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Diffusion • Diffusion is the relative movement of atoms in a solid, liquid or gas concerned here with diffusion in solids. • Is very important in 1. phase transformations 2. corrosion resistant coatings 3. separation of U235 by gaseous diffusion 4. permeability 5. impurity transistors 6. metal joining by diffusion bonding 7. radiation damage / defects - defect migration Fick's 1st Law : (no time dependence) dc J = - D dx we are { = - D ∇C in 3-dimensions} [J] = #/cm2s [C] = #/cc [D] = cm2/s • note - J along negative concentration gradient J1 - J 2 = ( ∂C ∂J ) ∆x = ∆x ∂t ∂x Fick's 2nd Law : C + f(x,t) ∂C = D ∇ 2C ∂t ∂C ∂J ∂ ∂C ∂ 2C or = − = − (−D ) = D 2 , ∂t ∂x ∂x ∂x ∂x if D≠f(x) which is true for most cases: ∂C = D ∇ 2C ∂t ⇒ diffusion eqn. Find the general and particular solutions {C(x,t)} to the diffusion equation using initial and boundary conditions. KL Murty page 1 J2 J1 1 ∆x 2 NE409/509 Examples 1. Thin-film solution {find C(x,t)} α -L x 0 -x L α is the amount in the thin layer at x=0 and L is very large ∂C ∂ 2C =D ∂t ∂x 2 x = 0 C → ∞ as t → 0 & x > 0 C → 0 as t → 0 3. Carburizing / Decarburizing Co x x2 note C = A exp(- 4Dt ) satisfies the Cs above equation Solution +∞ A is defined by ∫ C( x, t )dx = α −∞ ∴C = Cs - Cx x = erf( ) Cs - Co 2 Dt α x2 ) exp(− 4Dt 2 πDt 2. Pair of semi-infinite solids Cu -L Ni -x 0 x L x 2 2 ⌠ where erf(x) = ⌡exp(-ξ ) dξ πo C' α 0 -x C(x,t) = C 0 = initial concentration Cs = surface concentration for t = 0, C = Co at o ≤ x ≤ ∞ for t > o, C = Cs (constant) at x = 0 and C = Co at x = ∞ 0 erf (0) = 0 erf (∞) = 1 erf (x) = erf(-x) 0 x C' ∞ ( x − α) 2 )dα ∫ exp(− 4Dt 2 πDt 0 C' x C(x,t) = 2 [ 1 + erf( ] 2 Dt KL Murty page 2 NE409/509 4. Diffusion Experiment (self diffusion) α α is the radioisotope of the metal A A x 0 L x2 α ) C= exp(− 4Dt 2 πDt • T1 t o T1 ln C ln D(T) -Q/R t o T2 -1/4Dt o x 2 • T2 1/T QD D = Do exp(- RT) Mechanism (?) KL Murty page 3 NE409/509 Atomic Theory of Diffusion {Ref. Paul Shewmon, Diffusion in Solids , 2nd edition, TMS, 1989} D ⇒ atomic jump distance & frequency ⇔ random walk theory (text, Eq.7.25) Another way to do this is (see notes) : (no specific micromechanism !) Consider two adjacent crystal lattice planes 1 and 2 [Fig.] separated by α. a {if the planes are {110} type α = 2 since the direction x then is <110> type} n1, n2 : no. of atoms / unit area n1 | | | n2 | | | 1 n Γ δt is the number jumping from 2 1 | ______|____ X plane 1 to plane 2 1 ←α→ 2 Γ = number of jumps they make per sec. The atoms cannot stay in-between since there is no lattice plane between 1 and 2 The flux of atoms crossing a plane between 1 and 2 [J] : number of atoms 1 J = 2 (n1 - n2) Γ = (area) (time) . In terms of the concentrations [c, per unit volume] : c1α = n1 and c2α = n2 dc 1 1 J = 2 (c1 - c2) α Γ or J = 2 {-αdx } α Γ dc 1 1 Recalling Fick's first law {J = - D dx }, D = 2 α2 Γ or D = 6 α2 Γ This equation is similar to the one derived in the text using random-walk theory, 1 r2 = Γ t α2 (Eq.7.17) and also r2 = 6 D t (Eq.7.24), so