Ch05 ppts callister7e
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This chapter discusses diffusion in solids, including: 1) How diffusion occurs via vacancy or interstitial mechanisms and depends on structure and temperature. 2) Fick's laws of diffusion and how diffusion can be quantified by measuring flux. 3) Examples of diffusion applications like case hardening and doping semiconductors. 4) How the diffusion coefficient increases exponentially with temperature and can be used to solve problems involving steady-state and non-steady-state diffusion.
![Chapter 5 - 14
Diffusion and Temperature
• Diffusion coefficient increases with increasing T.
D = Do exp
−
Qd
RT
= 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]
D
Do
Qd
R
T](https://image.slidesharecdn.com/ch05pptscallister7e-130627171524-phpapp02/85/Ch05-ppts-callister7e-14-320.jpg)
Ch05 ppts callister7e
- 1. Chapter 5 - 1 ISSUES TO ADDRESS... • How does diffusion occur? • Why is it an important part of processing? • How can the rate of diffusion be predicted for some simple cases? • How does diffusion depend on structure and temperature? Chapter 5: Diffusion in Solids
- 2. Chapter 5 - 2 Diffusion Diffusion - Mass transport by atomic motion Mechanisms • Gases & Liquids – random (Brownian) motion • Solids – vacancy diffusion or interstitial diffusion
- 3. Chapter 5 - 3 • Interdiffusion: In an alloy, atoms tend to migrate from regions of high conc. to regions of low conc. Initially Adapted from Figs. 5.1 and 5.2, Callister 7e. Diffusion After some time
- 4. Chapter 5 - 4 • Self-diffusion: In an elemental solid, atoms also migrate. Label some atoms After some time Diffusion A B C D A B C D
- 5. Chapter 5 - 5 Diffusion Mechanisms Vacancy Diffusion: • atoms exchange with vacancies • applies to substitutional impurities atoms • rate depends on: --number of vacancies --activation energy to exchange. increasing elapsed time
- 6. Chapter 5 - 6 • Simulation of interdiffusion across an interface: • Rate of substitutional diffusion depends on: --vacancy concentration --frequency of jumping. (Courtesy P.M. Anderson) Diffusion Simulation
- 7. Chapter 5 - 7 Diffusion Mechanisms • Interstitial diffusion – smaller atoms can diffuse between atoms. More rapid than vacancy diffusion Adapted from Fig. 5.3 (b), Callister 7e.
- 8. Chapter 5 - 8 Adapted from chapter-opening photograph, Chapter 5, Callister 7e. (Courtesy of Surface Division, Midland-Ross.) • Case Hardening: --Diffuse carbon atoms into the host iron atoms at the surface. --Example of interstitial diffusion is a case hardened gear. • Result: The presence of C atoms makes iron (steel) harder. Processing Using Diffusion
- 9. Chapter 5 - 9 • Doping silicon with phosphorus for n-type semiconductors: • Process: 3. Result: Doped semiconductor regions. silicon Processing Using Diffusion magnified image of a computer chip 0.5mm light regions: Si atoms light regions: Al atoms 2. Heat it. 1. Deposit P rich layers on surface. silicon Adapted from chapter-opening photograph, Chapter 18, Callister 7e.
- 10. Chapter 5 - 10 Diffusion • How do we quantify the amount or rate of diffusion? • Measured empirically – Make thin film (membrane) of known surface area – Impose concentration gradient – Measure how fast atoms or molecules diffuse through the membrane ( )( ) sm kg or scm mol timeareasurface diffusingmass)(ormoles Flux 22 =≡≡J dt dM A l At M J == M = mass diffused time J ∝ slope
- 11. Chapter 5 - 11 Steady-State Diffusion dx dC DJ −= Fick’s first law of diffusionC1 C2 x C1 C2 x1 x2 D ≡ diffusion coefficient Rate of diffusion independent of time Flux proportional to concentration gradient = dx dC 12 12linearif xx CC x C dx dC − − = ∆ ∆ ≅
- 12. Chapter 5 - 12 Example: Chemical Protective Clothing (CPC) • 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. • If butyl rubber gloves (0.04 cm thick) are used, what is the diffusive flux of methylene chloride through the glove? • Data: – diffusion coefficient in butyl rubber: D = 110x10-8 cm2 /s – surface concentrations: C2 = 0.02 g/cm3 C1 = 0.44 g/cm3
