PHASE TRANSFORMATIONS MIME360 FINAL EXAM
FINAL EXAM, Dec 11th, 2007
DURATION: 3 HRS Version ϑ
CLOSED BOOK McGill ID ___________________________ Name
______
All students must have one of the following types of calculators: CASIO fx-115, CASIO fx-991, CASIO fx-570ms SHARP EL-520, or the SHARP EL-546. NON-REGULATION CALCULATORS WILL BE REMOVED AND NO REPLACEMENT CALCULATOR WILL BE PROVIDED.
McGill University Dept. of Mining, Metals and Materials Engineering
PHASE TRANSFORMATIONS MIME360 FINAL EXAM Date: Wednesday, December 11th, 2007 Time: 9 AM
Examiner: Prof. R.R. Chromik Associate Examiner: Prof. M. Hasan
Instructions 1.
Read the instructions carefully.
2.
Put your name and ID# on this page. Put your NAME on pages 2-11.
3.
Read each question completely before working on each part, (the parts may be interrelated).
4.
Please attempt every question. No parts are optional.
5.
Write your solutions in the space provided below the relevant question. Anything written outside the provided space will be ignored.
6.
For questions requiring calculations, you must SHOW YOUR WORK to receive full credit.
7.
Draw any diagrams as large as possible in the space provided.
8.
Label all diagrams, axes, curves etc.
9.
For concept type questions, the space provided roughly indicates the level of detail expected.
10. Phase diagrams, a periodic table, and other useful information are attached at the end of this examination book (pages i through iv).
P1
P2
Official USE ONLY P3 P4 P5
P6
P7
1
MIME360, Version ϑ
Name: _______________________________
1. Pd-Ti Phase Diagram (40 pts) – Use the attached Pd-Ti phase diagram to complete the following tasks. Assume that the dashed lines are true, experimentally confirmed phase boundaries (i.e. solid lines). a) Identify the five invariant points labeled on the diagram. Point Type of invariant point Shorthand Notation (if applicable) i ii iii iv v b) Sketch, for 1450 K, the G vs. composition curves for stable phases occurring from 55 to 100at% Ti.
c) Based on your sketch for (b) and the points labeled ‘A’ and ‘B’ on the diagram, what type of mixing does this indicate is occurring between Pd and Ti?
2
MIME360, Version ϑ
Name: _______________________________
d) Consider your answer to part (c) and the table of data below. Discuss the features of the phase diagram in terms of thermodynamic considerations and the Hume-Rothery rules. Crystal Structure Atomic Radius (nm) Electronegativity Pd FCC 0.14 2.2 HCP 0.14 1.54 αTi BCC 0.14 1.54 βTi
e) An alloy of 80at% Ti is being considered for aerospace applications. It is processed at 1000 K. Complete the table below regarding the composition of this alloy at this temperature. Name of Phase 1
at% Ti in Phase 1
weight fraction of alloy that is Phase 1
Name of Phase 2
at% Ti in Phase 2
weight fraction of alloy that is Phase 2
f) Sketch below the microstructures of the 80at% Ti alloy at 1000 K, and at 825 K. Assume equilibrium conditions in both cases.
1000 K
825 K 3
MIME360, Version ϑ
Name: _______________________________
2. Types of Growth (30 pts) The graph below (concentration versus position) shows β growing at the expense of α. (a) Describe the differences between interface and diffusion controlled growth. What conditions (e.g. thermodynamics, interface type, etc.) would lead to one case or the other? To help your discussion, use the equation below for the mobility of the interface, defining any variables you use to make your point.
⎧ A 2 n1ν1Vm2 ⎛ ΔSa ⎞⎫ ⎛ − ΔH a ⎟⎟⎬ exp⎜⎜ exp⎜⎜ M=⎨ N RT R ⎝ ⎠⎭ ⎝ RT a ⎩
(b) What good does it do to know if a transformation is diffusion or interface limited?
4
⎞ ⎟⎟ ⎠
MIME360, Version ϑ
Name: _______________________________
3. Diffusion in the Pd-Ti System (30 pts) Aerospace engineers are developing a process for Pd-Ti alloys in the range of 80 to 98at% Ti. They need diffusion data to understand and optimize the microstructure of their new alloys. Help your co-workers out by answering the questions below. a) Raphael Beaudoin needs to know the diffusion coefficient of Pd in αTi. Devise an experiment that would allow for this quantity to be measured.
b) After completing the experiment described in part (a), Pascale Sader reports diffusion coefficients at two temperatures (shown below). From this data, determine a pre-factor and activation energy. Temperature (K) Diffusion Coefficient (m2/s) 850 1.2x10-19 950 6.1x10-18
c) Petar Drianov needs to know the interdiffusion coefficient for PdTi3. Devise an experiment that would allow for this quantity to be measured.
