Atomic Models of Diffusivity
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Diffusion of Solute Atoms in BCC Crystal by the Interstitial Mechanism
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- Connection between jump rate, , and intersite jump distance, , and the correlation factor:
- Each interstitial site is associated with four nearest-neighbors
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- Lattice constant:
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- Consider concentration gradient and number of site-pairs that can contribute to flux across crystal plane
- Concentration gradient results in flux of atoms from three types of interstitial sites in plane
- : number of atoms in the plane per unit area
- Carbon concentration on each of the three sites:
- Jump rate of atoms from the type 1 and 3 sites between plan and :
- Contribution to the flux from the three sites:
- Convert to the number of atoms per unit volume:
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- Find the reverse flux by using a first-order expansion
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- Compare with Fick's law expression, , and total jump frequency, :
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Self-Diffusion in FCC Structure by Vacancy Mechanism
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- There are twelve nearest neighbors on an fcc lattice
- Vacancies randomly occupy sites and are associated with jump frequency,
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- : fraction of sites randomly occupied by vacancies
- Jump rate of host atoms:
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- Self-diffusivity with :
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- With uncorrelated vacancy diffusion, the vacancy diffusivity is
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- The vacancy diffusivity is related to the self-diffusivity
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- when the vacancies are in thermal equilibrium
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- : vacancy vibrational entropy
- : enthalpy of formation
Intrinsic Crystal Self-Diffusion with Schottky Defects
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- Predominant point defects are cation and anion vacancy complexes
- Self-diffusion occurs by a vacancy mechanism
- Defect-creation (Kroger-Vink notation)
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- Relation between free energy of formation, , and the equilibrium constant,
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- The activities correspond to anion and cation vacancies
- With dilute concentrations of vacancies, Raoult's law applies, and activities are equal to site fractions
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- A requirement of electrical neutrality is that the number of potassium vacancies is equal to the number of chlorine vacancies
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- Vacancy self-diffusion in a metal
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- : geometric factor
- : correlation factor
- Activation energy of self-diffusion
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Intrinsic Crystal Self-Diffusion with Frenkel Defects
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- Elecrical neutrality condition:
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- Activation energy of self-diffusivity of cations
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Extrinsic Crystal Self-Diffusion with Frenkel Defects
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- Extrinsic defects result from the addition of aliovalent solute
- Extrinsic cation-site vacancies are created by incorporation of through doping with
- Step 1: Two cation and two anion vacancies form
- Step 2: Single cation and two anions incorporated
- Cation and anionic vacancy populations relate to the site fraction of extrinsic Ca^{++} impurity
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- The equation can be solved to find the vacancy site fraction
- Two limiting cases of the behavior of
- Intrinsic: , then
- Extrinsic: , then
Self-Diffusion in Nonstochiometric Crystals
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- Oxidation of
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- Consider the sum of two reactions
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- A cation vacancy must be created with regard to every O atom added
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- Relationship between cation vacancy site fraction and oxygen gas pressure
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- Equilibrium constant of the reaction:
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- Electrical neutrality condition with oxidation-induced cation vacancies as dominant charged defects
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- Solve to find
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