crystal imperfection ch 4
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Chapter 4
Crystal Defects andNoncrystalline Structure–
Imperfection
ME 2105 – Dr. Lindeke
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In our pervious Lecture when
discussing Crystals we
ASSUMED PERFEC !RDER
In real materials we find:
Crystalline Defects or lattice irregularity
Most real materials have one or more “errors in perfection”
with dimensions on the order of an atomic diameter to many
lattice sites
De"ects can #e classi"ication$%& according to geo'etry
(point) line or plane*
+& di'ensions o" the de"ect
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alcohol. Mixing occurs on the
molecular scale.
We can defne thismixture/s!utin na "ei#ht r $atmic%
&asis
' simi!ar discussincan app!( t$mixtures% ) meta!s– ca!!ed a!!(s
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* +acancies,
-acant atmic sites in a structure.
* e!)-nterstitia!s,-extra atms psitined &et"een atmic sites.
int De)ects – in the s!idstate are mre predicta&!e
+acanc(
distrtin) p!anes
se!)-interstitia!distrtin) p!anes
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P!I, DEFECS
• he si'plest o" the point de"ect is a vacancy) or vacant lattice site&
• All crystalline solids contain vacancies&
• Principles o" ther'odyna'ics is used e-plain the necessity o" the
e-istence o" vacancies in crystalline solids&
• he presence o" vacancies increases the entropy (rando'ness* o"the crystal&
• he e.uili#riu' nu'#er o" vacancies "or a given .uantity o"
'aterial depends on and increases with te'perature as "ollows$ (an
Arrhenius 'odel*
N v= N exp(-Qv/kT)
Equilibrium no of vacancies
!otal no of atomic sites Energy required to form vacancy
T = absolute temperature in °Kelvin
k = gas or Boltzmann’s constant
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3" utcmes i) impurit( 6 added t hst '6,
* !id s!utin ) in ' i.e.7 randm dist. ) pintde)ects6
* !id s!utin ) in ' p!us partic!es ) a new phase usua!!( )r a !ar#er amunt ) 6
89
u&stitutina! s!ids!n.
e.#.7 Cu in :i6
nterstitia! s!ids!n.
e.#.7 C in ;e6
ecnd phase partic!e--di
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Solid solution of nickel in copper shown
along a (100) plane. This is asu&stitutina! solid solution with nickelatoms sustituting for copper atomson fcc atom sites.
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mper)ectins in !ids
Cnditins )r su&stitutina! s!id s!utin..6
• =ume – 9ther( ru!es – 1. ∆r atmic radius6 > 15?
– 2. rximit( in peridic ta&!e• i.e.7 simi!ar e!ectrne#atiities
– @. ame cr(sta! structure )r pure meta!s
– 4. +a!enc( eAua!it(• '!! e!se &ein# eAua!7 a meta! "i!! hae a #reater
tendenc( t diss!e a meta! ) hi#her a!enc( thanne ) !"er a!enc( it prides mre e!ectrns t the$c!ud%6
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mper)ectins in !ids
'pp!icatin ) =ume–9ther( ru!es – !id!utins
1. Wu!d (u predict
mre '! r '#t diss!e in Bn
2. Mre Bn r '!
in Cu
3a&!e n p. 107 !allister "e.
#lement $tomic !r%stal #lectro& 'alenceadius Structure nega&
(nm) tiit%
Cu 0.12F ;CC 1.G H2
C 0.01= 0.048 0.00'# 0.1445 ;CC 1.G H1'! 0.14@1 ;CC 1.5 H@C 0.125@ =C 1.F H2Cr 0.124G CC 1. H@;e 0.1241 CC 1.F H2:i 0.124 ;CC 1.F H2d 0.1@ ;CC 2.2 H2Bn 0.1@@2 =C 1. H2
Mre '! &ecause siIe is c!ser anda!. s hi#her – &ut nt t much&ecause ) structura! di
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mper)ectins in !ids
• Specication of composition
– "ei#ht percent100x
21
11
mm
mC
+=
m1 J mass ) cmpnent 1
100x21
1'
1
mm
m
nn
nC
+=
nm1 J num&er ) m!es ) cmpnent 1
– atm percent
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Wt. ? and 't. ? -- 'n examp!e
T!picall! "e "ork "it# a basis "eig#t ($%%g or $ kg) or moles
given& allo! b! "eig#t -- '% u* +% ,i
'%%.++
'.00 /
+%%'.12
01.' /
.++.01$ or 01.$
.++ '.12'.12
.+$.++ '.12
Cu
Ni
Cu
Ni
g n m
g m
g n m
g m
C
C
= =
= =
= ≈+
= ≈+
or +$.
