Kinetics Rate of change in response to thermodynamic forces Deviation from local equilibrium continuous change T heat flow temperature changes µ atom flow composition changes Deviation from global equilibrium discontinuous change ΔG (ΔF) phase change (or other change of structure)
Flow of Heat J Q T 2 T 1 Box of unit length, unit cross-section area Let: T 2 > T 1 (one-dimensional gradient) J Q = heat flow/unit area unit time (J/m 2 s) Ignore internal sources of heat From the Second Law: J Q = k dt dx Fourier s law of heat conduction k = thermal conductivity
Evolution of Temperature J Q 23 J Q 12 T 3 T 2 T 1 Let: T 3 > T 2 >T 1 (one-dimensional gradient) Q/ t = heat added/unit time (J/m 2 s) The net heat added to the center cell is: Q t = J Q Q [ 23 J 12 ]da = djq dv = dx d dx k dt dx dv J Q 12 = J Q 23 + djq dx dx Q t = E t = C V T t = k C V 2 T x 2 T t dv
Heat Conduction in 3-Dimensions A temperature gradient produces a heat flux: T = ( T/ x)e x + ( T/ y)e y + ( T/ z)e z J Q = J Q x e x + JQ y e y + Jq z e z How many thermal conductivities? In the most general case, 9 For a cubic or isotropic material, only need 1 For a cubic or isotropic material J Q = k T T t = k 2 T
Mechanisms of Heat Conduction J = (1/2)nev J Q+ J Q- v Energy must be transported through the solid Electrons Lattice vibrations - phonons Light - photons (usually negligible) J Q = J Q+ - J Q- T + > T - J Q > 0 Conduction is by a gas of moving particles Particles move both to left and right (J Q = J Q + -J Q- ) Particle energy increases with T If T decreases with x, particles moving right have more energy Net flow of heat to the right (J Q > 0)
Heat Conduction by Particles Transport of thermal energy Particle reaches thermal equilibrium by collisions Particle travels <l> = mean free path between collisions Particle transfers energy across plane Energy crossing plane reflects equilibrium <l x > upstream
Mechanisms of Heat Conduction J Q+ J Q- J = (1/2)nev J Q = J Q+ - J Q- Particles achieve thermal equilibrium by collision with one another Particles that cross at x were in equilibrium at x =x- l x l x = mean free path v J Q + = 1 2 nev x = 1 2 E v(t)v x = 1 2 E v[t(x l x )]v x E v J Q + = 1 2 E v v x 1 2 C v v x [ T( x l x )] = E v E v l x dt dx T dt dt l dx x = E v C v l dx x
Mechanisms of Heat Conduction J Q+ J Q- J = (1/2)nev J Q = J Q+ - J Q- Total heat flux: v J Q + = 1 2 E vv x 1 2 C dt vv x l x dx dt J Q = J Q + J Q = C v v x l x = k dt dx dx k = C v v x l x In three dimensions: k = 1 3 C vv l = 1 3 C vv 2 τ ( l = vτ)
Conduction of Heat J Q 23 J Q 12 T 3 T 2 T 1 J Q = k dt dx T t = k C V 2 T x 2 Thermal conductivity by particles: k = 1 3 C vv l = 1 3 C vv 2 τ - v = mean particle velocity - <l> = mean free path between collisions - τ = mean time between collisions Particles include - electrons - phonons
Heat Conduction by Electrons J = (1/2)nev k = 1 3 C vv l = 1 3 C vv 2 τ v Motion of electrons = electrical conductivity (ρ = ρ 0 + AT) k = LσT = LT ρ Wiedemann-Franz Law LT k = ρ 0 L A low temperature high temperature
Heat Conduction by Phonons k Crystal k = 1 3 C vv l = 1 3 C vv 2 τ Glass T Œ Scattering of phonons in imperfect crystals due to phonon-defect collisions <l> is the mean spacing between defects Phonon thermal conductivity is low Polygranular solids Defective solids Glasses High Θ D materials only have high k when they are nearly perfect Defective diamond films are no particular good
Kinetics Rate of change in response to thermodynamic forces Deviation from local equilibrium continuous change T heat flow temperature changes µ atom flow composition changes Deviation from global equilibrium discontinuous change ΔG (ΔF) phase change (or other change of structure)
Atom Diffusion J B 12 c c c 1 2 3 J B 23 g = g(t,p,c) g c T,P = µ (T,P,c) Mass flux in response to a gradient in chemical potential Assume constant T,P Flux of solute Evolution of composition J B = M d g = M dµ µ = M dc dx c dx c dx J B = nd dc dx c B t = J x = nd 2 c x 2 Fick s First Law Fick s Second Law
