ch2位错-22 位错的几何性质

1、1Ch2 Ch2 位错位错2.1 2.1 位错理论的产生位错理论的产生2.2 2.2 位错的几何性质位错的几何性质2.3 2.3 位错的弹性性质位错的弹性性质2.4 2.4 位错与晶体缺陷的相互作用位错与晶体缺陷的相互作用2.5 2.5 位错的动力学性质位错的动力学性质2.6 2.6 实际晶体中的位错实际晶体中的位错22.1 2.1 位错理论的产生位错理论的产生一、晶体的塑性变形方式一、晶体的塑性变形方式二、单晶体的塑性变形二、单晶体的塑性变形三、多晶体的塑性变形三、多晶体的塑性变形四、晶体的理论切变强度四、晶体的理论切变强度五、位错理论的产生五、位错理论的产生六、位错的基本知识六、位错的基本

2、知识32.2 2.2 位错的几何性质位错的几何性质一、位错的几何模型一、位错的几何模型二、柏格斯矢量二、柏格斯矢量三、位错的运动三、位错的运动四、位错环及其运动四、位错环及其运动五、位错与晶体的塑性变形五、位错与晶体的塑性变形六、割阶六、割阶4一、位错的几何模型一、位错的几何模型l、2两列原子已两列原子已完成了滑移完成了滑移,3、4、5各列原子虽各列原子虽开始滑移开始滑移,但还但还未达到平衡位置未达到平衡位置,6、7、8各列各列尚未滑移尚未滑移.这样这样,滑移面便分为已滑移面便分为已滑移区和未滑移区滑移区和未滑移区.已滑移区与末滑移区的界限已滑移区与末滑移区的界限(3、4、5列列),即即定义为

3、定义为位错位错.位错是线缺陷位错是线缺陷,位错线上成列的原子发生了有规则位错线上成列的原子发生了有规则的错排的错排. 位位错错与与滑滑移移5位错的类型位错的类型1.1.刃型位错：多余半原子面刃口（正、负）刃型位错：多余半原子面刃口（正、负） 位错线垂直位错线垂直b b2.2.螺型位错：螺旋面轴线（左、右）螺型位错：螺旋面轴线（左、右） 位错线平行位错线平行b b3.3.混合位错：介于以上两者混合位错：介于以上两者6刃型位错刃型位错刃型位错的基本结构特点刃型位错的基本结构特点, ,就是它的就是它的位错线位错线是是多余半多余半原子面与滑移面的交线原子面与滑移面的交线. .但是但是位错线位错线不只是

4、一列原子不只是一列原子, ,而是以而是以EFEF线为中心的一个管道线为中心的一个管道, ,其直径一般为其直径一般为3 34 4个个原子间距原子间距. .在此范围内在此范围内, ,原子发生严重的错排原子发生严重的错排. .7在滑移面上部在滑移面上部, ,位错线周围的原子因受到位错线周围的原子因受到压应力压应力而而向外偏离于平衡位置向外偏离于平衡位置; ;在滑移面下部在滑移面下部, ,位错线周围的位错线周围的原子因受原子因受拉应力拉应力也偏离平衡位置也偏离平衡位置. .8刃型位错的几何特征刃型位错的几何特征(1)(1)刃位错线与其滑移矢量刃位错线与其滑移矢量b b垂直垂直. .刃位错只有唯一的滑移

5、面刃位错只有唯一的滑移面, ,但但多余半原子面末端可以是任意形状多余半原子面末端可以是任意形状, ,故刃位错线可以为任意故刃位错线可以为任意形状的曲线形状的曲线, ,如下图所示如下图所示; ;(2)(2)刃位错可以人为地分为正和负两种刃位错可以人为地分为正和负两种. .当晶体倒置时当晶体倒置时, ,晶体中晶体中刃位错的正负符号随之颠倒刃位错的正负符号随之颠倒; ;(3)(3)当刃位错滑出晶体表面时当刃位错滑出晶体表面时, ,只在不垂直于位错线的晶体表面只在不垂直于位错线的晶体表面上出现滑移台阶上出现滑移台阶; ;(4)(4)刃位错周围产生了刃位错周围产生了体应变体应变与与切应变切应变. .91

