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Engineering 45. Solid State Diffusion-1. Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. Learning Goals - Diffusion. How Diffusion Proceeds How Diffusion Can be Used in Material Processing
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Engineering 45 Solid StateDiffusion-1 Bruce Mayer, PE Registered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu
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
InterDiffusion • In a SOLID Alloy Atoms will Move From regions of HI Concentration to Regions of LOW Concentration • Initial Condition • After Time+Temp
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
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)
Simulation of interdiffusion across an interface Substitutional Diff Simulation • Rate of substitutional diffusion depends on: • Vacancy concentration • Jumping Frequency
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.
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
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.
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
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
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
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
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:
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
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
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
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
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
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
WhiteBoard Work • Problem Similar to 5.9 • Hydrogen Diffusion Thru -Iron -Fe: = 7870 kg/m3