Binary phase diagrams and Gibbs free energy curves Binary solutions with unlimited solubility Relative proportion of phases (tie lines and the lever principle) Development of microstructure in isomorphous alloys Binary eutectic systems (limited solid solubility) Solid state reactions (eutectoid, peritectoid reactions) Binary systems with intermediate phases/compounds The iron-carbon system (steel and cast iron) Gibbs phase rule Temperature dependence of solubility Three-component (ternary) phase diagrams Reading: Chapters 1.5.1 – 1.5.7 of Porter and Easterling, Chapter 10 of Gaskell
Binary phase diagram and Gibbs free energy A binary phase diagram is a temperature - composition map which indicates the equilibrium phases present at a given temperature and composition. The equilibrium state can be found from the Gibbs free energy dependence on temperature and composition. We have discussed the dependence of G of a one component system on T:
We have also discussed the dependence of the Gibbs free energy from composition at a given T: G =XAGA + XBGB + Hmix -TSmix
Binary solutions with unlimited solubility Let’s construct a binary phase diagram for the simplest case: A and B components are mutually soluble in any amounts in both solid (isomorphoussystem) and liquid phases, and form ideal solutions. We have 2 phases – liquid and solid. Let’s consider Gibbs free energy curves for the two phases at different T T1 is above the equilibrium melting temperatures of both pure components: T1 > Tm(A) > Tm(B) → the liquid phase will be the stable phase for any composition.
Decreasing the temperature below T1 will have two effects: • GAliquid dan GBliquid will decrease more rapidly than GAsolid and GBsolid . WHY • The curvature of the G(XB) curves will decrease. WHY ?
Eventually we will reach T2 – the melting point pure component A, where GAliquid = GAsolid
For even lower temperature T3 < T2 = Tm(A) the Gibbs free energy curves for the liquid and solid phases will cross. As we discussed before, the common tangent construction can be used to show that for compositions near cross-over of Gsolid and Gliquid, the total Gibbs free energy can be minimized by separation into two phases.
As temperature decreases below T3, GAliquid and GBliquid continue to increase more rapidly than GAsolid and GBsolid Therefore, the intersection of the Gibbs free energy curves, as well as points X1 and X2 are shifting to the right, until, at T4 = Tm(B) the curves will intersect at X1 = X2 = 1 At T4 and below this temperature the Gibbs free energy of the solid phase is lower than the G of the liquid phase in the whole range of compositions –the solid phase is the only stable phase.
Based on the Gibbs free energy curves we can now construct a phase diagram for a binary isomorphous systems
Example of isomorphous system: Cu-Ni (the complete solubility occurs because both Cu and Ni have the same crystal structure, FCC, similar radii, electronegativity and valence). Liquidusline separates liquid from liquid + solid Solidusline separates solid from liquid + solid
In one-component system melting occurs at a well-defined melting temperature. In multi-component systems melting occurs over the range of temperatures, between the solidus and liquidus lines. Solid and liquid phases are in equilibrium in this temperature range.
Interpretation of Phase Diagrams For a given temperature and composition we can use phase diagram to determine: • The phases that are present • Compositions of the phases • The relative fractions of the phases Finding the composition in a two phase region: 1. Locate composition and temperature in diagram 2. In two phase region draw the tie line or isotherm 3. Note intersection with phase boundaries. Read compositions at the intersections. The liquid and solid phases have these compositions.
Interpretation of Phase Diagrams: the Lever Rule Finding the amounts of phases in a two phase region: 1. Locate composition and temperature in diagram 2. In two phase region draw the tie line or isotherm 3. Fraction of a phase is determined by taking the length of the tie line to the phase boundary for the other phase, and dividing by the total length of tie line The lever rule is a mechanical analogy to the mass balance calculation. The tie line in the two phase region is analogous to a lever balanced on a fulcrum.
Derivation of the lever rule: • All material must be in one phase or the other: Wα + WL = 1 • 2) Mass of a component that is present in both phases equal to the mass of the component in one phase + mass of the component in the second phase: WαCα + WβCβ = Co • 3) Solution of these equations gives us the Lever rule. Wβ = (C0 - Cα) / (Cβ- Cα) and Wα = (Cβ- C0) / (Cβ - Cα)
Composition/Concentration: weight fraction vs. molar fraction Composition can be expressed in Molar fraction, XB, or atom percent (at%) that is useful when trying to understand the material at the atomic level. Atom percent (at %) is a number of moles (atoms) of a particular element relative to the total number of moles (atoms) in alloy. For two -component system, concentration of element B in at. % is Where nmA and nmB are numbers of moles of elements A and B in the system.
