UNIVERSITY OF LJUBLJANA

FACULTY OF MATHEMATICS AND PHYSICS

SEMINAR II

Dislocations

Rudolf Šuligoj

Mentor: dr. Prelovšek

Ljubljana, 2006/2007

Abstract

Crystals are not perfect. They include impurities and different defects. Defects of crystal

lattice have an important influence on physical properties of metals. On the basis of their

extension we devide defects in onedimensional, twodimensional and threedimensional

defects. Their presence in metals is inevitable.

Dislocations are twodimensional defects. Their existance was confirmed with development of

electron microscopy around 1950. In this seminar I described different dislocations, force on

dislocation, velocity of dislocations, detection and observation of dislocations.

INDEX INTRODUCTION .........................................................................................................3

ELASTIC THEORY OF DISLOCATIONS..................................................................4

Elementary knowledge..............................................................................................4

Screw dislocation ......................................................................................................5

Force on a dislocation ...............................................................................................6

Motion of dislocations. Glide....................................................................................7

Relation between macroscopic deformation and dislocation glide........................8 Formation of dislocation with Frank-Read source....................................................9

Other dislocations....................................................................................................10

CRYSTAL DEFECTS DETECTION AND OBSERVATION...................................12

Research with ultrasound ........................................................................................12

Electron microscopy................................................................................................12

DENSITY OF DISLOCATIONS IN CRYSTALS AND THEIR STRENGTH .........14

CONCLUSION............................................................................................................15

LITERATURE.............................................................................................................16

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INTRODUCTION Metals have been interesting for people since they were discovered. At first people used them for tools, weapons and later for different constructions. Researchers also observed different processes in metals during their exposition to different cargo. During the 2nd World war they also discovered the importance of behaviour of metals at various temperatures. In the 19th century some dark lines on metals were observed. They appeared during the increase of the stress and stayed there also after stress was terminated. For this reason lines appered, when plastic deformation of metals was reached.

Figure 1: Edge dislocations in glide planes. Thickness of specimen is 200nm. [1] Scientists had various theories for the explanation of this phenomenon. It took quite some time to predict the right answer. In 1934, Polany, Orowan and Taylor have independently supposed imperfection build of crystals. Crystals contain line defects called dislocations. During decades, this idea has become more and more important in physics of metals. The real existence of dislocations was verified and confirmed with the development of electron microscopy around 1950. The electron mycroscopy enables separated dark shadows in several slide planes possible to see (Figure 1). [2] In Table 1 there are some data of critical shear stresses. Theoretical shear stress is the necessary stress to move all the atoms that belong to one crystal plain at the same time. Differences between theoretical and measured stresses are evidential. They are also explained with existence of dislocations. It is easier to move one dislocation than to move whole layer-plain at once. [3]

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Table 1: Calculated and measured shear stresses for two different metals. [3]

metal krσ [ 2m

N ] – calculated krσ [ 2mN ] – measured

aluminium 3,9 910⋅ 8 510⋅copper 6,6 910⋅ 5 510⋅

The effect of dislocations on reduction of stress can be explained by analogy with a carpet. First example is to take a rug at the end and drag it. Second example is to make along one edge a wrinkle and push it along the length of carpet. The necessary force for the first example is greater than one for the second example. Movement of caterpillar is also quite similar. [2] An animation giving a schematic view of the movement of an edge dislocation through a crystal is presented below (animation 1).

Animation 1: Schematic view of the movement of dislocation through a crystal. [4]

ELASTIC THEORY OF DISLOCATIONS

Elementary knowledge The matter can be elastically deformed, when it is exposed to external forces. The matter can be plasticaly deformed with the increase of the external force. Consequently, the shape of that matter is permanently changed. Dislocations are reason for the plastic deformations. They are defined with Burgers vector b.

Figure 2: left - edge dislocation; right – screw dislocation. [3]

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Burgers vector is defined by making a circle around dislocation in a crystal lattice. When we want to conclude the circle, we do not come back to the first atom. Thus is the b a vector directed from the first point to the last one (Figure 3). The whole circuit is called Burgers circuit.

