Microstructure and Properties II

1
Microstructure and Properties II

• MSE 27-302
• Fall, 2002 (2nd mini-course)
• Prof. A. D. Rollett
• http//neon.mems.cmu.edu/rollett/27302/ 27302.html

2
Course Content

• 27-302 is the second of a pair of (mini-)courses
  that describe the relationship between materials
  microstructure and properties.
• This course deals mainly with multi-phase
  microstructures. There is a strong emphasis on
  phase transformations as the basis for
  understanding the origin of (useful)
  microstructures.
• 27-301 dealt mainly with single phase
  microstructures and their properties.
• Multi-phase materials made through natural
  processes will be contrasted with (man-made)
  composite materials and biomaterials.
• Students are expected to learn a set of technical
  skills in addition to improving various
  attributes of scientist/engineers
  (communications, ethics, how to design
  experiments, )

3
Topics

• Where does microstructure come from? Phase
  transformations, kinetics of transformations, the
  Kolmogorov-Johnson-Mehl-Avrami equation.
• Properties of Composite materials background
  material on glass-ceramics for Lab 1.
• Phase transformations driving forces,
  thermodynamics of nucleation (precipitation
  reactions).
• Transformations kinetics of growth a simple TTT
  diagram. How to calculate and predict TTT and
  CCT diagrams.
• The role of interfaces in heterogeneous
  nucleation example of the Al-Cu system
  sequences of metastable precipitates.
• The age-hardening curve methods of measuring
  transformations. The similarities between
  mechanical hardness and magnetic hardness.
• Impact of precipitation on complex properties
  example of fatigue as a microstructure-sensitive
  property.
• More complex diffusive transformations example
  of Fe-C system for eutectoid reactions.
• Continuous transformations spinodal
  decomposition.
• Coarsening of two-phase structures effect of
  two-phase structures on creep properties
  (Ni-alloys as an example).
• Competition between transformation mechanisms
  discussion of non-diffusive transformations
  massive transformations, martensitic
  transformations exploitation of martensitic
  reactions for shape-memory alloys.
• The ultimate in complicated microstructures
  introduction to welding and joining.

4
Technical topics

• Technical Issues
• S olid state transformations
• Differences between transformations from the
  liquid state and transformations starting from
  the solid state the influence of crystalline
  structure
• Driving forces - should the reaction take place?
• Nucleation and growth (thermodynamics, kinetics)
  the rate at which reaction takes place
• Influence of defects on transformations
• Prediction of temperature-time-transformation
  (TTT) curves (and/or continuous-cooling-transforma
  tion, CCT, curves)
• Military transformations
• Precipitate coarsening
• These topics are some of the underpinnings for
  understanding various phenomena that are
  important for microstructure-property
  relationships.

5
Material Properties, Phenomena

• Examples of phenomena for which
  microstructure-property relationships are
  significant
• Age Hardening
• Shape memory effect, alloys
• Alloy optimization
• Multiphase materials and creep
• Energy absorption in structures
• Fatigue resistance
• Exploitation of nanostructured, amorphous
  materials
• Optimization of Materials Design
• All the technical topics are relevant to
  understanding and engineering the phenomena.
• Certain material systems are important examples.

6
Materials Systems

• Clearly there are too many material systems to
  study in one course. Certain systems are very
  useful as examples, however.
• Al-Cu precipitation, metastable phases, age
  hardening, effects of crystal structure,
  coarsening
• Fe-C-X (steel) allotropic transformations,
  eutectoids, military transformations, tempering,
  hardenability

7
Student Input for 302

• In 27-301, each student was required to make a
  short presentation in class.
• In 27-302, student input will be sought through
  discussion sessions. The objective is to learn
  how to apply the understanding of
  microstructure-property relationships to a
  specific system(s).
• The culmination of the student input exercise
  will be a discussion of the pros and cons and
  changing a given material (for a specific
  application).
• Discussions will be held in the second half of
  the Weds class.
• Next we discuss the sequence of steps required.

