chapter 5b crystal imperfections dislocations

8/20/2019 Chapter 5b Crystal Imperfections Dislocations

1/96

DISLOCATIONSDISLOCATIONS

Edge dislocation Screw dislocation

Dislocations in crystals

Introduction to DislocationsD. Hull and D.J. Bacon

Pergaon Press! O"#ord $%&'()

Recommended website

*tt+,--www.t#.uni/iel.de-atwis-aat-de#0en-

Further reading

Theory of Dislocations J. P. Hirt* and J. Lot*e

1c2rawHill! New 3or/ $%&4')

Advanced reading (comprehensive)

1ATE5IALS SCIENCE1ATE5IALS SCIENCE

66EN2INEE5IN2EN2INEE5IN2

Anandh Subramaniam & Kantesh Balani

1aterials Science and Engineering $1SE)

Indian Institute o# Tec*nology! 7an+ur 89'9%4

Email anandh!iit"#ac#in$ %R& home#iit"#ac#in'anandh

AN INT5OD:CTO53 EBOO7 AN INT5OD:CTO53 EBOO7

art of

http''home#iit"#ac#in'anandh'E*boo"#htm

A Learner’s GuideA Learner’s Guide


2/96

As seen ;e#ore dislocations are %D $line) de#ects

T*e role o# dislocations goes far beyond

T*ey +lay an i+ortant role in a ?ariety o# de#oration +rocesses li/e cree+! #atigue and #racture

T*ey can +lay a =constructi?e role> in crystal growt*

T*ey can +ro?ide s*ort circuit +at*s #or di##usion $+i+e di##usion)

:nderstanding t*e i+ortance o# dislocations in aterial ;e*a?iour cannot ;e

o?erstated@ *ence it is ?ery i+ortant to t*oroug*ly understand t*e structure and ;e*a?iour o# dislocations


3/96

T*e i+ortance o# understanding dislocations and t*eir e##ect on aterial ;e*a?iour cannot ;e o?erstated.

T*oug* o#ten t*e i+ortance o# dislocations in t*e conte"t o# +lasticde#oration ;y sli+ is *ig*lig*ted its role in aterials science is #ar greater.$T*e ne"t slide s*ows soe o# t*ese roles).

In t*is conte"t it is i+ortant to note t*at e?en in crystalline aterials t*ere arealternate ec*aniss o# +lastic de#oration $as s*own in an u+coing slide)

@ Twinning also ;eing an i+ortant one. T*e i+ortant t*ing to ;e /e+t in ind is t*e role o# dislocations in wea"ening

crystals $ta/en u+ a#ter t*e a;o?e entioned slides).

Dislocations

Plastic de#oration

@ +eranent de#oration t*at reains w*en all e"ternal loading-constraints are reo?ed Sli+ @ is a tec*nical ter re#erring to +lastic de#oration caused ;y dislocations t*e =#irst ste+> o# t*e +rocess is t*e sall sur#ace ste+ w*ic* is created w*en a dislocation lea?es a crystal Twinning

@ +rocess ;y w*ic* one +art o# t*e crystal gets related to anot*er +art! ;y a syetry o+erator $usually a irror) w*ic* is not a syetry o+erator o# t*e crystal.


4/96

Consider a dislocation in in an in#inite crystal

Ta/e into account #inite crystal e##ects

Consider interaction o# dislocations wit* ot*er de#ects

Pat* to understanding t*e role o# Dislocations in aterial ;e*a?iour

+tress fields$ strain fields$ energy etc#

Free surfaces$ grain boundaries etc#

! ! ! ! ! ...d ,, yy ,y ,, ,y E σ σ τ ε ε

Interactions with other dislocations$ interstitials$ precipitates etc#

Static and dynaic e##ects and interactions s*ould ;e included

- Dynaic e##ects include, (Altered) +tress field of a moving dislocation

Interactions evolving in time

Collecti?e ;e*a?iour and e##ects o# e"ternal constrains

&ong range interactions . collective behaviour . e,ternal constraints--

Though these points are

written as a se/uence

many of these have to be

considered in parallel

0ote the above step by step method may often not be the most practical one and there are techni/ues which ta"e up collective behaviour directly


5/96

Slip

5ole o# Dislocations

FractureFatigue Creep

Diffusion

(Pipe)

Structural

2rain ;oundary(low angle)

Inco*erent Twin

Seico*erent Inter#aces

Disc o# ?acancies edge dislocation

Crystal Growth(Screw dislocation)

0ote +tructural dislocations can also play a role in deformation and "inetic processes

and more122

Cree+ec*aniss in

crystallineaterials

Dislocation cli;

acancy di##usion

Crosssli+

2rain ;oundary sliding


6/96

Sli+

$Dislocationotion)

Plastic Deformation in Crystalline aterials

Twinning P*ase Trans#oration Cree+ 1ec*aniss

2rain ;oundary sliding

acancy di##usion

Dislocation climb

Ot*er 1ec*aniss

0ote lastic deformation in amorphous materials occur by other mechanisms including flow (viscous fluid) and

shear banding

T*oug* +lasticity ;y sli+ is t*e ost i+ortant ec*anis o# +lastic de#oration! t*ere areot*er ec*aniss as well (plastic deformation here means permanent deformation in theabsence of e,ternal constraints),

2rain rotation


7/96

Fea/ening o# a crystal ;y t*e +resence o# dislocations

To cause +lastic de#oration ;y s*ear ( all of plastic deformation by slip re/uire shear stresses at the microscopic scale- ) one can ?isualiGe a +lane o# atos

sliding +ast anot*er $#ig ;elow) T*is reuires stresses o# t*e order o# 2Pa $calculation in t*e ne"t slide)

But ty+ically crystals yield at stresses 1Pa

⇒ T*is i+lies t*at =soet*ing> ust ;e wea/ening t*e drastically It was +ostulated in %&9sK and con#ired ;y TE1 o;ser?ations in %&9s! t*at

t*e agent res+onsi;le #or t*is wea/ening are dislocations

E?en i# one does a +ure unia"ial tension test wit* t*e tension a"is along t*e G a"is! e"ce+t#or t*e *oriGontal and t*e ?ertical +lanes all ot*er +lanes =#eel> s*ear stresses on t*e

As to *ow t*is atoic sli+ is connected to acrosco+ic +eranent s*a+e c*anges will ;e

considered later K By Taylor! Orowan and Polyani


8/96

Plastic de#oration o# a crystal ;y s*ear

S

*

e

a

r

i

n

g

s

t

r

e

s

s

$

)

τ

D i s + l a c e e n t

Sinusoidal relations*i+

5ealistic cur?e

τm

Let us consider t*e s*earing o# an entire +lane o# atos o?er one anot*er@ causing +lastic de#oration ;y s*ear

+tarting configuration Final configuration

Entire row of atoms sliding past another

The shear stressdisplacement curve

loo"s as shown in the

diagram on the right


9/96

=

b

,+inm

π τ τ

8As a #irst a++ro"iation t*e stressdis+laceent cur?e can ;e written as

a ,33 == γ τ At sall ?alues o# dis+laceent

Hoo/e>s law s*ould a++ly

=

b

,m

π τ τ

8⇒ Mor sall ?alues o# "-;

a

b3m

π τ

8=Hence t*e a"iu s*ear

stress at w*ic* sli+ s*ould occur

8m

3τ

π :I# ; a

8m

, ,3

a b

π τ

= ÷

8m

3τ

π :

T*e t*eoretical s*ear stress will ;e int*e range 9 GPa~


10/96

8m3

τ π :

T*e s*ear odulus o# etals is in t*e range 89 %9 2Pa

D!S"#CA$!#%S

Actual s*ear stress is 9. %9 Pa (e,perimentally determined)

I.e. $S*ear stress)t*eoretical %99 × $S*ear stress)e"+eriental

Dislocations severely weaken the crystal

F*is/ers o# etals (single crystal free of dislocations$ Radius 45− 6

m) can approach theoretical shear strengthsF*is/ers o# Sn can *a?e a yield strengt* in s*ear %9 −8 2 (457 times bul" +n)

