Binary phase diagrams

Transcription

1 inary phase diagrams inary phase diagrams and ibbs free energy curves inary solutions with unlimited solubility Relative proportion of phases (tie lines and the lever principle) Development of microstructure in isomorphous alloys inary eutectic systems (limited solid solubility) Solid state reactions (eutectoid, peritectoid reactions) inary systems with intermediate phases/compounds The iron-carbon system (steel and cast iron) ibbs phase rule Temperature dependence of solubility Three-component (ternary) phase diagrams Reading: Chapters of Porter and Easterling, Chapter 10 of askell

2 inary phase diagram and ibbs free energy 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 ibbs free energy dependence on temperature and composition. We have discussed the dependence of of a onecomponent system on T: T P S 2 2 T P S T P cp T T We have also discussed the dependence of the ibbs free energy from composition at a given T: X X ΔH mix TΔS mix 0 X 1

3 inary solutions with unlimited solubility Let s construct a binary phase diagram for the simplest case: and components are mutually soluble in any amounts in both solid (isomorphous system) and phases, and form ideal solutions. We have 2 phases and solid. Let s consider ibbs free energy curves for the two phases at different T T 1 is above the equilibrium melting temperatures of both pure components: T 1 >T m () > T m () the phase will be the stable phase for any composition. solid solid T 1 solid 0 X 1 id X X RT X lnx X lnx

4 inary solutions with unlimited solubility (II) Decreasing the temperature below T 1 will have two effects: and and solid Why? will increase more rapidly than The curvature of the (X ) curves will decrease. solid Why? Eventually we will reach T 2 melting point of pure component, where solid solid T 2 solid solid 0 X 1

5 inary solutions with unlimited solubility (III) For even lower temperature T 3 <T 2 =T m () the ibbs free energy curves for the and solid phases will cross. solid T 3 solid solid 0 solid solid + X1 2 X X 1 s we discussed before, the common tangent construction can be used to show that for compositions near cross-over of solid and, the total ibbs free energy can be minimized by separation into two phases.

6 inary solutions with unlimited solubility (IV) s temperature decreases below T 3 to increase more rapidly than solid solid and and continue Therefore, the intersection of the ibbs free energy curves, as well as points X 1 and X 2 are shifting to the right, until, at T 4 =T m () the curves will intersect at X 1 =X 2 =1 T 4 solid solid solid 0 X 1 t T 4 and below this temperature the ibbs free energy of the solid phase is lower than the of the phase in the whole range of compositions the solid phase is the only stable phase.

7 inary solutions with unlimited solubility (V) ased on the ibbs free energy curves we can now construct a phase diagram for a binary isomorphous systems solid T 3 solid solid s T 1 l solid solid + T 2 s T T 1 T 2 l T 3 l T 4 T 4 s 0 X 1 T 5

8 inary solutions with unlimited solubility (VI) 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). Liquid Liquidus line Solidus line Solid solution Liquidus line separates from + solid MSE Solidus 3050, line Phaseseparates Diagrams and solid Kinetics, from Leonid Zhigilei + solid

9 inary solutions with unlimited solubility (VII) 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 us lines. Solid and phases are at equilibrium in this temperature range. Temperature L Liquidus + L Solidus solution solution + crystallites of solid solution polycrystal solid solution Composition, wt %

10 Interpretation of Phase Diagrams For a given temperature and composition we can use phase diagram to determine: 1) The phases that are present 2) Compositions of the phases 3) 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 and solid phases have these compositions. X X solid X

11 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 twophase region is analogous to a lever balanced on a fulcrum. W W Derivation of the lever rule: 1) ll material must be in one phase or the other: W + W β = 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: W C +W C =C o 3) Solution of these equations gives us the Lever rule. W = (C 0 -C ) / (C -C ) and W = (C - C 0 ) / (C -C )

12 Composition/Concentration: weight fraction vs. molar fraction Composition can be expressed in Molar fraction, X,oratom percent (at%)thatisusefulwhen trying to understand the material at the atomic level. tom 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 in at. % is C at n m n m n m 100 Where n m and n m are numbers of moles of elements and 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 in wt. % is C at C wt X m 100 m m 100 where m and m system. n m m are the weights of the components in the n m m where and are atomic weights of elements and.

