Binary Phase Diagrams

3.012 Fundamentals of Materials Science Fall 2005 Lecture 19: 11.23.05 Binary phase diagrams Today: LAST TIME .........................................................................................................................................................................................2 Eutectic Binary Systems...............................................................................................................................................................2 Analyzing phase equilibria on eutectic phase diagrams ...........................................................................................................3 Example eutectic systems.............................................................................................................................................................4 INVARIANT POINTS IN BINARY SYSTEMS ..........................................................................................................................................5 The phase rule and eutectic diagrams: eutectic invariant points .............................................................................................5 Other types of invariant points....................................................................................................................................................6 Congruent phase transitions........................................................................................................................................................7 INTERMEDIATE COMPOUNDS IN PHASE DIAGRAMS3 ........................................................................................................................8 EXAMPLE BINARY PHASE DIAGRAMS ...............................................................................................................................................9 DELINEATING STABLE AND METASTABLE PHASE BOUNDARIES: SPINODALS AND MISCIBILITY GAPS ........................................ 11 Conditions for stability as a function of composition ............................................................................................................. 11 SUPPLEMENTARY INFORMATION (NOT TO BE TESTED): TERNARY SOLUTION PHASE DIAGRAMS ............................................... 14 REFERENCES................................................................................................................................................................................... 16 Reading: W.D. Callister, Jr., “Fundamentals of Materials Science and Engineering,” Ch. 10S Phase Diagrams, pp. S67-S84. Supplementary Reading: Ternary phase diagrams (at end of today’s lecture notes) 3.012 Fundamentals of Materials Science Fall 2005 Last time Eutectic Binary 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. P = constant T = 800 L T G � � !+L X � X L L X X � ! "+L !+" XB XB ! Figure by MIT OCW. " 3.012 Fundamentals of Materials Science Fall 2005 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: L (C4 wt% Sn) Z j 300 Temperature (0C) �+L 200 L L 500 k � (18.3 wt% Sn) � �+L I m L (61.9 wt% Sn) 100 600 � � Eutectic structure 400 300 Primary � (18.3 wt% Sn) �+� 200 � (97.8 wt% Sn) Z' 0 0 (Pb) Eutectic � (18.3 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. Temperature (0F) • 3.012 Fundamentals of Materials Science Fall 2005 Example eutectic systems Solid-state crystal structures of several eutectic systems Component A Solid-state crystal Component B Solid-state crystal structure A structure B Bi rhombohedral Sn tetragonal Ag FCC Cu FCC (lattice parameter: (lattice parameter: 4.09 Å at 298K) 3.61 Å at 298K) Pb FCC Sn tetragonal 2800 Liquid 2600 1200 1000 Ag Cu T(0K) 1100 900 2400 CaO ss + L MgO ss + L MgO ss CaO ss 2200 MgO ss + CaO ss 2000 800 1800 700 600 Ag Temperature (0C) 1300 L 0.1 0.2 0.3 0.4 0.5 0.6 Xcm 0.7 0.8 0.9 Eutectic phase diagram for a silver-copper system. Figure by MIT OCW. Cu 1600 MgO 20 40 60 80 Wt % Eutetic phase diagram for MgO-CaO system. Figure by MIT OCW. • 100 Eutectic phase diagrams are also obtained when the solid state of a solution has regular solution behavior (can you show this?) CaO 3.012 Fundamentals of Materials Science Fall 2005 Invariant points in binary systems The phase rule and eutectic diagrams: eutectic invariant points • All phase diagrams must obey the phase rule. Does our eutectic diagram obey it? T L !