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**Solid State Diffusion-1**

Engineering 45 Solid State Diffusion-1 Bruce Mayer, PE Registered Electrical & Mechanical Engineer

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**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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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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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

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**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)

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**Substitutional Diff Simulation**

Simulation of interdiffusion across an interface Rate of substitutional diffusion depends on: Vacancy concentration Jumping Frequency

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**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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**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 Crack Resistant Shear Resistant

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**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. More on this later

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**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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**Concentration Profiles & Flux**

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

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**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 Cu flux Ni flux x C Position, x Fick’s Work (Fick, A., Ann. Physik 1855, 94, 59) lead to this Eqn (1st Law) for J

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**Fick’s First Law cont. Consider the Components of Fick’s 1st Law**

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

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**Fick’s First Law cont.2 Units for D D → m2/S One More CRITICAL Issue**

Cu flux Ni flux Units for D x C 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:

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**Diffusion and Temperature**

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

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**Diffusion Types Compared**

Some D vs T Data The Interstitial Diffusers C in α-Fe C in γ-Fe Substitutional Diffusers > All Three Self-Diffusion Cases The Interstitial Form is More Rapid 1500 1000 T(C) 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 1000 K/T 0.5 1.0 1.5

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**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 Concentration, C, in the box does not change w/time. J x (right) (left) 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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**Steady State Diffusion cont**

J x (right) (left) Since Box Cannot Be infinitely filled it MUST be the case: 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

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**Steady State → CONST SLOPE**

Example SS Diffusion C 1 = 1.2kg/m 3 2 = 0.8kg/m Carbon rich gas 10mm deficient x1 x2 5mm D=3x10-11 m2/s Steady State → CONST SLOPE Iron Plate Processed at 700 °C under Conditions at Right 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

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**Steady State → CONST SLOPE**

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

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**WhiteBoard Work Problem Similar to 5.9 Hydrogen Diffusion Thru -Iron**

-Fe: = 7870 kg/m3

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