13 Composition Conversions Weight % to tomic %: C at C wt C wt C wt 100 C at C wt C wt C wt 100 tomic % to Weight %: C wt C at C C at at 100 C wt C at C C at at 100 Of course the lever rule canbeformulatedforanyspecification of composition: M L = (X -X 0 )/(X -X L ) = (C at - C at o) / (C at -C at L) M = (X 0 -X L )/(X -X L ) = (C at - C at L) / (C at -C at L) W L = (C wt - C wt o) / (C wt -C wt L) W = (C wt o - C wt L) / (C wt -C wt L)

14 Phase compositions and amounts. n example. C o = 35 wt. %, C L = 31.5 wt. %, C = 42.5 wt. % Mass fractions: W L = S / (R+S) = (C -C o ) / (C -C L ) = 0.68 W = R / (R+S) = (C o -C L ) / (C -C L ) = 0.32

15 Development of microstructure in isomorphous alloys Equilibrium (very slow) cooling

16 Development of microstructure in isomorphous alloys Equilibrium (very slow) cooling Solidification in the solid + phase occurs gradually upon cooling from the us line. The composition of the solid and the 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 at the solidus line.

17 Development of microstructure in isomorphous alloys Non-equilibrium cooling

18 Development of microstructure in isomorphous alloys Non-equilibrium cooling Compositional changes require diffusion in solid and 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 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 phase, where diffusion is fast. verage Ni content of solid grains is higher. pplication of the lever rule gives us a greater proportion of 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.

19 inary solutions with a miscibility gap Let s consider a system in which the phase is approximately ideal, but for the solid phase we have H mix >0 T 1 T2 T 1 solid solid 0 X 1 0 X 1 T3 T 2 T T 1 T 2 solid 0 X 1 T X 1 t low temperatures, there is a region where the solid solution is most stable as a mixture of two phases 1 and 2 with compositions X 1 and X 2. This region is called a miscibility gap.

20 T T 1 T X 1 0 X 1 0 X 1

21 reg X X ΩX X RT X lnx X lnx

22 Eutectic phase diagram ForanevenlargerH mix the miscibility gap can extend into the phase region. In this case we have eutectic phase diagram. T1 T2 T 1 solid solid 0 X 1 0 X 1 T3 T 2 T T 1 T 2 1 +l 2 +l solid T X 1 0 X 1

23 Eutectic phase diagram with different crystal structures of pure phases similar eutectic phase diagram can result if pure and have different crystal structures. T 1 T2 T 1 0 X 1 0 X 1 T3 T 2 T T 1 +l +l T 2 T X 1 0 X 1

24 Temperature dependence of solubility There is limited solid solubility of in and in in the alloy having eutectic phase diagram shown below. T 1 T +l +l T X 1 0 X 1 The closer is the minimum of the ibbs free energy curve (X ) to the axes X = 0, the smaller is the maximum possible concentration of in phase. Therefore, to discuss the temperature dependence of solubility let s find the minimum of (X ). reg d dx X X ΩX X 0 - Minimum of (X ) RT X lnx X lnx

25 Temperature dependence of solubility (II) reg X X ΩX X RT X lnx X lnx - X X 1- X X RT1- X ln1- X 1 X lnx d dx - 2Ω X RT- ln1- X 1- X 1- X lnx X X - 2Ω X RT -ln 1-X lnx - X 1-X 1 2X RTln - RTlnX 0 if X is small (X 0) ( >> ) Minimum of (X ) X exp RT Solid solubility of in increases exponentially with temperature (similar to vacancy concentration)

26 Temperature dependence of solubility (III) +l +l T X 1 X exp RT X X vs. T T T vs. X T e T X supersaturation with phase separation α and β precipitation of -rich intermediate phase or compound

27 Eutectic systems - alloys with limited solubility (I) Solidus Liquidus Temperature, C Solvus + Copper Silver phase diagram Composition, wt% g Three single phase regions ( - solid solution of g in Cu matrix, = solid solution of Cu in g matrix, L - ) Three two-phase regions ( + L, +L, +) Solvus line separates one solid solution from a mixture of solid solutions. Solvus line shows limit of solubility

28 Eutectic systems - alloys with limited solubility (II) Lead Tin phase diagram Temperature, C Invariant or eutectic point Eutectic isotherm Composition, wt% Sn Eutectic or invariant point - Liquid and two solid phases coexist in equilibrium at the eutectic composition C E and the eutectic temperature T E. Eutectic isotherm - the horizontal solidus line at T E. Eutectic reaction transition between and mixture of two solid phases, + at eutectic concentration C E. The melting point of the eutectic alloy is lower than that of the components (eutectic = easy to melt in reek).

