Transcript Document 7498324

Introduction To Materials Science, Chapter 5, Diffusion
Chapter 5 Outline
Diffusion how atoms move in solids
Diffusion mechanisms
 Vacancy diffusion
 Interstitial diffusion
 Impurities
 Mathematics of diffusion
 Steady-state diffusion (Fick’s first law)
 Nonsteady-State Diffusion (Fick’s second law)
 Factors that influence diffusion
 Diffusing species
 Host solid
 Temperature
 Microstructure
5.4 Nonsteady-State Diffusion – Not Covered / Not
Tested
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Introduction To Materials Science, Chapter 5, Diffusion
What is diffusion?
Diffusion transport by atomic
motion.
Inhomogeneous material can become
homogeneous by diffusion. Temperature
should be high enough to overcome energy
barrier.
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Introduction To Materials Science, Chapter 5, Diffusion
Inter-diffusion vs. Self-diffusion
Concentration Gradient  Interdiffusion
(or Impurity Diffusion).
(Heat)
Before
After
Self-diffusion:
one-component material, atoms are of same
type.
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Introduction To Materials Science, Chapter 5, Diffusion
Diffusion Mechanisms (I)
Vacancy diffusion
Atom migration
Before
Vacancy migration
After
To jump from lattice site to lattice site, atoms need
energy to break bonds with neighbors, and to
cause the necessary lattice distortions during jump.
Therefore, there is an energy barrier.
Energy comes from thermal energy of atomic
vibrations
(Eav ~ kT)
Atom flow is opposite to vacancy flow direction.
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Introduction To Materials Science, Chapter 5, Diffusion
Diffusion Mechanisms (II)
Interstitial diffusion
Interstitial atom
before diffusion
Interstitial atom
after diffusion
Generally faster than vacancy diffusion
because bonding of interstitials to
surrounding atoms is normally weaker and
there are more interstitial sites than
vacancy sites to jump to.
Smaller energy barrier
Only small impurity atoms (e.g. C, H, O) fit
into interstitial sites.
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Introduction To Materials Science, Chapter 5, Diffusion
Flux of diffusing atoms, J.
Number of atoms diffusing through
unit area per unit time [atoms/(m2s)]
or
Mass of atoms diffusing through unit
area per unit time
[kg/(m2 s)]
Mass: J = M / (A t)  (1/A) (dM/dt)
J
A
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Introduction To Materials Science, Chapter 5, Diffusion
Steady-State Diffusion
Diffusion flux does not change with time
Concentration profile:
Concentration (kg/m3) vs. position
Concentration gradient: dC/dx (kg / m4)
dC C C A  C B


dx x x A  x B
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Introduction To Materials Science, Chapter 5, Diffusion
Steady-State Diffusion
Fick’s first law: J proportion to dC/dx
dC
J  D
dx
D=diffusion coefficient
Concentration gradient is ‘driving force’
Minus sign means diffusion is ‘downhill’:
toward lower concentrations
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Introduction To Materials Science, Chapter 5, Diffusion
Nonsteady-State Diffusion
(not tested)
Concentration changing with time
Fick’s second law
C
J

t
x
 2C
D 2
x
Find C(x,t)
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Introduction To Materials Science, Chapter 5, Diffusion
Atom needs enough thermal energy to break
bonds and squeeze through its neighbors.
Energy
Energy needed energy barrier
Called the activation energy Em (like Q)
Em Vacancy
Atom
Distance
Diagram for Vacancy Diffusion
Diffusion Thermally Activated Process
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Introduction To Materials Science, Chapter 5, Diffusion
Diffusion – Thermally Activated Process
Room temperature (kBT = 0.026 eV)
Typical activation energy Em (~ 1 eV/atom)
(like Qv)
Therefore, a large fluctuation in energy is
needed for a jump.
Probability of a fluctuation or frequency of
jump, Rj
E

R j  R0 exp   m

k
T
B 

R0 = attempt frequency proportional to
vibration frequency
Swedish chemist Arrhenius
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Introduction To Materials Science, Chapter 5, Diffusion
Calculate Activated Diffusion
1. Probability of finding a vacancy in an
adjacent lattice site (Chap. 4):
Q

times
P  Const.exp   v

k
T
B 
 fluctuation
2. Probability of thermal
Em


R j  R0 exp  

k
T
B 

The diffusion coefficient = Multiply
E
 exp   QV

D  Const.exp   m

kBT 
kBT 


Qd


D  D0 exp  

k
T
B 

Arrhenius dependence.
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Introduction To Materials Science, Chapter 5, Diffusion
Diffusion – Temperature Dependence
dC
J  D
dx
Diffusion coefficient is the measure of
mobility of diffusing species.
Q
  D exp   Qd

D  D0 exp   d



0
RT
kT




D0 – temperature-independent (m2/s)
Qd – the activation energy (J/mol or eV/atom)
R – the gas constant (8.31 J/mol-K)
or
kB - Boltzman constant ( 8.6210-5 eV/atom-K)
T – absolute temperature (K)
Qd  1 
ln D  ln D0   
R T 
Qd  1 
or log D  log D0 
 
2.3R  T 
Arrhenius Plots
(lnD) vs. (1/T) or (logD) vs. (1/T)
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Introduction To Materials Science, Chapter 5, Diffusion
Diffusion – Temperature Dependence (II)
Graph of log D vs. 1/T has slop of –Qd/2.3R,
intercept of ln Do
Qd  1 
log D  log D 0 
 
2.3R  T 
 log D1  log D 2 
Qd  2.3R 

 1 T1  1 T2 
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Introduction To Materials Science, Chapter 5, Diffusion
Diffusion – Temperature Dependence (III)
Arrhenius plot:
Diffusivity for metallic systems
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Introduction To Materials Science, Chapter 5, Diffusion
Diffusion of different species
Smaller atoms diffuse more readily
Diffusion faster in open lattices or in
open directions
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Introduction To Materials Science, Chapter 5, Diffusion
Diffusion: Role of the microstructure (I)
Self-diffusion coefficients for Ag
Depends on diffusion path
Grain boundaries and surfaces less restrictive
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Introduction To Materials Science, Chapter 5, Diffusion
Diffusion: Role of the microstructure (II)
Plots are from computer simulations
Initial positions are shown by the circles, paths are
shown by lines. See difference between mobility in
the bulk and in the grain boundary.
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Introduction To Materials Science, Chapter 5, Diffusion
Factors that Influence Diffusion
 Temperature - diffusion rate increases
very rapidly with increasing temperature
 Diffusion mechanism - interstitial is
usually faster than vacancy
 Diffusing and host species - Do, Qd is
different for every solute, solvent pair
 Microstructure - diffusion faster in
polycrystalline vs. single crystal materials
because of the rapid diffusion along grain
boundaries and dislocation cores.
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Introduction To Materials Science, Chapter 5, Diffusion
Summary
Make sure you understand
Activation energy
Concentration gradient
Diffusion
Diffusion coefficient
Diffusion flux
Driving force
Fick’s first and second
laws
Interdiffusion
Interstitial diffusion
Self-diffusion
Steady-state diffusion
Vacancy diffusion
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