Figure 3: way to Burgers vector b [5]

A screw dislocation is a dislocation where a direction of b is parallel to the dislocation line (Figure 2) and in the case where b is perpendicular to the dislocation line the dislocation is called edge dislocations (Figure 2). The elastic energy, stored in a volume V , is equal to the work done by the force per unit of volume, that is, ∑∫∫∫

ijijijduσ .

Expressing ijσ as a function of displacements1, the equation can be written as

∑ ∑ =∂∂

∂++

∂∂

j j ji

j

j

i

xxu

xu 0)(

2

2

2

µλµ or, in vectorial notation 0)()(2 =⋅∇∇++∇ uu vv µλµ .

If the medium is not immobile, one must take into account the kinetic energy 2

2

tu

∂∂ v

ρ ,

where ρ is the density, and one finds

2

22 )()(

tuuu

∂∂

=⋅∇∇++∇v

vv ρµλµ . This is an equation of elasticity and satisfies an

isotropis elastic medium containing a dislocation in motion. [5]

Screw dislocation Displacements and are equal for the screw dislocation (Figure 4). u changes

by

1u 2u 0 v

bv

when θ changes by π2 : πθ

2buv

v = . The elastic stresses corresponds to this

displacement. We use cylindrical coordinates represented in Figure 4. 1 In case of an isotropic body is δλδµσ jiijij u += 2 . λ and µ are Lame coefficients. µ is also

called the shear modulus and often denoted by G .

5

Figure 4: left - screw dislocation; right – torque. [2]

All stresses are zero accept

rbπµσσ θθ 233 == , where µ is the shear modulus.

These are two shear stresses. 3θσ lies in the radial planes, parallel to , 3Ox θσ 3 lies in the horizontal planes, perpendicular to the radius. Screw dislocation does not produce a volume dilatation in the material. They give rise exclusively to the shear stresses of cilindrical symmetry. Screw dislocations, created by displacement exclusively in the direction, are not stable, because the stresses generate a non-zero torque in the cylinder (Figure 4). Thus the total stress is

3x

)21(2 2

12

03 rr

rr

b+

−=πµσ θ .

If r is small compared to , the effect of the second term is negligible. Consequently, the stress-field of a screw dislocation not too close to the surface of the

body is described in practice by the relation

1r

rbπµ2

. [5], [6]

Force on a dislocation If we want to move a dislocation L in a direction from point to point N M , a force F per unit length, directed from towards N M , is needed. The necessary work for this is (Figure 6). FS

Figure 6: force on a dislocation [5]

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If we apply stress `σ , parallel to bv

, on the surface of the crystal, displacement is appeared (Figure 6). Work during the displacement b is equal bS`σ . The shear stress displaces L to left. Taking into account both work, we get for force per unit length

`σbF = . In general, the force F per unit length acting on a dislocation loop L is defined by change of work W with the position of L . Moreover, there are two different stresses which act on . The self-stresses S Sσ appear, when the dislocation L is created. The applied stresses `Sσ are all other stresses, which are a consequence of other imperfections in the crystal or of applied stress on the surface of the crystal. The work of stresses due to the creation of a dislocation loop L is

∫∫∫∫ +=S SS S dSbdSbW `

21 σσ

vv.

The factor 21 in upper equation is due to the proportionality of stresses Sσ to the

relative displacement xv . Stresses `Sσ are independent of xv . The work done by self-stresses is

∫∫ ∫∫∫ =S S S

b

S dSbxdbxdS σσ

vvvv

21

0

= . 1W

The self-stresses are zero on the outside surface of the crystal. They produce no work on a surface when dislocation loop is created. Consequently, the work is equal to the elastic energy.

1W

If varying forces are applied on a surface of elastic homogenous body, containing dislocations, the elastic energy of the body is modified by the quantity equal to the work of the applied forces. [5], [6] In Animation 2 one can see the reduction of dislocation speed, because of the interaction of stresses. Dislocations are in two parallel glide planes and affect one on another.

Animation 2: Dislocations slow down, because of the interaction of stresses. [7]

Motion of dislocations. Glide The speed with which a dislocation can move in a perfect crystal is still poorly known in most materials. A real crystal contains impurities or other dislocations that offer great resistance. Hence speed of dislocations is smaller than speed of sound.

v

7

Direct observation in the electron microscope shows that dislocations can glide at any speed. It depends on the stress applied on them (Figure 7).

