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Lecture on DIFFUSION IN SOLIDS Applications of Diffusion in Solids (besides nucleation and growth) Hard Facing (Carburizing of Steels) Tough Tools and Parts. Wear Facing of Gears, Wheels and Rails Chemical Tempering of Glass and Ceramics Toughened Ceramics (Corel Ware) Shard resistant safety glass

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Figure 4.41

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Thin Film Electronics (CMOS and Bipolar Transistors) Doping of Semiconductors Diffusion Bonding -- (Adhesives and cements for ceramic, metallic and polymer materials) Portland Cement as Bonding for Construction Solvents Cements for PVC Polymeric Piping Solders and Welds for Thermocouple Junctions Corrosion Protection Galvanizing, Electroplating, Anodizing, Inhibiting Gas (Chemical) Separation Processes – Diffusion membranes

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Diffusion is a RATE PROCESS Probability of Finding an atom with energy E *

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Fraction of atoms or molecules having energies greater than E * which is itself much greater than the average energy E.

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Arrhenius' equation for the rate of many chemical reactions

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Rewritten as linear functions of the reciprocal of the absolute temperature.

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ATOMIC DIFFUSION IN SOLIDS Diffusion can be defined as the mechanism by which matter is transported into or through matter. Two mechanisms for diffusion of atoms in a crystalline lattice: 1. Vacancy or Substitutional Mechanism. 2. Interstitial mechanism.

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Vacancy Mechanism Atoms can move from one site to another if there is sufficient energy present for the atoms to overcome a local activation energy barrier and if there are vacancies present for the atoms to move into. The activation energy for diffusion is the sum of the energy required to form a vacancy and the energy to move the vacancy.

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Interstitial Mechanism Interstitial atoms like hydrogen, helium, carbon, nitrogen, etc) must squeeze through openings between interstitial sites to diffuse around in a crystal. The activation energy for diffusion is the energy required for these atoms to squeeze through the small openings between the host lattice atoms.

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Steady-State Diffusion: Fick's First Law of Diffusion. For steady state conditions, the net flux of atoms is equal to the diffusivity times the concentration gradient.

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Diffusivity -- the proportionality constant between flux and concentration gradient depends on: 1. Diffusion mechanism. Substitutional vs interstitial. 2. Temperature. 3. Type of crystal structure of the host lattice. Interstitial diffusion easier in BCC than in FCC. 4. Type of crystal imperfections. (a) Diffusion takes place faster along grain boundaries than elsewhere in a crystal. (b) Diffusion is faster along dislocation lines than through bulk crystal. (c) Excess vacancies will enhance diffusion. 5. Concentration of diffusing species.

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Temperature Dependence of the Diffusion Coefficient D is the Diffusivity or Diffusion Coefficient ( m 2 / sec ) D o is the prexponential factor ( m 2 / sec ) Q d is the activation energy for diffusion ( joules / mole ) R is the gas constant ( joules / (mole deg) ) T is the absolute temperature ( K )

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Temperature Dependence of Diffusivity

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Non-Steady-State Diffusion : Fick's Second Law of Diffusion In words, The rate of change of composition at position x with time, t, is equal to the rate of change of the product of the diffusivity, D, times the rate of change of the concentration gradient, dC x /dx, with respect to distance, x.

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Second order partial differential equations are nontrivial and difficult to solve. Consider diffusion in from a surface where the concentration of diffusing species is always constant. This solution applies to gas diffusion into a solid as in carburization of steels or doping of semiconductors. Boundary Conditions For t = 0, C = C o at 0 x For t > 0 C = C s at x = 0 and C = C o at x =oo

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where C s = surface concentration C o = initial uniform bulk concentration C x = concentration of element at distance x from surface at time t. x = distance from surface D = diffusivity of diffusing species in host lattice t = time erf = error function

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Carburizing or Surface Modifying System: Species A achieves a surface concentration of C s and at time zero the initial uniform concentration of species A in the solid is C o. Then the solution to Fick's second law for the relationship between the concentration C x at a distance x below the surface at time t is given as where C s = surface concentration, C o = initial uniform bulk concentration C x = concentration of element at distance x from surface at time t. x = distance from surface D = diffusivity of diffusing species in host lattice t = time

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Carbon diffusion into Steel – Hard Facing

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Temperature Dependence of Diffusivity

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N-type and P-type Dopant diffusion into Silicon. The making of devices.

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Interdiffusion with Interface motion

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