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Chemical Engineering 378

Science of Materials Engineering

Lecture 9Steady-State Diffusion

Spiritual Thought

“When He answers yes, it is to give us confidence. When He answers no, it is to prevent error. When He withholds an answer, it is to have us grow through faith in Him, obedience to His commandments, and a willingness to act on truth. We are expected to assume accountability by acting on a decision that is consistent with His teachings without prior confirmation. We are not to sit passively waiting or to murmur because the Lord has not spoken. We are to act.”

-Richard G. Scott

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Materials Roadmap3

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Layered Silicates• Layered silicates (e.g., clays, mica, talc)

– SiO4 tetrahedra connected together to form 2-D plane

• A net negative charge is associated with each (Si2O5)2- unit

• Negative charge balanced by adjacent plane rich in positively charged cations

Fig. 12.13, Callister& Rethwisch 10e.

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• Kaolinite clay alternates (Si2O5)2- layer with Al2(OH)42+layer

Layered Silicates (cont)

Note: Adjacent sheets of this type are loosely bound to one another by van der Waal’s forces.

Fig. 12.14, Callister & Rethwisch 10e.

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Polymorphic Forms of CarbonDiamond– tetrahedral bonding of

carbon• hardest material known• very high thermal

conductivity– large single crystals –

gem stones– small crystals – used to

grind/cut other materials – diamond thin films

• hard surface coatings –used for cutting tools, medical devices, etc.

Fig. 12.16, Callister & Rethwisch 10e.

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Polymorphic Forms of Carbon (cont)

Graphite– layered structure – parallel hexagonal arrays of

carbon atoms

– weak van der Waal’s forces between layers– planes slide easily over one another -- good

lubricant

Fig. 12.17, Callister& Rethwisch 10e.

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• Vacancies-- vacancies exist in ceramics for both cations and anions

• Interstitials-- interstitials exist for cations

-- interstitials are not normally observed for anions because anions are large relative to the interstitial sites

Fig. 12.18, Callister & Rethwisch 10e.(From W.G. Moffatt, G.W. Pearsall, and J. Wulff, The Structure and Properties of Materials, Vol. 1, Structure, John Wiley & Sons, 1964. Reproduced with permission of Janet M. Moffatt.)

Point Defects in Ceramics (i)

CationInterstitial

CationVacancy

Anion Vacancy

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• Frenkel Defect-- a cation vacancy-cation interstitial pair.

• Shottky Defect-- a paired set of cation and anion vacancies.

• Equilibrium concentration of defects

Point Defects in Ceramics (ii)

ShottkyDefect:

FrenkelDefect

Fig. 12.19, Callister & Rethwisch 10e.(From W.G. Moffatt, G.W. Pearsall, and J. Wulff, The Structure and Properties of Materials, Vol. 1, Structure, John Wiley & Sons, 1964. Reproduced with permission of Janet M. Moffatt.)

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• Electroneutrality (charge balance) must be maintained when impurities are present

• Ex: NaCl

Imperfections in Ceramics

Na + Cl -

• Substitutional cation impurity

without impurity Ca2+ impurity with impurity

Ca2+

Na+

Na+Ca2+

cationvacancy

• Substitutional anion impurity

without impurity O2- impurity

O2-

Cl-

anion vacancy

Cl-

with impurity

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Ceramic Phase Diagrams

MgO-Al2O3 diagram:

Fig. 12.23, Callister & Rethwisch 10e. [Adapted from B. Hallstedt,“Thermodynamic Assessment of the System MgO–Al2O3,” J. Am. Ceram. Soc., 75[6], 1992, p.1502. Reprinted by permission of the American Ceramic Society.]

°

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Diffusion

Diffusion - Mass transport by atomic motion

• Interdiffusion - diffusion of atoms of one material into another material

• Self-diffusion – atomic migration in a pure metal

• Diffusion Mechanisms- Gases & Liquids – random (Brownian) motion- Solids – vacancy diffusion and interstitial

diffusion

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• Atoms tend to migrate from regions of high concentration to regions of low concentration.

Before diffusion

Figs. 5.1, Callister & Rethwisch 10e.

Diffusion

After diffusion

Concentration Profiles

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• Self-diffusion: Migration of host atoms in pure metals

Locations of 4 labeled atoms before diffusion

Diffusion

A

B

C

D

Locations of 4 labeled atoms after diffusion

AB

C

D

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Diffusion Mechanism I

• atoms and vacancies exchange positions• applies to host and substitutional impurity atoms • diffusion rate depends on:

-- number of vacancies-- activation energy to exchange.

increasing elapsed time

Vacancy Diffusion

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Diffusion Mechanism II

• Small, interstitial atoms move from one interstitial position to an adjacent one

More rapid than vacancy diffusionFig. 5.2 (b), Callister & Rethwisch 10e.

