![• Concentration Profile, C(x): [kg/m3
]
11
• Fick's First Law:
Concentration
of Cu [kg/m3]
Concentration
of Ni [kg/m3]
Position, x
Cu flux Ni flux
• The steeper the concentration profile,
the greater the flux
Jx
D
dC
dx
Diffusion coefficient [m2/s]
concentration
gradient [kg/m4]
flux in x-dir.
[kg/m2-s]
CONCENTRATION PROFILES & FLUX](https://image.slidesharecdn.com/diffusion-3-250218060402-770bc143/85/DIFFUSION-in-materials-and-its-basic-understandings-8-320.jpg)
![• Experimental result: x ~ t0.58
• Theory predicts x ~ t0.50
• Reasonable agreement!
18
B
B
B
B
B
B
B
B
B
B
B
B
B
B
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5 2 2.5 3
ln[x(mm)]
ln[t(min)]
Linear regression fit to data:
ln[x(mm)] 0.58ln[t(min)] 2.2
R2 0.999
DATA FROM DIFFUSION DEMO](https://image.slidesharecdn.com/diffusion-3-250218060402-770bc143/85/DIFFUSION-in-materials-and-its-basic-understandings-15-320.jpg)
![• Diffusivity increases with T.
• Experimental Data:
1000K/T
D (m2/s) C in -Fe
C
in
-Fe
A
l
i
n
A
l
C
u
i
n
C
u
Z
n
i
n
C
u
F
e
i
n
-
F
e
F
e
i
n
-
F
e
0.5 1.0 1.5 2.0
10-20
10-14
10-8
T(C)
1500
1000
600
300
D has exp. dependence on T
Recall: Vacancy does also!
19
pre-exponential [m2/s] (see Table 5.2,Callister 6e)
activation energy
gas constant [8.31J/mol-K]
DDoexp
Q
d
RT
diffusivity
[J/mol],[eV/mol]
(see Table 5.2,Callister 6e)
Dinterstitial >> Dsubstitutional
C in -Fe
C in -Fe Al in Al
Cu in Cu
Zn in Cu
Fe in -Fe
Fe in -Fe
Adapted from Fig. 5.7, Callister 6e. (Date for Fig. 5.7 taken from
E.A. Brandes and G.B. Brook (Ed.) Smithells Metals Reference
Book, 7th ed., Butterworth-Heinemann, Oxford, 1992.)
DIFFUSION AND TEMPERATURE](https://image.slidesharecdn.com/diffusion-3-250218060402-770bc143/85/DIFFUSION-in-materials-and-its-basic-understandings-16-320.jpg)
DIFFUSION in materials and its basic understandings
- 1. ISSUES TO ADDRESS... • How does diffusion occur? • Why is it an important part of processing? • How can the rate of diffusion be predicted for some simple cases? 1 • How does diffusion depend on structure and temperature? DIFFUSION IN SOLIDS
- 2. 2 • Glass tube filled with water. • At time t = 0, add some drops of ink to one end of the tube. • Measure the diffusion distance, x, over some time. • Compare the results with theory. to t1 t2 t3 xo x1 x2 x3 time (s) x (mm) DIFFUSION DEMO
- 3. 100% Concentration Profiles 0 Cu Ni 3 • Interdiffusion: In an alloy, atoms tend to migrate from regions of large concentration. Initially After some time 100% Concentration Profiles 0 Adapted from Figs. 5.1 and 5.2, Callister 6e. DIFFUSION: THE PHENOMENA (1)
- 4. 4 • Self-diffusion: In an elemental solid, atoms also migrate. Label some atoms After some time A B C D A B C D DIFFUSION: THE PHENOMENA (2)
- 5. 5 Substitutional Diffusion: • applies to substitutional impurities • atoms exchange with vacancies • rate depends on: --number of vacancies --activation energy to exchange. increasing elapsed time DIFFUSION MECHANISMS
- 6. 7 • 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. INTERSTITIAL SIMULATION
- 7. • Flux: 10 J 1 A dM dt kg m2 s or atoms m2 s • Directional Quantity • Flux can be measured for: --vacancies --host (A) atoms --impurity (B) atoms Jx Jy Jz x y z x-direction Unit area A through which atoms move. MODELING DIFFUSION: FLUX
- 8. • Concentration Profile, C(x): [kg/m3 ] 11 • Fick's First Law: Concentration of Cu [kg/m3] Concentration of Ni [kg/m3] Position, x Cu flux Ni flux • The steeper the concentration profile, the greater the flux Jx D dC dx Diffusion coefficient [m2/s] concentration gradient [kg/m4] flux in x-dir. [kg/m2-s] CONCENTRATION PROFILES & FLUX
