Physical Metallurgy Friday, January 28, 2011; 8:30 12:00 h

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1 Physical Metallurgy Friday, January 28, 2011; 8:30 12:00 h Always motivate your answers All sub-questions have equal weight in the assessment Question 1 Precipitation-hardening aluminium alloys are, after homogenisation at typically 550ºC, annealed to form precipitates in the structure. The diffusivity of Cu in Al at a temperature T is given by exp QD D = D0 RT, with D 0 = m 2 /s, Q D = 150 kj/mol and the gas constant R = J/molK. Consider an alloy of aluminium with 2% Cu. a) During annealing at 250 C the θ -phase forms, with a copper concentration van After annealing during a time t the precipitates can be seen as spheres with a radius r. In a schematic picture each precipitate is located in the middle of a sphere with radius R (R > r), in which there is no other precipitate. Give the value of R when it is given that r = 8 nm. b) Give an estimate of the time required for the necessary diffusion to take place for the formation of these precipitates. c) When measuring the strength of the material as a function of the annealing time a maximum is observed. Explain what happens in the microstructure during the increase of strength as a function of time and during the decrease of strength as a function of time. d) At an annealing temperature of 250 C the θ -phase is the first phase to form, while at a temperature of 200ºC the θ -phase forms first. What is the physical origin of this difference? Question 2 In modern steel grades, very often martensite is combined with retained austenite. Martensite (α ) forms when austenite is cooled rapidly in a martensitic transformation in which the forming martensite, which essentially is the BCC-phase, has the same composition as the austenite it is formed from. A certain fraction of the high-temperature phase austenite (γ, FCC) can be stabilised at room temperature ( retained austenite) if its carbon content x C is at least 1.0 wt.%. In order to achieve that, the formation of cementite (Fe 3 C) is suppressed in these materials. The binary Fe-C phase diagram is given below, and can be used as the relevant phase diagram for these materials. At 500ºC, the diffusivity of carbon in martensite is D α = m 2 /s and in austenite D γ = m 2 /s. a) Sketch the phase diagram for 500ºC < T < 950ºC and 0 < x C < 3.5 wt.% in the situation that cementite is prevented from forming. b) Sketch the free-energy curves for the BCC-phase (G α (x C )) and for the FCC-phase (G γ (x C )) at T = 500ºC for the same composition range as in question a). c) Consider a microstructure consisting of alternating platelets of austenite and martensite, both with a carbon content of 1.1 wt.%. Explain why carbon will diffuse from martensite to austenite when the material is annealed at 500ºC. page 1 of 5

2 d) The co-ordinate perpendicular to the platelets is denoted as z. At the α /γ-interface, the carbon redistributes quickly over martensite and austenite, locally establishing equilibrium. Assuming that the equilibrium concentrations are retained at the interface, give the carbon concentration profile as a function of z on both sides of the interface after 10 s annealing at 500ºC. e) The α /γ-microstructure exists in the absence of cementite. Describe the development of the microstructure and the carbon distribution when, starting from the condition sketched in question c), cementite does form at the α /γ-interface at 500ºC. Question 3 In a binary AB-alloy the elements A and B have diffusivities D given by exp QD D = D0 RT, with D 0 the pre-exponential factor, Q D the activation energy for diffusion, R the gas constant ( J/molK) and T the temperature. The table below gives values for these parameters for the elements A and B, of which the element B is interstitially dissolved in the crystal lattice formed by A. D 0 [m 2 /s] Q D [kj/mol] A B a) How do these values reflect the fact that B is an interstitial element? b) In a binary alloy with different values for the diffusivities of the two elements, the Kirkendall-effect occurs, related to changes in the local vacancy concentrations caused by diffusion. Does the Kirkendall-effect also take place in this alloy? Why (not)? page 2 of 5

