− ,
⊥
⊥
−
∥
∥
Δ( − ) Δ( − )
⊥
∥
∥
⊥
− ,
‖
‖
‖
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
(1 1 1) δ
‖
‖
.
‖
(1 1 1)
α
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
−
{10.1} = 5.77 × 10 K −1
−6
−
−
−6
{00.2} = 7.62 × 10 K −1 {10.1} −
{00.2}
{10.0} α−Zr {00.2} α−Zr
− = 3.23118
− = 5.14634
α
− (Å) = 3.23118 + 18.6439 × 10 ( (K) − 298)
−6
−6
− (Å) = 5.14634 + 39.2152 × 10 ( (K) − 298)
{10.0} α−Zr
{00.2} α−Zr
−
∆ {10.1}
− ( (K)) − 3.23118)
−
∆ {10.1} = − ( (K))
− ( (K)) + 3.23118
− ( (K)) = 2
− ( (K)) Å
−
Table 4-11: Total change in strain increments from ambient temperature for unreoriented hydrides in TD: ∆ {10.1} +
−
∆( − ), and reoriented hydrides in RD: ∆ {00.2} + ∆( − ), calculated based on the results
‖
‖
obtained using the pure lattice strain transformation strains. All strains are in units of 10 .
-3
T(C) Expansion Expansion Hydride Net Hydride Net Hydride Total Hydride Total
Strain Strain Strain: Strain: Strain: Strain:
Increment: Increment: ∆( − ) ∆( − ) ∆ − ∆ −
{ . }
{ . }
∆ − ∆ − ‖ ‖ + ∆( − ) + ∆( − )
{ . }
{ . }
‖
‖
25 0 0 0 0 0 0
225 1.151 1.524 0.87 –30.48 2.02 –29.28
300 1.583 2.096 3.12 –45.65 4.71 –43.55
400 2.171 2.858 3.64 –59.52 5.80 –56.66
454 2.479 3.269 4.97 –68.63 7.45 –65.36
504 2.757 3.650 5.81 –76.07 8.57 –72.42
© ANT International 2018
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(1 1 1) δ ‖
(1 1 1) δ
‖
(0 2 2) δ
(0 2 2) δ
(1 1 1) δ
(1 1 1) δ
‖
(1 1 1)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(1 1 1) δ
‖
‖
‖
(1 1 1)
‖
= 0.6
= 0.3 = 0.1
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(α + )/
1.488 × 10 −6 m ⁄ (mol H)
3
3
2.016 × 10 −6 m ⁄ (mol H)
α
α
λ a
α
−6
3
−6
3
1.633 × 10 m /(mol D) 1.763 × 10 m /(mol D)
−6
3
1.700 × 10 m /(mol D)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-95: Calculated expansion induced by hydrogen uptake at 25C based on Kearns factors fr = 0.6, f = 0.3, fz = 0.1 assuming
(a) PLTS (referred to as PLST throughout this text) and (b) PSTS (referred to as IPST throughout this text) hydride
phase transformation models (from [Hellouin de Menibus et al., 2013]).
α H/Zr = 1
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-96: Differential dilatometry curves estimated for a specimen containing 500 wppm hydrogen assuming hydride
transformation strains calculated according to (a) the PLTS and (b) the PSTS phase transformation models (from
[Hellouin de Menibus et al., 2013]).
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(Perovic)
Figure 4-97: Experimental (plain lines) and calculated (dotted lines) differential dilatometry curves in the axial direction for a sample
containing 500 wppm hydrogen with the calculated lines determined assuming the (a) PLTS and (b) PSTS hydride
transformation models (from [Hellouin de Menibus et al., 2013]).
4.6.5.1
α
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-98: Optical micrographs of hydride precipitates in the as-received (left) state and after hydride reorientation (right) under a
tensile load of 225 MPa (65 MPa for the L1 specimen) applied in the hoop direction (from [Vicente
Alvarez et al., 2012]).