- 13. Chapter 5 - 13 scm g 10x16.1 cm)04.0( )g/cm44.0g/cm02.0( /s)cm10x110( 2 5- 33 28- = − −=J Example (cont). 12 12- xx CC D dx dC DJ − − −≅= D tb 6 2 = glove C1 C2 skinpaint remover x1 x2 • Solution – assuming linear conc. gradient D = 110x10-8 cm2 /s C2 = 0.02 g/cm3 C1 = 0.44 g/cm3 x2 – x1 = 0.04 cm Data:
- 14. Chapter 5 - 14 Diffusion and Temperature • Diffusion coefficient increases with increasing T. D = Do exp − Qd RT = 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] D Do Qd R T
- 15. Chapter 5 - 15 Diffusion and Temperature Adapted from Fig. 5.7, Callister 7e. (Date 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 >> Dsubstitutional C in α-Fe C in γ-Fe Al in Al Fe in α-Fe Fe in γ-Fe 1000K/T D (m2 /s) C in α-Fe C in γ-Fe Alin Al Fe in α-Fe Feinγ-Fe 0.5 1.0 1.5 10-20 10-14 10-8 T(°C) 1500 1000 600 300
- 16. Chapter 5 - 16 Example: At 300ºC the diffusion coefficient and activation energy for Cu in Si are D(300ºC) = 7.8 x 10-11 m2 /s Qd = 41.5 kJ/mol What is the diffusion coefficient at 350ºC? −= −= 1 01 2 02 1 lnlnand 1 lnln TR Q DD TR Q DD dd −−==−∴ 121 2 12 11 lnlnln TTR Q D D DD d transform data D Temp = T ln D 1/T
- 17. Chapter 5 - 17 Example (cont.) − − = − K573 1 K623 1 K-J/mol314.8 J/mol500,41 exp/s)m10x8.7( 211 2D −−= 12 12 11 exp TTR Q DD d T1 = 273 + 300 = 573K T2 = 273 + 350 = 623K D2 = 15.7 x 10-11 m2 /s
- 18. Chapter 5 - 18 Non-steady State Diffusion • The concentration of diffucing species is a function of both time and position C = C(x,t) • In this case Fick’s Second Law is used 2 2 x C D t C ∂ ∂ = ∂ ∂Fick’s Second Law
- 19. Chapter 5 - 19 Non-steady State Diffusion Adapted from Fig. 5.5, Callister 7e. B.C. at t = 0, C = Co for 0 ≤ x ≤ ∞ at t > 0, C = CS for x = 0 (const. surf. conc.) C = Co for x = ∞ • Copper diffuses into a bar of aluminum. pre-existing conc., Co of copper atoms Surface conc., C of Cu atoms bar s Cs
- 20. Chapter 5 - 20 Solution: C(x,t) = Conc. at point x at time t erf (z) = error function erf(z) values are given in Table 5.1 CS Co C(x,t) ( ) −= − − Dt x CC Ct,xC os o 2 erf1 dye y z 2 0 2 − ∫π =
- 21. Chapter 5 - 21 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: use Eqn. 5.5 −= − − Dt x CC CtxC os o 2 erf1 ),(
- 22. Chapter 5 - 22 Solution (cont.): – t = 49.5 h x = 4 x 10-3 m – Cx = 0.35 wt% Cs = 1.0 wt% – Co = 0.20 wt% −= − − Dt x CC C)t,x(C os o 2 erf1 )(erf1 2 erf1 20.00.1 20.035.0),( z Dt x CC CtxC os o −= −= − − = − − ∴ erf(z) = 0.8125
- 23. Chapter 5 - 23 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 0.7970 z 0.8125 0.95 0.8209 7970.08209.0 7970.08125.0 90.095.0 90.0 − − = − −z z = 0.93 Now solve for D Dt x z 2 = tz x D 2 2 4 = /sm10x6.2 s3600 h1 h)5.49()93.0()4( m)10x4( 4 211 2 23 2 2 − − == =∴ tz x D
- 24. Chapter 5 - 24 • To solve for the temperature at which D has above value, we use a rearranged form of Equation (5.9a); )lnln( DDR Q T o 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)m10x6.2ln/sm10x3.2K)(ln-J/mol314.8( J/mol000,148 21125 −− − =T∴ Solution (cont.): T = 1300 K = 1027°C
- 25. Chapter 5 - 25 Example: Chemical Protective Clothing (CPC) • 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. • If butyl rubber gloves (0.04 cm thick) are used, what is the breakthrough time (tb), i.e., how long could the gloves be used before methylene chloride reaches the hand? • Data (from Table 22.5) – diffusion coefficient in butyl rubber: D = 110x10-8 cm2 /s
- 26. Chapter 5 - 26 Example (cont). Time required for breakthrough ca. 4 min glove C1 C2 skinpaint remover x1 x2 • Solution – assuming linear conc. gradient D tb 6 2 = Equation 22.24 cm0.0412 =−= xx D = 110x10-8 cm2 /s min4s240 /s)cm10x110)(6( cm)04.0( 28- 2 ===bt
- 27. Chapter 5 - 27 Diffusion FASTER for... • open crystal structures • materials w/secondary bonding • smaller diffusing atoms • lower density materials Diffusion SLOWER for... • close-packed structures • materials w/covalent bonding • larger diffusing atoms • higher density materials Summary
- 28. Chapter 5 - 28 Core Problems: Self-help Problems: ANNOUNCEMENTS Reading: