应用拉普拉斯变换和留数法求解常见非稳态扩散情况下的菲克定律  

Solution of Fick's Second Law in the Case of Common Diffusion

作　　者：丁锐[1,2] 桂泰江[2] 蒋建明[2] 余海斌[1] 

机构地区：[1]中国科学院海洋新材料与应用技术重点实验室,浙江省海洋材料与防护技术重点实验室,中国科学院宁波材料技术与工程研究所,浙江宁波315201 [2]海洋化工研究院有限公司,海洋涂料国家重点实验室,山东青岛266071

出　　处：《数学的实践与认识》2017年第1期271-279,共9页Mathematics in Practice and Theory

基　　金：青岛博士后应用研究项目基金2015307;青岛市科技计划项目2015-6-024-ZH

摘　　要：介绍了三维和一维扩散下的菲克定律,以及两类涉及到扩散的实际问题,即求扩散粒子通过曲面的扩散通量和求解扩散粒子的浓度分布.通过拉普拉斯变换和复变函数相关数学理论,求解了菲克扩散定律在无限长介质和有限长介质两种非稳态扩散情况下的解.粒子在无限长介质中的非稳态扩散和浓度分布可通过方程φ(z,t)=Φ·erfc(z/2DT^(1/2))表示.方程为余补高斯误差函数.粒子在有限长介质中的非稳态扩散和浓度分布可通过方程φ(z,t)=Φ+Φ·4/π∑_(n=1)^(+∞)((-1)~n)/(2n-1)cos[z/L(n-1/2)π]e^((D_t)/(L^2)(n-1/2)~2π~2)表示.该方程为无限加和形式,当n≥100000时,φ可以精确到小数点后6位,在方程的图像上不再能观察出由n的取值造成的误差.从方程的图像可得到粒子在扩散介质中达到饱和的时间或粒子扩散到z=0处的时间等具有重要物理意义的参数.This article described the Fick＇s Law of three-dimensional and one-dimensional diffusion,and two types of practical issues related to diffusion,which were solving diffusion fluxes of particles transferred through a surface and solving concentration distribution of diffusion particles.In this paper,the solutions of Fick diffusion law in unsteady situations,which were infinite medium and finite medium,were obtained by application of mathematical theory such as Laplace transform and complex functions.Unsteady diffusion and concentration distribution of particles in infinite media could be expressed by the equationφ（z,t） = Φ·erfc（z/2DT^（1/2）） which was complementary Gaussian error function.Unsteady diffusion and concentration distribution of particles in finite media could be described by equation ofφ（z,t） = Φ ＋ Φ·4/π ∑_（n=1）（＋∞）（-1）n/2n-1 cos[z/L（n-1/2）π]e^（-D_t/L^2（n-1/2）~2π~2）which was in form of unlimited plus.When n ≥ 100000,φ could be accurate to six decimal,and the error caused by n was no longer able to observe on image of the equation.Important parameters which had physical significant could be obtained by considering images of the kinetic equations,such as saturation time of particles in diffusion media and of particles reached the position of z = 0.

关 键 词：菲克定律 无限扩散 有限扩散 拉普拉斯变换 复变函数 

分 类 号：O552.2[理学—热学与物质分子运动论]