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-99: (a) Schematic of the sample geometry and experimental arrangements showing the azimuthal angle, ϕ, of the CCD
detector. (b) ambient temperature diffractograms for the as-received L2 specimen for two different azimuthal sections
corresponding to the directions referred to as and as explained further on in the text (from [Vicente
Alvarez et al., 2012]).
α
2θ
θ-2θ
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
α
{1 1 1} {2 2 0} α
{0 0 0 2} − = 0°
= 70°.
{0 0 0 2} − {1 1 1}
Figure 4-100: Intensities of the {0 0 0 2} − and {1 1 1} peaks for the L2 specimen for azimuthal angles, ϕ, ranging from the
axial to the hoop direction. The blue line corresponds to the as-received condition just after hydrogenation and
homogenization treatment whilst the yellow line corresponds to the result after hydride reorientation had occurred. In
this plot, arrows of the same size and colour indicate symmetry equivalent diffracting conditions of particular interest to
this study (from [Vicente Alvarez et al., 2012]).
= 0 α
(0 0 0 2) − ∥ (1 1 1)
= 90°
+
= 0
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
{0 0 0 2} − = 0°
{1 1 1} = 0
{1 1 1} = 90°
{0 0 0 2} −
α
{1 1 1}
{1 1 1}
= 0
(1 1 1)
∆ 111 15
−
∆ 111 =
(1 1 1)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Table 4-12: Intensity and d-spacing data in the as-received condition of the material and after hydride precipitation
under a tensile hoop stress. Only data of those precipitates formed in αZr grains of the most important
orientations are listed. A 225 MPa tensile stress was applied along the hoop direction during hydride
precipitation, resulting in some fraction of the total number of hydride precipitates to reorient in the radial
direction.
Sampl Processin [H], Experimental Intensity of ( ) d-spacing of ∆ of
e g schedule wppm condition ( ) , Å ( ) , *
= 0 = 70 = 90 = 0 = 70 Defined by
Equation 4-89
E1 Cold 55 3 As-received 7 1 9 3 2.7473 2.7395 2800 400
drawn
After 13 2 19 2 2.7468 2.7396 2600 400
reorientation
E2 “ 67 5 As-received 7.8 2 23.8 1 7.4 3 2.7466 2.7392 2700 200
After 21 3 45 1 3 3 2.7510 2.7400 4000 200
reorientation
L1 Cold 44 3 As- 2 0.5 7 0.6 2.7470 2.7405 2400 300
rolled received
16
After 3 0.2 8.7 2.7476 2.7408 2500 300
reorientation 0.2
L2 “ 130 5 As-received 15 2 35 2 10.5 2 2.7417 2.7403 500 200
After 34 2 53 6 5 3 2.7512 2.7427 3100 200
reorientation
≡ 10 −6
© ANT International 2018
∆ 111
(1 1 1)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-101: Variations of the d-spacing of the (1 1 1) peak of hydride precipitates for (a) and (b) orientations, for
the E2 and L2 samples as a function of temperature during the thermo-mechanical cycles as follows: 1-heating at
0 MPa; 2-cooling at 0 MPa; 3-loading; 4-heating at 225 MPa; 5-cooling at 225 MPa; 6-unloading. The value of the
room temperature unstressed d-spacing ( ) reported in [Santisteban et al., 2009; 2010] is indicated in the figure (from
[Vicente Alvarez et al., 2012]).
(1 1 1)
(1 1 1)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(1 1 1)
(1 1 1)
(1 1 1)
(1 1 1)
(1 1 1)
(1 1 1)
(1 1 1)
(1 1 1)
≈
(1 1 1)
(1 1 1)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
ϕ
ϕ
α
̅
̅
̅
̅
(1 0 1 0) − (1 0 1 1) − (1 1 2 0) − (0 0 0 2) − (1 0 1 2) −
̅
̅
̅
(1 0 1 3) − (1 1 2 2) − (2 0 2 1) −
(1 1 1) (2 2 0)
α
= 0°
2
=
1
̅
0, 180°; = 90° = 0° (1 0 1 1)
2 αZr
= 0, 180°; = 0° = 0°
2
1
= 90 to 0°
= 90, 270°; = 0° = 0°
1
2
α
α
ϕ
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-102: Crystallographic texture of the pressure tube material. (a) Schematic of the ideal orientations used in the description of
the αZr ODF. (b) Manifestation of the ideal components in the 2 = 0 section of the experimental αZr ODF. (c)
αZr pole figures calculated from the ODF, identifying the ideal components using symbols of different colours. The
experimental data are those from the L2 specimen (from [Vicente Alvarez et al., 2012]).