Weight percent(C, wt %) that is useful when making the solution. Weight percent is the weight of a particular component relative to the total alloy weight. For two component system, concentration of element B in wt. % is where mA and mB are the weights of the components in the system. where AA and AB are atomic weights of elements A and B.
Composition Conversions Weight % to Atomic %: Atomic % to Weight %:
Of course the lever rulecan be formulated for any specification of composition: ML = (XBα - XB0)/(XBα - XBL) = (Catα - Cato) / (Catα- CatL) Mα = (XB0 - XBL)/(XBα - XBL) = (Cat0 - CatL) / (Catα- CatL) WL = (Cwtα - Cwto) / (Cwtα- CwtL) WL = (Cwto - CwtL) / (Cwtα- CwtL)
Phase compositions and amounts. An example. Co = 35 wt. %, CL = 31.5 wt. %, Cα = 42.5 wt. % Mass fractions: WL = S / (R+S) = (Cα - Co) / (Cα- CL) = 0.68 Wα = R / (R+S) = (Co - CL) / (Cα- CL) = 0.32
Development of microstructure in isomorphous alloys Equilibrium (very slow) cooling
Development of microstructure in isomorphous alloys Equilibrium (very slow) cooling • Solidification in the solid + liquid phase occurs gradually upon cooling from the liquidus line. • The composition of the solid and the liquid change gradually during cooling (as can be determined by the tie-line method.) • Nuclei of the solid phase form and they grow to consume all the liquid at the solidus line.
Development of microstructure in isomorphous alloys Non-equilibrium cooling
Development of microstructure in isomorphous alloys Non-equilibrium cooling • Compositional changes require diffusion in solid and liquid phases • Diffusion in the solid state is very slow. The new layers that solidify on top of the existing grains have the equilibrium composition at that temperature but once they are solid their composition does not change. Formation of layered (cored) grains and the invalidity of the tie-line method to determine the composition of the solid phase. • The tie-line method still works for the liquid phase, where diffusion is fast. Average Ni content of solid grains is higher. Application of the lever rule gives us a greater proportion of liquid phase as compared to the one for equilibrium cooling at the same T. Solidus line is shifted to the right (higher Ni contents), solidification is complete at lower T, the outer part of the grains are richer in the low-melting component (Cu). • Upon heating grain boundaries will melt first. This can lead to premature mechanical failure.
Binary solutions with a miscibility gap Let’s consider a system in which the liquid phase is approximately ideal, but for the solid phase we have ∆Hmix > 0 ( A and B atoms dislike each other) At low temperatures, there is a region where the solid solution is most stable as a mixture of two phases α1 and α2 with compositions X1 and X2. This region is called a miscibility gap.
Eutectic phase diagram For an even larger ∆Hmix the miscibility gap can extend into the liquid phase region. In this case we have eutectic phase diagram.
Eutectic phase diagram with different crystal structures of pure phases A similar eutectic phase diagram can result if pure A and B have different crystal structures.
Eutectic systems - alloys with limited solubility (I) Three single phase regions (α - solid solution of Ag in Cu matrix, β = solid solution of Cu in Ag matrix, L - liquid) Three two-phase regions (α + L, β +L, α +β) Solvus line separates one solid solution from a mixture of solid solutions. Solvus line shows limit of solubility
Eutectic systems - alloys with limited solubility (II) Eutectic or invariant point- Liquid and two solid phases coexist in equilibrium at the eutectic composition CE and the eutectic temperature TE. Eutectic isotherm - the horizontal solidus line at TE. Eutectic reaction – transition between liquid and mixture of two solid phases, α + β at eutectic concentration CE. The melting point of the eutectic alloy is lower than that of the components (eutectic = easy to melt in Greek).
Eutectic systems - alloys with limited solubility (III) Compositions and relative amounts of phases are determined from the same tie lines and lever rule, as for isomorphous alloys For points A, B, and C calculate the compositions (wt. %) and relative amounts (mass fractions) of phases present.
Development of microstructure in eutectic alloys (I) Several different types of microstructure can be formed in slow cooling an different compositions. Let’s consider cooling of liquid lead – tin system as an example. In the case of lead-rich alloy (0-2 wt. % of tin) solidification proceeds in the same manner as for isomorphous alloys (e.g. Cu-Ni) that we discussed earlier. L → α+L → α
Development of microstructure in eutectic alloys (II) At compositions between the room temperature solubility limit and the maximum solid solubility at the eutectic temperature, β phase nucleates as the α solid solubility is exceeded upon crossing the solvus line.
Development of microstructure in eutectic alloys (III) Solidification at the eutectic composition No changes above the eutectic temperature TE. At TE all the liquid transforms to α and β phases (eutectic reaction).
