phase transformations in solids mime-360 final exam
Final Exam, Dec. 12th, 2006 2:00 PM
DURATION: 3 HRS
Student No.
Faculty standard calculator ONLY. Name McGill University Department of Mining, Metals and Materials Engineering
PHASE TRANSFORMATIONS IN SOLIDS MIME-360 FINAL EXAM – CLOSED BOOK Examiner: Prof. R.R. Chromik
Associate Examiner: Prof. S. Yue
Instructions 1.
Read the instructions carefully.
2.
Put your name and ID# on this page. Put your ID# on all of the other exam pages.
3
Read each question completely before working on each part, (the parts may be interrelated).
4.
The exam has two parts. For part ζ (pp. 2-8), complete all questions. For part η (pp. 9-10), complete four (4) out of six (6) questions.
5.
Write your solutions in the space provided below the relevant question. Anything written outside the provided space will be ignored.
6.
You may use the blank sides of this examination paper for rough work.
7.
Draw any diagrams as large as possible.
8.
Label all diagrams, axes, curves etc.
9.
Phase diagrams, a periodic table, graphs of erf(z) and erfc(z) and other useful information are attached at the end of this examination book (pp. i through v).
10. The gas constant, R = 8.314 J/mol/K. 11. Boltzmann’s constant, k = 8.26x10-5 eV/K = 1.38x10-23 J/K. 12. Avogadro’s number is 6.022x1023 13. e = 2.71828 and π = 3.14159 14. This examination, including the attachments, must be returned.
PART ζ
Student No. ____________________
1. (45 pts) Referring to the attached Pt-Zr phase diagram, answer the following questions. a) Five points are indicated by arrows on the diagram. For each point, indicate in the table below whether it is eutectic, eutectoid, peritectic, monotectic, catatectic, peritectoid or none of these. Also, if applicable, indicate the shorthand notation for the transformation type, labeling the appropriate phases involved.
POINT
TYPE (e.g. eutectic)
Short Hand Notation
1
2
3
4
5
b) An alloy of composition 93 at% is heated to 1200°C and allowed to reach equilibrium. When this alloy is cooled into the two phase region marked ‘b’, what type of diffusional transformation takes place and why?
2
PART ζ
Student No. ____________________
c) Consider that the transformation described for the 93at% alloy in part (b) takes place by homogeneous nucleation. Make a schematic plot of the nucleation rate versus temperature from the transformation temperature ( TTr ) to 830°C. Label the regions in your plot where the nucleation rate is high or low and provide reasons why this is the case.
d) The 93% alloy is allowed to equilibrate at 850°C. Using a ruler and the at%Zr axis, complete the following table. Name of at% Zr in Name of at% Zr in Fraction of Fraction of Phase 1 Phase 1 Phase 2 Phase 2 Phase 1 Phase 2
3
PART ζ
Student No. ____________________
2. (30 pts) CONGRATULATIONS, you have been hired as an engineer in the steel industry! Your first job is to conduct and maintain quality contol for the nitriding process. For this process, ammonia gas (NH3) dissociates at the anneal temperature and supplies a constant surface concentration of free nitrogen to diffuse into the steel. This modifies the mechanical and wear properties at the surface. a) You are supplied with a pre-factor, Do = 0.11 cm2/sec and an activation energy of Ea = 1.9 eV for nitrogen diffusion in AISI 316 steel. The nitriding process takes place at 525°C. Calculate the value of the diffusion coefficient at this temperature.
b) To obtain the desired enhancement in surface properties, a quality control check measures the concentration of nitrogen at a depth of 1 micron and requires it be 1015 atoms/cm3. If the steel part originally contains no nitrogen and the surface concentration of nitrogen is 1019 atoms/cm3, how many hours does the nitriding process take?
c) Three months after settling into your new job, an additional processing step is instituted that takes place at 450°C. Your boss, Mr. F.T. Wailes, wants to save some money and combine the nitriding process with this anneal. Based on your knowledge of diffusion and the fact that time is money, make an argument to your boss for or against this change in temperature. Also, based on the description of the nitriding process, what else might change due to the temperature change?