σln

clnvln

Figure 7: interdependence between velocity and applied stress [5]

A dislocation moving under a constant applied external stress cannot exceed the speed of sound . The speed of sound plays in elasticity the same role as the speed of light in electromagnetism. [5]

c

Relation between macroscopic deformation and dislocation glide Every dislocation contributes a little displacement iδ to total displacement of upper part of crystal (Figure8). In comparison of this with the dimensions of crystal ( L and

) is very small. For a glide, which is a consequence of a dislocation between and , displacement is

h b0=ix Lxi =

iδ =Lbxi

v

.

Figure 8: Relation between macroscopic deformation, ε , and dilsocation glide. [3]

We need to sum up all little displacements, to get total displacement∆ :

8

∑=∆ iδ = ∑=

N

iix

Lb

1)(v

.

N is number of dislocations, glided as a consequence of shear stress. Macroscopic shear deformation is

h∆

=ε = ∑=

N

iix

hLb

1)(

v

= xNhL

b ⋅⋅1 .

Let h and L be one unit of length, then follows ε = xb ⋅⋅ ρ , where ρ is equal to number of dislocations on unit of plane. Thus the relation between macroscopic deformation and velocity is

vbdtd

⋅⋅= ρε ,

where v is an average speed of dislocations. [3] One kind of dislocation motion is shown in animation below. A boundary unenables movement of dislocation. Hence dislocations pile up on another slip plane.

Animation 3: Pile up of dislocations. [7]

Formation of dislocation with Frank-Read source One of the main mechanisms for dislocation multiplication under stress is the Frank-Read mill or Frank-Read source. The operation of a Frank-Read source can be observed on a dislocation segment pinned at its ends. (Figure 9) [8]

a bd

c

Figure 9: Frank-Read source. Left figure from [2]. Right figure is a silicic crystal. It is made with infrared light (permeable for silicic) and dislocations are decorated with copper. [3]

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Every dislocation segment in a crystal tends to minimum (minimalize) length between two points. (figure 9) A dislocation stays bent only when it is exposed to external

shear stress, oτ =RbG . Dislocation bends in a bend with radius R . At the beginning

radius is equal to infinity ( ∞=R ), then it reduces to the minimum size R =2l and

increases again. Maximum external stress, stress for activity of Frank-Read source,

for radius 2l , that is needed, is:

lGb

FR2

=τ .

The amount of FRτ in a crystal of aluminium is , when 26102,9 −⋅ Nm ml µ8,0= , and . [2] nmb 28,0= PaG 10106,2 ⋅=

The whole action is shown in figure 9. A dislocation between PP ′ first bends. Radius increases and after some time part X and Y meet. Thus a dislocation loop is created. It spreads and goes away from the source PP ′ where a new bend is already seen. [5] Formation of dislocations with Frank-Read source is shown in animation 4.

Animation 4: Frank-Read source [7]

Other dislocations A partial dislocation is a phenomenon when Burgers vector is shorter from vector of

crystal lattice, tv

. Hence mtbvv

= where is small number. m

Sitting dislocation cannot glide, because their Burgers vector does not lie in glide plane.

Figure 10: Frank partial dislocation. [2].

A lack of atoms in separate atom layer is needed for formation of a Frank partial dislocation (Figure 10). Crystal lattice eventually collapses with an increase of lack. The only possible movement of this dislocation is climbing. For this reason the Frank partial dislocation is also sitting dislocation. Frank partial dislocation becomes larger with difusion of vacancies. [3]

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Climbing is a phenomenon, when dislocation moves to a different glide plain, where their movement is possible. Climbing of edge dislocation is shown in figure 11. Vacancies travel up, from to d , with diffusion. Consequently, a dislocation appears in other glide plain, where gliding is possible.

a

edge dislocation

Figure 11: Climbing of edge dislocation. [2]

Atomic planes are, in some crystals, put together in a certain sequence

(figure 12). Atoms C must go over wrinkle (atom ...ABCABC B ), if they want to glide. However, it is easier to go along vectors 2b

v and 3b

v between wrinkles. This way

is . CAC −− 2bv

and 3bv

are two partial dislocations originated from dislocation 1bv

. Partial dislocation like this one is called Schockley partial dislocation.

