Chapter 4 Point Defects and Diffusion in Solids.ppt

* 在数学中，误差函数（也称之为高斯误差函数）是一个非基本函数（即不是初等函数），其在概率论，统计学以及偏微分方程中都有广泛的应用。它的定义如下： * Vibration振动 * Analogous相似的 vibration振动 Fick’s Laws of Diffusion D is a function of T and c usually expressed in cm2/s Fick’s First Law (defines the diffusion coefficient D): Ficks laws of diffusion were derived by Adolf Fick (a German physiologist) in 1855. Fick’s Second Law (the Diffusion Equation): Recall species conservation For an isothermal system D is independent of c, thus and Fick’s First Law predicts concentration profile in an unsteady state diffusion process (concentration at x varies with time) Diffusion Equations for Special Geometries Measurement of Diffusion Coefficient in Solids Surface source method Diffusion couple method used to measure self-diffusion coefficient Concentration Profile of Diffusion Species : : The function erfc(ξ) of the dimensionless argument ξ is called the complementary error function, defined as 1 – erf(ξ), where erf(ξ) is the tabulated error function: erfc(0)=0 erfc(1)=0.5 erfc(2)=0.0068 With inexhaustible surface source source couple Characteristic Diffusion Depth: C(xdiff,t) = 0.5 C0 (source method) can be used to quickly estimate the (average) diffusion depth at a fixed time (and T) or for the estimation of the time needed for diffusion to certain depth erfc(1)=0.5 when x=2 Atomic Mechanisms of Diffusion in Solids The migration of self interstitials or substitutional impurity atoms in nearly all metals occurs by the vacancy mechanism. A Vacancy diffusion B Interstitial diffusion C Interstitialcy diffusion Typical interatomic potential function between two atoms where A, b, C and s are constants Jump of impurity atom in crystal with sc structure Energy Barrier and Vibration Frequency Jump Frequency by thermal vibration: Energy barrier for jump wave length λ: jump distance λ The impurity atom in the position of the barrier plane is analogous to a host atom in an interstitial site: with the interstitial fraction: exp(-εi/kT)