MM-501 Phase Transformation in Solids

Fall Semester-2015

Lecture No: 03Diffusion :

How do atoms move through solids?

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What is diffusion?• Material/Mass transport by atomic motion is called

Diffusion. OR• It is a transport phenomenon caused by the motion

of chemical species (molecules, atoms or ions), heat or similar properties of a medium (gas, liquid or solid) as a consequence of concentration (or, strictly, chemical potential) differences.

• In general, the species move from high concentration areas to low concentrations areas until uniform concentration is achieved in the medium.

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• Diffusion is easy in liquids and gases where atoms are relatively free to move around:

• In solids, atoms are not fixed at its position but constantly moves (oscillates) . So, Diffusion is difficult in solids due to bonding and requires, most of the time, external energy to mobilize the atoms.

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Mass transport can generally involve:

1. fluid flow – dominant in gases or liquids 2. viscous flow – flow of a viscous material, generally amorphous or

semi crystalline (e.g. glasses and polymers) due to the forces acting on it at that moment;

3. atomic diffusion – principal mechanism in solids and in static liquids (as occurs in solidification).

Atomic diffusion occurs during important processes such as:4. solidification of materials .5. precipitation hardening, e.g. Al-Cu alloys 6. annealing of metals to reduce excess vacancies & dislocations

formed during working7. manufacture of doped silicon, e.g. as used in many electronic

devices

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• For an active diffusion to occur, the temperature should be high enough to overcome energy barriers to atomic motion

• for atom to jump into a vacancy site, it needs enough energy (thermal energy) to break the bonds and squeeze through its neighbors and take the new position. The energy necessary for motion is Em called the activation energy for vacancy motion.

• At activation energy Em has to be supplied to the atom so that it could break inter-atomic bonds and to move into the new position.

Figure: Schematic representation of the diffusion of an atom from its original position into a vacant lattice site

Inhomogeneous materials can become homogeneous by

diffusion.

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Diffusion Mechanisms

How do atoms move between atomic sites?

For diffusion to occur:1. Adjacent site needs to be empty (vacancy or

interstitial).2. Sufficient energy must be available to break

bonds and overcome lattice distortion.

There are many diffusion mechanism to be observed but two possible mechanisms are considered:

1. Vacancy diffusion.2. Interstitial diffusion.

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1. Substitutional Diffusiona) Direct Exchangeb) Ring c) Vacancy

2. Interstitial Diffusion

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Vacancy Mechanism Atoms can move from one site to another if there is sufficient energy present for the atoms to overcome a local activation energy barrier and if there are vacancies present for the atoms to move into. The activation energy for diffusion is the sum of the energy required to form a vacancy and the energy to move the vacancy.

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Vacancy diffusion- An atom adjacent to a vacant lattice site moves into it.

Essentially looks like an interstitial atom: lattice distortion

First, bonds with the neighboring atoms need to be broken

From Callister 6e resource CD.

• To jump from lattice site to lattice site, atoms need energy to break bonds with neighbors, and to cause the necessary lattice distortions during jump. This energy comes from the thermal energy of atomic vibrations (Eav ~ o CT)

• Materials flow (the atom) is opposite the vacancy flow direction.

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Interstitial atoms like hydrogen, helium, carbon, nitrogen, etc) must squeeze through openings between interstitial sites to diffuse around in a crystal.

The activation energy for diffusion is the energy required for these atoms to squeeze through the small openings between the host lattice atoms.

Interstitial Mechanism

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Interstitial DiffusionMigration from one interstitial site to another (mostly for small atoms that can be interstitial impurities: (e.g. H, C, N, and O) to fit into interstices in host.

Carbon atom in Ferrite

Interstitial diffusion is generally faster than vacancy diffusion because bonding of interstitials to the surrounding atoms is normally weaker and there are many more interstitial sites than vacancy sites to jump to.

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Interstitial Diffusion-Animation

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• How do we quantify the amount or rate of diffusion? • Flux (J): No of atom s diffusing through unit area per unit

time OR Materials diffusion through unit area per unit time.

• Measured empirically– Make thin film (membrane) of known surface area– Impose concentration gradient– Measure how fast atoms or molecules diffuse through the membrane

smkgor

scmmol

timearea surfacediffusing mass) (or molesFlux 22J

M =mass

diffusedtime

J slope

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Temperature Dependence of the Diffusion Coefficient :

• Diffusion coefficient increases with increasing T.

