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Chapter 2: Diffusion in Solids2.1 2.2 2.3 2.4 Fundamental Equations of Diffusion Interstitial Diffusion Einstein Relation Atomic Mobility Substitutional Diffusion (Self-diffusion, Vacancy Diffusion, Darken Relation) Solutions to the Diffusion Equations

2.5

MMAT 305

Institute of Materials Science

DIFFUSION IN SOLIDS

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 by diffusing C or N to outer surfaceInstitute of Materials Science

DIFFUSION IN SOLIDSDiffusion: 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 & statisticsInstitute of Materials Science

DIFFUSION IN SOLIDS Interdiffusion: In an alloy or diffusion couple, atoms tendto migrate from regions of large to lower concentration.Initially (diffusion couple) After some time

Adapted from Figs. 5.1 and 5.2, Callister 6e.

Cu 100%

Ni

100%

0

Concentration Profiles

0 Concentration ProfilesInstitute of Materials Science

Correct Definition of Diffusion

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

MMAT 305

Institute of Materials Science

Correct Definition of Diffusion

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

Institute of Materials Science

2.1 Fundamental Equations of Diffusion

Institute of Materials Science

2.2 Interstitial Diffusion1 2

a)

atoms of parent lattice B interstitials

b) CB

axC B a ! (C xxX

Interstitial diffusion by random jumps in a concentration gradient.MMAT 305 Institute of Materials Science

2.2 Interstitial Diffusiona) C

0 J b)J1 J2

X X H X

X

0 c)J1

1

2

XJ2

HX area A

Figure 2.2.1 The derivation of Ficks second law.

MMAT 305

Institute of Materials Science

2.3 Einstein Relation Atomic Mobility

G2

(Gm1 3

a

MMAT 305

Institute of Materials Science

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

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.

MMAT 305

Institute of Materials Science

2.4 Substitutional Diffusion Self-diffusion: In an elemental solid, atomsalso migrate.Label some atoms After some time

C C A A D B B D

Institute of Materials Science

2.4 Substitutional DiffusionVacancy Diffusion

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. 2.16 The jumping of atoms in one direction can be considered as the jumping of vacancies in the other direction. Fig. 2.18 A flux of vacancies causes the atomic planes to move through the specimen. MMAT 305 Institute of Materials Science

2.4 Substitutional DiffusionSubstitutional Diffusion: applies to substitutional impurities atoms exchange with vacancies rate depends on: -- number of vacancies -- temperature -- activation energy to exchange.

i

r

i g l p

d timInstitute of Materials Science

2.4 Substitutional Diffusion

Initial state

Intermediate state

Final state

Energy

Activation energy

Also called energy barrier for diffusionInstitute of Materials Science

2.5 Solutions to the Diffusion Equations Copper diffuses into a bar of aluminum.S rf f e t .,

(x,t)

pr - xi ti

.,

f

b r pper t

to t

t3 t2 position, x

Adapted from Fig. 5.5, Callister 6e.

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

d2C dC =D 2 dt dxInstitute of Materials Science

2.5 Solutions to the Diffusion Equations Copper diffuses into a bar of aluminum.S rf ce conc., toms s of

Cs

C(x,t)

pre-existing onc.,

b r o of copper toms

o

to t

t3 t2 position, x

Adapted from Fig. 5.5, Callister 6e.

General solution:

x Co ! C(x, t) 1 erf 2 Dt Cs Co"error function"

Institute of Materials Science

2.5 Solutions to the Diffusion Equations Suppose we desire to achieve a specific concentration C1 at a certain point in the sample at a certain time C ( x , t ) C0 x ! 1 erf C s C0 2 Dt becomes C1 C0 x ! constant ! 1 erf Cs C0 2 Dt x2 ! constant Dt

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2.5 Solutions to the Diffusion Equations The experiment: record combinations oft and x that kept C constant.to t1 t2 t3 xo x1 x2 x3

C(x i i Co x i ! 1 erf = (constant here) Cs Co i

Diffusion depth given by:

x i w Dt iInstitute of Materials Science

2.5 Solutions to the Diffusion Equations 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? Key point 1: C(x,t500C) = C(x,t600C). Key point 2: Both cases have the same Co and Cs.

Result: Dt should be held constant. x C(x,t) Co (Dt) 5 = 1 erf 2Dt C C o s 5.3x Answer: 4.8x - 3m2/s 10hrs

C (Dt) 6

C

t5

- 4m2/s

(Dt)6 ! D5

!

hr

Note: values of D are provided here.Institute of Materials Science