V. Diffusion in Solids MECE 3345 Materials Science 1 VI. Diffusion in Solids copyright © 2008 by Li Sun

V. Diffusion in Solids MECE 3345 Materials Science 2 Examples Ink in water Smell Biology blood air Transpiration

V. Diffusion in Solids MECE 3345 Materials Science 3 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. Definition

V. Diffusion in Solids MECE 3345 Materials Science 4 concentration Ni Cu x = 0 t = t = t 1 t = ∞ NiCu alloy Ni Cu Cu in Ni Ni in Cu Ni-Cu

V. Diffusion in Solids MECE 3345 Materials Science 5 Vacancy diffusion Interstitial diffusion Diffusion Mechanism

V. Diffusion in Solids MECE 3345 Materials Science 6 Activation Energy GG Atom position Energy Vibration frequency: Nearest neighbor sites: z Interstitial diffusionVacancy diffusion

V. Diffusion in Solids MECE 3345 Materials Science 7 Vacancy Diffusion Applies to substitutional impurities Atoms must hop to open vacancy Rate depends on probability of overcoming migration activation energy, Q m p Q = Bexp(-H m /kT) And Probability of a vacancy p v = N v /N = exp(-H v /kT) Here is Diffusion Coefficient is: D = D 0 exp(-H d /kT) where H d = H m + H v

V. Diffusion in Solids MECE 3345 Materials Science 8 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, H m. Rate is defined by the Diffusion Coefficient D = D 0 exp(-H d /kT) where D 0 is a pre-exponential factor (m 2 /s) and H d = H m Interstitial Diffusion

V. Diffusion in Solids MECE 3345 Materials Science 9 Data solute  H m (kJmol -1 ) D 0 (10 -6 m 2 s -1 ) C842.0 N760.3 H130.1 Interstitial diffusion in Fe metalT m (K ) Q (kJmol -1 ) D 0 (10 -6 m 2 s -1 ) D(T m ) ( m 2 s -1 ) Al Cu Ni  -Fe Self-diffusion data for metals x = 0 isotope

V. Diffusion in Solids MECE 3345 Materials Science 10 Other Diffusion Mechanisms Direct exchange Zener ring

V. Diffusion in Solids MECE 3345 Materials Science 11 Fick’s First Law n1n1 n2n2 a x

V. Diffusion in Solids MECE 3345 Materials Science 12 Geometric Factor and Diffusion Coefficient Diffusion constantGeometric factor Average stay time at interstitial site Lattice parameter Simple cubic Einstein formula FCC BCC

V. Diffusion in Solids MECE 3345 Materials Science 13 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: Steady State Diffusion Position concentration x c A B Position x J n: number density

V. Diffusion in Solids MECE 3345 Materials Science 14 Position x J concentration x c t1t1 A B Fick’s Second Law concentration x c t2t2 A B n1n1 n2n2 xx x A

V. Diffusion in Solids MECE 3345 Materials Science 15 t ≤ 0, before diffusion start n(x) = n 0, 0≤ x≤ ∞ A B x = 0 n0n0 x = ∞ 0 nsns t > 0, diffusion start n (x=0) = n s, n (x = ∞) = n 0 x = 0 n0n0 x = ∞ A B 0 nsns Solution to the Fick’s second law, erf: Gaussian error function Position concentration x n 0 t=0 t increases n0n0 nsns Fixed surface concentration. Diffusion into a Semi-infinite Solid

V. Diffusion in Solids MECE 3345 Materials Science 16 Error function

V. Diffusion in Solids MECE 3345 Materials Science 17 Examples Steady state diffusion : Gas purification Pump Hydrogen gas purification can be achieved by pass mixed gas through palladium sheet. Assume the there is a hydrogen purification system with cross-sectional area of 0.2m 2, the palladium film thickness is 10mm hydrogen concentration on the two sides are 3kg/m 3 and 0.5kg/m 3. If the room temperature hydrogen diffusion coefficient in Pd is 1.0× m 2 /s and the activation energy is 27.8kJ/mol. Calculate the pure hydrogen generation speed (kg/hour) when the system works at 500 o C.

V. Diffusion in Solids MECE 3345 Materials Science 18 xAxA 0 C1C1 C2C2 diffusion direction C1C1 C2C2 xBxB What we know: C 1 = 3kg/m 3, C 2 = 0.5kg/m 3 Palladium film thickness d=10mm = 0.01m Diffusion area A=0.2m 2 300K = 1.0× m 2 /s Q d = 27.8kJ/mol Calculate how much hydrogen can diffuse through in 1hour at 500 o C. I.

V. Diffusion in Solids MECE 3345 Materials Science 19 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 1000 o C with 1.3 wt% carbon atmosphere. CsCs C0C0 c 0.2 wt % 1.3 wt % Position concentration x 0 t=0 C0C0 CsCs t 2 > t 1 t = t 1 Diffusion into a Semi-infinite Solid

V. Diffusion in Solids MECE 3345 Materials Science 20 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. D 0 : temperature independent parameter (m 2 /s). Q d : activation energy for diffusion (J/mol). R: gas constant 8.31J/mol  K T: absolute temperature

V. Diffusion in Solids MECE 3345 Materials Science 21 Arrhenius Plot (1/K) D (m 2 /s) Arrhenius Plot: log (rate) vs. From the slope the thermal activation energy can be calculated

V. Diffusion in Solids MECE 3345 Materials Science 22 Diffusion and Temperature D has exponential dependence on T D interstitial >> D substitutional C in  -Fe C in  -Fe Al in Al Fe in a-Fe Fe in g-Fe 1000 K/T D (m 2 /s) C in  -Fe C in  -Fe Al in Al Fe in  -Fe Fe in  -Fe T(  C)