Scattering contrast dependence on the difference in the thermal
expansion coefficient of phases in two-phase system
PSI (SANS-II) in May 2004,
DT706_1 (notation Dominique: 1)
0.012
1160K = 887degC
0.014
1107K = 834degC
An increase of the scattering intensity (see the Fig.) from the gamma prime precipitates at low and intermediate
temperatures in DT706 alloy (in situ, SINQ, May 2004) was observed during the temperature decrease. This
behavior is typical for all the measured samples and was observed also in the previous measurement of DT706 in
HMI Berlin, October 2003. The same characteristics exhibits also EI698VD superalloy (measured in OPctober
2004 in PSI).
Sum/moni
0.010
SumMon
0.008
0.006
0.004
precipitates originated
already slightly before
holding
0.002
400
600
800
1000
1200
1400
temperature (K)
Increase of the scattering intensity (and contrast) from the gamma prime
precipitates at low and intermediate temperatures in DT706 alloy (in situ,
SINQ, May 2004) during the temperature decrease.
At lower temperatures (below 700degC), the diffusion cannot play an important role during the relatively fast
cooling: a change of the chemical composition of the precipitates or matrix can be thus excluded to cause such a
change.
Diffusion scattering (decreasing the intensity of the transmitted beam when the temperature increases) can be
also excluded: its effect on transmission is a couple of percent only in the given temperature range.
As the scattering contrast coming from different compositions of gamma and gamma prime is relatively low, the
expansion/contraction of the lattice during heating/cooling (usually unimportant for the contrast) can play an
additional role here in the change of the contrast.
This effect is nearly linear with temperature in DT706 superalloy, even at very low temperatures, what further
supports the hypothesis about thermal expansion influence, which is in fact nearly linear (and difference of two
linear dependencies is again linear dependency).
The lattice parameter for DT706 thus surely changes differently for gamma and for gamma prime (different
thermal expansions coefficient) with temperature.
This effect is quite significant: it is around 17% in square scattering contrast when cooling from 700degC to
100degC.
Theoretical calculations follow:
Scattering contrast (assuming dependence of lattice parameters on temperature T and no influence of the
temperature on the composition) between precipitate and matrix in a two-phase system is
2
 T 
2
 b

b p 
b p 
  bm  3 1  31

 3 m  3
 a T  a T  
 a T  a T  b 
p
p
m 
 m

 m
2
Then, ratio of contrasts at temperature T and at room temperature TR is
 T 2
 R 2

 1
1 b p

 3
3
 am T  a p T  bm





2
2
 1
1 b p 

 3
3
 amR a pR bm 


Using approximation of linear and constant thermal expansion coefficients
http://en.wikipedia.org/wiki/Coefficient_of_expansion
m 
1
a mR
a m T   a mR
T  TR 
in the measured temperature range we arrive at
 T 2
 R 2


1


  T  T 1a 3
m
R
mR

1
r m T TR 1a pR 3
b p 
bm 

2
2
ratio of scattering contrasts at 700degC and at 100degC
 1
1 b p 

 3
3
a
bm 
 mR a pR

where r =αp/αm is the ratio of the thermal expansion coefficients of precipitate and matrix. The following graph is
a simulation of the influence of r and of ratio of scattering lengths sum on the ratio of scattering contrasts at
700degC and at 100degC:
F2
F3
F5
F4
1.2
sum(bP)/sum(bM) [ratio of sc.lengths over unit cell]
simulation:
0.9
0.959 (i.e. 70E9/73E9)
0.98
1.00
1.1
simulation:
matrix 3.559 A, precipitate 3.572 A at RT
1.0
DT706:
sum(bP)/sum(bM) [ratio of sc.lengths over unit cell]
estimated to be 0.959 (i.e. 70E9/73E9)
0.9
thermal expansion coefficient of matrix estimated
to be 15.39E-6 (valid for IN706 at 400degC)
0.8
0.6
0.8
1.0
1.2
1.4
ratio of thermal expansion coefficients of precipitate and matrix
In this simulations, lattice parameters of gamma and gamma prime of IN706 were used (see Appendix).
Thermal expansion coefficient of matrix was used according to the data sheet of IN706 at
http://www.specialmetals.com/products/inconelalloy706.htm
It can be seen that the nearer the ratio of scattering lengths sum is to 1.0, the larger influence of the theremal
expansion coefficient difference exists. The influence of misfit (and its evolution with temperature) on scattering
contrast is larger in that case.
The observed difference (DT706) 17% gives (for the realistic ratio of scattering lengths sum coming out from the
SANS experiment and its modeling&fitting using NOC) the ratio r=0.88.
With these values, the explanation of the observed behavior of DT706 during in-situ SANS using different
thermal expansions for gamma and for gamma prime is realistic (see the next graph giving the simulated
temperature dependence).
0.014
scattering contrast (rel. units)
0.012
0.010
0.008
F6
0.013*
(1/((15.39E-6*(X-373)+1)*3.559)^3 1/((0.88*15.39E-6*(X-373)+1)*3.572)^3*0.959 )^2
/ (1/(3.559)^3 - 1/(3.572)^3*0.959 )^2
0.006
0.004
0.002
400
600
800
1000
temperature (K)
Misfit evolution with temperature is:
r T T 1a

T T 1a
pR
m
R
mR
misfit  2 r mm T TRR 1a pR
  m T TR 1a mR
In our case, particularly:
0.016
0.014
0.012
Y Axis Title
0.010
F8 (aprecip-amatrix)
0.008
F7 misfit =
2*(aprecip-amatrix)/(aprecip+amatrix)
0.006
0.004
0.002
0.000
0
200
400
600
X Axis Title
800
1000
Appendix
Crystal Structure Data for Different Phases in Ni-base Alloys
Symbol

'
''

Phase
Ni-(Al,Cr,..)
Ni3Al
Ni3Nb
Ni3Ti
Crystal Structure
fcc (A1)
ordered fcc (L12)
bct (DO22)
Hexagonal (DO24)
Structure Type
Cu
AuCu3
Al3Ti
Ni3Ti
Space Group
Fm 3bar m
Pm 3bar m
I4 / mmm
P63 / mmc
Pearson Symbol
cF4
cP4
tI8
hP16
Space Gp. No.
225
221
139
194
Symbol

Atom Type
Ni
Al
Wyckoff no
4a
4a
x
y
0
0
0
0
No. Of Atoms per unit cell = 4
z
0
0
Al
Ni
1a
3c
0
0
0
1/2
No. Of Atoms per unit cell = 4
0
1/2
1
1
''
2a
2b
4d
0
0
0
0
0
1/2
No. Of Atoms per unit cell = 8
0
1/2
1/4
1
1
1

2a
2c
6g
6h
0
0
1/3
2/3
1/2
0
0.833
0.666
No. Of Atoms per unit cell = 16
0
1/4
0
1/4
1
1
1
1
Atom Positions
Structrue Type
Ni (Al)
Lattice Parameter : a = 0.3559 nm
Ni3Al
'
Lattice Parameter : a = 0.35720 nm
Ni3Nb
Ni3Ti
Nb
Ni (1)
Ni (2)
Lattice Parameter : a = 0.362 nm; c = 0.741 nm
Ti (1)
Ti (2)
Ni (1)
Ni (2)
Lattice Parameter : a = 0.51010 nm; c = 0.83067 nm;  = 120°
Tab. 1. Crystal Structure Data for Different Phases in Ni-base Alloy INCONEL 706.
occupancy
0.5
0.5