Transcript MECE3345
V. Diffusion in Solids MECE 3345 Materials Science VI . Diffusion in Solids copyright © 2008 by Li Sun 1 V. Diffusion in Solids MECE 3345 Materials Science Examples • Biology • Ink in water blood air Transpiration • Smell 2 V. Diffusion in Solids MECE 3345 Materials Science Definition Diffusion: The dispersal of material by random atom or molecular movement from regions of high concentration to those of lower concentration. The result of diffusion is to decrease the concentration non-uniformity. 3 V. Diffusion in Solids MECE 3345 Materials Science Ni-Cu Cu Ni Ni Cu Cu in Ni Ni in Cu NiCu alloy x=0 concentration 100 0 t=0 t = t1 t=∞ 4 V. Diffusion in Solids MECE 3345 Materials Science Diffusion Mechanism Vacancy diffusion Interstitial diffusion 5 V. Diffusion in Solids MECE 3345 Materials Science Activation Energy Energy G H TS G Vibration frequency: Nearest neighbor sites: z Atom position Interstitial diffusion G m H m TS m Jump Frequency : Vacancy diffusion Gm zX V v exp RT H m T S m H m D D 0 exp RT 6 V. Diffusion in Solids MECE 3345 Materials Science Vacancy Diffusion • Applies to substitutional impurities • Atoms must hop to open vacancy • Rate depends on probability of overcoming migration activation energy, Qm pQ = Bexp(-Hm/kT) And • Probability of a vacancy pv = Nv/N = exp(-Hv/kT) • Here is Diffusion Coefficient is: D = D0 exp(-Hd/kT) where Hd = Hm + Hv 7 V. Diffusion in Solids MECE 3345 Materials Science Interstitial Diffusion • • • • Applies to interstitial impurities (tend to be low concentration) Most interstitial sites are not occupied More rapid than vacancy diffusion Activation energy involves overcoming the migration activation energy, Hm. • Rate is defined by the Diffusion Coefficient D = D0exp(-Hd/kT) where D0 is a pre-exponential factor (m2/s) and Hd = Hm 8 V. Diffusion in Solids MECE 3345 Materials Science Data Interstitial diffusion in Fe Self-diffusion data for metals solute Hm(kJmol-1) D0(10-6m2s-1) C 84 2.0 N 76 0.3 H 13 0.1 metal Tm(K ) Q (kJmol-1) D0 (10-6m2s-1) D(Tm) (10-12m2s-1) Al 933 142 170 1.9 Cu 1356 200 31 0.59 Ni 1726 279 190 0.65 -Fe 1805 284 49 0.29 isotope x=0 9 V. Diffusion in Solids MECE 3345 Materials Science Other Diffusion Mechanisms Direct exchange Zener ring 10 V. Diffusion in Solids MECE 3345 Materials Science Fick’s First Law n 2 n1 a n2 n1 1 dn v : Jump Frequency dx : average tune if atome stay at the lattice site v Jump Frequency x a along x direction : J J 1 2 J 2 1 1 6 1 1 J a n1 n2 6 6 a 2 1 dn 6 dx 11 V. Diffusion in Solids MECE 3345 Materials Science Geometric Factor and Diffusion Coefficient J D dn J J 1 2 J 2 1 dx D Diffusion constant Einstein formula Simple cubic a a Average stay time at interstitial site Geometric factor D 1 1 6 FCC 2 2 1 6 Lattice parameter 2 1 12 2 : jump distance BCC a 3 a 2 1 8 12 V. Diffusion in Solids MECE 3345 Materials Science Steady State Diffusion concentration c Steady state: the state where the concentration profile does not change with time, which means at any time the concentration at certain spot in space is a constant. Fick’s first law: A B J D dn J dx Position Position n: number density x x 13 V. Diffusion in Solids MECE 3345 Materials Science Fick’s Second Law c concentration c concentration t1 A B x A t2 A B x x x J Position n2 n1 Position J 1 2 x J 2 1 14 V. Diffusion in Solids MECE 3345 Materials Science Diffusion into a Semi-infinite Solid t > 0, diffusion start n (x=0) = ns , n (x = ∞) = n0 A B n t ≤ 0, before diffusion start n(x) = n0, 0≤ x≤ ∞ A B ns n0 0 concentration ns ns x=0 n0 0 x=∞ x=0 Solution to the Fick’s second law n x (t ) n 0 n s n0 x=∞ t increases n0 t=0 0 Position x Fixed surface concentration. x 1 erf 2 Dt , erf: Gaussian error function 15 V. Diffusion in Solids MECE 3345 Materials Science Error function erf x 2 x 2 0 e d 16 V. Diffusion in Solids MECE 3345 Materials Science Examples Steady state diffusion : Gas purification Hydrogen gas purification can be achieved by pass mixed gas through palladium sheet. Pump Pump Assume the there is a hydrogen purification system with cross-sectional area of 0.2m2, the palladium film thickness is 10mm hydrogen concentration on the two sides are 3kg/m3 and 0.5kg/m3. If the room temperature hydrogen diffusion coefficient in Pd is 1.0×10-15m2/s and the activation energy is 27.8kJ/mol. Calculate the pure hydrogen generation speed (kg/hour) when the system works at 500oC. 17 MECE 3345 Materials Science I. What we know: C1= 3kg/m3, C2 = 0.5kg/m3 V. Diffusion in Solids Calculate how much hydrogen can diffuse through in 1hour at 500oC. Palladium film thickness d=10mm = 0.01m Diffusion area A=0.2m2 D @ 300K = 1.0×10-11m2/s Qd = 27.8kJ/mol diffusion direction C2 C1 C2 C1 0 xA xB 18 V. Diffusion in Solids MECE 3345 Materials Science Diffusion into a Semi-infinite Solid II. Calculate the carburizing time to achieve 0.45 wt% C at a position of 2mm into a 0.2 weight percent steel. Sample was maintained at 1000oC with 1.3 wt% carbon atmosphere. C0 0.2 wt % c concentration Cs 1.3 wt % Cs t = t1 t2 > t1 C0 0 t=0 Position x 1 .0 0 .8 e rf(z) 0 .6 0 .4 0 .2 0 .0 0 .0 0 .5 1 .0 1 .5 2 .0 19 V. Diffusion in Solids MECE 3345 Materials Science Diffusion Coefficient and Thermally Activated Process Thermally activated processes: The processes have exponentially increasing rates versus temperature. Vacancy formation, electrical conductivity, creep and diffusion are thermally activated processes. Q D D 0 exp d RT D0: temperature independent parameter (m2/s). Qd: activation energy for diffusion (J/mol). R: gas constant 8.31J/molK T: absolute temperature 20 V. Diffusion in Solids MECE 3345 Materials Science Arrhenius Plot Q rate C exp RT log( rate ) log( C ) Q 1 RT Arrhenius Plot: log (rate) vs. 1 T From the slope the thermal activation energy can be calculated 10-8 D (m2/s) 10-12 10-15 10-18 10-21 10-24 0 0.4 0.8 1.2 1000 T 1.6 2.0 2.4 (1/K) 21 V. Diffusion in Solids MECE 3345 Materials Science Diffusion and Temperature 10-8 300 600 1000 1500 D has exponential dependence on T D (m2/s) T(C) Dinterstitial C in -Fe C in -Fe 10-14 10-20 0.5 1.0 1.5 >> D substitutional Al in Al Fe in a-Fe Fe in g-Fe 1000K/T 22