexes or pseudomolecules exist in the liquid phase near the melting temperature. An appropriate estimation of the stoichiometry of complexes in the liquid alloy is usually made by an analysis of the physical properties and also from the phase diagram. For segregating systems, the size effects have an appreciable influence on their mixing properties. The magnitude of these effects increases together with a tendency of a system to phase separation 106 UNIVERSITY OF IBADAN LIBRARY (Novakovic et al., 2005). It was established in this study that size effect plays a prominent role in the energetics of Bi-In, Sb-Sn and Al-Si than it plays in Al-Zn, In-Pb and Ga-Tl liquid alloys and tends to pull the systems from segregation to order phase. The observed characteristic behaviour of Ga-Tl alloys as a function of concentration was attributed to the effect of the size ratio which is about 1.57 or higher. Furthermore, it is necessary to acknowledge the fact that the experimental val- ues of Scc(0) can be obtained theoretically from the observed free energy of mixing or activity data, respectively and such Scc(0) are usually termed experimental val- ues. The experimental values of Scc(0) used in all aspects of calculations in this study were obtained from the free energy of mixing. It was shown in this study that Scc(0) obtained via free energy of mixing compares well with the theoretical Scc(0) values. One generally notes that both the reported literature data and calculated theoretical values were in general good agreement. However, since theoretical de- termination of Scc(0) is of considerable interest to anyone trying to visualize the nature of atomic interactions in the mixtures, it would therefore be of high interest to obtain Scc(0) using the two approaches to be able to ascertain which of the two give consistent agreement with the theoretical Scc(0) values. A theoretical approach to determine the mixing properties and activities of liq- uid binary alloys using analytical expressions based on a statistical thermodynamic theory has been presented without recourse to input from thermodynamic measure- ments. The potential of the approach has been verified to be quite convenient and reliable based on the fact that theoretical results obtained conveniently reproduced the experimental results obtained by Hultgren et al.,(1973), Katayama et al.,(2003) and Kostov et al.,(2007). The approach can be adopted for determining thermo- dynamic properties of liquid alloys, which might not be available in the literature. For instance, the Gibbs free energies of mixing of liquid Al-Si and Ga-Tl alloys de- termined by the current approach agree with the experimental data, while all the mixing properties and activities of both In-Pb and Sb-Sn calculated compare well 107 UNIVERSITY OF IBADAN LIBRARY with the experiment. Thus the approach is a good and very useful method to cut down the cost and time for obtaining thermodynamic data (mixing properties and activities) of binary liquid alloys from theoretical calculations, rather than from experiment and thereby enhancing the development of new alloys. 5.2 Limitations to this study The major hindrance which tends to limit the scope of this study was non-availability or complete lack of recent and sufficient experimental data which would have facil- itated comparison of the calculated theoretical results with experiment at investi- gated temperatures. Another challenge faced was lack of enough fund or support to get experimental data on high temperature liquid alloys. It should however be noted that the main drawback of the theoretical approach presented in section 3.4 to obtaining thermodynamic data of binary liquid alloys is that it still depends on availability of experimental data to validate the theoretical results obtain using the method. 5.3 Recommendations In the light of these limitations and the experience acquired during this study, the following suggestions are recommended for further studies: 1. From technological point of view, binary liquid alloys are of practical impor- tance in the present production practice. Therefore, an important task for the present scientific investigations should be a good theoretical understanding of the energetics of the interrelationship between thermodynamic and structural behaviour of these materials due to complications and difficulty associated with the high-temperature experiments. 2. Adequate fund and technical assistance is required and future work should be geared towards identifying binary liquid alloys for which no thermody- 108 UNIVERSITY OF IBADAN LIBRARY namic data presently exist in the literature with a view to determine their thermodynamic data as calculation of thermodynamic properties of ternary and quaternary systems relies heavily on their constituent binary systems (Zivkovic et al., 2011) and could thus serve as a guide towards predicting thermodynamic data for ternary and higher components liquid alloys. 3. It would also be highly desirable to extend the present understanding of concentration-concentration fluctuations in the long-wavelength limits, Scc(0) and the chemical short-range order parameter, α1 in explaining the energetics of mixing of liquid binary alloy in terms of chemical ordering and phase sepa- ration to ternary and quaternary liquid alloys in future studies. As increasing attention is recently being shifted towards higher components liquid alloys due to their potential usefulness in industrial applications (Gomidzelovic et al., 2007; Zivkovic et al., 2011). 4. It is believed that with a significant increase in activity in this emerging area of research, our understanding of the electronic, structural and thermody- namic properties of ternary and quaternary liquid alloys still at the early stages will continue to evolve in years to come. 5.4 Contributions The contributions that this work has made in the area of high temperature appli- cations of liquid alloys are many. The study identified the inadequacies of many of the models in the literature used for studying thermodynamic properties of liquid systems. The suitability of a formalism which relates both the thermodynamic and dynamical properties within the frame of the Quasi-Lattice theory has been estab- lished in this thesis for a proper description of the alloying behaviours of liquid binary alloys. This was presented in section 4.1 of this thesis. The quasi-chemical approximation model for strongly interacting systems used to investigate order- ing and glass formation tendencies in Ga-Zn, Ga-Mg and Al-Ga binary alloys at 109 UNIVERSITY OF IBADAN LIBRARY different temperatures through the study of their thermodynamic functions was re- ported in section 4.2 of this thesis. In spite of some shortcomings of the FACM as reported by Akinlade (1997), yet the model as used here has successfully explained the thermodynamics properties of Bi-Pb and Sb-Sn alloys. This might perhaps be connected with the fact that their size ratios are just close to 1, which can be neglected. This was given in section 4.3 of this thesis. It has also been shown through this study that the experimental Scc(0) obtained via free energy of mixing compare well with the theoretical Scc(0) values as all the experimental values of Scc(0) used in all aspects of the calculations were obtained from the free energy of mixing. This can be found in chapters 3, 4 and 5 of the thesis. Finally, I have pre- sented and tested an approach for determining the mixing properties and activities data of liquid binary systems. The approach is economic, cost effective, environ- mentally friendly and thus, the data obtained using this approach could as well be adopted as a basis for comparison with some future critical experimental results and aid in developing new alloys. It is these contributions that one hopes will go to further the understanding of the thermodynamic properties and the energetics of alloy formations in binary liquid alloys. 110 UNIVERSITY OF IBADAN LIBRARY References Akinlade, O. (1994). A study of segregation and local order in Na-In liquid alloys. Z. Metallkunde. 85 p487-491. Akinlade, O. (1995). Ordering phenomena in Na-Ga and Na-Sn molten alloys. P- hys. Chem. Liq. 29 p9-21. Akinlade, O. (1997). Higher order conditional probabilities and thermodynamics of binary molten alloy. Modern Physics Letters 11(2-3) p93-106. Akinlade, O. (1998). Energetic effects in some molten alloys of arsenic. Physica B 245 p330-336. Akinlade, O., Anusionwu, B. C. and Hussain, L. A. (1998). Structure and Ordering in Cu-Si and Ni-Si molten alloys. Z. metallkd. 