Diffusion

Question 1

Derivation force for solidication (undercooling, assuming near Tm, delta H = R)

Answer 1

DeltaG = deltaH-TdeltaS
at Tm, deltaG = deltaH - TmdeltaS = 0
deltaS = deltaH / Tm
assume deltaH = L (constant
deltaG = L - T(L/Tm)
deltaG = L*deltaT/Tm

Question 2

Fick’s first law of diffusion?

Answer 2

J = -D dC/dx

Question 3

Intrinsic diffusivity terms? (for interstitial)

Answer 3

(see Chap 2 pg 1)
D = 1/6 * jump frequency * distance for jump
jump frequency = nearby sites * vibrating frequency (attempts per second) * exp(-deltaG_m/RT)
deltaG_m is the activation energy to move positions
exp(-deltaG_m) term is the fraction of attempts that are successful

D = D_0exp(-deltaH_m/RT)

Question 4

Derivation for Homogenization.
What is the relaxation time?

Answer 4

(See Chap 2 pg 1)
Relaxation time is the time it takes for the amplitude to decrease by 1/e (1/2.72)

at t = 2tau, amplitude has decreased by 1/e^2 –> exponential decay!

Question 5

Derivation for Carburization

Answer 5

(See Chap 2 pg 1)
C = Cs - (Cs-C0)erf(x/2sqrt(Dt))

Question 6

Fick’s Second Law? and Derivation

Answer 6

dC/Dt = Dd^2C/dx^2
(See Chap 2 pg 1 for derivation)

Question 7

Intrinsic Diffusivity Terms (for self-diffusion)?

Answer 7

Need to add a vacancy concentration term that depends on temperature

jump frequency = atom vibration * number of neighbors * vacancy concentration * exp(-deltaG_m/RT)

vacancy concentration = exp(-deltaG_v/RT)

D = D_0exp(-(deltaH_m + delta_H_v/RT)

Question 8

What is the Kirkendall Effect? Derivation of J_v?

Answer 8

When there are two alloys interdiffusing, they will have different fluxes, and hence a vacancy flux is needed to counteract this. This manifests as the interface moving with respect to the original.

J_v = (D_A - D_B)dC/dx

Question 9

What is Darken’s First Equation? Derivation?

Answer 9

velocity of the lattice plane because of the Kirkendall Effect

(See Chap 2 pg 2)

J_v = C_0*v = (D_A - D_B)dC/dx
x = Ca/c0
v = (D_A-D_B)dx_a/dx

Question 10

What is Darken’s Second Equation?

Answer 10

Interdiffusion coefficient D = X_BD_A + X_AD_B

Question 11

What are other driving forces of diffusion?

Answer 11

Diffusion can occur without concentration gradient.

Impurity atoms are larger than the lattice, and hence there is a strain. They might want to go to grain boundaries, interfaces, and dislocations

Hence, the driving force is chemical potential, not concentration

Grain boundaries: Db = D_b0 exp(-Qb/RT)
Free Surface = Ds = Ds0 exp(-Qs/RT)

Ds > Db > Dbulk

Db is most relevant, since there is a lot more

Question 12

What is surface free energy?

Answer 12

gamma = [J/m^2]

extra energy when a surface is created.

Question 13

What is laplace pressure?

Answer 13

Pressure percieved by the system due to surface tension.

For a spherical droplet, P = 2*gamma/r

P_laplace = F/A

change in volume* P_laplace = gamma * change in area