Interstitial Diffusion

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Chapter-opening photograph, Chapter 5, Callister & Rethwisch 10e. (Courtesy ofSurface Division, Midland-Ross.)

• Case Hardening:

- Example of interstitial diffusion- Outer surface selectively hardened by

diffusing carbon atoms into surface- Presence of C atoms makes

iron (steel) harder

• Example: Case hardened gear- Case hardening improves wear

resistance of gear - Resulting residual compressive

stresses improve resistance to fatigue failure

Processing Using DiffusionCase hardened region

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• Doping – Diffusion of very small concentrations of atoms of an impurity (e.g., P) into the semiconductor silicon.

• Process:

3. Result is P dopedsemiconductorregions

silicon

Processing Using Diffusion

2. Heat treatthe sample todrive in P

1. Deposit P richlayers on surface

silicon

Diffusion in Semiconducting Devices

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Rate of Diffusion• Diffusion is a time-dependent process.• Rate of Diffusion- expressed as diffusion flux, J

M =mass

diffusedtime

• Measured experimentally– Use thin sheet (or membrane) – cross-sectional area A– Impose concentration gradient across sheet– Measure mass of diffusing species (M) that passes through

the sheet over time period (t)

SS Diffusion: Fick’s First Law20

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Steady-State Diffusion

Fick’s first law of diffusionC1

C2

x

C1

C2

x1 x2

D = diffusion coefficient

Rate of diffusion (or flux) independent of timeFlux (J) proportional to concentration gradient:

C = concentrationx = diffusion direction

C

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Diffusion Example

• Methylene chloride is a common ingredient of paint removers. Besides being an irritant, it also may be absorbed through skin. When using this paint remover, protective gloves should be worn.

• Lets investigate whether butyl rubber gloves (0.04 cm thick) commonly found in the kitchen can be used as protective gloves.

• Note: The maximum allowable flux for a 150 lb person is less than 3.5 x 10-7 g/cm2/s

• Compute the diffusion flux of methylene chloride through the gloves.

Chemical Protective Clothing (CPC)

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CPC Example (cont.)

gloveC1

C2

skinpaintremover

x1 x2

• Solution– diffusion flux of methylene chloride assume linear conc. gradient

x2 – x1 = 0.04 cm

Data:C1 = 0.44 g/cm3

C2 = 0.02 g/cm3

D = 110x10-8 cm2/s

Note: this is more than 30 times the allowable flux. Unsafe to use these gloves for paint removal.

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Influence of Temperature on Diffusion

• Diffusion coefficient increases with increasing T

D = Do exp −QdRT

= pre-exponential [m2/s]= diffusion coefficient [m2/s]

= activation energy [J/mol] = gas constant [8.314 J/mol-K]= absolute temperature [K]

DDoQdRT

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transform data

D

Temp = T

ln D

1/T

Influence of Temperature on Diffusion (cont.)

take natural log of both sides

26Influence of Temperature on Diffusion (cont.)

Adapted from Fig. 5.6, Callister & Rethwisch 10e. (Data for Fig. 5.7 taken from E.A. Brandes and G.B. Brook (Ed.) Smithells Metals Reference Book, 7th ed., Butterworth-Heinemann, Oxford, 1992.)

D has exponential dependence on T

Dinterstitial >> DsubstitutionalC in α-FeC in γ-Fe

Al in AlFe in α-FeFe in γ-Fe

1000K/T

D (m2/s)

0.5 1.0 1.510-20

10-14

10-8T(°C)15

00

1000

600

300

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Influence of Temperature on Diffusion (cont.)

Subtracting equation at T1 from equation at T2

Derive an equation relating the diffusion coefficients at two temperature T1 and T2 using the equation derived on slide 19.

Take the exponential of each side to get the final equation

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At 300°C the diffusion coefficient and activation energy for Cu in Si are

D1(300°C) = 7.8 x 10-11 m2/sQd = 41.5 kJ/mol

Compute the diffusion coefficient D2 at 350°C?

Example Diffusion Problem

T1 = 273 + 300 = 573K

T2 = 273 + 350 = 623K

D2 = 15.7 x 10-11 m2/s

Chemical Engineering 378��Science of Materials Engineering��Spiritual ThoughtMaterials RoadmapLayered SilicatesLayered Silicates (cont)Polymorphic Forms of CarbonPolymorphic Forms of Carbon (cont)Point Defects in Ceramics (i)Point Defects in Ceramics (ii)Imperfections in CeramicsCeramic Phase DiagramsDiffusionDiffusionDiffusionDiffusion Mechanism IDiffusion Mechanism IIProcessing Using DiffusionProcessing Using DiffusionRate of DiffusionSS Diffusion: Fick’s First LawSteady-State DiffusionDiffusion Example CPC Example (cont.)Influence of Temperature on DiffusionSlide Number 25Influence of Temperature on Diffusion (cont.)Slide Number 27Slide Number 28