89 p27-31. Akinlade O., Hussain, L. A., Awe, O. E. (2003). Thermodynamics of liquid Al-In, Ag-In and In-Sb alloys from a four atom cluster model. Z. Metallkd 94 (12) p1276. Akinlade, O., Singh, R. N., Sommer, F. (1998). Thermodynamic investigation of viscosity in Cu-Bi and Bi-Zn liquid alloys. J. Alloys and Cmpds. 267 p195-198. Akinlade, O., Singh, R. N., Sommer, F. (2000). Thermodynamics of liquid Al-Fe alloys. J. Alloys and Cmpd 299 p163-168. Akinlade, O. and Singh, R. N. (2002). Bulk and surface properties of liquid In-Cu alloys. J. Alloys and Cmpd 333 p84-90. Alblas, B. P., Van der Lugt, W., Visser, E. G. and De Hosson, J. Th. M. (1982). Thermodynamic calculations for the liquid systems Na-K, K-Cs and Li- Pb. Physica 114B p59-66. Aljarrah, M., Aghaulor, U. and Medraj, M. (2007). Thermodynamic assessment of the Mg-Zn-Sr system. Intermetallics 15(2) p93-97. Aljarrah, M. and Medraj, M. (2008). Thermodynamic modelling of the Mg-Ca, Mg-Sr, Ca-Sr and Mg-Ca-Sr systems using the modified quasi-chemical model. Computer Coupling of Phase Diagrams and Thermochemistry 111 UNIVERSITY OF IBADAN LIBRARY 32 p240-251. Alonso, J. A. and March, N. H. (1990). Electrons in Metals and Alloys, (London, Academic). Andrew, Y. Y. and Goncharov, A. V. (2005). thermodynamic calculation and ex- perimental investigation of the surface enrichment of electrochemically activated Al-Me (Sn, In, Zn) alloys. Electrochimica Acta 50 p2629-2637. Anusionwu, B. C. (2006). Thermodynamic and surface properties of Sb-Sn and In- Sn liquid alloys Pramana J. Phys. 67(2) p319-330. Anusionwu, B. C. and Adebayo, G. A. (2001). Quasi-chemical studies of ordering in the Cu-Zr and Cu-Si melts. J. Alloys and Cmpd. 329 p162-167. Anusionwu, B. C., Madu, C. A, and Orji, C. E. (2009). Theoretical studies of mu- tual diffusivities and surface properties in Cd-Ga liquid alloys. Pramana J. Phys. 72(6) p951-967. Ashcroft, N. W. and Stroud, D. (1978). Solid state Physics (Academic Press, New York). 33 p1. Awe, O. E., Akinlade, O. and Hussain, L. A. (2005). Conditional probabilities and thermodynamic properties of liquid Ag-Au, Cd-Pb and Ga-Zn alloy. J. Alloys and Cmpd. 387 p256-259. Awe, O. E., Akinlade, O. and Hussain, L. A. (2006). Bulk and surface properties of liquid Al-Mg, Au-Sn and Mg-Tl compound forming alloys. Surface Science 600 p2122-2128. Bhatia, A. B. and March, N. H. (1975). Size effects, peaks in concentration fluctua- tions and liquidus curves of Na-Cs. Journal of Physics F: Metal Physics. 5(6) p1100. Bhatia, A. B. and Hargrove, W. H. (1974). Concentration fluctuations and thermo- dynamic properties of some compound forming binary molten system. Phys. Rev. B 10(8) p3186-3196. Bhatia, A. B. and Singh, R. N. (1982). Short range order and concentration fluctu- ations in regular and compound molten alloys. Phys. Chem. Liq. 11 112 UNIVERSITY OF IBADAN LIBRARY p285-313. Bhatia, A. B. and Singh, R. N. (1984). Quasi-chemical lattice model for compound forming molten alloys. Phys. Chem. Liq. 13 p177-190. Bhatia, A. B., Thornton, D. E. (1970). Structural Aspects of the Electrical Resis- tivity of Binary Alloys. Phys. Rev. B, 2 p3004-3012. Brown, R. D. Jr., (2000). U. S. Geological Survey Minerals Yearbook. p13.1-13.4. Busch G. and Guntnerodt H. N. (1974) in: Ehrenreich H., Seitz F., Turnball D. (Eds.), Solid State Physics,Academic, New York, Vol. 29 p1235. Cai Anhu, Sun Guoxiong and Pan Ye (2006). Evaluation for the parameters related with glass forming ability of bulk metallic glasses. Materials and Design 27 p479-488. Cai Anhui, Xiong Xiang, Liu Yong, Tan JingYing, Zhou Yong and AnWeike (2008). Estimation of Gibbs free energy difference in Pd-based bulk metallic gla- sses China Foundary: Research and Development 5(2) p124-127. Cartier, A. and Barriol, J. (1976). Theorie statistique des alliages metalliques bina- ires liquides a l’aide d’une methode de groupes de sitesAnalyse de l’ordre local Physica B 81(1) p35-45. Chen, X., Li, H. and Ding, X. (2002). A new thermodynamic calculation method for binary alloys. J. Mater. Sci. Technol. 18 p237-241. Chou, K. C. and Wang J. J. (1987). Calculating activities from the phase-diagram involving an intermediate compound using its entropy of formation. Met- allurgical and Materials Transactions A 18(2) p323-326. Chou, K. C.(1990). A new treatment for calculating activities from phase-diagrams containing solid-solution. Calphad 14(3) p275-282. Cowley, J. M. (1950). An approximate theory of order in alloys. Phys. Rev. 77 p667-675. Dalgic, S., Dalgic, S. and TOMAK, M. (1998) The Structure of Liquid Alloys with Chemical Short Range Order. Tr. J. of Physics 22 p505-510. Darken, L. S. (1948). Diffusion, Mobility and their interrelation through free ene- 113 UNIVERSITY OF IBADAN LIBRARY rgy in binary metallic systems. Trans. Inst. Min. Metall. Eng. 175 pp. 184-201. Darken, L. S. (1967). Thermodynamics of binary metallic solutions. Transaction Metallurgical Sociey (AIME), 239, p80-89. Dubinin, N. E. (2003). Thermodynamics of Alkali Metals Melts. J. Optoelectron- ics and Adv. Mater. 5(5) p1259-1262. Eustathopoulos N., nicholas M. and Drevet B. (1999). Wettability at High temper- atures, Vol. 3, Pergamon Materials Series, Oxford, UK. Faber, T. E. (1972). Introduction to the Theory of Liquid Metals, Cambridge Uni- versity Press, U. K.. Flory, P. J. (1942). Thermodynamics of high polymer solutions. J. Chem. Phys. 10 p51-61. Frost, B. R. T. (1981). Materials for fluid fuel reactors. J. Nuclear Mater. 100 (1-3) p128-131. Ganesh, P. and Widom, M. (2006). Signature of nearly icosahedral structures in liquid and supercooled liquid copper. Phys. Rev. B 74 p134205. Godbole, R. P., Jha, S. A., Milanarun, M. and Mishra, A. K. (2004). Thermodyna- mics of liquid Cu-Mg alloys. J. Alloys and Compds 363 p182-188. Gomidzeloviz L., Zivkovic D., trbac N. and Zivkovic Z. (2007). Calculation of mix- ing enthalpies for ternary Au-In-Sb alloys. Journal of the University of Chemical Technology and Metallurg 42(2), p207-210. Guggenheim, E. A. (1952). Mixtures. Oxford University Press, Oxford. Guggenheim, E. A. (1976). Transactions of the Metallurgical Society of the AIME. 239 p90. Hafner, J., Jaswal, S. S., Tegze, M., Pflugi, A., Krieg, J., Oelhafen, P. (1988). The atomic and electronic structure of metallic glasses: search for a str- ucture-induced minimum in the density of states.Journal of Physics F: Metal Physics 18(12) p2583604. Hardy, H. K. (1953). Acta. Metall. 1 p202. 114 UNIVERSITY OF IBADAN LIBRARY Harrison, W. A. (1966). Pseudopotential in the theory of metals, Benjamin, New York. Heine, V. (1970). Solid State Physics,Academic Press, New York. p419. Hildebrand, J. H. and Scott, R. L. (1950). The solubility of Nonelectrolytes, 3rd edition, Van Nortrand Reinhold, New York, N. Y. Hoshino, K. and Young, W. H.(1981). On the entropy of mixing of the liquid Na- Pb alloy. J. Phys. F11L7. Hultgren, R., Desai, P. D., Hawkins, D. T., Gleiser, M., and kelly, K. K. (1973). Se- lected Values of the Thermodynamic Properties of Binary Alloys, Amer- ica Society for Metals, Material Park, OH. Iida, T. and Guthrie, R. I. L. (1993). The Physical Properties of Liquid Metals. (Clarendon Press, Oxford). Islam, F. and Medraj, M. (2004). Thermodynamic modelling of Mg-Ca and Al-Ca binary systems. CSME Forum p921-929. Iwata, K., Matsumiya, T., Sawada, H., Kawakami, K. (2003). Prediction of thermo- dynamic properties of solute elements in Si solutions using first-principles calculations. Acta Mater. 51(2) p551559. Jank, W. and Hafner, J. (1988). The Electronic Structure of Liquid germanium. Europhysics Letters 7 p623. Jha, N. and Mishra, A. K. (2001). Thermodynamic and surface properties of liquid Mg-Zn alloys. J. alloys and cmpd. 329 p224-229. Kaban, I., Hoyer, W., Il’inskii, A., Slukhovskii, O., Slyusarenko, S. (2003). Short- range order in liquid silver-tin alloys. J. Non-Cryst. Solids 331 p254-262. Kanth B. K. and Chakrabarti S. K. (2009). Heat Of mixing of some binary liquid Alloys. Scientific World 7(7) p96-99. Katayama, I., Shimazawa, K., Zivkovic, D., Manasijevic, D., Zivkovic, Z. and Iida, T. (2003). Activity measurements of Ga in liquiid Ga-Tl allys by EMF method with zirconia solid electrolyte. Z. Metallkde 94 p1-4. Kostov, A., Zivkovic, D. and Freidrich, B. (2007). Thermodynamic predicting of 115 UNIVERSITY OF IBADAN LIBRARY Si-Me (Me = Ti, Al) binary systems. J. Mining and Metallurgy 43 B p29-38. Kostov, A., Freidrich, B. and Zivkovic, D. (2008) Thermodynamic calculations in alloys Ti-Al, Ti-Fe, Al-Fe and Ti-Al-Fe. J. Mining and Metallurgy 44B p49-61. Lad Kint, N., Raval, K. G. and Pratap, Arun (2004) Estimation of Gibbs free en- ergy difference in bulk metallic glass forming alloys J. Non-Cryst. Solids 334-335 p259-262. Liu, Y. Q. and Fan, Z. (2002). Application of thermodynamic calculation to the aluminium alloy design for Semi-Solid Metal Processing. Material Sci- ence Forum. 