Dislocation Model of Strain Anisotropy - IIT Kanpur Ungar/II... ·  · 2015-05-12Dislocation Model...

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Dislocation Model of Strain Anisotropy Tamás Ungár Department of Materials Physics, Eötvös University Budapest Budapest, Hungary

Transcript of Dislocation Model of Strain Anisotropy - IIT Kanpur Ungar/II... ·  · 2015-05-12Dislocation Model...

Dislocation Model of

Strain Anisotropy

Tamás Ungár

Department of Materials Physics, Eötvös University Budapest

Budapest, Hungary

What is strain-anisotropy?

anisotropy:

Breadths in a

Williamson-Hall plot

are anisotropic

in terms of hkl indices

Ball-milled WC Gillies, D.C. & Lewis, D.

Powder Metallurgy, 11 (1968) 400.

strain-anisotropy:

global increase

of breadths

Ball-milled WC Gillies, D.C. & Lewis, D.

Powder Metallurgy, 11 (1968) 400.

Evaluation of strain by

ignoring

strain anisotropy

100

300

1

Ball-milled WC Gillies, D.C. & Lewis, D.

Powder Metallurgy, 11 (1968) 400.

Brief history of strain-anisotropy

4 consecutive papers of A.J.C. Wilson and coworkers:

1) A.R Stokes & A.J.C. Wilson:

The diffraction of X rays by distorted crystal aggregates - I

Proc. Phys. Soc. 56 (1944) 174-181

2) A.J.C. Wilson:

The diffraction of X-rays by distorted-crystal aggregates. II.

Diffraction by bent lamellae

Acta Cryst. 2 (1949) 220-222

3) J.N. Eastabrook & A.J.C. Wilson:

The Diffraction of X-Rays by Distorted-Crystal Aggregates III

Remarks on the Interpretation of the Fourier Coefficients

Proc. Phys. Soc. B 65 (1952) 67-75

4) A.J.C. Wilson:

The diffraction of X-rays by distorted-crystal aggregates. IV

Diffraction by a crystal with an axial screw dislocation

Acta Cryst. 5 (1952) 318-322

A.R Stokes & A.J.C. Wilson:

The diffraction of X rays by distorted crystal aggregates - I

Proc. Phys. Soc. 56 (1944) 174-181

. . . It is found that the "apparent strain" is given by η . . .

where

2 = A + BH

and

2222

222222

)( lkh

lklhkhH

Strain is hkl dependent

Unfortunately:

. . . This equation is verified within the rather large experimental error

for metal filings and wire.

Details of the experimental work will be published elsewhere. . . .

2 = A + BH

“Details of the experimental work”

were never published

The next appearance of strain anisotropy:

structure refinement by the Rietveld method using

neutron diffraction data

Caglioti G, Paoletti A, Ricci FP. Nucl. Instrum. 3 (1958) 223

introduce the term: strain anisotropy

strain anisotropy is disturbing in Rietveld structure refinement

First suggestion to make use of strain anisotropy for dislocation analysis:

P. Klimanek & R. Kuzel:

X-ray Diffraction Line Broadening Due to Dislocations in

Non-Cubic Materials. I. General Considerations and the

Case of Elastic Isotropy Applied to Hexagonal Crystals

J. Appl. Cryst. (1988). 21, 59-66

R. Kuzel & P. Klimanek:

X-ray Diffraction Line Broadening Due to Dislocations in

Non-Cubic Materials. II. The Case of Elastic Anisotropy

Applied to Hexagonal Crystals

J. Appl. Cryst. (1988). 21,363-368

R. Kuzel & P. Klimanek:

X-ray Diffraction Line Broadening Due to Dislocations in

Non-Cubic Crystalline Materials. III. Experimental Results for

Plastically Deformed Zirconium

J. Appl. Cryst. (1989). 22, 299-307

Slip systems from X-ray line broadening in hexagonal crystals:

R. Kužel jr., P. Klimanek, J. Appl. Cryst., 22 (1989) 299-307.

Orientation factor

Pioneering wok,

however,

NOT easy to implement

Strain-anisotropy is a general feature

in line broadening

Copper deformed by Equal Channel Angular Pressing (ECAP)

0 4 8 12

0.00

0.02

0.04

K [1/nm ]

FW

HM

[ 1

/nm

]