= pre-exponential [m2/s]

= diffusion coefficient [m2/s]

= activation energy [J/mol or eV/atom]

= gas constant [8.314 J/mol-K]

= absolute temperature [K]

DDo

Qd

R

T

With conc. gradient fixed, higher D means higher flux of mass transport.

T RQ

- D = D dolnln

T RQ

- D = D do exp

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• Diffusivity increases with T.

• Experimental Data:

1000K/T

D (m2/s) C in -Fe

C in -Fe

Al in Al

Cu in Cu

Zn in CuFe in -Fe

Fe in -Fe

0.5 1.0 1.5 2.010-20

10-14

10-8T(C)15

0010

00

600

300

D has exp. dependence on TRecall: Vacancy does also!Dinterstitial >> Dsubstitutional

C in -FeC in -Fe Al in Al

Cu in Cu

Zn in CuFe in -FeFe in -Fe

Diffusion and Temperature

ln D ln D0 QdR

1T

log D log D0 Qd

2.3R1T

Note:

pre-exponential [m2/s]activation energy

gas constant [8.31J/mol-K]

DDoExp QdRT

diffusivity[J/mol],[eV/mol]

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Steady-State Diffusion

dxdCDJ

Fick’s first law of diffusionC1

C2

x

C1

C2

x1 x2

D diffusion coefficient(be careful of its unit)

Rate of diffusion independent of timeFlux proportional to concentration gradient =

dxdC

12

12 linear ifxxCC

xC

dxdC

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Diffusivity -- depends on: 1. Diffusion mechanism. Substitutional vs interstitial.2. Temperature. 3. Type of crystal structure of the host lattice. 4. Type of crystal imperfections.

(a) Diffusion takes place faster along grain boundaries than elsewhere in a crystal. (b) Diffusion is faster along dislocation lines than

through bulk crystal. (c) Excess vacancies will enhance diffusion.

5. Concentration of diffusing species.

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Microstructural Effect on Diffusion:• If a material contains grains, the grains will act as diffusion

pathways, along which diffusion is faster than in the bulk material.

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Physical Aspect of D1. D is the indicator of how fast atom moves.2. In liquid state, D reaches similar level regardless of structure.3. In solid state, D shows high sensitivity to temperature and

structure. 4. Absolute temperature and Tm are what we should care about.

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Example: At 300ºC the diffusion coefficient and activation energy for Cu in Si are

D(300ºC) = 7.8 x 10-11 m2/sQd = 41.5 kJ/mol

What is the diffusion coefficient at 350ºC?

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1lnln and 1lnlnTR

QDDTR

QDD dd

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11lnlnln TTR

QDDDD d

transform data

D

Temp = T

ln D

1/T

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Example (cont.)

K 5731

K 6231

K-J/mol 314.8J/mol 500,41exp /s)m 10 x 8.7( 211

2D

1212

11exp TTR

QDD d

T1 = 273 + 300 = 573 K

T2 = 273 + 350 = 623 K

D2 = 15.7 x 10-11 m2/s

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• Steel plate at 7000C with geometry shown:

• Q: In steady-state, how much carbon transfers from the rich to the deficient side?

Adapted from Fig. 5.4, Callister 6e.C1 =

1.2kg/m3

C2 = 0.8kg/m3

Carbon rich gas

10mmCarbon deficient

gas

x1 x20 5mm

D=3x10-11m2/s

Steady State = straight line!

Example: Steady-state Diffusion

J DC2 C1x2 x1

2.410 9 kgm2s

Knowns: C1= 1.2 kg/m3 at 5mm (5 x 10–3 m) below surface.

C2 = 0.8 kg/m3 at 10mm (1 x 10–2 m) below surface.

D = 3 x10-11 m2/s at 700 C.

700 C

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• Concentration profile,C(x), changes with time.

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• To conserve matter: • Fick's First Law:

• Governing Eqn.:

Concentration, C, in the box

J (right)J (left)

dx

dCdt =Dd2C

dx2

dx

dCdt

J D dCdx orJ (left)J (right)

dJdx

dCdt

dJdx

D d2Cdx2

(if D does not vary with x)

equate

Non-Steady-State DiffusionIn most real situations the concentration profile and the concentration gradient are changing with time. The changes of the concentration profile is given in this case by a differential equation, Fick’s second law.

