清华大学：《材料科学基础》课程教学资源（PPT课件讲稿）Chapter 6. 4 Non-Steady-State Di ffus ion

36.4 Non-Steady-State Diffusion ≠0 (for one dimensional diffusion only ◆|.| nfinite system Formula ac a2C = D at at t=0 C=C for x>0 C=C for x<0 (2 let a=x (Boltzman transforma tion)(3)

§6.4 Non-Steady-State Diffusion ( ) (for one dimensional diffusion only) ◆ Ⅰ. Infinite system (1) 2 2 x C D t C   =   let (Boltzman transforma tion ) (3) t x  = (2) for 0 at 0 for 0 2 1 =  = =  C C x t C C x 1. Formula

ac do dc d42t√td22t ac dC 2Cd1dC、11d2C ax dn dn t dn t t d22 put(4)n()dC_Ddc 2t dnt dx do let d put(6)in(5) du d D一→ da d 2D →ln +4 4D u= Ae 4D

(4) d 1 1 d ) d 1 d ( d d ) 1 ( d d ) 2 ( d d ) 2 ( d d 2 2 2 2        C t t C x t C t C x C t C t t C x t C =  =   =   = − = −   ( ) ( ) (5) d d d d 2 put 4 in 1 2 2    C t C D t － = (6) d d let  C u = ( ) ( ) (7) 4 ln d d d 2 d 2 put 6 in 5 4 2 2 Ae D u A D u u u D u u D       − =  = − +  － =  − =

put(7)in(6 dc d Ae 4D C=AL e 4Dda+B let (8) 2√D2√Dt C=Aeds+B t=0 C for x>0 from (9) C1=A"+B +B B

( ) ( ) (8) 2 2 let d d d put 7 in 6 0 4 4 2 2 Dt x D C A e B Ae C D D = = = + =  − −        ( ) 0 for 0 d 9 1 0 2 = =  =  +  − t C C x C A e  B   ( ) (10) 2 2 2 from 9 1 2 1 2 2 1    C C A C C B C A B C A B −  = +  = = −  + =  +

put(10)in(9) +0 C1、2 2 let erf(5)=esds erf error function +C1 C 2) erf(5)

( ) ( ) d (11) 2 ) 2 ( 2 put 10 in 9 0 2 1 2 1 2      − − − + = e C C C C C ( ) erf : error function d 12 2 let erf( ) 0 2       − = e )erf( ) (13) 2 ( 2 2 1 2 1  C C C C C − − + =

2√D x 1-erfl 2√D

      = − − Dt x C C C C s s x 2 erf 0       = − − − Dt x C C C C s x 2 1 erf 0 0

2. Discussion ① Application: given t,D→C(x,0) x/2√D x→2=x/2√D→ookp→erf(2)→C(x,) 2 C-x curve (concentration penetration curve) x=0→5=0→erf()=0 与时间无关 at x=0. the concentration is invariable

2. Discussion: ① Application: ② C－x curve. (concentration penetration curve) given t, D  C(x,t) x = 0  = 0  erf() = 0 2 1 2 0 C C C +  = x  = x / 2 Dt  lookup  erf()  C(x,t)  = x / 2 Dt 与时间无关  at x = 0, the concentration is invariable

+∞ ac dc as Ox ds a 2 C+O C-x curve is symmetrical.(x=0, C 2 0

1 1 : C C x C C = =  = + +     2 2 1 d 2 d 2 2 1   − = −   =    − Dt e C C x C x C C－x curve is symmetrical. ( ) 2 0, C1 C2 x C + = = 0 t 1 t 2 t C1 C2 C 0 x 2 C1 +C2

≡ Couple

③ Parabolic grow law √√t f(c) ∴x2=k(c)Dt 4 16t

③ Parabolic grow law x k c Dt f c t x t x ( ) ( ) 2  = =    =   t 4t 9t 16t C C1 C0 x

Variation of expression (13) Derf (5) Co[l-erf(5)]-Clll-erf(5)]+Cl C =1-erf() 1)fC1=0C=C0(-er(5) b) if Co=0 C= Cerf(5) ⑤ approximation f5=0.5erf()≈0.5 ∴atx2=DtC=0

④ Variation of expression (13) 1 erf( ) [1 erf( )] [1 erf( )] )erf( ) 2 ( 2 1 1 1 1 1 1 2 1 2     = − − −  = − − − + − + − + = C C C C C C C C C C C C C   0 2 at 0.5 if 0.5 erf( ) 0.5  x = Dt C = C   =   ( ) ) if 0 erf( ) ) if 0 1 erf( ) 0 1 1 0   b C C C a C C C  = =  = = − ⑤ approximation