Diffusion J B 12 J B 23 J B = nd dc dx c c c 1 2 3 The diffusivity (D) is the material property that governs diffusion Diffusion in solids requires Atoms jump from position to position Atom jumps result in net flux Diffusion mechanism depends on atom site Interstitial Substitutional
Jumps of Interstitial Atoms G ΔG m Atom motion Must overcome barrier ΔG m to move from site to site (~ 1eV) Attempts with vibrational frequency ν e ~ 10 14 /sec Number of jumps per unit time ω = (# attempts/time)(probability of jump/attempt) = ν e exp ΔG m kt = ν exp Q (of the order of 10 5 /sec at room T) m kt x
Jumps of Substitutional Atoms For a substitutional atom to jump There must be a neighboring vacancy to permit the jump The atom must overcome its barrier and jump ω = [P(vacant site)][p(jump given vacant site)] = [c v ]ν exp Q m kt = exp Q v kt ν exp Q m kt = ν exp (Q v + Q m ) kt
Concentration-Driven Diffusion a J 12 J 21 c 1 c 2 J = J 12 J 21 = nd dc dx Diffusion by interchange of atoms on adjacent planes J 12 = N 1 ω x = nc 1 aω x J = J 12 J 21 = naω x (c 1 c 2 ) N 1 = atoms of type 1 per unit area n = atom sites per unit volume a = volume per unit area of plane ω x = jumps in the x-direction/time = 1 6 naω(c 1 c 2 ) = 1 6 na2 ω dc dx D = 1 6 a2 ω
Concentration-Driven Diffusion a J 12 J 21 c 1 c 2 J = J 12 J 21 = nd dc dx Diffusion by interchange of atoms on adjacent planes Random-walk diffusion Atoms do not jump preferentially in either direction Net flux because there are more atoms on plane 1 than on 2 Diffusivity governs random-walk diffusion D = 1 6 a2 ω = 1 6 a2 ν exp Q D kt D = D 0 exp Q D kt Q Q D = m Q V + Q m interstitial substitutional
Random-Walk Diffusion X Let an atom move by random steps of length a Its position at time t is X(t) X(t) = 0 X 2 (t) = X X = na 2 = ωa 2 t (X and -X are equally likely) x 2 = 1 3 X 2 = 1 3 ωa2 t = 2Dt x = x 2 1/ 2 = 2Dt
Application: Homogenization time c c x Solidification usually results in chemical heterogeneities Represent it with a sinusoid of wavelength, λ Composition should homogenize when applex > λ/2 The approximate time necessary is: t > t h = 1 2 λ = λ2 2D 2 8D t h = 1 λ 2 exp Q D 8D 0 kt Homogenization time - increases with λ 2 - decreases exponentially with T
Application: Service Life of a Microelectronic Device Si P Microelectronic devices have built-in heterogeneities Can function only as long as these doped regions survive To estimate the limit on service life. t s Let doped island have dimension, λ Device is dead when applex ~ λ/2, hence t s << t h = 1 2 λ = λ2 2D 2 8D t h = 1 λ 2 exp Q D 8D 0 kt Service life - decreases with miniaturization (λ 2 ) - decreases exponentially with T
Influence of Microstructure on Diffusivity Interstitial species Usually no effect from microstructure Stress may enhance diffusion Substitutional species Raising vacancy concentration increases D Quenching from high T Solutes Irradiation Defects provide short-circuit paths Grain boundary diffusion Dislocation core diffusion
Adding Vacancies Increases D ln(d) t D eq D = c v ν exp Q m kt Quench from high T Rapid cooling freezes in high c v D decreases as c v evolves to equilibrium Add solutes that promote vacancies High-valence solutes in ionic solids Mg ++ increases vacancy content in Na + Cl - Ionic conductivity increases with c Mg Large solutes in metals Interstitials in metals Processes that introduce vacancies directly Irradiation Plastic deformation
Grain Boundary Diffusion ln(d) D 0 D 0 B -Q decreasing D grain size 1/kT -Q B d D = D 0 exp Q D kt = D 0 exp (Q m + Q V ) kt δ D GB D 0 exp Q m d kt Grain boundaries have high defect densities Effectively, vacancies are already present Q D ~ Q m Grain boundaries have low cross-section Effective width = δ Areal fraction of cross-section: δ A GB A = Kδd d 2 δ d δ D GB 0 D 0 d