(1 1 1) (2 2 0)
(1 1 0)
(1 1 1)
(1 1 1)
(1 1 1)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-103: (a) Hydride pole figures determined from the intensity of the (1 1 1) and (2 2 0) peaks for the L2 sample in the as-
received condition; (b) Pole figures determined by Vicente Alvarez and co-workers [Vicente Alvarez et al., 2011] for
hydride precipitates contained in commercial pressure tube material (from [Vicente Alvarez et al., 2012]).
α
α
α
α
α
α
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
≈ 1 × 10 3
(1 1 1) ‖ (0 0 0 1) −
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
Figure 4-104: (a) Predicted hydride pole figure obtained by applying the orientation relationships {0 0 0 2} − ∥ (1 1 1) δ and
[1 1 2 ̅ 0] α−Zr ∥ (1 1 ̅ 0) to the experimental αZr ODF given in Figure 4-102b. The location of the poles associated
with hydride precipitation in the ideal αZr orientations are identified by symbols of different colours. (b) Predicted
(1 1 1) intensity profiles for hydrides precipitated in αZr grains having ideal orientations corresponding to the
azimuthal angles shown in Figure 4-100 (from [Vicente Alvarez et al., 2012]). (In (a) this is the circle at the junction of
the axial and hoop directions.)
(1 1 1)
α
α
(1 1 1)
= 0°
α
α-
α
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-105: Modelling of the (1 1 1) intensity registered for (a) as-received and (b) reoriented hydrides. The purple data points
connected by a purple line correspond to the experimental values. The coloured lines are the results of calculations
using different precipitation probabilities for the two main matrix orientations and , weighted in terms of
the isotropic probability, the latter set equal to one (from [Vicente Alvarez et al., 2012]).
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
α
α
(1 1 1)
{0 0 0 2} − ∥ (1 1 1) δ
̅
{1 0 1 7} α−Zr ∥ (1 1 1) δ ≈
{0 0 0 2} −
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
α
Figure 4-106: Schematics showing the effects of grain morphology and hydride–hydride interactions on the probability for hydride
precipitation in grains belonging to three of the idealized αZr texture orientations identified in Figure 4-102. The
idealized hydride cluster arrangements making up the macroscopic circumferential (top) and radial (bottom) hydride
clusters are indicated. Of the hydride cluster arrangements shown, (a) and (f) are energetically unfavourable (from
[Vicente Alavarez et al., 2012]).
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-107: Schematics of suggested stacking arrays of microscopic hydride plates in pressure tube alloy material as viewed, (a) in
a longitudinal (axial) section and, (b) in a transverse (hoop or circumferential) section. In this schematic all the arrays
are shown in grains with their normals oriented at ~ 20 to the basal pole direction in each case (from
[Perovic et al., 1983]).
α
1: 5: 50
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
(1 1 1)
∆ 111
(1 1 1)
(1 1 1)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-108: Schematic of the orientations and relative magnitudes of the in-plane compressive stresses within circumferential and
radial hydrides. Compressive stresses reduce the inter-planar spacing of the (1 1 1) planes along the direction of
compression, whilst stretching the d-spacing perpendicular to the load.
p >
(1 1 1)
p (1 1 1)
p Tilted
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
| | > | |
∆ 111 ≈ 2600 ± 400
| | ≈ | |
∆ 111 ≈ 500 ± 200
o
o
α
o
o
=
o
= 97 = 0.35
(1 1 1)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(1 1 1)
>
<
(1 1 1)
→ =
∆ / → > 0
→ < 0
4.6.5.2
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
-2
– 2
= 0 = 70°
= 0°
2
B
(0 0 0 2) − = 0°
= 70° (0 0 0 2) −
(1 1 1)
(1 1 1) (1 1 1)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-109: (a) Details of typical Debye rings captured experimentally showing indexing of the first four rings. (b) Ideal texture
components of the αZr phase for this pressure tube material. (c) Diffractograms obtained from the two azimuthal
sections shown in (a): the red curve gives information from hydrides precipitated in grains of both and
orientations (from [Vicente Alvarez et al., 2012]).
