DIFFUSION IN SOLIDS FICKS LAWS KIRKENDALL EFFECT ATOMIC MECHANISMSDiffusion in SolidsP.G. Shewmon McGraw-Hill, New York (1963)

ArH2Movable piston with an orificeH2 diffusion directionAr diffusion directionPiston motionPiston moves in the direction of the slower moving species

ABInert Marker thin rod of a high melting material which is basically insoluble in A & BKirkendall 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 jumpsConcentration / chemical potentialElectricGradientMagneticStress

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 / m2 / s]

Ficks I lawNo. of atoms crossing area A per unit timeCross-sectional areaConcentration gradientMatter transport is down the concentration gradientDiffusion coefficient/ diffusivityAFlow direction As a first approximation assume D f(t)

Ficks first law

Diffusivity (D) f(A, B, T)D = f(c)D f(c)C1C2Steady state diffusionx Concentration

DiffusionSteady state J f(x,t)Non-steady state J = f(x,t)D = f(c)D = f(c)D f(c)D f(c)

Ficks II lawJxJx+xxFicks first lawD f(x)

RHS is the curvature of the c vs x curve+ve curvature c as t ve curvature c as t LHS is the change is concentration with time

Solution to 2o de with 2 constants determined from Boundary Conditions and Initial Condition Erf () = 1 Erf (-) = -1 Erf (0) = 0 Erf (-x) = -Erf (x)u Exp( u2) 0Area

ABApplications based on Ficks II lawx Concentration Cavg tt1 > 0 | c(x,t1)t2 > t1 | c(x,t1)t = 0 | c(x,0)A & B welded together and heated to high temperature (kept constant T0)Fluxf(x)|tf(t)|xNon-steady state If D = f(c) c(+x,t) c(-x,t) i.e. asymmetry about y-axis C(+x, 0) = C1 C(x, 0) = C2C1C2 A = (C1 + C2)/2 B = (C2 C1)/2Determination of Diffusivity

Temperature dependence of diffusivityArrhenius type

Applications based on Ficks II lawCarburization 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 CH4 (g) 2H2 (g) + C (diffuses into steel)x 0C1CS C(+x, 0) = C1 C(0, t) = CS A = CS B = CS C1

Approximate formula for depth of penetration

ATOMIC MODELS OF DIFFUSION1. Interstitial Mechanism

2. Vacancy Mechanism

3. Interstitialcy Mechanism

4. Direct Interchange and Ring

Interstitial Diffusion12 At T > 0 K vibration of the atoms provides the energy to overcome the energy barrier Hm (enthalpy of motion) frequency of vibrations, number of successful jumps / time

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) Hm enthalpy of motion of atom frequency of successful jumpsAs derived for interstitial diffusion

Calculated and experimental activation energies for vacancy Diffusion

ElementHfHmHf + HmQAu9780177174Ag9579174184

Interstitial DiffusionSubstitutional Diffusion D (C in FCC Fe at 1000C) = 3 1011 m2/s D (Ni in FCC Fe at 1000C) = 2 1016 m2/s

DIFFUSION PATHS WITH LESSER RESISTANCEQsurface < Qgrain boundary < QlatticeExperimentally determined activation energies for diffusion Core of dislocation lines offer paths of lower resistance PIPE DIFFUSIONLower 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 polycrystal1/T Log (D) SchematicPolycrystalSingle crystal Increasing Temperature Qgrain boundary = 110 kJ /mole QLattice = 192 kJ /mole