Diffusion Thermally activated process CC512 Chapter 5 Diffusion Thermally activated process Thermal production of point defects Point defects and solid-state diffusion Steady state diffusion Alternate diffusion paths 5 - 1

Thermally activated processes For a large number of processes in MS&E, Rate rises exponentially with temperature Examples: diffusivity, creep rate, electrical conductivity of semiconductors Mechanical analog: the box must overcome an increase in potential energy, ΔE, in order to move from one stable position to another 5 - 2

Activation energy _ Indication of the mechanism of the process Example: Calculate the activation energy of the oxidation process of a Mg alloy Data: the oxidation rate (r1) at 300º C = 1.05 x 10-8 kg/(m4.s); The rate r2 at 400º C = 2.95 x 10-4 kg/(m4.s) Q = [8.314(J/mol)x ln[(2.95x10-4)/(1.05x10-8)] / [(673 – 573)/(673x573)] = 328.4 kJ/mol 5 - 3

Thermal production of point defects At a given temperature, -the thermal energy of given material is constant -the thermal energy of individual atom varies over a wide range of energy_ Maxwell-Boltzmann dist. -A certain fraction of atoms have sufficient thermal energy to produce the point defects -This fraction increases exponentially with temperature Thermal production of vacancies in Al -the overall thermal expansion (ΔL/L) greater than the lattice parameter expansion (Δa/a) -the difference is due to the production of vacancies Example_ Vacancy fraction at 400C = 2.29 x 10-5 (EV = 0.76 eV) Calculate the fraction at 600C: C = (nV/nsites)exp(EV/kT) = 11.2 (k = 8.62x10-5 eV/K) nV/nsites = 8.82 x 10-4 5 - 4

Point Defects and Solid State Diffusion -Migration of molecules (or atoms) from high to low concentration region at sufficiently high temperatures -Example_ a drop of ink in water at RT; ink molecules move from high to low conc. region until reaching a uniform concentration -Easy to observe in gaseous (i.e., perfume) or liquid states due to high molecular mobility -Difficult to observe in solid state because high energy is required for an atom to move in a tightly squeezed state. -Atomic migration can only occur through vacancy mechanism at sufficiently high temperatures -Net migration of atoms (from high conc. to low conc. region) results from a series of random walk motions (jumps) of atoms. 5 - 5

Interdiffusion_ A - B atoms system Random walk motion of atoms (at sufficiently high temperature) -Probability of randomly “walking” (atomic jumps) in any direction is equal -High conc. of A- atoms on the left-side of couple (i.e., concentration gradient of A-atoms) cause such random motion (jumps) to produce a net flow of A- atoms into solid B. -Likewise, B- atoms move into solid A, i.e., an interdiffusion occurs. Temperature, say, 800 ºC 5 - 6

Fick’s Laws _ formal mathematical treatment of diffusion 5 - 7

Fick’s 2nd Law Application_ Carburization Gradient of flux in a unit volume = concentration variation with time Application_ Carburization : Diffusion of carbon atom in steel from carbon rich environment 5 - 8

Diffusion coefficient_ Diffusivity Diffusivity obeys an Arrhenius equation Note that carbon atoms diffuse in steel by Interstitialcy mechanism. 5 - 9

Diffusivity_ non-metallic systems -Smaller ionic species (e.g., Al+3) diffuse much more readily through the material -The Arrhenius behavior of ionic diffusion is similar to that of conductivity of certain ceramic semiconductor (e.g., ZnO) -This ionic (not electron) transport mechanism is responsible for semiconducting behavior of such ceramics. 5 - 10

Example Example 5.4 The carbon environment (a hydro-carbon gas) is used to set the surface carbon content (cs) at cs = 1.0 wt.%. The initial carbon content of steel (c0) is c0 = 0.2 wt.%. Using the error function table, calculate how long it would take at 1000C to reach a carbon content of 0.6 wt.% at a distance of 1mm from the surface. 5 - 11

Steady-State Diffusion -Diffusion through thin membrane; -Surface concentration are fixed at both surfaces : Ch = const.; Cl = const. -For long times, A linear concentration profile is established Concentration remains constant regardless of diffusion time , Steady state diffusion, i.e., mass-transport unchanging with time Example 5.7 A 5mm thick sheet of Pd membrane with cross-section area A = 0.2m2 is used as a steady state diffusion membrane for purifying hydrogen. If the hydrogen conc. on the high pressure side (impure gas) is 0.3 kg/m3 And the diffusion coefficient for hydrogen in Pd is 1.0x10-8m2/s, Calculate the mass of hydrogen being purified per hour. Solution: J = -D[-(ch – cl)/x0] kg/m2.s Total mass m = J x A x 3600s = 0.432 x 10-3 kg/h 5 - 12

Alternate Diffusion Paths Possible diffusion paths: -Volume diffusion -Surface diffusion -Grain boundary diffusion -Qv> Qgb > Qs Self-diffusion coefficient for Ag 5 - 13