29 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 Temperature, C C Composition, wt% Sn For points,, and C calculate the compositions (wt. %) and relative amounts (mass fractions) of phases present.

30 Temperature, C C Composition, wt% Sn

31 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 lead tin system as an example. Temperature, C 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 +L Composition, wt% Sn

32 Development of microstructure in eutectic alloys (II) t compositions between the room temperature solubility limit and the maximum solid solubility at the eutectic temperature, phase nucleates as the solid solubility is exceeded upon crossing the solvus line. L Temperature, C +L + Composition, wt% Sn

33 Development of microstructure in eutectic alloys (III) Solidification at the eutectic composition No changes above the eutectic temperature T E. t T E all the transforms to and phases (eutectic reaction). Temperature, C Composition, wt% Sn L +

34 Development of microstructure in eutectic alloys (IV) Solidification at the eutectic composition Compositions of and phases are very different eutectic reaction involves redistribution of Pb and Sn atoms by atomic diffusion (we will learn about diffusion in the last part of this course). This simultaneous formation of and 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 phase, the light layers are the tin-reach phase.

35 Development of microstructure in eutectic alloys (V) Compositions other than eutectic but within the range of the eutectic isotherm Primary phaseisformedinthe + L region, and the eutectic structure that includes layers of and phases (called eutectic and eutectic phases) is formed upon crossing the eutectic isotherm. L + L + Temperature, C Composition, wt% Sn

36 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 phase and the eutectic structure. lthough the eutectic structure consists of two phases, it is a microconstituent with distinct lamellar structure and fixed ratio of the two phases.

37 How to calculate relative amounts of microconstituents? Eutectic microconstituent forms from 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 phase (18.3 wt% Sn) and the eutectic structure (61.9 wt% Sn): W e =P/(P+Q) (eutectic) W =Q/(P+Q) (primary) Temperature, C Composition, wt% Sn

38 Temperature, C Composition, wt% Sn

39 How to calculate the total amount of phase (both eutectic and primary)? Fraction of phase determined by application of the lever rule across the entire + phase field: W = (Q+R) / (P+Q+R) ( phase) W =P/(P+Q+R) ( phase) Temperature, C Composition, wt% Sn

40 T 0 X 1

41 Monotectic system T + L 1 Limit of immiscibility Upper solidus Monotectic point L 1 L 1 + L 2 Lower us Lower solidus L 2 Monotectic isotherm 0 + L 2 + +L 2 X 1 F.N. Rhines, Phase Diagrams in Metallurgy, 1956

42 inary solutions with H mix < 0 - ordering If H mix <0bonding 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 H mix can become larger than X X and the ibbs free energy curve for the solid phase can become steeper than the one for. T 1 T2 T 1 α 0 X 1 0 α X 1 T3 T 2 T T 1 T 2 α 0 α` X 1 T 3 t low temperatures, strong attraction between unlike atoms can lead to the formation of ordered phase `. 0 ` X 1

43 inary solutions with H mix < 0 - intermediate phases If attraction between unlike atoms is very strong, the ordered phase may extend up to the. T X 1 In simple eutectic systems, discussed above, there are only two solid phases ( and 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 and.

44 T X 1

45 T X 1

46 H mix <0 - tendency to form high-melting point intermediate phase T T 0 X 1 Increasing negative H mix T 0 ` X X 1

47 Phase diagrams with intermediate phases: example Example of intermediate solid solution phases: in Cu-Zn, and are terminal solid solutions,, are intermediate solid solutions.