Called Fick’s second law

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Fick's Second Law of Diffusion

In words, the rate of change of composition at position x with time, t, is equal to the rate of change of the product of the diffusivity, D, times the rate of change of the concentration gradient, dCx/dx, with respect to distance, x.

x dC d D

x dd =

t dC d xx

Non-Steady-State Diffusion

Co

Cs

position, x

C(x,t)

to t1 t2t3

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• Copper diffuses into a bar of aluminum.

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• General solution:

"error function"Values calibrated in Table 5.1, Callister 6e.

C(x,t) CoCs Co

1 erf x2 Dt

pre-existing conc., Co of copper atomsSurface conc., Cs of Cu atoms bar

Co

Cs

position, x

C(x,t)

to t1 t2t3 Adapted from Fig.

5.5, Callister 6e.

Example: Non Steady-State Diffusion

t3>t2>t1

Fig. 6.5: Concentration profiles nonsteady-state diffusion taken at three different times

C0=Before diffusion

For t=0, C=C0 at 0x

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Non-steady State Diffusion• Sample Problem: An FCC iron-carbon alloy initially

containing 0.20 wt% C is carburized at an elevated temperature and in an atmosphere that gives a surface carbon concentration constant at 1.0 wt%.

• If after 49.5 h the concentration of carbon is 0.35 wt% at a position 4.0 mm below the surface, determine the temperature at which the treatment was carried out.

• Solution tip: use Eqn.

Dtx

CCCtxC

os

o

2erf1),(

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Solution (cont.):

– t = 49.5 h x = 4 x 10-3 m– Cx = 0.35 wt% Cs = 1.0 wt%– Co = 0.20 wt%

Dtx

CCC)t,x(C

os

o

2erf1

)(erf12

erf120.00.120.035.0),( z

Dtx

CCCtxC

os

o

erf(z) = 0.8125

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Solution (cont.):We must now determine from Table 5.1 the value of z for which the error function is 0.8125. An interpolation is necessary as follows

z erf(z)0.90 0.7970z 0.81250.95 0.8209

7970.08209.07970.08125.0

90.095.090.0

z

z 0.93

Now solve for D

Dtxz

2

tzxD 2

2

4

/sm 10 x 6.2s 3600

h 1

h) 5.49()93.0()4(

m)10 x 4(

4211

2

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2

2

tzxD

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• To solve for the temperature at which D has above value, we use a rearranged form of Equation (5.9a);

)lnln( DDRQTo

d

from Table 5.2, for diffusion of C in FCC Fe

Do = 2.3 x 10-5 m2/s Qd = 148,000 J/mol

/s)m 10x6.2 ln/sm 10x3.2 K)(ln-J/mol 314.8(

J/mol 000,14821125

T

Solution (cont.):

T = 1300 K = 1027°C

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T RQ

- D = D do exp

T RQ

- D = D dolnln

WhereD is the Diffusivity or Diffusion Coefficient ( m2 / sec )Do is the prexponential factor ( m2 / sec )Qd is the activation energy for diffusion ( joules / mole )R is the gas constant ( joules / (mole deg) )T is the absolute temperature ( K )

Temperature Dependence of the Diffusion Coefficient

OR

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End of Lecture

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Experimental Determination of Diffusion Coefficient

Tracer method• Radioisotopic tracer atoms are deposited at surface of solid by e.g. electro deposition• isothermal diffusion is performed for a given time t, often quartz ampoules are used (T<1600°C)• Sample is then divided in small slices either mechanically, chemically or by sputtering techniques• Mechanically: for diffusion length of > 10 µm; D>10-11 cm2/s• Sputtering of surface: for small diffusion length (at low temperatures) 2nm …10µm, for the range D = 10-21 …10-12 cm2/s

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Experimental Determination of Diffusion Coefficient

• Example: Diffusion of Fe in Fe3Si• From those figures thediffusion constant can bedetermined with an accuracyof a few percent• Stable isotopes can be used as

well, when high resolution SIMS is used

• This technique is moredifficult

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Diffusion Data

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• Copper diffuses into a bar of aluminum.• 10 hours processed at 600 C gives desired C(x).• How many hours needed to get the same C(x) at 500 C?

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(Dt)500ºC =(Dt)600ºCs

C(x,t) CoC Co

=1 erf x2Dt

• Result: Dt should be held constant.

• Answer: Note: values of D are provided.