5
MIME360, Version ϑ
Name: _______________________________
4. TTT Diagrams and JMA Kinetics (20 pts) Kinetics for transformations that occur by diffusion can be modeled, (isothermally) by the Johnson-Mehl-Avrami (JMA) equation: n
f = 1 − exp(−kt )
Using the attached TTT diagram and the above equation, determine the time to isothermally transform 35% austenite at 650 °C.
6
MIME360, Version ϑ
Name: _______________________________
5. (25 pts) Congratulations, you have landed a job in the steel industry! You have been put in charge of their carburization operations. For a steel with 0.10wt% C, the process exposes the part to a constant 0.90wt% C at the surface for 31 hours. The temperature is maintained at 1100°C, where the steel is austenite. The company provides you with the following diffusion data. Diffusing Species Host Metal Do (m2/s) Qd (kJ/mol) C 6.2x10-7 80 αFe -5 C 2.3x10 148 γFe a. Calculate the carbon concentration at a depth of 4 mm into the steel part.
b. Draw schematic concentration profiles of carbon content versus position from the steel surface to the interior for t = 0, t = 15 hrs and t = 31 hrs.
7
MIME360, Version ϑ
Name: _______________________________
c. Discuss how a concentration profile for carbon relates to the formation of a “casing” or carburized layer. Are the kinetics the same, different or related in some other way?
DO NOT WRITE BELOW THIS LINE ON THIS PAGE
8
MIME360, Version ϑ
Name: _______________________________
6. Nucleation (25 pts) Heterogeneous nucleation of a solid from a liquid has a change in Gibbs free energy of:
{
}
ΔGhet = − 43 πr 3 ΔGV + 4πr 2 γ SL S (θ ) , where the shape factor, S (θ ) = (2 + cos θ )(1 − cos θ ) / 4 2
a. Define all of the variables in the first equation, including units.
b. Derive expressions for (i) the critical radius and (ii) the energy of the critical radius.
9
MIME360, Version ϑ
Name: _______________________________
c. Make a plot of the change in Gibbs free energy for homogeneous and heterogeneous nucleation as a function of embryo/nuclei size. Label key points on the graph and quickly discuss the differences between the two cases and how it affects the kinetics.
DO NOT WRITE BELOW THIS LINE ON THIS PAGE
10
MIME360, Version ϑ
Name: _______________________________
7. (30 pts) Give an example of a material from one of the labs whose microstructure was modified. Discuss and, also, explain the thermodynamic and kinetics of the process using the diagram below, a modified version from the course concept map. (Other possible discussion points: What did the atoms do and where did they go? What phases nucleated and grew? How did they grow? How did the microstructure change and what effect did it have on properties? What engineering process or application was explored by this example?) ENERGY
REACTION COORDINATE
11
i
ii iii
i
B
A
v
iv
ii
Selected Solutions to Fick’s Second Law Carburization Type
800 austenite
Equilibrium transformation
⎛ x ⎞ C(x , t ) = C s − (C s − C 0 )erf ⎜ ⎟ ⎝ 2 Dt ⎠ Tracer (radio-isotope) Type C (x, t ) =
⎛ − x2 ⎞ Q ⎜ ⎟ exp (π D t )12 ⎜⎝ 4 D t ⎟⎠
Homogenization
−t ⎛ πx ⎞ C = C + β 0 sin⎜ ⎟ exp ⎝ l ⎠ τ l2 τ= 2 π D
700 pearlite 600
500
95% transformation
Temp (ºC) 400
bainite
5% 300 transformation
M
200
M50
Diffusion Couple between C1 and C2
⎛ C + C2 ⎞ ⎛ C1 − C2 ⎞ ⎛ x ⎞ ⎟−⎜ ⎟erf ⎜ C =⎜ 1 ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ( Dt ) ⎠
M90
100
martensite 1
10
100
103
Time (s)
iii
104
105
erfc(z) = 1 – erf(z) erf (-z) = - erf (z)
Other possibly useful constants, equations and relations:
The gas constant, R = 8.314 J/mol/K. Boltzmann’s constant, kb = 8.62x10-5 eV/K = 1.38x10-23 J/K. Avogadro’s number is 6.022x1023 e = 2.71828 and π = 3.14159 D = Do exp (-Q / RT)
( r )3 − rο3 = kt
D = Do exp (-Q / kb T)
D 2 − Do2 = kt
J B = − DB
ΔG =
2γ Vm r
∂C B ∂ 2C B = DB ∂t ∂x 2
∂C B ∂x
iv
DURATION: 3 HRS Version ϑ
CLOSED BOOK McGill ID ___________________________ Name
______
All students must have one of the following types of calculators: CASIO fx-115, CASIO fx-991, CASIO fx-570ms SHARP EL-520, or the SHARP EL-546. NON-REGULATION CALCULATORS WILL BE REMOVED AND NO REPLACEMENT CALCULATOR WILL BE PROVIDED.
