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Material Sciences and Engineering, MatE271 1

Material Sciences and Engineering MatE271 1Week6

Diffusion

Atomic Motion in Solids

Material Sciences and Engineering MatE271 Week 6 2

Diffusion in Materials

A. Atoms must be able to move around (diffusion)

Diffusion occurs in solids, liquids and gases:

� Redistribution of non-uniform chemical species (impurity diffusion or interdiffusion)

� Random atomic movement can also occur inchemically uniform materials (self diffusion)

Q. How do changes in microstructure and chemicalcomposition actually occur?



Diffusion is �driven� by Nonuniformity

A B

Distance (x)

Conc

entr

atio

n of

�A�

Time

TemperatureA B

Distance (x)Co

ncen

trat

ion

of �

A�Concentration Profile


Interdiffusion forming a solid solution

Initial

Intermediate time

Much longer time

What is this time scale?



Diffusion?

Diffusion is necessary for:� Redistribution of chemical species� Physical changes in microstructure� Densification of powder compacts� Deformation at high temperature (creep)� Formation of solid state reaction products� One kind of conductivity in ceramics (ionic)


Atoms would not move around because

there would be no places for them to move to

(all sites would be occupied)-- �locked in place�

Diffusion:

Perfect crystal:

� Point defects must be present in a crystal to permit atomic movement (diffusion)

In a way, atomic diffusion is actually the movement of defects.



Diffusion Mechanisms

I. Vacancy diffusiono Only adjacent atoms can move into a vacancyo Vacancy moves in opposite direction of atomic

motiono Rate depends on concentration of vacancies

Atomic flux

Vacancy flux


II. Interstitial Diffusiono Atom can move into any adjacent empty

interstitial position (usually smaller atoms)o Rate depends on concentration of interstitial

atoms� (Usually faster than vacancy diffusion)




o Would you expect vacancy or interstitial diffusion to be faster?

o Why?



Net migrationafter n jump

� After many random jumps by an atom, it�s displacment� can be calculated by the theory of �random walks�

Diffusion occurs by random jumps



- The rate of diffusion is characterized by describing atomic

fluxes at particular locations in the material

- Critical quantities

J = atomic flux (atoms/m2-s, kg/ m2-s)

(dc/dx) = concentration gradient

(atoms/m4)

D = diffusion coefficient (m2/s)

Quantitative Description of Diffusion

area

J

c

dc/dx x

Fick�s first law: J = - D (dc/dx)


Interrelating the quantities

� Fick�s first law: J = - D (dc/dx)

(negative sign indicates that the direction of diffusion flux

is �down� the concentration gradient from high to low

concentration)

� For steady state diffusion (local flux doesn�t change

with time), Fick�s First Law can be solved directly



Hydrogen (H) gas can be purified by passing atomic hydrogen through a thin sheet of palladium (Pa) at 700oC in a paladium diffusion cell. If the impure hydrogen gas is maintained at 1 atm on one side of a 1 mm thick Pd sheet (A=1 m2), and the pressure on the purified side is maintained at 0.1 atm by pumping, what is the mass of the hydrogen purified in 1 hr? Assume steady state conditions. The concentration of H2 at 1 atm is 9.0x10-3 gm/cm3 and D(H) in Pa is 1.2x10-6 cm2/sec.

Example


Non-steady state diffusion

The diffusion flux at a particular point varies with time

� � (There is a net accumulation or depletion of the diffusing species at a given location)

� � i.e., local concentration of diffusing species changes with time as diffusion proceeds

� � This is the most common situation

What is this time scale?




� Fick�s Second Law governs

� Many solutions exist for particular geometries (initial and boundary conditions)

� Diffusion from a constant source into an semi-infinite solidBC-1: For t = 0, C = C0 at 0 ≤ x ≤ ∞ BC-3: C = C0 at x = ∞

BC-2: t > 0, C = Cs at x = 0

(Cx - C0) = 1 - erf x(Cs-C0) 2√Dt

∂C = D ∂2C∂t ∂x2 x

C(∞,t)=Co

C(x, t)=Cx ?

C(0,t)=Cs

C(x,0)=Co



x = 0

Cs

C0

to< t1 < t2 < t3

t1t2

t3

Cx

x

C

C(0,t)=Cs C(∞,t)=Co

C(x,0)=Co

to

C(x, t)=Cx ?