(1 1 1)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(0 0 0 2) −
(0 0 0 2) − ∥ (1 1 1)
̅
̅
[1 2 1 0] − ‖ [1 1 0] = 0°
{0 0 0 2} − (1 1 1)
= 70° {0 0 0 2} −
(1 1 1)
α
α
α
α
= 70°
(1 1 1)
(1 1 1)
(1 1 1)
Figure 4-110: Details of diffractograms recorded at 270C during cooling of the L2 specimen with and without an applied uniaxial
tensile stress, showing the effect of stress on hydride precipitation (from [Vizcaíno et al., 2014]). Plots (a) and (b) show
diffractograms obtained from the azimuthal sections shown in Figure 4-99, with hydride precipitates clearly visible
under load but absent in the unloaded condition. Plot (c) shows diffractograms obtained by integrating the counts
registered around the full diffraction ring, in which case the hydride phase present under load becomes almost
invisible.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-111: Evolution of selected crystallographic variables during the first heating and cooling cycle of the L2 specimen (from
[Vizcaíno et al., 2014]). Data obtained from the diffractogram of: (a) (1 1 1) peak area, (b) c lattice
parameter of the hexagonal αZr phase. Inflection points of the curves indicate the temperatures at which hydride
dissolution ends (TSSD), and hydride precipitation begins (TSSP). (c) Inflection points are more sharply defined after
subtraction of the thermal expansion component of the c lattice parameter.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(1 1 1)
(1 1 1)
{0 0 0 2} −
1 ∙ ≈
5 H
wppm
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-112: Effect of external tensile stress on hydride precipitation within grains of orientation of the L2 specimen: (a)
(1 1 1) peak area; this was used to determine the amount of hydrogen present as hydride, (b) c-axis strain; this was
also used to give the amount of hydrogen in solution in the αZr phase (from [Vizcaíno et al., 2014]). The applied
tensile stress moves the TSSP temperatures towards higher values. In addition, it increases the fraction of hydride
precipitates forming in grains with orientation. The concentration of hydrogen in solution in grains with
orientation is seemingly not affected by the application of the tensile stress.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(1 1 1)
α +
+
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-113: Effect of external tensile hoop stress on TSSD and TSSP temperatures for all specimens investigated. Results are
separately identified for hydrides precipitated in αZr grains of the and + orientations (from
[Vizcaíno et al., 2014]). Purple symbols correspond to precipitation in the orientation whilst blue symbols
denote precipitation in the + orientations. Open symbols correspond to TSSP temperatures whilst
filled symbols correspond to TSSD temperatures.
+
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
+
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Figure 4-114: Dependence of the amount of hydride phase (in wppm (wt ppm in plot legend) of hydrogen in hydrides) precipitated
within grains of the orientation as a function of applied stress for all specimens investigated (from
[Vizcaíno et al., 2014]). Application of a tensile stress along the c axis of the grains largely increased the number of
hydrides precipitated within such grains.
(α/(α + )
ℎ
= ( ) ( ) ( )
= −
=
=
ℎ
ℎ
= − =
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α
(α + )/
ℎ ≡ −ℎ −ℎ
2’
∆ ̅ ℎ ∆ ̅ ℎ Δ ̅
, 2
, 2
,
= ( ) ( ) (− )
ℎ
∆ ̅ Δ ̅
̅ −
ℎ
∆ ̅ = −
̅ −
Δ ̅ = −V
,
α = ⁄ −ℎ
(α + )/ , 2 −ℎ
∆ ̅ ℎ
, 2
ℎ
∆ ̅ Δ ̅
ℎ
∆ ̅
ℎ
∆ ̅ , 2 ∆ ̅ ℎ
, 1
α
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
=
⁄
1
,
ℎ
ℎ
∆ ̅ , 2 ∆ ̅ ,
∆ ̅ ℎ ∆ ̅ ℎ Δ ̅
,
,
, = ( ) ( ) (− )
, 2
ℎ
∆ ̅ , 1 ,
ℎ
∆ ̅ , 2
∆ ̅ ℎ ∆ ̅ ℎ Δ ̅
, 1
, 1 = , ( ) ( ) (− )
, 1
, ( ∆ ̅ ℎ )
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
+ .