T*e t*eoretical s*ear stress will ;e in

t*e range 9 GPa~


11/96

As we *a?e seen ;e#ore dislocations can +lay di?erse /inds o# role in aterialsstructure and its ;e*a?iour

Per*a+s t*e ost i+ortant o# t*ese is t*e wea/ening o# t*e crystal in t*e +resence o#dislocations

Mro a slide ;e#ore we /now t*e +at* to understanding t*e role o# dislocations inaterials in?ol?es t*eir interactions wit* ot*er dislocations and de#ects +resent in t*eaterial $and t*e e?olution o# t*e syste wit* tie-de#oration)8 This path will include the 9hardening: of the crystal$ i#e strengthening of the

wea"ened crystal

8 In this conte,t it will be noted that many dislocations will interact with each other

and there will be a strengthening effect

As late as %&9 t*e reason ;e*ind t*is wea/ening o# t*e crystal was not clear (to imagine that this was the post Relativity$ post ;uantum


12/96

T*e analogy usually gi?en to understand t*e role o# dislocations in wea/ening a crystal is t*eone o# =+ulling a car+et>.

I# one tries to +ull an entire car+et $a long and wide one)! ;y sliding it against t*e #loor! t*ee##ort reuired is large.

Howe?er! i# a =;u+> is ade in t*e car+et $as in t*e #igure in t*e #ollowing slide) and t*is ;u+ is o?ed

across t*e lengt* o# t*e car+et! t*en t*e car+et o?es #orward ;y a sall distance $as +ro?ided ;y t*e ;u+).

T*e #orce reuired to o?e t*e ;u+ will ;e considera;ly sall as co+ared to t*e #orcereuired to +ull t*e entire car+et.

By creating and o?ing a series o# ;u+s successi?ely t*e car+et can ;e o?ed #orward =;it ;y ;it>. (3raphic on ne,t slide)#


13/96

T * e = c a

r + e t . + u

l l i n g > a

n a l o g y


14/96

T*e olterra Dislocation

T*e continuu conce+t o# a dislocation $and ot*er de#ects) was +ro+osed ;y ito olterra in %&9.

His ideas and calculations ;ased on *is ideas +redate t*eir a++lication to crystals. Howe?er!continuu calculations ;ased on olterra>s idea are used e?en today to understand t*e ;e*a?iour o#

dislocations in crystals. Continuu calculations o# stress #ields! dis+laceent #ields etc. related to dislocations are #ound to ;e

?alid to wit*in a #ew atoic s+acing $i.e. t*e continuu descri+tion #ails only wit*in a;out atoicdiaeters-Burgers ?ector).

In t*is c*a+ter t*e stress #ields o# dislocations s*own are ;ased on elastic continuu t*eories.

Continuu i+lies t*at we are not =worried> a;out atos (Antonym → Discretum)# T*is is a =strange> as+ect, we *a?e used elasticity t*eory to calculate t*e stress #ields o# t*e ?ery $ost i+ortant) agent res+onsi;le #or

+lastic de#oration

=ontinuum description of a dislocation


15/96

Deformations of a hollow cylinder showing the formation of various defects (>olterra constructions)

erfect cylinder

+crew dislocation

Edge dislocations

D i s l o c a t i o n s D

i s c l i n a t i o

n s

Disclinations


16/96

I# one loo/s at a sa+le o# Aluinu under a TE1! one usually #inds cur?ed dislocationlines ⇒ :sually dislocations *a?e a i"ed c*aracter and Edge and +crew dislocations aret*e ideal e,tremes.

Not only t*is! t*e c*aracter o# t*e dislocation $i.e t*e +ercentage o# screw and +ercentageo# edge c*aracter) will c*ange #ro +osition to +osition along t*e dislocation line.

Howe?er! under s+ecial circustances Pure Edge! Pure Screw or a 1i"ed Dislocationwit* a #i"ed +ercentage o# edge c*aracter can #or.(e#g# in 3e+i epita,ial films on +i substrate 65° misfit dislocations form* i#e# the dislocation lines are straight with

the angle between b and t being 65° )

8 more about these aspects will be considered later

T*e edge dislocation is easier to ?isualiGe and *ence any o# t*e conce+ts regardingdislocations will ;e illustrated using t*e e"a+le o# t*e +ure edge dislocation.

DG

D!S"#CA$!#%S

!'D SC()

Dislocation linesTE< micrograph

showing dislocation lines


17/96

Slipped

part

of the

crystal

Unslipped

part

of the

crystal

Dislocation can ;e consideredas a ;oundary ;etween t*esli++ed and t*e unsli++ed

+arts o# t*e crystal lying o?era sli+ +lane

- this is ?ust a way of visuali@ation and often the slipped and unslipped regions may not be distinguished


18/96

A dislocation *as associated wit* it two ?ectors,

linendislocatiot*ealongnt ?ectorunit tangeAt →

?ector BurgersT*e ; →

T*e Burgers ?ector is li/e t*e =SO:L o# a dislocation>. It =can ;e de#ined> e?en i#

t*ere is no dislocation in t*e crystal $it is t*e s*ortest lattice translation ?ector #or a

#ull-+er#ect dislocation)! it de#ines e?ery as+ect o# t*e dislocation $its stress #ields! energy!

etc.) and e"+resses itsel# e?en in t*e =deat*> o# t*e dislocation $i.e. w*en t*e dislocation

lea?es t*e crystal and creates a ste+ o# *eig*t =;>).

Burgers ?ector is t*e s*ortest translation ?ector $#or a #ull-+er#ect dislocation) and can ;e

deterined ;y t*e Burgers circuit $coing u+).

Hence! Burgers ?ector is an in?ariant #or a crystal! w*ile t*e line ?ector is not. I#

one loo/s at a transission electron icrogra+* s*owing a dislocation line inAluiniu! it will not ;e straig*t− i.e. t*e line ?ector is not #i"ed $usually).

Dislocation linesTE< micrograph

showing dislocation lines


19/96

Burgers *ector dge dislocation

Deterination o# Burgers ?ector in a dislocated crystal using 5ig*t Hand Minis* to Start 5ule $5HMS)

In a +er#ect crystal a/e a circuit $e.g. as in t*e #igure s*own, ' atoic ste+s to rig*t! R

down! ' le#t 6 R u+). T*e circuit is Right anded . Do t*e sae in t*e sae in t*e dislocated crystal. T*e =issing lin/> $using soe con?entionli/e 5HMS) is t*e Burgers ?ector.

5HMS,

5ig*t Hand Minis* to Start con?ention Note: the circuit is drawn away from the dislocation line


20/96

Soe odels o# Edge Dislocation

ideo, Edge Dislocation

1odel using agnetic ;alls

(not that accurate2)


21/96

T*e edge dislocation is NOT t*e =e"tra *al#+lane>! it is neither t*e =issing *al#+lane>@ it is t*e line ;etween t*e =e"tra> and t*e =issing> *al#+lanes.

T*e regions #ar away #ro t*e dislocation line are +er#ect @ all t*e =de#oration> isconcentrated around t*e dislocation line.

Howe?er! t*e stress #ield o# t*e dislocation is a =long range> #ield.

:nderstanding t*e Edge dislocation

Note, T*e Burgers ?ector *as ;een drawn away #or t*e

dislocation line $soeties it ay ;e drawn closeto dislocation line #or con?enience).

T*e dislocation line is ;etween t*e =issing> and=e"tra> *al#+lane.


22/96

dge dislocation

t

;

Dislocation line

O#ten to ?isualiGe t*e edge dislocation! only t*e e"tra =*al#>+lane and sli+ +lane ares*own. T*e reaining crystal is *idden away.

T*e intersection o# t*e e"tra *al#+lane and sli+ +lane can ;e ?isualiGed as t*e dislocation

line $one o# t*e two +ossi;le directions is re+resents t*e line ?ector s*own in ;luecolour).