111 200 220 311 222 400

Strain anisotropy

Williamson-Hall plot

0 20 40 60 80 100 120 140 160-3

-2

-1

0

10

8

6

4

321

L[nm]{400}{222}{311}{220}{200}{111}ln

A

g2 [1/nm]

Warren-Averbach plot of inert-gas condensed copper

Strain anisotropy

0 20 40 60 80 100 120 140 160-3

-2

-1

0

10

8

6

4

321

L[nm]{400}{222}{311}{220}{200}{111}

ln A

g2 [1/nm]

Warren would have suggested to evaluate this plot as:

Structural investigations of submicrocrystalline metals obtained by

high-pressure torsion deformation R. Kužel, Z. Matej, V. Cherkaska, J. Pešicka, J. Cıžek, I. Procházka, R.K. Islamgaliev, Journal of Alloys and Compounds, 2005

Williamson-Hall plot

Structural investigations of submicrocrystalline metals obtained by

high-pressure torsion deformation R. Kužel, Z. Matej, V. Cherkaska, J. Pešicka, J. Cıžek, I. Procházka, R.K. Islamgaliev, Journal of Alloys and Compounds, 2005

Williamson-Hall plot

SPD:

Severe Plastic Deformation

Soft magnetism in mechanically alloyed nanocrystalline materials T. D. Shen, R. B. Schwarz, and J. D. Thompson, PHYSICAL REVIEW B 72, 14431 (2005)

Fe80Cu20 (at. %)

Strain anisotropy

2 4 6 8 10

-2

-1

0

K [ 1/nm ]

log

A(L

)

L [ nm ]

5

14

23

33

42

51

61

111 200 220 311 222 400 420 331 422

Warren-Averbach plot

Revealing the powdering methods of black makeup in Ancient Egypt

by fitting microstructure based Fourier coefficients to the whole

x-ray diffraction profiles of galena (PbS)

T. Ungár, P. Martinetto, G. Ribárik, E. Dooryhée, Ph. Walter, M. Anne, J.Appl.Phys. 91 (2002) 2455

Simple model based on

dislocations

T

g b

T g

b

gb 0

dislocation is visible dislocation is invisible

strong contrast weak contrast

gb = 0

strong line broadening weak line broadening

Analogous with transmission electron microscopy (TEM)

K.Shiramine, Y.Horisaki, D.Suzuki, S.Itoh, Y.Ebiko, S.Muto, Y. Nakata, N.Yokoyama,

Threading dislocations in InAs quantum dot structure, Journal of Crystal Growth, 205 (1999) 461-466

g = 004 g = 222 g = 222

ln An = ln - S

nA 22g2L2 < > 2

,gL

Fundamental equation of line broadening

Warren & Averbach (1952):

size Fourier coefficients

mean square strain

for dislocations [Krivoglaz, Wilkens]:

ln(Re/L)

C is the contrast factor of dislocations

C = C (g,b,l,cij)

4

2bC

2

,gL

2

,gL

1) The dislocation contrast factors can be evaluated numerically.

2) For polycrystals or when almost all slip systems are populated

averaging over the permutations of hkl can be done

3) Average contrast factors for cubic crystals:

= h00 (1-qH2) ,

where H2=(h2k2+ h2l2+ k2l2) / (h2+k2+l2)2

4) q depends on the a) elastic constants, and on the

b) dislocation character, e.g. edge or screw

CC

Az=2c44/(c11-c12)

fcc: a/2<110>[111]

bcc: a/2<111>[110]

copper

q=2.37

q=1.61

iron

q=2.6

q=1.3

2.5 5.0 7.5 10.0 12.5 15.0 17.50.0

0.1

0.2

0.3

0.4 screw; edge

Co

ntr

ast

facto

rs

K [ 1/nm ]

110 200 211 220 222 400 411

Ferritic steel

0 2 4 6

0.00

0.02

0.04

{400}

{311}

{222}

{220}

{200}

{111}

FW

HM

[ 1

/nm

]

K C 1/2

[1/nm ]

modified Williamson-Hall plot

Copper deformed by Equal Channel Angular Pressing (ECAP)

