DIFFUSION IN SOLIDS

DIFFUSION IN SOLIDS

ISSUES TO ADDRESS...

• How does diffusion occur?

• Why is it an important part of processing?

• How can the rate of diffusion be predicted for some simple cases?

• How does diffusion depend on structure and temperature?

Gear from case-hardened steel (C diffusion)

http://web.mse.uiuc.edu/courses/mse280/

Diffusion

Mass transport by atomic motion

Mechanisms

Gases & Liquids – random (Brownian) motion

Solids – vacancy diffusion or interstitial diffusion Cause (Driving force)

Concentration gradient

• Glass tube filled with water.

• At time t = 0, add some drops of ink to one end of the tube.

• Measure the diffusion distance, x, over some time.

t _o t ₁ t ₂ t ₃

x _o x ₁ x ₂ x ₃

Diffusion: gases and liquids

time (s)

x (mm)

100%

Concentration Profiles 0

Cu Ni

• Interdiffusion: in alloys, atoms tend to migrate from regions of large concentration.

Initially After some time

100%

Concentration Profiles 0

Interdiffusion

• Selfdiffusion: In an elemental solid, atoms also migrate.

Label some atoms After some time

A B C

D A

B C

D

Selfdiffusion

This can be observed on metal surface under UHV conditions

with scanning tunneling microscopy

Concentration of Vacancies at temp T

The two processes have an activation energy (eV/atom).

• applies to substitutional impurities

• atoms exchange with vacancies

• rate depends on

(1) number of vacancies

(2) activation energy to exchange.

Vacancy diffusion or interstitial diffusion

Vacancy

Interstitial

T K

E

i i e

N c n

 



More rapid than vacancy diffusion

Interstitial diffusion

Concentration gradient

• 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" planes from shearing.

-hard to crack: C atoms put the surface in compression.

Processing using Diffusion

• Doping Silicon with P for n-type semiconductors:

• Process:

1. Deposit P rich

layers on surface.

2. Heat it.

3. Result: Doped semiconductor regions.

silicon silicon

magnified image of a computer chip

0.5mm

light regions: Si atoms

light regions: Al atoms

Processing using Diffusion

• Flux:

• Flux can be measured for:

- vacancies

- host (A) atoms - impurity (B) atoms

• Empirically determined:

– Make thin membrane of known surface area – Impose concentration gradient

– Measure how fast atoms or molecules diffuse through the membrane

x-direction

Unit area A through

which atoms move.

Modeling rate of diffusion: flux

A = Area of flow

d dt

Q _m ^{ 1} A ^

• Concentration Profile, C(x): [kg/m

]

• Fick's First Law:

Concentration of Cu [kg/m3]

Concentration of Ni [kg/m3]

Position, x Cu flux Ni flux

Steady-state Diffusion

D=diffusion coefficient

D = m

/s

dx

D dC

Q _{m , x} ^{ }

• Steady State: concentration profile not changing with time.

• Apply Fick's First Law:

The slope, dC/dx, must be constant (i.e., doesn't vary with position)

• since Q

= Q

then

Steady-State Diffusion

C(x)

Q

Q

Q

= Q

dx D dC Q _{m , x} ^{ }

right

left dx

dC dx

dC _

Steady-State Diffusion

C

C

x C

C

x

x

Rate of diffusion independent of time

1 2

1

linear 2

if x x

C C

x C dx

dC



 



 

dx

D dC

Q _{m , x} ^{ }

C 1 = 1.2 kg/ m 3

C 2 = 0 .8k g/m 3

Carbon rich

gas

10 m m

Carbon deficient

gas

x1 x2 0 5m

m

D=3x10-11m2/s Steady State = straight line!

• Steel plate at 700

C with geometry shown:

In steady-state, how much carbon transfers from the rich to the deficient side?

Example: C Diffusion in steel plate

J  D C ₂  C ₁

x ₂  x ₁  2.4  10 ^9 kg m ² s

Knowns:

C

= 1.2 kg/m

at 5mm (5 x 10

m) below surface.

C

= 0.8 kg/m

at 10mm (1 x 10

m) below surface.

D = 3 x10

m

/s at 700 C.

