ENGR-45_Lec-07_Diffusion_Fick-2.ppt 1 Bruce Mayer, PE Engineering-45: Materials of Engineering Bruce Mayer, PE Registered Electrical & Mechanical Engineer Engineering 45 Solid State Diffusion-2

ENGR-45_Lec-07_Diffusion_Fick-2.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

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 3 Bruce Mayer, PE Engineering-45: Materials of Engineering Recall Fick’s FIRST Law.  Fick’s 1st Law Position, x Cu fluxNi flux xx CC Where –J  Flux in kg/m 2 s or at/m 2 s –dC/dx = Concentration GRADIENT in units of kg/m 4 or at/m 4 –D  Proportionality Constant (Diffusion Coefficient) in m 2 /s  In the SteadyState Case J = const So dC/dx = const –For all x & t  Thus for ANY two points j & k Concen., C

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 4 Bruce Mayer, PE Engineering-45: Materials of Engineering NONSteady-State Diffusion  In The Steady Case  In The NONSteady, or Transient, Case the Physical Conditions Require In The Above Concen-vs-Position Plot Note how, at x  1.5 mm, Both C and dC/dx CHANGE with Time

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 5 Bruce Mayer, PE Engineering-45: Materials of Engineering NONSteady State Diffusion Math  Consider the Situation at Right  Box Dimensions Width =  x Height = 1 m Depth = 1 m –Into the slide  Box Volume, V =  x11 =  x  Now if  x is small Can Approximate C(x) as Concentration, C, in the Box J (right) J (left) xx  The Amount of Matl in the box, M

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 6 Bruce Mayer, PE Engineering-45: Materials of Engineering NONSteady State Diffusion cont  or  Material ENTERING the Box in time  t  For NONsteady Conditions Concentration, C, in the Box J (right) J (left) dx  So Matl ACCUMULATES in the Box  Material LEAVING the Box in time  t

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 7 Bruce Mayer, PE Engineering-45: Materials of Engineering NONSteady State Diffusion cont.2  So the NET Matl Accumulation  Adding (or Subtracting) Matl From the Box CHANGES C(x)  With V = 11  x Concentration, C, in the Box J (right) J (left) xx  Partials Req’d as C = C(x,t)

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 8 Bruce Mayer, PE Engineering-45: Materials of Engineering NONSteady State Diffusion cont.3  In Summary for CONSTANT D  Now, And this is CRITICAL, by TAYLOR’S SERIES Concentration, C, in the Box J (right) J (left) xx  so

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 9 Bruce Mayer, PE Engineering-45: Materials of Engineering NONSteady State Diffusion cont.4  After Canceling  Now for very short  t Concentration, C, in the Box J (right) J (left) xx  Finally Fick’s SECOND LAW for Constant Diffusion Coefficient Conditions

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 10 Bruce Mayer, PE Engineering-45: Materials of Engineering Comments of Fick’s 2nd Law  The Formal Statement  This Leads to the GENERAL, and much more Complicated, Version of the 2 nd Law Concentration, C, in the Box J (right) J (left) xx  This Assumes That D is Constant, i.e.;  In many Cases Changes in C also Change D

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 11 Bruce Mayer, PE Engineering-45: Materials of Engineering Example – NonSS Diffusion  Example: Cu Diffusing into a Long Al Bar  The Copper Concentration vs x & t pre-existing conc., C o of copper atoms Surface conc., C s of Cu atoms bar C o C s position, x C(x,t) t o t 1 t 2 t 3  The General Soln is Gauss’s Error Function, “erf”

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 12 Bruce Mayer, PE Engineering-45: Materials of Engineering Comments on the erf  Gauss's Defining Eqn  z is just a NUMBER Thus the erf is a (hard to evaluate) DEFINITE Integral  Treat the erf as any other special Fcn  Some Special Fcns with Which you are Familiar: sin, cos, ln, tanh These Fcns used to be listed in printed Tables, but are now built into Calculators and MATLAB  See Text Tab 5.1 for Table of erf(z)