0 10 20 30 40 50-2.0

-1.5

-1.0

-0.5

0.0 {400}{311}{200}

{220}{222}

{111}

10

8

6

4

3

L [nm]

ln A

+

L

' W(g

)

g2 C [ nm-2 ]

modified Warren-Averbach method

Inert-gas condensed copper

Average crystallite size in the

inert-gas condensed copper specimen

classical Warren-Averbach analysis

200 – 400 reflections

<Lo> 7 nm

modified Warren-Averbach analysis

all reflections

<Lo> 18 nm

TEM size: 18 nm

T.Ungár, S.Ott, P.G.Sanders, A.Borbély, J.R.Weertman, Acta Materialia, 10, 3693-3699 (1998)

Soft magnetism in mechanically alloyed nanocrystalline materials T. D. Shen, R. B. Schwarz, and J. D. Thompson, PHYSICAL REVIEW B 72, 14431 (2005)

Fe80Cu20 (at. %)

0 5 10 15 20 25

-2

-1

0

422331

420

222311

220

400

111

200

L [nm]

5

9

14

19

23

28

33

37

42

47

51

56

61

log

A(L

)

K2C [ nm

-2 ]

modified Warren-Averbach method

Ball-milled galena (PbS) T. Ungár, P. Martinetto, G. Ribárik, E. Dooryhée, Ph. Walter, M. Anne, J.Appl.Phys. 91 (2002) 2455

TEM Dislocations structure in ball-milled PbS (galena) P. Martinetto, J. Castaing, and P. Walter, P. Penhoud and P. Veyssiere, J. Mater. Res., Vol. 17, No. 7, Jul 2002

1014 m-2

nanocrystalline SiC

sintered at 1800 oC by 5.5 GPa J. Gubicza, S. Nauyoks, L. Balogh, J. Labar, T.W. Zerda, T. Ungár,

J. Mater. Res. 22, 1314-1321 (2007)

0 4 8 120.00

0.01

0.02

422

331

420

400222

311

220

200

111

FW

HM

[1/n

m]

K [1/nm]

0 5 10 15 20 250.00

0.01

0.02

K2C [1/nm]

FW

HM

[1/n

m]

422400

420331

311

220 222

200111

Twinning Twins + dislocations

Williamson-Hall plot

modified Williamson-Hall plot

100 nm

in single crystals the concept of

average contrast factors

does NOT work

each reflection needs an individual contrast factor

Diffraction on a single grain in an

MgSiO3 perovskite

P. Cordier, T. Ungár, L. Zsoldos, G. Tichy,

Dislocations creep in MgSiO3 perovskite

at conditions of the Earth's uppermost lower mantle,

Nature, 428 (2004) 837-840.

Lower

mantle

Transition

zone

Upper

mantle

3 GPa

1100°C

13 GPa

1400°C

23 GPa

1600°C

135 GPa

3500°C

Depth

100 km

410 km

520 km

670 km

2900 km

P, T

Olivine (Mg, Fe)2SiO4

Wadsleyite (Mg, Fe)2SiO4

Ringwoodite (Mg, Fe)2SiO4

Pyroxenes (Mg, Fe)SiO3

(Ca, Mg, Fe)2Si2O6

Perovskite (Mg, Fe, Al)(Si, Al)SiO3-x

CaSiO3

Magnesiowustite (Mg, Fe)O

Garnets (Mg, Fe, Ca)3

Al2Si3O12

Garnets (Mg, Fe, Ca)3(Al, Si)2Si3O12

Schematic composition of Earth's mantle

1 mm

-0,05 0,00 0,05

1E-3

0,01

0,1

1

I/IMax

K [1/nm]

hkl K [1/nm]

110 2.92

120 4.57

121 4.79

022 4.99

Strain anisotropy

2 4 6 8 10

0.004

0.006

0.008

0.010B

read

ths

[1/

nm]

K [ 1/nm ]

FWHM

Integral Breadth

120

K

[ 1

/nm

]

K [ 1/nm ]

006

123

023

211

121

022112

021

111

110

Strain anisotropy

Williamson-Hall plot

Measured dislocation contrast factors: Cm

2 4 6 8 100,00

0,08

0,16

121

211

006

123

023

022021

120

112

111

110

Cm

K [ 1/nm ]