8
Materials Design

• The sequence of steps leading towards the
  discussion of materials design
• Each student chooses an application for which a
  material is critical in at least one component
  (Oct. 23rd, Weds).
• The application is analyzed to determine which
  component is materials-critical (Oct. 30th,
  Weds).
• The material is analyzed to determine its
  microstructure and likely processing history
  (Oct. 30th, Weds).
• The microstructure-property relationships are
  analyzed (Nov. 13th, Weds).
• Possible changes to the microstructure are
  analyzed for their effect on properties (Nov.
  20th, Weds).
• Discussion between a pro-change group and a
  status-quo group on the merits of optimization
  of the material (Nov. 25th, Monday).
• Each student writes up a report on materials
  optimization.

9
Applications

• Stents
• Sutures
• Bone substitute
• Stealth aircraft (Low Observable materials)
• Nuclear reactors (fuels)
• Solar cells
• Light weight armor (ceramic armor)

10
302 Jeopardy 1
1. Rank is sum of the rank of the quantities on
each side
4. -RT lnX0/Xe.
Q1. How is the rank of a property tensor
determined from the rank of each related quantity?
Q4. What is the formula for the driving force
for precipitation in a simple 2-phase system?
2. Free energy
5. No nucleation barrier
Q2. What thermodynamic quantity should we use to
predict whether or not a reaction will occur?
Q5. Name a key difference between discontinuous
and continuous phase transformation.
3. 2-fold symmetry axes (diads)
6. Approximately 3 times the yield stress.
Q3. Which symmetry element is found on lt110gt
directions in fcc materials?
Q6. How much greater is the hardness than the
yield stress (same units)?
11
302 Jeopardy 2
1. Proportional to undercooling
4. -Hf (?T/Tmelt).
Q1. How is driving force related to undercooling?
Q4. What is the formula for the driving force
for solidification?
2. Difference between the temp. at which the
composition intersects the solvus (liquidus) and
the current temp.
5. Two phases in a composite generally
expand/contract at different rates with ?T.
Q2. How is the undercooling defined?
Q5. What is one cause of residual stress in a
composite material?
3. Einstein notation
6. Differentiate the total energy.
Q3. What is the name for the convention that
states that repeated indices are summed over?
Q6. How do we determine the point at which an
energy release rate is zero?
12
302 Jeopardy 3
1. Balance between rates of adding surface
energy and gaining free energy from transformation
4. Because large interfacial energies mean high
barriers to nucleation (and heterogeneous sites,
if available, operate first).
Q1. How does one determine the barrier to
nucleation?
Q4. Why is homogeneous nucleation only observed
in a few cases?
2. In precipitation of pro-eutectoid ferrite,
the thermodynamic term involves the log of a
ratio of terms in (1-X).
5. 16pg3/?GV2.
Q2. Why is the driving force for a eutectoid
decomposition small compared to decomposition of
a simple solid solution (e.g. pro-eutectoid
decomposition of austenite)?
Q5. What is the formula for the critical free
energy of nucleation?
6. It is a volumetric energy and is subtracted
off the chemical free energy for transformation.
3. 2g/?GV
Q3. What is the formula for the critical radius?
Q6. What is the role of elastic energy in
nucleation?
13
302 Jeopardy 4
4. The rate increases because of increasing
driving force but then decreases because of
decreasing diffusion rate.
1. 16pg3/?GV-?GS 2
Q1. What is the free energy barrier in the
presence of an elastic energy?
Q4. Why does the growth rate first increase with
undercooling and then decrease?
2. Al2Cu platelets aligned with 100 planes.
5. D ?2C ?C/?t.
Q2. What effect does elastic anisotropy have on
precipitation in the Al-Cu system?
Q5. What is the diffusion equation (w/o source
terms)?
3. Matching of close-packed planes, e.g.
110bcc// 111fcc
6. Linearized gradients.
Q3. What impact does atomic matching have on the
orientation relationship between parent and
product phases?
Q6. What approximation can we make to solve the
diffusion equation for ppt growth in 1D?
14
302 Jeopardy 5
4. Solute diffuses from small precipitates to
large ones.
1. The change in concentration around one
precipitate affects the concentration around
adjacent precipitates.
Q1. What is the cause of impingement of
concentration fields?
Q4. What causes coarsening of precipitates?
2. Grain boundaries act as short circuit
diffusion paths for transport of solute to
precipitates.
5. ltR3(t)gt - ltR3(t0)gt k t.
Q2. Why do precipitates grow more rapidly on
grain boundaries than in the bulk (at low
temperatures)?
Q5. What is the relationship between radius and
time for coarsening?
3. Decreasing radius of a precipitate raises its
solubility.
6. x ?C0/ (Cb - Ce) v(Dt).
Q3. What does the Gibbs-Thomson effect do to
precipitates?
Q6. What is the relationship between ppt size
and time for diffusion controlled growth in 1D?
15
Office hours, CAs

• Office hours will be as in 301 330-5 Monday,
  1130-1230 Weds/Fri.
• The CA for the Lab is Ms. Mitra Taheri.