48 Phase diagrams for systems containing compounds For some systems, 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 intermetallic compound This diagram can be thought of as two joined eutectic diagrams, for Mg-Mg 2 Pb and Mg 2 Pb-Pb. In this case compound Mg 2 Pb (19 %wt Mg and 81 %wt Pb) can be considered as a component. sharp drop in the ibbs free energy at the compound composition should be added to ibbs free energy curves for the existing phases in the system.

49 Stoichiometric and non-stoichiometric compounds Compounds which have a single well-defined composition are called stoichiometric compounds (typically denoted by their chemical formula). Compound with composition that can vary over a finite range are called non-stoichiometric compounds or intermediate phase (typically denoted by reek letters). T T + l + l + l + +l + + l+ +l + 0 X 1 0 X 1 non-stoichiometric stoichiometric Common stoichiometric compounds: compound composition, at% compound composition, at%

50 Eutectoid Reactions The eutectoid (eutectic-like in reek) 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 ( + in the example below) Looks as V on top of a horizontal tie line (eutectoid isotherm) in the phase diagram. Cu-Zn Eutectoid

51 Eutectic and Eutectoid Reactions Temperature l l l Eutectic temperature Eutectoid temperature Eutectoid composition Eutectic composition Composition The above phase diagram contains both an eutectic reaction and its solid-state analog, an eutectoid reaction

52 Temperature l l l Eutectoid temperature Composition

53 Peritectic Reactions peritectic reaction - solid phase and phase will together form a second solid phase at a particular temperature and composition upon cooling - e.g. L + 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. Temperature Peritectoid is a three-phase reaction similar to peritectic but occurs from two solid phases to one new solid phase ( + = ).

54 Temperature

55 Example: The Iron Iron Carbide (Fe Fe 3 C) 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.

56 Phases in Fe Fe 3 C Phase Diagram -ferrite - solid solution of C in CC Fe Stable form of iron at room temperature. The maximum solubility of C is wt% Transforms to FCC -austenite at 912 C -austenite - solid solution of C in FCC Fe The maximum solubility of C is 2.14 wt %. Transforms to CC -ferrite at 1395 C Is not stable below the eutectic temperature (727 C) unless cooled rapidly -ferrite solid solution of C in CC Fe The same structure as -ferrite Stable only at high T, above 1394 C Melts at 1538 C Fe 3 C (iron carbide or cementite) This intermetallic compound is metastable, it remains as a compound indefinitely at room T, but decomposes (very slowly, within several years) into -Fe and C (graphite) at C Fe-C solution

57 C is an interstitial impurity in Fe. It forms a solid solution with phases of iron Maximum solubility in CC -ferrite is limited (max wt% at 727 C) - CC has relatively small interstitial positions Maximum solubility in FCC austenite is 2.14 wt% at 1147 C - FCC has larger interstitial positions Mechanical properties: Cementite is very hard and brittle - can strengthen steels. Mechanical properties of steel depend on the microstructure, that is, how ferrite and cementite are mixed. Magnetic properties: -ferrite is magnetic below 768 C, austenite is non-magnetic Classification. Three types of ferrous alloys: Iron: less than wt % C in ferrite at room T Steels: wt % C (usually < 1 wt % ) -ferrite + Fe 3 C at room T Examples of tool steel (machine tools for cutting other metals): Fe + 1wt % C + 2 wt% Cr Fe + 1 wt% C + 5 wt% W + 6 wt % Mo Stainless steel (food processing equipment, knives, petrochemical equipment, etc.): wt% Cr, ~$1500/ton Cast iron: wt % (usually < 4.5 wt %) heavy equipment casing

58 Eutectic and eutectoid reactions in Fe Fe 3 C Eutectic: 4.30 wt% C, 1147 C L + Fe 3 C Eutectoid: 0.76 wt%c, 727 C (0.76 wt% C) (0.022 wt% C) + Fe 3 C Eutectic and eutectoid reactions are very important in heat treatment of steels

59 Development of Microstructure in Iron - Carbon alloys Microstructure depends on composition (carbon content) and heat treatment. In the discussion below we consider slow cooling in which equilibrium is maintained. Microstructure of eutectoid steel (I)