23/96

Dislocation can ;e considered as t*e ;oundary ;etween t*e sli++ed and t*eunsli++ed +arts o# t*e crystal lying o?er a sli+ +lane.

T*e intersection o# t*e e"tra *al#+lane o# atos wit* t*e sli+ +lanede#ines t*e dislocation line (for an edge dislocation)#

Direction and agnitude o# sli+ is c*aracteriGed ;y t*e Burgers ?ector o# t*e dislocation(A dislocation is born with a Burgers vector and e,presses it even in its death2)#

T*e Burgers ?ector can ;e deterined ;y t*e Burgers Circuit. 5ig*t *and screw $#inis* to start) con?ention is usually used #or deterining t*e

direction o# t*e Burgers ?ector. As t*e +eriodic #orce #ield o# a crystal reuires t*at atos ust o?e

#ro one euili;riu +osition to anot*er ⇒ b ust connect onelattice +osition to anot*er (for a full dislocation)#

Dislocations tend to *a?e as sall a Burgers ?ector as +ossi;le.

Dislocations are noneuili;riu de#ects and would lea?e t*e crystal i# gi?en ano++ortunity.


24/96

Screw dislocation

0otes

The figure shows a Right anded +crew (R+) dislocation (R+ is structurally distinct from &+)#

As for the edge dislocation the Burgers circuit has to be drawn far away from the dislocation line#

Sli+ Plane


25/96


26/96

1odel o# Screw Dislocation

T*oug* it is di##icult to understand anyt*ing #ro t*e +*oto o# t*e odel

1i d di l ti Di l i i h i d d d h


27/96

As we *ad noted! e"ce+t in s+ecial circustances! dislocations *a?e i"ed edge andscrew c*aracter.

In a cur?ed dislocation t*e edge and screw c*aracter c*ange #ro +oint to +oint.

Ty+ically in a dislocation loo+ only =+oints> *a?e +ure edge or +ure screw c*aracter Edge, b ⊥ tScrew, b t.

1i"ed dislocations

b

Dislocations with mi,ed edge and screw character

b

t ectors de#ining a dislocation

?eEdge

−?eEdge

5HS

LHSli+ Plane

Red line is the loop

L t id = t > # l


28/96

Pure EdgePure screw

S

E

E,cept for points + and E the remaining portion of the dislocation line has a mi,ed character

Let us consider a =uarter> o# a loo+

Ed d S t t* = l> t t t* ## ti B t


29/96

Edge and Screw co+onents, t*e =usual> way to get t*e e##ecti?e Burgers ?ector

T*e b ?ector is resol?ed into co+onents, = +arallel to t> @ screw co+onent and= +er+endicular to t> @ edge co+onent

Co+onents o# t*ei"ed dislocation at P

Screw Co+onent

Edge co+onent $ )b +in θ

$ )b =os θ

Edge component

+crew component

Ed d S t di## t t i li t* i t ti # t* ## ti * l# l


30/96

Edge and Screw co+onents, di##erent way to ?isualiGe t*e orientation o# t*e e##ecti?e *al#+lane

Instead o# resol?ing t*e b ?ector i# t*e t ?ector is resol?ed to #ind t*e edge and screw co+onents

Mor an edge dislocation t*e e"tra *al#+lane contains t*e t ?ector @ ;y resol?ing t*e t ?ector t*e edge co+onent o# t*e t ?ector t+Sinθ lies in t*e e##ecti?eU *al#+lane $Migure ;elow)

-0ote For a mi,ed dislocation there is no distinct 9half*plane:

Edge and Screw co+onents t*e e##ecti?e *al#+lane, a =crude> anology to understand t*e orientation o# t*e e"tra *al# +lane


31/96

Edge and Screw co+onents t*e e##ecti?e *al# +lane, a crude anology to understand t*e orientation o# t*e e"tra *al# +lane

Assue water is #lowing #ro le#t to rig*t onto a rigid cur?ed wall $in red colour ;elow). T*e green +ortiono# t*e wall =#eels> only s*ear stresses! w*ile t*e aroon +ortion #eels only noral stresses $o# agnitude ;).

A +oint 1 $in t*e cur?ed segent) #eels ;ot* noral and s*ear stresses. T*e e##ecti?e +art w*ic* #eelsnoral stresses is oriented ?ertically wit* agnitude ;Sinθ.

M

1otion o# Dislocations


32/96

Two /inds o# otion o# a dislocation are +ossi;le, 2lide and Cli;. Mirst we consider glide otion. Dislocations may o?e under an e"ternally a++lied #orce $resulting in stress

inside t*e aterial o#ten casually re#erred to =a++lied stress>). At t*e local le?el s*ear stresses on t*e sli+ +lane can only dri?e dislocations. T*e iniu stress reuired to o?e a dislocation is called t*e PeierlsNa;arro

$PN) stress or t*e Peierls stress or t*e Lattice Mriction stress (i#e the e,ternally 9applied stress: may even be purely tensile but on the slip plane shear stresses must act in order to move the dislocation)#

Dislocations ay also o?e under t*e in#luence o# ot*er internal stress #ields (e#g#those from other dislocations$ precipitates$ those generated by phase transformations etc#)# Dislocations are attracted to #reesur#aces $and inter#aces wit* so#ter aterials)

and ay o?e ;ecause o# t*is attraction @ t*is #orce is called t*e Iage Morce. In any case the eierls stress must be e,ceeded for the dislocation to move#

T*e ?alue o# t*e Peierls stress is di##erent #or t*e edge and t*e screw dislocations. T*e #irst ste+ o# +lastic de#oration can ;e considered as t*e ste+ created w*ent*e dislocation o?es and lea?es t*e crystal.@ One sall ste+ #or t*e dislocation! ;ut a giant lea+ #or +lasticityU.

F*en t*e dislocation lea?es t*e crystal a ste+ o# *eig*t =;> is created @ wit* it

all t*e stress and energy stored in t*e crystal due to t*e dislocation is relie?ed.

1otion o# Dislocations

Clic/ *ere to /now ore a;out Peierls StressClic/ *ere to /now ore a;out Peierls Stress


33/96

otion of dge

dislocation

Conser?ati?e(3lide)

Nonconser?ati?e(=limb-)

Mor edge dislocation, as b ⊥ t @ t*ey de#ine a +lane @ the slip plane# Cli; in?ol?es addition or su;traction o# a row o# atos ;elow t*e *al# +lane

V ?e cli; W cli; u+ @ reo?al o# a +lane o# atos

V −?e cli; W cli; down @ addition o# a +lane o# atos. I+ortance o# cli;, cli; +lays an i+ortant role in any ways.

I# an edge dislocation is =stuc/> at soe o;stacle on a particular slip plane! t*en it cancontinue to glide i# cli; can cli; =a;o?e> t*at sli+ +lane. T*is way cli; +lays ani+ortant role in #acilitating continued sli+. acancy concentration in t*e crystal can decrease due to ?e cli;.


34/96

Edge Dislocation 2lide

Sur#ace ste+(atomic dimensions)


35/96

1otion o# ascrew

dislocationleading to aste+ o# b

2ra+*ics ideo, 1otion o# Screw Dislocation2ra+*ics ideo, 1otion o# Screw Dislocation

0ote +chematic diagrams

F* t* di l ti l t* t l t*


36/96

Sur#ace ste+

X F*en t*e dislocation lea?es t*e crystal! t*estress #ield associated wit* it is relie?ed.

X Howe?er! it costs soe energy to create t*ee"tra sur#ace corres+onding to t*e ste+.

Are t*ese ste+s ?isi;leY

T*ese ste+s ;eing o# atoic diensions are not ?isi;le in o+tical icrosco+es. Howe?er! i# anydislocations o+erate on t*e sae sli+ +lane t*en a ste+ o# n; $n %99s%999s) is created w*ic* can e?en ;eseen in an o+tical icrosco+e (called the slip lines).