Development of microstructure in eutectic alloys (IV) Solidification at the eutectic composition Compositions of α and β phases are very different → eutectic reaction involves redistribution of Pb and Sn atoms by atomic diffusion This simultaneous formation of α and β phases result in a layered (lamellar) microstructure that is called eutectic structure. Formation of the eutectic structure in the lead-tin system. In the micrograph, the dark layers are lead reach α phase, the light layers are the tin-reach β phase.
Development of microstructure in eutectic alloys (V) Compositions other than eutectic but within the range of the eutectic isotherm Primary α phase is formed in the α + L region, and the eutectic structure that includes layers of α and β phases (called eutectic α and eutectic β phases) is formed upon crossing the eutectic isotherm.
Development of microstructure in eutectic alloys (VI) Microconstituent – element of the microstructure having a distinctive structure. In the case described in the previous page, microstructure consists of two microconstituents, primary α phase and the eutectic structure. Although the eutectic structure consists of two phases, it is a microconstituent with distinct lamellar structure and fixed ratioof the two phases.
How to calculate relative amounts of microconstituents? Eutectic microconstituent forms from liquid having eutectic composition (61.9 wt% Sn) We can treat the eutectic as a separate phase and apply the lever rule to find the relative fractions of primary α phase (18.3 wt% Sn) and the eutectic structure (61.9 wt% Sn): We = P / (P+Q) (eutectic) Wα’ = Q / (P+Q) (primary)
How to calculate the total amount of α phase (both eutectic and primary)? Fraction of α phase determined by application of the lever rule across the entire α + β phase field: Wα= (Q+R) / (P+Q+R) (α phase) Wβ= P / (P+Q+R) (β phase)
Binary solutions with ∆Hmix < 0 - ordering If ∆Hmix < 0 bonding becomes stronger upon mixing → melting point of the mixture will be higher than the ones of the pure components. For the solid phase strong interaction between unlike atoms can lead to (partial) ordering → |∆Hmix| canbecome larger than |ΩXAXB| and the Gibbs free energy curve for the solid phase can become steeper than the one for liquid. At low temperatures, strong attraction between unlike atoms can lead to the formation ofordered phase α`.
Binary solutions with ∆Hmix < 0 - intermediate phases If attraction between unlike atoms is very strong, the ordered phase may extend up to the liquid. In simple eutectic systems, discussed above, there are only two solid phases (α and β) that exist near the ends of phase diagrams. Phases that are separated from the composition extremes (0% and 100%) are called intermediate phases. They can have crystal structure different from structures of components A and B.
Phase diagrams with intermediate phases: example Example of intermediate solid solution phases: in Cu-Zn, α and η are terminal solid solutions, β, β’, γ, δ, ε are intermediate solid solutions.
lPhase diagrams for systems containing compounds For some sytems, instead of an intermediate phase, an intermetallic compound of specific composition forms. Compound is represented on the phase diagram as a vertical line, since the composition is a specific value. When using the lever rule, compound is treated like any other phase, except they appear not as a wide region but as a vertical line
This diagram can be thought of as two joined eutectic diagrams, for Mg-Mg2Pb and Mg2Pb-Pb. In this case compound Mg2Pb(19%wt Mg and 81%wt Pb) can be considered as a component. A sharp drop in the Gibbs free energy at the compound composition should be added to Gibbs free energy curves for the existing phases in the system
Eutectoid Reactions The eutectoid (eutectic-like in Greek) reaction is similar to the eutectic reaction but occurs from one solid phase to two new solid phases. Eutectoid structures are similar to eutectic structures but are much finer in scale (diffusion is much slower in the solid state). Upon cooling, a solid phase transforms into two other solid phases (δ ↔ γ + ε in the example below) Looks as V on top of a horizontal tie line (eutectoid isotherm) in the phase diagram.
Eutectic and Eutectoid Reactions The above phase diagram contains both an eutectic reaction and its solid-state analog, an eutectoid reaction
Peritectic Reactions A peritectic reaction - solid phase and liquid phase will together form a second solid phase at a particular temperature and composition upon cooling - e.g. L + α ↔ β These reactions are rather slow as the product phase will form at the boundary between the two reacting phases thus separating them, and slowing down any further reaction. Peritectoid is a three-phase reaction similar to peritectic but occurs from two solid phases to one new solid phase (α + β = γ).
Example: The Iron–Iron Carbide (Fe–Fe3C) Phase Diagram In their simplest form, steels are alloys of Iron (Fe) and Carbon (C). The Fe-C phase diagram is a fairly complex one, but we will only consider the steel part of the diagram, up to around 7% Carbon.