第三章固體材料中的擴散

1、第三章固體材料中的擴散chapter3 the diffusion in solid materials本章基本問題：1. 菲克第一定律的含義和各參數(shù)的量綱。2. 能根據(jù)一些較簡單的擴散問題中的初始條件和邊界條件。運用菲克第二定律求解。3. 柯肯達(dá)耳效應(yīng)的起因，以及標(biāo)記面漂移方向與擴散偶中兩組元擴散系數(shù)大小的關(guān)系。4. 互擴散系數(shù)的圖解方法。5. “下坡擴散”和“上坡擴散”的熱力學(xué)因子判別條件。6. 擴散的幾種機制，著重是間隙機制和空位機制。7. 間隙原子擴散比置換原子擴散容易的原因。8. 計算和求解擴散系數(shù)及擴散激活能的方法。9. 影響擴散的主要因素。questions for chapte

2、r 31. what is the meaning of fick's first law?2. how to solve the problems by fick，s second law?3 what is the the kirkendall effect?4. how to explain diffusion coefficient schematically?5. what is the diffusion driving force;6. what are diffusion mechanisms，especially interstitial and vacancy me

3、chanisms7. what is the reason that interstitial diffusion is easier than substitutional diffusion?8 what are the methods to compute diffusion coefficient and diffusion activation energy?9. what are main factors affecting diffusion?the field of diffusion studies in metals is of great practical, as we

4、ll as theoretical importance. by diffusion one means the movements of atoms within a solution. in general, our interests lie in those atomic movements that occur in solid solutions. this chapter will be devoted in particular to the study of diffusion in substitutional solid solutions and atomic move

5、ments in interstitial solid solutions.diffusion is a process of mass transport that involves the movement of one atomic species into anothe匸3-1擴散方程sec.3.1 diffusion equations1 菲克第一定律fick，s first lawdiffusion can be modeled as the jumping of atoms from one plane to anothe匚(1) 第一定律描述：單位時間內(nèi)通過垂直于擴散方向的某一

6、單位面積截面的擴散物質(zhì) 流量(擴散通量 j flax or the rate of diffusion)與濃度梯度(concentration gradient)成正比。the rate of diffusion is proportional to the concentration gradient.(2) 表達(dá)式:j=-d(dc/dx)o (c溶質(zhì)原子濃度;d擴散系數(shù) diffusivity or diffusion coefficiento)(3) 適用條件：穩(wěn)態(tài)擴散，dc/dt=0o濃度及濃度梯度不隨時間改變。fick's first law assumes that the

7、 concentration gradient is independent of time.2 菲克第二定律 fick，s second law'一般：dc/8t=d(ddc/dx)/ dx'二維：(1) 表達(dá)式丨特殊：ac/dc/sx2'穩(wěn)態(tài)擴散：dc/dt=0, dj/dx=0 (2) 適用條件彳i 非穩(wěn)態(tài)擴散：acwo, ajwo(ac/at=-sj/ax)0assume that diffusivity, d is independent of c, the rate of change in concentration with time, dc/dt is

8、 proportional to the rate at which the concentration gradient changes with distance in a given direction, d2c/dx23擴散第二定律的應(yīng)用(1) 誤差函數(shù)解適用條件：無限長棒和半無限長棒。表達(dá)式：c=c!-(ci-c2)erf(x/2 vdt) (t無限長棒)。在滲碳條件2 c:x,t皿的濃度；g：表面含碳量;c2：鋼的原始含碳量。(2) 正弦解cx=cp-aosin( n x/a )cp: 均成分；ao：振幅cmax- cp；入：枝晶間距的一半。對于均勻化退火，若要求枝晶中心成分偏析振

9、幅降低到1/100,則： c(a /2,t)- cp/( cmax- cp)=exp(-兀 2dt/ x2)=l/100oexperimental work has shown that the atoms in facc-ccntcrcd cubic, body-ccntcrcd cubic, and hexagonal metals move about in the crystal lattice as a result of vacancy motion. let it now be assumed that the jumps are entirely random; that is

10、, the probability of jumping is the same for all of the atoms surrounding a given vacancy. this statement implies that the jump rate does not depend on the concentration.concentrotion of solute (a otoms). 1 est this endconcentrotion of solute (a atoms) highest this endfig. 3.1 hypothetical single cr

11、ystal with a concentration gradientxo oooo eo / oooooooeoo 一 oooo ooooo oplane yfig.3.2 atomistic view of section of the hypothetical crystal of fig. 3.1fig31 represents a single crystal bar composed of a solid solution of a and b atoms in which the composition of the solute varies continuously alon

12、g the length of the bar, but is uniform over the cross-section. for the sake of simplifying the argument, the crystal structure of the bar is assumed to be simple cubic with a <100> direction along the axis of the bar. it is further assumed that the concentration is greatest at the right end o

13、f the bar and least at the left end, and that the macroscopic concentration gradient dna/dx applies on an atomic scale so that the difference in composition between two adjacent transverse atomic planes is:dxwhere a is the interatomic, or lattice spacing (see fig.3.2). let the mean time of stay of a

14、n atom in a lattice side be t . the average frequency with which the atoms jump is therefore 1/ t . in the simple cubic lattice pictured in fig.3.2, any given atom, such as that indicated by the symbol x, can jump in six different directions: right or left, up or down, or into or out of the plane of

15、 the pape匚 the exchange of a atoms betweentwo adjacent transverse atomic planes, such as those designated x and y in fig.3.2, will now be considered of the six possible jumps that an a atom can make in either of these planes, only one will carry it over to the other indicated plane, so that the aver

