Defects in Solids

1
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 ?
2
Diffusional Processes
diffusion coefficient
concentration gradient
Pd
flux
H2
response ? driving force
CO
Ficks first law (similar to Ohms
law) phenomenological
CO2
H
Applies under steady state conditions c(x) ? f(t)
Units of D
x
hydrogen separation membrane
3
Diffusional Processes
Non steady-state c(x) f(t)
Continuity requirements
Ficks second law
if D ? f(x)
4
Atomistics of diffusion
planes of atoms tracer species with a
concentration gradient c concentration /cm3 n
/cm2 density in the plane n c?ao n1
on plane (1) n2 on plane (2)
Net from plane (1) to plane (2)
Flux from plane (1) to plane (2)
(½ jump to the left)
Flux from plane (2) to plane (1)
5
Mechanisms of Diffusion
Vacancy
net transport
vacancy to right atom to left
Interstitial (self or impurity)
6
Atomistics from Mechanism
fraction of atoms that participate
jump distance
G
probability that an atom will jump into an
available site
geometric constant 1/( nearest neighbor
sites)
probability that a nearest neighbor site is
vacant (available) for jumping into
Evaluate terms
atom vibrates at frequency nD Debye frequency
w
crystallographic sites
G
success rate of jumping
DGm
position
7
Atomistics from Mechanism
fraction of atoms that participate
P and N differ depending on mechanism
probability that a nearest neighbor site is
vacant (available) for jumping into

• Substitutional impurity
• P concentration of vacancies
• N fixed, lt 1
• Interstitial impurity
• P 1
• N fixed, lt 1

• Vacancy
• P concentration of vacancies
• N 1
• Interstitial
• P 1
• N concentration of interstitial atoms

defect concentrations ??
8
Classic Diffusion Problem
Expose a solid material to a gas phase and
observe diffusion into the solid
surface concentration
cs
Gas
Solid
c
t ?
t 0
co
initial concentration

0 ? 1
x
Boundary conditions
Solution
at t 0, c(x) co 0 ? x ? ? t gt 0,
c(x0) cs
t ? ? ? erf(0) ? 0 ? c(x) ? cs
Characteristic diffusion distance time set
argument 1, 1-erf(1) 0.157