60 Microstructure of eutectoid steel (II) When alloy of eutectoid composition (0.76 wt % C) is cooled slowly it forms perlite, a lamellar or layered structure of two phases: -ferrite and cementite (Fe 3 C) The layers of alternating phases in pearlite are formed for the same reason as layered structure of eutectic structures: redistribution C atoms between ferrite (0.022 wt%) and cementite (6.7 wt%) by atomic diffusion. Mechanically, pearlite has properties intermediate to soft, ductile ferrite and hard, brittle cementite. In the micrograph, the dark areas are Fe 3 C layers, the light phase is - ferrite

61 Microstructure of hypoeutectoid steel (I) Compositions to the left of eutectoid ( wt % C) hypoeutectoid (less than eutectoid -reek) alloys. + + Fe 3 C

62 Microstructure of hypoeutectoid steel (II) Hypoeutectoid alloys contain proeutectoid ferrite (formed above the eutectoid temperature) plus the eutectoid perlite that contain eutectoid ferrite and cementite.

63 Microstructure of hypereutectoid steel (I) Compositions to the right of eutectoid ( wt % C) hypereutectoid (more than eutectoid -reek) alloys. + Fe 3 C + Fe 3 C

64 Microstructure of hypereutectoid steel

65 Microstructure of hypereutectoid steel (II) Hypereutectoid alloys contain proeutectoid cementite (formed above the eutectoid temperature) plus perlite that contain eutectoid ferrite and cementite.

66 How to calculate the relative amounts of proeutectoid phase ( or Fe 3 C) and pearlite? pplication of the lever rule with tie line that extends from the eutectoid composition (0.75 wt% C) to ( +Fe 3 C) boundary (0.022 wt% C) for hypoeutectoid alloys and to ( +Fe 3 C) Fe 3 C boundary (6.7 wt% C) for hipereutectoid alloys. Fraction of phase is determined by application of the lever rule across the entire ( +Fe 3 C) phase field:

67 Example for hypereutectoid alloy with composition C 1 Fraction of pearlite: W P = X / (V+X) = (6.7 C 1 ) / ( ) Fraction of proeutectoid cementite: W Fe3C = V / (V+X) = (C ) / ( )

68 The most important phase diagram in the history of civilization pearlite naturally formed composite: hard & brittle ceramic (Fe 3 C) + soft CC iron (ferrite)

69 Reconstruction of aroque organs (TRUESOUND EU project) aroque-style organ reconstructed in Örgryte Nya Kyrka, othenburg, Sweden Components of reed pipes MRS ulletin Vol 32, March 2007,

70 Reconstruction of aroque organs (TRUESOUND EU project) nondestructive measurements of composition on ~30 organs produced from 1624 to 1880 (neutron, X-ray diffraction, SIMS) Zn is soluble in Cu Pb is insoluble in Cu and is distributed non-uniformly in the alloy Cu Zn phase diagram Cu Pb phase diagram

71 Reconstruction of aroque organs (TRUESOUND EU project) Surprisingly constant Zn concentration: ~26 wt.% from 1624 to 1790 ~32.5 wt.% from 1750 ll historical tongues and shallots are made of α-brass (fcc solid solution of Zn in Cu)

72 Reconstruction of aroque organs (TRUESOUND EU project) efore 1738 Zn was only available as zinc oxide (ZnO) or zinc carbonate (ZnCO 3 ) and brass was produced by mixing solid Cu and gas-phase Zn in a process called cementation: pieces of Cu, coal, and ZnCO 3 are heated to above T b of Zn (907ºC) but below T m of Cu (1083ºC) ZnCO 3 is reduced and evaporated Zn diffuses into hot solid Cu Heating much above 907ºC was not economical temperature was kept slightly above this temperature experienced craftsman can estimate temperature from the color of a furnace within ~20ºC the process was taking place at ~920ºC Zn concentration in brass is determined by solidus concentration maximum solubility of Zn in solid Cu (26 wt.% at 920ºC) 1738: patent on distilling metallic Zn from ZnCO 3, 1781: patent on alloying brass from metallic Zn concentration of Zn in Cu is no longer limited by the solubility MSE 3050, at Phase 920ºC, Diagrams but is 32.5 and Kinetics, wt.% - maximum Leonid Zhigilei solubility after crystallization.