16、age frequency with which an a atom jumps from x to y is 1/6 t . the number of these atoms that will jump per second from plane x to plane y equals the total number of the atoms in plane x times the average frequency with which an atom jumps from plane x to plane y. the number of solute atoms in plan

17、e x equals the number of solute atoms per unit volume (the concentration g) times the volume of the atoms in plane x, (aa), so that flux of solute atoms from plane x to plane y is丿xt)，=*(/)32where 丿 二 flux of solute atoms from plane x to plane y per unit cross-sectiont = mean time of stay of a solut

18、e atom at a lattice sitena = number of a atoms per unit volume a = lattice constant of crystalthe concentration of a atoms in plane y may be written:3.3(譏=g+(a)學(xué)ax3.4where na is the concentration at plane x、and a is the lattice constant, or distance between planes x and y. the rate at which a atoms

19、move from plane / to x is thusdx 6rwhere jy-x represents the flux of a atoms from plane y to plane x. because the flux of solute atoms from right to left is not the same as that from left to right, there is a net flux (designated by the symbol j) which can be expressed mathematically as follows:細(xì))-5

20、+(。)如dxa6r3.5or3.6a dna6r dxsince the cross-sectional area was chosen to be a unit area. notice that in eq.3-6, the flux (j) of a atoms is negative when the concentration gradient is positive (concentration of a atoms increases from left to right in fig.3.2). this result is general for diffusion in

21、an ideal solution; the diffusion flux is down the concentration gradient. notice that if one considers the flow of b atoms instead of a atoms, the net flux willbe from left to right, in agreement with a decreasing concentration of the b component as one moves from left to right. again, the flux (in

22、this case of b atoms) is down the concentration gradient.let us now make the substitution6r3.7in the equation for the net flux, which gives:j =-ddnadx3.8this equation is identical with that first proposed by adolf fick in 1855 on theoretical grounds for diffusion in solutions- in this equation, call

23、ed fick s first law, j is the flux, or quantity per second, of diffusing matter passing normally through a unit area under the action of a concentration gradient dn/dx. the factor d is known as the diffusivity, or the diffusion coefficient.3-2擴散的微觀機制sec. 3.2 mechanisms of diffusionthe mechanism of d

24、iffusion determines the energy barrier that must be overcome (i.e., the activation energy q) for the process to occu匚 since energy is supplied thermally, the higher the temperature, the greater the probability that large numbers of atoms will have sufficient energy to overcome the energy barrier and

25、 the more rapid will be the diffusion process-the lattice geometry also affects the diffusion coefficient through the preexponential constant.interstitial confusion mechanism: when a small interstitial atom or ion is present in the crystal structure, the atom or ion moves from one interstitial site

26、to another. no vacancies are required for this mechanism.i間隙一間隙；（1）間隙機制 平衡位置一間隙一間隙：較困難; i間隙一篡位一結(jié)點位置。（間隙固溶體川間隙原子的擴散機制。）vacancy exchange mechanism of diffusion: in self-diflusion and diffusion involving substitutional atoms, an atom leaves its lattice site to fill a nearby vacancy( thus creating a new

27、 vacancy at the original lattice site). as diffusion continues, we have countercurrent flows of atoms and vacancies, called vacancy diffusion.'方式：原子躍遷到與之相鄰的空位；(2)空位機制條件：原子近旁存在空位。（金屬和直換固溶體中原子的擴散。）(3)換位機制壞形換位 zener ring mechanism for diffusion(所需能量較高。)in general, the activation energies for vacanc

28、y-assisted diffusion qv are higher than those for interstitial diffusion the reason is that the former mechanism requires energy to both form a vacancy and move an atom into the vacancy, while in the latter case energy is needed only to move the interstitial atom into the interstitial site.3-3擴散的驅(qū)動力

29、sec. 3.3 the driving force for diffusion（1）擴散的驅(qū)動力對于多元休系，設(shè)n為組元i的原子數(shù)，則在等溫等壓條件卜組元i原子的自由能可 用化學(xué)位表示：擴散的驅(qū)動力為化學(xué)位梯度，即f=刀” i /sx.負(fù)號表示擴散驅(qū)動力指向化學(xué)位降低的方向。（2）擴散的熱力學(xué)因子組元i的擴散系數(shù)可表示為di=ktbi（ + 方加 yi/lnxi）其中，（dlnyi/dlnxi）稱為熱力學(xué)因子。當(dāng)（1+刀加必刃曲丿0吋，口0,發(fā)生上坡擴散。（3）上坡擴散概念：原子曲低濃度處向高濃度處遷移的擴散。驅(qū)動力：化學(xué)位梯度。其它引起上坡擴散的因素：彈性應(yīng)力的作用一大直徑原了跑向點陣的受

30、拉部分，小直徑原了跑向點陣的受壓 部分。晶界的內(nèi)吸附：某些原子易富集在晶界上。電場作用：人電場作用可使原子按一定方向擴散。3-4影響擴散系數(shù)的因素sec. 34 factors affecting diffusion coefficient1 溫度 temperatured=doexp(-q/rt)可以看出，溫度越高，擴散系數(shù)越大。the kinetics of process of diftusion are strongly dependent on temperature. the diffusion coefficient d is related to temperature by an arrhenus-type equation.2原子鍵力和晶體結(jié)構(gòu)dependence on bonding and crystal structure原了鍵力越強，擴散激活能越高；致密度低的結(jié)構(gòu)中擴散系數(shù)大（舉例：滲碳選擇在 奧氏體區(qū)進行）；在対稱