材料科学基础英文版课件 (13)

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Chapter 6

Diffusion in Solids

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Diffusion - Introduction

A phenomenon of material transport by atomic migration The mass transfer in macroscopic level is implemented by the motion of atoms in microscopic level Self-diffusion and interdiffusion (or impurity diffusion) Topics: mechanisms of diffusion, mathematics of diffusion, effects of temperature and diffusing species on the rate of diffusion, and diffusion of vacancy-solute complexes

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Demonstration of diffusion

Before heat treatment

After heat treatment

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Diffusion – Mechanisms (1)

Two mechanisms:

Vacancy diffusion Interstitial diffusion

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Diffusion – Mechanisms (2)

Vacancy diffusion In substitutional solid solutions, the diffusion (both self-diffusion and interdiffusion) must involve vacancies

For self-diffusion, the activation energy is vacancy formation energy + vacancy migration energy.

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Diffusion – Mechanisms (3)

Interstitial diffusion In interstitial solid solutions, the diffusion of interstitial solute atoms is the migration of the atoms from interstitial site to interstitial site

Position of interstitial atom after diffusion

The activation energy is the migration energy of the interstitial atom.

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Mathematics of Diffusion (1)

Steady-state diffusion

– Time-dependent process, the rate of mass transfer is expressed as a diffusion flux (J)

M J= At

In differential form

Mass transferred through a crosssectional area Diffusion time Area across which the diffusion occurs

J

=

1 A

d M d t

J = Mass transferred through a unit area per unit time (g/m2 s))

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Mathematics of Diffusion (2)

Concentration profile does not change with time – steady-state diffusion

e.g. the diffusion of atoms of a gas through a metal plate

concentration gradient = dC/dx

dC/dx

J

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Mathematics of Diffusion (3)

For steady-state diffusion, the diffusion flux is proportional to the concentration gradient The mathematics of steady-state diffusion in one dimension is given by

J

=

D

d C d x

where D is the diffusion coefficient (m2/s), showing the rate of diffusion

Minus sign indicates the diffusion is down the concentration gradient Negative For this case, the unit of C is in mass per unit volume, e.g. g/m3

Fick’s first law

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Mathematics of Diffusion (4)

Nonsteady-state diffusion

The diffusion flux at a particular point varies with time.

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Mathematics of Diffusion (5)

The diffusion equation is represented by

C t

=

( D x

C ) x

Fick’s second law C is a function of x and t

If D is independent of the composition, the above equation changes to

C t

=

D

2

C

2

x

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Unit area cross-section C = mass per unit volume (concentration)

Volume of the box: 1 dx

dC/dt = mass increase in the box per unit volume per unit time Mass decrease in the box per unit volume per unit time

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Mathematics of Diffusion (6) Solutions of diffusion equation

According to error function solutions for diffusion equation, the solution

for these profiles can be given by

C x = A + B erf (

x 2 Dt

)

A and B are constants and erf(z) is the error function, defined as

erf ( z ) =

2

π

∫

z

e

y2

dy

0

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FFF

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Mathematics of Diffusion (7)

Boundary conditions: For t>0, Cx=Cs at x=0 Cx=Co at x= ∞ Therefore

C s = A + B erf ( C o = A + B erf (

0 2 Dt ∞ 2 Dt

) )

A=Cs Co=A+B B=-(Cs-Co)

C x = C s (C s C o ) erf (

x 2 Dt

)

C x Co x = 1 erf ( ) Cs Co 2 Dt

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Mathematics of Diffusion (8)

Boundary conditions: For t=0, Cx=C1 at x<0 Cx=C2 at x>0 Therefore

C 1 = A + B erf ( ∞ ) C 2 = A + B erf ( ∞ )

C1=A-B A= (C1+C2)/2 C2=A+B B=-(C1-C2)/2

(C1 + C 2 ) (C1 C 2 ) x erf ( ) Cx = 2 2 2 Dt

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Mathematics of Diffusion (9)

x

2 Dt 2 Dt Boundary conditions: For t=0, Cx=0 at x<-h; Cx=Co at -hh A-B-C=0 A + B erf ( ∞ ) + C erf ( ∞ ) = 0 A+B-C=Co A + B erf ( ∞ ) + C erf ( ∞ ) = C o A + B erf ( ∞ ) + C erf ( ∞ ) = 0 A+B+C=0 Cx Co x+h x h = [erf ( ) erf ( )] 2 2 Dt 2 Dt

C x = A + B erf (

x+h

) + C erf (

x h

)

A=0 B=Co/2 C=-Co/2

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Mathematics of Diffusion (10)

C x Co x = 1 erf ( ) Cs Co 2 Dt

When Cx reaches a certain value at a particular position at different temperatures

C x Co = constant Cs Co

Therefore

x2 = constant 4 Dt

For example, if the same diffusion effect is obtained at two different temperatures T1 and T2, there is D1t1=D2t2

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Mathematics of Diffusion (11)

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