Lecture 19: 11.23.05 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)

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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 " + L

" � L L � ! X X X X ! + "

Figure by MIT OCW.

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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 600 300 j � L

� + L L 500

� (18.3 k F) C) 0 0 wt% Sn) 200 � + L 400 � I � m L (61.9 wt% Sn) Eutectic structure 300 Temperature ( Temperature Temperature ( Temperature Primary � (18.3 wt% Sn) � + � 100 200 � (97.8 wt% Sn)

Eutectic � (18.3 wt% Sn) 100 Z'

0 0 20 60 80 100

(Pb) C4 (40) (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.

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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 L

CaO ss + L 1300 Liquid 2600 MgO ss + L

1200 C) 0 2400 1100 MgO ss CaO ss

Ag Cu K) 0 1000 2200 T( Temperature ( Temperature 900 MgO ss + CaO ss 2000 800

700 1800

600 Ag 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Cu 1600 20 40 60 80 100 Xcm MgO CaO Eutectic phase diagram for a silver-copper system. Wt % Eutetic phase diagram for MgO-CaO system. Figure by MIT OCW. Figure by MIT OCW.

• Eutectic phase diagrams are also obtained when the solid state of a solution has regular solution behavior (can you show this?)

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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 !

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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 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)

o 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

1200 � + L L F) C) 0 0 β + � � 0 600 β P 598 C

5600C � + α Temperature ( Temperature Temperature ( Temperature E 1000 α α + L β + α 500

60 70 80 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 (5980 C, 78.6 wt% Zn), respectively.

Figure by MIT OCW.

. Note that each single-phase field is separated from other single-phase fields by a two- phase field.

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Congruent phase transitions

• Congruent phase transition: complete transformation from one phase to another at a fixed composition

LS T G

XB G

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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

AB + B A + AB2 2

A AB2 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 L Al2O3 + L 1800 0 C) 1785 MA +L 0 0 MA +L 1770 1700 23 MnO + L

Temperature ( Temperature 1600 8 23

15200 3 2310 2310 O 2 1500

MA + Al2O3 MnO + MA

1400 Al MnO +

20 40 60 80 MnO Al2O3 WI%

System MnO + Al2O3 (MA = MnO + Al2O3) Figure by MIT OCW.

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Example binary phase diagrams

• Phase diagrams in this section are © W.C. Carter, 2002: Graphics were produced by Angela Tong, MIT.

1500

1400

1300

1200 C

o 1100

1000

900 (Au,Ni)

T(Temperature), T(Temperature), 800

700

600

500

400

300 0 10 20 30 40 50 60 70 80 90 100 Au Ni Atomic percent nickel Phase diagram for Gold-Nickel showing complete solid solubility above about 800oC and below about 950oC. The miscibility gap at low temperatures can be understood with a regular solution model. Figure by MIT OCW.

800 Al 2 Liquid LiAl Li 700

600 C o Alpha 500 Beta

400 T (temperature) 300

200

100 Beta 0 10 20 30 40 50 60 70 80 90 100 Al Ll Zintl crystal Structure of LiAl (� phase)4 Atomic Percent Lithium

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.

Lecture 19 – Binary phase diagrams Figure by MIT OCW. 9 of 16 11/23/05 3.012 Fundamentals of Materials Science Fall 2005

1100

1000

900

800 C o 700 AuSn 600 AuSn2

T (temperature) T 500

AuSn4 400

300

200

100 0 10 20 30 40 50 60 70 80 90 100 Au Sn Atomic Percent Tin Phase diagram of Gold-Tin has seven distinct phases, three peritectics, two eutectics, and one eutectoid reactions.

Figure by MIT OCW.

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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.

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INSIDE SPINODAL: SYSTEM UNSTABLE OUTSIDE SPINODAL: SYSTEM STABLE TO SMALL COMPOSITION TO SMALL COMPOSITION FLUCTUATIONS FLUCTUATIONS

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Figure by MIT OCW.

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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

Bi-Sn

C 183 0 Eutectic / c t Pb-Sn Eutectic 139 125 Bi-Pb Eutectic Triple Eutectic at 960C

Bi n of ctio Mass s Fra Pb Frac Mas tion of Pb

Bi Mass Fraction of Sn 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 equivalent binary phase diagram (2 components present while 3rd 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:

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0 = A B X

1 = B X B C XB = 1 XB = 0

• Similar to binary phase diagrams, tie lines are used to identify the compositions and phase fractions in multi-phase regions of the ternary diagram:

�

�+ε � + �

� + � + ε

ε ε + � � X XG B

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.

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References

1. McCallister, W. D. Materi als Scienc e and Engineering: An I ntroducti on (John Wiley & Sons, Inc., New York, 2003).

2. Lupis, C. H. P. C hemic al T hermodynamics of Materials (Prentice-Hall, New York, 1983).

3. Bergeron, C. G. & Risbud, S. H. I ntroducti on to Phase Equilibria i n C eramics (American Ceramic Society, Westerville, OH, 1984).

4. Lindgren, B. & Ellis, D. E. Molecular Cluster Studies of Lial with Different Vacancy Structures. Journal of Physics F-Met al Phy sics 13, 1471-1481 (1983). 5. Carter, W. C. (2002). 6. Mortimer, R. G. Physical Chemistry (Academic Press, New York, 2000).

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