McGill University Dept. of Mining, Metals and Materials Engineering
PHASE TRANSFORMATIONS MIME360 FINAL EXAM Date: Wednesday, December 11th, 2007 Time: 9 AM
Examiner: Prof. R.R. Chromik Associate Examiner: Prof. M. Hasan
Instructions 1.
Read the instructions carefully.
2.
Put your name and ID# on this page. Put your NAME on pages 2-11.
3.
Read each question completely before working on each part, (the parts may be interrelated).
4.
Please attempt every question. No parts are optional.
5.
Write your solutions in the space provided below the relevant question. Anything written outside the provided space will be ignored.
6.
For questions requiring calculations, you must SHOW YOUR WORK to receive full credit.
7.
Draw any diagrams as large as possible in the space provided.
8.
Label all diagrams, axes, curves etc.
9.
For concept type questions, the space provided roughly indicates the level of detail expected.
10. Phase diagrams, a periodic table, and other useful information are attached at the end of this examination book (pages i through iv).
P1
P2
Official USE ONLY P3 P4 P5
P6
P7
1
MIME360, Version ϑ
Name: _______________________________
1. Pd-Ti Phase Diagram (40 pts) – Use the attached Pd-Ti phase diagram to complete the following tasks. Assume that the dashed lines are true, experimentally confirmed phase boundaries (i.e. solid lines). a) Identify the five invariant points labeled on the diagram. Point Type of invariant point Shorthand Notation (if applicable) i ii iii iv v b) Sketch, for 1450 K, the G vs. composition curves for stable phases occurring from 55 to 100at% Ti.
c) Based on your sketch for (b) and the points labeled ‘A’ and ‘B’ on the diagram, what type of mixing does this indicate is occurring between Pd and Ti?
2
MIME360, Version ϑ
Name: _______________________________
d) Consider your answer to part (c) and the table of data below. Discuss the features of the phase diagram in terms of thermodynamic considerations and the Hume-Rothery rules. Crystal Structure Atomic Radius (nm) Electronegativity Pd FCC 0.14 2.2 HCP 0.14 1.54 αTi BCC 0.14 1.54 βTi
e) An alloy of 80at% Ti is being considered for aerospace applications. It is processed at 1000 K. Complete the table below regarding the composition of this alloy at this temperature. Name of Phase 1
at% Ti in Phase 1
weight fraction of alloy that is Phase 1
Name of Phase 2
at% Ti in Phase 2
weight fraction of alloy that is Phase 2
f) Sketch below the microstructures of the 80at% Ti alloy at 1000 K, and at 825 K. Assume equilibrium conditions in both cases.
1000 K
825 K 3
MIME360, Version ϑ
Name: _______________________________
2. Types of Growth (30 pts) The graph below (concentration versus position) shows β growing at the expense of α. (a) Describe the differences between interface and diffusion controlled growth. What conditions (e.g. thermodynamics, interface type, etc.) would lead to one case or the other? To help your discussion, use the equation below for the mobility of the interface, defining any variables you use to make your point.
⎧ A 2 n1ν1Vm2 ⎛ ΔSa ⎞⎫ ⎛ − ΔH a ⎟⎟⎬ exp⎜⎜ exp⎜⎜ M=⎨ N RT R ⎝ ⎠⎭ ⎝ RT a ⎩
(b) What good does it do to know if a transformation is diffusion or interface limited?
4
⎞ ⎟⎟ ⎠
MIME360, Version ϑ
Name: _______________________________
3. Diffusion in the Pd-Ti System (30 pts) Aerospace engineers are developing a process for Pd-Ti alloys in the range of 80 to 98at% Ti. They need diffusion data to understand and optimize the microstructure of their new alloys. Help your co-workers out by answering the questions below. a) Raphael Beaudoin needs to know the diffusion coefficient of Pd in αTi. Devise an experiment that would allow for this quantity to be measured.
b) After completing the experiment described in part (a), Pascale Sader reports diffusion coefficients at two temperatures (shown below). From this data, determine a pre-factor and activation energy. Temperature (K) Diffusion Coefficient (m2/s) 850 1.2x10-19 950 6.1x10-18
c) Petar Drianov needs to know the interdiffusion coefficient for PdTi3. Devise an experiment that would allow for this quantity to be measured.
