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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 1 Bruce Mayer, PE Engineering-45: Materials of Engineering Bruce Mayer, PE Registered Electrical & Mechanical Engineer Engineering 45 Solid State Diffusion-1

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 2 Bruce Mayer, PE Engineering-45: Materials of Engineering 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

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 3 Bruce Mayer, PE Engineering-45: Materials of Engineering InterDiffusion In a SOLID Alloy Atoms will Move From regions of HI Concentration to Regions of LOW Concentration Initial Condition After Time+Temp

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 4 Bruce Mayer, PE Engineering-45: Materials of Engineering 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 28 Si (92.5% Abundance) with one or both of 29 Si → 4.67% Abundance 30 Si → 3.10% Abundance

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 5 Bruce Mayer, PE Engineering-45: Materials of Engineering Diffusion Mechanisms Substitutional Diffusion Applies to substitutional impurities Atoms exchange position with lattice-vacancies Rate depends on: –Number/Concentration of vacancies (N v by Arrhenius) –Activation energy to exchange (the “Kick-Out” reaction)

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 6 Bruce Mayer, PE Engineering-45: Materials of Engineering Substitutional Diff Simulation Simulation of interdiffusion across an interface Rate of substitutional diffusion depends on: Vacancy concentration Jumping Frequency

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 7 Bruce Mayer, PE Engineering-45: Materials of Engineering Interstitial Diff Simulation Applies to interstitial impurities More rapid than vacancy diffusion. 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.

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 8 Bruce Mayer, PE Engineering-45: Materials of Engineering Diffusion in Processing Case1 Example: CASE Hardening Diffuse carbon atoms into the host iron atoms at the surface. Example of interstitial diffusion is a case Hardened gear. 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 Shear Resistant Crack Resistant

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 9 Bruce Mayer, PE Engineering-45: Materials of Engineering Doping Silicon with Phosphorus for n-type semiconductors: Process: 1. Deposit P rich layers on surface. 2. Heat it. 3. Result: Doped semiconductor regions. Diffusion in Processing Case2

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 10 Bruce Mayer, PE Engineering-45: Materials of Engineering Modeling Diffusion - Flux 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 Flux is a DIRECTIONAL Quantity

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 11 Bruce Mayer, PE Engineering-45: Materials of Engineering Concentration Profiles & Flux Consider the Situation Where The Concentration VARIES with Position i.e.; Concentration, say C(x), Exhibits a SLOPE or GRADIENT The concentration GRADIENT for COPPER Concentration of Cu (kg/m 3 ) Concentration of Ni (kg/m 3 ) Position, x Cu fluxNi flux xx CC

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 12 Bruce Mayer, PE Engineering-45: Materials of Engineering Fick’s First Law of Diffusion 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 (1 st Law) for J Position, x Cu fluxNi flux xx CC

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 13 Bruce Mayer, PE Engineering-45: Materials of Engineering Fick’s First Law cont. Consider the Components of Fick’s 1 st Law Position, x Cu fluxNi flux xx CC J Mass Flux in kg/m 2 s Atom Flux in at/m 2 s dC/dx = Concentration Gradient In units of kg/m 4 or at/m 4 D Proportionality Constant Units Analysis

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 14 Bruce Mayer, PE Engineering-45: Materials of Engineering Fick’s First Law cont.2 Units for D Position, x Cu fluxNi flux xx CC D → m 2 /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:

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 15 Bruce Mayer, PE Engineering-45: Materials of Engineering Diffusion and Temperature Diffusion coefficient, D, increases with increasing T → D(T) by: D DoDo exp – QdQd RT = pre-exponential constant factor [m 2 /s] = diffusion coefficient [m 2 /s] = activation energy [J/mol or eV/atom] = gas constant [8.314 J/mol-K] = absolute temperature [K] D DoDo QdQd R T

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 16 Bruce Mayer, PE Engineering-45: Materials of Engineering Diffusion Types Compared Some D vs T Data 1000 K/T D (m 2 /s) C in -Fe C in -Fe Al in Al Fe in -Fe Fe in -Fe T( C) The Interstitial Diffusers C in α -Fe C in γ-Fe Substitutional Diffusers > All Three Self- Diffusion Cases The Interstitial Form is More Rapid

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 17 Bruce Mayer, PE Engineering-45: Materials of Engineering 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, J x Concentration, C, in the box does not change w/time. J x(right) J x(left) x For Steady State the Above Situation may, in Theory, Persist for Infinite time To Prevent infinite Filling or Emptying of the Box

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 18 Bruce Mayer, PE Engineering-45: Materials of Engineering Steady State Diffusion cont Since Box Cannot Be infinitely filled it MUST be the case: J x(right) J x(left) x Now Apply Fick’s First Law Thus, since D=const Therefore While C(x) DOES change Left-to-Right, the GRADIENT, dC/dx Does NOT –i.e. C(x) has constant slope

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 19 Bruce Mayer, PE Engineering-45: Materials of Engineering Example SS Diffusion Iron Plate Processed at 700 °C under Conditions at Right Find the Carbon Diffusion Flux Thru the Plate For SS Diffusion dC/dx f(x) i.e., The Gradient is Constant For const dC/dx C 1 = 1.2kg/m 3 C 2 = 0.8kg/m 3 Carbon rich gas 10mm Carbon deficient gas x1x1 x2x2 0 5mm D=3x m 2 /s Steady State → CONST SLOPE

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 20 Bruce Mayer, PE Engineering-45: Materials of Engineering Expl SS Diff cont Thus the Gradient Use In Fick’s 1 st Law or C 1 = 1.2kg/m 3 C 2 = 0.8kg/m 3 Carbon rich gas 10mm Carbon deficient gas x1x1 x2x2 0 5mm D=3x m 2 /s Steady State → CONST SLOPE

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ENGR-45_Lec-06_Diffusion_Fick-1.ppt 21 Bruce Mayer, PE Engineering-45: Materials of Engineering WhiteBoard Work Problem Similar to 5.9 Hydrogen Diffusion Thru -Iron -Fe: = 7870 kg/m 3

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