3 c) The free energy of the alloy is well described by the regular-solution model, with the parameter Ω equal to 6.5 J/mol. Is the tracer diffusivity for B-elements larger or smaller than the value following from the table? Why? d) It can easily be derived that the thermodynamic factor F for regular solutions is given by Ω F = 1 2 xb ( 1 xb ), with x B the fraction of element B. This means that the effect of RT the thermodynamics on the diffusivity decreases with increasing temperature. Explain the physical (so not the numerical) reasons for this. Question 4 A certain metallic alloy can, at fixed composition, assume three different crystal structures, which are indicated by the letters α, β and γ. The relation between the free energy G for the three phases and the temperature T is given in the figure below for a temperature range above room temperature. The temperatures at which the free energy for two phases is equal are indicated as resp. T 0 αβ en T 0 αγ. All grain-boundary energies can be assumed to be independent of the temperature. G γ β α a) After a long anneal at T > T 0 αβ, during which the α-phase is formed, the metal is cooled down. Which phase transformation takes place if the cooling rate is very low? b) In which temperature range can the γ-phase form during cooling? c) Between α- and γ-grains coherent grain boundaries can exist, but the grain boundaries between α and β are always incoherent. How do these facts influence the possible formation of the γ-phase? d) Assume that the alloy has been cooled to room temperature in such a way that the microstructure is formed by just the γ-phase. Then, it is reheated fast to a certain temperature T, at which it is held for a long time. Describe the evolution of the phases in the microstructure when this annealing is performed (1) at T < T 0 αγ, (2) at T 0 αγ < T < T 0 αβ, (3) at T > T 0 αβ. T page 3 of 5

4 Answers Question 1 a) The equilibrium volume fraction of precipitates can be calculated by the lever rule as being equal to 5.2%, taking the solubility of copper in the matrix at 250ºC as approximately 0.3% (Porter & Easterling, fig. 5.25). The volume of the precipitate should therefore occupy a volume fraction of 0.06 of the larger sphere, which implies R = 21 nm. b) The given values lead to D = m 2 /s at T = 250ºC. Using the equations given in section leads to t = 92 h. c) During the increase in strength the precipitates form and their number density and volume fraction increases. During the decrease in strength coarsening of the precipitates takes place. d) The meta-stable θ -phase has a lower free energy than the supersaturated matrix at 200ºC, but at 250ºC its free energy has become higher than that of the matrix. Question 2 a) In the absence of cementite, the two-phase region α+γ is extended to lower temperatures. The two-phase regions α+fe 3 C and γ+fe 3 C do not exist. b) The two free-energy curves have the regular shape, and are situated such that the common tangent indicates the two-phase region to exist between approximately 0.3 wt.% (the equilibrium composition for α) and 3.5 wt.% (the equilibrium composition for γ). The free energy curves of these two phases are independent of cementite having formed or not. c) The chemical potential for carbon in the α-phase containing 1.1 wt.% carbon is higher than the chemical potential for carbon in the γ-phase of the same composition. d) The interface composition in each of the two phases will be the equilibrium value as found in question b). In the α-phase, the carbon concentration will gradually increase with increasing distance from the interface, with a width of approximately Dt = 4.5 µm. In the γ-phase the concentration will gradually decrease with a width of 0.07 µm. At large distance from the interface the carbon concentration will be the original 1.1% in each of the two phases. e) If cementite forms, it will grow into both phases, eventually leaving the α-phase with a low carbon concentration (the solubility of carbon in the α-phase in equilibrium with cementite is practically zero, see the phase diagram) and entirely consuming the γ-phase. Question 3 a) The activation energy is relatively low, indicating that no vacancies are needed for diffusion of B. page 4 of 5

5 b) The Kirkendall-effect is only found if there is a difference in diffusion fluxes of substitutional elements, causing a net flux of vacancies. Diffusion of interstitials has no effect on the vacancies, so the Kirkendall-effect will not occur. c) As can be derived with the equation given in question d), the thermodynamic factor is larger than 1, and the tracer diffusivity is therefore lower than the diffusivity that was given in the table. d) The thermodynamic factor reflects the tendency of mixing of A and B due to the negative enthalpy of mixing. At higher temperatures this effect becomes weaker because of the increasing influence of the entropy term in the free energy. Question 4 a) If the cooling rate is very low, it will be possible for the structure to follow the equilibrium, meaning that the β-phase is formed below T 0 αβ. b) If cooling is faster, the γ-phase can form before the β-phase does, but only at T < T 0 αγ. c) The given types of grain boundaries favour nucleation of γ from α, and make nucleation of β from α more difficult. Formation of the meta-stable γ-phase is therefore favoured by these grain boundaries. d) (1) The α-phase cannot form from the γ-phase in this temperature range. Only β can form, and eventually this will be the only phase in the microstructure. (2) In principle it is possible that the α-phase forms from γ in this temperature range, but formation of the β-phase is because of the lower free energy of the β-phase. Eventually the microstructure will consist of the β-phase. (3) In principle it is possible that the β-phase forms from γ in this temperature range, but formation of the α-phase is subject to a larger driving force because of the lower free energy of the α-phase. Eventually the microstructure will consist of the α-phase. page 5 of 5