4
PART ζ
Student No. ____________________
3. (30 pts) a) Describe the differences between Fick’s 1st and 2nd laws. Under what circumstances would one need to use the 2nd rather than the 1st or vice versa?
b) Empirically, the diffusion coefficient is found to have the form shown below. Atomistically, what is the exponential function describing? Why does it vary with temperature in this fashion?
⎛−Q⎞ D = D0 exp⎜ ⎟ ⎝ kT ⎠
c) For vacancy diffusion, the pre-factor may be written as shown below. Describe each variable and how it affects diffusivity. ⎛ ΔS m + ΔS f ⎞ ⎟⎟ Do = f nc λ2 ν o exp⎜⎜ k ⎝ ⎠
5
PART ζ
Student No. ____________________
4. (20 pts) For this problem, refer to the attached Al-Zn binary phase diagram. a) In the space below, draw schematically the Gibbs free energy versus wt%Zn for the stable phases at 500ºC.
b) On the diagram in part (a), label with an arrow the driving force for any possible transformations for an alloy of 30 wt%Zn at 500°C. c) BONUS (5pts) An alloy with 77.7 wt%Zn is heated to 277ºC. What is the specific name for the reaction that occurs at this point?
DO NOT WRITE BELOW THIS LINE ON THIS PAGE
6
PART ζ
Student No. ____________________
5. (30 pts) Non-equilibrium solidification of a two component system with complete miscibility in the liquid and solid state leads to a cored microstructure. Answer the following questions about this phenomenon. a) What consideration(s) about diffusion are used to explain non-equilibrium solidification?
b) During solidification, what experimental variable could be modified to prevent coring and how would it be changed to do this?
Problem 5 continues on the next page.
DO NOT WRITE BELOW THIS LINE ON THIS PAGE
7
PART ζ
Student No. ____________________
c) Non-equilibrium solidification could not be prevented for two specimens of the same material, ending up with a cored microstructure in both. Below is a plot of the variation in composition with position for the two samples. Both samples are annealed together for a time of t = q 2 (π 2 D) . Calculate the new peaks for the sinusoidal variation for sample 1 and sample 2. Show your work in the space below.
C
Sample 1 βo = 30
C Sample 2 βo = 17
l
q
x
8
PART η (Answer 4 out of 6 at 10 pts each)
Student No. ____________________
1. When determining a phase diagram by the cooling curve method, one plots temperature versus time and discovers phase transformations from inflections and plateaus in the curve. Why do these features occur and why do they reflect phase transformations?
2. You are given a metallographic specimen and asked to characterize it. You don’t have any clue what it contains chemically, but because you did the steel microstructure lab, you can characterize the microstructure! List five features that you can characterize by observation with a light optical microscope.
3. In terms of its microstructure and mechanical properties, compare quenched martensite to martensite tempered at 600ºC?
DO NOT WRITE BELOW THIS LINE ON THIS PAGE
9
PART η (Answer 4 out of 6 at 10 pts each)
Student No. ____________________
4. What are the differences between the surface and the interior of a carburized specimen of 1020 steel?
5. You are given two specimens of aluminium that have been subjected to different amounts of cold work. Propose an annealing experiment that could determine which specimen was cold worked more.
6. In the wire demo, the heated piano wire was cooled at a nominally constant rate and changed from a caternary shape (curved) back to a straight line. As this shape change progressed, the wire hesitated (stopped moving) for a moment. Describe why the wire hesistated and what it has to do with phase transformations.
DO NOT WRITE BELOW THIS LINE ON THIS PAGE
10
i
ii
erfc(z) = 1 – erf(z) erf (-z) = - erf (z) Selected Solutions to Fick’s Second Law
iii
Carburization Type
⎛ x ⎞ C(x , t ) = C s − (C s − C 0 )erf ⎜ ⎟ ⎝ 2 Dt ⎠ Tracer (radio-isotope) Type
⎛ − x2 ⎞ Q ⎟⎟ C (x, t ) = exp⎜⎜ 1 2 4 D t (π D t ) ⎠ ⎝ Homogenization
−t ⎛ πx ⎞ C = C + β 0 sin⎜ ⎟ exp ⎝ l ⎠ τ
l2 τ= 2 π D
Diffusion Couple between C1 and C2
⎛ C1 + C2 ⎞ ⎛ C1 − C2 ⎞ ⎛ x ⎞ ⎟−⎜ ⎟erf ⎜ C =⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ( Dt ) ⎠
iv
v
DURATION: 3 HRS
Student No.