700 C

Example: Chemical Protection Clothing

Methylene chloride is a common ingredient of paint removers. Besides being an irritant, it also may be absorbed through skin. When using, protective gloves

should be worn.

If butyl rubber gloves (0.04 cm thick) are used, what is the diffusive flux of methylene chloride through the glove?

Data:

D in butyl rubber: D = 110 x10

cm

/s

surface concentrations: C

= 0.44 g/cm

C

= 0.02 g/cm

Diffusion distance: x

– x

= 0.04 cm

glove C

C

^skin

paint

remover x

x

= 1.16 x 10

g cm

s J

= -D C

- C

x

- x

tb 6D

2



Concentration profile C(x) changes with time

• To conserve matter: • Fick's First Law:

• Governing Eqn.:

Non-Steady-State Diffusion

C(x)

Q

Q

• Fick's Second Law

dt dC dx

Q Q _{m r m l}



 _, 

,

dx D dC Q _{m , x} ^{ } dt

dC dx

dQ _{m  }

2 2

dx C D d

dx

dQ _{m  }

2 2

dx C D d

dt

dC _

• Diffusivity increases with T exponentially (so does Vacancy conc.).

Diffusion and Temperature

D ^ D _o exp æ

è ç ö ø ÷

 E _ACT RT

= pre-exponential [m

/s]

= diffusion coefficient [m

/s]

= activation energy [J/mol or eV/atom]

= gas constant [8.314 J/mol-K]

= absolute temperature [K]

D D _o

E _ACT

R

T

• Experimental Data:

Dinterstitial >> Dsubstitutional C in -Fe

C in -Fe Al in Al Cu in Cu Zn in Cu Fe in -Fe Fe in -Fe

1000 K/ T D (m

/s)

0.5 1.0 1.5

10

10

10

₁₅ ⁰⁰ ₁₀ ⁰⁰ ₆₀ ⁰ ₃₀ ⁰ T (C)

Callister & Rethwisch 3e.

from E.A. Brandes and G.B. Brook (Ed.) Smithells Metals Reference Book,

7th ed., Butterworth-Heinemann, Oxford, 1992.)

Example: Comparing Diffusion in Fe

Is C in fcc Fe diffusing faster than C in bcc Fe?

fcc-Fe: D

=2.3x10

(m

/s) E

=1.53 eV/atom

T = 900 C D=5.9x10

(m

/s) bcc-Fe: D

=6.2x10

(m

/s) E

=0.83 eV/atom

T = 900 C D=1.7x10

(m

/s)

• FCC Fe has both higher activation energy E

and D

(holes larger in FCC).

BCC and FCC phase exist over limited range of T (at varying %C).

Hence, at same T, BCC diffuses faster due to lower E

.

Cannot always have the phase that you want at the %C and T you want!

1000 K/T D (m

/s)

0.5 1.0 1.5

10

10

10

^{15 T(C)}

00 10 00 60 0 30 0

Vacuum cases

1) Gas permeation through chamber’s walls

The pressure gradient depends on the vacuum inside the chamber and is a steady state situation, after initial pumpdown

1° Fick’s law

The initial concentration of gas si decreasing with time since the gas molecules are pumped away. (outgassing)

2° Fick’s law

2) Gas permeation from a solid located inside the chamber

Relative concentration for a sheet of thickness d and different values of Dt/d

dx D dC Q

^{ }

2 2

dx C D d

dt

dC _

Material outgassing

The role for UHV chambers

Not all materials are good for UHV

At RT, the flux stabilizes after few hours

After 20 h, the flux from untreated 3mm thick inox is 1 x 10

mbar L/(s cm

) So for surface area of 20000 cm

, the flux is 2 x 10

mbar L/s

Pumping speed S = Q/p’ = 250 l/s p’ = 2x10

/250 = 8 x10

mbar

Inox machined and degreased after 24 h, flux 5 x 10

mbar L/(s cm

) So for surface area of 20000 cm

, the flux is 1 x 10

mbar L/s

p’ = 1x10

/250 = 4 x10

mbar

In my lab 3 x 10

after two days Not included all evaporators, manipulators, etc

example

Material outgassing

Flux composition: H

0 40%

CO 20%

CO

12%

So to remove H

O and decrease the outgassing flux from material you need to

The entire chamber and instruments for 48 h

BAKE OUT