Diffusion

Diffusion means atoms moving and changing places. This happens

in solids and liquids, exactly in the same way that an unpleasant

smell moves from one part of a room to another even without wind,

then dissipates after some time.

Partial Molar Free Energy=Chemical Potential

Gi i

i P1 i Po P

Ficks First Law

atomic flux is proportional to the chemical potential gradient

Ji Dxi

d

d

for an ideal solution

Ji Dxi

d

d

DxCi

d

d

Ji DC

kT

xi

d

d

For an ideal solution

i o kT ln Ci

Substitution into the derivative term with only composition varying with x

Ji DxCi

d

d

The driving force for diffusion is the chemical potential gradient.

For diffusion due to a pressure gradient (as in sintering)

JD

k TC

xPd

d

Preview: In steel, strength goes up but toughness goes down with C.

Design application: Carburization of a transmission gear.

Attributes needed?

How to do it.

Attributes needed?

How to do it.

“Carburizing”

2 CO CO2 + C (in solution)

Carbon diffuses from the surfaceInto the steel

Mechanism of diffusion in solidsDiffusion of an individual

atom is random and

probabilistic. Thus, the

interstitial black atom has

an equal probability of

moving up, down, left or

right.

Mechanism of diffusion in solidsIf we have a concentration

gradient, overall atomic

movement is not random,

even though individual

atom movement is.

In this example, since some of the black atoms will move to

the right on average, the concentration of black atoms will increase

on the right and decrease on the left. This is similar

to the dissipation of the concentrated unpleasant odor in the room.

Mechanisms of diffusion in solids

At equilibrium, there will

be a random distribution of

black atoms, and no

concentration gradient.

A common heat treatment for cast metal alloys is

homogenization, or heating for a long time at high temperature

to allow the chemical segregation to even out in this way.

Mechanism of diffusion in solids

Diffusion Couples

Eventually Cu, Ni

x

C

Concentration gradient = dC/dx

is the slope of the curves.

Mass transport, or Flux

• Flux is a measurement of the number of atoms per unit

area that cross a particular plane per unit time.

M = mass, A = area, t = time

Fick’s first law

• We saw that the concentration gradient affects the direction of diffusion. It also affects the rate of diffusion. For simplicity let us assume the concentration gradient is a constant, then we can say:

• where D is the “diffusivity”, which depends on temperature and

• which atom is diffusing in which material. – This says that the mass transport is directly proportional to the

concentration gradient.

Steady state diffusion

dC/dx = concentration gradient = slope = constant = C/x = (CA-CB)/(xA-xB)

ALE

What affects diffusion rates?

• Diffusion is a thermally-activated process. This means that it is accelerated by temperature. Using a higher temperature will result in more diffusion in a given time if there is a composition gradient present (could accomplish the same effect by prolonging the time to diffuse at a lower temperature)

• Time: straightforward result of Fick’s First Law:

J is mass/(time x area) or atoms/(time xarea). Total mass diffusing across a plane of area A is just J x t, if J is constant. Otherwise M = Jdt

What affects diffusion rates?

• Temperature: Comes in mainly through the diffusivity,

D:

ALE: Take the natural log of both sides of Eqn 6.8,

and construct a plot that would be useful for looking at D(T)

ALE - Answer

)ln(1

ln

)ln(ln

od

do

DTR

QD

RT

QDD

y = m x + b

Nonsteady-State Diffusion

• Most practical diffusion situations are nonsteady-state. That is, the concentration gradient = f(time), i.e. dC/dx

changes.

Gear surface

Gas

Gear interior

Nonsteady-State diffusion

• To understand these problems, we need to solve a differential equation called Fick’s Second Law:

• It is clear that the concentration gradient can vary with time through the term dC/dt.

• We are not going to get into this level of detail in MY2100.

D 1018 m

2

s

Ci10

21

m3

x 107

m

C x t( )Ci

2 D t exp

x m( )2

4 D t

x

x 107 9.9 10

8 107

00

1 1022

2 1022

3 1022

C x 100

s C x 10

1s

C x 102

s

x

The solution to the PDE is obtained by conversion to an ODE.

x t( )x

2 D t

Define

Use the Chain Rule

Cd

d td

d

D

ddx

d

d Cd

d xd

d

2

Cd

d

2

Cd

d

2

D 1018 m

2

s

Ci10

21

m3

x 107

m

C x t( )Ci

2 D t exp

x m( )2

4 D t

x

x 0 108 10

6

0 5 108

0

5 1021

1 1022

C x 101

s C x 10

2s

C x 103

s

x

Semiconductor

Dopant Rich Layer, Ci , ∆x

C1 x t( )

Ci

2

2 D t exp

x m 108

m 2

4 D t

x

C3 x t( )

Ci

4

2 D t exp

x m 10( )8

3 m 2

4 D t

xC2 x t( )

Ci

3

2 D t exp

x m 10( )8

2 m 2

4 D t

x

00

1 1021

2 1021

3 1021

C x 102

s C1 x 10

2s

C2 x 102

s C3 x 10

2s

x0

0

5 1021

1 1022

C x 101

s C1 x 10

1s

C2 x 101

s C3 x 10

1s

x

We can add up the solutions from thin films displaced throughout the volume

C x t( ) C surface C initial C surface erfx

2 D t

2 0 21

0

10.995

0.995

erf ( )

22

2 COCO2 +C(Fe)

2at% C at surface0.2at% initially in low carbon steelD=10^(-12) m^2/s

What is the time required to get 1at% at 10^-4 m?C x t( ) Csurface

Cinitial Csurface erfx

2 D t

2 1 0 1 21

0

1

erf ( )

1 20.2 2

0.556

C x t( ) C surface

C initial C surface erfx

2 D t

erf 0.54( ) 0.555

0.54

0.5410

4

1 1012 t

34293

36009.526 hr