Stress and Texture - Stanford University

Stress and Texture

Strain

Two types of stresses:

microstresses – vary from one grain to another on a microscopic scale.

macrostresses – stress is uniform over large distances.

Usually:

macrostrain is uniform –produces peak shift

microstrain is nonuniform –produces peak broadening

tan22d

db

Applied and Residual Stress

Plastic flow can also set up residual stress.

Loaded below elastic limit

Loaded beyond elastic limit

Unloaded

Shaded areas show regions plastically strained

Methods to Measure Residual Stress

X-ray diffraction.

Nondestructive for the measurements near the surface: t < 2 m.

Neutron diffraction.

Can be used to make measurements deeper in the material, but the minimum volume that can be examined is quite large (several mm3) due to the low intensity of most neutron beams.

Dissection (mechanical relaxation).

Destructive.

General Principles

Consider a rod of a cross-sectional area A stressed in tension by a force F.

0,, zxyA

FStress:

Stress y produces strain ey:

0

0

L

LL

L

L f

y

e

L0 and Lf are the original and final lengths of the bar.The strain is related to stress as:

yy Ee

L increases D decreases so:

0

0

D

DD

D

D f

zx

ee

yzx eee for isotropic material

– Poisson’s ratiousually 0.25 < < 0.45

x-rays

General Principles

This provides measurement of the strain in the z direction since:

0

0

d

ddnz

e

y

y

Then the required stress will be:

0

0

d

ddE ny

Diffraction techniques do not measure stresses in materials directly

• Changes in d-spacing are measured to give strain• Changes in line width are measured to give

microstrain• The lattice planes of the individual grains in the

material act as strain gauges

General Principles

we do not know d0 !

0

0

d

ddE ny

Elasticity

In general there are stress components in two or three directions at right angles to one another, forming biaxial or triaxial stress systems.

Stresses in a material can be related to the set of three principal stresses 1, 2 and 3.

To properly describe the results of a diffraction stress measurement we introduce a coordinate systems for the instrument and the sample. These two coordinate systems are related by two rotation angles and f.

By convention thediffracting planes arenormal to L3

Li – laboratory coordinate systemSi – sample coordinate system

Elasticity

In an anisotropic elastic material stress tensor ij is related to the strain tensor ekl as:

klijklij C e

where Cijkl is elastic constants matrix.

Similarly:

klijklij S e

where Sijkl is elastic compliance matrix.

For isotropic compound:

kkijijijEE

e

1

where ij is Kroenecker’s delta, “kk” indicates the summation 11 + 22 + 33

Elasticity

Or we can write it as:

1212

3131

2323

22113333

33112222

33221111

2

1

,2

1

,2

1

,1

,1

,1

e

e

e

e

e

e

E

E

E

where

12

E

shear modulus

Elasticity

Lets relate emn in one coordinate system to that in another system through transformation matrix:

S

ij

SL

nj

SL

mi

L

mn MM ee

The transformation matrix is:

fff

fff

cos0sin

sinsincoscossin

sincossincoscosSLM

so that we find

fefe

efe

fefe

eef

2sinsin2sincos

cossinsin

sin2sinsincos

2313

2

33

22

22

2

12

22

11

3333

SS

SS

SS

S

ij

SL

j

SL

i

L MM

Elasticity

In terms of stresses:

0

0

2313332211

33

2

33

2

2212

2

1133

2sinsincos1

1sinsin2sincos

1

d

dd

EE

EE

SSSSS

SSSSSL

f

f

ff

fff

e

Biaxial and Triaxial Stress Analysis

Biaxial stress tensor is in the form:

000

00

00

22

11

since stress normal to a free surface must be zero:

For our tensor lets define:

0jijn

fff2

22

2

11 sincos SSS

Then the equation for strain becomes:

SSSL

EEd

dd2211

2

0

0sin

1

e f

f

f

The sin2 Method

Stress 33 is zero, but strain e33 is not zero. It has finite value given by the Poisson contractions due to 11 and 22:

SSSSS

E2211221133

eee

ee

f

f2

33 sin1

E

S

SL

Then strain equation can be written as:

ee

ff

f

f

2

0

0

0

0

0

33

sin1

Ed

dd

d

dd

d

dd

S

n

nSL

The sin2 Method

We make ingenious approximation (by Glocker et al. in 1936):

dn, di and d0 are very nearly equal to one another,

(di – dn) is small compared to d0,

unknown d0 is replaced by di or dn with negligible error.

