April 18, 2023 1

Scattering contrast dependence on thermal-expansion-coefficient difference

of phases in two-phase systemP. Strunz1,2, R. Gilles3, D. Mukherji4, M. Hofmann5, D. del Genovese4, J.

Roesler4, M. Hoelzel3 and V. Davydov1

1Nuclear Physics Institute, CZ-25068 Řež near Prague (strunz@ujf.cas.cz)2Research Centre Řež, CZ-25068 Řež near Prague, Czech Republic

3TU München, ZWE FRM-II, Lichtenbergstr. 1, D-85747 Garching, Germany4TU Braunschweig, IfW, Langer Kamp 8, D-38106 Braunschweig, Germany5TU Darmstadt c/o FRM II, Lichtenbergstr. 1, D-85747 Garching, Germany

Project supported by the European Commission under the 6th Framework Programme through the Key Action: Strengthening the European Research Area, Research Infrastructures. Contract n°: RII3-CT-2003-505925 '

Outline Observation of SANS

intensity increase Theory Simulation

Experiment• diffraction• SANS

Prospective application

April 18, 2023 2

SANS – tool for microstructural characterizationSANS – tool for microstructural characterization

Microstructural characterization: essential part in any alloy development

Neutron scattering: increasingly complementing XRD, SEM, TEM

Microstructural characterization: essential part in any alloy development

Neutron scattering: increasingly complementing XRD, SEM, TEM

scattering caused by γ’ precipitates (ordered fcc -

L12 crystal structure) coherently embedded in γ

matrix (crystal structure fcc - A1)

scattering caused by γ’ precipitates (ordered fcc -

L12 crystal structure) coherently embedded in γ

matrix (crystal structure fcc - A1)

1E-3 0.01 0.10.01

0.1

1

10

100

1000

10000

azimuthal average

Q (Å-1)d

/d

(cm

-1sr

-1)

DA ST MST MST-1

DT706 superalloy (wt.%: Fe=22, Cr=18, Nb=2.9, Ti=1.9, Al=0.55, C=0.03, Ni = balance)

DT706 superalloy (wt.%: Fe=22, Cr=18, Nb=2.9, Ti=1.9, Al=0.55, C=0.03, Ni = balance)

April 18, 2023 3

SANS data intensity in “low”-temperature regionSANS data intensity in “low”-temperature region

Increase of the integral SANS intensity from γ' precipitates at low and intermediate temperatures during the temperature decrease

DT706 (SINQ, SANS-II) 17% increase, nearly linear

Increase of the integral SANS intensity from γ' precipitates at low and intermediate temperatures during the temperature decrease

DT706 (SINQ, SANS-II) 17% increase, nearly linear

400 600 800 1000 1200 14000.002

0.004

0.006

0.008

0.010

0.012

0.014

heating

formation of ' precipitates

PSI (SANS-II), DT706

1107

K =

834

°C10

61K

= 7

88°C

Integral intensity

Inte

gral

inte

nsity

(re

l. un

its)

temperature (K)

Possible cause

volume fraction change of γ’

change in the size distribution of γ’ precipitates

γ’ scattering contrast change

Possible cause

volume fraction change of γ’

change in the size distribution of γ’ precipitates

γ’ scattering contrast change

April 18, 2023 4

scattering contrast changescattering contrast change

scattering length densities (SLD) m,p=[bm,p]/am,p3 (matrix, precip.)

[bm], [bp] not changed but am, ap change with temperature

Can it significantly change the scattering contrast?

scattering length densities (SLD) m,p=[bm,p]/am,p3 (matrix, precip.)

[bm], [bp] not changed but am, ap change with temperature

Can it significantly change the scattering contrast?

Answer: yes, under certain circumstances

Circumstances (fulfilled in superalloys)

low Δ with respect to high volume fraction (to make SANS visible)

Answer: yes, under certain circumstances

Circumstances (fulfilled in superalloys)

low Δ with respect to high volume fraction (to make SANS visible)

)1(

2

33

22

Ta

b

Ta

bTTT

m

m

p

pmp

April 18, 2023 5

Theory – scattering contrastTheory – scattering contrast

Scattering contrast of a two-phase systemScattering contrast of a two-phase system