Sur#ace ste+s $sli+ lines)?isi;le in a Scanning

Electron 1icrogra+* Sli+ lines $w*ic* are crystallogra+*ic ar/ers)=re#lecting across> a twin ;oundary in Cu

Dislocations lea?ing t*e sli+ +lane


37/96

As it was o;ser?ed t*e =#irst ste+> o# +lastic de#oration is t*e otion o# a dislocationlea?ing t*e crystal $or to soe ot*er inter#ace ;ounding t*e crystal) @ leading to t*e#oration o# a ste+.

Mor continued +lastic de#oration it is necessary t*at dislocations continue to o?e andlea?e t*e crystal. Hence! any i+edients to t*e otion o# a dislocation will lead to=*ardening> o# t*e crystal and would =stall> +lastic de#oration (the pinning of adislocation).

Once a dislocation *as ;een +inned it can eit*er =;rea/ down t*e ;arrier> or =;y+ass> t*e ;arrier.

By+assing t*e ;arrier can ta/e +lace ;y ec*aniss li/e, Cli; Cross Sli+ Mran/5ead ec*anis Z.

In cli; and cross sli+ t*e dislocation lea?es-c*anges its =current> sli+ +lane and o?es toanot*er sli+ +lane t*us a?oiding t*e ;arrier

Howe?er! t*ese +rocesses $cli; and cross sli+) can occur inde+endent o# t*e +inning o#

t*e dislocation

Dislocations lea?ing t*e sli+ +lane


38/96

Dislocation lea?ing-c*angingt*e sli+ +lane

Screw dislocation

Edge dislocation Cli;

Cross Sli+

0on*conservative-

involves mass transport

=onservative

-=onservative climb is also possible22 8 by motion of prismatic edge loop on the slip plane

In climb an edge dislocation moves to an ad?acent parallel plane$ but in cross slip a screw dislocation

moves to a plane inclined to the original plane#


39/96

Climb of dge Dislocation

Positi?e cli; Removal of a row of atoms

Negati?e cli; Addition of a row of atoms

5eo?al o# a row o# atos leads to a decrease in ?acancy concentration in t*e crystal and negati?e cli;leads to an increase in ?acancy concentration in t*e crystal.

Screw dislocation, Cross Sli+


40/96

Screw dislocation, Cross Sli+

T*e dislocation is s*own crosssli++ing #ro t*e ;lue +lane to t*e green +lane

Let t*e dislocation ;e o?ing on SP% $as t*e resol?ed s*ear stress is a"iu on Sli+Plane% $SP%)).

T*e #igures ;elow s*ow t*e cross sli+ o# a screw dislocation line #ro SP% to Sli+ +lane8

$SP8). T*is ay occur i# t*e dislocation is =+inned> in sli+ +lane%. Mor suc* a +rocess to occur t*e 5esol?ed S*ear Stress on SP8 s*ould ;e at least greatert*an t*e Peierls stress(often stresses higher than the eierls stress has to be overcome due to the presence of other stress fields)#

It is to ;e noted t*at SP% 6 SP8 are $usually) crystallogra+*ically eui?alent! i.e. i# SP% is$%%%)CCP Crystal t*en SP8 can ;e $%%%)CCP Crystal.

How does +lastic de#oration ;y sli+


41/96

How does +lastic de#oration ;y sli+occurY

Munda C*ec/

T*e #irst ste+ o# +lastic de#oration ;y sli+ $at t*e #undaental le?el) is t*e otion o# a

dislocation lea?ing t*e crystal.

By e"ternally a++lied #orce $or soe ot*er eans) stress *as to ;e =generated> wit*in t*ecrystal.

T*e sli+ +lane s*ould #eel s*ear stresses.

T*e s*ear stress s*ould e"ceed t*e =Critical 5esol?ed S*ear Stress $C5SS)> or Peierls

stress. T*e dislocation s*ould lea?e t*e crystal creating a sur#ace ste+ o# *eig*t =;>.

T*e +rocess a*ead o# t*is w*ic* leads to an ar;itrary s*a+e c*ange is co+licated and we will

deal wit* a +art o# it later.

F*ere can a dislocation line endY


42/96

Dislocation line cannot end inside t*e crystal (abruptly) T*e dislocation line,

• Ends on a #ree sur#ace o# t*e crystal

• Ends on an internal sur#ace or inter#ace• Closes on itsel# to #or a loo+• Ends in a node

A node is t*e intersection +oint o# ore t*an two dislocations T*e ?ectoral su o# t*e Burgers ?ectors o# dislocations eeting at a

node W 9

F*ere can a dislocation line endY

F*at a;out t*e introduction o# a uarter +lane o# atos doesn>t t*e dislocation line end inside t*e crystalY

As seen in t*e #igure ;elow t*ere are two sections to t*e dislocation line ending on #ree sur#ace o# t*e crystal and *encenot inside t*e crystal.

Munda C*ec/

Positi?e and Negati?e dislocations


43/96

As we *a?e seen w*en t*ere are two are ore ED2E dislocations in a sli+ +laneone o# t*e is assigned a ?e sign and t*e ot*er one a −?e sign (done arbitrarily)

In t*e case o# screw dislocations t*e 5ig*t Handed Screw $5HS) Dislocation is+tructurally Distinct #ro t*e Le#t Handed Screw $LHS) Dislocations In t*e case o# 5HS dislocation as a cloc/wise circuit $Burgers) is drawn t*ena *elical +at* leads into t*e +lane o# t*e

Positi?e and Negati?e dislocations

Energy o# dislocations


44/96

Energy o# dislocations

T*e +resence o# a dislocation distorts t*e ;onds and costs energy to t*e crystal. Hence!dislocations *a?e distortion energy associated wit* t*e

T*e energy is e"+ressed as Energy +er unit lengt* o# dislocation line

@ :nits, [J-\ Edge @ Co+ressi?e and tensile stress #ields

Screw @ S*ear stress #ields T*e energy o# a dislocation can a++ro"iately ;e calculated #ro linear elastic t*eory.

T*e distortions are ?ery large near t*e dislocation line and t*e linear elastic descri+tion#ails in t*is region @ called t*e Core o# t*e dislocation (estimates of this region range from b to Cb depending on the crystal in /uestion). T*e structure and energy o# t*e core*as to ;e co+uted t*roug* ot*er et*ods and t*e energy o# t*e core is a;out %-%9 t*etotal energy o# t*e dislocation.

T*e #orula gi?en ;elow gi?es reasona;le a++ro"iation o# t*e dislocation energy.

Energy o# dislocation Elastic Nonelastic $Core)E ~E/10

8%I

8

d E 3bElastic Energy o# a dislocation - unit lengt* 2 @ $µ) s*ear odulus

; @ ,b,

As it costs energy to +ut a dislocation in a crystal,


45/96

⇒ Dislocations will *a?e as sall a b as +ossi;le

Dislocations(in terms of lattice translation)

Mull

Partial

b @ Mull lattice translation

b @ Mraction o# lattice translation

gy + y Dislocations tend to *a?e as sall a b as +ossi;le T*ere is a line tension associated wit* t*e dislocation line Dislocations ay dissociate into Partial Dislocations to reduce t*eir energy

8%I8

d E 3b

8

9 8 ln( $% )

edge

d 3b E

bγ

π ν + ÷ −

Anot*er #orula #or t*e energy $Edge dislocation)

γ 9 siGe o# t*e control ?olue R9;

8

9 8 ln(

screw

d

3b E

b

γ

π

+ ÷

Core contri;ution

Dissociation o# dislocations


46/96

Dissociation o# dislocations

Consider t*e reaction, 8b @ b b

C*ange in energy,

Initial energy ;e#ore s+litting into +artials, 2$8;)8-8 W 82;8

Energy a#ter s+litting into +artials, 8[2$;)8-8\ W 2$;)8

5eduction in energy W 82;8 2;8 W 2;8.

⇒ T*e reaction would ;e #a?ora;le.