Figure 12: Left-Burgers vector 1b

vof dislocation and two partial dislocations 2b

v

and 3bv

. [2] Right-Atom planes [9]

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CRYSTAL DEFECTS DETECTION AND OBSERVATION The search for dislocations and other defects is mostly performed on patterns that must not be destructed. Today the best copies of defined surface of body are gained with electron microscopy. However, this kind of microscopy is limited on a laboratory. We often need to check a pattern greater dimensiones, which is not possible to take in a room ( for example a bridge, plane and other constructions). In these cases, we investigate material with other methods: with ultrasound, making a replica or investigation with penetrant. Last two methods are used for larger defects than dislocations. The combination of uper methods is needed to maintain all significant structures.

Research with ultrasound This research enables to detect a crystal defect several meters in depth of material. Uncover of defect base on interaction of ultrasound waves with crystal defects like dislocation. In this case, the repulsion form the acoustical boundary is important. This method is not good for crystal with large grains, when wave length is comparable with the grain size. The acoustical impedance, cZ ρ= , is needed to estimate the quantity of rebound ultrasound energy from surface and the quantity of ultrasound energy that goes throuh surface. There are two types of ultrasound methods. In case of transmission ultrasound method, two ultrasound heads are needed. One head is transmitter and the second one is receiver. When a receiver detects lower intensity of waves from the expected one, we know that waves came across defect on their way, consequently the part of waves was rebounded. The second method is rebound method. In this we use only one ultrasound head. From the shape of returned signal conclusions like largeness and shape of defect can be made. For example, dislocations reduce the distance between the signals. For gas bubble are characteristic widened signals.

Electron microscopy The use of transmission electron microscope (TEM) make it possible to see objects to the order of a few angstrom (10-10 m). For example, you can study small details in the cell or different materials down to near atomic levels. The possibility for high magnifications has made the TEM a valuable tool in materials research. (Figure 12) A "light source" at the top of the microscope emits the electrons that travel through vacuum in the column of the microscope. Electromagnetic lenses focus the electrons into a very thin beam. The electron beam then travels through the specimen you want to study. Depending on the density of the material present, some of the electrons are scattered and disappear from the beam. At the bottom of the microscope the unscattered electrons hit a fluorescent screen, which gives rise to a "shadow image" of the specimen with its different parts displayed in varied darkness according to their density. The image can be studied directly by the operator or photographed with a camera. [10] Transmission electron microscopy reveals the interior of the specimen, because the electron beam goes through the sample. It displays structure: the size, shape, and the distribution of the phases that make up the material. It also gives the composition: the distribution of the elements, including segregations if present. It shows the crystallography: the crystal structure of the phases and the character of the crystal defects. [11]

12

Figure 12: scheme of electron microscope [11]

There are two imaging modes. In case of a bright field electron microscopy imaging mode images are based on transmitted beam. Thus dislocations appear dark (figure 1). On the other hand, dark field imaging mode uses specific Bragg diffracted electrons to image the region from which they originate. The diffraction conditions surrounding a dislocation are different, because the atomic planes near dislocation are strained. [1] Figure 13 is made with STM (Scanning Tunneling Microscope) and shows Burgers circuit and edge dislocation, where Burgers vector is perpendicular to the dislocation line.

Figure 13: Burgers circuit and edge dislocation. Figure is made with SEM. [12]

Screw dislocation looks like in an STM image (Figure 14): A step begins at the dislocation core.

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Figure 14: Screw dislocations. Figure is made with SEM. [12]

DENSITY OF DISLOCATIONS IN CRYSTALS AND THEIR STRENGTH In perfect crystals, density of dislocations is between 102 and 103 on . Density is higher in normal crystals, between 10

2cm7 and 108 on . Unit for density is ,

because it expresses ratio between the length of dislocation line and unit of volume. [2]

2cm 2cm

In figure 15 there is complex dislocation tagle. A crack emits dislocation on thermal cyclings.