16
Exam Rules

• No books no lecture notes no computers
• One cheat sheet with notes (both sides if you
  like) hand in the the cheat sheet with the exam
  paper/book. You must write the notes yourself
  copying and pasting is OK, but not literal cut
  and paste. The idea of the cheat sheet is for
  you to go through the course material and extract
  the most important ideas, equations, etc.
• Calculator OK (but not a device, such as a Palm
  Pilot, in which you can store lecture notes etc.)

17
27-302, Labs

• Lab 1 Investigation of precipitation in
  glass-ceramics. Purpose to demonstrate the
  effect of phase transformation on mechanical and
  optical properties.
• Lab 2 Short experiments on crystallization of
  amorphous metals, magnetic domain imaging and age
  hardening curves.

18
Calendar 302
Please consult the separate file posted on the
website.
19
Topic List 302

• Where does microstructure come from? Phase
  transformations, kinetics of transformations, the
  Kolmogorov-Johnson-Mehl-Avrami equation.
• Properties of Composite materials rule of
  mixtures. Background material on glass-ceramics
  for Lab 1.
• Phase transformations driving forces,
  thermodynamics of nucleation (precipitation
  reactions).
• Transformations kinetics of growth a simple TTT
  diagram.
• The role of interfaces in heterogeneous
  nucleation example of the Al-Cu system
  sequences of metastable precipitates.
• The age-hardening curve methods of measuring
  transformations.
• Impact of precipitation on complex properties
  example of fatigue as a microstructure-sensitive
  property.
• More complex diffusive transformations example
  of eutectoid reactions.
• Continuous transformations spinodal
  decomposition.
• Coarsening of two-phase structures effect of
  two-phase structures on creep properties
  (Ni-alloys as an example).
• Competition between transformation mechanisms
  discussion of non-diffusive transformations
  massive transformations, martensitic
  transformations exploitation of martensitic
  reactions for shape-memory alloys.
• Parallels between magnetic and mechanical
  hardness.
• The ultimate in complicated microstructures
  introduction to welding and joining not
  addressed in 2001.
• Cellular structures foams, wood, bread(!), bone,
  composites.
• Guest Lecture (Prof. E. Towe) quantum dot
  structures in semiconductors.

20
Sample problem 1

• KJMA equation (transformation kinetics) an
  alloy is recrystallized at 2 different
  temperatures, 400 and 500C. The KJMA exponent
  is found to be 2. By interpolating the f vs time
  data, the time required for 50 recrystallization
  is found to be 30s and 5m, respectively.
  Estimate the activation energy for the process.
• Answer use the form of the equation from the
  homeworkt-ln(.5)/k1/ngt k -ln(.5)/ tn
  Assume kk0exp-Q/RT gt -Q/RTln(k/k0)So,
  -QRT1ln(k1/k0) RT2ln(k2/k0)-Q/R(1/T1-1/T2)
  ln(k1/k2)QR ln(t12/t22) /(1/T1-1/T2)Q 8.31
  ln(302/3002) (1/673 - 1/773)Q 199,086 J/mole

21
Sample Problem 2

• Composites a certain type of (cheap) plywood is
  made up of two thin outer sheets of a high
  density wood with a lower density filler material
  inside. If the modulus of the cladding is 10
  Gpa, and each sheet is 2mm thick, and the modulus
  of the filler layer is 100 Mpa with a thickness
  of 10mm, what is the stiffness of the plywood,
  measured through the thickness?
• Apply elementary isostress theory. Modulus
  VA(1/EA) VB(1/EB) 1/14 (4/10 10/0.1)
  GPa 0.14 GPa.
• The composite is dominated by the softer filler
  layer because of the loading method. It would be
  much stiffer if it were loaded on its edge.