Dislocations ay dissociate to reduce t*eir energy

Note t*at t*is e"a+le is considered #or illustration +ur+ose only (here a full dislocation is not splitting into partials)

Stress Mields o# Dislocations

dge dislocationThe self stresses


47/96

An edge dislocation in an in#inite ;ody *as co+ressi?e stress #ield a;o?e (the region ofthe e,tra half*plane) and tensile stress #ield ;elow (the region of the missing half*plane)t*e sli+ +lane

T*ese stress #ields will ;e altered in a #inite ;ody Asyetric +osition o# t*e dislocation in t*e crystal will also alter t*e stress #ield

descri;ed ;y t*e standard euations (as listed below) T*e core region is ignored in t*ese euations (which hence have a singularity at , 5$ y 5)

$Core ;eing t*e region w*ere t*e linear t*eory o# elasticity #ails)

O;?iously a real aterial cannot ;ear suc* =singular> stresses

T*e interaction o# t*e stress #ields o# t*e dislocations wit*, $i) t*ose originating #roe"ternally a++lied #orces and $ii) ot*er internal stress #ields @ deterines t*e otion o#dislocation @ leading to any as+ects o# ec*anical ;e*a?iour o# aterials

Stress Mields o# Dislocations

1ore a;out stress #ields o# dislocations1ore a;out stress #ields o# dislocations

8 8

8 8 8

$ )

8 $% ) $ )

+=

− + ,,

3b y , y

, y

σ

π ν 8 8

yy 8 8 8

$ ) 8 $% ) $ )

− −=

− +3b y , y

, yσ

π ν

8 8

"y 8 8 8

$ ) W

8 $% ) $ ) ,y

3b , , y

, yσ τ

π ν

− −=

− +

stress #ields

The material is considered isotropic (two elasticconstants only* E . ν or 3 . ν )

8 in reality crystals are anisotropic w#r#t to the

elastic properties

gThe self stresses

dge dislocation


48/96

Note t*at t*e region near t*e dislocation *as stresses o# t*e order o# 2Pa

T*ese stresses are t*e sel# stresses o# t*e dislocation and a straig*t dislocation line cannoto?e under t*e action o# sel# stresses alone (in an infinite body)

A dislocation interacts with other defects in the material via these 9long range: stress fields


49/96

Note t*at t*e stresses near t*e dislocation line reac*es ?alues in 2Pa. 3ield stressis usually in %99s o# 1Pa. How does a aterial not yield under t*e e##ect o# suc**ig* stressesY

Munda C*ec/

T*e +redoinant ec*anis o# +lastic de#oration is sli+! w*ic* in?ol?es otion o# a

large nu;er o# dislocations $ultiately lea?ing inter#aces). T*e stresses descri;ed *ere are due to t*e =dislocation itsel#> $t*e causati?e agent #or +lastic

de#oration ;y sli+) and t*ey are t*e elastic stress fields associated wit* t*e dislocation.

Interaction ;etween dislocations dge dislocation


50/96

Here we only consider elastic interactions ;etween edge dislocations on t*e sae sli+ +lane.

T*is can lead to Attracti?e and 5e+ulsi?e interactions.

To understand t*ese interaction we need to consider Positi?e and Negati?e edgedislocations. I# a single edge dislocation is +resent in a aterial it can ;e called eit*er +ositi?e or negati?e. I# two $or ore) dislocations are +resent on t*e sae sli+ +lane! wit*t*e e"tra *al#+lane on two di##erent sides o# t*e sli+ +lane! t*en one o# t*e is +ositi?eand t*e ot*er negati?e.

T*e +icture $region o# attraction and re+ulsion) gets a little =detailed> i# t*e twodislocations are ar;itrarily oriented. [See Stress Mields o# Dislocations\

Interaction ;etween dislocations

Positi?e edge dislocation Negati?e edge dislocation

ATT5ACTION Can coe toget*er and cancel one anot*er

5EP:LSION

Dislocations in CCP Crystals


51/96

Sli+ syste @ ^%%9! _%%%` Per#ect dislocations can s+lit into +artials (+hoc"ley partials considered first) to reduce

t*eir energy

T*e dissociation into +artials lea?es a Stac/ing Mault ;etween t*e two +artials on t*e sli+ +lane

T*e two +artials re+el eac* ot*er and want to ;e as #ar as +ossi;le @ ;ut t*is leads to alarger #aulted area $leading to an increase in energy) @ de+ending on t*e stac/ing #aultenergy t*ere will ;e an euili;riu se+aration ;etween t*e +artials

T*e S*oc/ley +artial *as Burgers ?ector o# t*e ty+e, $%-4) [8%%\ ty+e.T*is is an i+ortant ?ector in t*e CCP crystals! as ?ectors o# t*is #aily connect B site to C site and ?ice?ersa.

Mor a +ure edge dislocation in a CCP crystal t*e =e"tra *al#+lane> consists o# two atoic +lanes. T*e +artial dislocations consist o# one =e"tra> atoic +lane eac* $;ut t*e Burgers?ector o# t*e +artial is not +er+endicular to t*e dislocation line as in t*e case o# t*e +er#ect edge dislocation).

Dislocations in CCP Crystals

- ill be considered in the topic on GD defects

Figures in coming slides

CCP


52/96

8

%8%%

4b =

%%8%

4b =

%

%%%9

8b =

$%%%)

$%%%)

+lip plane

$%%%)

%[%%9\

8

÷ $%%%)

%[%8%\

4

÷

$%%%)

%[8%%\

4

÷ → +

;%8 $;8

8 ;8)

b

CCP

S*oc/ley Partials

$%%%)

+ome of the atoms are omitted for clarity$ Full vectors (blue vector) . (green vector) shown in the left figure

$%%%)

%[%8%\

4 ÷

$%%%)

%[8%%\

4

÷

8 88 8 8

8

% 8 % 4 %S S

4 4 4b

+ += = = ÷ ÷ ÷ ÷

r

$;88 ;8) W %-4 %-4 W %-

Energy of the dislocation is proportional to bG#

As the energy of the system is reduced on dissociation into partials the perfect dislocation will split into two partials#

Per#ect EdgeDislocation andS*oc/ley Partials

8%% %8%

Pure edge dislocation CCP


53/96

The e,tra* Hhalf plane consists of two 9planes: of atoms2

S* /l P ti l


54/96

S*oc/ley Partials

erfect edge dislocation (9full: Burgers

vector) with two atomic 9e,tra*half: planes

artial dislocations each with one atomic

9e,tra*half: plane

Clic/ *ere to see *ow two dislocations gi?e rise to one dislocationClic/ *ere to see *ow two dislocations gi?e rise to one dislocation

Here one dislocation s+lits into two +artials

Dislocation loo+s and Mran/ Partial dislocations


55/96

Dislocation loo+s and Mran/ Partial dislocations

Mored ;y insertion or reo?al o# a disc o# atos #ro t*e $%%%) +lane. T*e $%%%) crystal +lane consists o# t*ree atoic +lanes and t*e lattice translation ?ector

along [%%%\ *as a agnitude o# √ $t*e distance ;etween t*e atoic +lanes along [%%%\ is

√-). T*e +ac/ing along t*is direction is, ABCABCABCZ 5eo?al o# a disc @ Intrinsic #ault

Insertion o# a disc @ E"trinsic #ault bMran/ Partial W [%%%\-

As b is not on a sli+ +lane $a e;er o# t*e _%%%` #aily) t*e dislocation cannot o?e

conser?ati?ely $i.e. wit*out ass trans+ort) @ is a Sessile Dislocation $as o++osed to a2lissile dislocation $w*ic* can o?e! e.g. t*e S*oc/ley +artials)). E"cess ?acancies $uenc*edin or #ored ;y irradiation) can #or an intrinsic #ault $t*ese

ay *a?e *e"agonal s*a+e in soe cases). T*is s*ows t*at a dislocation loo+ can *a?e a co+letely edge c*aracter ;ut ne?er a

co+letely screw c*aracter.

