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 • Figure 5.1 shows as formed • 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. Self-diffusion • 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.)

12. 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

13. INTERSTITIAL DIFFUSION • Applies to interstitial impurities. • More rapid than vacancy diffusion. • Simulation: --shows the jumping of a smaller atom (gray) from one interstitial site to another in a BCC structure. The interstitial sites considered here are at midpoints along the unit cell edges. (Courtesy P.M. Anderson) 7

14. 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, J = M / (At) • In differential form: J = (1/A)(dM/dt) A: area across which diffusion is occuring t: elapsed diffusion time

15. 5.3 Steady-State Diffusion (Contd.) • If the diffusion flux does not change with time  steady-state diffusion • Example: • Diffusion of a gas through a plate of metal • Concentration (or pressure) of diffusing species on both side are held constant • Concentration profile: Concentration versus position • Assumed linear concentration profile as shown in figure (b)

16. 5.3 Steady-State Diffusion (Contd.) • Concentration gradient • Slope at a particular point on the concentration profile curve • Concentration gradient = dC / dx • For linear concentration shown in figure 5.4b: Conc. Gradient = DC/Dx = (CA – CB) / (xA – xB) • Fick’s first law: For steady-state diffusion, the flux is proportional to the concentration gradient J = -D(dC/dx) D: diffusion coefficient (sq. m per second ) -ve sign: direction of diffusion from a high to a low concentration

17. 5.3 Steady-State Diffusion (Contd.)

18. 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

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

20. 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

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

24. 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

25. Factors That Influence Diffusion

26. 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

27. 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

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