Download - Chapter 2 Diffusion

Institute of Materials Science

Chapter 2: Diffusion in Solids

2.1 Fundamental Equations of Diffusion

2.2 Interstitial Diffusion

2.3 Einstein Relation – Atomic Mobility

2.4 Substitutional Diffusion (Self-diffusion, Vacancy Diffusion, Darken Relation)

2.5 Solutions to the Diffusion Equations

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WHY STUDY DIFFUSION?

• Materials often heat treated to improve properties

• Atomic diffusion occurs during heat treatment

• Depending on situation higher or lower diffusion rates desired

• Heat treating temperatures and times, and heating or cooling rates can be determined using the mathematics/physics of diffusion

Example: steel gears are “case-hardened” bydiffusing C or N to outer surface

DIFFUSION IN SOLIDS


• 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

Diffusion: Material transport by atomic or particle transport from region of high to low concentration (???)

DIFFUSION IN SOLIDS


100%

Concentration Profiles0

Cu Ni

• 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 IN SOLIDS


Correct Definition of Diffusion

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Fig. 2.1 Free energy and chemical potential changes during diffusion. (a) and (b) ‘down-hill’ diffusion. (c) and (d) ‘up-hill’ diffusion. (e) therefore A atoms move from (2) to (1), therefore B atoms move from (1) to (2). (f) therefore A atoms move from (1) to (2), therefore B atoms move from (2) to (1).

12AA

21BB

12BB

21AA


Correct Definition of Diffusion

Diffusion: Material transport by atomic transport from regions of high to low

CHEMICAL POTENTIALS


2.1 Fundamental Equations of Diffusion



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Interstitial diffusion by random jumps in a concentration gradient.



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Figure 2.2.1 The derivation of Fick’s second law.


2.3 Einstein Relation – Atomic Mobility

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2.4 Substitutional Diffusion(Self-diffusion, Vacancy diffusion, Darken Relation)

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Interdiffusion and vacancy flow. (a) Composition profile after interdiffusion of A and B. (b) The corresponding fluxes of atoms and vacancies as a function of position x. (c) The rate at which the vacancy concentration would increase or decrease if vacancies were not created or destroyed by dislocation climb.


• Self-diffusion: In an elemental solid, atoms also migrate.

Label some atoms After some time

A

B

C

DA

B

C

D

2.4 Substitutional Diffusion


2.4 Substitutional DiffusionVacancy Diffusion

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Fig. 2.16 The jumping of atoms in one direction can be considered as the jumping of vacancies in

the other direction.

Fig. 2.17 (a) before, (b) after: a vacancy is absorbed at a jog on an edge dislocation (positive climb). (b) before, (a) after: a vacancy is created by negative climb of an edge dislocation. (c) Perspective drawing of a jogged edge dislocation.

Fig. 2.18 A flux of vacancies causes the atomic planes to move through the specimen.


Substitutional Diffusion:

• applies to substitutional impurities• atoms exchange with vacancies• rate depends on: -- number of vacancies -- temperature -- activation energy to exchange.

increasing elapsed time




• Also called energy barrier for diffusion

Initial state Final stateIntermediate state

Energy Activation energy


• Copper diffuses into a bar of aluminum.

• Boundary conditions:For t = 0, C = C0 at x > 0For t > 0, C = Cs at x = 0

C = C0 at x = ∞

pre-existing conc., Co of copper atoms

Surface conc., Cs of Cu atoms bar

Co

Cs

position, x

C(x,t)

tot1

t2t3 Adapted from

Fig. 5.5, Callister 6e.

dCdt

=Dd2C

dx2



• Copper diffuses into a bar of aluminum.

• General solution:

"error function"

C(x,t) Co

Cs Co

1 erfx

2 Dt

pre-existing conc., Co of copper atoms

Surface conc., Cs of Cu atoms bar

Co

Cs

position, x

C(x,t)

tot1

t2t3 Adapted from

Fig. 5.5, Callister 6e.



• Suppose we desire to achieve a specific concentration C1 at a certain point in the sample at a certain time

Dt

xerf

CC

CtxC

s 21

),(

0

0

Dt

xerf

CC

CC

s 21constant

0

01

becomes

constant 2

Dt

x



• The experiment: record combinations of t and x that kept C constant.

to

t1

t2

t3

xo x1 x2 x3

• Diffusion depth given by:

xi Dti

C(xi,ti) CoCs Co

1 erfxi

2 Dti

= (constant here)



• 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?

(Dt)500ºC =(Dt)600ºCs

C(x,t) CoC Co

=1 erfx

2Dt

• Result: Dt should be held constant.

• Answer:Note: valuesof D areprovided here.

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

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

t500(Dt)600

D500

110hr

4.8x10-14m2/s

5.3x10-13m2/s 10hrs


Download - Chapter 2 Diffusion

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