Fran" partial loop bounding a stac"ing fault in == crystal


56/96

Intrinsic

E"trinsic

Stac/ing #aults

Two ;rea/s introduced intot*e stac/ing seuence

Per#ect region

Maulted region

Mran/ +artial loo+s wit* bW%- [%%%\

J444K

These lines are pro?ection of (444) planes

BCC P d di l ti%

[%%8\ ÷ Di l ti li t


57/96

BCC Pure edge dislocation

$%%9)

% [% % %\8

÷

$%%%)$%%9)!

[%%8\8

÷

$%%%)

$%%9)

Sli+ +lane

E"tra *al# +lane

Burger>s ?ector

Dislocation line ?ector


58/96

Moration o# dislocations $in t*e ;ul/ o# t*e crystal)


59/96

Due to accidents in crystal growt* #ro t*e elt. 1ec*anical de#oration o# t*e crystal. Nucleation o# dislocation.

Hoogenous nucleation o# a dislocation reuired *ig* stresses $2-%9). Stress concentrators in t*e crystal can aid t*e +rocess.

Dislocation density increases due to +lastic de#oration ainly ;y ulti+lication o# +ree"isting dislocations.

Dislocation density re#ers to t*e lengt* o# dislocation lines in a ?olue o#aterial @ *ence t*e units are [-\ (it is better not to cancel the 9m: in the numerator and the denominator and write as 'mG as the

units m'm7

is more physical2)# Annealed crystal, dislocation density $ρ) %94 %9%9 -. Cold wor/ed crystal, ρ %9%8 %9%( -. As t*e dislocation density increases t*e crystal ;ecoes stronger $ore a;out t*is later).

Ty+ical ?alues o# Dislocation Density

0ote in this conte,t it is noteworthy that screw dislocations can actually play a role in crystal

growth 8 =onstructive role of dislocations

Burgers ?ectors o# dislocations in cu;ic crystals


60/96

1onoatoic MCC ^%%91onoatoic BCC ^%%%

1onoatoic SC ^%99

NaCl ty+e structure ^%%9

CsCl ty+e structure ^%99

DC ty+e structure ^%%9

Crystallography determines the Burgers 3ector

fundamental lattice translational vector lying on the slip plane

“Close packed volumes tend to remain close packed,close packed areas tend to remain close packed &close packed lines tend to remain close packed”

A rule of thumb can be evolved as follows

=lose pac"ed in this conte,t implies 9better bonded:

Sli+ systes


61/96

Crystal Slip plane4s5 Slip direction

MCC _%%%` ^%%9

HCP $999%) ^%% 89

BCC 0ot close pac"ed

_%%9`! _%%8`! _%8` ^%%%

0o clear choice of slip plane

⇒ Fa?y sli+ linesAnisotro+ic sli+ ;e*a?iour

A co;ination o# a sli+ direction $b) lying on a sli+ +lane is called a sli+ syste. T*is isdescri;ed in ters o# a #aily o# directions and a #aily o# +lanes.

In close +ac/ed crystals it is a close +ac/ed direction lying on a close +ac/ed +lane.

In BCC crystals t*ere are any +lanes wit* siilar +lanar atoic density @ t*ere is noclear c*oice o# sli+ +lane.

T*ere ig*t ;e ore t*an one active sli+ syste in soe crystals (e#g# B== crystals below)#8 the active slip system gives rise to plastic deformation by slip#

E?en i# t*ere is only one sli+ syste is acti?e at low te+erature! ore sli+ systes ay ;ecoe acti?e at *ig* te+eratures @ +olycrystalline aterials w*ic* are ;rittle at roo

te+erature ay ;ecoe ductile at *ig* te+eratures.

Jogs and 7in/s Defect in a defect2


62/96

A straig*t dislocation line can *a?e a ;rea/ in it o# two ty+es, A


63/96

Edge dislocation Screw Dislocation

Jog Edge c*aracter Edge c*aracter 7in/ Screw C*aracter Edge c*aracter

Jogs and 7in/s, C*aracter Ta;le

Jogs and 7in/s in a screw dislocation will *a?e edge c*aracter.

Jog in a Edge dislocation *as Edge c*aracter and 7in/ in a edge dislocation *as screwc*aracter.

Jogs


64/96

T*e +resence o# a


65/96

Two straig*t dislocation can intersect to lea?e Jogs and 7in/s in t*e dislocation line T*ese e"tra segents in a dislocation line cost energy and *ence reuire wor/ done ;y t*e

e"ternal #orce ⇒ lead to *ardening o# t*e aterial

(Additional stress as compared to the stress re/uired to glide the dislocation line isre/uired to form the Oog'Pin")

Mour ty+es o# interactions are considered ne"t.

EdgeEdge Intersection% erpendicular Burgers vector


66/96

T*e


67/96

Bot* dislocations are /in/ed Edge Dislocation% (Burgers vector b% ) → 7in/ $Screw c*aracter) → Lengt* b8 Edge Dislocation8 (Burgers vector b8 ) → 7in/ $Screw c*aracter) → Lengt* b% T*e /in/s can glide

EdgeScrew Intersection erpendicular Burgers vector


68/96

Edge Dislocation (Burgers vector b% ) → Jog $Edge C*aracter) → Lengt* b8 Screw Dislocation (Burgers vector b8 ) → 7in/ $Edge C*aracter) → Lengt* b%

Screw Screw Intersection( erpendicular Burgers vector


69/96

I+ortant #ro +lastic de#oration +oint o# ?iew Screw Dislocation (Burgers vector b% ) → Jog $Edge C*aracter) → Lengt* b8 Screw Dislocation (Burgers vector b8 ) → Jog $Edge C*aracter) → Lengt* b/ Bot* t*e


70/96

T*e stress #ield o# a dislocation can interact wit* t*e stress #ield o# +oint de#ects. De#ects associated wit* tensile stress #ields are attracted towards t*e co+ressi?e region

o# t*e stress #ield o# an edge dislocation (and vice versa)#

Solute atos can segregate in t*e core region o# t*e edge dislocation (formation of the=ottrell atmosphere) 8 *ig*er stress is now reuired to o?e t*e dislocation (the system is in alow energy state after the segregation and higher stress is re/uired to 9pull: the dislocation out of the energy well)#

Hig*er #ree?olue at t*e core o# t*e edge dislocation aids t*is segregation +rocess. De#ects associated wit* s*ear stress #ields $*a?ing a nons+*erical distortion #ield e.g.

interstitial car;on atos in BCC Me) can interact wit* t*e stress #ield o# a screw

dislocation.

acancies are attracted to t*e co+ressi?e regions o# an edge dislocation and are re+elled#ro tensile regions


71/96

+tress values in 3a

σ""

Position o# t*e Dislocation line → into t*e +lane

Tensile Stresses

Co+ressi?e Stresses

5 stress lineacancies $

)

A t t r a c t e d

R e p e l l e

d

0o interaction

#ro tensile regions T*e ;e*a?iour o# su;stitutional atos saller t*an t*e +arent atos is siilar to t*at o#

t*e ?acancies Larger su;stitutional atos are attracted to t*e tensile region o# t*e edge dislocation and

are re+elled #ro t*e co+ressi?e regions Interstitial atos $associated wit* co+ressi?e stress #ields) are attracted towards t*e

tensile region o# t*e edge dislocation and are re+elled #ro t*e co+ressi?e region o# t*estress #ield

Suary o# edge dislocation +oint de#ect interactions


72/96

Point De#ect Tensile 5egion Co+ressi?e 5egion

acancy 5e+elled Attracted

Interstitial Attracted 5e+elled

Saller su;stitutional ato 5e+elled Attracted

Larger Su;stitutional atos Attracted 5e+elled

3ield Point P*enoenon


73/96

3ield Point P*enoenon

+chematic

T*e interaction o# interstitial car;on atos wit* edge dislocations (interaction of stress fields of dislocations with solute atoms) @ leading to t*eir segregation to t*e core o# t*eedge dislocations is res+onsi;le #or t*e 3ield Point P*enoenon seen in t*e tensile test o#ild steel s+eciens

Interstitial Atom at the core

1odel ade o# agnetic ;alls(core segregation)

1ore a;out t*is in t*e c*a+ter on +lasticity

DislocationPreci+itate Interactions


74/96

Dislocations can interact wit* t*e stress #ields o# +reci+itates. 1o?ing dislocations can,

A glide t*roug* co*erent +reci+itates @ s*earing t*e +reci+itate.B ;ow around inco*erent +reci+itates! lea?ing loo+s as t*ey ;y+ass two +reci+itatesw*ic* act li/e +inning centres #or t*e dislocations (8 leading to an increase in thedislocation density)#

Bot* t*ese +rocesses need an a++lication o# *ig*er stresses $assuing an *arder +reci+itate) @ lead to t*e strengt*ening o# t*e aterial.