5
MIME360, Version ϑ
Name: _______________________________
4. TTT Diagrams and JMA Kinetics (20 pts) Kinetics for transformations that occur by diffusion can be modeled, (isothermally) by the Johnson-Mehl-Avrami (JMA) equation: n
f = 1 − exp(−kt )
Using the attached TTT diagram and the above equation, determine the time to isothermally transform 35% austenite at 650 °C.
6
MIME360, Version ϑ
Name: _______________________________
5. (25 pts) Congratulations, you have landed a job in the steel industry! You have been put in charge of their carburization operations. For a steel with 0.10wt% C, the process exposes the part to a constant 0.90wt% C at the surface for 31 hours. The temperature is maintained at 1100°C, where the steel is austenite. The company provides you with the following diffusion data. Diffusing Species Host Metal Do (m2/s) Qd (kJ/mol) C 6.2x10-7 80 αFe -5 C 2.3x10 148 γFe a. Calculate the carbon concentration at a depth of 4 mm into the steel part.
b. Draw schematic concentration profiles of carbon content versus position from the steel surface to the interior for t = 0, t = 15 hrs and t = 31 hrs.
7
MIME360, Version ϑ
Name: _______________________________
c. Discuss how a concentration profile for carbon relates to the formation of a “casing” or carburized layer. Are the kinetics the same, different or related in some other way?
DO NOT WRITE BELOW THIS LINE ON THIS PAGE
8
MIME360, Version ϑ
Name: _______________________________
6. Nucleation (25 pts) Heterogeneous nucleation of a solid from a liquid has a change in Gibbs free energy of:
{
}
ΔGhet = − 43 πr 3 ΔGV + 4πr 2 γ SL S (θ ) , where the shape factor, S (θ ) = (2 + cos θ )(1 − cos θ ) / 4 2
a. Define all of the variables in the first equation, including units.
b. Derive expressions for (i) the critical radius and (ii) the energy of the critical radius.
9
MIME360, Version ϑ
Name: _______________________________
c. Make a plot of the change in Gibbs free energy for homogeneous and heterogeneous nucleation as a function of embryo/nuclei size. Label key points on the graph and quickly discuss the differences between the two cases and how it affects the kinetics.
DO NOT WRITE BELOW THIS LINE ON THIS PAGE
10
MIME360, Version ϑ
Name: _______________________________
7. (30 pts) Give an example of a material from one of the labs whose microstructure was modified. Discuss and, also, explain the thermodynamic and kinetics of the process using the diagram below, a modified version from the course concept map. (Other possible discussion points: What did the atoms do and where did they go? What phases nucleated and grew? How did they grow? How did the microstructure change and what effect did it have on properties? What engineering process or application was explored by this example?) ENERGY
REACTION COORDINATE
11
i
ii iii
i
B
A
v
iv
ii
Selected Solutions to Fick’s Second Law Carburization Type
800 austenite
Equilibrium transformation
⎛ x ⎞ C(x , t ) = C s − (C s − C 0 )erf ⎜ ⎟ ⎝ 2 Dt ⎠ Tracer (radio-isotope) Type C (x, t ) =
⎛ − x2 ⎞ Q ⎜ ⎟ exp (π D t )12 ⎜⎝ 4 D t ⎟⎠
Homogenization
−t ⎛ πx ⎞ C = C + β 0 sin⎜ ⎟ exp ⎝ l ⎠ τ l2 τ= 2 π D
700 pearlite 600
500
95% transformation
Temp (ºC) 400
bainite
5% 300 transformation
M
200
M50
Diffusion Couple between C1 and C2
⎛ C + C2 ⎞ ⎛ C1 − C2 ⎞ ⎛ x ⎞ ⎟−⎜ ⎟erf ⎜ C =⎜ 1 ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ( Dt ) ⎠
M90
100
martensite 1
10
100
103
Time (s)
iii
104
105
erfc(z) = 1 – erf(z) erf (-z) = - erf (z)
Other possibly useful constants, equations and relations:
The gas constant, R = 8.314 J/mol/K. Boltzmann’s constant, kb = 8.62x10-5 eV/K = 1.38x10-23 J/K. Avogadro’s number is 6.022x1023 e = 2.71828 and π = 3.14159 D = Do exp (-Q / RT)
( r )3 − rο3 = kt
D = Do exp (-Q / kb T)
D 2 − Do2 = kt
J B = − DB
ΔG =
2γ Vm r
∂C B ∂ 2C B = DB ∂t ∂x 2
∂C B ∂x
iv