[bm], [bp] usually unknown, but temperature independent

known [b]alloy

c … volume fraction of γ’ precipitates

[bm], [bp] usually unknown, but temperature independent

known [b]alloy

c … volume fraction of γ’ precipitates

)1(

2

33

22

Ta

b

Ta

bTTT

m

m

p

pmp

)8(

,)(1

12

3

2

cTv

b

Ta

b

TcT

c

alloy

p

p

)5(

11

,

33 Ta

c

Ta

ccTv

pm

c

the average unit cell volume the average unit cell volume

April 18, 2023 6

Theory - integral SANS intensity Theory - integral SANS intensity

when a part of the assymptotic (Porod) region is used: the shape of the scattering curve cannot change (Porod law) only the dependence of the specific interface and sample

thickness on the temperature has to be taken into account =>

when a part of the assymptotic (Porod) region is used: the shape of the scattering curve cannot change (Porod law) only the dependence of the specific interface and sample

thickness on the temperature has to be taken into account =>

where all T-independent parameters are in the constant C2

the ratio (ap/νc1/3)2 is only marginally temperature dependent =>

temperature dependence of intensity driven by numerator in the scattering contrast form:

where all T-independent parameters are in the constant C2

the ratio (ap/νc1/3)2 is only marginally temperature dependent =>

temperature dependence of intensity driven by numerator in the scattering contrast form:

)11(

,2

2

32 T

cTv

TaCTI

c

p

cTv

b

Ta

b

c

alloy

p

p

,3

April 18, 2023 7

Scattering contrast simulationScattering contrast simulation

using lattice parameters of γ and γ’ determined in DT706 the temperature dependence for various [Σb]p (fixed [Σb]alloy) volume fraction fixed (c=0.1)

using lattice parameters of γ and γ’ determined in DT706 the temperature dependence for various [Σb]p (fixed [Σb]alloy) volume fraction fixed (c=0.1)

increasing / decreasing temperature dependence determines which SLD (precipitate or matrix) is smaller

strong correlation “curve shape” – “magnitude of the scattering contrast”

increasing / decreasing temperature dependence determines which SLD (precipitate or matrix) is smaller

strong correlation “curve shape” – “magnitude of the scattering contrast”

0 100 200 300 400 500 600 700 800 900 10001E14

1E15

1E16

1E17

1E18

1E19

1E20

c=0.1

3.55E-12

3.42E-12

3.366E-12

3.38E-12

3.25E-12

balloy

=3.34076E-12 cmb

p [cm]

3.36E-12

3.372E-12

3.35E-12

3.33E-12

3.0E-123.8E-12

2(T) simulation

2 (T

) (

cm-4)

temperature (°C)

April 18, 2023 8

Scattering contrast simulationScattering contrast simulation

volume-fraction change simulation: change of the curve due to [Σb]p change can be nearly

equivalently achieved by changing c

volume-fraction change simulation: change of the curve due to [Σb]p change can be nearly

equivalently achieved by changing c

=> [Σb]p and volume

fraction of precipitates are correlated parameters

=> [Σb]p and volume

fraction of precipitates are correlated parameters

0 100 200 300 400 500 600 700 800 900 10001E18

1E19

1E20c=0.10

c=0.10 3.25E-12

balloy

=3.34076E-12 cm bp [cm]

3.33E-12

3.0E-12

2(T) simulation

2 (T

) (

cm-4)

temperature (°C)

3.25E-12c=0.75

c=0.5 3.25E-12

c=0.25 3.25E-12

c=0.01 3.25E-12

April 18, 2023 9

Experimental and results - Diffraction experimentExperimental and results - Diffraction experiment

DT706 samples in-situ at elevated temperatures at FRM-II (SPODI and

StressSpec)

Initial heat treatment: solution treatment step at 1080°C for 2 h (dissolve γ’) stabilization step at 835°C for 10 h (new population of γ’

precipitates)

In situ measurement: temporary stops (≤2 h) during the temperature decrease

(700, 600, 500, 400, 300, 200, 100°C, RT)

DT706 samples in-situ at elevated temperatures at FRM-II (SPODI and

StressSpec)

Initial heat treatment: solution treatment step at 1080°C for 2 h (dissolve γ’) stabilization step at 835°C for 10 h (new population of γ’

precipitates)

In situ measurement: temporary stops (≤2 h) during the temperature decrease

(700, 600, 500, 400, 300, 200, 100°C, RT)

April 18, 2023 10

Experimental and results - Diffraction peaksExperimental and results - Diffraction peaks

