mae 381 exam 2

What is Linear Density?

# of atoms per unit length, who’s centers lie on the direction vector

LD = (atoms centered on direction vector) / (length of direction vector)

What is Planar Density?

PD = (# of atoms on plane) / (Area of plane)
Same PD = equivalent planes

What are the characteristics of a Single crystal?

Crystal structure extends throughout entire piece

What characteristic are most crystalline solids?

Polycrystaline

What is a polycrystaline composed of?

Collection of small crystals called grains

What are grains separated by?

Grain boundaries

What does isotropic mean?

Measured properties are independent of direction

What does Anisotropic mean?

Properties depend on direction (occurs in single crystals)

Why does aluminum have a different elastic modulus depending on the direction but we use 69 GPa as the value?

Single crystals of aluminum are anisotropic, causing them to have a different elastic modulus upon direction.

Everyday aluminum materials are polycrystaline, meaning there are millions of grains going in one direction vs the other. This averages out to one value (69)

Polycrystalline materials have the ability to behave how?

Isotropically

What is Bragg’s Law

n = order of reflection (integer)

lambda = wavelength of x-ray

d_hkl = distance between h, k, l planes = a / sqrt(h²+k²+l²) (for cubic crystal structures)

Provide an example of imperfections playing a role in material properties

Semiconductors in integrated circuits often ‘doped’ with small number of impurity atoms to alter electrical properties

How are vacancies/empty lattice sites in crystal structures created?

As temperature increases, vibration increases and atoms eventually have energy to move around

All crystalline solids contain what?

Vacancies

How to obtain the equilibrium number of vacancies (N_v)?

= N * e^ (-Qv/kT)

N = total number of atomic sites

Qv = Energy required to form a vacancy

k = Boltzmann constaint

T = Temperature in Kelvin

A vacancy exerts what type of strain on the surrounding crystal lattice?

Tensile strain

How to calculate N, the total number of atomic sites?

N = (N_a * density) / Atomic Weight all times the volume

Vacancy formation causes material to what?

To expand

Why do vacancies exist?

Gibbs free energy - amount of energy available to do work at constant temp and pressure.

Minimized when system is at chemical equilibrium (constant pressure and temp)

What is the gibbs function?

ΔG = ΔH - TΔS

What does a negative ΔH mean?

the process is energetically favorable

What does a positive ΔS mean?

the process is entropically favorable

Does ΔH increase or decrease with the formation of vacancies?

Increases

Why does ΔH increase with the formation of vacancies?

There is an increase in internal energy because of the breaking of interatomic bonds

Are vacancies energetically favorable?

No

Does ΔS increase or decrease with the formation of vacancies?

ΔS increases. Disorder increases

What does gibbs free energy tell us?

• if a spontaneous reaction is possible

• no indication of speed/rate of reaction

Why can’t diamond decompose to graphite even though it has a lower gibbs free energy?

There is an energy barrier that is really large preventing it from breaking down

Why is it impossible for a pure metal to have just one element?

There is always impurity atoms

Impurities are added to metals to create what?

Alloys

What is the difference between alloys and compounds?

Alloys:
- Can be mixed in arbitrary amounts

• random but uniform distribution

Compounds:

• have a given stoichiometry

• atoms have a fixed position relative to e/o

What does the addition of impurity atoms result in?

Formation of a solid solution

Describe solvent vs. Solute

Solvent: element/compound present in the greatest amt

Solute: less present than the solvent

What are the two types of impurity point defects in solid solutions?

Substitutional and Interstitial

What are the characteristics of substitutional impurity atoms?

• The solute/impurity atoms replace the host atoms

What does a smaller substitutional atom cause?

Tensile strain on surrounding lattice

What does a larger substitutional atom cause?

Compressive strain on the surrounding lattice

What is the purpose of the hume-rothery rules?

To express the amount of impurity atoms that can be dissolved to form a solid solution

What are the four hume-rothery rules?

