Defects in Solids

1. Defects in Solids • 0-D or point defects • vacancies, interstitials, etc. • control mass diffusion • 1-D or linear defects • dislocations • control deformation processes • 2-D or planar defects • grain boundaries, surfaces, interfaces • 3-D or volume defects • voids, secondary components (phases) concentrations 

2. cH Diffusional Processes diffusion coefficient concentration gradient Pd flux H2 response  driving force + CO Fick’s first law (similar to Ohm’s law) phenomenological + CO2 H Applies under steady state conditions c(x)  f(t) Units of D x hydrogen separation membrane

3. A B t = 0 CA CB t > 0 CA CB Diffusional Processes Non steady-state: c(x) = f(t) Continuity requirements Fick’s second law if D  f(x)

4. Random walk (1) (2) ao x 3-D D Atomistics of diffusion planes of atoms tracer species with a concentration gradient c = concentration #/cm3 n = #/cm2 # density in the plane n = cao n1 = on plane (1) n2 = on plane (2) Net from plane (1) to plane (2) Flux from plane (1) to plane (2) (½ jump to the left) Flux from plane (2) to plane (1)

5. Mechanisms of Diffusion Vacancy net transport vacancy to right atom to left Interstitial (self or impurity)

6. Atomistics from Mechanism fraction of atoms that participate jump distance G probability that an atom will jump into an available site geometric constant ~ 1/(# nearest neighbor sites) probability that a nearest neighbor site is vacant (available) for jumping into Evaluate terms atom vibrates at frequency nD = Debye frequency w crystallographic sites G success rate of jumping DGm position empty occupied

7. Atomistics from Mechanism fraction of atoms that participate P and [N] differ depending on mechanism probability that a nearest neighbor site is vacant (available) for jumping into • Substitutional impurity • P = concentration of vacancies • [N] = fixed, < 1 • Interstitial impurity • P = 1 • [N] = fixed, < 1 • Vacancy • P = concentration of vacancies • [N] = 1 – P  1 • Interstitial • P = 1 – [N]  1 • [N] = concentration of interstitial atoms defect concentrations 

8. Classic Diffusion Problem Expose a solid material to a gas phase and observe diffusion into the solid surface concentration cs Solid Gas c t  t = 0 co initial concentration } 0  1 x Boundary conditions: Solution: at t = 0, c(x) = co 0  x   t > 0, c(x=0) = cs t    erf(0)  0  c(x)  cs Characteristic diffusion distance & time: set argument = 1, 1-erf(1) = 0.157