ff 2sin1

Ed

dd

n

n

n

n

d

ddE f

f

2sin1

• The stress in a surface can be determined by measuring the d-spacing as a function of the angle between the surface normal and the diffracting plane normal

• Measurements are made in the back-reflection regime (2 180°) to obtain maximum accuracy

The sin2 Method

ff 2sin1

Ed

dd

n

n

The sin2 Method

Lets assume that stresses in zx plane are equal. This is referred to as an equal-biaxial stress state. We can write sample frame stress as:

000

00

00

No stress dependence on f

2sin

1

Ed

dd

n

n

Biaxial or uniaxial stress Triaxial stresses present Texture present

sin2 sin2 sin2

> 0 < 0

The sin2 Method

nn dd

Ed

2sin1

Slope of the plot is:

nd

E

d

1

sin2

Generally and E are well-known constants

d(3

11)(Å

)

sin2()

150.00 152.40 154.80 157.20 159.60 2

Inte

nsi

ty

Diffractometer Method

sin

sin

R

D

Diffractometer Method

The effect of sample or -axis displacement can be minimized if a parallel beam geometry is used instead of focused beam geometry.

Measurement of Line Position

Sample must remain on the diffractometer axis as is changed (even if the sample is large)

Radial motion of the detector to achieve focussing must not change the measured 2

L-P factor may vary significantlyacross a (broad) peak

Absorption will vary when 0

Measurement of peak positionoften requires fitting the peakwith a parabola:

cottan1

sin

2cos12

2

LPA

ba

ba3

2

222 10

2312

32

12

22222

IIb

IIa

Measurements of Stress in Thin Films

Thin films are usually textured. No difficulty with moderate degree of preferred orientation.

Sharp texture has the following effects:

Diffraction line strong at = 0 and absent at = 45o.

If material anisotropic E will depend on direction in the specimen. Oscillations of d vs sin2.

Measurements of Stress in Thin Films

In thin films we have a biaxial stress, so:

ff 2sin1

Ed

dd

n

n

If we have equal-biaxial stress then it is even simpler:

2sin

1

Ed

dd

a

aan

hkl

n

hklhkl

n

n

We can calculate for any unit cell and any orientation. For cubic:

222222cos

lkhlkh

llkkhha

Symbols h, k and l define (hkl) - oriented film

Texture Analysis

The determination of the lattice preferred orientation of the crystallites in a polycrystalline aggregate is referred to as texture analysis.

The term texture is used as a broad synonym for preferred crystallographic orientation in a polycrystalline material, normally a single phase.

The preferred orientation is usually described in terms of pole figures.

{100} poles of a cubic crystal

The Pole Figures

Let us consider the plane (h k l) in a given crystallite in a sample. The direction of the plane normal is projected onto the sphere around the crystallite.

The point where the plane normal intersects the sphere is defined by two angles: pole distance a and an azimuth b.

The azimuth angle is measured counter clock wise from the point X.

The Pole Figures

Let us now assume that we project the plane normals for the plane (h k l) from all the crystallites irradiated in the sample onto the sphere.

Each plane normal intercepting the sphere represents a point on the sphere. These points in return represent the Poles for the planes (h k l) in the crystallites. The number of points per unit area of the sphere represents the pole density.

Random orientation Preferred orientation

The Pole Figures

The Stereographic Projection

As we look down to the earth The stereographic projection

The Stereographic Projection

The Pole Figures

The Pole Figures

The Pole Figures

We now project the sphere with its pole density onto a plane. This projection is called a pole figure.

A pole figure is scanned by measuring the diffraction intensity of a given reflection with constant 2 at a large number of different angular orientations of a sample.

A contour map of the intensity is then plotted as a function of the angular orientation of the specimen.

The intensity of a given reflection is proportional to the number of hkl planes in reflecting condition.

Hence, the pole figure gives the probability of finding a given (h k l) plane normal as a function of the specimen orientation.

If the crystallites in the sample have random orientation the contour map will have uniform intensity contours.

The Pole Figures

(1 0 0)

(1-11)

(1 1 1)(1 -1 1)

(-1 -1 1)

Rotation Axis f

Rotation

Axis

(1 0 0)

{1 1 1}

out-of-plane

direction

The Pole Figures

(1 1 1)

(0 0 1)

(0 1 0)

(1 0 0)

Rotation Axis f

Rotation

Axis

(1 1 1)

out-of-plane

direction

{1 0 0}

Texture Measurements

Requires special sample holder which allows rotation of the specimen in its own plane about an axis normal to its surface, f, and about a horizontal axis, c.