6、0Screw dislocationNow we go on and learn a new thing: There is a second basic type of dislocation, called screw dislocation. Its atomistic representation is somewhat more difficult to draw - but a Burgers circuit is still possible:You notice that for no particularly good reason here we chose to go c

7、lock-wise.11If you imagine a walk along the non-closed Burges circuit which you keep continuing round and round, it becomes obvious how a screw dislocation got its name. q It also should be clear by now how Burgers circuits are done. 12螺型位错螺型位错 ABCDABCD为为滑移面滑移面, ,EFEF以右为已滑移区以右为已滑移区, ,EFEF以左为未滑移区以左为未

8、滑移区, ,它们分界的地方就是它们分界的地方就是位错线位错线. .这种位错是由上下两部这种位错是由上下两部分晶体相对旋转而形成的分晶体相对旋转而形成的, ,所以叫所以叫螺型位错螺型位错. .螺型位错螺型位错也是也是一个半径大约为一个半径大约为3-43-4个原子间距的管道个原子间距的管道. .13 图图2-42-4表明了表明了BCBC、EFEF间间, ,上、下两个原子面上的原子相互错动上、下两个原子面上的原子相互错动的情况的情况. .EFEF与与BCBC之间即之间即位错线区位错线区, ,OOOO可以看作是可以看作是位错中心位错中心. .螺螺型位错形成以后型位错形成以后, ,所有原来所有原来与位错

9、线相垂直的晶面与位错线相垂直的晶面, ,都将由都将由平平面面变成以位错线为中心轴的变成以位错线为中心轴的螺旋面螺旋面, ,如图如图2-52-5所示所示. .1415螺型位错的几何特征螺型位错的几何特征(1)(1)螺位错线与其沿路矢量螺位错线与其沿路矢量b b平行平行, ,故故纯螺位错只能是直线纯螺位错只能是直线; ;(2)(2)包含有螺位错线的面必然包含滑移矢量包含有螺位错线的面必然包含滑移矢量b.b.因此因此, ,对于连续介对于连续介质质, ,螺位错可以有无穷多个滑移面螺位错可以有无穷多个滑移面. .但是但是, ,在晶体中滑移面只在晶体中滑移面只能在晶体的密排面上进行能在晶体的密排面上进行,

10、 ,故晶体中的故晶体中的螺位错只有有限个滑螺位错只有有限个滑移面移面; ;(3)(3)根据螺蜷面的不同根据螺蜷面的不同, ,螺位错可分右和左两种螺位错可分右和左两种, ,左螺和右螺左螺和右螺不不会因为晶体位置的颠倒而改变会因为晶体位置的颠倒而改变; ;(4)(4)当螺位错滑出晶体时当螺位错滑出晶体时, ,只只在不平行于位错线的晶体表面出现在不平行于位错线的晶体表面出现滑移台阶滑移台阶; ;(5)(5)螺位错没有多余半原子面螺位错没有多余半原子面, ,它周围它周围只引起切应变而无体应变只引起切应变而无体应变. .16 汽相或溶液中生长出的晶体表面台阶汽相或溶液中生长出的晶体表面台阶( (即螺位错

11、即螺位错):):如如果有一条螺位错线在晶体表面露头果有一条螺位错线在晶体表面露头, ,在露头处的晶面在露头处的晶面上必然形成上必然形成一个台阶一个台阶, ,这个台阶不会因复盖了一层原这个台阶不会因复盖了一层原子而消失子而消失, ,这样这样, ,螺位错露头处就为晶体生长提供了有螺位错露头处就为晶体生长提供了有利条件利条件, ,使之能在过饱和度不高使之能在过饱和度不高( (只有只有1%,1%,根据理论计根据理论计算应高达算应高达50%)50%)的蒸汽压下或溶液中连续不断地生长的蒸汽压下或溶液中连续不断地生长. .1718We already know enough by now, to deduc

12、e some elementary properties of dislocations which must be generally valid1. A dislocation is one-dimensional defect because the lattice is only disturbed along the dislocation line (apart from small elastic deformations which we do not count as defects farther away from the core). The dislocation l