Seico*erent +reci+itates *a?e inter#acial is#it dislocations w*ic* +artially relie?e t*e

co*erency strains. (These dislocations are structural dislocations)#Co*erent +reci+itate note t*at t*e lattice

+lanes are continuous across t*e +reci+itate

- Though the word coherent is used as an ad?ective for the precipitate* actually what is meant is that the interface iscoherent (or semi*coherent if we tal" about a semi*coherent precipitate)

SeiCo*erent inter#ace

Clic/ *ere to /now ore a;out inter#acesClic/ *ere to /now ore a;out inter#aces

A 2lide t*roug* co*erent +reci+itates


75/96

Preci+itate +articleb

;

Double Ended Mran/5ead SourceB Bow around inco*erent +reci+itates


76/96

A dislocation can ;e +inned ;etween two inco*erent +reci+itate +articles $or in ot*erways)! t*us *indering t*e otion o# t*e dislocation.

Mor otion o# t*e dislocation! leading to +lasticity t*e dislocation *as to ;y+ass t*e +inned

segent! under t*e action o# t*e a++lied stress $s*ear stress on t*e sli+ +lane dri?es t*eotion). T*e dislocation ta/es a series o# con#igurations $as s*own t*e in t*e #igures) under t*e

action o# t*e a++lied stress @ leading to t*e #oration o# a dislocation loo+ (leaving theoriginal pinned segment)#

T*is leads to an increase in dislocation density (one of the mechanisms by which

dislocation density increases with plastic deformation)# As t*e original segent is retained t*e =source> $Mran/5ead source) can o+erate

re+eatedly #oring a loo+ eac* tie. As t*e +ree"isting loo+s would o++ose t*e #oration o# t*e ne"t loo+ $re+ulsi?e stresses

dislocations o# t*e sae sign)! *ig*er stresses are reuired to o+erate t*e source eac* tie. Till t*e #oration o# t*e *al#loo+ $seicircle)! increasingly *ig*er stresses are reuired.

A#ter t*is t*e +rocess occurs down*ill in stresses. T*e a"iu stress $τa") reuired to o+erate t*e source t*us corres+onds to t*e

#oration o# t*e *al#loo+ $wit* radius r in).

a"

in

I

8

3b

r

τ

Initial con#iguration A B


77/96

Dislocation line segent +innedat A and B ;y +reci+itates

6 Pinning could 4also5 be caused by7

Dislocation in t*e +lane o# t*e +a+er intersects dislocations in ot*er +lanes

Anc*ored ;y i+urity atos or +reci+itate +articles

Dislocation lea?es t*e sli+ +lane at A and B

A++lication o# stress on dislocation segent τ


78/96

A B

Force - τ b

τ

Line tensionBowing

As the dislocation line gets curved the energy of the system increases⇒ wor" has to be done by

e,ternal stresses to cause this e,tension# &ine tension (opposes the shear stress on the slip plane ( τ )# At a given stress there might be a

balances of forces leading to a curved geometry of the dislocation line# Further e,tension of the dislocation line occurs by increasing the stress#

8

8

%I 3bengthenergy ' l n Dislocatio γ =

dsblinendislocatioon Force τ =

θ γ θ

γ d8

dsin8

= forcetension &ine

dsbd τ θ γ =Mor euili;riu in cur?ed con#iguration

r

3b

8τ

τa"

⇒ r in

L%

and − segents coe toget*er and annul eac* ot*er


79/96

τ b

τ b

τ b

I n c r e a s i n g s t r e s s

semicircle8

corresponds to ma,imum stress re/uired

to e,pand the loop

After this decreasing stress is re/uired to e,pand the loop

Direction of dislocation motion

is ⊥ to the dislocation line

(e,cept at A and B)

8

%

(

Original segent

New loo+ created

8

Fran9 (ead dislocation source ;


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Fran9:(ead dislocation source ;

Can o+erate #ro a single source +roducing a loo+ eac* tie

T*is loo+ +roduces a sli+ o# %b eac* tie on t*e sli+ +lane

T*e a"iu ?alue o# s*ear stress reuired is w*en t*e ;ulge ;ecoes a seicircle$r in W L-8) @ τa" 2;-L⇒ τ as L i.e. T*e longest segents o+erate #irstV F*en t*e long segents get io;iliGed s*orter segents o+erate wit* increasing stress

⇒ wor" hardening

I# t*e dislocation loo+s /ee+ +iling u+ on t*e sli+ +lane t*e ;ac/ stress will o++ose t*ea++lied stress

F*en t*e bac"*stress τa" t*e source will cease to o+erate

Dou;le ended M5 sources *a?e ;een o;ser?ed e"+erientally t*ey are not #reuent ⇒ ot*erec*aniss ust e"ist

a" -3b &τ α = α W 9. #or edge dislocations and α W %. #or screw dislocations.

Dislocation Mree sur#ace Interaction @ Conce+t o# Iage Morces


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A dislocation near a #ree sur#ace $in a seiin#inite ;ody) e"+eriences a #orce towards t*e#ree sur#ace! w*ic* is called t*e iage #orce. T*is is a ty+e o# Con#igurational Morce (i#e# force e,perienced when the energy is lowered by a change in configuration of a system)

T*e #orce is called an =iage #orce> as t*e #orce can ;e calculated assuing an negati?e*y+ot*etical dislocation on t*e ot*er side o# t*e sur#ace (figure below)# T*e #orce o#attraction ;etween t*e dislocations $ 6 −) is gi?es t*e iage #orce. T*e aterial +ro+erties are identical t*roug*out.

I# t*e iage #orce e"ceeds t*e Peierls stress t*en t*e dislocation can s+ontaneously lea?et*e crystal! wit*out a++lication o# e"ternal stresses

Hence! regions near a #ree sur#ace and nanocrystals can ;ecoe s+ontaneouslydislocation #ree. In nanocrystals due to t*e +ro"iity o# ore t*an one sur#ace! anyiages *a?e to ;e constructed and t*e net #orce is t*e su+er+osition o# t*ese iage #orces.

A hypothetical negative dislocation is assumed to e,ist across the free*surface for the calculation of the force (attractive)

e,perienced by the dislocation in the pro,imal presence of a free*surface

d 3b F image

)%$(

8

ν π −−=

Doain de#orations in Nanocrystals in t*e +resence o# dislocations


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Dislocation near a #ree sur#ace in a seiin#inite ;ody can de#or t*e sur#ace. T*is is asall de#oration as s*own in t*e #igure ;elow #or t*e case o# an edge dislocation.