396-402 p717-722. Liu, Y. Q., Das, A. and Fan, Z. (2004). Thermodynamic Prediction of Mg-Al-M (M = Zn, Mn, Si) alloy compositions amenable to semisolid metal process- ing. Material Science and Technology 20 p35-41. Liu, D. Y., Sun, W. S., Zhang, H. F. and Hu Z. Q. 2004). Preparation, thermal sta- bility and magnetic properties of Fe-Co-Ni-Zr-Mo-B bulk metallic glass. Intermetallics 12 p1149-1152. Lucas, L. D. (1984). Tension Superficielle. Techniques de l’lngenieur, From M67F. Lupis, C. H. P. and Elliot, J. F. (1967). Prediction of enthalpy and entropy interact- ion coefficients by the central atoms theory. Acta Metallurgica 15 (2) p265-276. McGreevy, R. L. and Pusztai, L. (1988). Reverse Monte Carlo simulation: a new technique for the determination of disordered strutures. Mol. Sim. 1 p359-367. Müller, S. (2003). Bulk and surface ordering phenomena in binary metal alloys. J. Phys: Condens. matt. 15 R1429-R1500. Novakovic, R., Ricci, E., Giuranno, D. and Gnecco, F. (2002). Surface properties of Bi-Pb liquid alloys. Surface Science 515 p377-389. Novakovic, R., Ricci, E., Muolo, M. L., Giuranno, D. and Passerone, A. (2003). 116 UNIVERSITY OF IBADAN LIBRARY On the Application of Modelling to Study the Surface and Interfacial Phenomena in Liquid Alloy-Ceramic Substrate Systems. Intermetallics 11(11-12) p1301-1311. Novakovic, R., Muolo, M. L., Passerone, A. (2004). Bulk and surface properties of liquid X-Zr (X = Ag, Cu)) compound forming alloys. Surface Science 549 p281-293. Novakovic, R. and Zivkovic, D. (2005). Thermodynamics and surface properties of liquid Ga-X (X = Sn, Zn) alloys. J. Mater. science 40 p2251-2257. Novakovic, R. and Tanaka, T. (2006). Bulk and surface properties of Al-Co and Co-Ni liquid alloys. Physica B 371 p223-231. Novakovic, R., Giuranno, D., Ricci E and and Lanata, T. (2008). Surface and tran- sport properties of In-Sn liquid alloys. Surface Science 602 p1957-1963. Osman, S. M. and Singh, R. N. (1995). Description of concentration fluctuations in liquid binary mixtures with nonadditive potentials. Phys. Rev. E 15 (1) p332-338. Pettifor, G (1993). Electron Theory of Crystal Structure in Structure of Solids ed. V Gerold (Weinheim: VCH). Popescu, C. A. and Taloi, D. (2007). Thermodynami calculations in liquid Al-Sn alloys systems. U. P. B. Sci. Bull., Series B, Vol(69) No.(1) p78-83. Prasad, L. C. and Mikula, A. (2000a). Concentration Fluctuations and Interfacial Adhesion at the Solid-Liquid Interface between Al2O3 and Al-Sn Liquid Alloys. High Temp. Mater. Processes, 19(1) p61. Prasad, L. C., Mikula, A. (2000b). Effect of temperature on intermetallic assoia- tions in Sb-Zn liquid alloys. J. Alloys Cmpd 299 p175-182. Prasad, L. C. and Mikula, A. (2006). Thermodynamics of liquid Al-Sn-Zn alloys and concerned binaries in the light of soldering characteristics. Physica B 373 p64-71. Prasad, L. C., Singh, R. N., Singh, V. N. and Singh, G. P. (1998). Correlation bet- ween bulk and surface properties of AgSn liquid alloys. J. Phys. Chem. 117 UNIVERSITY OF IBADAN LIBRARY B 102(6) p921-926. Rao, Y. K. and Belton, G. R. (1981). in:Chemistry Metallurgy-A tribute to Carl Wagner, edited by N. A. Gocken. Metall. Soc. of AIME, Warrendale, Pa. p75-96. Ratner, B. D., Hoffman, A. S., Schoen, F. J. and Lemons, J. E. (1996). (Eds.) Bio- materials Science: An Introduction to Materials in Medicine, Academic Press. Richardson, F. D. (1974). Physical Chemistry of Melts in Matallurgy, Vol. 1, Aca- demic Press, London. p135. Saida J. and Inoue A. (2003). Quasicrystals from glass devitritication J. Non-Cry- stalline Solids 317(1-2) p97-105. Senkov, O. N. and Scott J. M. (2004). Formation and thermal stability of Ca-Mg- Zn and Ca-Mg-zn-Cu bulk metallic glasses. Mater. Lett. 58 p1375-1378. Shimoji, M. (1977). Liquid Metals, London Academy, London. Singh, R. N. (1987). Short-range and concentration fluctuations in binary molten alloys. Canadian Journal of Physics 65 p309-325. Singh, R. N. (1993). Higher order conditional probabilities and short range order in molten alloys. J. Phys. Chem. 25 p251-267. Singh, R. N. and March, N. H., (1995) in: J. H. Westbrook and R. L. Fleischer (Eds.), Intermetallic Compounds, Principles and Practice, vol. 1, John Wiley Sons, New York. p661-686. Singh, R. N. and Mishra, I. K. (1988). Conditional probabilities and thermodyn- amics of binary molten alloys. Phys. Chem. Liq. 18 p303-319. Singh, R. N. and Sommer, F. (1992a). Temperature dependence of the thermodyn- amic functions of strongly interacting liquid alloys. J. Phys. Condens. Matter 4 p5345-5358. Singh, R. N. and Sommer, F. (1992b). A simple model for demixing binary liquid alloys. Z. Metallkd 83 p533. Singh, R. N. and Singh, K. K., (1995). Thermodynamic properties of some molten 118 UNIVERSITY OF IBADAN LIBRARY alloys. Mod. Phys. Lett. B9 p1729. Singh, R. N. and Sommer, F. (1997). Segregation and immiscibility in liquid binary alloys. Rep. Prog. Phys. 60 p57-150. Singman, C. N., (1984). Atomic Volume and Allotropy of the Elements. J. chem. Edu., 61 (137). Sommer, F. (1997). Heat capacity of liquid alloys, in: Chang Y. A., Sommer F. (Eds.), Thermodynamics of Alloy Formation, The Minerals, Metals and Materials Society, Warrendale. p99. Sommer, F., Lee, J. J., Predel, B., (1983). Thermodynamic studies of liquid alumi- num-calcium, aluminum-strontium, magnesium-nickel and calcium-nickel alloys. Z. Metallkde 7492) p100-104. Takana, T. and Gokcen, N. A. (1995). Excess thermodynamic properties of dilute solutions. J. Phase Equilibria 16(1) p10-15. Tao, D. P. (2000). A new model of thermodynamics of liquid mixtures and its application to liquid alloys. Thermochimica Acta 363 p105-113. Tao, D. P. (2003). Prediction of the thermodynamic properties of binary continu- ous solid solutions by infinite dilute activity coefficients. Thermochimica Acta 408 p67-74. Tao, D. P., Yang, B. and Li, D. F. (2002). Prediction of the thermodynamic pro- perties of quinary liquid alloys by modified coordination equation. Fluid Phase Equilibria 193 p167-177. Thakor, P. B., Gajjar, P. N., Jani, A. R. (2002). Structural study of liquid rare earth metals from charged hard sphere reference fluid. Condens. Matt. Phys. 3(31) p493-501. Thakur, A., Negi, N. S. and Ahluwalia, P. K. (2005). Electrical resistivity of NaPb compound-forming liquid alloy using ab initio pseudopotentials Pramana J. Phys. 65(2) p349-358. Visser, E. G., Van der Lugt, W. and De Hosson, J Th M. (1980). Thermodynamic calculations for liquid alloys with an application to sodium-caesium. J. 119 UNIVERSITY OF IBADAN LIBRARY Phys. F: Metal Phys. 10 p1681-1692. Wagner, C. N. J. (1985). Rapidly quenched metals (Amsterdam:Elsevier) p405. Warren, B. E. (1969). X-ray diffraction, Addison-Wesley, Reading, MA. p227. Waseda, Y. (1980). The Structure of Non-Crystalline Materials, Liquids and Amor- phous Solids, McGraw-Hill, New York. Witusiewicz, V. T. and Sommer, F. (2000). Estimation of the excess entropy of mixing and the excess heat capacity of liquid alloys. J. Alloys compds 312 p228-237. Yang, H. W., Tao, D. P. and Zhou, Z. H. (2008). Prediction of the mixing entha- lpies of binary liquid alloys by molecular interaction volume model. Acta Metall. Sin (Engl. Lett.) 21(5) p336-340. Yang, H. W., Tao, D. P., Yang, X. M. and Yuan, Q. M. (2009). Prediction of the formation enthalpies of Bi-Cd-Ga-In-Pb-Sn-Zn liquid alloys by binary infinitely dilute enthalpies. J. Alloys Cmpds 2(480) p625-628. Yong, J. Lv and M. Chen (2011). Thermophysical Properties of Undercooled Al- loys: An Overview of the Molecular Simulation Approaches International Journal of Molecular Sciences. 12 p278-316. Young, W. H. (1992). Structural and thermodynamic properties of nearly free elec- trons (NFE) liquid metals and binary alloys. Report Progress in Physics 55 p1769. Zivkovic, D., Balanovic, L., Manasijevic, D., Mitovski, A., Kostov, A., Gomidzelovic, L. and Zivkovic, Z. (2011). Calculation of thermodynamic properties in quarternary Ni-Cr-Co-Al systems. Journal of the University of Chemical Technology and Metallurgy 46(1) p95-98. 