+L ! "+L !+" XB ! " 3.012 Fundamentals of Materials Science Fall 2005 Other types of invariant points • Other transformations that occur in binary systems at a fixed composition and temperature (for constant pressure) are given titles as well: o o Two 2-phase regions join into one 2-phase region:  Eutectic: L <-> (α + β) (upper two region is liquid)  Eutectoid: γ <-> (α + β) (upper region is solid) One 2-phase region splits into two 2-phase regions:  Peritectic: (α + L) <-> β (upper two-phase region is solid + liquid)  Peritectoid: (α + γ) <-> β (upper two-phase region is solid + solid) β+L 700 β+� � β 600 L 5980C P �+α 5600C E β+α 500 60 70 1200 Temperature (0F) Temperature (0C) �+L 1000 α 80 α+L 90 Composition (wt % Zn) A region of the copper-zinc phase diagram that has been enlarged to show eutectoid and peritectic invariant points , labeled E (5600C, 74 wt% Zn) and P (5980C, 78.6 wt% Zn), respectively. Figure by MIT OCW.  Note that each single-phase field is separated from other single-phase fields by a twophase field. 3.012 Fundamentals of Materials Science Fall 2005 Congruent phase transitions • Congruent phase transition: complete transformation from one phase to another at a fixed composition LS G T SL XB XB ! G XB 3.012 Fundamentals of Materials Science Fall 2005 Intermediate Compounds in phase diagrams3 • Stable compounds often form between the two extremes of pure component A and pure component B in binary systems- these are referred to as intermediate compounds. AB2 + L L B+L A+L T AB2 + B A + AB2 AB2 A B Mol% Figure by MIT OCW. • When the intermediate compound melts to a liquid of the same composition as the solid, it is termed a congruently melting compound. Congruently melting intermediates subdivide the binary system into smaller binary systems with all the characteristics of typical binary systems. • Intermediate compounds are especially common in ceramics, as the pure components may form unique molecules at intermediate ratios. Shown below is the example of the system MnO-Al2O3: 20500 2000 X 1900 18500 1800 1700 1600 23 MnO + L 15200 8 23 10 23 MnO MnO + MA 20 17700 MA +L 1500 1400 Al2O3 + L MA +L 17850 MnO + Al2O3 Temperature (0C) L 40 60 MA + Al2O3 80 WI% System MnO + Al2O3 (MA = MnO + Al2O3) Al2O3 Figure by MIT OCW. 3.012 Fundamentals of Materials Science Fall 2005 Example binary phase diagrams • Phase diagrams in this section are © W.C. Carter, 2002: Graphics were produced by Angela Tong, MIT. 1500 1400 1300 T(Temperature), oC 1200 1100 1000 900 (Au,Ni) 800 700 600 500 400 300 0 10 20 Au 30 40 50 60 70 80 90 100 Ni Atomic percent nickel o Phase diagram for Gold-Nickel showing complete solid solubility above about 800 C and below o about 950 C. The miscibility gap at low temperatures can be understood with a regular solution model. Liquid Li2Al LiAl 800 Figure by MIT OCW. T (temperature) oC 700 600 Alpha Alpha 500 Beta 400 300 200 100 Zintl crystal Structure of LiAl (� phase)4 0 Al 10 20 30 Beta 40 50 60 70 Atomic Percent Lithium 80 90 100 Ll Phase diagram for light metals Aluminum-Lithium. There are five phases illustrated; however the BCC Lithium end-member phase shows such limited Aluminum solubility that it hardly appears on this plot. There are two eutectics and one peritectoid reactions. The LiAl(�) and Li2Al intermetallic phases are ordered compounds where the atoms order on sublattices, thus changing the symmetry of the material. The ordered phases show limited solubility where, for instance, the extra Li will occupy sites where an Al usually sits, or occupies an interstitial position. Figure by MIT OCW. 3.012 Fundamentals of Materials Science Fall 2005 1100 1000 900 T (temperature) oC 800 700 AuSn 600 AuSn2 500 AuSn4 400 300 200 100 0 Au 10 20 30 40 50 60 Atomic Percent Tin 70 80 90 100 Sn Phase diagram of Gold-Tin has seven distinct phases, three peritectics, two eutectics, and one eutectoid reactions. Figure by MIT OCW. 3.012 Fundamentals