CHAPTER 6: DIFFUSION IN SOLIDS Diffusion- Steady and Non-Steady State Gear from case-hardened steel (C diffusion) Diffusion - Mass transport by atomic motion ISSUES TO ADDRESS... • How does diffusion occur? Mechanisms • Gases & Liquids – random (Brownian) motion • Why is it an important part of processing? • Solids – vacancy diffusion or interstitial diffusion • How can the rate of diffusion be predicted for some simple cases? • How does diffusion depend on structure and temperature? MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 Simple Diffusion Inter-diffusion • Interdiffusion: In alloys, atoms tend to migrate • Glass tube filled with water. from regions of large concentration. • At time t = 0, add some drops of ink to one end This is a diffusion couple. of the tube. Initially After some time • Measure the diffusion distance, x, over some time. • Compare the results with theory. Adapted from Figs. 6.1 - 2, Callister 6e. 100% 0 Concentration Profiles 2 3 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 1 Self-diffusion Substitution-diffusion:vacancies and interstitials • applies to substitutional impurities • Self-diffusion: In an elemental solid, atoms also migrate. • atoms exchange with vacancies Number (or concentration*) • rate depends on (1) number of vacancies; of Vacancies at T (2) activation energy to exchange. − ΔE n k T Label some atoms After some time i B ci = =e N C • kBT gives eV A * see web handout for derivation. D B ΔE is an activation energy for a particular process (in J/mol, cal/mol, eV/atom). 4 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 Substitution-diffusion Inter-diffusion across Interfaces Vacancy Diffusion: • applies to substitutional impurities • atoms exchange with vacancies • rate depends on (1) number of vacancies; (2) activation energy to exchange. • Rate of substitutional diffusion depends on: - vacancy concentration - frequency of jumping. (Courtesy P.M. Anderson) increasing elapsed time Why should interstitial diffusion be faster than by vacancy mode of diffusion? MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 2 Diffusion Mechanisms Processing using Diffusion • Interstitial diffusion – smaller atoms diffuse between atoms. • Case Hardening: Fig. 6.0, Callister 6e. -Diffuse carbon atoms into the (courtesy of Surface host iron atoms at the surface. Div., Midland-Ross.) -Example of interstitial diffusion is a case hardened gear. • Result: The "Case" is -hard to deform: C atoms "lock" planes from shearing. -hard to crack: C atoms put the surface in compression. More rapid than vacancy diffusion MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 Processing using Diffusion Modeling rate of diffusion: flux • Doping Silicon with P for n-type semiconductors: • Flux: • Process: 1. Deposit P rich layers on surface. • Directional Quantity silicon • Flux can be measured for: Fig. 18.0, - vacancies 2. Heat it. Callister 6e. - host (A) atoms 3. Result: Doped - impurity (B) atoms semiconductor A = Area of flow regions. • Empirically determined: – Make thin membrane of known surface area diffused – Impose concentration gradient mass J ∝ slope silicon – Measure how fast atoms or molecules diffuse M through the membrane time MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 3 Steady-state Diffusion: J ~ gradient of c Steady-State Diffusion • Concentration Profile, C(x): [kg/m3] • Steady State: concentration profile not changing with time. Cu flux Ni flux Adapted from Fig. 6.2(c) Concentration Concentration of Cu [kg/m3] of Ni [kg/m3] Position, x • Fick's First Law: D is a constant! dC • Apply Fick's First Law: Jx = −D dx dC dC • If Jx)left = Jx)right , then = dx dx left right • The steeper the concentration profile, the greater the flux! • Result: the slope, dC/dx, must be constant (i.e., slope doesn't vary with position)! MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 Steady-State Diffusion Example: Chemical Protection Clothing dC Rate of diffusion independent of time J ~ • Methylene chloride is a common ingredient of paint removers. dx Besides being an irritant, it also may be absorbed