120 UNIVERSITY OF IBADAN LIBRARY APPENDIX A PUBLISHED WORK 121 UNIVERSITY OF IBADAN LIBRARY This article was originally published in a journal published by Elsevier, and the attached copy is provided by Elsevier for the author’s benefit and for the benefit of the author’s institution, for non-commercial research and educational use including without limitation use in instruction at your institution, sending it to specific colleagues that you know, and providing a copy to your institution’s administrator. All other uses, reproduction and distribution, including without limitation commercial reprints, selling or licensing copies or access, or posting on open internet sites, your personal or institution’s website or repository, are prohibited. For exceptions, permission may be sought for such use through Elsevier’s permissions site at: http://www.elsevier.com/locate/permissionusematerial UNIVERSITY OF IBADAN LIBRARY Journal of Non-Crystalline Solids 353 (2007) 1167–1171 www.elsevier.com/locate/jnoncrysol Bulk and dynamic properties in Al–Zn and Bi–In liquid alloys using a theoretical model Y.A. Odusote a,*, L.A. Hussain b, O.E. Awe c a Department of Physics, Olabisi Onabanjo University, Ago-Iwoye, Nigeria b Vice Chancellor’s Office, Lagos State University, Ojoo, Lagos, Nigeria c Department of Physics, University of Ibadan, Ibadan, Nigeria Received 28 January 2006; received in revised form 14 October 2006 Available online 6 February 2007 Abstract A formalism that connects thermodynamic and dynamic properties such as viscosity and diffusion coefficient has been used to calcu- late the concentration dependence of the free energy of mixing, concentration–concentration fluctuations in the long-wavelength limit and the concentration dependence of diffusion as well as viscosity in Al–Zn and Bi–In binary liquid alloys at various temperatures. Our calculations show that a reasonable degree of chemical order exists in Bi–In system, while Al–Zn liquid alloy was observed to exhibit a liquid miscibilty gap or phase separation over the concentration range. It can be concluded that size effect plays a more prominent role in the energetics of Bi–In than it plays in Al–Zn alloy. This fact is substantiated by predicted dynamic properties of both systems.  2007 Elsevier B.V. All rights reserved. Keywords: Alloys; Liquid alloys and liquid metals 1. Introduction extensively on thermodynamic information (both experi- mentally and theoretically) on phase separating systems. Different kinds of theoretical models proposed to They have equally mentioned different kinds of theoretical explain the concentration dependence of the thermody- methods that could be used to explain the mechanism for namic properties of liquid binary alloys and by so doing, the energetics of liquid alloys. extracting useful microscopic information on them has Modelling of thermodynamic, structural and surface attracted the attention of physicists, chemists and metallur- properties of liquid alloys usually requires data regarding guists for a long time [1–5]. In this regard, a substantial their structures and the relevant forces that describes the effort has been directed towards the understanding of the interatomic interactions, i.e. the energetics of the system. mechanism for the phenomenon of compound formation This can be done by different theoretical approaches such in liquid binary alloys [1,6]. as the electron theory [8] and the Bhatia and Hargrove In addition, their energetics are reasonably well under- [2] formalism put in a more tractable form by [1]. stood theoretically. On the contrary, relatively little is Singh and Sommer [9,10] established an empirical model known and written on the other class of liquid alloys that which provides a possible way of relating two of the exhibit segregation (i.e. preference of like atoms as nearest dynamics properties, as diffusion coefficient (D) and viscos- neighbours) or on the liquid alloys endowed with a misci- ity (g) with thermodynamic properties like the concentra- bility gap. Singh and Sommer [7] have documented tion–concentration fluctuations in the long wavelength limit, Scc(0) and the free energy of mixing, GM RT for liquid bin- * ary alloys. These quantities can then be connected in termsCorresponding author. Tel.: +234 805 220 3325; fax: +234 803 714 2446. of entropic and enthalpic contributions to the free energy E-mail address: yisau24@yahoo.co.uk (Y.A. Odusote). of mixing. However, no practical application was made 0022-3093/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.12.023 A UuNt IVh EoRr'SsIT Y pOeF rIBsAoDnANa lL I B RcAoRpYy 1168 Y.A. Odusote et al. / Journal of Non-Crystalline Solids 353 (2007) 1167–1171 to a particular liquid alloy in their paper so as to know the where k1 and k2 are functions of the size and shape of the limit of suitability of the model. constituent particles. Thus, we define the quantity / as In this paper, an attempt is made to apply the theory in cAcB [9,10] and show that it could be used to model the structure / ¼ ð3ÞSccð0Þ of liquid binary alloys and to facilitate our understanding of id the energetics of the liquid alloys that have been investigated. and the ideal value of Sccð0Þ as Here, we apply the model to Al–Zn and Bi–In liquid Sidccð0Þ ¼ cAcB ð4Þ alloys. Our choice of Al–Zn and Bi–In was however influ- enced by the availability of all the required thermodynamic Eq. (3) can be rewritten as data for both systems. Moreso, due to the difficulties in g ¼ go/; ð5Þ performing high temperature surface and viscosity mea- surements, in the literature only a few reference data for with   metals [11,12] are available. For binary systems they are ¼ KBT cA þ cBscarce, and in the case of complex alloys a nearly complete go : ð6ÞDm k2 k1 lack of data is evident. Therefore, it is necessary to estimate the missing values of the viscosity data by theoretical mod- Ordinarily, one would expect that go be linear in cA, so els. The dynamic properties of Al–Zn and Bi–In liquid that the deviation of g  cA isotherm from the additPive rule alloys are calculated in the framework of Quasi-Lattice of mixing can be attributed to the factor / (i.e. g ¼ icigi). Theory (QLT) with the aim to analyse existing thermody- This factor has been extensively reported in the literature namic data and use these data as the input for the fitted for a number of binary liquid alloys [6,7]. parameter calculations, and to explain the ordering and Thermodynamic investigation of / have been performed phase separating phenomena in liquid binary alloys. In [15] for the more general case in which one then introduces XA addition, available information on phase diagrams and the entropic contribution (i.e. the size ratio c ¼ X ,B other thermodynamic properties indicate the existence of XB > XA), where X is the atomic volume and the enthalpic liquid miscibility gap in Al–Zn system. Its Scc(0) exhibit a contributions (via the interchange energy, x). They tendency for phase separation over the concentration obtained an expression for / given by range, whereas the Scc(0) for Bi–In alloy exhibits com- / ¼ 1 cAcBf ðc;W Þ; ð7Þ pound formation tendency. Based on the above mentioned where facts, we deemed it fit to investigate these two systems to widen our theoretical understanding of liquid binary alloys. 2c2W  ðc  1Þ2ðcA þ ccBÞ The layout of the paper is as follows: In Section 2, we f ðc;W Þ ¼ ; ð8ÞðcA þ ccBÞ3 discuss the basic theory used for the calculations. In the   x next section, we present our results and discussion and W ¼ XA ; ð9Þ finally, we make our conclusions. KBT with 2. Basic theory w ¼ ZðAB  ðAA þ BBÞ=2Þ: ð10Þ We propose to apply the Quasi-Lattice Theory (QLT) of Eq. (10) is known as the interchange energy, here we re- liquid mixtures which is based on the ideas underlying the mind ourselves that AB, AA, and BB are the energies for Guggenheim theory [13] in which one assumes the existence AB, AA and BB pairs of atoms respectively and Z is the of a binary mixtures consisting of NA = NcA and NB = NcB coordination number of the liquid alloys. Obviously, if atoms of elements A and B respectively. We assume also x < 0, there is tendency to form unlike atom pairs, and if that a small gradient of composition cA is maintained in x > 0, like atoms tend to pair together. x = 0, however, an equilibrium condition along the x-direction by the shows that atoms in the mixture are perfectly disordered application of a force FA, such that for dilute alloy, it is [1]. By taking together Eqs. (5), (7)–(9), we express Dg g aso possible to write [14]: DgcA ¼ cAcBf ðc;W Þ: ð11ÞF A ¼ KBT d ln ; ð1Þ gx od where T is the temperature and K , the Boltzmann’s con- The factor f(c,W) which introduces both the entropicB stant. In the framework of the compound formation model and enthalpic effects is responsible for the characteristic the stoichiometry of associates can be ascertained by the behaviour of Dg for a given binary alloy. In the light of viscosity, g. By using Eq. (1), it has been shown [9,10] that the result obtained in [9,10], it is readily shown that we can relate Scc(0) and diffusion coefficient, Dm