1. Atomic size factor: difference in atomic radii should be less than ±15% (larger difference causes too much lattice strain)

2. Crystal structure must be the same for visible solid solubility

3. Electronegativity: large difference leads to intermetallic compound instead of solid solution

4. Valences: higher tendency to dissolve another metal with higher valences

Describe interstitial impurity atoms

They fill in the void between the host atoms

What are the two types of interstitial sites in FCC and BCC structures?

Tetrahedral and Octahedral (depend on the coordination number of the impurity)

Describe the tetrahedral interstitial site

• surrounded by 4 host atoms

• tetrahdron is formed when connecting centers of surrounding atoms

• CN = 4

Describe the octahedral interstitial site

• surrounded by 6 host atoms

• octahedron is formed when connecting centers of surrounding atoms

• CN = 6

How does different metals having a high APF correlate to interstitial sites and impurities?

Interstitial sites and impurities are small because the high atomic packing factor causes them to have to be small

Describe a self interstitial

• an interstitial atom is the same as the host atom, forcing its way into a space that should be empty

• infrequent

Why do interstitials still exist in a material at equilibrium

after a bunch of interstitials, the material is ‘desensitized’ to the new impurities as compared to when the first impurity was added. (easier to fight off covid variant if you grew antibodies to regular covid)

Describe dislocations

Linear defect around which some atoms are misaligned

What is an edge dislocation

Extra portion of a plane terminates within the crystal (look at the middle top three dots)

What is the burgers vector?

The magnitude and direction of lattice distortion associated with a dislocation
(draw from finish to start position) (magnitude needed to closes the gap)

What direction is the mode of edge dislocation?

Parallel to the applied shear stress

Describe screw dislocation

A spiral path around dislocation line

Burgers vector direction for screw dislocation

What is the energy of a dislocation?

E = G/2 | burgers vector |² (d)

G = shear modulus

d = length

Dislocations favor what size burger vectors?

small sized

What are most dislocations a mixture of?

Screw and edge dislocation

Burgers vector for mixed dislocation

Same magnitude, shift amount doesn’t change because of the type of shift

List the steps of nucelation

1. nucleation of crystals

2. Crystal growth

3. irregular boundaries form as crystals grow and merge together

4. grain boundaries as theyd appear in a microscope

What is a grain boundary?

A boundary separating two regions with different crystallographic orientation

What happens when defects are introduced?

It pushes the atom into an unstable energy state

What is the relationship between higher angle and disalignment?

There’s more disalignment with a higher angle

Why are grain boundaries more chemically reactive?

atoms sitting right at the grain boundary are forced into irregular, mismatched bonding positions, making them unstable energy wise

Why do impurities tend to segregate around grain boundaries?

The impurities tend to migrate to the boundaries because there is more room at the boundary for them as opposed to in between crystal grains. This relieves the overall lattice strain

What is a tilt boundary?

a small angle grain boundary that can be described as a series of edge dislocations

Describe the formation of tilt boundaries during recovery

The dislocations match and eventually cancel oout, then lining up to lower the internal energy of the material

Describe the beginning process of recovery dislocations

• occurs during annealing ( heating up metal to reduce its stresses)

• reduces strain energy

What is a characteristic of external surface’s bonds?

One side of the surface’s atoms are not bonded to anything, which are broken/unsatisfied bonds

• increases surface energy

What are phase boundaries?

• separates different phases in multi-phase materials

List different volume defects

• Pores (e.g casting)

• cracks (e.g quench)

• foreign inclusions (big clumps of impurities)

• usually a result of processing

What are the different ways to specify composition

1. weight percent

2. atom percent (number of atoms or moles)

Describe diffusion

Net movement of substance from region of high concentration to low concentration region

Why does diffusion happen?

• comes back to gibbs free energy

• increasing entropy → decrease in gibbs free energy

• Spontaneous

List qualities of the entropy of mixing

• higher probability of random distribution → higher entropy

• low probability that components will separate → low entropy for unmixed state

How does this entropy of mixing occur?

for T > 0 Kelvin, atoms are in constant motion and vibrate around equilibrium point

What conditions must atoms satisfy to move to different lattice positions?