13、ine thus can be described at any point by a line vector t(x,y,z). 2. In the dislocation core the bonds between atoms are not in an equilibrium configuration, i.e. at their minimum enthalpy value ,H; they are heavily distorted. The dislocation thus must possess energy (per unit of length) and entropy

14、,S.193. Dislocations move under the influence of external forces which cause internal stress in a crystal. The area swept by the movement defines a plane, the glide plane, which always (by definition) contains the dislocation line vector. 4. The movement of a dislocation moves the whole crystal on o

15、ne side of the glide plane relative to the other side. 5. (Edge) dislocations could (in principle) be generated by the agglomeration of point defects: self-interstitial on the extra half-plane, or vacancies on the missing half-plane.20Now we will turn to a more formal description of dislocations tha

16、t will include all possible cases, not just the extreme cases of pure edge or screw dislocations.21Mixed dislocation除除刃型位错刃型位错和和螺型位错螺型位错之外之外, ,还有一种形式更为普遍还有一种形式更为普遍的位错的位错, ,其滑移矢量既不平行于也不垂直于位错线其滑移矢量既不平行于也不垂直于位错线, ,而与位错线相交成任意角度而与位错线相交成任意角度, ,这种位错即为这种位错即为混合位错混合位错. .2223Volterra Construction and Consequen

17、cesWe now generalize the present view of dislocations as follows:q 1. Dislocation lines may be arbitrarily curved - never mind that we cannot, at the present, easily imagine the atomic picture to that.q 2. All lattice vectors can be Burgers vectors, and as we will see later, even vectors that are no

18、t lattice vectors are possible. A general definition that encloses all cases is needed.24 In 1907,Volterra, coming from the mechanics of the continuum (even crystals havent been discovered yet), had defined all possible basic deformation cases of a continuum (including crystals) and in those element

19、ary deformation cases the basic definition for dislocations was already contained!q The following shows Volterra basic deformation modes - three can be seen to produce edge dislocations in crystals, one generates a screw dislocation.25Volterras TubesHow can we obtain an arbitrary deformation of an a

20、rbitrary body by just repeating and combining some basic deformation procedures? The illustrations shows Volterras answer to this question: Take a cylinder of a material, cut it along some wall, shift the surfaces of the cut in all ways that - after welding the walls together again (including taking

21、 out or adding material) - will lead to different deformation states. qAs Volterra showed, there is a limited and rather small number of possible independent cuts + shifts. All other cuts plus some deformation can always be expressed as a linear superposition of the elementary cuts.26qHere are the e

22、lementary cuts. The first one just shows the cut, the next three ones correspond to dislocations - i.e. a real dislocation produces exactly the strain field generated by the cut and shift procedure. 27qThe last three cuts corresponds to special defects called disclinations旋转位错，向错旋转位错，向错 that are mor

23、e elementary than dislocations, but are not observed in real crystals (except, maybe, in grain boundaries and liquid crystal). They do however, appear in two-dimensional lattices, e.g. in the flux-line lattice of a superconductor.28How to define dislocations in a very general way by Volterra knife V

24、olterras insight gives us the tool to define dislocations in a very general way. We use the Volterra knife which has the property that you can make any conceivable cut into a crystal with ease (in your mind). So lets produce dislocations with the Volterra knife:29Make a cut, any cut, into the crysta

25、l using the Volterra knife.The cut does not have to be on a flat plane (the picture shows a flat cut just because it is easier to draw). The cut is by necessity completely contained within a closed line, the red cut line (most of it on the outside of the crystal). That part of the cut line that is i

26、nside the crystal will define the line vector t of the dislocation to be formed.302. Move the two parts of the crystal separated by the cut relative to each other by a translation vector of the lattice; allowing elastic deformation of the lattice in the general area of the dislocation line. q The tr

27、anslation vector chosen will be the Burgers vector b of the dislocation to be formed. The sign will depend on the convention used. Shown are movements leading to an edge dislocations (left) and a screw dislocation (right).31q It will always be necessary for obvious reasons if your chosen translation

28、 vector has a component perpendicular to the plane of the cut. q Shown is the case where you have to fill in material - always preserving the structure of the crystal that was cut, of course.q Left: After cut and movement. Right: After filling up the gap with crystal material3. Fill in material or t