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Cnertin# et"een, Wt? and 't?6
$ 2$
$ 2 2 $
2 $2
$ 2 2 $
$ $$
$ $ 2 2
2 22
$ $ 2 2
$%%
$%%
$%%
$%%
C AC
C A C A
C AC C A C A
C AC
C A C AC A
C C A C A
∗= ×∗ + ∗
∗= ×∗ + ∗
∗= ×
∗ + ∗∗
= ×∗ + ∗
Cnerts )rm"t? t 't? 'i
is atmic"ei#ht6
Cnerts )rmat? t "t? 'iis atmic"ei#ht6
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nterstitia! s!id s!utin app!ies t car&n in K-irn. 3he car&n atm is sma!! enu#h t ft
"ith sme strain in the interstice r penin#6amn# adacent ;e atms in this imprtantstee! structure. 3his unit-ce!! structure can &ecmpared "ith that sh"n in ;i#ure @.4&.N
ut the interstitia! s!u&i!it( is Auite !" since the siIe mismatch ) the site tthe radius ) a car&n atm is n!( a&ut 1/4
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andom* sustitution solid solutioncan occur in +onic !r%stalline
materials as well. ,ere of -i inMg. The 2O arrangement isuna/ected. The sustitution occurs
among -i2H
and Mg2H
ions.
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$ sustitution solid solution of $l2@ in Mg
is not as simple as the case of -i in Mg.The requirement of charge neutralit% in theoerall compound permits onl% two $l@H ionsto ll eer% threeMg2H acant sites* leaing
oneMg2H
acanc%.
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+ron oxide* Fe1O x with x P 0.05* is an
example of a nonstoichiometric compound.Similar to the case of Figure .2* oth Fe2H and Fe@H ions occup% the cation sites* withone Fe2H acanc% occurring for eer% twoFe@H ions present.
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• Frenkel Defect
--a cation is out of place.
• Shottky Defect --a paired set of cation and anion vacancies.
• !uili"riu# concentration of defectskT / QDe~ −
fro# $.%. &offatt %.$. (earsall
and ). $ulff The Structure and
Properties of Materials *ol. 1
Structure )ohn $iley and Sons
+nc. p. ,.
De)ects in Ceramictructures
Shottky
Defect
Frenkel
Defect
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'nd,* s!ip &et"een cr(sta! p!anes resu!t "hen dis!catinsme7* this mtin prduces permanent p!astic6de)rmatin.
're ca!!ed Dis!catins,
chematic ) Binc =C6,
* &e)re de)rmatin * a)ter tensi!e e!n#atin
s!ip steps "hichare the ph(sica!
eidence ) !ar#enum&ers )dis!catinss!ippin# a!n#the c!se packedp!ane Q0001R
Line De)ects
'dapted )rm ;i#. .F7 !allister "e.
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Linear De)ects Dis!catins6
– 're ne-dimensina! de)ects arund "hich atms aremisa!i#ned
• Ed#e dis!catin, – extra ha!)-p!ane ) atms inserted in a cr(sta!
structure
– b (the berger’s vector is ⊥ perpendicu!ar6 tdis!catin !ine
• cre" dis!catin, – spira! p!anar ramp resu!tin# )rm shear de)rmatin
– b is || para!!e!6 t dis!catin !ine
!urger’s vector" b# is a measure of latticedistortion and is measured as a distance along theclose pac$ed directions in the lattice
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Ed#e Dis!catin
;i#. 4.@7 !allister "e.
