Engineering 45
Solid State
Diffusion-1
Bruce Mayer, PE
Registered Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Learning Goals - Diffusion
 How Diffusion Proceeds
 How Diffusion Can be Used in
Material Processing
 How to Predict The RATE Of Diffusion
Be Predicted For Some Simple Cases
• Fick’s FIRST and second Laws
 How Diffusion Depends On Structure
And Temperature
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
InterDiffusion
 In a SOLID Alloy Atoms will Move From regions of HI
Concentration to Regions of LOW Concentration
 Initial Condition
 After Time+Temp
100%
Engineering-45: Materials of Engineering
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0
Concentration
Profiles
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
SelfDiffusion
 In an Elemental Solid Atoms are NOT in Static
Positions; i.e., They Move, or DIFFUSE
 Label Atoms
 After Time+Temp
C
A
D
B
 How to Label an ATOM?
• Use a STABLE ISOTOPE as a tag
– e.g.; Label 28Si (92.5% Abundance) with one or both of

29Si
→ 4.67% Abundance

30Si
→ 3.10% Abundance
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Diffusion Mechanisms
 Substitutional Diffusion
• Applies to substitutional impurities
• Atoms exchange position with lattice-vacancies
• Rate depends on:
– Number/Concentration of vacancies (Nv by Arrhenius)
– Activation energy to exchange (the “Kick-Out” reaction)
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Substitutional Diff Simulation
 Simulation of
interdiffusion
across an interface
 Rate of
substitutional
diffusion
depends on:
• Vacancy
concentration
• Jumping Frequency
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Interstitial Diff Simulation
 Applies to interstitial
impurities
 More rapid than
vacancy diffusion.
 Simulation shows
• the jumping of a smaller
atom (gray) from one
interstitial site to another
in a BCC structure. The
interstitial sites
considered here are at
midpoints along the unit
cell edges.
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Diffusion in Processing Case1
 Example: CASE Hardening
• Diffuse carbon atoms into the host
iron atoms at the surface.
• Example of interstitial diffusion
is a case Hardened gear.
 Result: The "Case" is
• hard to deform: C atoms "lock"
xtal planes to reduce shearing
• hard to crack: C atoms put the
surface in compression
Shear
Resistant
Engineering-45: Materials of Engineering
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Crack
Resistant
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Diffusion in Processing Case2
• Doping Silicon with Phosphorus for n-type
semiconductors:
• Process:
1. Deposit P rich
layers on surface.
silicon
2. Heat it.
3. Result: Doped
semiconductor
regions.
silicon
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Modeling Diffusion - Flux
 Flux is the Amount of Material Crossing a
Planar Boundary, or area-A, in a Given Time
 Flux is a DIRECTIONAL
Quantity
x-direction
Unit Area, A, Thru
Which Atoms
Move
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Concentration Profiles & Flux
 Consider the Situation Where The
Concentration VARIES with Position
• i.e.; Concentration, say C(x), Exhibits a SLOPE or
GRADIENT
Cu flux Ni flux
x
Concentration
of Cu (kg/m3)
C
Concentration
of Ni (kg/m3)
Position, x
 The concentration GRADIENT for COPPER
C
Concentrat ion Grad 
x
Engineering-45: Materials of Engineering
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and
dC
lim ConGrad  
x 0
dx
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Fick’s First Law of Diffusion
 Note for the Cu Flux
• Proceeds in the
POSITIVE-x
Direction (+x)
• The Change in C is
NEGATIVE (–C)
 Experimentally
Adolph Eugen Fick
Observed that FLUX
is Proportional to the
Concentration Grad
Engineering-45: Materials of Engineering
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Cu flux Ni flux
x
C
Position, x
• Fick’s Work (Fick, A.,
Ann. Physik 1855,
94, 59) lead to this
Eqn (1st Law) for J
 dC 
J  D 