Key point 1: C(x,t500C) = C(x,t600C).Key point 2: Both cases have the same Co and Cs.

t500(Dt)600D500

110hr4.8x10-14m2/s

5.3x10-13m2/s 10hrs

Processing Question

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

tot1t2

t3xo x1 x2 x3

• Diffusion depth given by: xi Dti

C(xi,ti) Co

Cs Co1 erf xi

2 Dti

= (constant here)

Diffusion Analysis

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Non steady-state diffusion

From Fick’s 1st Law: dxdcDJ

Take the first derivative w.r.t. x:

dxdcD

dxd

dxdJ

Conservation of mass:

i.e. flux to left and to right has to correspond to concentration change.dxdJ

dxJJ

dtdc lr

Sub into the first derivative:

dxdcD

dxd

dtdc Fick’s 2nd law

JrJl

dx c = conc. inside box

Partial differential equation. We’ll need boundary conditions to solve…

In most practical cases steady-state conditions are not established, i.e. concentration gradient is not uniform and varies with both distance and time. Let’s derive the equation that describes non steady-state diffusion along the direction x.

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EX: NON STEADY-STATE DIFFUSION• Copper diffuses into a bar of aluminum (semi infinite solid).

• General solution:

"error function"Values calibrated in Table 5.1, Callister 6e.

C(x,t) CoCs Co

1 erf x2 Dt

pre-existing conc., Co of copper atomsSurface conc., Cs of Cu atoms bar

Adapted from Fig. 5.5, Callister 6e.

From Callister 6e resource CD.

Co

Cs

position, x

C(x,t)At to, C = Co inside the Al bar

to

At t > 0, C(x=0) = Cs and C(x=∞) = Co

t1t2 t3

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If it is desired to achieve a specific concentration C1

i.e.

os

o

os

o

CCCC

CCCtxC 1),(

constant

which leads to:

Dtx

2constant

Known for given system

Specified with C1

1

1

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PROCESSING QUESTION• 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?

(Dt)500ºC =(Dt)600ºCs

C(x,t) CoC Co

=1 erf x2Dt

• 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)600D500

110hr4.8x10-14m2/s

5.3x10-13m2/s 10hrs

Adapted from Callister 6e resource CD.

Dt2

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Diffusion: Design ExampleDuring a steel carburization process at 1000oC, there is a drop in carbon concentration from 0.5 at% to 0.4 at% between 1 mm and 2 mm from the surface (g-Fe at 1000oC).

– Estimate the flux of carbon atoms at the surface.Do = 2.3 x 10-5 m2/s for C diffusion in -Fe.Qd = 148 kJ/molr-Fe = 7.63 g/cm3

AFe = 55.85 g/mol

– If we start with Co = 0.2 wt% and Cs = 1.0 wt% how long does it take to reach 0.6 wt% at 0.75 mm from the surface for different processing temperatures?

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T (oC) t (s) t (h)300 8.5 x 1011 2.4x108

900 106,400 29.6950 57,200 15.9

1000 32,300 9.01050 19,000 5.3

Need to consider factors such as cost of maintaining furnace at different T for corresponding times.

27782 yrs!

Diffusion: Design Example Cont’d

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A Look at Diffusion Bonding

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Introduction

• Diffusion bonding is a method of creating a joint between similar or dissimilar metals, alloys, and nonmetals.

• Two materials are pressed together (typically in a vacuum) at a specific bonding pressure with a bonding temperature for a specific holding time.

• Bonding temperature– Typically 50%-70% of the melting temperature of the most

fusible metal in the composition– Raising the temperature aids in the interdiffusion of atoms

across the face of the joint.

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How does diffusion bonding work?

• Bonding pressure– Forces close contact between the edges of the

two materials being joined.– Deforms the surface asperities to fill all of the

voids within the weld zone .– Disperses oxide films on the materials, leaving

clean surfaces, which aids the diffusion and coalescence of the joint.

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How does diffusion bonding work?

• Holding Time– Always minimized

• Minimizing the time reduces the physical force on the machinery.

• Reduces cost of diffusion bonding process.• Too long of a holding time might leave voids in the weld

zone or possibly change the chemical composition of the metal or lead to the formation of brittle intermetallic phases when dissimilar metals or alloys are being joined.

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How does diffusion bonding work?

• Sequence for diffusion bonding a ceramic to a metal– a) Hard ceramic and soft metal

edges come into contact.– b) Metal surface begins to yield

under high local stresses.– c) Deformation continues mainly

in the metal, leading to void shrinkage.

