Lecture2 Diffusion SOLID STATE

8/22/2019 Lecture2 Diffusion SOLID STATE

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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 coolingrates can be determined using the mathematics/physics of

diffusion

Example: steel gears are case-hardened by

diffusing C or N to outer surface

DIFFUSION IN SOLIDS

Courtesy of P. Alpay


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ISSUES TO ADDRESS...

Atomic mechanisms of diffusion

Mathematics of diffusion

Influence of temperature and diffusing species on

Diffusion rate

DIFFUSION IN SOLIDS



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DIFFUSION

Phenomenon of material transport by atomic or particletransport 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 & statisticsCourtesy of P. Alpay


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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.

to

t1

t2

t3

xo x1 x2 x3time (s)

x (mm)

DIFFUSION DEMO



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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.

Fig. 5.0,

Callister 6e.

(Fig. 5.0 is

courtesy of

SurfaceDivision,

Midland-

Ross.)

PROCESSING USING DIFFUSION (1)

--hard to crack: C atoms put

the surface in compression.

Courtes of P. Al a


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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

semiconductorregions.

silicon

silicon

magnified image of a computer chip

0.5mm

light regions: Si atoms

light regions: Al atoms

Fig. 18.0,

Callister 6e.


Process

Courtes of P. Al a


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Driving force ofsintering process = reduction of energy associated with particles surfaces

Reducing scattering losses implies control of ceramic microstructure

Porosity = 0

Transparent material

(if no refractive index mismatch between grains)

Porosity 0

Translucent material

Two processesin competition:

densification and coarsening

Green

densification

grain growthcoarsening

raw powder synthesis

pressing or casting

T

emperature


C S


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Ceramic Sintering

Factors affecting

solid-state sintering: . Temperature, pressure. Green density

. Uniformity of green microstructure

. Atmosphere composition

. Impurities

. Particles size and size distribution

Sintering mechanisms


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Self-diffusion: In an elemental solid, atomsalso migrate.

Label some atoms After some time

A

B

C

DA

B

C

D

DIFFUSION: THE PHENOMENA (1)

Courtes of P. Al a

B*: self diffusion mobility

Self diffusion coefficient in a simple cubic lattice

: interatomic distancev: jump frequency


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(12.6)

f: correlation coefficient

Di*: true self-diffusion coefficient

Tracer atom studies may not give the true self-diffusion coefficient, Di*


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100%

Concentration Profiles0

Cu Ni

Interdiffusion: In an alloy or diffusion couple, atoms tendto 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 (2)

Courtes of P. Al a


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FICKS LAW AND DIFFUSIVITY OF

MATERIALS


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DEFINITION OF FLUXES FICKS FIRST LAW


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If the concentration of component A is given in mass units, the rate equation

for diffusion is:

(12.1)

WAx: mass flux of A in the x-direction, kg (of A) m-2s-1

A: mass concentration of A, kg (of A) m-3 (of total material)DA: diffusion coefficient, or diffusivity of A, m

2s-1

(intrinsic diffusion coefficient)

In the materials field, the rate equation is usually written in terms of molar

concentrations:

(12.2)

jAx: molar flux of A in the x-direction, mol (of A) m-2

s-1

CA: molar concentration of A, mol (of A) m-3 (of total material)


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Fick s first law of diffusion states that species A diffuses in the

direction of decreasing concentration of A, similarly as heat flows by

conduction in the direction of decreasing temperature, and momentum istransferred in viscous flow in the direction of decreasing velocity.

Eq. (12.1) and (12.2) are convenient forms of the rate equation whenthe density of the total solution is uniform (in solid or liquid solutions, or in

dilute gaseous mixture).

When the density is not uniform:

(12.3)

(12.4)

: density of the entire solution, kg/m-3A*: mass fraction of A

C: local molar concentration in the

solution at the point where the gradient ismeasured


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DIFFUSION IN SOLIDS

1. Self-diffusion

Self diffusion data apply to homogenous alloys in which there is no gradientin chemical composition.


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DIFFUSION IN SOLIDS

2. Diffusion under the influence of a composition gradient

Vacancy mechanism

Ring mechanism

Interstitialcy mechanism


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Flux of gold atoms (according to an observer sitting on a plane moving

with the solids velocity):

Total flux of gold atoms (according to an observer sitting on an unattachedplane in space):

(12.7)

The accumulation of gold as a function of time within a unit volume:

(12.8)


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(12.11)

(12.10)

(12.9)

Accumulation of nickel in the same unit volume:


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If the vacancy concentration within the unit volume is constant, then the

volume is approximately constant and:

(12.13)

(12.12)

Only in terms of the gold concentration gradient:

(12.14)


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Accumulation of gold solely in terms of diffusion coefficients and

concentration gradients:

(12.15)

Ficks first law in its simplest form, for gold:

(12.16)

D indicates that the flux is proportional to the concentration gradient butside-steps the issue of bulk flow.