Ed#e Dis!catin
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3enition of the 4urgers ector* b*relatie to an edge dislocation. (a) +nthe perfect cr%stal* an mS n atomicstep loop closes at the starting point.() +n the region of a dislocation* thesame loop does not close* and theclosure ector (b represents themagnitude of the structural defect.For the edge dislocation* the 4urgers
ector is perpendicu!ar to thedislocation line.
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Screw dislocation. The spiral stacking of cr%stal planes leads to the 4urgers ector eing parallel to
the dislocation line.
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Mixed dislocation. This dislocation has oth edge
and screw character with a single 4urgers ectorconsistent with the pure edge and pure screwregions.
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4urgers ector for thealuminum oxide structure.The large repeat distancein this relatiel% complex
structure causes the4urgers ector to eroken up into two (for2O ) or four (for $l@H )
partial dislocations* each
representing a smaller slip
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mper)ectins in !ids
Dis!catins are isi&!e in 36 e!ectrn micr#raphs
'dapted )rm ;i#. 4.7 !allister "e.
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Dis!catins T Cr(sta! tructures
* tructure, c!se-
packed p!anes T directins are pre)erred.
ie" nt t"
c!se-packedp!anes.
c!se-packed p!ane &ttm6c!se-packed p!ane tp6
c!se-packed directins
* Cmparisn amn# cr(sta! structures, ;CC, man( c!se-packed p!anes/directinsU =C, n!( ne p!ane7 @ directinsU CC, nne $super-c!se% man( $near c!se%
* pecimens that"ere tensi!e
tested.
M# =C6
'! ;CC6
tensi!e directin
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• 8ne case is a t"in &undar( p!ane6 – Essentia!!( a reVectin ) atm psitins acrss the
t"innin# p!ane.
• tackin# )au!ts – ;r ;CC meta!s an errr in 'C'C packin# seAuence – Ex, 'C''C
!anar De)ects in !ids
'dapted )rm ;i#. 4.G7 !allister "e.
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Simple iew of the surface of a cr%stallinematerial.
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$ more detailed model of the elaorate ledgelikestructure of the surface of a cr%stalline material.#ach cue represents a single atom. =From 9. >.
,irth and ?. M. >ound* . Chem. h(s. 2* 1812(1
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T%pical opticalmicrograph of a grain
structure* 100S. Thematerial is a low&caronsteel. The grainoundaries hae een
lightl% etched with achemical solution so thatthe% reBect lightdi/erentl% from the
polished grains* there%
giing a distinctiecontrast. (From Meta!s=and&k* Cth ed.* 'ol. "D't!as ) Micrstructures )
ndustria! '!!(s*
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Simple grain&oundar%structure. This is termed atilt oundar% ecause it is
formed when two adEacentcr%stalline grains are tiltedrelatie to each other % afew degrees (). The
resulting structure isequialent to isolated edgedislocations separated %the distance G* where isthe length of the 4urgersector* b. (From 5. T. ead*Dis!catins in Cr(sta!s*Mc?raw&,ill 4ook !ompan%*-ew ;ork* 1
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he ledge /rowth leads to structures with /rain
0oundries he shape and average si1e or
dia'eter o" the grains "or so'e polycrystalline
speci'ens are large enough to o#serve with the
unaided eye& (Macrosocipic e-a'ination*
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Specimen for the calculation of the grain&siIenumer* ? is dened at a magnication of 100S.This material is a low&caron steel similar to that
shown in Figure .1C. (From Meta!s =and&k* Cthed.* 'ol. "D 't!as ) Micrstructures ) ndustria!'!!(s* $merican Societ% for Metals* Metals >ark*,* 1
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* Xse)u! up t ∼2000Y ma#nifcatin 6.