Faculty standard calculator ONLY. Name McGill University Department of Mining, Metals and Materials Engineering
PHASE TRANSFORMATIONS IN SOLIDS MIME-360 FINAL EXAM – CLOSED BOOK Examiner: Prof. R.R. Chromik
Associate Examiner: Prof. S. Yue
Instructions 1.
Read the instructions carefully.
2.
Put your name and ID# on this page. Put your ID# on all of the other exam pages.
3
Read each question completely before working on each part, (the parts may be interrelated).
4.
The exam has two parts. For part ζ (pp. 2-8), complete all questions. For part η (pp. 9-10), complete four (4) out of six (6) questions.
5.
Write your solutions in the space provided below the relevant question. Anything written outside the provided space will be ignored.
6.
You may use the blank sides of this examination paper for rough work.
7.
Draw any diagrams as large as possible.
8.
Label all diagrams, axes, curves etc.
9.
Phase diagrams, a periodic table, graphs of erf(z) and erfc(z) and other useful information are attached at the end of this examination book (pp. i through v).
10. The gas constant, R = 8.314 J/mol/K. 11. Boltzmann’s constant, k = 8.26x10-5 eV/K = 1.38x10-23 J/K. 12. Avogadro’s number is 6.022x1023 13. e = 2.71828 and π = 3.14159 14. This examination, including the attachments, must be returned.
PART ζ
Student No. ____________________
1. (45 pts) Referring to the attached Pt-Zr phase diagram, answer the following questions. a) Five points are indicated by arrows on the diagram. For each point, indicate in the table below whether it is eutectic, eutectoid, peritectic, monotectic, catatectic, peritectoid or none of these. Also, if applicable, indicate the shorthand notation for the transformation type, labeling the appropriate phases involved.
POINT
TYPE (e.g. eutectic)
Short Hand Notation
1
2
3
4
5
b) An alloy of composition 93 at% is heated to 1200°C and allowed to reach equilibrium. When this alloy is cooled into the two phase region marked ‘b’, what type of diffusional transformation takes place and why?
2
PART ζ
Student No. ____________________
c) Consider that the transformation described for the 93at% alloy in part (b) takes place by homogeneous nucleation. Make a schematic plot of the nucleation rate versus temperature from the transformation temperature ( TTr ) to 830°C. Label the regions in your plot where the nucleation rate is high or low and provide reasons why this is the case.
d) The 93% alloy is allowed to equilibrate at 850°C. Using a ruler and the at%Zr axis, complete the following table. Name of at% Zr in Name of at% Zr in Fraction of Fraction of Phase 1 Phase 1 Phase 2 Phase 2 Phase 1 Phase 2
3
PART ζ
Student No. ____________________
2. (30 pts) CONGRATULATIONS, you have been hired as an engineer in the steel industry! Your first job is to conduct and maintain quality contol for the nitriding process. For this process, ammonia gas (NH3) dissociates at the anneal temperature and supplies a constant surface concentration of free nitrogen to diffuse into the steel. This modifies the mechanical and wear properties at the surface. a) You are supplied with a pre-factor, Do = 0.11 cm2/sec and an activation energy of Ea = 1.9 eV for nitrogen diffusion in AISI 316 steel. The nitriding process takes place at 525°C. Calculate the value of the diffusion coefficient at this temperature.
b) To obtain the desired enhancement in surface properties, a quality control check measures the concentration of nitrogen at a depth of 1 micron and requires it be 1015 atoms/cm3. If the steel part originally contains no nitrogen and the surface concentration of nitrogen is 1019 atoms/cm3, how many hours does the nitriding process take?
c) Three months after settling into your new job, an additional processing step is instituted that takes place at 450°C. Your boss, Mr. F.T. Wailes, wants to save some money and combine the nitriding process with this anneal. Based on your knowledge of diffusion and the fact that time is money, make an argument to your boss for or against this change in temperature. Also, based on the description of the nitriding process, what else might change due to the temperature change?
4
PART ζ
Student No. ____________________
3. (30 pts) a) Describe the differences between Fick’s 1st and 2nd laws. Under what circumstances would one need to use the 2nd rather than the 1st or vice versa?
b) Empirically, the diffusion coefficient is found to have the form shown below. Atomistically, what is the exponential function describing? Why does it vary with temperature in this fashion?