for liquid Dg HM alloys [6,7] by ¼  ; ð12Þ  go RT g ¼ KBT cA þ cB cAcB ; ð2Þ R is the universal gas constant. It is observed that using the Dm k2 k1 Sccð0Þ formulae given above, we do not have sufficient information A UuNt IVh EoRr'SsIT Y pOeF rIBsAoDnANa lL I B RcAoRpYy Y.A. Odusote et al. / Journal of Non-Crystalline Solids 353 (2007) 1167–1171 1169 on thermodynamic quantities to relate it to g. In order to Dm S id ccð0Þ achieve this, we make use of the general expression for the ¼ ; ð21ÞDid Sccð0Þ Gibb’s free energy of mixing GM where Dm is the mutual diffusion coefficient and Did is the GM ¼ TSM þ HM; ð13Þ intrinsic diffusion coefficient for an ideal mixture, given as where SM is the entropy of mixing and HM is the enthalpy Did ¼ cADB þ cBDA; ð22Þ of mixing. Using Guggenheim’s theory of mixtures [13], QLT makes it possible for one to write an explicit expres- with DA and DB being the self-diffusion coefficients of pure sion for G as components A and B, respectively. For ideal mixing,M   Sccð0Þ ! Sidccð0Þ, i.e. Dm ! Did; for ordered alloys, GM id¼ c lnW þ c lnð1 WÞ þ xc W X ; ð14Þ Sccð0Þ < Sccð0Þ, i.e. Dm > Did; and similarly for segregation, RT B A A A K T D DB mm < Did. The highest peak of D as a function of compo-id ¼ cB lnW þ cA lnð1 WÞ þ cAWW ; ð15Þ sition indicate the presence of maximum chemical order in molten alloy system as well as the composition of the most with likely associates to be formed in the liquid phase [6]. ¼ ccW Bþ : ð16ÞcA ccB 3. Results and discussion And from GM as in Eq. (14) or (15), Scc(0) can easily be cal- The basic theory explained in Section 2, have been used culated from standard relationship in terms of free energy to investigate the composition dependence of the bulk ther- of mixing,   modynamics and dynamic properties such as diffusion and2 1G viscosity for the two systems Al–Zn and Bi–In liquid alloys.ð Þ ¼ oScc 0 RT M ; ð17Þ It is understood from the theory that there are some oc2 T ;P ;N parameters that need to be fitted in order to carry out or in terms of activity, a and a , the present calculations. These parameters are the size  A B   ratio, c, the coordination number, Z and the interchange1 1 ð Þ ¼ ð  Þ oaA ¼ oaB ð energy, x. For uniformity, the coordination number, Z inScc 0 1 c aA caB ð  Þ 18Þoc o 1 c the liquid phase was chosen as 10 [19,20]. We observed thatT ;P ;N T ;P ;N the choice of Z does not significantly affect our results. In as addition, the size ratio, c should have been obtained from cAcB experimental density measurements but we chose to treatSccð0Þ ¼ : ð19Þ 1 c c f ðc;W Þ it as a free parameter in our calculations simply becauseA B there are no experimental measurements for the densities Once Scc(0) is fitted from Eq. (19), then all other parame- at the temperatures for which thermodynamics data are ters could be calculated. available. Once suitable values of these parameters are cho- The degree of order and segregation in the melt can be sen, the theory allows to calculate other quantities men- quantified by another important microscopic function, tioned above. The values of the fitted parameters are known as Warren–Cowley short range order parameter presented in Table 1. (CSRO), a1, [16,17]. The parameter a1 is related to the Scc(0) by 3.1. Concentration fluctuations in the long-wavelength Sccð0Þ 1þ a1 limit, the free energy of mixing and the chemical short-range¼  ð : ð20ÞcAcB 1 Z  1Þa order parameter1 For equiatomic composition, the chemical short range or- Due to the difficulties associated with diffraction exper- der (CSRO) parameter, a1, is found to be 1 6 a1 6 1. iments, the theoretical calculation of Scc(0) is of great inter- The negative values of this parameter indicate the ordering est when investigating the nature of interaction as well as in the melt, and complete ordering is showed by amin1 ¼ 1. the structure of binary liquid alloys [21]. The mixing behav- On the contrary, the positive values of a1 indicate segrega- iour of liquid binary alloys can be inferred from the devia- tion, whereas the phase separation takes place if amax1 ¼ 1. tion of Scc(0) from the ideal value, Sidccð0Þ. The presence of In addition,the mixing properties of binary molten met- chemical order is indicated as Sccð0Þ < Sidccð0Þ, on the con- als forming alloys can as well be analysed at the micro- scopic scale in terms of the quantity / defined in Eq. (3) Table 1 known as diffusion. The formalism relation that connects Values of the parameters for Al–Zn and Bi–In liquid alloys diffusion and Scc(0) [9] combines the Darken’s thermody- Alloy T (K) Z W c namic equation for diffusion [18] with the basic thermody- RT namic relation in the form [10]: Al–Zn 1000 10 0.800 0.950 Bi–In 900 10 0.900 1.135 A UuNt IVh EoRr'SsIT Y pOeF rIBsAoDnANa lL I B RcAoRpYy 1170 Y.A. Odusote et al. / Journal of Non-Crystalline Solids 353 (2007) 1167–1171 Table 2 Computed chemical short range order parameter (a1) for Al–Zn and Bi–In alloys at T = 1000 K and 900 K respectively CAl,Bi (a1)Al–Zn (a1)Bi–In 0.1 0.0697 0.0696 0.2 0.0625 0.0614 0.3 0.0551 0.0534 0.4 0.0476 0.0453 0.5 0.0399 0.0375 0.6 0.0322 0.0298 0.7 0.0244 0.0222 0.8 0.0164 0.0145 0.9 0.0083 0.0073 Fig. 1. Concentration fluctuations in the long-wavelength limit (S (0) and gests a tendency of complex formation i.e. preference forcc Sidccð0ÞÞ vs. concentration for Al–Zn and Bi–In liquid alloys at 1000 K and unlike atoms pairing as nearest neighbour. 900 K respectively. The solid line denotes theoretical values while the The plot of the concentration dependence of GMRT for the triangle and the cross denote experimental values for Al–Zn and Bi–In two alloys are given in Fig. 2. It is interesting to observe respectively. The dot denotes the ideal values Sidccð0Þ. CAl and CBi are the Al that the computed values are in quite excellent agreement and Bi concentrations in the alloy. with experiment. From the result, the Gibbs energies of mixing GM for Al–Zn and Bi–In liquid alloys, have values trary, if Sccð0Þ > Sidccð0Þ, the segregation and demixing in of about 0.4986RT and 0.9344RT respectively. This liquid alloys take place. shows that the tendency of compound formation in the A perusal of Fig. 1, the plot of concentration fluctua- liquid phases is weaker in Al–Zn than in Bi–In alloy. tions in the long-wavelength limit Scc(0) versus concentra- The same mixing behaviour can be deduced from the tion for the systems studied show that the fitted values of Warren–Cowley short-range order parameter, parameters yield a good representation of the experimental a1, Table 2. The positive values of the parameter a1, for data. The experimental values of Scc(0) for the two systems Al–Zn support a tendency towards segregation and the were derived from the experimental free energy of mixing negative values of a1, for Bi–In system at all concentrations Eq. (17), the theoretical Scc(0) was obtained using Eq. confirm a tendency of complex formation. (19) and thermodynamic data were taken from [22]. The Scc(0) values for the Al–Zn system at T = 1000 K 3.2. Dynamic properties: diffusion and viscosity (Fig. 1) clearly indicate that S ð0Þ > Sidcc ccð0Þ in the whole concentration range. This implies a tendency for homoco- From the ordering point of view, the computed chemical ordination, i.e segregation of preference for like atom Al– diffusion ratio (Eq. (21)) seems to be a more realistic Al and Zn–Zn tend to pair as nearest neighbours. The parameter than the Warren–Cowley short range order Scc(0) curve exhibits the maximum value of about 0.409 parameter, a1, [21] since Eq. (21) does not take into consid- at CAl = 0.50. For Bi–In system, the Scc(0) values show eration the coordination number as an input and thus the that S idccð0Þ < Sccð0Þ over the concentration range. This sug- inherent problems related to its estimation are avoided. The relationship between the Scc(0) and diffusion expressed by the ratio of the mutual and self-diffusion coefficient DmD ,id indicates the mixing behaviour of alloys, i.e. phase separa- tion tendency or segregation in Al–Zn, for DmD < 1 over theid concentration range, Table 3. The values DmD > 1 for Bi–Inid Table 3 Concentration dependence of DmD for Al–Zn and Bi–In alloys atid T = 1000 K and 900 K respectively C Dm DmAl,Bi DidAl DZn idBiIn 0.1 0.8510 1.1493 0.2 0.7393 1.2751 0.3 0.6631 1.3746 0.4 0.6209 1.4443 0.5 0.6112 1.4805 Fig. 2. Concentration dependence of GmRT for Al–Zn and Bi–In liquid alloys 0.6 0.6324 1.4792 at 1000 K and 900 K respectively. The solid line denotes theoretical values 0.7 0.6824 1.4358 while the triangle and the cross denote experimental values for Al–Zn and 0.8 0.7623 1.3334 Bi–In respectively. CAl