29、ake some out, if necessary.32q Since the displacement vector was a translation vector of the lattice, the surfaces will fit together perfectly everywhere - except in the region around the dislocation line defined as by the cut line. A one-dimensional defect was produced, defined by the cut line (= l

30、ine vector t of the dislocation) and the displacement vector which we call Burgers vector b. q It is rather obvious (but not yet proven) that the Burgers vector defined in this way is identical to the one defined before. This will become totally clear in the following paragraphs.4. Restore the cryst

31、al by welding together the surfaces of the cut. 33From the Voltaterra construction of a dislocation, we can not only obtain the simple edge and screw dislocation that we already know, but any dislocation. Moreover, from the Volterra construction we can immediately deduce a new list with more propert

32、ies of dislocations:More properties of dislocations341. The Burgers vector for a given dislocation is always the same, i.e. it does not change with coordinates, because there is only one displacement for every cut. On the other hand, the line vector may be different at every point because we can mak

33、e the cut as complicated as we like.2. Edge- and screw dislocations (with an angle of 90 or 0, resp., between the Burgers- and the line vector) are just special cases of the general case of a mixed dislocation with has an arbitrary angle between b and t that may even change along the dislocation lin

34、e. The illustration shows the case of a curved dislocation that changes from a pure edge dislcation to a pure screw dislocation.35 We are looking at the plane of the cut (sort of a semicircle centeredin the lower left corner). Blue circles denote atoms just below,red circles atoms just above the cut

35、. Up on the right the dislocation isa pure edge dislocation, on the lower left it is pure screw. In between it is mixed. In the next page this dislocation is shown moving in ananimated illustration.36The picture is animated the dislocation can be seen as it moves out of the crystal, thus reversing t

36、he cut-and-displace procedure that created it. 373. The Burgers vector must be independent from the precise way the Burgers circuit is done since the Volterra construction does not contain any specific rules for a circuit. This is easy to see, of course:Two arbitrary alternative Burgers circuits. Th

37、e colors serve to make it easierto keep track of the steps384. A dislocation cannot end in the interior of an otherwise perfect crystal (try to make a cut that ends internally with your Volterra knife), but only at a crystal surface an internal surface or interface (e.g. a grain boundary) a dislocat

38、ion knot on itself - forming a dislocation loop. 395. If you do not have to add matter or to take matter away (i.e. involve interstitial or vacancies), the Burgers vector b must be in the plane of the cut which has two consequences: The cut plane must be planar; it is defined by the line vector and

39、the Burgers vector. The cut plane is the glide plane of the dislocation; only in this plane can it move without the help of interstitials or vacancies. 40The glide plane is thus the plane spread out by the Burgers vector b and the line vector t.6. Plastic deformation is promoted by the movement of d

40、islocations in glide planes. This is easy to see: Extending your cut produces more deformation and this is identical to moving the dislocation! 7. The magnitude of b (= b) is a measure for the strength of the dislocation, or the amount of elastic deformation in the core of the dislocation. A not so

41、obvious, but very important consequence of the Voltaterra definition is 8. At a dislocation knot the sum of all Burgers vectors is zero, Sb = 0, provided all line vectors point into the knot or out of it. A dislocation knot is simply a point where three or more dislocations meet. A knot can be const

42、ructed with the Volterra knife as shown below.41Statement 8. can be proved in two ways: Doing Burgers circuits or using the Voltaterra construction twice. At the same time we prove the equivalence of obtaining b from a Burgers circuit or from a Voltaterra construction.q Lets look at a dislocation kn

43、ot formed by three arbitrary dislocations and do the Burgers circuit - always taking the direction of the Burgers circuit from a right hand rule42qLets look at a dislocation knot formed by three arbitrary dislocations and do the Burgers circuit - always taking the direction of the Burgers circuit fr

44、om a right hand ruleqSince the sum of the two individual circuits must give the same result as the single big circuit, it follows:b1 = b2 + b3 qOr, more generally, after reorienting all t -vectors so that they point into the knot: Si bi = 0 43Now lets look at the same situation in the Voltaterra con