Reflection Transmission

Schulz Reflection Method

In the Bragg-Brentano geometry a divergent x-ray beam is focused on the detector.

This no longer applies when the sample is tilted about c.

Advantage: rotation in around c in the range 40o < c < 90o

does not require absorption correction.

Field and Merchant Reflection Method

The method is designed for a parallel incident beam.

Defocusing Correction

As sample is tilted in c, the beam spreads out on a surface.

At high c values not all the beam enters the detector.

Need for defocusing correction.

Change in shape and orientation of the irradiated spot on a sample surface for different sample inclinations as a function of tilt angle a and Bragg angle 2. The incident beam is cylindrical.

Intensity correction for x-ray pole figure determination in reflection geometry. Selected reflections for quartz.

U.F. Kocks, C.N. Tomé, and H.-R. Wenk, “Texture and Anisotropy” 2000.

Absorption Correction

Diffracted intensities must be corrected for change in absorption due to change in a.

dxeabI

dI x

D

b

sin1sin10

sin

aba oo 90,90

Integrate 0 < x <

substitute

aa

coscos1

0abIID

We interested in the intensity at angle a, relative to intensity at a = 90o:

aa

aacotcot1

90

o

D

D

I

IS

a – volume fraction of a specimen containing particles having correct orientation for reflection of the incident beam.b – fraction of the incident energy which is diffracted by unit volume

Pole Figure Measurement

Pole figure diffractometer consists of a four-axis single-crystal diffractometer.

Rotation axes:

, w, c, and f

Example: Rolled Copper

Texture measurements were performed on Cu disk Ø = 22 mm, t = 0.8 mm. Four pole figures (111), (200), (220) and (311) were collected using Schulz reflection method. Background intensities were measured next to diffraction peaks with offset 2 = 4o. Defocusing effects were corrected using two methods:

• measured texture free sample• calculated (FWHM of the peaks at = 0o is required – obtained from -2 scan.

• X’Pert Texture program was used for quantitative analysis.


• Measure -2 scan in order to determine the reflections used for the pole figure measurements.• Use FWHM of the peaks to calculate the defocusing curve.


Pole figure intensities include background. Correction for background radiation is performed by

measuring the intensity vs. -tilt next to the diffraction peak.

Background Correction


Experimental pole figures are corrected for background intensities. Either experimental or theoretical defocusing correction curve is applied.

Corrections

Measured and calculated defocusing corrections for Cu(220) pole figure


2D representation of Cu(220) pole figure as (a) measured and (b) corrected. The most noticeable effect is at higher -tilt angles.

Corrections

(a) (b)


Four pole figures have been measured. Symmetry of rolling process is obtained

from the pole figures:• pole figure is symmetrical around

f = 0o and f = 90o. The symmetry is called orthorhombic

sample symmetry.

Pole Figure Measurements


X’Pert Texture calculates ODF When ODF is available X’Pert Texture can

calculate pole figures and inverse pole figures for any set of (hkl).

Orientation Distribution Function Calculation

Measured Calculated

Inverse Pole Figures

Inverse Pole Figures

Engineering Cu surfaces for the electrocatalyticconversion of CO2

X-ray pole figures for:

(A) Cu on Al2O3(0001); Cu(200)

intensities are shown.

(B) Cu on Si(100); Cu(111)


(C) Cu on Si(111); Cu(111)


(D) Inverse pole figure shows

highest intensities for the (751)

plane, indicating that Cu on

Si(111) is predominantly

oriented in the [517] direction

out-of-plane.

Cu(200) Cu(111)

Cu(111)

TaN Thin Film

w (deg)

-1.0 -0.5 0.0 0.5 1.0

Inte

nsity (

cp

s)

0

1000

2000

3000

4000

5000

6000

FWHM=0.72 deg

2 (deg)

41 42 43 44

Inte

nsity (

cps)

101

102

103

104

105

20

0 T

aN

20

0 M

gO

MgO (001)

TaN (001)

n

f (deg)

0 50 100 150 200 250 300 350

Inte

nsity (

cps)

MgO 220

TaN 220

TaN Thin Film

f-scan of TaN (202) and MgO (202) reflections.

TaN Thin Film

TEM reveals additional structure.

TaN Thin Film

XRD 002 pole figure of TaN film

TaN Thin Film

54

TaN Thin Film

Atomic Force Microscopy images of TaN films prepared under different N2 partial pressure

pN=2 mTorr pN=2.5 mTorr pN=4 mTorr

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Stress and Texture - Stanford University