of Materials Science Fall 2005 Delineating stable and metastable phase boundaries: spinodals and miscibility gaps • We saw last time that a homogeneous solution can spontaneously decompose into phase-separated mixtures when interactions between the molecules are unfavorable. The example we started with was the regular solution model for the free energy, where the enthalpy of mixing two components might be positive (i.e., in terms of total free energy, unfavorable). Conditions for stability as a function of composition • For closed systems at constant temperature and pressure, the Gibbs free energy is minimized with respect to fluctuations in its other extensive variables. Thus, if we allow the composition of a binary system to vary, the system will move toward the minimum in the free energy vs. XB curve: • If the system is at the minimum in G, then we can write: • We can perform a Taylor expansion for a fluctuation in Gibbs free energy, assuming the only variable that can vary is composition (XB): o If we examine the consequence of a fluctuation in composition near the extremum point of the free energy curve, the first derivative of G is zero. If we assume the third-order (and higher) terms are negligible, then the condition for stable equilibrium is controlled by the value of the second derivative. 3.012 Fundamentals of Materials Science Fall 2005 G XB ! INSIDE SPINODAL: SYSTEM UNSTABLE TO SMALL COMPOSITION FLUCTUATIONS OUTSIDE SPINODAL: SYSTEM STABLE TO SMALL COMPOSITION FLUCTUATIONS A B 3.012 Fundamentals of Materials Science Fall 2005 Figure by MIT OCW. 3.012 Fundamentals of Materials Science Fall 2005 Supplementary Information (not to be tested): Ternary solution phase diagrams • A 3-component analog to the binary phase diagram is also commonly encountered in materials science & engineering problems. For a 3 component system, a triangular 2D phase equilibrium map can be used to represent the stable phases as a function of composition in the 3-component ternary solution: 327 271 Bi-Pb Peritectic 232 tc/0C Bi-Sn Eutectic 183 Pb-Sn Eutectic 125 Bi-Pb Eutectic Triple Eutectic at 960C c ti s F ra Mas Bi 139 f Bi on o Pb Mass Fraction of Sn Mass F r a ct io n o f Pb Sn 3-Dimensional Depiction of Temperature-Composition Phase Diagram of Bismuth, Tin, and Lead at 1atm. The diagram has been simplified by omission of the regions of solid solubility. Each face of the triangular prism is a two-component temperature-composition phase diagram with a eutectic. There is also a peritectic point in the Bi-Pb phase diagram. Figure by MIT OCW. • 3D depictions are necessary to show all 3 composition variables and temperature at fixed pressure (temperature is shown in the vertical axis)- in this arrangement, each face of the triangular column is the rd equivalent binary phase diagram (2 components present while 3 component is not present). • A single horizontal slice from the 3D ternary construction provides the phase equilibria as a function of composition for a fixed value of temperature and pressure: Fall 2005 A X B =0 3.012 Fundamentals of Materials Science =1 ! X B " # B C XB = 0 XB = 1 • Similar to binary phase diagrams, tie lines are used to identify the compositions and phase fractions in multi-phase regions of the ternary diagram: XR � �+ε �+� �+�+ε ε ε+� XG � XB Figure by MIT OCW. o The phase rule allows 3-phase equilibria to lie within fields of the ternary diagram, unlike the binary systems, where 3-phase equilibria are confined to invariant points. 3.012 Fundamentals of Materials Science Fall 2005 References 1. 2. 3. 4. 5. 6. McCallister, W. D. (John Wiley & Sons, Inc., New York, 2003). Lupis, C. H. P. (Prentice-Hall, New York, 1983). Bergeron, C. G. & Risbud, S. H. (American Ceramic Society, Westerville, OH, 1984). Lindgren, B. & Ellis, D. E. Molecular Cluster Studies of Lial with Different Vacancy Structures. 1471-1481 (1983). Carter, W. C. (2002). Mortimer, R. G. Physical Chemistry (Academic Press, New York, 2000). Materi als Scienc e and Engineering: An I ntroducti on C hemic al T hermodynamics of Materials I ntroducti on to Phase Equilibria i n C eramics Journal of Physics F-Met al Phy sics , 13