Chapter 5: Diffusion (2)

Transcription

1 Chapter 5: Diffusin () ISSUES TO ADDRESS... Nn-steady state diffusin and Fick s nd Law Hw des diffusin depend n structure? Chapter 5-1

2 Class Eercise (1) Put a sugar cube inside a cup f pure water, rughly draw the sugar cncentratin prfile inside the water fr i) time is zer (i.e., right after the sugar cube was placed in water), ii) after a shrt time (e.g., 30 sec) and iii) after very lng time C t = 0 t is small t is lng Chapter 5 -

3 Class Eercise () Given Fick s 1 st Law in 1D, J D dc d 1) Eplain the physical meaning fr each f the terms in the equatin and their unit ) if bth flu J and diffusin cefficient D are cnstants, prve the cncentratin change linearly with psitin Chapter 5-3

4 Class Eercise (3) If diffusin cefficient at different temperatures (i.e., D 1 at T 1, D at T, D 3 at T 3, D 4 at T 4 ) are measured, describe hw t mathematically determine diffusin activatin energy Q d D D 0 ep Qd RT lnd lnd 0 Qd 1 ln D ln D0 R T ln D 1 ln D ln D 3 ln D 4 Slpe = Q d R T1 T T3 T4 1/T Chapter 5-4

5 Nn-Steady State Diffusin General case: The cncentratin f diffusing species is a functin f psitin, time, and, maybe, cncentratin C = C(, t, c) Fick s Secnd Law C t D C If D is a cnstant (i.e., des nt change with cncentratin) C t D C Chapter 5-5

6 Special Eample fr Nn-steady State Diffusin Cpper diffuses int a bar f aluminum. Surface cnc., C s f Cu atms C s bar pre-eisting cnc., C f cpper atms Adapted frm Fig. 5.5, Callister & Rethwisch 8e. C C Fick s nd law D C(, t)? t Bundary cnditins at t = 0, C = C fr 0 at t > 0, C = C S fr = 0 (cnstant surface cnc.) C = C fr = Chapter 5-6

7 Slutin: C,t C C C s 1 erf C(,t) = Cnc. at pint at time t erf (z) = errr functin z e y 0 dy erf(z) values are given in Table 5.1 C S C(,t) C Adapted frm Fig. 5.5, Callister & Rethwisch 8e. Chapter 5-7

8 Eample f a Special Case fr Nn-steady State Diffusin Prblem: An FCC irn-carbn ally initially cntaining 0.0 wt% C is carburized at an elevated temperature and in an atmsphere that gives a surface carbn cncentratin cnstant at 1.0 wt%. If after 49.5 h the cncentratin f carbn is 0.35 wt% at a psitin 4.0 mm belw the surface, determine the diffusin cefficient. Knwing cncentratin at time t and psitin satisfy the fllwing relatinship: C(, t) C C C s 1 erf t = 49.5 h C = 0.35 wt% C = 0.0 wt% = m C s = 1.0 wt% Chapter 5-8

9 Slutin (cnt.): C(,t ) C C C s 1 erf t = 49.5 h = m C = 0.35 wt% C s = 1.0 wt% C = 0.0 wt% C(, t) C C C s erf 1 erf( z) erf(z) = Chapter 5-9

10 Slutin (cnt.): We must nw determine the value f z fr which the errr functin is If as in Table 5.1 in tetbk, tabulated data are given, an interplatin can be used t btain the apprimate value z erf(z) z z z 0.93 Nw slve fr D z D 4z t D 4z t (4 10 (4)(0.93) 3 m) (49.5 h) 1h 3600 s m /s Chapter 5-10

11 Diffusin Length C,t C C C s 1 erf When C C s C C 0.5 C C C 0 s C C s C C erf 1 erf z Diffusin (characteristic) length C C S C S C 0 C 0 0 t 1 t t The depth (length) within which significant diffusin has happened Chapter 5-11

12 Class Eercise Knwing diffusin length Diffusin cefficient fr P and Cu in Si at 1000C is cm /sec and cm /sec, respectively. If putting Cu and P n surface f Si, please estimate the depth within silicn that is impacted by P and Cu diffusin after 30 min at 1000C, respectively 14 P 10 cm cm 4nm Cu 0.09cm m 6 cm / sec1800sec / sec1800sec At the same temperature fr same time, Cu wuld diffuse (penetrate) much deeper than P int Si! As Cu is detrimental t Si devices, peple have t apply a diffusin barrier (e.g., Ru r TaN) in the semicnductr industry when using Cu as metal cnductr fr Si micr-chips Chapter 5-1

13 Eample Prblem () Prblem: The diffusin cefficient fr cpper in Al at 500 and 600 C are and m /s, respectively. Determine the apprimate time at 500 C that will prduce the same diffusin result (i.e., diffusin length) as a 10 hur heat treatment at 600 C. Nte that diffusin length Same diffusin result, therefre, same diffusin length, i.e., In this case, D 1 (600 C) = m /s, D (500 C) = m /s t 1 (600 C) = 10 h t D1 ( 500 C) t1 D 1 D1t h It wuld need a much lnger time at lwer temperature t prduce the same (diffusin) result as that btained at a higher temperature Chapter 5-13

14 Structure Effects n Diffusin Diffusin FASTER fr... Mre pen crystal structures Materials w/secndary bnding Smaller diffusing atms Lwer density materials Diffusin SLOWER fr... Clse-packed structures Materials with all cvalent bnding Larger diffusing atms Higher density materials Chapter 5-14

15 Hmewrk Read chapter 5 f Calister 8ed and give an hnr statement cnfirming yu finished the required reading Callister 8ed, 5.1, 5.3(a), 5.6, 5.11, 5.16, 5. Chapter 5-15

16 Calister 8ed, Briefly eplain the difference between selfdiffusin and interdiffusin. Calister 8ed, 5.3(a) 5.3 (a) Cmpare interstitial and vacancy atmic mechanisms fr diffusin. Chapter 5 -

17 Calister 8ed, 5.6 The purificatin f hydrgen gas by diffusin thrugh a palladium sheet was discussed in Sectin 5.3. Cmpute the number f kilgrams f hydrgen that pass per hur thrugh a 5-mm-thick sheet f palladium having an area f 0.0 m at 500C. Assume a diffusin cefficient f m /s, that the cncentratins at the high- and lw-pressure sides f the plate are.4 and 0.6 kg f hydrgen per cubic meter f palladium, and that steady-state cnditins have been attained. Chapter 5 -

18 Calister 8ed, Determine the carburizing time necessary t achieve a carbn cncentratin f 0.45 wt% at a psitin mm int an irn carbn ally that initially cntains 0.0 wt% C. The surface cncentratin is t be maintained at 1.30 wt% C, and the treatment is t be cnducted at 1000C. Use the diffusin data fr γ-fe in Table 5. Chapter 5 -

19 Calister 8ed, Cite the values f the diffusin cefficients fr the interdiffusin f carbn in bth α-irn (BCC) and γ-irn (FCC) at 900 C (frm Table 5.). Which is larger? Eplain why this is the case. Chapter 5 -

20 Calister 8ed, The diffusin cefficients fr silver in cpper are given at tw temperatures: (a) Determine the values f D 0 and Q d. (b) What is the magnitude f D at 875 C? T ( C) D (m /s) Chapter 5 -