– d) The bond is formed

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Advantages of diffusion bonding

• Properties of parent materials are generally unchanged.

• Diffusion bonding can bond similar or dissimilar metals and nonmetals.

• The joints formed by diffusion bonding are generally of very high quality.

• The process naturally lends itself to automation.

• Does not produce harmful gases, ultraviolet radiation, metal spatter or fine dusts.

• Does not require expensive solders, special grades of wires or electrodes, fluxes or shielding gases.

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Summary II

1. Diffusion is just one of many mechanisms for mass transport.

2. Electrical field can produce mass transport.

3. Magnetic field can produce mass transport.

4. Combination of fields can produce mass transport such as electrochemical transport.

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Application: Homogenization time

Solidification usually results in chemical heterogeneities– Represent it with a sinusoid of wavelength, λ– Composition should homogenize when, x > λ/2– The approximate time necessary is:

Homogenization time- increases with λ2- decreases exponentially with T

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Application:Service Life of a Microelectronic Device

•Microelectronic devices– have built-in heterogeneities– Can function only as long as these doped regions survive

• To estimate the limit on service life, ts– Let doped island have dimension, λ– Device is dead when, x ~ λ/2, hence

Service life- decreases with miniaturization (λ2)- decreases exponentially with T

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Influence of Microstructure on Diffusivity

Interstitial species– Usually no effect from microstructure– Stress may enhance diffusion

Substitutional species– Raising vacancy concentration increases D

• Quenching from high T• Solutes• Irradiation

– Defects provide “short-circuit” paths• Grain boundary diffusion• Dislocation “core diffusion”

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Adding Vacancies Increases D

• Quench from high T– Rapid cooling freezes in high cv– D decreases as cv evolves to equilibrium

• Add solutes that promote vacancies– High-valence solutes in ionic solids

• Mg++ increases vacancy content in Na+Cl-• Ionic conductivity increases with cMg

– Large solutes in metals– Interstitials in metals

• Processes that introduce vacancies directly– Irradiation– Plastic deformation

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Grain Boundary Diffusion

• Grain boundaries have high defect densities– Effectively, vacancies are already present– QD ~ Qm

• Grain boundaries have low cross-section– Effective width = δ– Areal fraction of cross-section:

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Diffusion – Thermally Activated Process (I)

In order for atom to jump into a vacancy site, it needs to posses enough energy (thermal energy) to break the bonds and squeeze through its neighbors. The energy necessary for motion, Em, is called the activation energy for vacancy motion.

At activation energy Em has to be supplied to the atom so that it could break inter-atomic bonds and to move into the new position.

Schematic representation of the diffusion of an atom from its original position into a vacant lattice site.

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Diffusion – Thermally Activated Process (II)

The average thermal energy of an atom (kBT = 0.026 eV for room temperature) is usually much smaller that the activation energy Em (~ 1 eV/atom) and a large fluctuation in energy (when the energy is “pooled together” in a small volume) is needed for a jump.

The probability of such fluctuation or frequency of jumps, Rj, depends exponentially from temperature and can be described by equation that is attributed to Swedish chemist

Arrhenius :

where R0 is an attempt frequency proportional to the frequency of atomic vibrations.

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Diffusion – Thermally Activated Process (III)For the vacancy diffusion mechanism the probability for any atom in a solid to

move is the product of the probability of finding a vacancy in an adjacent lattice site (see Chapter 4):

and the probability of thermal fluctuation needed to overcome the energy barrier for vacancy motion

The diffusion coefficient, therefore, can be estimated as

Temperature dependence of the diffusion coefficient, follows the Arrhenius dependence.

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Diffusion – Temperature Dependence (I)Diffusion coefficient is the measure ofmobility of diffusing species.

D0 – temperature-independent preexponential (m2/s)Qd – the activation energy for diffusion (J/mol or eV/atom)R – the gas constant (8.31 J/mol-K or 8.62x105 /atom-KT – absolute temperature (K)

The above equation can be rewritten as

The activation energy Qd and preexponential D0, therefore, can be estimated by plotting lnD versus 1/T or logD versus 1/T. Such plots are Arrhenius plots.

