MATERIALS SCIENCE

SSP 2412 DIFFUSION IN SOLIDS

Prof. Dr. Samsudi Sakrani

Physics Dept. Faculty of Science

Universiti Teknologi Malaysia

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Contents

• Atomic mechanisms of diffusion

• Mathematics of diffusion

• Influence of temperature and diffusing species on

Diffusion rate

DIFFUSION IN SOLIDS

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WHAT IS DIFFUSION? Phenomenon of material transport by atomic or particle

transport from region of high to low concentration

• What forces the particles to go from left to right?

• Does each particle “know” its local concentration?

• Every particle is equally likely to go left or right!

• At the interfaces in the above picture, there are more particles going right than left this causes an average “flux” of particles to the right!

• Largely determined by probability & statistics 3

• 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.

DIFFUSION DEMO

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• Interdiffusion: In an alloy or “diffusion couple”, atoms tend

to migrate from regions of large to lower concentration.

Initially (diffusion couple) After some time

100%

Concentration Profiles0

Adapted from

Figs. 5.1 and

5.2, Callister

6e.

DIFFUSION: THE PHENOMENA (1)

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• Self-diffusion: In an elemental solid, atoms

also migrate.

Label some atoms After some time

A

B

C

D

DIFFUSION: THE PHENOMENA (2)

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Conditions for diffusion:

• there must be an adjacent empty site

• atom must have sufficient energy to break bonds with its

neighbors and migrate to adjacent site (“activation” energy)

DIFFUSION MECHANISMS Diffusion at the atomic level is a step-wise migration of atoms from

lattice site to lattice site

Higher the temperature, higher is the probability that an atom will have

sufficient energy

hence, diffusion rates increase with temperature

Types of atomic diffusion mechanisms:

• substitutional (through vacancies)

• interstitial

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Substitutional Diffusion:

• applies to substitutional impurities

• atoms exchange with vacancies

• rate depends on:

-- number of vacancies

-- temperature

-- activation energy to exchange.

DIFFUSION MECHANISMS

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ACTIVATION ENERGY FOR DIFFUSION

• Also called energy barrier for diffusion

Initial state Final state Intermediate state

Energy Activation energy

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• Simulation of

interdiffusion

across an interface:

• Rate of substitutional

diffusion depends on: -- vacancy concentration

-- activation energy (which is

related to frequency of jumping).

(Courtesy P.M. Anderson)

DIFFUSION SIMULATION

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(Courtesy P.M. Anderson)

• Applies to interstitial impurities.

• More rapid than vacancy

diffusion (Why?).

• Interstitial atoms smaller and

more mobile; more number of

interstitial sites than vacancies

INTERSTITIAL SIMULATION

• 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.

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• Case Hardening: -- Example of interstitial

diffusion is a case

hardened gear.

-- Diffuse carbon atoms

into the host iron atoms

at the surface.

• Result: The "Case" is --hard to deform: C atoms

"lock" planes from shearing.

PROCESSING USING DIFFUSION (1)

--hard to crack: C atoms put

the surface in compression.

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• Doping Silicon with P for n-type semiconductors:

1. Deposit P rich

layers on surface.

2. Heat it.

3. Result: Doped

semiconductor

regions.

silicon

silicon

Fig. 18.0,

Callister 6e.

PROCESSING USING DIFFUSION (2)

• Process

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• Flux: amount of material or atoms moving past a unit area in unit time

Flux, J = M/(A t)

• Directional Quantity

• Flux can be measured for: --vacancies

--host (A) atoms

--impurity (B) atoms

MODELING DIFFUSION: FLUX

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• Concentration Profile, C(x): [kg/m3]

• Fick's First Law:

Concentration

of Cu [kg/m3]

Concentration

of Ni [kg/m3]

Position, x

Cu flux Ni flux

• The steeper the concentration profile,

the greater the flux!

Adapted from

Fig. 5.2(c),

Callister 6e.

CONCENTRATION PROFILES & FLUX

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• Steady State: Steady rate of diffusion from one end to the other.

Implies that the concentration profile doesn't change with time. Why?

• Apply Fick's First Law:

• Result: the slope, dC/dx, must be constant

(i.e., slope doesn't vary with position)!