through skin. When using, protective gloves should be worn. • If butyl rubber gloves (0.04 cm thick) are used, what is the C1 C1 Fick’s first law of diffusion diffusive flux of methylene chloride through the glove? • Data: dC -8 2 J = −D – D in butyl rubber: D = 110 x10 cm /s C C2 C = 0.44 g/cm3 C = 0.02 g/cm3 2 dx – surface concentrations: 1 2 – Diffusion distance: x2 – x1 = 0.04 cm x1 x2 D ≡ diffusion coefficient glove x C1 2 t = l b 6D dC ΔC C −C paint skin C - C g 2 1 J = -D 2 1 = 1.16 x 10-5 if linear ≅ = remover x 2 dx x x x x2 - x1 cm s Δ 2 − 1 C2 x1 x2 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 4 Example: C Diffusion in steel plate Example: Diffusion of radioactive atoms • Steel plate at 7000C with geometry shown: • Surface of Ni plate at 10000C contains 50% Ni63 (radioactive) and 50% Ni (non-radioactive). Adapted from Fig. • 4 microns below surface Ni63 /Ni = 48:52 700 C 5.4, Callister 6e. • Lattice constant of Ni at 1000 C is 0.360 nm. Knowns: • Experiment shows that self-diffusion of Ni is 1.6 x 10-9 cm2/sec 3 C1= 1.2 kg/m at 5mm (5 x 10–3 m) below surface. What is the flux of Ni63 atoms through a plane 2 µm below surface? 3 63 63 C2 = 0.8 kg/m at 10mm (4Ni /cell)(0.5Ni /Ni) (4Ni /cell)(0.48Ni /Ni) –2 C = C2 = (1 x 10 m) below surface. 1 9 3 (0.36x10−9m)3 /cell (0.36x10− m) /cell -11 2 27 63 3 41.15x1027Ni63 /m3 D = 3 x10 m /s at 700 C. = 42.87x10 Ni /m = 27 63 3 13 2 (41.15 − 42.87)x10 Ni /m = −(1.6x10− m /sec) • Q: In steady-state, how much carbon transfers € 6 € (4 − 0)x10− m from the rich to the deficient side? 20 63 2 = +0.69x10 Ni /m •s 2 63 How many Ni63 atoms/second through cell? J •(0.36nm) = 9 Ni /s € MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 € Where can we use Fick’s Law? Non-Steady-State Diffusion Fick's law is commonly used to model transport processes in • Concentration profile, • foods, C(x), changes w/ time. • clothing, • biopolymers, • pharmaceuticals, • porous soils, • semiconductor doping process, etc. • To conserve matter: • Fick's First Law: Example The total membrane surface area in the lungs (alveoli) may be on the order of 100 square meters and have a thickness of less than a millionth of a meter, so it is a very effective gas-exchange interface. 2 2 CO2 in air has D~16 mm /s, and, in water, D~ 0.0016 mm /s • Governing Eqn.: 14 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 5 Non-Steady-State Diffusion: another look Non-Steady-State Diffusion: C = c(x,t) • Concentration profile, concentration of diffusing species is a function of both time and position 2 ∂c ∂c C(x), changes w/ time. Fick's Second "Law" D ≈ 2 ∂t ∂x • Rate of accumulation C(x) • Copper diffuses into a bar of aluminum. ∂C ∂C ∂Jx ∂Jx dx = Jx − Jx dx → dx = Jx − (Jx + dx) = − dx ∂t + ∂t ∂x ∂x Cs C J C B.C. at t = 0, C = Co for 0 ≤ x ≤ ∞ • Using Fick’s Law: ∂ ∂ x ∂ ∂ Fick’s = − = − −D at t > 0, C = C for x = 0 (fixed surface conc.) ∂t ∂x ∂x ∂x 2nd Law S C = Co for x = ∞ 2 ∂c ∂ ∂c ∂c Fick's Second "Law" = D ≈ D Adapted from Fig. 6.5, • If D is constant: 2 Callister & Rethwisch 3e. ∂t ∂x ∂x ∂x 14 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 Non-Steady-State Diffusion Example: Non-Steady-State Diffusion FCC iron-carbon alloy initially containing 0.20 wt% C is carburized at an elevated • Cu diffuses into a bar of Al. CS temperature and in an atmosphere that gives a surface C content at 1.0 wt%. If after 49.5 h the concentration of carbon is 0.35 wt% at a position 4.0 mm below the surface, what temperature was treatment done? C(x,t) Solution 2 ∂c ∂c C(x,t) −Co 0.35 − 0.20 x Fick's Second "Law": ≈ D C = = 1− erf = 1− erf(z) ∴ erf(z) = 0.8125 2 o C −C 1.0 − 0.20 2 Dt ∂t ∂x s o Using Table 6.1 find z where erf(z) = 0.8125. Use interpolation. z erf(z) • Solution: 0.90 0.7970 z − 0.90 0.8125 − 0.7970 = So, z = 0.93 z 0.8125 0.95 − 0.90 0.8209 − 0.7970 0.95 0.8209 "error function” Values calibrated in Table 6.1 x x2 Now solve for D z = D = z 2 Dt 2 2 −y 2 4z t erf (z) = e dy x2 (4 x 10−3 m)2 1 h ∫ 0 ∴D = = = 2.6 x 10−11 m2 /s π 2 2 4z t (4)(0.93) (49.5 h) 3600 s 15 MatSE 280: Introduction to Engineering Materials ©D.D.