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Physical Metallurgy Questions Chapter 1

Question 1

The figure displays the phase diagram of Al-Zn. The fraction of Zn is given by x.

a) Draw (schematically) the relevant free-energy curves G(x) at 277°C. b) Draw (schematically) the relevant free-energy curves G(x) at 370°C. c) Draw (schematically) the relevant free-energy curves G(x) at 400°C. d) Consider the hypothetical case that the G(x)-curve for (Zn)-phase is shifted along

the G-axis in such a way that the melting point of pure Zn would be 370°C. Sketch for that case the part of the phase diagram for 330°C < T < 430°C and 0.80 < x < 1.

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e) As can be seen on the Al-rich side of the phase diagram, below 277°C the

solubility of Zn in Al, xs, gradually increases with increasing temperature. Also above 277°C this increase persists, but the rate of increase, dxs/dT, changes abruptly at 277°C. Why does this discontinuous change of slope occur?

Question 1 a)

b)

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c)

d) The boundaries of the FCC-L two-phase region will be extrapolated towards lower

temperatures; the L-HCP two-phase region will be shifted by 50°C. This will lead to a eutectic at a composition of about 91% Zn and a temperature of about 350°C.

e) Below 277°C the solubility is determined by the FCC- and HCP-phase, above this temperature by the two minima in the free-energy curve for the FCC-phase.

Question 2 a) When a piece of steel is left to cool down naturally from a temperature in the austenitic

region, a temperature increase is observed due to the release of so-called latent heat when the phase transformation to ferrite takes place. Why is energy released during this transformation?

b) Does the effect mentioned in question a) always occur during transformation from a high-

temperature phase to a low-temperature phase, or can it also be reversed (i.e. energy absorption instead of release)?

Question 2 a) At the transition temperature T0 the free energies of the two phases are equal. Since

austenite is stable at higher temperatures, it has a higher entropy than ferrite. This means that the enthalpy of austenite is also higher, since G = H − TS is equal at T0. Therefore, energy is released when austenite transforms into ferrite.

b) Because of the entropy difference, a transformation from a high-temperature phase into a low-temperature phase always gives a heat release.

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Question 3 The picture below shows the phase diagram for the binary Au-Pt system.

a) Sketch, as accurately as possible, the relevant free-energy curves G(xPt) at a temperature

of 1270°C. b) Do the same at T = 600°C. c) Give the fractions and compositions of the phases in the microstructure if an alloy with 40

at.% Pt is annealed for a long time at T = 600°C. Question 3 a) At T = 1270°C, both the solid phase (FCC) and the liquid phase have a free-energy curve

with a single minimum (no miscibility gap). At this temperature these two curves have a common tangent that gives the equilibrium values of approximately 15% Pt for the liquid and 41% Pt for the solid.

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b) At T = 600°C, the liquid phase is not relevant anymore, and the solid phase has a free-energy curve with a miscibility gap, with a lowest tangent giving the equilibrium compositions 19% Pt and 98% Pt.

c) The compositions of the Pt-rich and Pt-poor solid phase are 98% and 19% Pt, and the fractions 0.26 and 0.74.

Question 4

Consider the binary phase diagram for Cr-Ni alloys, given above. The three phases involved are the liquid phase (indicated as L), the FCC phase (indicated as (Ni)) and the BCC phase (indicated as (Cr)).

a. Sketch the three curves of the free energy G as a function of xNi (the atomic fraction of Ni), indicating the essential details, for T = 1600°C.

b. Do the same for T = 1100°C. c. Give a plot of the (estimated) equilibrium fractions of the three phases for an alloy of

composition Cr-20%Ni as a function of decreasing temperature, starting at 1900ºC.

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d. Why is continuous mixing of Ni and Cr possible in the liquid state (for T > 1895ºC), but not in the solid state?

Question 4c

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Question 5

Consider the binary phase diagram for Cr-Nb alloys, given above.

a. Sketch the relevant curves of the free energy G as a function of xNb, the atomic fraction of Nb, indicating the essential details, at T = 1400°C.

b. Do the same at T = 1650°C. c. Is the value of ∆Hmix for the liquid state positive or negative? d. What is the meaning of the “extra” line at 1580°C (in the (Cr)+NbCr2-region) and

1620°C (in the NbCr2+(Nb)-region), crossing the NbCr2 phase region? e. Why is this line at a constant temperature in both the two-phase regions?

