Intergranular δ-hydride nucleation and orientation in zirconium alloys

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Acta Materialia 59 (2011) 7010–7021 www.elsevier.com/locate/actamat

Intergranular d-hydride nucleation and orientation in zirconium alloys W. Qin a,⇑, N.A.P. Kiran Kumar b, J.A. Szpunar a, J. Kozinski a,c a

Department of Mechanical Engineering, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5A9 b Department of Metals and Materials Engineering, McGill University, Montreal, Quedec, Canada H3A 2B2 c Faculty of Science and Engineering, York University, Toronto, Ontario, Canada M3J 1P3 Received 1 March 2011; received in revised form 13 July 2011; accepted 26 July 2011 Available online 24 August 2011

Abstract A theoretical understanding of the intergranular d-hydrides is still lacking, although hydride-related degradation of the mechanical properties of zirconium alloys has been studied for many years. In this paper a thermodynamic model is developed to analyze the nucleation and orientation of intergranular d-hydrides. The results show that the grain boundary structure and zirconium grain orientation simultaneously govern hydride precipitation. An electron backscatter diffraction study of hydrided Zircaloy-4 provides direct evidence supporting theoretical predictions. A criterion is proposed to reveal the inherent relation between hydride precipitation at the grain boundaries and in the zirconium grains. Stress-induced susceptibility to hydride precipitation at the radial grain boundaries of zirconium alloy tubes is theoretically analyzed. This work provides a general framework based on which the correlation between the grain boundary structure, the grain orientation, the stress and their effects on the nucleation and orientation of the intergranular d-hydrides can be clarified. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Grain boundaries; Hydrides; Nucleation; Orientation

1. Introduction Zirconium (Zr) alloys are used for structural components in the nuclear power industry because of their low neutron absorption, high ductility and stability. Hydrides precipitate when the solubility limit of hydrogen in Zr alloys is exceeded [1–3]. Hydrides have received much attention because of their intrinsic embrittling effects that may serve as fracture initiators [1–7]. Both intergranular and the intragranular hydrides are observed [4–10]. Compared with the intra-grain region, grain boundaries (GB) may be preferred sites for hydride formation. Some works have shown that intergranular hydrides could play a more important role in hydride-induced rupture compared with intragranular hydrides, because cracking is more likely to take place in intergranular hydrides due to the non-coincidence of the slip systems of the intergranular hydrides and ⇑ Corresponding author.

E-mail address: [email protected] (W. Qin).

the surrounding grains, which means that intergranular hydrides suffer higher strain incompatibilities and stresses [11]. Moreover, intergranular hydrides could cause rapid intergranular fracture under tensile stress [12]. However, the literature on intergranular hydrides is still very limited, and the inherent relation between intergranular and intragranular hydride formation is unknown to date. Furthermore, hydride orientation is an important factor affecting the performance of Zr alloys. Radially orientated hydrides can drastically reduce the ductility of Zr alloy tubes [13]. Intragranular hydrides preferentially precipitate on certain habit planes, such as the frequently reported (0 0 0 1) and f1 0 1 7g planes [14–16]. Nevertheless, the crystallographic orientation relationship between intergranular hydrides and the Zr matrix is still controversial [8,9,17,18]. In addition, it is known that hydride orientation is dependent on applied stress. Although Puls [19] and Ells [20] have studied this problem in detail in their previous work, they only considered intragranular hydrides and did not account for the influence of GB. A lack of

1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.07.054

W. Qin et al. / Acta Materialia 59 (2011) 7010–7021

knowledge about intergranular hydrides may be an important reason why hydrogen embrittlement of Zr alloys is not yet completely understood, although it has been studied for many years. In this paper we will try to establish a comprehensive model using a thermodynamic approach by which the intrinsic factors that affect the formation and orientation of intergranular hydrides are clarified. The paper is organized as follows. First, a thermodynamics-based model is used to study hydride nucleation at GB, and using this the GB structures and the grain orientation that favor hydride precipitation are determined. Secondly, electron backscatter diffraction (EBSD) analysis of the crystallographic orientation of hydrided Zircaloy-4 was carried out, and the experimental results obtained are compared with theoretical prediction. Thirdly, lattice matching between the Zr matrix and intergranular hydrides and its effects on the precipitation of intergranular hydride are analyzed. Fourthly, a criterion is proposed to determine the hydride precipitation sites. Finally, the stress-induced reorientation of intergranular hydrides is discussed and structural modifications to reduce the formation and radial orientation of intergranular hydrides in Zr alloy tubes are introduced. 2. Thermodynamic model of intergranular hydride nucleation