73 The ibbs phase rule (I) Let s consider a simple one-component system. In the areas where only one phase is stable both pressure and temperature can be independently varied without upsetting the phase equilibrium there are 2 degrees of freedom. P solid gas T long the lines where two phases coexist in equilibrium, only one variable can be independently varied without upsetting the two-phase equilibrium (P and T are related by the Clapeyron equation) there is only one degree of freedom. t the triple point, where solid and vapor coexist any change in P or T would upset the three-phase equilibrium there are no degrees of freedom. In general, the number of degrees of freedom, F, in a system that contains C components and can have Ph phases is given by the ibbs phase rule: F C Ph 2

74 The ibbs phase rule (II) Let s now consider a multi-component system containing C components and having Ph phases. thermodynamic state of each phase can be described by pressure P, temperature T, and C - 1 composition variables. The state of the system can be then described by Ph(C-1+2) variables. ut how many of them are independent? The condition for Ph phases to be at equilibrium are: T =T =T = - Ph-1 equations P =P =P = - Ph-1 equations μ μ α α μ μ β β μ μ γ γ Ph-1 equations - Ph-1 equations C sets of equations Therefore we have (Ph 1)(C + 2) equations that connect the variables in the system. The number of degrees of freedom is the difference between the total number of variables in the system and the minimum number of equations among these variables that have to be satisfied in order to maintain the equilibrium. F Ph(C 1) (Ph 1)(C 2) F C Ph 2 - ibbs phase rule C Ph 2

75 The ibbs phase rule example (an eutectic systems) F C Ph P const 2 F C Ph 1 C 2 F 3 Ph In one-phase regions of the phase diagram T and X can be changed independently. In two-phase regions, F = 1. If the temperature is chosen independently, the compositions of both phases are fixed. Three phases (L,, ) are in equilibrium only at the eutectic point in this two component system (F = 0).

76 Multicomponent systems (I) The approach used for analysis of binary systems can be extended to multi-component systems. wt. % wt. % wt. % Representation of the composition in a ternary system (the ibbs triangle). The total length of the red lines is 100% : X +X +X C =1

77 Multicomponent systems (II) The ibbs free energy surfaces (instead of curves for a binary system) can be plotted for all the possible phases and for different temperatures. The chemical potentials of,, and C of any phase in this system are given by the points where the tangential plane to the free energy surfaces intersects the,, and C axis.

78 Multicomponent systems (III) three-phase equilibrium in the ternary system for a given temperature can be derived by means of the tangential plane construction. γ α β For two phases to be in equilibrium, the chemical potentials should be equal, that is the compositions of the two phases in equilibrium must be given by points connected by a common tangential plane (e.g. l and m). The relative amounts of phases are given by the lever rule (e.g. using tie-line l-m). three phase triangle can result from a common tangential plane simultaneously touching the ibbs free energies of three phases (e.g. points x, y, and z). Eutectic point: four-phase equilibrium between,,, and

79 n example of ternary system The ternary diagram of Ni-Cr-Fe. It includes Stainless Steel (wt.% of Cr > 11.5 %, wt.% of Fe > 50 %) and Inconel tm (Nickel based super alloys). Inconel have very good corrosion resistance, but are more expensive and therefore used in corrosive environments where Stainless Steels are not sufficient (Piping on Nuclear Reactors or Steam enerators).

80 nother example of ternary phase diagram: oil water surfactant system Drawing by Carlos Co, University of Cincinnati Surfactants are surface-active molecules that can form interfaces between immiscible fluids (such as oil and water). large number of structurally different phases can be formed, such as droplet, rod-like, and bicontinuous microemulsions, along with hexagonal, lamellar, and cubic crystalline phases.

81 Make sure you understand language and concepts: Summary Common tangent construction Separation into 2 phases Eutectic structure Composition of phases Weight and atom percent Miscibility gap Solubility dependence on T Intermediate solid solution Compound Isomorphous Tie line, Lever rule Liquidus & Solidus lines Microconstituent Primary phase Solvus line, Solubility limit ustenite, Cementite, Ferrite Pearlite Hypereutectoid alloy Hypoeutectoid alloy Ternary alloys ibbs phase rule Elements of phase diagrams: Eutectic (L + ) L L Eutectoid ( + ) Peritectic ( +L) + L Peritectoid ( + ) Compound, n m useful link: