1. Chapter 5- 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? 1 How does diffusion depend on structure and temperature? CHAPTER 5: DIFFUSION IN SOLIDS

2. Chapter 5- Diffusion Diffusion - Mass transport by atomic motion Mechanisms Gases & Liquids – random (Brownian) motion Solids – vacancy diffusion or interstitial diffusion

3. Chapter 5- Why Study Diffusion ? Diffusion plays a crucial role in… –Alloying metals => bronze, silver, gold –Strengthening and heat treatment processes Hardening the surfaces of steel –High temperature mechanical behavior –Phase transformations Mass transport during FCC to BCC –Environmental degradation Corrosion, etc.

4. Chapter 5-2 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. Compare the results with theory. DIFFUSION DEMO

5. Chapter 5- How do atoms move in Solids ? Why do atoms move in Solids ? Diffusion, simply, is atoms moving from one lattice site to another in a stepwise manner –Transport of material by moving atoms Two conditions are to be met: –An empty adjacent site –Enough energy to break bonds and cause lattice distortions during displacement What is the energy source ? –HEAT ! What else ? –Concentration gradient !

6. Chapter 5- 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

7. Chapter 5-3 Interdiffusion: In an alloy, atoms tend to migrate from regions of large concentration. InitiallyAfter some time Adapted from Figs. 5.1 and 5.2, Callister 6e. DIFFUSION: THE PHENOMENA (1)

8. Chapter 5-4 Self-diffusion: In an elemental solid, atoms also migrate. Label some atoms (use isotopes)After some time DIFFUSION: THE PHENOMENA (2)

9. Chapter 5- More examples in 3-D !

10. Chapter 5- Diffusion Mechanisms (I) Energy is needed to generate a vacancy, break bonds, cause distortions. Provided by HEAT, kT ! Atom moves in the opposite direction of the vacancy !

11. Chapter 5- Diffusion Mechanisms (II) Much faster than vacancy diffusion, why ? Smaller atoms like B, C, H, O. Weaker interaction with the larger atoms. More vacant sites, no need to create a vacancy ! Interstitial Diffusion

12. Chapter 5-5 Substitutional Diffusion: applies to substitutional impurities atoms exchange with vacancies rate depends on: --number of vacancies --activation energy to exchange. Diffusion Mechanisms (III)

13. Chapter 5- Case Hardening: --Diffuse carbon atoms into the host iron atoms at the surface. --Example of interstitial diffusion is a case hardened gear. Result: The "Case" is --hard to deform: C atoms "lock" planes from shearing. --hard to crack: C atoms put the surface in compression. 8 Fig. 5.0, Callister 6e. (Fig. 5.0 is courtesy of Surface Division, Midland- Ross.) PROCESSING USING DIFFUSION (1)

14. Chapter 5- Doping silicon with phosphorus for n-type semiconductors: Process: 3. Result: Doped semiconductor regions. silicon Processing Using Diffusion magnified image of a computer chip 0.5 mm 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.

15. Chapter 5- 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 M = mass diffused time J  slope

16. Chapter 5- Diffusion Flux: 10 Directional Quantity (anisotropy ?) Flux can be measured for: --vacancies --host (A) atoms --impurity (B) atoms MODELING DIFFUSION: FLUX Material Diffusion is a time-dependent process ! RATE OF MATERIAL TRANSPORT

17. Chapter 5- Concentration Profile, C(x): [kg/m 3 ] 11 Fick's First Law: The steeper the concentration profile, the greater the flux! Concentration gradient is the DRIVING FORCE ! Adapted from Fig. 5.2(c), Callister 6e. CONCENTRATION PROFILES & FLUX

18. Chapter 5- Concentration Gradient

19. Chapter 5- Steady State: the concentration profile doesn't change with time. 12 Apply Fick's First Law: Result: the slope, dC/dx, must be constant (i.e., slope doesn't vary with position)! If J x ) left = J x ) right, then STEADY STATE DIFFUSION Why is the minus sign ?

20. Chapter 5- Steel plate at 700º C 13 Q: How much carbon transfers from the rich to the deficient side? Adapted from Fig. 5.4, Callister 6e. EX: STEADY STATE DIFFUSION

21. Chapter 5 - 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 = 110 x10 -8 cm 2 /s –surface concentrations: C 2 = 0.02 g/cm 3 C 1 = 0.44 g/cm 3

22. Chapter 5 - Example (cont). glove C1C1 C2C2 skin paint remover x1x1 x2x2 Solution – assuming linear conc. gradient D = 110 x 10 -8 cm 2 /s C 2 = 0.02 g/cm 3 C 1 = 0.44 g/cm 3 x 2 – x 1 = 0.04 cm Data:

23. Chapter 5 - Temperature Dependency ! Atom has to break bonds and “squeeze” thru => activation energy, E m ≈ 1 eV. What is the probability to find a vacancy at a nearest site ? COMBINE

24. Chapter 5 - Temperature Effect ! The diffusion depends on temperature because; a- # of vacancies in the vicinity b- thermally activated successful jumps

25. Chapter 5 - Diffusion and Temperature Diffusion coefficient increases with increasing T. D  DoDo exp        QdQd RT = pre-exponential [m 2 /s] = diffusion coefficient [m 2 /s] = activation energy [J/mol or eV/atom] = gas constant [8.314 J/mol-K] = absolute temperature [K] D DoDo QdQd R T

26. Chapter 5 - 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 D interstitial >> D substitutional C in  -Fe C in  -Fe Al in Al Fe in  -Fe Fe in  -Fe 1000 K/T D (m 2 /s) C in  -Fe C in  -Fe Al in Al Fe in  -Fe Fe in  -Fe 0.51.01.5 10 -20 10 -14 10 -8 T(  C) 1500 1000 600 300