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 13 Bruce Mayer, PE Engineering-45: Materials of Engineering Comments on the erf cont.  1-erf(z) appears So Often in Physics That it is Given its Own Name, The COMPLEMENTARY Error Function:  Recall The erfc Diffusion solution  Notice the Denom in this Eqn  This Qty has SI Units of meters, and is called the “Diffusion Length” The Natural Scaling Factor in the efrc

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 14 Bruce Mayer, PE Engineering-45: Materials of Engineering

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 15 Bruce Mayer, PE Engineering-45: Materials of Engineering Example  D = f(T)  Given Cu Diffusing into an Al Bar  At given point in the bar, x 0, The Copper Concentration reaches the Desired value after 10hrs at 600 °C The Processing Recipe  Get a New Firing Furnace that is Only rated to 1000 °F = 538 °C To Be Safe, Set the New Fnce to 500 °C  Need to Find the NEW Processing TIME for 500 °C to yield the desired C(x 0 )

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 16 Bruce Mayer, PE Engineering-45: Materials of Engineering Example  D = f(T) cont  Recall the erf Diffusion Eqn  For this Eqn to be True, need Equal Denoms in the erf  Since C S and C o have NOT changed, Need  Since by the erf

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 17 Bruce Mayer, PE Engineering-45: Materials of Engineering Example  D = f(T) cont.2  Now Need to Find D(T)  As With Xtal Pt- Defects, D Follows an Arrhenius Rln –Q d  Arrhenius Activation Energy in J/mol or eV/at –R  Gas Constant = 8.31 J/mol-K = 8.62x10 -5 eV/at-K –T  Temperature in K  Find D 0 and Q d from Tab 5.2 in Text For Cu in Al –D 0 = 6.5x10 -5 m 2 /s –Q d = 136 kJ/mol Where –D 0  Temperature INdependent Exponential PreFactor in m 2 /s

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 18 Bruce Mayer, PE Engineering-45: Materials of Engineering Example  D = f(T) cont.3  Thus D(T) for Cu in Al  Thus for the new 500 °C Recipe  In this Case D 600 = 4.692x m 2 /s D 500 = 4.152x m 2 /s  Now Recall the Problem Solution  This is 10x LONGER than Before; Should have bought a 600C fnce

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 19 Bruce Mayer, PE Engineering-45: Materials of Engineering Find D Arrhenius Parameters  Recall The D(T) Rln  Applied to the D(T) Relation  Take the Natural Log of this Eqn  This takes the form of the slope-intercept Line Eqn:

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 20 Bruce Mayer, PE Engineering-45: Materials of Engineering Find D(T) Parameters cont  And, Since TWO Points Define a Line If We Know D(T1) and D(T2) We can calc D 0 Q d  Quick Example D(T) For Cu in Au at Upper Right  Slope, m =  y/  x xx yy  x = ( )x1000/K = K -1  y = ln(3.55x ) − ln(4x ) = − 7.023

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 21 Bruce Mayer, PE Engineering-45: Materials of Engineering Find D(T) Parameters cont.2  By The Linear Form  in the (x,y) format x 1 = y 1 = ln(4x ) = −  So b  Now, the intercept, b  Pick (D,1/T) pt as (4x10 -13,0.8)  Finally D 0

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 22 Bruce Mayer, PE Engineering-45: Materials of Engineering Diff vs. Structure & Properties  Faster Diffusion for Open crystal structures Lower melting Temp materials Materials with secondary bonding Smaller diffusing atoms Cations Lower density materials  Slower Diffusion for Close-packed structures Higher melting Temp materials Materials with covalent bonding Larger diffusing atoms Anions Higher density materials

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 23 Bruce Mayer, PE Engineering-45: Materials of Engineering Diffusion Summarized  Phenomenon: Mass Transport In Solids  Mechanisms Vacancy InterChange by KickOut Interstitial “squeezing”  Governing Equations Fick's First Law Fick's Second Law  Diffusion coefficient, D Affect of Temperature Q d & D 0 –How to Determine them from D(T) Data

ENGR-45_Lec-07_Diffusion_Fick-2.ppt 24 Bruce Mayer, PE Engineering-45: Materials of Engineering WhiteBoard Work  Problem 5.28 Ni Transient Diffusion into Cu