- 9. • Steady State: the concentration profile doesn't change with time. 12 • Apply Fick's First Law: • Result: the slope, dC/dx, must be constant (i.e., slope doesn't vary with position)! Jx(left) = Jx(right) Steady State: Concentration, C, in the box doesn’t change w/time. Jx(right) Jx(left) x J x D dC dx dC dx left dC dx right • If Jx)left = Jx)right , then STEADY STATE DIFFUSION
- 10. • Steel plate at 700C with geometry shown: 13 • Q: How much carbon transfers from the rich to the deficient side? J D C2 C1 x2 x1 2.4 10 9 kg m2 s C1 = 1.2kg/m 3 C2 = 0.8kg/m 3 Carbon rich gas 1 0 m m Carbon deficient gas x1 x2 0 5 m m D=3x10-11m2/s Steady State = straight line! EX: STEADY STATE DIFFUSION
- 11. • Concentration profile, C(x), changes with time. 14 • To conserve matter: • Fick's First Law: • Governing Eqn.: Concentration, C, in the box J(right) J(left) dx dC dt = D d2C dx2 dx dC dt J D dC dx or J(left) J(right) dJ dx dC dt dJ dx D d2C dx2 (if D does not vary with x) equate NON STEADY STATE DIFFUSION
- 12. • Copper diffuses into a bar of aluminum. 15 • General solution: "error function" Values calibrated in Table 5.1, Callister 6e. C(x,t) Co Cs Co 1 erf x 2 Dt pre-existing conc., Co of copper atoms Surface conc., Cs of Cu atoms bar Co Cs position, x C(x,t) to t1 t2 t3 Adapted from Fig. 5.5, Callister 6e. EX: NON STEADY STATE DIFFUSION
- 13. • Copper diffuses into a bar of aluminum. • 10 hours at 600C gives desired C(x). • How many hours would it take to get the same C(x) if we processed at 500C? 16 (Dt)500ºC =(Dt)600ºC s C(x,t) Co C Co = 1 erf x 2Dt • Result: Dt should be held constant. • Answer: Note: values of D are provided here. Key point 1: C(x,t500C) = C(x,t600C). Key point 2: Both cases have the same Co and Cs. t500 (Dt)600 D500 110hr 4.8x10-14m2/s 5.3x10-13m2/s 10hrs PROCESSING QUESTION
- 14. 17 • The experiment: we recorded combinations of t and x that kept C constant. to t1 t2 t3 x o x 1 x 2 x3 • Diffusion depth given by: xi Dti C(xi,ti) Co Cs Co 1 erf xi 2 Dti = (constant here) DIFFUSION DEMO: ANALYSIS
- 15. • Experimental result: x ~ t0.58 • Theory predicts x ~ t0.50 • Reasonable agreement! 18 B B B B B B B B B B B B B B 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 ln[x(mm)] ln[t(min)] Linear regression fit to data: ln[x(mm)] 0.58ln[t(min)] 2.2 R2 0.999 DATA FROM DIFFUSION DEMO
- 16. • Diffusivity increases with T. • Experimental Data: 1000K/T D (m2/s) C in -Fe C in -Fe A l i n A l C u i n C u Z n i n C u F e i n - F e F e i n - F e 0.5 1.0 1.5 2.0 10-20 10-14 10-8 T(C) 1500 1000 600 300 D has exp. dependence on T Recall: Vacancy does also! 19 pre-exponential [m2/s] (see Table 5.2,Callister 6e) activation energy gas constant [8.31J/mol-K] DDoexp Q d RT diffusivity [J/mol],[eV/mol] (see Table 5.2,Callister 6e) Dinterstitial >> Dsubstitutional C in -Fe C in -Fe Al in Al Cu in Cu Zn in Cu Fe in -Fe Fe in -Fe Adapted from Fig. 5.7, Callister 6e. (Date for Fig. 5.7 taken from E.A. Brandes and G.B. Brook (Ed.) Smithells Metals Reference Book, 7th ed., Butterworth-Heinemann, Oxford, 1992.) DIFFUSION AND TEMPERATURE
- 17. 20 Diffusion FASTER for... • open crystal structures • lower melting T materials • materials w/secondary bonding • smaller diffusing atoms • cations • lower density materials Diffusion SLOWER for... • close-packed structures • higher melting T materials • materials w/covalent bonding • larger diffusing atoms • anions • higher density materials SUMMARY: STRUCTURE & DIFFUSION
Editor's Notes
- #14: Typically, we let the experiment described on slide 2 run during the class; at this point in the class, the results are available for the class to look at.