5 - 8

9 - 12

1 - 4

13 - 20

Conceivable Burgers vectors: 1 - 20

Measured and calculated dislocation contrast factors: Cm , Ccalc

2 4 6 8 10

0.00

0.08

0.16

006

121

123

023

022

021

211

120

112

111

110

Dis

loca

tion

Con

tras

t Fac

tors

K [ 1/nm ]

Ccalc

Cm

The only Burgers vectors

that survived the search-match

by comparing the measured and calculated

dislocation contrast factors

[100]

[010]

Contrast factors for hexagonal crystals:

hk.l = hk.0 [1 + q1x + q2 x2 ]

x =(2/3)(l/ga)2

C C

0,0 0,5 1,0 1,5 2,0 2,5

0,0

0,2

0,4

0,6 BE

PrE

PYE

PR2E

PR3E

PY2E

PY3E

PY4E

S1

S2

S3

Cav

2l2/3(ag)

2

I. C. Dragomir and T. Ungár J. Appl. Cryst., 2002, 35, 556-564.

The parabolas for the eleven sub-slip-systems in Ti

vs. x=(2/3)(l/ga)2

Mg deformed at different temperatures between RT and 300oC

Williamson-Hall plot for deformation at 200oC:

2 4 6 8 100,002

0,004

0,006

0,008

0,010

20.3

11.2

20.110.310.210.100.210.0

Bre

adth

s [1

/nm

]

K [1/nm]

Integral breadths

FWHM

The q parameters for Ti

Sub-Slip

System

hk.0 q1 q2

BE 0.20227 -0.101142 -0,102625

PrE 0.35387 -1.19272 0.355623

Pr2E 0.04853 3.6161928 1.2264112

Pr3E 0.10247 2.017177 -0.616631

PYE 0.3118 -0.894009 0.1833109

PY2E 0.09227 1.299046 0.3972469

PY3E 0.09813 1.894120 -0.365739

PY4E 0.09323 1.5270212 0.146150

S1 0.1444 0.59492 -0.710368

S2 0.41873 1.25714 -0.94015

S3 3.61x10-6 165366 -98611

C

- Two experimental parameters: q1(m) and q2

(m)

- Volume fractions of active slip systems: hi

= 1 ih

a) Edge dislocations:

Major slip

systems

sub-slip-systems Burgers vector Slip plane Burgers vector

types

Basal BE <2-1-10> {0001} a

Prismatic PrE <-2110> {01-10} a

PrE2 <0001> {01-10} c

PrE3 <-2113> {01-10} c + a

Pyramidal PyE <-12-10> {10-11} a

Py2E <-2113> {2-1-12} c + a

PyE3 <-2113> {11-21} c + a

PyE4 <-2113> {10-11} c + a

b) Screw dislocations:

sub-slip-systems Burgers vector Burgers vector

types

S1 <2-1-10> a

S2 <-2113> c + a

S3 <0001> c

Three Burgers vector types

The measuremets provide 3 equations:

(1) =

(2) =

(3) = 1

i = 1, 2 or 3 for <a>, <c> or <c+a> dislocations,

respectively

)(1mq

3

1

)(1

2)(0.

1

i

ii

ihki qbCh

P

)(2mq

3

1

)(2

2)(0.

1

i

ii

ihki qbCh

P

3

1iih

# sub-slip-systems

<a> 1/3< 110> 4

<c> 1/3<0001> 2

<c+a> 1/3< 113> 5

2

2

Fitting of measured and theoretical q parameters

For a particular Burgers vector type or i value:

the weights of the sub-slip-systems are:

1 or 0 in all possible combinations

accepted solution: if all hi > 0

not accepted solution: if any of hi < 0

0 50 1000

100

200

[%]

<a><c>

<c+a>

Num

ber

of solu

tio

ns

hi

Bar diagram of the solution matrix of the hi fractions:

in the as cast Mg specimen

The volume fractions of the Burgers vector types,

<a>, <c> and <c+a>,

vs. the temperature of deformation in Mg:

0 100 200 300

0

50

100

as cast

[10

-13 m

-2 ]

[%]

<a>

<c>

<c+a>hi

T [°C]

0

20

40

60

Mathis, K., et al. Acta mater. 52, 2004, 2889–2894.

Below about 100 oC:

<a> type dislocations in basal slip

+ twinning

Above about 100 oC:

twinning is replaced by <c+a> dislocations

(energetically more favorable than twinning)

Thank you for your attention