Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege

[email protected] • ENGR-45_Lec-06_Diffusion_Fick-1.ppt1

Bruce Mayer, PE Engineering-45: Materials of Engineering

Bruce Mayer, PERegistered Electrical & Mechanical Engineer

[email protected]

Engineering 45

Solid Solid StateState

Diffusion-Diffusion-11



Learning Goals - DiffusionLearning 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



InterDiffusionInterDiffusion In a SOLID Alloy Atoms will Move From regions of HI

Concentration to Regions of LOW Concentration Initial Condition After Time+Temp

100%

Concentration Profiles0

Cu Ni100%

Concentration Profiles0



SelfDiffusionSelfDiffusion In an Elemental Solid Atoms are NOT in Static

Positions; i.e., They Move, or DIFFUSE Label Atoms After Time+Temp

A

B

C

DA

B

C

D

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



Diffusion MechanismsDiffusion 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)

increasing elapsed time



Substitutional Diff SimulationSubstitutional Diff Simulation Simulation of

interdiffusion across an interface

Rate of substitutional diffusion depends on:• Vacancy

concentration

• Jumping Frequency



Interstitial Diff SimulationInterstitial 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.



Diffusion in Processing Case1Diffusion 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

ShearResistant

CrackResistant



• Doping Silicon with Phosphorus for n-type semiconductors:• Process:1. Deposit P rich layers on surface.

2. Heat it.

3. Result: Doped semiconductor regions.

silicon

siliconmagnified image of a computer chip

0.5mm

light regions: Si atoms

light regions: Al atoms

Diffusion in Processing Case2Diffusion in Processing Case2



Modeling Diffusion - FluxModeling Diffusion - Flux Flux is the Amount of Material Crossing a

Planar Boundary, or area-A, in a Given Time

J

1A

dMdt

kg

m2s

or

atoms

m2s

x-direction

Unit Area, A, Thru Which Atoms

Move

Flux is a DIRECTIONAL Quantity

J x

J y

J z x

y

z



Concentration Profiles & FluxConcentration Profiles & Flux Consider the Situation Where The

Concentration VARIES with Position• i.e.; Concentration, say C(x), Exhibits a SLOPE or

GRADIENT

The concentration GRADIENT for COPPER

Concentrationof Cu (kg/m3)

Concentrationof Ni (kg/m3)

Position, x

Cu flux Ni fluxx

C

dx

dC

x

Cx

ConGradlimandGradion Concentrat0



Fick’s First Law of DiffusionFick’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

• Fick’s Work (Fick, A., Ann. Physik 1855, 94, 59) lead to this Eqn (1st Law) for J

Position, x

Cu flux Ni fluxx

C

dx

dCDJ



Fick’s First Law cont.Fick’s First Law cont. Consider the

Components of Fick’s 1st Law

Position, x

Cu flux Ni flux

x

C

dx

dCDJ

J • Mass Flux in kg/m2•s• Atom Flux in at/m2•s

dC/dx = Concentration Gradient

• In units of kg/m4 or at/m4

D Proportionality Constant• Units Analysis

kg

mm

Sm

kg

dCJdxD

3

2

1



Fick’s First Law cont.2Fick’s First Law cont.2 Units for D

Position, x

Cu flux Ni flux

x

C

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

kg

mm

Sm

kg

dCJdxD

3

2

1

dx

dCDJ

The NEGATIVE Sign Indicates:



Diffusion and TemperatureDiffusion and Temperature

Diffusion coefficient, D, increases with increasing T → D(T) by:

DDo exp

–

Qd

RT

= 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]

D

DoQd

R

T



Diffusion Types ComparedDiffusion Types Compared

Some D vs T Data

1000 K/T

D (m2/s) C in -Fe

C in -Fe

Al in Al

Fe in -Fe

Fe in -Fe

0.5 1.0 1.510-20

10-14

10-8

T(C)

1500

1000

600

300

The Interstitial Diffusers• C in α-Fe

• C in γ-Fe

Substitutional Diffusers > All Three Self-Diffusion Cases• The Interstitial

Form is More Rapid



STEADY STATE DiffusionSTEADY 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)J x(left)

x

For Steady State the Above Situation may, in Theory, Persist for Infinite time• To Prevent infinite Filling or Emptying of the Box



Steady State Diffusion contSteady State Diffusion cont Since Box Cannot

Be infinitely filled it MUST be the case:

J x(right)J x(left)

x

rightxleftx JJ

Now Apply Fick’s First Law

rightxleftx J

dx

dCDJ

Thus, since D=const

Therefore• While C(x) DOES

change Left-to-Right, the GRADIENT, dC/dx Does NOT– i.e. C(x) has

constant slope

rightleft dx

dC

dx

dC



Example SS DiffusionExample SS Diffusion Iron Plate Processed at

700 °C under Conditions at Right

Find the Carbon Diffusion Flux Thru the Plate

For SS Diffusion

2

21

1xx dx

dCDxJ

dx

dCDxJ

dC/dx f(x)• i.e., The Gradient is

Constant

For const dC/dx

C1 = 1.2kg/m

3

C2 = 0.8kg/m

3

Carbon rich gas

10mm

Carbon deficient

gas

x1 x20 5mm

D=3x10-11 m2/s

Steady State →CONST SLOPE

12

12

21 xx

CC

x

C

dx

dC

dx

dC

xx



Expl SS Diff contExpl SS Diff cont Thus the Gradient

Use In Fick’s 1st Law

or

C1 = 1.2kg/m

3

C2 = 0.8kg/m

3

Carbon rich gas

10mm

Carbon deficient

gas

x1 x20 5mm

D=3x10-11 m2/s

Steady State →CONST SLOPE

4211 /80103 mkgsmJ

dx

dCDJ

x

x

43

3

8080

005.001.0

2.18.0

mkgm

mkg

m

mkg

x

C

smgJ

smkgJ

x

x

2

29

4.2

104.2



WhiteBoard WorkWhiteBoard Work

Problem Similar to 5.9• Hydrogen Diffusion Thru -Iron

-Fe: = 7870 kg/m3

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Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege

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