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Question 5

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a&b. Essential are the common tangents. At T = 1650°C, there is a eutecticum, so the three

phases L, BCC and NbCr2 have one common tangent. c. This cannot be said with certainty. The fact that the intermetallic phase NbCr2 forms is

an indication that ∆Hmix is negative, but the phase equilibriums that are seen in the phase diagram van exist in the same manner if ∆Hmix > 0.

d. The ordering/disordering transition for the phase NbCr2. e. Because the NbCr2-phase has a constant equilibrium composition throughout each of

the two-phase regions.

Question 6 a) If you keep your wet clothes on after cycling through the rain without a rain suit, you get

cold when your clothes dry (i.e. when the water evaporates from your clothes). Explain why, considering the enthalpy, the entropy and the free energy involved.

b) During natural cooling of steel from the high-temperature austenitic phase region, often a

temperature increase is observed when the austenite transforms into ferrite or pearlite. See for instance the picture below, or the CCT-diagrams on page 348 of the book of Porter & Easterling. Explain why this phenomenon (called recalescence) occurs, considering the enthalpy, the entropy and the free energy involved.

Question 6 a) When water evaporates, it assumes a state of higher entropy. Since the process will take

place close to the equilibrium conditions, the free energies of the two phases will be equal, and therefore the enthalpy of the gas is higher than that of the water. The process thus absorbs energy (is endothermic), which it extracts from your body.

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b) Recalescence is due to the same principle as described in question a), but now the low-temperature phase, with the lower entropy, is formed and therefore energy is released.

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Question 7 Consider the Co-Cu phase diagram given below. The crystal structure of both pure elements in the solid phase is FCC.

a) Is the enthalpy of mixing for Cu-Co in the solid state positive or negative? Why? b) Sketch, as accurately as possible, the relevant curves for free-energy G as a function of the

copper concentration xCu at T = 1085ºC. c) A sample is made of the binary alloy Co-Cu, containing 50 at.% of each of the elements.

Give the phases that this sample will contain after a long annealing treatment at T = 1085ºC (long enough to reach equilibrium) and give quantitative values for the composition and volume fraction of these phases.

Question 7 a) The enthalpy of mixing is positive, since a two-phase region exists: the equilibrium state

consists of a Cu-rich and a Co-rich phase.

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b) Essential points: (i) there is only one solid-state curve, namely the one for FCC, displaying the two-minima shape shown in figure 1.15d; (ii) the solid-state curve and the liquid curve have the same value for pure Cu.

c) The Co-rich phase contains 11.8% Cu and has a volume fraction of 53.2%, the Cu-rich

phase contains 93.5%Cu and has a volume fraction of 46.8%.

Question 8 Below, the iron-carbon phase diagram is given, the basis for the production of steel. Five phases appear in this diagram: α (BCC), γ (FCC), δ (BCC), L (liquid), θ (cementite, Fe3C). a) Sketch the curves of the free energy as a function of composition for all phases at the

temperature of 800ºC. b) Should α and δ be considered as two different phases? Why (not)?

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c) Consider an Fe-C alloy with 0.3 wt.% carbon, which is initially annealed at 800ºC. Give the equilibrium compo-sitions of both phases.

d) Below 727ºC the equilibrium state

consists of the α-phase in equilibrium with cementite. The formation of cementite can however be quite slow. Describe quantitatively (with estimated numbers for the phase compositions and phase fractions) the quasi-equilibrium that will form when the material (question c)) is cooled from 800ºC to 700ºC in the case that the cementite does not form.

Question 8 a) Points to consider: (i) α and δ have the same G-curve, namely the one for the bcc-phase;

(ii) the G-curve for Fe3C is very sharp; (iii) α and γ have a common tangent, as well as γ and θ, in between γ is stable.

b) The two phases both have the bcc-structure, so are actually the same phase. c) The equilibrium composition for the α-phase is approximately 0.013 wt.%, for the γ-phase

it is 0.4 wt.%. d) Extend the α+γ-line below T = 727ºC. The equilibrium composition for the α-phase is

then 0.025 wt.%, for the γ-phase approximately 0.9 wt.%. The phase fractions are then 0.31 for γ and 0.69 for α.

