1. DIFFUSION IN SOLIDS  FICK’S LAWS  KIRKENDALL EFFECT  ATOMIC MECHANISMS Diffusion in Solids P.G. Shewmon McGraw-Hill, New York (1963)

2. ArH2H2 Movable piston with an orifice H 2 diffusion direction Ar diffusion direction Piston motion Piston moves in the direction of the slower moving species

3. AB Inert Marker – thin rod of a high melting material which is basically insoluble in A & B Kirkendall effect  Materials A and B welded together with Inert marker and given a diffusion anneal  Usually the lower melting component diffuses faster (say B) Marker motion

4. Diffusion  Mass flow process by which species change their position relative to their neighbours  Driven by thermal energy and a gradient  Thermal energy → thermal vibrations → Atomic jumps Concentration / chemical potential ElectricGradient Magnetic Stress

5.  Assume that only B is moving into A  Assume steady state conditions → J  f(x,t) (No accumulation of matter)  Flux (J) (restricted definition) → Flow / area / time [Atoms / m 2 / s]

6. Fick’s I law No. of atoms crossing area A per unit time Cross-sectional area Concentration gradient Matter transport is down the concentration gradient Diffusion coefficient/ diffusivity A Flow direction  As a first approximation assume D  f(t)

7. Fick’s first law

8.  Diffusivity (D) → f(A, B, T) D = f(c) D  f(c) C1C1 C2C2 Steady state diffusion x → Concentration →

9. Diffusion Steady state J  f(x,t) Non-steady state J = f(x,t) D = f(c) D  f(c)

10. Fick’s II law JxJx J x+  x xx Fick’s first law D  f(x)

11. RHS is the curvature of the c vs x curve x → c → x → c → +ve curvature  c ↑ as t ↑  ve curvature  c ↓ as t ↑ LHS is the change is concentration with time

12. Solution to 2 o de with 2 constants determined from Boundary Conditions and Initial Condition  Erf (  ) = 1  Erf (-  ) = -1  Erf (0) = 0  Erf (-x) = -Erf (x) u → Exp(  u 2 ) → 0  Area

13. A B Applications based on Fick’s II law x → Concentration → C avg ↑ t t 1 > 0 | c(x,t 1 ) t 2 > t 1 | c(x,t 1 ) t = 0 | c(x,0) A & B welded together and heated to high temperature (kept constant → T 0 ) Flux f(x)| t f(t)| x Non-steady state  If D = f(c)  c(+x,t)  c(-x,t) i.e. asymmetry about y-axis  C(+x, 0) = C 1  C(  x, 0) = C 2 C1C1 C2C2  A = (C 1 + C 2 )/2  B = (C 2 – C 1 )/2 Determination of Diffusivity

14. Temperature dependence of diffusivity Arrhenius type

15. Applications based on Fick’s II law Carburization of steel  Surface is often the most important part of the component, which is prone to degradation  Surface hardenting of steel components like gears is done by carburizing or nitriding  Pack carburizing → solid carbon powder used as C source  Gas carburizing → Methane gas CH 4 (g) → 2H 2 (g) + C (diffuses into steel) x → 0 C1C1 CSCS  C(+x, 0) = C 1  C(0, t) = C S  A = C S  B = C S – C 1

16. Approximate formula for depth of penetration

17. ATOMIC MODELS OF DIFFUSION 1. Interstitial Mechanism

18. 2. Vacancy Mechanism

19. 3. Interstitialcy Mechanism

20. 4. Direct Interchange and Ring

21. Interstitial Diffusion 1 2 12 HmHm  At T > 0 K vibration of the atoms provides the energy to overcome the energy barrier  H m (enthalpy of motion)  → frequency of vibrations, ’ → number of successful jumps / time

22. 1 2 Vacant site     c = atoms / volume  c = 1 /  3  concentration gradient dc/dx = (  1 /  3 )/  =  1 /  4  Flux = No of atoms / area / time = ’ / area = ’ /  2 On comparison with

23. Substitutional Diffusion  Probability for a jump  (probability that the site is vacant). (probability that the atom has sufficient energy)   H m → enthalpy of motion of atom  ’ → frequency of successful jumps As derived for interstitial diffusion

24. Element HfHf HmHm  H f +  H m Q Au9780177174 Ag9579174184 Calculated and experimental activation energies for vacancy Diffusion

25. Interstitial Diffusion Substitutional Diffusion  D (C in FCC Fe at 1000ºC) = 3  10  11 m 2 /s  D (Ni in FCC Fe at 1000ºC) = 2  10  16 m 2 /s

26. DIFFUSION PATHS WITH LESSER RESISTANCE Q surface < Q grain boundary < Q lattice Experimentally determined activation energies for diffusion  Core of dislocation lines offer paths of lower resistance → PIPE DIFFUSION Lower activation energy automatically implies higher diffusivity  Diffusivity for a given path along with the available cross-section for the path will determine the diffusion rate for that path

27. Comparison of Diffusivity for self-diffusion of Ag → single crystal vs polycrystal 1/T → Log (D) → Schematic Polycrystal Single crystal ← Increasing Temperature  Q grain boundary = 110 kJ /mole  Q Lattice = 192 kJ /mole