CHAPTER 5: DIFFUSION IN SOLIDS

1. CHAPTER 5:DIFFUSION IN SOLIDS 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? 1

2. Chapter 5: DIFFUSION • Why study Diffusion? • Heat-treated to improve their properties. • Heat-treatment almost always involve atomic diffusion. • desired results depends on diffusion rate • Heat-treatment temperature, time, and/or rate of heating/cooling can be predicted by the mathematics of diffusion • Steel gear  Case hardened to improve hardness and resistance to fatigue  diffusing excess carbon or nitrogen into outer surface layer.

3. 5.1 Introduction • Diffusion: The phenomenon of material transport by atomic motion. Many reactions and processes that are important in the material treatment rely on the mass transfer: • Either with a specific solid ( at microscopic level ) • Or from a liquid, a gas, or another solid phase. • This chapter covers: • Atomic mechanism • Mathematics of diffusion • Influence of temperature and diffusing species of the diffusion rate

4. 5.1 Introduction (Contd.) • Phenomenon of diffusion • Explained using diffusion couple, formed by joining bars of two different materials having intimate contact • Copper and Nickel diffusion couple • Atom locations and concentration • Heated for an extended period at an elevated temperature ( but below melting temperature of both ) and cooled to room temperature.

5. DIFFUSION • Interdiffusion: In an alloy, atoms tend to migrate from regions of large concentration. After some time Initially 3

6. 5.1 Introduction (Contd.) • Chemical analysis reveals • Alloy region • Variation of concentration • Atoms migrated or diffused into one another • Interdiffusion or impurity diffusion • Atoms of one metal diffuses into another • Net drift of atoms from high to lower concentration

7. DIFFUSION • • Self-diffusion:In an elemental solid, atoms • also migrate. • All atoms exchanging positions are of same type • No compositional Diffusion in pure metal • changes Label some atoms After some time 4

8. 5.2 Diffusion Mechanism • Atoms in solids are in constant motion rapidly changing positions. • Diffusion is just the stepwise migration of atoms from a lattice site to other lattice site. • Two conditions for movement: 1. There must be an empty adjacent site 2. Atom must have sufficient energy to break bonds with neighbor atoms Atomic vibration (Section 4.7): • Every atom is vibrating very rapidly about its lattice position within the crystal • At any instant, not all vibrate with same frequency and amplitude. • Not all atoms have same energy • Same atom may have different level of energy at different time • Energy increases with temperature

9. 5.2 Diffusion Mechanism (Contd.) • Several different models for atomic motion • Two dominate for metallic diffusion • VACANCY DIFFUSION • Involves interchange of an atom from a normal lattice position to an adjacent vacant lattice site or vacancy • Necessitates presence of vacancies • Diffusing atoms and vacancies exchange positions  they move in opposite directions • Both self- and inter-diffusion occurs by this mechanism

10. DIFFUSION MECHANISMS Vacancy Diffusion: • applies to substitutional impurities • atoms exchange with vacancies • rate depends on: --number of vacancies --activation energy to exchange. 5

11. 5.2 Diffusion Mechanism (Contd.) • INTERSTITIAL DIFFUSION • Atoms migrate from an interstitial position to a neighboring one that is empty • Found for interdiffusion of impuries such as hydrogen, carbon, nitrogen, and oxygen  atoms small enough to fit into interstitial positions. • Host or substitutional impurity atoms rarely have insterstitial diffusion • Interstitial atoms are smaller and thus more mobile  interstitial diffusion occurs much more rapidly then by vacancy mode • There are more empty interstitial positions than vacancies  interstitial atomic movement have greater probability

12. Diffusion Simulation • Simulation of interdiffusion across an interface: • Rate of substitutional diffusion depends on: --vacancy concentration --frequency of jumping. (Courtesy P.M. Anderson)

13. Diffusion Mechanisms • Interstitial diffusion – smaller atoms can diffuse between atoms. Adapted from Fig. 5.3 (b), Callister 7e. More rapid than vacancy diffusion

14. Processing Using Diffusion • Case Hardening: --Diffuse carbon atoms into the host iron atoms at the surface. --Example of interstitial diffusion is a case hardened gear. Adapted from chapter-opening photograph, Chapter 5, Callister 7e. (Courtesy of Surface Division, Midland-Ross.) • Result: The presence of C atoms makes iron (steel) harder.

15. 5.3 Steady-State Diffusion • The quantity of an element that is transported within another is a function of time  diffusion is a time-dependent process. • Diffusion flux (J) • Rate of diffusion or mass transfer • Defined as “mass or number of atoms (M) diffusing through and perpendicular to a unit cross-sectional area of solid per unit time. • Mathematically, In differential form: A: area across which diffusion is occuring t: elapsed diffusion time

16. C1 C1 C2 x1 x2 C2 x Steady-State Diffusion Rate of diffusion independent of time • Flux proportional to concentration gradient = Fick’s first law of diffusion D diffusion coefficient

17. 5.3 Steady-State Diffusion (Contd.)

18. 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: C1= 0.44 g/cm3 C2= 0.02 g/cm3

19. Data: D = 110x10-8 cm2/s C1 = 0.44 g/cm3 C2 = 0.02 g/cm3 x2 – x1 = 0.04 cm Example (cont). • Solution – assuming linear conc. gradient glove C1 paint remover skin C2 x1 x2

20. 5.4 Nonsteady-State Diffusion • Most practical diffusion situations are non-steady • Non-steady • Diffusion flux and the concentration flux at some particular point of solid vary with time • Net accumulation or depletion of the diffusing species • Figure shown concentration profile at three different times

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

22. Solution for Semi-infinite Solid with constant surface concentration • Assumptions • Initial concentration C0 • X = 0 at the surface and increases with distance into the solid • Initial time = 0 • Boundary conditions • For t = 0, C = Co at 0  x   • For t > 0, C = Cs (Constant surface concentration) at x=0 C = C0 at x =  • Solution • erf ( ) : Gaussian error function • Values given in Table 5.1

23. NON STEADY STATE DIFFUSION • Copper diffuses into a bar of aluminum. • General solution: "error function" Values calibrated in Table 5.1, Callister 6e. 15

26. 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

27.  erf(z) = 0.8125 Solution (cont.): • t = 49.5 h x = 4 x 10-3 m • Cx = 0.35 wt% Cs = 1.0 wt% • Co = 0.20 wt%

28. z erf(z) 0.90 0.7970 z 0.8125 0.95 0.8209 Now solve for D 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= 0.93

29. from Table 5.2, for diffusion of C in FCC Fe Do = 2.3 x 10-5 m2/s Qd = 148,000 J/mol  T = 1300 K = 1027°C Solution (cont.): • To solve for the temperature at which D has above value, we use a rearranged form of Equation (5.9a);

30. EXAMPLE PROBLEM • 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? Key point 1:C(x,t500C) = C(x,t600C). Key point 2:Both cases have the same Co and Cs. • Dt should be held constant. Note: values of D are Given here. • Answer: 16

31. Factors That Influence Diffusion

32. Factors That Influence Diffusion (Contd.) • DIFFUSING SPECIES • Magnitude of diffusion coefficient (D)  indicative of the rate at which atoms diffuse • D depends on both the diffusing species as well as the host atomic structure • Self-diffusion Fe in a-Fe 3.0E(-21) m2/s Vacancy Diffusion Inter-diffusion C in a-Fe 2.4E(-12) m2/s Interstitial Diffusion • Interstitial is faster than vacancy diffusion

33. Factors That Influence Diffusion (Contd.) TEMPERATURE • Temperature has a most profound influence on the coefficients and diffusion rate • Example: Fe ina-Fe (Table 5.2) 500oC D=3.0E(-21) m2/s 900oC D=1.8E(-15) m2/s  approximately six orders

34. DIFFUSION AND TEMPERATURE • Diffusivity increases with T. • Experimental Data: D has exp. dependence on T Recall: Vacancy does also! 19