and CBi are the Al and Bi concentrations in the 0.9 0.8683 1.2021 alloy. The experimental data are from [22]. A UuNt IVh EoRr'SsIT Y pOeF rIBsAoDnANa lL I B RcAoRpYy Y.A. Odusote et al. / Journal of Non-Crystalline Solids 353 (2007) 1167–1171 1171 calculations. We investigated the bulk and dynamic prop- erties in Al–Zn and Bi–In liquid alloys using a theoretical approach with special interest on their bulk thermody- namic properties such as free energy of mixing, concentra- tion fluctuations in long-wavelength limit, chemical short range order parameter and the concentration dependence of diffusion and viscosity. Our theoretical investigation of bulk properties of Al–Zn liquid alloy is substantiated by the experimental data, obtained at 1000 K [22]. The order- ing in Al–Zn liquid phase has been analysed in terms of the microscopic functions, Scc(0), and CSRO (a1). The calcu- lated values of these functions indicate the existence of liquid miscibility gap or segregation in the melt, on the Fig. 3. Concentration dependence of viscosity Dg Eqs. (11) and (12) for Al– contrary, the presence of chemical order was observed forgo Zn and Bi–In at 1000 K and 900 K respectively. The solid line are for Bi–In at 900 K. The bulk thermodynamic and dynamic calculated Dgg . The triangle and the cross denote experimental values foro properties of the two liquid alloys have been explained  HMRT for Al–Zn and Bi–In respectively. CAl and CBi are the Al and Bi to a reasonable extent and the role of size effects notwith- concentrations in the alloy. The experimental data for heat of mixing are taken from [22]. standing the non-availability of the experimental viscosity data. The results obtained in this work further confirm liquid phase at all concentrations suggest a tendency for the applicability of this theoretical approach for a compound formation in the melt. proper description of mixing properties of binary liquid As to the other aspect of our calculations which is to alloys. investigate how well the theoretical formulation in Eqs. (11) and (12) compare with experiment, we show in References Fig. 3, our results for Dgg and  HM RT , but due to the lack ofo experimental data for viscosity at temperatures of interest [1] R.N. Singh, Can. J. Phys. 65 (1987) 309. it was not possible to compare our calculated results for [2] A.B. Bhatia, W.H. Hargrove, Phys. Rev. B 10 (1974) 3186. viscosity with experimental values. However, a perusal of [3] R.N. Singh, K.K. Singh, Mod. Phys. Lett. B 9 (1995) 1729. Dg [4] O. Akinlade, Phys. Chem. Liq. 29 (1995) 9.Fig. 3, reveals that computed g for Al–Zn show a negativeo [5] O. Akinlade, Z. Metallkd. 85 (1995) 487. deviation while its  HMRT is characterized by liquid miscibil- [6] R.N. Singh, N.H. March, in: J.H. Westbrook, R.L. Fleischer (Eds.), ity gaps and exhibit large positive heat of mixing, HM. This Intermetallic Compounds, Principles and Practice, vol. 1, John Wiley fact is substantiated by our earlier submission of the exis- Sons, New York, 1995, p. 661. tence of a liquid miscibility gap in Al–Zn liquid alloy. [7] R.N. Singh, F. Sommer, Rep. Prog. Phys. 60 (1997) 57–150. Dg [8] N.W. Ashcroft, D. Stroud, Solid State Phys. 33 (1978) 1.On the other hand, the computed g for Bi–In exhibito [9] R.N. Singh, F. Sommer, J. Phys. Condens. Mat. 4 (1992) 5345. positive deviation and its  HMRT shows negative deviation [10] R.N. Singh, F. Sommer, Z. Metallkd. 83 (1992) 533. which confirm that Bi–In liquid alloy is an ordered alloy. [11] L.D. Lucas, Tension Superficielle, Techniques de l’lngenieur, 1984, In order to discuss the disagreement between the calculated from M67 (F). values of Dg and HMg RT for the two systems, it is important one [12] B.J. Keene, Int. Mat. Rev. 38 (4) (1993) 157. o [13] E.A. Guggenheim, Mixtures, Oxford University Press, Oxford, 1952. looks at the parameters used to fit Scc(0) and hence to carry [14] O. Akinlade, R.N. Singh, F. Sommer, J. Alloys Comp. 267 (1998) out the calculations. It is observed that the fitted value of c 195. for Bi–In is 1.135 and 0.950 for Al–Zn (Table 1). The calcu- [15] S.M. Osman, R.N. Singh, Phys. Rev. E 15 (1995) 332. lated values of c for Bi–In is closer to that determined at the [16] J.M. Cowley, Phys. Rev. 77 (1950) 667. melting point than that of Al–Zn. One reasonable conclu- [17] B.E. Warren, X-ray Diffraction, Addison-Wesley, Reading, MA, 1969, p. 227. sion drawn from this is that the size effect plays a prominent [18] L.S. Darken, Trans. Metall. Soc. AIME 175 (1948) 184. role in the energetics of Bi–In than it plays in Al–Zn alloy [19] B.C. Anusionwu, G.A. Adebayo, J. Alloys Compd. 329 (2001) 162. and thus pulling the system from segregation to order phase. [20] Y. Waseda, The Structure of Non-Crystalline Materials, Liquids and Amorphous Solids, McGraw-Hill, New York, 1980. 4. Conclusions [21] R. Novakovic, M.L. Muolo, A. Passerone, Surf. Sci. 549 (2004) 281– 293. [22] R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K.K. Kelly, A Quasi-Lattice Theory has been utilised to obtain the Selected Values of the Thermodynamic Properties of Binary Alloys, fitted parameters, that are assumed to be invariant in all American Society for Metals, Material Park, OH, 1973. A UuNt IVh EoRr'SsIT Y pOeF rIBsAoDnANa lL I B RcAoRpYy ARTICLE IN PRESSCorrespondi E-mail addre 1Present add Ojoo, Lagos, N 0921-4526/$ - se doi:10.1016/j.phPhysica B 403 (2008) 2629–2633 www.elsevier.com/locate/physbThermodynamic properties of some gallium-based binary alloys O.E. Awea, Y.A. Odusoteb,, O. Akinladec, L.A. Hussaina,1 aDepartment of Physics, University of Ibadan, Ibadan, Nigeria bDepartment of Physics, Olabisi Onabanjo University, Ago-Iwoye, Nigeria cDepartment of Physics, University of Agriculture, Abeokuta, Nigeria Received 18 November 2007; received in revised form 12 December 2007; accepted 19 January 2008 ARY R Abstract We have studied the concentration dependence of the free energy of mixing, concentration–concentration fluctuations in the long- wavelength limit, the chemical short-range order parameter, the enthalpy and entropy of mixing of Ga–Zn, Ga–Mg and Al–Ga binary alloys at different temperatures using a quasi-chemical approximation for compound forming binary alloys and that for simple regular alloys. From the study of the thermodynamic quantities, we observed that thermodynamic properties of Ga–Zn and Al–Ga exhibit positive deviations from Raoultian behaviour, while Ga–Mg exhibits negative deviation. Hence, this study reveals that both Ga–Zn and Al–Ga are segregating systems, while chemical order exists in Ga–Mg alloy in the whole concentration range. Furthermore, our investigation indicate that Al–Ga binary alloy have a tendency to exhibit ideal mixture behaviour in the concentration range 0pcAlp0:30 and 0:7pcAlp1. r 2008 Elsevier B.V. All rights reserved. Keywords: Chemical order; Homocoordination; Interaction energy; Raoultian behaviour IBADAN LIBF 1. Introduction Many investigations have been reported in literature on liquid binary alloy systems which are of importance from both the scientific and also the technological points of view. An accurate knowledge of the thermodynamic properties and phase diagrams of the alloy systems are essential to establish a respectable understanding between the experi- mental results, theoretical approaches and empirical models for liquid alloys with a miscibility gap. The interatomic interactions and the related energies of the bond between the A and B component atoms of a binary alloy play an essential role in understanding the mixing behaviour of two metals. Because of this, the energetically preferred heterocoordination of A–B atoms as nearest neighbours over self-coordination A–A and B–B, or vice versa lead to the classification of all binary alloys UNIVERSITY Ong author. Tel.: +234 8052203325. ss: yisau24@yahoo.co.uk (Y.A. Odusote). ress: Vice Chancellor’s Office, Lagos State University, igeria. e front matter r 2008 Elsevier B.V. All rights reserved. ysb.2008.01.026into two distinct groups: short-range ordered [1–5] or segregating (demixing) alloys [6–9]. In this study, Ga–Zn, Ga–Mg and Al–Ga binary alloys are the alloys of interest. Our choice of these alloys arises from the fact that the three alloys have various industrial applications. Alloy systems containing semiconducting components such as gallium which is a group IIIB element like aluminium have been the subject of an increasing attention in the semiconductor production for their important application in the solid-state electronic devices and as a useful thermometric liquid [10]. Also, magnesium alloys offer lightweight alternatives to conventional metal- lic