Jx D

dC

dx

dC

dxleft

dC

dxright

• If Jx)left = Jx)right , then

STEADY STATE DIFFUSION

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• Steel plate at

700C with

geometry

shown:

• Q: How much

carbon transfers

from the rich to

the deficient side?

Adapted from

Fig. 5.4,

Callister 6e.

EX: STEADY STATE DIFFUSION

Note: Steady state does not set in instantaneously. 17

STEADY STATE DIFFUSION: ANOTHER PERSPECTIVE

• Hose connected to tap; tap turned on

• At the instant tap is turned on, pressure is high at the tap end, and 1 atmosphere at the other end

• After steady state is reached, pressure linearly drops from tap to other end, and will not change anymore

Tap end Flow end

Pressure Increasing time

Steady state

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• Concentration profile,

C(x), changes

w/ time.

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

• Governing Eqn.:

NON STEADY STATE DIFFUSION

Fick’s second law

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• Copper diffuses into a bar of aluminum.

• Boundary conditions:

For t = 0, C = C0 at x > 0

For t > 0, C = Cs at x = 0

C = C0 at x = ∞

Co

Cs

position, x

C(x,t)

tot1

t2t3 Adapted from

Fig. 5.5,

Callister 6e.

EX: NON STEADY STATE DIFFUSION

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• Copper diffuses into a bar of aluminum.

• General solution:

"error function"

Values calibrated in Table 5.1, Callister 6e.

Co

Cs

position, x

C(x,t)

tot1

t2t3 Adapted from

Fig. 5.5,

Callister 6e.

EX: NON STEADY STATE DIFFUSION

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• Suppose we desire to achieve a specific concentration C1

at a certain point in the sample at a certain time

PROCESS DESIGN EXAMPLE

Dt

xerf

CC

CtxC

s 21

),(

0

0

Dt

xerf

CC

CC

s 21constant

0

01

becomes

constant 2

Dt

x

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• The experiment: record combinations of

t and x that kept C constant.

• Diffusion depth given by:

C(xi, t i ) Co

Cs Co1 erf

xi

2 Dt i= (constant here)

DIFFUSION DEMO: ANALYSIS

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• Experimental result: x ~ t0.58

• Theory predicts x ~ t0.50 • Reasonable agreement!

DATA FROM DIFFUSION DEMO

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• Copper diffuses into a bar of aluminum.

• 10 hours at 600C gives desired C(x).

• How many hours would it take to get the same C(x)

if we processed at 500C, given D500 and D600?

• Result: Dt should be held constant.

• Answer: Note: values

of D are

provided here.

Key point 1: C(x,t500C) = C(x,t600C).

Key point 2: Both cases have the same Co and Cs.

PROCESSING QUESTION

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• Diffusivity increases with T.

DIFFUSION AND TEMPERATURE

• Remember vacancy concentration: NV = N exp(-QV/kT)

• QV is vacancy formation energy (larger this energy,

smaller the number of vacancies)

• Qd is the activation energy (larger this energy, smaller

the diffusivity and lower the probability of atomic diffusion)

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ACTIVATION ENERGY FOR DIFFUSION

• Also called energy barrier for diffusion

Initial state Final state Intermediate state

Energy Activation energy

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• Experimental Data:

D has exp. dependence on T

Recall: Vacancy does also! Dinterstitial >> Dsubstitutional

C in -FeC in -Fe Al in Al

Cu in Cu

Zn in Cu

Fe in -FeFe in -Fe

Adapted from Fig. 5.7, Callister 6e. (Date for Fig. 5.7 taken from E.A.

Brandes and G.B. Brook (Ed.) Smithells Metals Reference Book, 7th

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

DIFFUSION AND TEMPERATURE

NOTE: log(D) = log(D0) – Qd/(RT)

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Diffusion FASTER for...

• open crystal structures

• lower melting T materials

• materials w/secondary

bonding

• smaller diffusing atoms

• lower density materials

Diffusion SLOWER for...

• close-packed structures

• higher melting T materials

• materials w/covalent

bonding

• larger diffusing atoms

• higher density materials

SUMMARY: STRUCTURE & DIFFUSION

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