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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 ([email protected])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