22
Sample Problems 3

• Nucleation in a problem on solidification, the
  latent heat is 50,000 J/mole and the melting
  point (liquidus) is 850C. The molar volume is
  106 m-3. No appreciable nucleation is observed
  in a carefully controlled experiment in which
  only homogeneous nucleation can occur. What is
  the volumetric driving force for an undercooling
  of 50C?
• Answer - use the expression for driving force
  where the latent heat (enthalpy) is known.
  ?GV(?H?T/Te)/Vm 50,000.50/(850273)/10-6
  2.23.109 J.m-3
• Based on this information, what is the apparent
  interfacial energy?
• Answer assume that ?G 60kT at the point where
  nucleation occurs 601.38.10-23112316pg3/3
  /(2.23.109)2so, g 3v(0.276) 0.65 J.m-2.

23
Sample Problem 4

• Precipitate growth rates for a precipitate that
  is pure element B, and a solvus line described by
  log10(XB) 2.853 - 2.875.103/T, where XB is the
  composition in atomic , what is the growth rate
  at T600C for a matrix composition X0B1.5 1
  minute after nucleation has taken place? Assume
  1D growth (e.g. of a slab of precipitate
  nucleated on a grain boundary). The pre-factor
  and activation energy for diffusion of B in A are
  7.4.10-5 m2.s-1 and Q217.2 kJ.mole-1,
  respectively.
• Answer - first calculate the equilibrium
  concentration of matrix (alpha) in equilibrium
  with the precipitate (beta) XB 0.36Then the
  growth rate is given by v?X/2(Xb-Xe) v(D/t)
  (1.5-0.36)/2/(100-0.36)v(7.4.10-5exp-217,200/8.
  31/873/60) 2.0 10-12 m.s-1, or 7nm per hour!
• Pretty slow!

24
Sample Problem 5

• Coherency Loss show how the following
  expression can be derived for the critical size
  of a precipitate at which coherency is
  lost. rcrit 3?g/4µd2.
• Answer recall that ?Gcoherent 4µd2 4pr3/3
  4pr2gcoherent ?Gnon-coherent
  4pr2gnon-coherentAt the transition size, the two
  free energies will be the same, and above this
  size, the precipitate with incoherent interface
  will have the lower energy. Therefore we can
  write that 4µd24prcrit3/3 4prcrit2gcoherent
  4prcrit2gnon-coherentWrite
  ?g(gnon-coherent - gcoherent)Thus rcrit
  3?g/4µd2.

25
Sample Problem 6

• Coherency Loss, contd. for the problem outlined
  in number 5, given a (cubic) precipitate with
  lattice parameter 3.9 Å, and a matrix with a3.8
  Å, shear modulus µ45GPa, and an observed loss of
  coherency at r5nm, what difference in
  interfacial energy would you estimate for
  incoherent versus coherent interfaces?
• Answer turn the equation around and estimate
  the difference rcrit 3?g/4µd2 ltgt ?g
  rcrit 4µd2 /3.The misfit ?a/a 0.1/3.8
  0.0263.Thus ?g 5.10-9 4 45.109 0.02632 /3
  0.21 J.m-2.
• This is a reasonable value.

26
Sample Problem 7

• Spinodal Decomposition how can we represent the
  phenomenology of spinodal decomposition? One key
  is to postulate a function for the dependence of
  free energy on composition. The simplest form
  that will yield a G(X) curve with a central
  hump is this G(X) 25,000 4(X-0.5)4 -
  (X-0.5)2 J. mole-1
• Based on this constitutive description, we can
  now ask, for example, what the limits of the
  chemical spinodal are?
• Answer differentiate the formula to find the
  curvature and set it equal to zero to locate the
  inflection points d2G/dX2 25,000
  443(X-0.5)2 - 2 0 48(X-0.5)2 2 X
  0.5 v(2/48) 0.704 or 0.296
• We can also easily obtain the miscibility gap
  because of the symmetry of the function about
  X0.5 dG/dX0 gtdG/dX 25,000 44(X-0.5)3 -
  2(X-0.5) 0 gt(X-0.5)2 1/8 gt X 0.146 or
  0.854