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Diffusion – Temperature Dependence (II)

Graph of log D vs. 1/T has slop of –Qd/2.3R, intercept of ln Do

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Diffusion – Temperature Dependence (III)

Arrhenius plot of diffusivity data for some metallic systems

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Diffusion of different species

Smaller atoms diffuse more readily than big ones, and diffusion is faster in open lattices or in open directions

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Diffusion: Role of the microstructure (I)

Self-diffusion coefficients for Ag depend on the diffusion path.

In general, the diffusivity is greater through lessrestrictive structural regions – grain boundaries, dislocation cores, external surfaces.

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Diffusion: Role of the microstructure (II)

The plots (opposite) are from the computer simulation by T. Kwok, P. S. Ho, and S. Yip.

Initial atomic positions are shown by the circles, trajectories of atoms are shown bylines.

We can see the difference between atomic mobility in the bulk crystal and in the grain boundary region.

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Exercise1. A thick slab of graphite is in contact with a 1mm thick sheet of steel. Carbon steadily diffuses through the steel at 925C. The carbon reaching the free surface reacts with CO2 gas to form CO, which is then rapidly pumped away.

Determine the carbon concentration, C2, adjacent to the free surface, and the find the carbon flux in the steel, given that the reaction velocity for C+CO22CO is =3.010-6cm/sec.

At 925C, the solubility of carbon in the steel in contact with graphite is 1.5wt% and the diffusivity of carbon through steel is D=1.710-7cm2/sec. The equilibrium solubility of carbon in steel, Ceq, is 0.1wt% for the CO/CO2 ratio established at the surface of the steel.

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ExercisePe l

D1.76The Péclet number is

Note: The value of the Péclet number suggests mixed kinetic behavior is expected.

C2 Ceq C0 Ceq

1 lD

0.11.5 0.111.76

, [wt%]

C2 0.61wt%.

The carbon concentration in the steel at the free surface, C2, is

The steady-state flux is Jss = 1.51 10-6 [wt% C cm/s]

Jss = 1.18 10-7 [g/ cm2-s]

Divide by the density of steel, =12.8 cm3/100g to obtain the steady-state flux of carbon

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Exercise2. Two steel billets—a slab and a solid cylinder—contain 5000ppm residual H2 gas. These billets are vacuum annealed in a furnace at 725C for 24 hours to reduce the gas content. Vacuum annealing is capable of maintaining a surface concentration in the steel of 10ppm H2 at the annealing temperature.

Estimate the average residual concentration of H2 in each billet after vacuum annealing, given that the diffusivity of H in steel at 725C is DH=2.2510- 4 cm2/sec.

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Exercise15 cm

2h=1

0 cm

2h=10 cm

15 cm

10 cm

15 cm

15 cm

2h = 10 cm

2h =

10

cm

Rectangular and cylindrical slabs of steel

10 cm

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Given:t=24 hr=86400 sCo= Initial Concentration= 5000 ppmCs= Surface concentration= 10 ppmDH= 2.25x10-4 cm2/sC1= average residual concentration=?

We know that:(C1-Co) / (Cs-Co) = Constant(z) and also

X (Dt) or x = Constant x (Dt) or Constant(z) = (Dt) / x2

Now we can write:(C1-Co) / (Cs-Co) = (Dt) / x2 or C1= (Co-Cs) x (f) + Cs

Therefore, For slab:C1= (Co-Cs) x (flong x fshort x fshort) + Cs

and For Cylinder:C1= (Co-Cs) x (flong x fshort) + Cs

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STRUCTURE & DIFFUSIONDiffusion FASTER for...

• open crystal structures

• lower melting T materials

• materials with secondary bonding

• smaller diffusing atoms

• lower density materials

Diffusion SLOWER for...

• close-packed structures

• higher melting T materials

• materials with covalent bonding

• larger diffusing atoms

• higher density materials

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Factors that Influence Diffusion: Summary

" Temperature - diffusion rate increases very rapidly with increasing temperature

" Diffusion mechanism - interstitial is usually faster

than vacancy

" Diffusing and host species - Do, Qd is different for

every solute, solvent pair

" Microstructure - diffusion faster in polycrystalline vs. single crystal materials because of the accelerated diffusion along grain boundaries and dislocation cores.

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Concepts to remember• Diffusion mechanisms and phenomena.

– Vacancy diffusion.– Interstitial diffusion.

• Importance/usefulness of understanding diffusion (especially in processing).

• Steady-state diffusion.• Non steady-state diffusion.• Temperature dependence.• Structural dependence (e.g. size of the diffusing

atoms, bonding type, crystal structure etc.).

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Thanks