1. Solid State Diffusion-1
Engineering 45
Solid State Diffusion-1
Bruce Mayer, PE
Registered Electrical & Mechanical Engineer

2. 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

3. InterDiffusion
In a SOLID Alloy Atoms will Move From regions of HI Concentration to Regions of LOW Concentration
Initial Condition
After Time+Temp

4. SelfDiffusion
In an Elemental Solid Atoms are NOT in Static Positions; i.e., They Move, or DIFFUSE
Label Atoms
After Time+Temp
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

5. 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)

6. Substitutional Diff Simulation
Simulation of interdiffusion across an interface
Rate of substitutional diffusion depends on:
Vacancy concentration
Jumping Frequency

7. 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.

8. 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
Crack Resistant
Shear Resistant

9. Diffusion in Processing Case2
• Doping Silicon with Phosphorus for n-type semiconductors:
• Process:
1. Deposit P rich
layers on surface.
More on this lateR
2. Heat it.
3. Result: Doped
semiconductor
regions.
More on this later

10. Modeling Diffusion - Flux
Flux is the Amount of Material Crossing a Planar Boundary, or area-A, in a Given Time
x-direction
Unit Area, A, Thru Which Atoms Move
Flux is a DIRECTIONAL Quantity

11. Concentration Profiles & Flux
Consider the Situation Where The Concentration VARIES with Position
i.e.; Concentration, say C(x), Exhibits a SLOPE or GRADIENT
Concentration of Cu (kg/m3)
Concentration of Ni (kg/m3)
Position, x
Cu flux Ni
flux x C
The concentration GRADIENT for COPPER

12. 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
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

13. Fick’s First Law cont. Consider the Components of Fick’s 1st Law
Cu flux Ni
flux
Consider the Components of Fick’s 1st Law x C
Position, x
In units of kg/m4 or at/m4
D  Proportionality Constant
Units Analysis
J 
Mass Flux in kg/m2•s
Atom Flux in at/m2•s
dC/dx = Concentration Gradient

14. Fick’s First Law cont.2 Units for D D → m2/S One More CRITICAL Issue
Cu flux Ni
flux
Units for D x C
Position, x
D → m2/S
One More CRITICAL Issue
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
The NEGATIVE Sign Indicates:

15. Diffusion and Temperature
Diffusion coefficient, D, increases with increasing T → D(T) by: D = Do
exp æ è ç ö ø ÷ – Qd R T D
= pre-exponential constant factor [m2/s]
= diffusion coefficient [m2/s]
= activation energy [J/mol or eV/atom]
= gas constant [8.314 J/mol-K]
= absolute temperature [K] Do Qd R T

16. Diffusion Types Compared
Some D vs T Data
The Interstitial Diffusers
C in α-Fe
C in γ-Fe
Substitutional Diffusers > All Three Self-Diffusion Cases
The Interstitial Form is More Rapid
1500
1000
T(C)
600
300
10-8
D (m2/s)
C in g-Fe
C in a-Fe
10-14
Fe in a-Fe
Fe in g-Fe
Al in Al
10-20
1000 K/T
0.5
1.0
1.5

17. 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
Concentration, C, in the box does not change w/time. J x
(right)
(left)
For Steady State the Above Situation may, in Theory, Persist for Infinite time
To Prevent infinite Filling or Emptying of the Box

18. Steady State Diffusion contJ x
(right)
(left)
Since Box Cannot Be infinitely filled it MUST be the case:
Now Apply Fick’s First Law
Therefore
While C(x) DOES change Left-to-Right, the GRADIENT, dC/dx Does NOT
i.e. C(x) has constant slope
Thus, since D=const

19. Steady State → CONST SLOPE
Example SS Diffusion C 1
= 1.2kg/m 3 2
= 0.8kg/m
Carbon
rich
gas
10mm
deficient x1 x2
5mm
D=3x10-11 m2/s
Steady State → CONST SLOPE
Iron Plate Processed at 700 °C under Conditions at Right
Find the Carbon Diffusion Flux Thru the Plate
For SS Diffusion
For const dC/dx
 dC/dx  f(x)
i.e., The Gradient is Constant

20. Steady State → CONST SLOPE1
= 1.2kg/m 3 2
= 0.8kg/m
Carbon
rich
gas
10mm
deficient x1 x2
5mm
D=3x10-11 m2/s
Steady State → CONST SLOPE
Expl SS Diff cont
Thus the Gradient
Use In Fick’s 1st Law or

21. WhiteBoard Work Problem Similar to 5.9 Hydrogen Diffusion Thru -Iron
-Fe:  = 7870 kg/m3