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

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

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

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

 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]

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)

Fick’s first law

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

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

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

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

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

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

Temperature dependence of diffusivity Arrhenius type

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

Approximate formula for depth of penetration

ATOMIC MODELS OF DIFFUSION 1. Interstitial Mechanism

2. Vacancy Mechanism

3. Interstitialcy Mechanism

4. Direct Interchange and Ring

Interstitial Diffusion 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

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

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

Element HfHf HmHm  H f +  H m Q Au Ag Calculated and experimental activation energies for vacancy Diffusion

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

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

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