Materials Science and Engineering/Derivations/Kinetics

Atomic Models of Diffusivity

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Metals

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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
 
 
  • Lattice constant:  
    •  
 
 
  • 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:  
 
    • Find the reverse flux by using a first-order expansion
 
    • Find the net flux
 
 
    • Compare with Fick's law expression,  , and total jump frequency,  :
 
 
  

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,  
 
 
  •  : fraction of sites randomly occupied by vacancies
  • Jump rate of host atoms:
 
  • Self-diffusivity with  :
 
 
  • With uncorrelated vacancy diffusion, the vacancy diffusivity is
 
  • The vacancy diffusivity is related to the self-diffusivity
 
  •   when the vacancies are in thermal equilibrium
 
    •  : vacancy vibrational entropy
    •  : enthalpy of formation
  
  

Ionic Solids

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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)
 
 
    • Relation between free energy of formation,  , and the equilibrium constant,  
 
 
 
    • The activities correspond to anion and cation vacancies
    • With dilute concentrations of vacancies, Raoult's law applies, and activities are equal to site fractions
 
    • A requirement of electrical neutrality is that the number of potassium vacancies is equal to the number of chlorine vacancies
 
    • Vacancy self-diffusion in a metal
 
    •  : geometric factor
    •  : correlation factor
    • Activation energy of self-diffusion
 

Intrinsic Crystal Self-Diffusion with Frenkel Defects

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  • Frenkel pair formation
 
  •  
 
  • Elecrical neutrality condition:
 
  • Activation energy of self-diffusivity of cations
 

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
 
 
    • 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  
 
    • Consider the sum of two reactions
 
 
 
    • A cation vacancy must be created with regard to every O atom added
 
    • Relationship between cation vacancy site fraction and oxygen gas pressure
 
 
    • Equilibrium constant of the reaction:
 
    • Electrical neutrality condition with oxidation-induced cation vacancies as dominant charged defects
 
    • Solve to find  
 
    • Activation energy