alloys and consequently, research on Mg alloys is fuelled nowadays by the need for low-density materials that suit aerospace and automobile industries [11] and as a result of which the world consumption of magnesium alloys in the automobile industry has been on the increase in the last decade [12]. Moreover, aluminium and its alloys are used in many aspects of modern life, from soda cans and house- hold foil to automobiles and aircraft in which we travel. Al-based alloys have also been found useful as additives in various fuel formulations for propellants, explosives, ARTICLE IN PRESS 2630 O.E. Awe et al. / Physica B 403 (2008) 2629–2633 incendiaries or pyrotechnics [13]. Large quantities of zinc are used to produce die castings. In addition, all the essential experimental data that are required for the calculation of the thermodynamic properties are available. We observed that both Ga–Zn and Al–Ga systems are characterized by positive interaction energies indicating segregation in the melts, while Ga–Mg alloy is character- ized by a negative interchange energy indicating ordering in the melt. This characteristic behaviour is likely to be a reflection of the interplay of the energetic and structural re-adjustment of the constituent elemental atoms. The long-wavelength limit of the concentration fluctua- tions, Sccð0Þ, the Warren–Cowley chemical short-range order parameter (CSRO), a1, the enthalpy of mixing, HM and the entropy of mixing, SM of Ga–Zn and Al–Ga molten alloys deviate positively from the ideality and thus both alloys belong to the class of liquid alloys that exhibits a preference towards homocoordination (segregation), while Ga–Mg deviate negatively from ideality suggesting a preference towards heterocoordination. In this work, we study the composition dependence of thermodynamic properties of these binary alloys by using a quasi-chemical approximation model [14] for compound forming binary alloys and that for simple regular alloys. The energetics of mixing as well as the positive deviation from Raoultian behaviour was discussed for the various thermodynamic quantities calculated. The structure of this paper is as follows. In Section 2, we present the model descriptions for calculating the thermo- dynamic properties of the alloys. This is followed by results and discussion in Section 3 and our conclusions at the end of the paper. 2. Model descriptions The general mathematical expression treats an alloy as a pseudoternary mixture of A atoms, B atoms and AmBn (m and n are small integers) group of atoms or clusters with the stoichiometry of intermetallics present in the solid state, all in chemical equilibrium with one another. The absence of clusters in the melt reduces the model to the quasi- chemical approximation for regular solutions. The grand partition function X of a binary molten alloy AB, which consists of NA ¼ Nc and NB ¼ Nð1 cÞ atoms of elements A and B, respectively, where the total number of atoms, N, is equXal to NA þ NB, can be expressed as X ¼ qNA ðTÞqNBA B ðTÞ exp½ðmANA þ mBNB  EÞ=kBT , E (1) where qNi ðTÞ and mi are atomic partition function and chemical potential of ith components ði ¼ A;BÞ, kB is Boltzmann’s constant, T is the absolute temperature and E is the configurational energy of the alloy. The quasi- chemical model (QCM) is utilized to determine the probable chemical complexes existing in a liquid binary UNIVERSITY OFalloy. Detailed discussion of the model are given in Ref. [14]. After doing some algebra [15], the solution of Eq. (1) is given as the ratio of the activity coefficient g ðg ¼ gA=gBÞ; gA and gB are activity coefficients of A and B atoms, respectively) for the compound ðAmBnÞ forming alloys as ¼ Zð1 cÞ ðbþ 2c  1Þln g : (2) 2c ðb 2c þ 1Þ with b ¼ f1þ 4cð1 cÞðZ2  1Þg1=2, (3) where     2o 2P D  P D Z2 ¼ AB AB AA AA  PBBDBBexp exp zkBT kBT (4) and the interchange or ordered energy, o is expressed as o ¼ Z½D 1AB  2ðDAA þ DBBÞ. (5) DAB, DAA, DBB are the interaction parameters and Dij is the change in the energy of the ij bond in the complex AmBn. Pij is the probability that the ij bond is a part of the complex. Pij may be expressed as P ¼ cm1ð1 cÞn1AB ½2 cm1ð1 cÞn1, (6) P ¼ cm2ð1 cÞn½2 cm2AA ð1 cÞn; mX2, (7) PBB ¼ cmð1 cÞn2½2 cmð1 cÞn2; mX2. (8) We now try to obtain an expression for the excess Gibbs free energy of mixing GxsM defined by GxsM ¼ GM  RTfc ln c þ ð1 cÞ lnð1 cÞg. (9) GM being the free energy of mixing. G xs M can be ob- tained from Eq. (2) by using the thermodynamic relation- ship [3] Gxs Z c M ¼ Z ½ln sþ ð2k 1BTÞ ðPAADAA  PBBDBBÞdx RT 0 þ f. (10) Here Z is the coordination number, c is the concentration of atom A and R is the universal gas constant and 1 ð1 cÞðbþ 2c  1Þ ln s ¼ ln 2 cðb 2c þ 1Þ . (11) The constant f in Eq. (10) is determined from the requirement that GxsM ¼ 0 at c ¼ 0 and 1. One observes that for Dij ¼ 0, the expression reduces to the regular solution expression [16]. The concentration–concentration fluctuations in the long-wavelength limit, Sccð0Þ, has emerged as an important microscopic function to understand the mixing behaviour of liquid alloys in terms of compound formation and phase segregation [14]. The Sccð0Þ can be expressed by GM, or the IBADAN LIBRARY ARTICLE IN PRESS O.E. Awe et al. / Physica B 403 (2008) 2629–2633 2631 I activity, ai ði ¼ A;BÞ, as 1   ð Þ ¼ q 2 1GM Scc 0 RT ¼ qaA c a qc2 B A A T ;P; qcN A T ;P;N q 1¼ aBcAaB qð1 . (12)cAÞ T ;P;N For ideal mixing the energy parameters, o, given in Eq. (5) is equal to zero, and Eq. (12) becomes Sidccð0Þ ¼ cAcB. (13) Substituting Eq. (9) for GM into Eq. (12), we obtain cð1 cÞ Sccð0Þ ¼ . (14) 1þ Z=2ð1=b 1Þ Eq. (12) is usually utilized to obtain the experimental values of Sccð0Þ from the measured activity or the free energy of mixing data [17]. The mixing behaviour of liquid binary alloys can be inferred from the deviation of Sccð0Þ from Sidccð0Þ. The presence of chemical order is indicated by Sccð0ÞoSidccð0Þ; on the contrary, if S idccð0Þ4Sccð0Þ, the segregation and demixing in liquid alloys take place. The Warren–Cowley short-range order parameter, a1 [18,19] can further be used to gain insight into the local arrangement of atoms in the molten alloys. This parameter a1 is expressed in term of b-function (Eq. (5)), as ¼ b 1a1 . (15)bþ 1 The enthalpy of mixing, HM within the QCM can be obtained from the standard thermodynamic relation [20] ¼  qGMHM GM T . (16)qT P From the expression for GMin Eq. (9), we obtain 2ð 2o8RTc 1 cÞ2 exp   ¼  ZkBT 1 do  oHM ðb 1þ 2cÞð1þ bÞðbþ 1 2cÞ kB dT T (17) and thus the entropy of mixing, ¼ HM  GMSM . (18) T 3. Results and discussion Based on the model descriptions in Section 2, the mixing behaviour of Ga–Zn, Ga–Mg and Al–Ga binary alloys UNIVERSITY OF Table 1 Fitted interaction parameters for the systems System T ðKÞ Z o ðeVÞ do=dT  103 Ga–Zn 700 10 0.0323 1.240 Ga–Mg 923 10 0.3932 9.215 Al–Ga 1073 10 0.0150 0.250have been defined. The values of the relevant parameters used to obtain our results are presented in Table 1. It is important to add that keeping these fitted parameters, which gives the best representation of the observed Gibbs free energy of mixing GM data unchanged in our calculations, one can then proceed using these fixed values to compute such properties as the Sccð0Þ, a1, the enthalpy of mixing, HM and the entropy of mixing, SM and thus, forming a basis to elucidate the energetics of the alloys. Using Eq. (9), we have calculated the free energy of mixing for these alloys. The experimental data given as symbols in Fig. 1 were obtained from Ref. [21]. A perusal of Fig. 1 shows that our interaction parameters give a good representation of the experimental values of the free energy of mixing for the alloys Ga–Zn, Ga–Mg and Al–Ga at temperatures of 700, 923 and 1073K, respectively. The good agreement obtained for GM as shown in the figure for the three systems we have worked on, gives us the confidence to use the fitted energy parameters to study the nature of ordering in the liquid alloys. A comparison of the figure for the three binary alloys reveal that the Gibbs free energy of mixing are almost symmetric around the equiatomic composition ðcGa;Al ¼ 0:5Þ, with Ga–Mg ½ðGM=RTÞc ¼ 2:08 exhibiting a higher tendency forc compound formation and is the most interacting of the three alloys; Al–Ga ½ðGM=RTÞc ¼ 0:652 is more inter-c acting than Ga–Zn alloy ½ðGM=RTÞc ¼ 0:569.c For proper analysis of the nature of ordering in the melts, it is important to critically consider the results for the structure related quantities. From this point of view the first quantity investigated is the concentration fluctuations Sccð0Þ. The deviation of Sccð0Þ from ideal value Sidccð0Þ ¼ cAcB is an essential parameter in order to visualize the nature of atomic interactions in the mixture. If, at a given composition Sccð0ÞbSidccð0Þ, then there is a tendency for segregation and vice versa for heterocoordination. BADAN LIBRARYFig. 