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1、1Chapter 7 Diffusion in SolidWhat is Diffusion?Issues to Address?How does diffusion occur?Why is it an important part of processing?How can the rate of diffusion be predicted for some simple cases?How does diffusion depend on structure and temperature?DIFFUSIONDIFFUSION7.1 7.1 Introduction- - Diffus
2、ion of Diffusion of ArAr/He/He?Consider a box containing an impermeable partition that divides the box into equal volumes. On one side, we have pure argon (Ar) gas; on the other side, we have pure helium (He) gas. Explain what will happen when the partition is opened? ?What will happen when we drop
3、a red ink in water?What will happen if we replace the Ar side with a Cu single crystal and the He side with a Ni single crystal?Diffusion in Solid Diffusion in Solid - - Cu/NiCu/NiDiffusion of copper atoms into nickel. Eventually, the copper atoms are randomly distributed throughout the nickel.2What
4、 is Diffusion?the phenomenon of material transport by atomic motion.?Diffusion: Mass transport A atomsB atoms?Inhomogeneous materials can become homogeneous by diffusion. For an active diffusion to occur, the temperature should be high enough to overcome energy barriers to atomic motion.?扩散是指一个系统由非均
5、化不平衡状态向均化平衡状态转化 而引起粒子迁移的现象。What is the driving force for diffusion?Difference of concentration cause diffusion.?Concentration gradient?Chemical potential gradient?Stress gradient?扩散本质: 原子和分子的热运动导致其在浓度梯度 下的定向迁移,宏观上则表现为物质的定向输运。Diffusion in Solid?低温不明显，高温明显；?各向异性?扩散速度慢?Diffusion in solid is obvious at
6、high temperature, not obvious at low temperature.?Diffusion in gas and liquid is isotropic, but anisotropic in solid.?Diffusion rate in solid is slow.Diffusion in a solid Glass tube filled with water. At time t = 0, add some drops of ink to one end of the tube. Measure the diffusion distance, x, ove
7、r some time. Compare the results with theory. tot1t2t3xox1x2x3time (s)x (mm)Diffusion Demo7.2 Rate of Diffusion (Ficks First Law) & Composition Profile (Ficks Second Law)Illustration of the concentration gradient3The flux of diffusing atoms, J, is used to quantify how fast diffusion occurs.The flux
8、during diffusion is defined as the number of atoms passing through a plane of unit area per unit timeDiffusion flux, JDiffusion coefficient, DConcentration gradient Flux:J =1 AdM dtkgm2s oratomsm2s Directional Quantity Flux can be measured for: -vacancies -host (A) atoms -impurity (B) atomsJxJyJzxyz
9、x-directionUnit area A through which atoms move.Modeling Diffusion: Flux Steady State: the concentration profile doesnt change with time.12 Apply Ficks First Law: Result: the slope, dC/dx, must be constant (i.e., slope doesnt vary with position)!Jx(left) = Jx(right)Steady State:Jx= DdCdxdC dx left=d
10、C dx right If Jx)left= Jx)right, thenSteady State Diffusion: Ficks First LawConcentration, C, in the box doesnt change w/time.Jx(right)Jx(left)x Concentration Profile, C(x): kg/m3 Ficks First Law:Concentration of Cu kg/m3Concentration of Ni kg/m3Position, xCu fluxNi flux The steeper the concentratio
11、n profile, the greater the flux!Jx= DdC dxDiffusion coefficient m2/sconcentration gradient kg/m4flux in x-dir. kg/m2-sConcentration Profiles & FluxRate of Diffusion (Ficks First Law)?Ficks first law - The equation relating the flux of atoms by diffusion to the diffusion coefficient and the concentra
12、tion gradient.xcDJx=ycDJy=zcDJz=)(zckycjxciDJkJ jJ iJzyx+=+=?The minus sign in the equation means that diffusion is down the concentration gradient.?Diffusion coefficient (D) - A temperature-dependent coefficient related to the rate at which atoms, ions, or other species diffuse.?The concentration g
13、radient is often called the driving forcein diffusion?Concentration gradient - The rate of change of composition with distance in a nonuniform material, typically expressed as atoms/cm3cm or at%/cm.0= tc0= xJx4Nonsteady-State Diffusion: Ficks second law?In most real situations the concentration prof
14、ile and the concentration gradient are changing with time. The changes of the concentration profile is given in this case by a differential equation, Ficks second law. Concentration profile, C(x), changes with time. To conserve matter: Ficks First Law: Governing Eqn.:Concentration, C, in the boxJ(ri
15、ght)J(left)dxdC dt=Dd2Cdx2 dx=dCdtJ= DdCdxorJ(left)J(right)dJdx=dCdtdJdx=Dd2Cdx2(if D does not vary with x)equateNon Steady State DiffusionFicks Second Law21元体积dxdydz时间内沿x方向扩散t流入的粒子数：tdydzJx流出的粒子数：tdydzdxxJJx x)(+净粒子数：tdxdydzxJtdydzdxcJJtdydzJJxx xxx=+=)(同理，在y、z方向分别为：tdxdydzyJJy y=tdxdydzzJJz z=对整个元体积：tdxdydzzJ yJxJzyx +若时间内粒子浓度变化，则在dxdydz 元体积中粒子变化为tccdxdydz += zJ yJxJ tczyx24 +=222222 ),( zc yc xcDtctzyx一维系统：22)( xcDxJ tcx =球对称扩散： += rc rrcDtc222菲克第二定律的数学表达式 描述在不稳定扩散条件下，在介质中各点作为时间 函数的扩散物质聚集的过