45、struction:q We make a first cut with a Burgers vector b1 (the green one in the illustration below). q Now we make a second cut in the same plane that extends partially beyond the first one with Burgers vector b2 (the red line). We have three different kinds of boundary lines: red and green where the

46、 cut lines are distinguishable, and black where they are on top of each other. And we have also produced a dislocation knot!44qObviously the displacement vector for the black line, which is the Burgers vector of that dislocation, must be the sum of the two Burgers vectors defined by the two cuts: b

47、= b1 + b2. So we get the same result, because our line vectors all had the same flow direction (which, in this case, is actually tied to which part of the crystal we move and which one we keep at rest).45If we produce a dislocation knot by two cuts that are not coplanar but keep the Burgers vector o

48、n the cut plane, we produce a knot between dislocations that do not have the same glide plane. As an immediate consequence we realize that this knot might be immobile - it cannot move. q A simple example is shown below (consider that the Burgers vector of the red dislocation may have a glide plane d

49、ifferent from the two cut planes because it is given by the (vector) sum of the two original Burgers vectors!).46We can now draw some conclusion about how dislocations must behave in circumstances not so easy to see directly:q Lets look at the glide plane of a dislocation loop. We can easily produce

50、 a loop with the Volterra knife by keeping the cut totally inside the crystal (with a real knife that could not be done). In the example the dislocation is an edge dislocationq The glide plane; always defined by Burges and line vector, becomes a glide cylinder! The dislocation loop can move up or do

51、wn on it, but no lateral movement is possible.47Sign of Burgers and Line VectorsLets look at a dislocation loop in cross-section. After the cut along the red line, the lower half was moved to the right by b. Two edge dislocation are visible if we look at a cross-section taken through the middle of t

52、he loop. A Burgers circuit now would give Burgers vectors of different signs - or does it?48We can ask the same question in a different way: From the Volterra construction we know that the Burgers vector - including the sign - must be the same everywhere. But the dislocations shown in the cross sect

53、ion look reversed - we would certainly assign different signs just looking at the picture. How is this contradiction to be solved?49Solution to Exercise The problem is solved easily by doing one simple thing: Look at the dislocation loop from aboveqAfter assigning a direction of t, it is defined for

54、 the whole loop. At the places where we took the cross-section, it is actually the sign of t that is reversed! The Burgers vector thus must be the other way around if it is to be constant for the local t. It is important to realize that we only can be unambiguous if we know that we are looking at on

55、e and the same 同一个同一个dislocation. The cross-section by itself does not tell us that fact; it just as well could show two unconnected single dislocations. In this case we would assign Burgers vectors with different signs because we automatically would take the line direction to be the same.50二、二、Burg

56、ers vector b(一一)确定柏氏矢量确定柏氏矢量b b的方法的方法(二二)柏氏矢量柏氏矢量b b的物理意义的物理意义(三三)柏氏矢量柏氏矢量b b的守恒性的守恒性(四四)柏氏矢量柏氏矢量b b的表示方法的表示方法51二、二、How to seek out Burgers vector bNow we add a new property. The fundamental quantity defining an arbitrary dislocation is its Burgers vector b. Its atomistic definition follows from a B

57、urgers circuit around the dislocation in the real crystal which is illustrated below52qLeft picture: Make a closed circuit that encloses the dislocation from lattice point to lattice point (later from atom to atom). You obtain a closed chain of the base vectors which define the lattice. q Right pict

58、ure: Make exactly the same chain of base vectors in a perfect reference lattice. It will not close. q The special vector needed for closing the circuit in the reference crystal is by definition the Burgers vector b.53But beware! As always with conventions, you may pick the sign of theBurgers vector

59、at will. q In the version given here (which is the usual definition), the closed circuit is around the dislocation, the Burgers vector then appears in the reference crystal. q You could, of course, use a closed circuit in the reference crystal and define the Burgers vector around the dislocation. Yo

60、u also have to define if you go clock-wise or counter clock-wise around your circle. You will always get the same vector, but the sign will be different! And the sign is very important for calculations! So whatever you do, stay consistent!.54(一一)确定柏氏矢量确定柏氏矢量b b的方法的方法1.规定位错线的正向规定位错线的正向(如出纸面为正如出纸面为正);