 dx 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Fick’s First Law cont.
 Consider the
Components of
Fick’s 1st Law
dC
J D
dx
 J
• Mass Flux in kg/m2•s
• Atom Flux in at/m2•s
 dC/dx =
Concentration
Gradient
Engineering-45: Materials of Engineering
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Cu flux Ni flux
x
C
Position, x
• In units of kg/m4
or at/m4
 D  Proportionality
Constant
• Units Analysis
1
kg m3
D  Jdx
 2 m
dC
m S kg
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Fick’s First Law cont.2
 Units for D
3
1
kg m
D  Jdx
 2 m
dC
m S kg
m2/S
 D→
 One More
CRITICAL Issue
dC
J D
dx
 The NEGATIVE
Sign Indicates:
Engineering-45: Materials of Engineering
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Cu flux Ni flux
x
C
Position, x
• Flux “Flows”
DOWNHILL
– i.e., Material Moves From
HI-Concen to LO-Concen
• The Greater the
Negative dC/dx the
Greater the Positive J
– i.e.; Steeper Gradient
increases Flux
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Diffusion and Temperature
 Diffusion coefficient, D, increases
with increasing T → D(T) by:
 Qd 

D  Do exp–
RT


D = diffusion coefficient [m2/s]
Do = pre-exponential constant factor [m2/s]
Qd = activation energy [J/mol or eV/atom]
R = gas constant [8.314 J/mol-K]
T = absolute temperature [K]
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Diffusion Types Compared
 The Interstitial
Diffusers
300
600
1000
10-8
1500
 Some D vs T Data
• C in γ-Fe
D (m2/s)
 Substitutional
Diffusers > All
Three SelfDiffusion Cases
10-14
10-20
0.5
1.0
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• C in α-Fe
T(C)
1.5
1000 K/T
• The Interstitial
Form is More Rapid
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
STEADY STATE Diffusion
 Steady State → Diffusion Profile, C(x) Does
NOT Change with TIME (it DOES change w/ x)
• Example: Consider 1-Dimensional,
X-Directed Diffusion, Jx
J x(left)
J x(right)
x
Concentration, C, in the box does not change w/time.
 For Steady State the Above Situation may,
in Theory, Persist for Infinite time
• To Prevent infinite Filling or Emptying of the Box
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Steady State Diffusion cont
 Since Box Cannot
Be infinitely filled it
MUST be the case:
J x left  J x
right
 Now Apply Fick’s
First Law
J x left
dC
 D
 Jx
dx
right
 Thus, since D=const
Engineering-45: Materials of Engineering
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J x(left)
J x(right)
x
dC
dx
left
dC

dx
right
 Therefore
• While C(x) DOES
change Left-to-Right,
the GRADIENT,
dC/dx Does NOT
– i.e. C(x) has
constant slope
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Example SS Diffusion
 Iron Plate Processed at
700 °C under
Carbon
rich
Conditions at Right
 Find the Carbon
Diffusion Flux Thru the
Plate
gas
dC
dx
 J x2    D
x1
  dC/dx  f(x)
• i.e., The Gradient is
Constant
Engineering-45: Materials of Engineering
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dC
dx
Carbon
deficient
gas
0 x1 x2
 For SS Diffusion
J x1    D
Steady State →
CONST SLOPE
x2
D=3x10-11 m2/s
 For const dC/dx
dC
dC
C C 2 C1



dx x1 dx x 2 x
x 2  x1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
Expl SS Diff cont
 Thus the Gradient
C 0.8  1.2 kg m3

0.01  0.005m
x
Carbon
rich
gas
kg m3
 80
 80 kg m 4
m
Steady State →
CONST SLOPE
Carbon
deficient
gas
0 x1 x2
D=3x10-11 m2/s
 Use In Fick’s 1st Law
dC
J x  D
dx
J x   3 10 11 m 2 s  80kg / m 4


Engineering-45: Materials of Engineering
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 or

J x  2.4 109 kg m 2  s
J x  2.4 g m 2  s
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt
WhiteBoard Work
 Problem Similar to 5.9
• Hydrogen Diffusion Thru -Iron
-Fe:  = 7870 kg/m3
Engineering-45: Materials of Engineering
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-45_Lec-06_Diffusion_Fick-1.ppt