Figure 15: Complex dislocation tagle. A crack emits dislocations on thermal cyclings. [1]

The strength of metals is related to the density of dislocations. When dislocations start to move or formate, plasticity is reached. Dislocations also do not go back to the previous position, but stay there or move on if applied stress is not terminated. Plasticity is the macroscopic outcome from the combination of many dislocation elementary properties at the micro scale. Boundary between elasticity and plasticity is different for each metal. The stress necessary to reach plasticity depends on the density of defects in metals. Metals with higher density need also higher applied stress to move or formate a dislocation, because all defects including dislocations disable movement and also formation of other defects. Consequently, the strength of perfect crystals is lower than the strength of real crystals, because for one dislocation it is easier to move or formate in metals with more space or lower density. The duty of blacksmith is to create as high density as possible by hitting the metal with hammer. Result of this is a metal with higher strength.

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Dislocations are not the only defects in metals. Their crystal lattice contains many other defects which are divided by their largeness in four groups: • Zero-dimensional defects also point defects (their largeness is ). m1010−

• One-dimensional defects also dislocations (their largeness is betweeen and ).

m1010−

m510−

• Two-dimensional defects (their largeness is betweeen and ). m810− m210−

• Three-dimensional defects (their largeness is betweeen and ). m710− m010The existence of defect in metals is evident in figure 16. It can be seen, that real

extension (ll∆ ) of certain material is larger than the calculated (

aa∆ ) on and that the

difference increases with the increase of temperature. The mentiond defects interact one with another. Diffusion [13] of vacancies enables certain dislocation to climb. This phenomenon becomes important in processes with higher temperature.

Figure 16: Dependence between extension of material and temperature. Real

extension,ll∆ , is higher than calculated,

aa∆ . [15]

temperature [ ] Co

aa∆

ll∆

310−⋅

∆

∆

aa

ll

CONCLUSION Dislocations are not to be neglected in metals. Especially, they are crusual, when a metal is exposed to external stresses. They are the reason why theoretical shear stress is differrent from the measured one. The force acting on a dislocation increases with applied stress. Every dislocation contributes a little displacement to total displacement of one part of crystal. In these days we should be aware of dislocations. Their presence in materials also needs to be investigated. Although todays life is very fast, we need to take enough time when we choose and develop different materials. In 2nd World war some sheeps sink because of different behaviour of the material at lower temperatures. The right choice of material is of the greatest significance in build of nuclear reactor. Damages caused by radiation are not wanted, because they cause changes in the material properties. In reactor there is a major causer of defects neutron radiation. Consequences are point defects and dislocations. [9]

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Defects are not wanted, but in some cases they are necessary. This is especially in cases, when porosity is needed. [14] In aeronautics, a crystal defect like dislocation becomes very important. We should be aware of dislocations before take off of plane. On ten kilometers hight it is already too late to discover them because they are crucial to the happy ending of the flight. LITERATURE

[1] mse.uta.edu/dislocations.ppt (05.01.2007)[2] J. Vojvodič Tuma, Mehanske lastnosti kovin, Fakulteta za gradbeništvo in geodezijo (2002) [3] V. Marinković, Fizikalna metalurgija II, Naravoslovnotehniška fakulteta (1999) [4] http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_5/illustr/ a5_1_1.html (05.01.2007)

[5] J. Friedel, Dislocations, Pergamon Press (1964) [6] I. Kovacs, L. Zsoldos, Dislocations and Plastic Deformation, Akademiai Kiado (1973) [7] http://www.gpm2.inpg.fr/axes/plast/MicroPlast/ddd/index.html(5.1.07)

[8] http://zig.onera.fr/DisGallery/ (05.01.2007)[9] S. Spaić, Fizikalna metalurgija, Univerzitetna tiskarna v Ljubljani

(2002) [10] http://nobelprize.org/educational_games/physics/microscopes/

tem/index.html (01.05.2006) [11] http://cmm.mrl.uiuc.edu/techniques/tem.htm (05.01.2007)[12] http://www.iap.tuwien.ac.at/www/surface/STM_Gallery/ dislocations.html (05.01.2007)[13] http://www.materials.ox.ac.uk/teaching/diffusion/ (01.05.2006) [14] www.s-sc.ce.edus.si/igor_lah/graslide6.ppt (01.05.2006)

[15] C. Kittel, Uvod u fiziku čvrstog stanja, Wiley (1970)

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