α
+
α
+
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
(0.08 ± 0.02)C/MPa
+ (0.04 ± 0.01)C/MPa
α
α
(5.1468 ± 0.001) Å
+
and
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
0.3 1
(1 1 1)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
α/(α + )
α
ℎ
, ( ∆ ̅ , 1 )
(TSSP1) ( (TSSP1) )
( 1) = 2.473 × 10 4 ( 1) = −25 840
(TSSP1) ∆ ̅ ℎ Δ ̅
, 1 = (TSSP1) ( ) ( ) (− )
∆ (TSSP1)
, 1 ( = 0) ≡ TSSP1 =
ℓ ( )
(TSSP1)
∆ ̅ ℎ
∆ (TSSP1) + − Δ ̅
, 1 ( ) =
ℓ ( )
(TSSP1)
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
∆ ̅ ℎ − Δ ̅
, 1 ( ) − 1 =
ℓ ( )
(TSSP1)
ℎ
∆ ̅ Δ ̅
≲ 1 = 1 Δ , 1 ( )
̅ −
[ − ]
∆ , 1 ( ) = , 1 ( ) − 1 =
ℓ ( A(TSSP1) )
∆ , 1 ( ) − < 0 ℓ ( A(TSSP1) )
α
ℎ
∆ ̅ ℎ ( ) − ∆ ̅ , 2 (0)
, 2
= 1
∆ ̅ ℎ ( ) − ∆ ̅ ℎ (0) + ̅ − [ − ]
∆ , 2 ( ) = , 2 , 2
ℓ ( A(TSSP2) )
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
̅ −
∆ ̅ ℎ ( ) − ∆ ̅ ℎ (0) + [ − ]
∆ , ( ) = , ,
ℓ ( (TSSD) )
= 1.
∆ , 1 ( )
= 130 ( 1) = 2.473 × 10 4
=
3
V
225 MPa = 0.0542 ̅ − = 23.387 Å /(mol Zr) ≡
6
3
14.092 × 10 m /(mol Zr) = 1.47
(PLST) = 0.0749 (IPST) = 0.1843
∆ , 1 ( )
6
14.092 10 −6 × 225 × 10 × [1.47 × 0.0542 − 0.0749]
∆ , 1 ( ) = = −0.2360C
1.47 × 8.3144 × ℓ ( 130 )
2.473×10 4
ℎ
Δ ̅ − ∆ ̅ = 10.30
6
14.092 10 −6 × 225 × 10 × [1.47 × 0.0542 − 0.1843]
∆ , 1 ( ) = = 5.172C
1.47 × 8.3144 × ℓ ( 130 )
2.473×10 4
Δ ̅ − ∆ ̅ ℎ = −225.7
0.0010C/MPa 0.0230C/MPa
0.08 ± 0.02)C/MPa
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
6
14.092 10 −6 × 225 × 10 × [−0.0749]
∆ , 1 ( ) = = 3.702C
1.47 × 8.3144 × ℓ ( 130 )
2.473×10 4
∆ ̅ ℎ = −161.6 J/(mol H)
6
14.092 10 −6 × 225 × 10 × [−0.1843]
∆ , 1 ( ) = = 9.110C
1.47 × 8.3144 × ℓ ( 130 )
2.473×10 4
ℎ
∆ ̅ = −397.5 J/(mol H)
0.0165C/MPa 0.0405C/MPa
Copyright © Advanced Nuclear Technology International Europe AB, ANT International, 2019.
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