材料科學(xué)基礎(chǔ)英文版課件_(13)-文庫吧
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【正文】 case, the unit of C is in mass per unit volume, . g/m3 Minus sign indicates the diffusion is down the concentration gradient Fick’s first law The mathematics of steadystate diffusion in one dimension is given by For steadystate diffusion, the diffusion flux is proportional to the concentration gradient Negative Mathematics of Diffusion (4) ? Nonsteadystate diffusion The diffusion flux at a particular point varies with time. Mathematics of Diffusion (5) The diffusion equation is represented by )(xCDxtC???????Fick’s second law C is a function of x and t If D is independent of the position, the above equation changes to 22xCDtC?????Mass decrease in the box per unit volume per unit time dC/dt = mass increase in the box per unit volume per unit time Volume of the box: 1?dx C = mass per unit volume (concentration) Unit area crosssection Mathematics of Diffusion (6) ? Solutions of diffusion equation )2(e r fDtxBACx ??A and B are constants and erf(z) is the error function, defined as dyezz y? ??022)(e r f?According to error function solutions for diffusion equation, the solution for these profiles can be given by FFF Mathematics of Diffusion (7) For t0, Cx=Cs at x=0 Cx=Co at x=?? Therefore )20(e r fDtBAC s ??A=Cs )2(e r foDtBAC ???Co=A+B B=(CsCo) )2(e r f)(sDtxCCCCosx ??? )2(e r f1osoDtxCCCC x ????Boundary conditions: Mathematics of Diffusion (8) For t=0, Cx=C1 at x0 Cx=C2 at x0 Therefore )2(e r f2 )(2 )( 2121DtxCCCCCx????)(e r f1 ???? BAC C1=AB )(e r f2 ??? BACC2=A+B A= (C1+C2)/2 B=(C1C2)/2 Boundary conditions: Mathematics of Diffusion (9) x For t=0, Cx=0 at xh。 Cx=Co at hxh。 Cx=0 at xh Boundary conditions: )]2(e r f)2([ e r f2 oDthxDthxCCx????0)(e r f
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