Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Engineering 45 Solid StateDiffusion-1 Bruce Mayer, PE Registered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. Learning Goals - Diffusion • How Diffusion Proceeds • How Diffusion Can be Used in Material Processing • How to Predict The RATE Of Diffusion Be Predicted For Some Simple Cases • Fick’s FIRST and second Laws • How Diffusion Depends On Structure And Temperature

3. InterDiffusion • In a SOLID Alloy Atoms will Move From regions of HI Concentration to Regions of LOW Concentration • Initial Condition • After Time+Temp

4. SelfDiffusion • In an Elemental Solid Atoms are NOT in Static Positions; i.e., They Move, or DIFFUSE • Label Atoms • After Time+Temp • How to Label an ATOM? • Use a STABLE ISOTOPE as a tag • e.g.; Label 28Si (92.5% Abundance) with one or both of • 29Si → 4.67% Abundance • 30Si → 3.10% Abundance

5. Diffusion Mechanisms • Substitutional Diffusion • Applies to substitutional impurities • Atoms exchange position with lattice-vacancies • Rate depends on: • Number/Concentration of vacancies (Nv by Arrhenius) • Activation energy to exchange (the “Kick-Out” reaction)

6. Simulation of interdiffusion across an interface Substitutional Diff Simulation • Rate of substitutional diffusion depends on: • Vacancy concentration • Jumping Frequency

7. Applies to interstitial impurities More rapid than vacancy diffusion. Interstitial Diff Simulation • Simulation shows • the jumping of a smaller atom (gray) from one interstitial site to another in a BCC structure. The interstitial sites considered here are at midpoints along the unit cell edges.

8. Example: CASE Hardening Diffuse carbon atoms into the host iron atoms at the surface. Example of interstitial diffusion is a case Hardened gear. ShearResistant CrackResistant Diffusion in Processing Case1 • Result: The "Case" is • hard to deform: C atoms "lock" xtal planes to reduce shearing • hard to crack: C atoms put the surface in compression

9. Diffusion in Processing Case2 • Doping Silicon with Phosphorus for n-type semiconductors: • Process: 1. Deposit P rich layers on surface. More on this lateR 2. Heat it. 3. Result: Doped semiconductor regions.

10. Flux is the Amount of Material Crossing a Planar Boundary, or area-A, in a Given Time x-direction Unit Area, A, Thru Which Atoms Move Modeling Diffusion - Flux • Flux is a DIRECTIONAL Quantity

11. Consider the Situation Where The Concentration VARIES with Position i.e.; Concentration, say C(x), Exhibits a SLOPE or GRADIENT Cu flux Ni flux Concentrationof Cu (kg/m3) Concentrationof Ni (kg/m3) Position, x x C Concentration Profiles & Flux • The concentration GRADIENT for COPPER

12. Note for the Cu Flux Proceeds in the POSITIVE-x Direction (+x) The Change in C is NEGATIVE (–C) Experimentally Adolph Eugen Fick Observed that FLUX is Proportional to the Concentration Grad Fick’s Work (Fick, A., Ann. Physik 1855, 94, 59) lead to this Eqn (1st Law) for J x C Fick’s First Law of Diffusion Cu flux Ni flux Position, x

13. Consider the Components of Fick’s 1st Law x C Fick’s First Law cont. Cu flux Ni flux Position, x • In units of kg/m4or at/m4 • D  Proportionality Constant • Units Analysis • J  • Mass Flux in kg/m2•s • Atom Flux in at/m2•s • dC/dx = Concentration Gradient

14. Units for D x C Fick’s First Law cont.2 Cu flux Ni flux Position, x • D → m2/S • One More CRITICAL Issue • Flux “Flows” DOWNHILL • i.e., Material Moves From HI-Concen to LO-Concen • The Greater the Negative dC/dx the Greater the Positive J • i.e.; Steeper Gradient increases Flux • The NEGATIVE Sign Indicates:

15. æ ö ç ÷ = D Do exp è ø Qd = diffusion coefficient [m2/s] – R T = pre-exponential constant factor [m2/s] = activation energy [J/mol or eV/atom] = gas constant [8.314 J/mol-K] = absolute temperature [K] Diffusion and Temperature • Diffusion coefficient, D, increases with increasing T → D(T) by: D Do Qd R T

16. Some D vs T Data Diffusion Types Compared • The Interstitial Diffusers • C in α-Fe • C in γ-Fe • Substitutional Diffusers > All Three Self-Diffusion Cases • The Interstitial Form is More Rapid T(C) 1500 1000 600 300 10-8 D (m2/s) C in g-Fe C in a-Fe 10-14 Fe in a-Fe Fe in g-Fe Al in Al 10-20 1000K/T 0.5 1.0 1.5

17. J J x (left) x (right) x Concentration, C, in the box does not change w/time. STEADY STATE Diffusion • Steady State → Diffusion Profile, C(x) Does NOT Change with TIME (it DOES change w/ x) • Example: Consider 1-Dimensional, X-Directed Diffusion, Jx • For Steady State the Above Situation may, in Theory, Persist for Infinite time • To Prevent infinite Filling or Emptying of the Box

18. Since Box Cannot Be infinitely filled it MUST be the case: J J x (left) x (right) x Steady State Diffusion cont • Now Apply Fick’s First Law • Therefore • While C(x) DOES change Left-to-Right, the GRADIENT, dC/dx Does NOT • i.e. C(x) has constant slope • Thus, since D=const

19. Iron Plate Processed at 700 °C under Conditions at Right 3 = 1.2kg/m 3 1 C = 0.8kg/m 2 C Carbon Steady State →CONST SLOPE rich gas Carbon deficient gas D=3x10-11 m2/s x1 x2 0 10mm 5mm Example SS Diffusion • Find the Carbon Diffusion Flux Thru the Plate • For SS Diffusion • For const dC/dx •  dC/dx  f(x) • i.e., The Gradient is Constant

20. Thus the Gradient 3 = 1.2kg/m 3 1 C = 0.8kg/m 2 C Carbon Steady State →CONST SLOPE rich gas Carbon deficient gas D=3x10-11 m2/s x1 x2 0 10mm 5mm Expl SS Diff cont • Use In Fick’s 1st Law • or

21. WhiteBoard Work • Problem Similar to 5.9 • Hydrogen Diffusion Thru -Iron -Fe:  = 7870 kg/m3