Largest accessible anglular range (separation of the γ and γ’ peaks)

reflection 311 (StressSpec) and 331 (SPODI)

Figs.: γ and γ’ double peaks recorded at StressSpec and SPODI

instrumental profile deconvoluted using ProfEdgeReal program

γ’ peaks: 10% of γ peaks

Largest accessible anglular range (separation of the γ and γ’ peaks)

reflection 311 (StressSpec) and 331 (SPODI)

Figs.: γ and γ’ double peaks recorded at StressSpec and SPODI

instrumental profile deconvoluted using ProfEdgeReal program

γ’ peaks: 10% of γ peaks

103.0 103.5 104.0 104.5 105.0 105.5 106.0 106.5 107.00

2000

4000

6000

8000

10000

DT706StressSpec311100°C

Measured intensity Fit Deconvoluted profile

Mea

sure

d an

d fit

ted

data

(n/

100s

)

2 (°)

135 136 137 138 139 140 141 1420

50

100

150

200

250

300

350

400

450

DT706SPODI331400°C

Measured intensity Fit Deconvoluted profile

Mea

sure

d an

d fit

ted

data

(n

/100

s)

2 (°)

April 18, 2023 11

Experimental and results – lattice parametersExperimental and results – lattice parameters

Approximation of lattice parameter by quadratic polynomial am(T) = 3.585155 + 4.5891E-5×T + 2.1355E-8×T2 [matrix]

ap(T) = 3.598196 + 4.1247E-5×T + 1.7052E-8×T2 [γ']

Approximation of lattice parameter by quadratic polynomial am(T) = 3.585155 + 4.5891E-5×T + 2.1355E-8×T2 [matrix]

ap(T) = 3.598196 + 4.1247E-5×T + 1.7052E-8×T2 [γ']

0 200 400 600 800 1000 12003.58

3.59

3.60

3.61

3.62

3.63

3.64

3.65

3.66

3.67

0 200 400 600 800 1000 12003.58

3.59

3.60

3.61

3.62

3.63

3.64

3.65

3.66

3.67

0.000

0.001

0.002

0.003

0.004

0.005

0.006

fit

latt

ice

para

met

er (

Å)

Temperatutre (°C)

DT706, SPODI and StressSpec results

misfit

fit

mis

fit (

dim

en

sio

nle

ss)

Combination of the data obtained form both SPODI and StressSpec

=> the evolution of the lattice parameters and misfit (RT-835°C)

Combination of the data obtained form both SPODI and StressSpec

=> the evolution of the lattice parameters and misfit (RT-835°C)

April 18, 2023 12

Experimental and results – SANS integral intensityExperimental and results – SANS integral intensity

SANS II, SINQPorod region of the

scattering curve: sample-to-detector

distance 5m λ = 4.55 ÅQ = 0.01-0.08 Å-1

I(T) corrected for background and transmission

SANS II, SINQPorod region of the

scattering curve: sample-to-detector

distance 5m λ = 4.55 ÅQ = 0.01-0.08 Å-1

I(T) corrected for background and transmission

The weighted fit using the derived theory and the analytical approximation of am(T) and ap(T) from neutron diffraction

The fitted parameters are C2, cR and [Σb]p.

The weighted fit using the derived theory and the analytical approximation of am(T) and ap(T) from neutron diffraction

The fitted parameters are C2, cR and [Σb]p.

0 100 200 300 400 500 600 700 800 9000.0100

0.0105

0.0110

0.0115

0.0120

0.0125

Data: Data1_SumoverMonCorrModel: ScatteringIntensityWeighting: I w = 1/(data1_errcorr)^2 Chi^2/DoF = 1.54271R^2 = 0.96364 const 2.649E-22 ±2.133E-20cR 0.14372 ±2163.1Sbp 3.09932E-12 ±6.15067E-10Agp 3.598196 ±0Bgp 4.1247E-5 ±0Cgp 1.7052E-8 ±0 measured integral intensity

fit confidence bands

PSI (SANS-II), DT706

Inte

gral

inte

nsity

(ne

utro

ns /

mon

itor

coun

t)

temperature (°C)

April 18, 2023 13

Integral SANS data evaluation and discussionIntegral SANS data evaluation and discussion

[Σb]p and cR parameters are correlated

Nevertheless, the resulting ΔρR is very insensitive to the input value of cR

[Σb]p and cR parameters are correlated

Nevertheless, the resulting ΔρR is very insensitive to the input value of cR

=> scattering contrast (ΔρR)2 can be determined without a non-trivial measurement of composition of the individual phases