In nanocrystals (e#g# the "ind shown in the figure below) t*e doain can ;end-;uc/le in

t*e +resence o# dislocations (the figure shows the effect of an edge dislocation in a plate8 a screw dislocation leads to the twisting of * for e,ample * a cylindrical domain) T*is is elastic de#oration in t*e +resence o# dislocations in a nanoaterial

Hence! we can *a?e reversible plastic deformation due to elasticity222

%9;

Dislocation in a t*in =+late> o#Al leading to its ;ending

5ole o# dislocations in crystal growt* =onstructive role of dislocations2


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Crystals grown under low su+ersaturation $%) t*e growt* rate is considera;ly#aster t*an t*at calculated #or an ideal crystal

In an ideal crystal sur#ace t*e di##iculty in growt* arises due to di##iculty in t*enucleation o# a new onolayer

I# a screw dislocation terinates on t*e sur#ace o# a crystal t*en addition o#atos can ta/e +lace around t*e +oint w*ere t*e screw dislocation intersects t*esur#ace (the step) @ leading to a s+iral $actually s+rial *eli") growt* staircase +attern

2rowt* s+iral on t*e sur#ace o# crystalliGed +ara##in wa"

5ole o# dislocations in +*ase trans#oration


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Dislocations can act as *eterogeneous nucleation sites during +*ase trans#oration. Dislocations ay dictate t*e orientation and or+*ology o# t*e second +*ase.

T*e stain associated wit* t*e dislocation ay ;e +artly relie?ed ;y t*e #oration o# a

second +*ase. T*e strain associated wit* t*e trans#oration $t*e Es*el;y strain) ay ;e accoodated

;y +lastic #low $ediated ;y dislocations).

1ental Picture o# a dislocation


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Let us consider t*e ?arious ways o# understanding t*e dislocation @ T*e di##erent +ers+ecti?es

Association wit* translational syetry

As a line de#ect

Distortion o# ;onds @ region o# *ig* energy

Increase in entro+y o# t*e syste

Mree ?olue at t*e core @ +i+e di##usion

Core o# dislocation 6 its geoetry @ Peierls Stress Stress 6 Strain #ields

Interaction wit* ot*er de#ects

5ole in sli+

5ole as a structural de#ect

Munda C*ec/ F*y are dislocations noneuili;riu de#ectsY


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It is clear #ro t*e a;o?e euation t*at i# a con#iguration gi?es an entro+y ;ene#it $i.e. ∆S is +ositi?e) t*en t*at state will ;e sta;iliGed at soe te+erature$e?en i# t*e ent*al+y cost is ?ery *ig* #or t*at con#iguration)

In t*e +resent case, it costs an energy o# 2;8-8 +er unit lengt* o# dislocation line

introduced into t*e crystal ;ut! t*is gi?es us a con#igurational entro+y ;ene#it $ast*is dislocation can e"ist in any e/uivalent +ositions in t*e crystal)

T*is i+lies t*at t*ere ust ;e te+erature w*ere dislocations can ;ecoe sta;lein t*e crystal $ignoring t*e c*ange in t*e energy cost wit* te+erature #or now)

%nfortunately this temperature is above the melting point of all "nown materials

Hence! dislocations are not sta;le t*erodynaic de#ects in aterials

- Including positional$ electronic$ rotational . vibrational multiplicity of states

V T*e energy reuired to create 7in/s and Jogs o# lengt* =;> is /0@ t*ese can ;e created ;y t*eral #luctuations

∆2 W ∆H − T ∆S ?e #or dislocations

Munda C*ec/ How are crystals wea/enedY T*e two ste+ +rocess


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As we *a?e seen t*e +rocess o# sliding $;y s*ear) o# an =entire +lane> o# atos can ;e reduced toa 9line*wise: +rocess ;y dislocationsV t*is leads to a s*ear stress reduction o# a #ew orders o# agnitude

T*is +ro;le can ;e #urt*er ;ro/en down $in diension and energy) to t*e #oration andigration o# /in/ +airs along t*e dislocation line (usually occurs to screw components of dislocations in B==metals)

V t*is can #urt*er lead to t*e reduction in stress reuired #or dislocation otion

=ontinued1


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V In BCC etals and 2e t*erally assisted #oration o# /in/ +airs can cause sli+ at stresses

τ ^ τPN =ontinued1


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D!S"#CA$!#%S


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5ando Structural

Distinct #ro =2eoetrically Necessary Dislocations> $2ND)

As entioned ;e#ore, Structural de#ects +lay a ?ery di##erent role in aterial ;e*a?iour as co+ared to 5ando Statistical De#ectsU (non*structural)

Structural dislocations ay ;e associated wit* a ;oundary and *ence t*eir role in +lasticity ay ;e ?ery di##erent #ro rando $statistically storedU) dislocations

O#ten a related ter 2eoetrically Necessary Dislocations is used in literature(we have intentionally avoided the use of the term here)

Structural dislocations include, Dislocations at low angle grain ;oundaries $will ;e discussed along wit* ot*er 8D de#ects) @ res+onsi;le #or a tilt or twist

Dislocation at seico*erent inter#aces@ res+onsi;le #or atc*ing is#it $;etween ad


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Tilt

1is#it

Structural dislocations can accoodate Twistbetween two crystals


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T*ere are two distinct uestions we can as/,

f%

I# you already *a?e a dislocation *ow to deterine t*e Burgers ?ectorYf8 F*at deterines t*e Burgers ?ectorY

T*e answer to f% is ;y constructing a Burgers circuit

T*e answer to f8 is, Crystallogra+*y @ t*e Burgers ?ector is t*e s*ortest latticetranslation ?ector $#or a +er#ect-#ull dislocation)

Sol3ed

?ample

In a cu;ic crystal a dislocation line o# i"ed c*aracter lies along t*e [%%8\direction. Burgers ?ector W [%%9\. F*at are t*e edge and screw co+onents o#


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pt*e Burgers ?ectorY F*ic* is t*e sli+ +laneY


94/96

$on [%%9\)

$on [%%8\)

Sol3ed

?ample

In a CCP crystal is t*e dislocation reaction s*own ;elow #easi;le energeticallyYF*at is t*e signi#icance o# t*e ?ectors on t*e 5HS o# t*e reactionY


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p

[ ] [ ]% % %

%%9 8% % %8%8 4 4

→ +

88 8

8

%

% % %S S

8 8b

+= = ÷ ÷

T*is is o# t*e #or ;%

@ ;8

;

T*e dislocation reaction is #easi;le i#,8 8 8

% 8 Jb b b

> + As the energy of a dislocation (per unit length of the dislocation line is proportional to bG

88 8 8

8

8

8 % % %S S

4 4b

+ += = ÷ ÷

% % % %

8 4 4 J

> + = ÷

⇒ t*e dislocation reaction is #easi;le (i#e# the full dislocation can lower its energy by splitting into partials)

88 8 8

8

J

% 8 % %S S

4 4b

+ += = ÷

÷

T*e ?ectors on t*e 5HS lie on t*e $%%%) close +ac/ed +lane in a CCPcrystal and t*ey connect B to C sites and C to B sites res+ecti?ely.Eui?alent ?ectors (belonging to the same family) are s*own in t*e#igure on t*e rig*t.

Sol3ed

?ample

F*at is t*e iage #orce e"+erienced ;y an edge dislocation at a distance o# %99;#ro t*e #ree sur#ace o# an sei in#inite Al crystalY Is t*is #orce su##icient to o?et* di l ti i t* t t* P i l M $ P i l St × ;) 8 × %9 ( N-


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t*e dislocation gi?en t*at t*e Peierls Morce $W Peierls Stress × ;) W 8. × %9−( N-

8

Iage( $% )

3b F

d π ν

−=

−

Data #or Al, a9 W (.9( ]! Sli+ syste, ^%%9_%%%`! ; W √8a9-8 W 8.'4 ]! 2 W 84.%' 2Pa! ν W 9.('

& %9J

Iage

$84.%' %9 )$8.'4 %9 )&.% %9 -

( $% 9.J(')$%99)

F 0 m

π

−−− × ×= = ×

−

8

Iage( $% )%99

3b F

bπ ν

−=

− −?e sign i+lies an attraction towards t*e #ree sur#ace

As Miage MPeierls ⇒ t*at t*e dislocation will s+ontaneously o?e to t*e sur#ace (creating a step) under t*e

action o# t*e iage #orce! wit*out t*e a++lication o# an e"ternally a++lied stress.

chapter 5b crystal imperfections dislocations

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