that D = 6 α2 Γ • note : no specific mechanisms of atom jumping assumed • KL Murty page 4 NE409/509 Diffusion Mechanisms Vacancy Interstitial Interstitialcy Ring Self-diffusion via Vacancy Mechanism Γ is proportional to the number of nearest neighbors [β] and the probability of finding the lattice point vacant, i.e., the vacancy concentration [Cv] so that Γ = β Cv ω, where ω is the atomic jump frequency, ω = νD e- (Em / kT) 1 1 ∴ D = 6 α2 Γ = 6 α2 β Cv νD e- (Em/kT) Cv = e- (Ef / kT) β D = 6 α2 νD e- (Em/ kT) e- (Ef / kT) • For diffusion of vacancies, there is no need for Cv and only Em appears in the expression for Dv • z L Or DL = DSD = 6 α2 νD e- (ED/ kT) = Do e- (ED/ kT) , ED= Ef + Em KL Murty page 5 NE409/509 i i ⇒ similarly for interstitial diffusion : Di = Do e-(Em /kT) Melting Point, ÞC Short-Circuit Diffusion • Surface (Qs < QD) • Grain-Boundary (QGB < QD) surface lnD • Dislocations (Pipe) (QGB ≈ Q⊥) self (lattice) GB / Dislocation QGB ≈ 0.35 QD 1/T KL Murty page 6 NE409/509 Diffusion in Ionics 1 e.g. Schottky-type : ED = Em + 2 Es NaCl + Ca NaCl with Ca++ lnD at low temperatures, where extrinsic vacancies dominate, ED = Em ⇐ vacancy concentration is independent of temperature (athermal vacancies) ⇒ thus can determine both Es and Em QD Qm 1/T In metals, it is not easy to determine Ef and Em separately 1. Can get Ef from (a) Simmons-Balluffi experiment, or (b) Quenched in Resistivity 2. Annealing of quenched-in vacancies ⇒ Em Quench from high temperature , determine ∆ρο ; heat to T for t ; quench and determine ∆ρ ∆ρ (t) = ∆ρo e-t/τ where τ(Τ) is the time for a vacancy to anneal out to sink l2 τ ∝ D where l is the sink distance and Dv ∝ e-Em/kT v β {note : Dv = 6 α2 e-Em/kT no Cv appears since T is low enough thta the quenched-in vacancies from ‘high’ temperature is far larger} 40ÞC Example : Bauerle & Koehler in Au 60ÞC Em 1 1 τ1 Dv(T2) = D (T ) = exp{- k (T - T ) } τ2 v 1 2 1 ln(∆ρ/∆ρο) Tq=700ÞC Em = 0.82 eV (18.9 kCal/mole) t, hours KL Murty page 7 NE409/509 Group Work (solutions) For the example below, determine (a) β and α (in cm), and (b) lattice diffusivity, DL in cm2/s at 500 ˚C.* QD β DL = 2 D α2 νD exp(), RT β = no. of positions an atom can jump to, and D in the denominator is the dimension (1, 2 or 3) for Cu (fcc) along <110> : β = 12, α = Group 1. 12 a = 2.556x10-8 cm, 2 50000 DL = 6 (2.556x10-8) 2 1013 exp(- 1.987x773 ) = 9.547x10-17 cm2/sec for Cu (fcc) along <100> : β = 6, α = a = 3.615x10-8 cm Group 2. 6 50000 DL = 6 (3.615x10-8) 2 1013 exp(- 1.987x773 ) = 9.519x10-17 cm2/sec 3 for Cu (fcc) along <111> : β = 8, α = 2 a = 3.131x10-8 cm Group 3. 8 50000 DL = 6 (3.131x10-8) 2 1013 exp(- 1.987x773 ) = 9.521x10-17 cm2/sec Group 4. for Zr (hcp) along c-axis : β = 2, α = c = 1.593x3.231 Å = 5.147x10-8 cm 2 70000 DL = 2 (5.147x10-8) 2 1013 exp(- 1.987x773 ) = 4.270x10-22 cm2/sec note ‘2’ in the denominator - since the atom jumps are 1-D (!) * Cu : fcc, a = 3.615 Å, QD = 50 kCal/mole c νD = 1013 per sec a Zr : hcp, a = 3.231Å, c/a = 1.593, QD = 70 kCal/mole, QD = 60 kCal/mole KL Murty page 8 NE409/509