• must have availible empty spot to move to

• must have enough energy to break existing bonds (comes from thermal lattice vibrations)

• more motion = higher temp

What are the two types of diffusion mechanisms in metals?

• vacancy diffusion

• interstitial diffusion

Describe vacancy diffusion

• atoms move from normal lattice position to an adjacent vacancy

• # of atoms that can move into empty site = coordination nuber

• moving atom can be host atom (self diffusion) or impurity atom (interdiffusion)

Describe Interstitial diffusion

• atoms move from one interstitial position to neighboring one that is empty

• atoms are “small” enough to fit into interstitial site

• common for: hydrogen, carbon, nitrogen, oxygen

• some lattice strain is acceptable

How does temperature affect diffusion?

Higher temperature leads to faster diffusion because there is more energy available, and more vacancies

Why is interstitial diffusion faster than vacancy diffusion?

• interstitial atoms are smaller and more mobile

• weaker bonding

• smaller energy barrier

• way more interstitial sites than vacancies

What is diffusion flux?

Rate of diffusion, J = M/A(t)

m = mass

A = area

t = time

When is the flux proportional to the concentration gradient dC/dx?

When steady state diffusion is in a single direction

J = -D(dC/dx) ← Fick’s first law

D = diffusion coefficient

Why does one vacancy/atom not care about the concentration gradient?

• bceause it isn’t a net flow of atoms/vacancies, will randomly walk through crystal

What does steady state mean in the context of diffusion flux?

condition where the diffusion flux does not change with time

dC/dX = ΔC/ΔX = Cb - Ca / Xb - Xa

What is fick’s second law and what is it used for?

• used for nonsteady diffusion

• ∂C/∂t = D( ∂²C/∂x²)

What is case hardening and why is it used?

• the Interstitial diffusion of carbon into a metal's surface layer at high temperatures.

• Used to increase hardness of surface while keeping interior impact resistant

What is the schematic for case hardening?

Cx - Co / Cs - Co = 1 - erf(x/2sqrt(Dt))
Cx = concentration at depth x after time t

C = C0 at t = 0

C = Cs at x = 0

C = Co at x = infinity

What are factors that influence diffusion?

• Diffusing species and host material determines type of diffusion

• Temperature - diffusion coefficient is temperature dependent

• D = D0 e^(-Qd/RT)

D0 = temperature independent constant

Qd = activation energy for diffusion

R = Gas constant (8.314)

T = Temp in kelvin

How do you find Qd and D0 experimentally?

Take ln of D = D0 exp equation, slope is -Qd/2×3R, intercept is LogD0.

Where else does diffusion occur?

Occurs along surface and grain boundaries

What causes faster diffusion?

• more open structure

• Dsurface > Dgrainboundary > Dvolume(lattice)

What is faster diffusion a result of?

Lower activation energy

Qsurface < Q grain boundary < Q volume (lattice)

What happens when an element lands on a metal?

1. atoms sprint along outside surface b/c little resistance

2. atoms plunge down into the grain boundaries to bypass lattice and get to the materials

3. the atoms seep sideways into the lattice

What are the qualities of polycrystalline diffusion?

1. volume/lattice diffusion dominant at high temperatures

2. grain boundary diffusion dominant at low temps

What is engineering stress

instantaneous load divided by the initial area ( F/A)

What is engineering strain

Change in length divided by initial length

A stress-strain curve can be created by

Applying a gradually increasing displacement and measuring the force sensor

Elastic deformation is —?

Not permanent (returns to original shape when released)

What happens when you have a material that has a non-linear elastic region?

a secant modulus (slope between stress 1 and origin) or a tangent modulus can be defined (slope at stress 2)

ex: grey cast iron, concrete, many polymers, elastomers

When force is applied to a material in one direction what happens

In one direction it will elongate (tension), and will contract in the direction perpendicular to the stress