⎛−Q⎞ D = D0 exp⎜ ⎟ ⎝ kT ⎠
c) For vacancy diffusion, the pre-factor may be written as shown below. Describe each variable and how it affects diffusivity. ⎛ ΔS m + ΔS f ⎞ ⎟⎟ Do = f nc λ2 ν o exp⎜⎜ k ⎝ ⎠
5
PART ζ
Student No. ____________________
4. (20 pts) For this problem, refer to the attached Al-Zn binary phase diagram. a) In the space below, draw schematically the Gibbs free energy versus wt%Zn for the stable phases at 500ºC.
b) On the diagram in part (a), label with an arrow the driving force for any possible transformations for an alloy of 30 wt%Zn at 500°C. c) BONUS (5pts) An alloy with 77.7 wt%Zn is heated to 277ºC. What is the specific name for the reaction that occurs at this point?
DO NOT WRITE BELOW THIS LINE ON THIS PAGE
6
PART ζ
Student No. ____________________
5. (30 pts) Non-equilibrium solidification of a two component system with complete miscibility in the liquid and solid state leads to a cored microstructure. Answer the following questions about this phenomenon. a) What consideration(s) about diffusion are used to explain non-equilibrium solidification?
b) During solidification, what experimental variable could be modified to prevent coring and how would it be changed to do this?
Problem 5 continues on the next page.
DO NOT WRITE BELOW THIS LINE ON THIS PAGE
7
PART ζ
Student No. ____________________
c) Non-equilibrium solidification could not be prevented for two specimens of the same material, ending up with a cored microstructure in both. Below is a plot of the variation in composition with position for the two samples. Both samples are annealed together for a time of t = q 2 (π 2 D) . Calculate the new peaks for the sinusoidal variation for sample 1 and sample 2. Show your work in the space below.
C
Sample 1 βo = 30
C Sample 2 βo = 17
l
q
x
8
PART η (Answer 4 out of 6 at 10 pts each)
Student No. ____________________
1. When determining a phase diagram by the cooling curve method, one plots temperature versus time and discovers phase transformations from inflections and plateaus in the curve. Why do these features occur and why do they reflect phase transformations?
2. You are given a metallographic specimen and asked to characterize it. You don’t have any clue what it contains chemically, but because you did the steel microstructure lab, you can characterize the microstructure! List five features that you can characterize by observation with a light optical microscope.
3. In terms of its microstructure and mechanical properties, compare quenched martensite to martensite tempered at 600ºC?
DO NOT WRITE BELOW THIS LINE ON THIS PAGE
9
PART η (Answer 4 out of 6 at 10 pts each)
Student No. ____________________
4. What are the differences between the surface and the interior of a carburized specimen of 1020 steel?
5. You are given two specimens of aluminium that have been subjected to different amounts of cold work. Propose an annealing experiment that could determine which specimen was cold worked more.
6. In the wire demo, the heated piano wire was cooled at a nominally constant rate and changed from a caternary shape (curved) back to a straight line. As this shape change progressed, the wire hesitated (stopped moving) for a moment. Describe why the wire hesistated and what it has to do with phase transformations.
DO NOT WRITE BELOW THIS LINE ON THIS PAGE
10
i
ii
erfc(z) = 1 – erf(z) erf (-z) = - erf (z) Selected Solutions to Fick’s Second Law
iii
Carburization Type
⎛ x ⎞ C(x , t ) = C s − (C s − C 0 )erf ⎜ ⎟ ⎝ 2 Dt ⎠ Tracer (radio-isotope) Type
⎛ − x2 ⎞ Q ⎟⎟ C (x, t ) = exp⎜⎜ 1 2 4 D t (π D t ) ⎠ ⎝ Homogenization
−t ⎛ πx ⎞ C = C + β 0 sin⎜ ⎟ exp ⎝ l ⎠ τ
l2 τ= 2 π D
Diffusion Couple between C1 and C2
⎛ C1 + C2 ⎞ ⎛ C1 − C2 ⎞ ⎛ x ⎞ ⎟−⎜ ⎟erf ⎜ C =⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ( Dt ) ⎠
iv
v