The accumulation of gold within the unit volume:

(12.17)

The unsteady state equation for unidirectional diffusion in solids (Ficks


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The unsteady-state equation for unidirectional diffusion in solids (Ficks

second law):

(12.18)

Equation (12.18) is used to obtain analytical solutions for diffusion problems

(12.19)

DAu and DNi: Intrinsic diffusion coefficients

D : Interdiffusion, mutual diffusion, or chemical diffusion


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DIFFUSION IN SOLIDS

3. Darkens equation

(12.20)

nAx: flux of A atoms passing through an unit area in the x-direction

nA: the number of A atoms per unit volume

BA: mobility of A atoms in the presence of the energy gradient

N0: Avogadros numberGA: partial molar free energy of A (chemical potential of A)_


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(12.21)

ax: activity of A

(12.22)

KB: Boltzmanns constant = R/N0

XA: Mole fraction of A

aA= XA (for a thermodynamically ideal solution)

(12.23)

NERNST-EINSTEIN EQUATION

Gibb D h ti


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Gibbs Duhem equation:

(12.24)

And only if BA = BA* and BB = BB*, then Eq. (12.19) can be written as:

(12.25)

If the system is ideal:

(12.26)


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DIFFUSION IN SOLIDS


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DIFFUSION IN SOLIDS

4. Temperature dependence of diffusion in solids

(Arrhenius equation)

(12.27)

According to Zener, if the jump process is an activated one, jump frequency

can be described by:(12.28)

0= vibrational frequency of the atom in the latticeZ = coordination number

G: = free energy of activation required for the atom to jump from one site into the next


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(12.29)


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Sherby and Simnad developed a correlation equation for predicting self-


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Sherby and Simnad developed a correlation equation for predicting self

diffusion data in pure metals:

(12.30)

K0 depends only on the crystal structure

V = normal valence of the metal

TM = absolute melting point

D0 is approximated as 1 10-4 m2 s-1 for estimation purposes

(12.31)

DIFFUSION IN CERAMIC MATERIALS


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DIFFUSION IN CERAMIC MATERIALS

The general equation for the transport of the species in the x-direction


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under the influence of an electric field and a concentration gradient in the

x-direction:

(12.32)

In the absence of a concentration gradient, the flux is:

(12.33)

The current density:


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The current density:

(12.34)

(12.35)

The electrical conductivity :

(12.36)

(12.37)

This equation is known as the extended Nernst-Einstein equation


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(12.39)

(12.38)

ti: transference number of species i

For NaCl:

(12.40)


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For CoO:


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Thermodynamic factor can be calculated by:

PO2: standard oxygen pressure, 1 atm


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DIFFUSION IN ELEMENTAL SEMICONDUCTORS


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DIFFUSION IN LIQUIDS


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1. Liquid state diffusion theories

Hydrodynamical theory:

The force on a sphere moving at steady-state in laminar flow:

and since B = V/F,

(12.41)

Stokes Einstein equation

(12 42)


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(12.42)

Sutherland Einstein equation

(12 43)


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(12.43)

A is a constant

M = atomic weight or molecular weight

TM = melting point

D* = self-diffusion coefficient

VM

= molar (or atomic) volume

(12.44)

If it is assumed that the atoms in the liquid are in a cubic array, then for materials

generally the simple theory predicts that:

Hole theory: This theory presumes the existence of holes or vacancies

randomly distributed throughout the liquid and providing ready diffusion


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paths for atoms and ions. The concentration of these holes would have to

be very great in order to account for the volume increase upon melting,thus resulting in much higher diffusion rates in liquids than in solids just

below the melting point.

Eyring theory:

(12.45)

If the liquid is considered quasi-crystalline, and the atoms are in a cubic

configuration, then an expression relating D* and results:

Fluctuation theory:


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Fluctuation theory:

(12.46)

(12.47)

In this theory, the extra volume (over that of the solid) in the liquid is

distributed evenly throughout the liquid, making the average nearest

neighbour distance increase.

Reyniks fluctuation model:


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DIFFUSION IN LIQUIDS


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2. Liquid diffusion data

Diffusion in liquid metals:


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Diffusion in molten salts and silicates:


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Diffusion in common liquids:


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DIFFUSION IN GASES


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Based on the kinetic theory of gases, self-diffusivity of spherical A atomsdiffusing in pure A can be predicted by:

For the diffusivity of two unequal size spherical atoms A and B, kinetic

theory predicts:

KB: Boltzmanns constant, 1,3810-16 ergs molecule-1 K-1

(12.48)

(12.49)

Chapman Enskog theory:

F t i


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For monoatomic gases (12.50)

The collision integral can be calculated by:


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where A = 1.06036, B = 0.15610, C = 0.19300, D = 0.47635, E = 1.03587,

F = 1.52996, G = 1.76474, and H = 3.89411.

(12.51)

(12.52)

(12.53)


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(12.54)


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DIFFUSION THROUGH POROUS MEDIA


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effective

diffusivity

void

fraction

tortuosity

Tortuosity is a number greater than one and it ranges from values of 1.5 -2.0

for unconsolidated particles, to as high as 7 or 8 for compacted particles.

(12.55)

An analysis of the flow of molecules through a cylindirical pore radius r

under a concentration gradient yields


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(12.56)


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Knudsen diffusion coefficient

mean free path of the gas

collision diameter of the molecule, cm

concentration of the molecules, cm-3

If r and are of the same order of magnitude, or if the pore diameter isonly one order larger, then Knudsen diffusion is presumably important.

If the ratio DAB/DK is small, then the ordinary diffusion is the rate

determinig step. Conversely, ifDAB/DK is large, it is presumed that Knudsen

diffusion controls


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(12.57)

diffusion controls.

Lecture2 Diffusion SOLID STATE

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