(Cx - C0) = 1 - erf x(Cs-C0) 2√Dt



Example

For some applications (e.g. gears), it is necessary to harden the surface of a steel (Fe-C alloy) above that of its interior. One way of accomplishing this is by increasing the surface concentration of carbon in the steel (as we will see later) using a process termed carburizing. In carburizing the steel piece is exposed, at elevated temperature, to an atmosphere rich in a hydrocarbon gas, such as methane (CH4).

Surface Treatment of Steel:


Example: Surface Treatment of Steel:

Consider on such alloy that initially has a uniform carbon concentration of 0.25 wt% and is to be treated at 950° C. If the concentration of carbon at the surface is suddenly brought to and maintained at 1.20 wt%, how long will it take to achieve a carbon content of 0.80 wt% at a position 0.5 mm below the surface? The diffusion coefficient for C in Fe at this temperature is 1.6 x 10-11 m2/sec. Assume piece is semi-infinite.



Factors that Influence Diffusion

I. Diffusing Species� magnitude of diffusion coefficient, D - indicates

the rate at which atoms diffuse� both diffusing species and host material

influence the coefficient� Relative sizes of atoms� �Openess� of lattice� Ionic charges



For example:

� For the host species of iron:- Self diffusion at 500°C (Fe moving in Fe)

D = 1.1 x 10-20 m2/s (vacancy diffusion)

- Carbon interdiffusion at 500°C (C moving in FeD = 2.3 x 10-12 m2/s (interstitial diffusion)

� Atomic Size/Mechanism

This shows the contrast between rates of

vacancy and interstitial diffusion




II. Temperature� very strong effect on the diffusion coefficient:

� A large activation energy results in a small D

D = Do exp −Q d

RT� �

� �

Do = T independent preexponential Qd = the activation energy for diffusion (J/mol, or eV/atom)

R = the gas constant, 8.31 J/mol - K or 8.662 x 10-5 eV/ atom − KT = absolute temperature, (K)

ln D = ln Do − −Qd

R1T

� �

� �

Plot ln D vs 1/T - get straight line(to measure activation energy and Do)


Example:

o Using data from Table 5.2, compute the diffusion coefficient of C in α−Fe and γ−Fe at 900ºC.



Solution:

D = Do exp −Q d

RT� �

� �

Do = T independent preexponential Qd = the activation energy for diffusion (J/mol, or eV/atom)

R = the gas constant, 8.31 J/mol - K or 8.662 x 10-5 eV/ atom − KT = absolute temperature, (K)

o C in α−Fe (BCC) at 900ºC D = 6.2x10-7 m2/sec exp (-0.83 eV/atom / (1173K)(8.62x10-5 eV/atom-K)D = 1.7x10-10 m2/sec

o C in γ−Fe (FCC) at 900ºCD = 2.3x10-5 m2/sec exp (-1.53 eV/atom / (1173K)(8.62x10-5 eV/atom-K)D = 5.9x10-12 m2/sec


What does this tell you about interstitial sites in BCC and FCC?�.

� BCC more open than FCC for interstitial diffusion�i.e. it is easier to move from one interstitial site to another in BCC

� But it does not say anything about the sizeor number of interstitial sites in each�.actually, as you will see, FCC has bigger (and more) interstitial sites



(Besides through volume of the crystal)

� Atomic migration often occurs more rapidly along so-called �short circuiting paths�

� Dislocations

� Grain boundaries

� External surfaces

� However, there is usually small total area

for this to occur - so not always important

Other Diffusion Paths


Volume, grain boundary and surface diffusion

surface

Grain boundary

volume

Ag in Ag



Diffusion and Materials Processing

o Properties of materials are altered through diffusion

� steelmaking� sintering� semiconductor doping

o �Heat treatment� is used to allow these to occur over a reasonable time frame.


Summary� Recognize various imperfections in crystals

� Point imperfections

� Impurities

� Line imperfections (dislocations)

� Bulk imperfections

� Define various diffusion mechanisms

� Identify factors controlling diffusion processes



Reading Assignment

Shackelford 2001(5th Ed)

� Read Chapter 5, pp 158-181

diffusion - iowa state universitynano.engineering.iastate.edu/courses/mate271/week6.pdf ·...

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