1. Free energy of mixing, GM=RT versus concentration for Ga–Zn, Ga–Mg and Al–Ga liquid alloys at 700, 923 and 1073K, respectively. The solid line denotes theoretical values while the cross, triangle and star denote experimental values for Ga–Zn, Ga–Mg and Al–Ga, respectively. cGa and cAl are the Ga and Al concentrations in the alloy. The experimental data are from Ref. [21]. ARTICLE IN PRESS 2632 O.E. Awe et al. / Physica B 403 (2008) 2629–2633 Fig. 3. Calculated Warren–Cowley short-range order parameter a1 versus concentration for Ga–Zn, Ga–Mg and Al–Ga liquid alloys at 700, 923 and 1073K, respectively. cGa and cAl are the Ga and Al concentrations in the alloy. Fig. 4. Enthalpy of mixing, HM=RT versus concentration for Ga–Zn, Ga–Mg and Al–Ga liquid alloys at 700, 923 and 1073K, respectively. The solid line denotes theoretical values while the cross, triangle and star denote experimental values for Ga–Zn, Ga–Mg and Al–Ga, respectively. cGa and cAl are the Ga and Al concentrations in the alloy. The experimental data are from Ref. [21]. IBADAN LIBRARYIt is obvious from Eq. (12) that Sccð0Þ can be obtained directly from the experimental Gibbs energy of mixing or from the activity data. This is usually referred to as an experimental Sccð0Þ in literature. We have used Eq. (14) to determine the computed Sccð0Þ for these liquid alloys, while their measured Sccð0Þ were obtained by numerical differ- entiation of the Gibbs free energy of mixing data taken from Ref. [21]. It is important to add that Eq. (13) was used to calculate Sidccð0Þ. Our results for the concentration fluctuations as a function of composition for the three systems are as shown in Fig. 2. From the figure, it is clear that the calculated Sccð0Þ4Sidccð0Þ for Ga–Zn and Al–Ga, while calculated S idccð0ÞoSccð0Þ for Ga–Mg. This implies a tendency for homocoordination in Ga–Zn and Al–Ga alloys, i.e. like atoms Ga–Ga, Zn–Zn or Al–Al tend to pair as nearest neighbours and a presence of heterocoordination in Ga–Mg, i.e. unlike atoms Ga–Mg tend to pair as nearest neighbours. A close look at Fig. 2 shows that in the region of 0pcAlp0:30 and 0:7pcAlp1, it is observed that the calculated Sccð0Þ values for Al–Ga liquid alloy almost attain ideal values. This indicates that Al–Ga alloy have tendency to exhibit ideal behaviour for these concentrations. To have a clear understanding into the local arrange- ment of atoms in the molten alloys. We have used Eq. (15) to obtain the Warren–Cowley short-range order parameter a1. The positive values of a1 (Fig. 3) in the whole concentration range for both Ga–Zn and Al–Ga alloys are sufficient indicators of the presence of homocoordina- tion (segregation) in the liquid alloys, while negative values of a1 for Ga–Mg indicates chemical order in this concentration range. In addition, a1 is much less than unity for both Ga–Zn and Al–Ga alloys and a pointer that the alloys are weakly segregating alloys. It is noted that varying the value of Z does not have any significant effect on a12c curves, the only effect is to vary the position of the maxima while the overall features remain unchanged. ITY OFFig. 2. Concentration–concentration fluctuations Sccð0Þ versus concentra- tion for Ga–Zn, Ga–Mg and Al–Ga liquid alloys at 700, 923 and 1073K, respectively. The solid line denotes theoretical values while the cross, triangle and star denote experimental values for Ga–Zn, Ga–Mg and Al–Ga, respectively. The dot denotes the ideal values. cGa and cAl are the Ga and Al concentrations in the alloy. UNIVERSIt is seen from Eq. (16) for HM that in order to obtain a good fit to this parameter, we need to incorporate the temperature dependence of the interaction parameters; any other approximation would be invalid. Using Eqs. (17) and (18), we have ascertained the variation in temperature parameters from the measured values of HM and SM, the results are shown in Table 1. From the fitted enthalpy of mixing and entropy of mixing in Figs. 4 and 5, respectively, we note that the fits obtained for the three alloys compared quite well with the experimental data. The values obtained (Table 1) show that the temperature dependence of the energy parameters are quite small. Also, one notes that the enthaply of mixing for Ga–Zn and Al–Ga alloys are symmetric and positive (typical of segregating systems) in agreement with the work of Novakovic et al. [20] on Ga–Zn and Singh et al. [6] on gallium-based alloys, although their Gibbs free energy of mixing are indicative of a very weakly interacting system. On the contrary, both the free energy of mixing, GM and the enthalpy of mixing, ARTICLE IN PRESS O.E. Awe et al. / Physica B 403 (2008) 2629–2633 2633 Fig. 5. Entropy of mixing, SM=R versus concentration for Ga–Zn, Ga–Mg and Al–Ga liquid alloys at 700, 923 and 1073K, respectively. The solid line denotes theoretical values while the cross, triangle and star denote experimental values for Ga–Zn, Ga–Mg and Al–Ga, respectively. cGa and cAl are the Ga and Al concentrations in the alloy. The experimental data are from Ref. [21]. I R HM exhibit negative deviations from Raoultian behaviour for Ga–Mg alloy. In addition, the entropy of mixing (Fig. 5) for the three alloys investigated is almost symmetric and positive around the equiatomic composi- tion. Generally, one observes that both the reported literature data and theoretical values obtained using the interaction parameters and their temperature dependence are in very good agreement. 4. Conclusions The thermodynamic properties of the three binary alloys, namely Ga–Zn, Ga–Mg and Al–Ga have been dis- cussed in terms of the free energy of mixing, GM=RT , the concentration fluctuations in the long-wavelength limit, Sccð0Þ, chemical short-range order parameter, a1, as a function of compositions on the basis of the theory in Section 2. Positive deviations from Raoultian behaviour was observed in the thermodynamic properties of Al–Ga and Ga–Zn which agrees with the result of Awe et al. [10] using four-atom-cluster-model (FACM), while Ga–Mg exhibits negative deviation. An indication that both Ga–Zn and Al–Ga are segregating systems, while a reasonable VERSITY OF degree of chemical order exists in Ga–Mg alloy across the whole concentration range. The enthalpy of mixing and entropy of mixing for the three alloys studied are positive and symmetric around equiatomic composition, except for the enthalpy of mixing, HM of Ga–Mg alloy that shows a negative deviation from Raoultian behaviour. We conclude that in the glass-forming composition range which usually lies far away from the stoichiometric composition, com- puted Sccð0Þ almost attain ideal values for Al–Ga: i.e. Al–Ga exhibits ideal behaviour for the composition 0pcAlp0:30 and 0:7pcAlp1. Y References [1] A.B. Bhatia, W.H. Hargrove, Phys. Rev. B 10 (1974) 3186. [2] R. Novakovic, E. Ricci, D. Giuranno, F. Gnecco, Surface Science 515 (2002) 377. [3] A.B. Bhatia, R.N. Singh, Phys. Chem. Liq. 11 (1982) 285. [4] R.N. Singh, N.H. March, in: J.H. Westbrook, R.L. Fleischer (Eds.), Intermetallic Compounds, Principles and Practice, vol. 1, Wiley, New York, 1995, p. 661. [5] R. Novakovic, M.L. Muolo, A. Passerone, Surface Science 549 (2004) 281. [6] R.N. Singh, F. Sommer, Rep. Progr. Phys. 60 (1997) 57. [7] R.N. Singh, J. Phys. Chem. 25 (1993) 251. [8] L.C. Prasad, A. Mikula, High Temp. Mater. Process 19 (1) (2000) 61. [9] R. Novakovic, E. Ricci, M.L. Muolo, D. Giuranno, A. Passerone, Intermetallic 11 (2003) 1301. [10] O.E. Awe, O. Akinlade, L.A. Hussain, J. Alloys Compds. 387 (2005) 256. [11] F. Islam, M. Medraj, in: Proceedings of CSME, Forum, 2004, pp. 921–929. [12] O.E. Awe, O. Akinlade, L.A. Hussain, Surface Science 600 (2006) 2122–2128. [13] M. Schroenitz, E.L. Dreizin, J. Mater. Res. 18 (8) (2003) 1827. [14] R.N. Singh, Canad. J. Phys. 65 (1987) 309. [15] L.C. Prasad, R.N. Singh, V.N. Singh, G.P. Singh, J. Phys. Chem. B 102 (1998) 921. [16] R.N. Singh, I.K. Mishra, Phys. Chem. Liq. 18 (1988) 303. [17] O. Akinlade, Phys. B 245 (1998) 330. [18] B.C. Anusionwu, Pramana J. Phys. 67 (2) (2006) 319. [19] Y.A. Odusote, L.A. Hussain, E.O. Awe, J. Non-Cryst. Solids 353 (2007) 1167. [20] R. Novakovic, D. Zivkovic, J. Mater. Sci. 40 (2005) 2251. [21] R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K.K. Kelly, in: Selected Values of the Thermodynamic Properties of Binary Alloys, American Society for Metals, Materials Park, OH, 1973. BADAN LIBRAI UN