EMA5001 Lecture 4 Self-Diffusion & Vacancy Diffusion

EMA5001 Lecture 4Self-Diffusion &Vacancy Diffusion

Self-Diffusion (1)A* Self-DiffusionAIntroducing radioactive atoms A*, thenmeasure penetration at different time A* and A chemically identical AssumingA “Successful” jump frequency depend on Thermal activation:Portion of atoms with energy higher thanequilibrium by Gm Availability of vacancy Xv Vacancy concentrationEMA 5001 Physical Properties of MaterialsAAZhe Cheng (2016)AAGAAAAAA*AA GmAAAAAAA*AAA*A A* and A have the same jump frequency Each jump is unrelated to the previous jumpeven for the case of jumping into a vacancy1DA DA 26AAAA 4 Self-Diffusion & Vacancy Diffusion2

Self-Diffusion (2) Continue from p. 2“Successful” substitutional diffusion jump frequency depend on Thermal activation: : Thermal vibration frequency Gm : Portion of atoms with energy higher than equilibrium by Gm RT exp Availability of vacancy Xv : Probability of one site is vacant (i.e., vacancy molar fraction)Therefore, Gm Gm exp X Xexp v vRTRT If the vacancies are in thermodynamic equilibrium, Gv X v X veq exp RT in which Gv is the free energy change (positive value) needed to create vacanciesEMA 5001 Physical Properties of MaterialsZhe Cheng (2016)4 Self-Diffusion & Vacancy Diffusion3

Self-Diffusion (3) Continue from p.3We have D D 1 2AA6 Gm X v exp RT Gv X v X veq exp RT Therefore, diffusion coefficient isDA 1 Gv Gm 2 1 2 Gm Gv exp exp vexp 6 RT RT6RT 1 S S v H m H v DA 2 v exp mexp 6RRT DefineFrequency factor1 S S v D0 2 v exp m 6R QSD H m H vSubstitutional diffusion activation energyWe have Q DA D0 exp SD RT EMA 5001 Physical Properties of MaterialsRepresentation of “self-diffusion”coefficient has similar formality as thatfor interstitial diffusion coefficientZhe Cheng (2016)4 Self-Diffusion & Vacancy Diffusion4

Measurement of Self-Diffusion usingRadioactive Tracer ElementA* ExperimentRadio active atoms sputtered onto surfaceweld together into a diffusion coupleMeasure concentration profile with timeand temperatureAACt1 Illustrate concentration profile andhow it changes with timet1 t2 t 3t2t30 What is the analytical concentrationprofile?Similar to special case solution to Fick’s 2ndLaw of “spin on dopant”If M is total amount of radio active atomsEMA 5001 Physical Properties of MaterialsZhe Cheng (2016)x Mx2 C B x, t exp 2 Dt 4 Dt 4 Self-Diffusion & Vacancy Diffusion5

Considerations on e earth) -La11931.3102.610.442BCCRb3122339.415.25.8(Alkali ansition metal)Zr2125134273.515.525FCCAl933170142.018.31.9 05496.035.53.6x10-4BCCDiamondEMA 5001 Physical Properties of MaterialsZhe Cheng (2016)4 Self-Diffusion & Vacancy Diffusion6

Diffusion of Vacancy Considerations An atom jumps into a vacancy can beA A Aviewed as a vacancy jumps to aA A V Aneighboring atom Vacancy always surrounded by atoms A A AVacancy diffusion similar to interstitialdiffusionSimilar to derivation of interstitial diffusionWe haveA1Dv v 26AAssuming vacancy thermal vibration frequencythe same as atoms, thenAAAAAVAAAAAVAAAAAA Gm v exp RT EMA 5001 Physical Properties of MaterialsZhe Cheng (2016)4 Self-Diffusion & Vacancy Diffusion7

Diffusion of Vacancy vs.Substitutional Atoms Continue from p. 71 Gm 2 expDv v vRT6 1 Gm 2 1 2 S H m Therefore, Dv v exp v exp m exp 6 6 RT R RT Diffusion coefficient of vacancy vs. substitutional atomFor self-diffusion DA DA 1 2 X v exp Gm 6 RT Gm The relationship between jump frequency is X v exp v X vRT Since the jump distance is the sameTherefore,We have11 1 DA DA 2 v X v 2 v 2 X v Dv X v66 6 DDv AAs Xv 1, Dv DAXvEMA 5001 Physical Properties of MaterialsZhe Cheng (2016)4 Self-Diffusion & Vacancy Diffusion8

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 4 Self-Diffusion & Vacancy Diffusion Diffusion of Vacancy vs. Substitutional Atoms Continue from p. 7 2 Therefore, Diffusion coefficient of vacancy vs. substitutional atom For self-diffusion 2 The relationship between jump frequency is Since the jump distance is the same