Defects in Solids• 0-D or point defects

– vacancies, interstitials, etc.– control mass diffusion

• 1-D or linear defects– dislocations– control deformation processes

• 2-D or planar defects– grain boundaries, surfaces, interfaces

• 3-D or volume defects– voids, secondary components (phases)

concentrations

Diffusional Processes

hydrogen separation membrane

Pd

H2

+ CO

+ CO2

H

x

cH

cJ D

x

0

c

x

0J

Fick’s first law (similar to Ohm’s law)phenomenological

Applies under steady state conditionsc(x) f(t)

flux

diffusion coefficient

cJ D

x

#

area time

#vol

length

response driving force

concentration gradient

2length

time

Units of D

Diffusional Processes

# vol

time

/

Continuity requirements

Non steady-state: c(x) = f(t)

Fick’s second law

BA

t = 0CA

CB

t > 0

CA

CB

c J

t x

c

Dx x

if D f(x)

c cD

t x x

2

2

c cD

t x

2

~length

Dtime

2

# vol

length

/

Atomistics of diffusion

planes of atomstracer species with a concentration gradientc = concentration #/cm3

n = #/cm2 # density in the plane n = cao

n1 = on plane (1)

n2 = on plane (2)

Flux from plane (2) to plane (1)

1 2 1½ J n 2 1

o

c c c

a x

Random walk

ao

(1) (2)

x

(½ jump to the left)

Flux from plane (1) to plane (2)

2 1 2½ J n

Net from plane (1) to plane (2)

1 2 2 1 1 2½ J J n n

1 2 2o

neta

J c c

2½ net oc

J ax

D

3-D21

6 oD a

Mechanisms of Diffusion

Vacancy

Interstitial (self or impurity)

net transport

vacancy to rightatom to left

Atomistics from Mechanism

2oD a P N

geometric constant~ 1/(# nearest neighbor sites)

jump distance

probability that a nearest neighbor site is vacant (available) for jumping into

fraction of atoms that participate

probability that an atom will jump into an available site

Evaluate terms

G

position

crystallographic sites

occupied empty

Gm

atom vibrates at frequencyD = Debye frequency

success rate of jumping

exp m

b

G

k T

exp mD

b

G

k T

Atomistics from Mechanism

P and [N] differ depending on mechanism

A. Vacancy• P = concentration

of vacancies• [N] = 1 – P 1

B. Interstitial• P = 1 – [N] 1

• [N] = concentration of interstitial atoms

probability that a nearest neighbor site is vacant (available) for jumping into

fraction of atoms that participate

C. Substitutional impurity• P = concentration

of vacancies

• [Nimp] = fixed, < 1

D. Interstitial impurity

• P = 1 – [Nimp] 1

• [Nimp] = fixed, < 1

defect concentrations exp f

b

G

k T

( )exp m f

ob

G GD D

k T

expo

b

QD

k T

Classic Diffusion ProblemExpose a solid material to a gas phase and observe diffusion into the solid

Gas Solid

c

x

cs

co initial concentration

surface concentration

Boundary conditions:

2

2

c cD

t x

at t = 0, c(x) = co 0 x

t > 0, c(x=0) = cs

Solution:( )

1 erf2

o

s o

c x c x

c c Dt

Characteristic diffusion distance & time: set argument = 1, 1-erf(1) = 0.157

t erf(0) 0 c(x) cs

}0 1

2l D

t = 0

t