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Question 9

The above figure gives the free-energy curves for an AB-alloy for two phases (a solid phase α and the liquid phase L) for two different temperatures: the dashed lines are for the temperature T1, the solid lines for T2. a) On the basis of these four curves, which temperature will be the highest, T1 or T2? Why? b) Is it also possible to derive from just the two α-curves which temperature is the highest?

How/why?

c) Sketch the phase diagram of the AB-alloy as realistically and complete as possible. d) Can the ideal-solution model describe or predict a phase diagram of this type? Explain. Question 9

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a) The temperature T2 is the highest, because (i) the liquid phase (the one having the highest entropy) has lowered its free-energy curve relative to the α-curve, (ii) the double minimum in the α-curve can be present only at lower temperatures.

b) Yes, see reason (ii) in question a). c) The phase diagram will be of the type shown below, with the melting temperature of A

being above T2 and the melting temperature of B below. On the basis of the given curves, it cannot be decided if there is a eutectic in the phase diagram; it may be that the miscibility gap in the α-phase disappears at a lower temperature.

d) No, the ideal-solution model assumes ∆Hmix = 0, and therefore the miscibility gap cannot

be reproduced.

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Question 10

The above figure gives a part of the ternary phase diagram for Al-Ti-Ni at 400ºC. a) What are the colours of one-phase regions, of two-phase regions and of three-phase

regions? b) There are no four-phase regions in this graph. Are they physically possible for this

system? Explain why (not). c) Consider the alloy Al-25%Ti-25%Ni. What are the equilibrium phases and their

compositions for this alloy at the given temperature? d) Sketch, as detailed as possible, the relevant free-energy curves for the binary Al-Ni alloy

as a function of Ni-content. e) Sketch, as detailed as possible, the relevant free-energy curves for the binary Al-Ti alloy

as a function of Ti-content. Question 10 a) One-phase regions are blue, two-phase regions white and three-phase regions yellow. b) According to Gibbs phase rule for ambient pressure, four phases can only be in

equilibrium at a fixed temperature and fixed compositions, so no regions but a point. c) The composition is in the three-phase region Ni2Al3, Ti6Ni7Al17 and TiNi0.25Al2.75. The

exact compositions are given by the corner points of the triangle, slightly deviating from the stoichiometric compositions.

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d)

e)

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Question 11

The above figure gives the binary phase diagram for Au-V. The crystal structure of pure gold is FCC, that of pure vanadium is BCC. a) Sketch, as detailed as possible, the relevant free-energy curves for T = 1385ºC. b) Sketch, as detailed as possible, the relevant free-energy curves for T = 1280ºC for the

composition range with more than 50% V. c) By choosing the appropriate conditions, the formation of the three intermetallic phases

Au4V, Au2V and AuV3 can be suppressed. Sketch the phase diagram in the absence of these three phases.

Question 11 a)

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b)

c)

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Question 12

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a) The figure shows a measurement of the temperature of a steel sample during “natural cooling”, which means that no forced cooling rate is imposed; the material is left in air to cool down by itself. What happens in the material around 700ºC, where an anomalous effect in the temperature is seen?

b) Explain why a temperature increase takes place during the event of question a), based on

the thermodynamics of the phenomenon. c) Suppose that the same material is heated up by providing a constant amount of heat per

unit of time, intended to heat the material from room temperature to 1000ºC in 150 s. Sketch the temperature as a function of time during heating.

Question 12

a) The phenomenon is called recalescence and takes place as a result of the austenite-to-ferrite transformation.

b) The high-temperature phase austenite has a higher entropy than ferrite (which is why

austenite is stable at high temperature). Therefore around the equilibrium temperature also the enthalpy of austenite is higher than that of ferrite. This enthalpy difference is released as heat during the phase transformation and causes the temperature increase.

c) During heating the same phenomenon causes a decrease in temperature.