27
Sample Problem 7 graph

• A plot of G(X) 4(X-0.5)4 - (X-0.5)2

Chemical Spinodal
Miscibility Gap
28
Sample Problem 8

• Heterogeneous Nucleation versus Homogeneous
  Consider problem 5.5 from PE and estimate the
  ratio between the homogeneous and heterogeneous
  nucleation rates. The critical free energy for
  homogeneous nucleation is 10-19 J and the
  temperature is 500C. Assume that the effective
  grain boundary thickness is 0.4nm and the grain
  size 25µm gAA 500, gAB 600 mJ.m-2.
• Answer First calculate the contact angle gAA
  2gAB cosq q cos-1 (gAA/ 2gAB) 53.1
• Then calculate the shape factor, S(q)S(q) 0.5
  (2 cosq)(1 - cosq)2 0.208
• The ratio in nucleation rates is given by PE Eq.
  5.25 Nhet/Nhomo C1/C0 exp-(?Ghomo-?Ghetero)
  /kT

29
Sample Problem 8, contd.

• Nhet/Nhomo C1/C0 exp(?Ghomo-?Ghetero)/kT
  ?/D exp(?Ghomo- S(q) ?Ghomo)/kT ?/D
  exp((1- S(q) ?Ghomo)/kT 0.4/25,000
  exp(1-0.208)10-19/(1.38. 10-23 773) 0.027
• Note the sign of the exponential which gives a
  large number. The ratio of the (effective) grain
  boundary thickness to grain size decreases the
  ratio quite significantly. In practical terms,
  heterogeneous nucleation is most significant at
  (or adjacent) to the nucleation sites
  (boundaries, dislocations etc.).

gab
a
Grainboundary in alpha
q
gaa
b
30
Sample Problem, no. 9

• From Dieter, p219 (adapted)
• Question Al-4Cu (by wt.) has a yield stress of
  600MPa. Estimate the particle size and spacing.
• Solution recognize that this stress relates to
  age hardening beyond the peak hardness.
  Therefore use the Orowan bowing stress to
  estimate the stress. s ltMgt tcrss ltMgt
  Gb/l
• G27.6GPa b0.25nm ltMgt3.1spacing
  3.127,6000.25.10-9/ 600 35.7 nm
• Now we must estimate the volume fraction of
  particles for which we use the phase diagram,
  assuming that we are dealing with the equilibrium
  phase, q, which is 54 w/o Cu, and the a in
  equilibrium with it, 0.5 w/o Cu.
• Wt. Al (54-4)/(54-0.5) 93.5 wt. q
  4-0.5/(54-0.5)6.5
• Volume of a 93.5gm/2.7 gm/cm3 34.6 cm3
• Volume of q 6.5/ 4.443 gm/cm3 1.5 cm3
• Volume fraction of a 0.96 volume fraction of q
  0.04.
• Use l4r(1-f)/3f (slide 22) r
  30.0435.7/4/(1-0.04) 1.12 nm.

31
Cheating Policy

• Students are referred to the University Policy
  About Cheating and Plagiarism (Organization
  Announcement No. 297, 6116/80). It shall be the
  policy in this course to discourage cheating to
  the extent possible, rather than to try to trap
  and to punish. On the other hand, in fairness to
  all concerned, cheating and plagiarism will be
  treated severely.
• Cheating includes but is not necessarily limited
  to
• 1.Plagiarism, explained below.
• 2.Submission of work that is not the
  student's own for reports or quizzes.
• 3.Submission or use of falsified data.
• Plagiarism includes (but is not limited to)
  failure to indicate the source with quotation
  marks or footnotes, where appropriate, if any of
  the following are reproduced in the work
  submitted by a student
• 1.A graph or table of data.
• 2. Specific language.
• 3.Exact wording taken from the work,
  published or unpublished, of another person."

32
Test, Exams, Grading Policy

• Homeworks 1 per week 100 points
• Quizzes 1 per week 20 points
• Exams two see weighting below
• Grading Policy A gt 90 B gt 80 C gt
  70 D gt 55
• The instructor will request an Oral exam in
  borderline cases.
• Weighting ()Homeworks 15Quizzes 5Lab 30E
  xams 50
• Notes the distribution between the two exams is
  to be determined. The quizzes are mainly there
  to encourage students to stay on top of the
  material. The 30 weighting for the Lab (or
  Project) reflects the number of units assigned to
  the Lab part of the class.