27. Chapter 5 - Diffusivity increases with T. Experimental Data: D has exp. dependence on T Recall: Vacancy does also! 19 Adapted from Fig. 5.7, Callister 6e. (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.) DIFFUSION AND TEMPERATURE

28. Chapter 5 - Example: At 300ºC the diffusion coefficient and activation energy for Cu in Si are D(300ºC) = 7.8 x 10 -11 m 2 /s Q d = 41.5 kJ/mol What is the diffusion coefficient at 350ºC? transform data D Temp = T ln D 1/T D  DoDo exp        QdQd RT

29. Chapter 5 - Example (cont.) T 1 = 273 + 300 = 573 K T 2 = 273 + 350 = 623 K D 2 = 15.7 x 10 -11 m 2 /s

30. Chapter 5- Fick’s Second Law ; Non-steady state Diffusion In most practical cases, J (flux) and dC/dx (concentration gradient) change with time (t). –Net accumulation or depletion of species diffusing How do we express a time dependent concentration? Concentration at a point x Changing with time ? Flux, J, changes at any point x !

31. Chapter 5- How do we solve this partial differential equation ? Use proper boundary conditions: –t=0, C = C 0, at 0 ≤ x ≤ ∞ –t>0, C = C s, at x = 0 C = C 0, at x = ∞

32. Chapter 5 - Non-steady State Diffusion Adapted from Fig. 5.5, Callister 7e. B.C. at t = 0, C = C o for 0  x   at t > 0, C = C S for x = 0 (const. surf. conc.) C = C o for x =  Copper diffuses into a bar of aluminum. pre-existing conc., C o of copper atoms Surface conc., C of Cu atoms bar s C s

33. Chapter 5 - Solution: C(x,t) = Conc. at point x at time t erf (z) = error function erf(z) values are given in Table 5.1 CSCS CoCo C(x,t)C(x,t)

34. Chapter 5 - 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

35. Chapter 5 - Solution (cont.): –t = 49.5 h x = 4 x 10 -3 m –C x = 0.35 wt%C s = 1.0 wt% –C o = 0.20 wt%  erf(z) = 0.8125

36. Chapter 5 - 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 zerf(z) 0.900.7970 z0.8125 0.950.8209 z  0.93 Now solve for D

37. Chapter 5 - To solve for the temperature at which D has above value, we use a rearranged form of Equation (5.9a); from Table 5.2, for diffusion of C in FCC Fe D o = 2.3 x 10 -5 m 2 /s Q d = 148,000 J/mol  Solution (cont.): T = 1300 K = 1027°C

38. Chapter 5 - 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 (t b ), 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 = 110 x10 -8 cm 2 /s

39. Chapter 5 - Example (cont). Time required for breakthrough ca. 4 min glove C1C1 C2C2 skin paint remover x1x1 x2x2 Solution – assuming linear conc. gradient Equation 22.24 D = 110 x 10 -8 cm 2 /s Given in web chapters !

40. Chapter 5- Copper diffuses into a bar of aluminum. 15 General solution: “Gaussian error function" Adapted from Fig. 5.5, Callister 6e. EX: NON STEADY STATE DIFFUSION C(x,t)  C o C s  C o  1  erf x 2 Dt       C (x, t) = concentration at any time and position !

41. Chapter 5- Concentration profile, C(x), changes w/ time. 14 To conserve matter: Fick's First Law: Governing Eqn.: NON STEADY STATE DIFFUSION

42. Chapter 5-17 The experiment: we recorded combinations of t and x that kept C constant. Diffusion depth given by: = (constant here) DIFFUSION DEMO: ANALYSIS

43. Chapter 5- Experimental result: x ~ t 0.58 Theory predicts x ~ t 0.50 Reasonable agreement! 18 DATA FROM DIFFUSION DEMO

44. Chapter 5- 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? 16 Result: Dt should be held constant. Answer: Note: values of D are provided here. Key point 1: C(x,t 500C ) = C(x,t 600C ). Key point 2: Both cases have the same C o and C s. Example 5.3

45. Chapter 5- Size Impact on Diffusion Smaller atoms diffuse faster

46. Chapter 5- Fast Tracks for diffusion ! eg. self-diffusion of Ag : -Areas where lattice is pre- strained can allow for faster diffusion of atoms -Less energy is needed to distort an already strained lattice !

47. Chapter 5- Important Temperature - diffusion rate increases with increasing temperature (WHY ?) Diffusion mechanism – interstitials diffuse faster (WHY ?) Diffusing and host species - Do, Qd is different for every solute - solvent pair Microstructure - grain boundaries and dislocation cores provide faster pathways for diffusing species, hence diffusion is faster in polycrystalline vs. single crystal materials. (WHY ?)

48. Chapter 5-20 Diffusion FASTER for... open crystal structures lower melting T materials materials w/secondary bonding smaller diffusing atoms cations lower density materials Diffusion SLOWER for... close-packed structures higher melting T materials materials w/covalent bonding larger diffusing atoms anions higher density materials SUMMARY: STRUCTURE & DIFFUSION WHY ?

49. Chapter 5- Reading: Chapter 5 and Chapter 6 Go over the exercises in Chapter 5, using your book and notes Core Problems: 5.1, 5.2, 5.3, 5.6, 5.13, 5.22, 5.30 Self-help Problems (not bonus !): 5.9-5.12, 5.16-5.21, 5.23-5.29 0 ANNOUNCEMENTS Due date: 03-18-2008