Binary Systems: Miscibility Gaps and Eutectics

3.012 Fundamentals of Materials Science Fall 2005 Lecture 18: 11.09.05 Binary systems: miscibility gaps and eutectics Today: LAST TIME .........................................................................................................................................................................................2 Free energy and phase diagrams of ideal binary solutions ......................................................................................................2 BINARY SOLUTIONS WITH LIMITED MISCIBILITY IN THE SOLID STATE: MISCIBILITY GAPS ...........................................................4 The Regular Solution Model, part I ............................................................................................................................................4 Phase diagram of a regular solution ..........................................................................................................................................7 EUTECTIC SYSTEMS ..........................................................................................................................................................................8 Free energy diagrams of eutectic systems..................................................................................................................................8 Analyzing phase equilibria on eutectic phase diagrams .........................................................................................................11 Example eutectic systems ..........................................................................................................................................................12 REFERENCES ...................................................................................................................................................................................14 Reading: Supplementary Reading: Lupis, Chemical Thermodynamics of Materials, Ch VII, ‘Binary Phase Diagrams,’ pp. 204-219 3.012 Fundamentals of Materials Science Fall 2005 Last time Free energy and phase diagrams of ideal binary solutions Key points: 1. Common tangents to free energy curves define composition regions where phase separation (twophase equilibria) occurs. 2. Phase separation lowers the overall free energy by splitting the homogenous system into a weighted mix of two separate phases, which each have lower free energy than the starting homogeneous phase. 3. Compositions of the phases in two-phase regions are given by the tangent points, and the amount of each phase is determined by the lever rule. Two-phase equilibrium introduces the phase fraction, determined by the lever rule: • nSA + n BS f = mole fraction solid in whole system = nA + nB S n LA + n BL f = mole fraction liquid in whole system = nA + nB L f S + f L =1 ! 3.012 Fundamentals of Materials Science • Fall 2005 Free energy diagrams directly relate to binary phase diagrams: Key points: 1. Free energy diagrams define the structure of the phase diagram. 2. Compositions of two-phase regions, which are tangent points on the free energy diagram, become phase boundaries on the phase diagram. 3. The lever rule is directly applied to tie lines on the phase diagram, to determine the amount of each phase present in two-phase regions. o The ideal solution is a special case- not all materials form binary solutions that are miscible at all compositions and all temperatures, particularly in the solid state. Materials that have incompatible crystal structures cannot form homogeneous solid solutions- they are forced to phase separate into an A-rich solid phase and a B-rich solid phase, to preserve their stable crystalline forms. Today we will examine the phase diagrams of these and other common binary materials systems. 3.012 Fundamentals of Materials Science Fall 2005 Binary solutions with limited miscibility in the solid state: Miscibility gaps The Regular Solution Model, part I • What happens if the molecules in the solution interact with a finite energy? The entha lpy of mixing will mix < 0 ) or disfavoring ( "H mix > 0 ) mixing of the two now have a finite value, either favoring ( "H components. The simplest model of a solution with finite interactions is called the regular solution model: ! ! o Let the enthalpy of mixing take on a finite value given by: o We take the entropy of mixing to be the same as in the ideal solution. This gives a total free energy of mixing which is: 3.012 Fundamentals of Materials Science o Fall 2005 We can also use the equations above to determine the enthalpy, entropy, and Gibbs free energy of the regular solution: "G mix,rs = G rs # G unmixed G rs = "G mix,rs + G unmixed = $X A X B + RT [ X A ln X A + X B ln X B ] + µAo X A + µBo X B $ #G rs ' S = "& ) = "R[ X A ln X A + X B ln X B ] % #T ( P,n rs ! H rs = G rs + TS rs = "X A X B + µAo X A + µBo X B o ! solution model describes the liquid phase of many real systems such as Pb-Sn, GaThe regular Sb, and!Tl-Sn, and some solid solutions. Today we will analyze the behavior of a system with this free energy function; in a few lectures we will show how the given forms of the enthalpy and entropy of mixing arise from consideration of molecular states (using statistical mechanics). o Let’s look at the behavior of regular solutions graphically. The plots below are all prepared for a binary system where the standard state chemical potentials are assumed to be approximately invariant with temperature (i.e., the endpoints of the free energy curves are not moving with temperature- approximately valid over certain temp. ranges for some solids): J mole J µBo = "1,000 mole µAo = "2,000 ! 3.012 Fundamentals of Materials Science • The free energy of the system varies both with the value of Ω and with temperature: o As a function of temperature at a fixed positive value of Ω: Fall 2005 3.012 Fundamentals of Materials Science Fall 2005 Phase diagram of a regular solution !Gmix,RS XB !Gmix,RS T XB XB ! !Gmix,RS XB 3.012 Fundamentals of Materials Science Fall 2005 Eutectic systems • It is commonly found that many materials are highly miscible in the liquid state, but have very limited mutual miscibility in the solid state. Thus much of the phase diagram at low temperatures is dominated by a 2-phase field of two different solid structures- one that is highly enriched in component A (the α phase) and one that is highly enriched in component B (the β phase). These binary systems, with unlimited liquid state miscibility and low or negligible solid state miscibility, are referred to as eutectic systems. o The behavior just described, where the two components are completely miscible at high temperatures in the liquid state and phase-separated into two solids at low temperatures would be represented by a phase diagram as follows: L T ! !+" " XB ! o • Now the question is, what happens in the region between high and low temperatures? Let’s examine how the phase diagram of a eutectic system develops. Free energy diagrams of eutectic systems • Suppose we have a binary solution of A and B. The molar free energy for each phase can be diagrammed for as a function of composition, as we did for the simple ideal solution:(1) 3.012 Fundamentals of Materials Science Fall 2005 P = constant � T L G � !+L Liquid ! "+L " !+" XB XB ! P = constant T = 900 L T � G � !+L � X X ! L "+L " !+" XB XB ! P = constant T = 800 � L T G � !+L X � X L L X X � XB ! "+L !+" XB ! Figure by MIT OCW. " 3.012 Fundamentals of Materials Science � Fall 2005 � L T G !+L P = Constant T=T+ EU X� X L XL X� ! "+L !+" X� XB ! Figure by MIT OCW. " 3.012 Fundamentals of Materials Science Fall 2005 � G � P = constant T = TEU X� X� X� Figure by MIT OCW. L T !+L "+L " ! !+" (© W.C. Carter 2002, p. 203) XB ! Analyzing phase equilibria on eutectic phase diagrams • Next term, you will learn how these thermodynamic phase equilibria intersect with the development of microstructure in materials: 3.012 Fundamentals of Materials Science Fall 2005 L (C4 wt% sn) Z Temperature (0C) �+L 200 L L � (18.3 wt% Sn) 500 k �+L I � m L (61.9 wt% Sn) 100 600 � � Eutectic structure 400 300 Primary � (18.3 wt% Sn) �+� Temperature (0F) f 300 200 � (97.8 wt% Sn) Z' 0 0 (Pb) Eutectic � (18.03 wt% Sn) 20 60 C4 (40) 100 80 100 (Sn) Composition (wt% Sn) Schematic representations of the equilibrium microstructures for a lead-tin alloy of composition C4 as it is cooled from the liquid-phase region. Figure by MIT OCW. Example eutectic systems Solid-state crystal structures of several eutectic systems Component A Solid-state crystal Component B Solid-state crystal 3.012 Fundamentals of Materials Science Fall 2005 structure A rhombohedral FCC (lattice parameter: 4.09 Å at 298K) FCC Bi Ag Pb structure B tetragonal FCC (lattice parameter: 3.61 Å at 298K) tetragonal Sn Cu Sn 2800 Liquid 2600 1200 Ag Cu T(0K) 1100 1000 900 700 CaO ss + L MgO ss + L 2400 MgO ss CaO ss 2200 MgO ss + CaO ss 2000 800 600 Ag Temperature (0C) 1300 L 1800 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Xcm Cu 1600 MgO 20 Eutectic phase diagram for a silver-copper system. Figure by MIT OCW. 40 60 80 100 Wt % Eutetic phase diagram for MgO-CaO system. Figure by MIT OCW. CaO 3.012 Fundamentals of Materials Science Fall 2005 References 1. 2. 3. 4. Carter, W. C. 2002. 3.00 Thermodynamics of Materials Lecture Notes. McCallister, W. D. 2003. Materials Science and Engineering: An Introduction. John Wiley & Sons, Inc., New York. Lupis, C. H. P. 1983. Chemical Thermodynamics of Materials. Prentice-Hall, New York. Bergeron, C. G., and S. H. Risbud. 1984. Introduction to Phase Equilibria in Ceramics. American Ceramic Society, Westerville, OH.