EMA500 Lecture Interstitial Diffusion & Fick s st Law Prof. Zhe Cheng Mechanical & Materials Engineering Florida International University
Substitutional Diffusion Different possibilities Exchange Mechanis Vacancy Mechanis A A A A A A A A A A May occur on surface Higher activation energy Far ore coon in solids Lower activation energy Features of substitutional diffusion via vacancy echanis Size of diffusing atos/ions: siilar to surrounding atrix ato/ion size Successful juping rate deterined by Frequency of jup attepts (theral activation), & Availability of vacancy (concentration)
3 Interstitial Diffusion Atos/ions at interstitial sites -D 3-D (Octahedral) Interstitial sites Features FCC CC http://www.tf.uni-kiel.de/atwis/aat/def_en/kap_/illustr/t_3_3.htl Size of diffusing atos/ions - Much saller than surrounding (atrix) atos/ions C diffusing through Fe: r C ~ 70 p vs. r Fe ~ 40 p Cu diffusing through Si: r Cu+ = ~77 p vs. r Si ~ 0 p http://en.wikipedia.org/wiki/atoic_radius; http://en.wikipedia.org/wiki/ionic_radius Successful juping rate for dilute solid solution - Deterined by theral activation alone as neighboring interstitials typically are rarely occupied
Interstitial Diffusion as a Rando up Process () 4 Dilute interstitial solid solution of in A - average successful jup frequency (in unit of sec - ) of interstitial ato n area (nuber) density of in plane () n area (nuber) density of in plane () () () In unit tie, flux of, fro plane () to () fro plane () to () n n Net flux of along x direction: ( n n ) in unit of - sec - x Introduce as jup distance of plane () fro (). For siple cubic lattice, volue concentration (in unit of -3 ) C at position () n n C () C () and () will be
Interstitial Diffusion as a Rando up Process () 5 Continue fro p4 C () C Since both lattice constant and Then () Define diffusion coefficient C ( ) C () C x C () C () D C ( ) C () are very sall, C C x () C () D in unit of /sec x C We have D C x Fick s st Law
Other Considerations Diffusion C x Ipacts of lattice structures D Other cubic lattice (e.g., FCC, CC) - still applicable (need soe derivation) Non-cubic - D different along different crystal directions Ipacts of solute concentration D Assuption: D constant for sall difference in concentration Over large concentration range: D different for different solute concentration D C in -Fe at 000 o C is.5 0 - /s at 0.5wt% carbon and 7.7 0 - /s at.4wt% Estiation of interstitial atos jup frequency -Fe, FCC structure, lattice constant 0.37 n, jup distance n At 000 o C, D =.5 0 - /s. Answer: ) What is estiated successful jup frequency? ) One in how any attepts succeeds with a jup? Successful jup frequency: C =. x0 9 /s Lattice vibration frequency: C =.38 0-3 /K (000+73) K /(.3 0-34 s) =. x0 3 /s (fro kt = h) C / C = 0-4, one in 0,000 jup successful a / 0.
Theral Activation of Interstitial Diffusion () 7 Activation energy Migration energy barrier G Proportion of high energy atos Portion of atos with energy higher than ean/equilibriu energy by G ( for igration) G exp RT up frequency - Theral vibration frequency (successful) jup frequency will be G G exp RT G x
8 S D exp R Define Frequency factor D Theral Activation of Interstitial Diffusion () up frequency G exp RT Diffusion coefficient D H TS Therefore, D exp RT 0 Interstitial diffusion activation energy H exp RT S exp R Q ID H We have D D 0 exp QID RT Diffusion coefficient increases exponentially with T!
Theral Activation of Interstitial Diffusion (3) 9 Soe data In CC α-fe Obtaining Q and D 0 fro easured D We have Or Interstitial ato D 0 ( /s) Q (k/ol) D C 84. N 0.3 7. H 0. 3.4 Porter et al. Phase Transforations in Metals & Alloys, 3 rd Ed (009), CRC Press, p. 73 D 0 exp Q RT Q ln D ln D0 R T Q lg D lg D0.3R T lgd lgd 0 Slope = Q.3R /T