=> scattering contrast (ΔρR)2 can be determined without a non-trivial measurement of composition of the individual phases

Tab. 1. The results of the fit for various cR cR, vol. fraction at RT

[b]p (10-12 cm)

C2 (10-22 cm4 neutrons/ monitor count)

pR (109 cm-2)

[b]m (10-12 cm)

mR (109 cm-2)

R (109 cm-2)

(R)2 (1019 cm-4)

0.05 3.0744 2.674 65.942 3.35463 72.733 -6.792 4.613 0.10 3.0877 2.661 66.227 3.36858 73.036 -6.809 4.636 0.14372 3.09932 2.649 66.476 3.38085 73.302 -6.826 4.659 0.15 3.1010 2.648 66.512 3.38262 73.340 -6.828 4.662 0.20 3.1144 2.635 66.800 3.39674 73.646 -6.846 4.687

April 18, 2023 14

Temperature dependence of the scattering contrastTemperature dependence of the scattering contrast

most probable and extreme values of cR

most probable and extreme values of cR

0 100 200 300 400 500 600 700 800 900

3.6x1019

3.8x1019

4.0x1019

4.2x1019

4.4x1019

4.6x1019

4.8x1019

5.0x1019

scattering contrast whenc

R=0.14327

cR=0.05

cR=0.20

temperature evolution of the scattering contrast

scat

terin

g co

ntra

st (

R)2 (

cm-4

)

temperature (°C)

scattering contrast of γ’ in γ matrix, DT706

scattering contrast of γ’ in γ matrix, DT706

April 18, 2023 15

The expressions for SANS scattering contrast dependence on temperature (no phase-composition changes) <= difference in thermal expansions of γ and γ’ in Ni superalloys.

Simulation: this difference is the determining factor for the (Δρ)2 temperature dependence

The hypothesis proven by experiment on a Ni-Fe-base alloy DT706. The evolution of lattice parameters of both phases obtained from the in-situ wide angle neutron diffraction. The theoretical scattering contrast dependence was then successfully fitted to the measured SANS integral intensity.

The magnitude of (ΔρR)2 is firmly connected with the particular shape of the SANS integral intensity temperature dependence => used for the determination of the scattering contrast without the knowledge of the compositions of the individual phases

Investigation of superalloys with no scattering contrast at RT

The expressions for SANS scattering contrast dependence on temperature (no phase-composition changes) <= difference in thermal expansions of γ and γ’ in Ni superalloys.

Simulation: this difference is the determining factor for the (Δρ)2 temperature dependence

The hypothesis proven by experiment on a Ni-Fe-base alloy DT706. The evolution of lattice parameters of both phases obtained from the in-situ wide angle neutron diffraction. The theoretical scattering contrast dependence was then successfully fitted to the measured SANS integral intensity.

The magnitude of (ΔρR)2 is firmly connected with the particular shape of the SANS integral intensity temperature dependence => used for the determination of the scattering contrast without the knowledge of the compositions of the individual phases

Investigation of superalloys with no scattering contrast at RT

SummarySummary

April 18, 2023 16

The authors are indebted to SINQ (PSI Villigen, Switzerland) and FRM II (TU Muenchen, Germany) for providing the beam time at the SANS-II facility and diffractometers StressSpec and SPODI

NMI3 support is acknowledged as well (6th Framework Programme ‘Strengthening the European Research Area, Research Infrastructures’ - contract no. RII3-CT-2003-505925

We thank the sample environment group of FRM II (A. Schmidt and A. Pscheidt) for support during the high-temperature experiment

The authors are indebted to SINQ (PSI Villigen, Switzerland) and FRM II (TU Muenchen, Germany) for providing the beam time at the SANS-II facility and diffractometers StressSpec and SPODI

NMI3 support is acknowledged as well (6th Framework Programme ‘Strengthening the European Research Area, Research Infrastructures’ - contract no. RII3-CT-2003-505925

We thank the sample environment group of FRM II (A. Schmidt and A. Pscheidt) for support during the high-temperature experiment

AcknowledgmentsAcknowledgments

April 18, 2023 17

April 18, 2023 18

volume fraction temperature dependencevolume fraction temperature dependence

)6(

111

1

3

3

3

3

mR

pR

p

m

R a

a

Ta

Ta

c

Tc