* !ishin# remes sur)ace )eatures e.#.7scratches6* Etchin# chan#es reVectance7 dependin# ncr(sta! rientatin since di
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ince Zrain
&undaries...* are p!aner imper)ectins7* are mre suscepti&!e t etchin#7* ma( &e reea!ed as dark !ines7
* re!ate chan#e in cr(sta!rientatin acrss&undar(.
curtes( ) L.C. mithand C. rad(7 the:atina! ureau )tandards7 Washin#tn7
DC n" the :atina!nstitute ) tandardsand 3echn!#(7Zaithers&ur#7 MDN.6
8ptica! Micrscp(
'3M #rain
siIe num&er
- J 2? & 1
num&er ) #rains/in2 at 100x
ma#nifcatin
;e-Cr a!!(&6
#rain &undar(
sur)ace #re
p!ished sur)ace
a6
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ASM (A'erican Society "or testing and Materials*
2ISUAL C3ARS (3$%%x) eac# "it# a number
Quick an4 eas! 5 use4 6or steel
ASM has prepared several standard co'parison charts) all having di""erentaverage grain si1es& o each is assigned a nu'#er "ro' % to %4) which is ter'ed
the grain si1e nu'#er5 the larger this nu'#er) the s'aller the grains&
, 6 + " 7% ,o. o6 grains/s7uare inc#
8rain size no.
:83E, 3he '3M #rain siIe is re!ated rre!ates6 a #rain area $T 100x M$?-+F+!$T+-
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Determinin# Zrain iIe7 usin# a
micr#raph taken at @00x
• We cunt 14 #rainsin a 1 in2 area nthe @00x6 ima#e
• 3 reprt '3M#rain siIe "eneeded a measure) : at 100x nt@00x
• We need acnersin methd[
( ) ( )( ) ( ) ( )
( ) ( )
( )( ) ( )
2
$2$%%
9 is mag. o6 image
, is measure4 grain count at 9
no" solve 6or 8&
log( ) 2 log log $%% $ log 2
log 2log +$
log 2log $+ 2log %% +
$ :.1 1%.%$
G
M
M
M
m
M N
N M G
N M G
G
− = ÷
+ − = −
∴
+ −= +
+ −= + = ≅
;r this same materia! h" man(
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;r this same materia!7 h" man(Zrains "u!d expect /in2 at 100x 't
50x
$ 1 $ 2
2 2
1 $
2 2
(at $%%x)2 2 $21 grains/in
,o"* #o" man! grain "oul4 ; expect at 0%x<$%% $%%
, 2 $210%
, $212 0$2 grains/in
G
M
M
N
M
− −
−
= = ≅
= ∗ = ÷ ÷
= =
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>t $%%x
T di i l h ti i i f
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Two&dimensional schematics gie a comparison of(a) a cr%stalline oxide and () a non&cr%stallineoxide. The non&cr%stalline material retains short&range order (the triangularl% coordinated uildinglock)* ut loses long&range order (cr%stallinit%).This illustration was also used to dene glass in!hapter 1 (Figure 1.C).
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4ernal model of an amorphous metal structure. Theirregular stacking of atoms is represented as aconnected set of pol%hedra. #ach pol%hedron is
produced % drawing lines etween the centers ofadEacent atoms. Such pol%hedra are irregular inshape and the stacking is not repetitie.
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$ chemical impurit% suchas -aH is a glassmodier* reaking up therandom network and
leaing nonridgingox%gen ions. =From 4. #.5arren* . 'm. Ceram.c. 24* 8@2 (1
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Schematic illustration ofmedium&range ordering in a!aJSi2 glass. #dge&sharing
!a octahedra hae een
identied % neutron&di/ractionexperiments. =From >. ,.?askell et al.* :ature @50* 2"@(1
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ummar(
• int7 Line7 ur)ace and +!umetric de)ects exist ins!ids.
• 3he num&er and t(pe ) de)ects can &e aried andcntr!!ed – T cntr!s acanc( cnc.
– amunt ) p!astic de)rmatin cntr!s \ ) dis!catins
– Wei#ht ) char#e materia!s determine cncentratin )su&stitutina! r interstitia! pint ]de)ects^
• De)ects a
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