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to that of the other. In this paper three representative GB types, schematically depicted in Fig. 2a–c, are used to analyze the nucleation of possible intergranular d-hydrides. 2.2. Critical nucleation energy of intergranular hydrides A treatment of the influence of the habit plane on precipitate nucleation at GB has been proposed in a classic paper by Johnson et al. [27]. We will develop this treatment so that it can be applied to study the formation of dhydrides at the GB of a-Zr alloys. In the work of Johnson et al. [27] the strain energy resulting from nucleus–matrix lattice matching was neglected, based on the assumption that vacancies at the nucleus–matrix interface reduce the strain energy to a negligible level. However, for hydride nucleation in Zr alloys experiments have provided convincing evidence that hydrides are coherent with the matrix in the initial stage of precipitation [14,28]. The strain energy associated with matrix–hydride accommodation therefore needs to be considered. Based on the work of Johnson et al. [27] and considering the strain energy effect the critical nucleation energies of hydrides at GB, DGi (i = a, b or c), are obtained by (Appendix A): (a) GB plane not parallel to the habit planes of two adjacent grains (Fig. 2a)

2.1. Possible shapes of intergranular hydride nuclei Zr hydride has three crystal structures: c-hydride (ZrH1.0, face-centered tetragonal (fct), c/a > 1), d-hydride (ZrH1.66, face-centered cubic (fcc)) and e-hydride (ZrH2, fct, c/a < 1) [1,21]. The attention of this paper is focused on the d-hydride formed in a-Zr alloys, because d phase is more common than the c and e phases under practical reactor conditions [22,23]. Due to the preferred matching of atom patterns and spacings at d-hydride/a-Zr interfaces there is a crystallographic orientation relationship between the two phases [14–16]. Transmission electron microscopy (TEM) observations show that a thin plate-shaped d-hydride with a planar facet incorporating an a-Zr matrix in the habit plane is a typical morphological feature [14,24–26]. The shape of intragranular d-hydrides is schematically illustrated in Fig. 1. Experiments have confirmed that intergranular hydrides retain their plate shape [14,24]. However, due to different orientations of two adjacent Zr grains, when the intergranular hydride plane is parallel to the habit plane of one grain it cannot generally be parallel

DGa ¼

16pr3ad 3ðDGchem þ DGGB strain Þ

f 2 a

¼ 1  1:5 cos w þ 0:5 cos3 w

with f a ð1Þ

(a)

(b)

(c)

{hkil}α-Zr //{h'k'l'}δ-ZrH1.66 {hkil}α-Zr {h'k'l'}δ-ZrH1.66 {hkil}α-Zr {hkil}α-Zr //{h'k'l'}δ-ZrH1.66 Fig. 1. Schematic drawing of the morphology of intragranular d-hydrides.

Fig. 2. Possible shapes of the intergranular hydride nuclei: (a) GB not lying on the habit planes (two abutting spherical caps); (b) GB lying on the habit plane of one of the two adjacent grains (one spherical cap truncated by a facet); (c) GB lying on the habit planes of two grains (both spherical caps truncated by facets).

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(b) GB plane parallel to the habit plane of one of the two adjacent grains (Fig. 2b) DGb ¼

16pr3ad

f with f b 2 b 3ðDGchem þ DGGB strain Þ 1 ¼ ð2  6 cos w þ 2 cos3 w þ 3 cos /  cos3 /Þ 4

ð2Þ

(c) GB plane parallel to the habit planes of two adjacent grains (Fig. 2c) DGc ¼

16pr3ad

f with f c 2 c 3ðDGchem þ DGGB strain Þ 1 ¼ ð3 cos w þ cos3 w þ 3 cos /  cos3 /Þ 2

ð3Þ

where DGchem is the absolute value of chemical free energy of transformation per unit volume and DGGB strain is the strain energy per unit volume for hydride formation at GB. According to a Wulff construction for the shape of the nucleus [27] the angles w and / are cos1(raa/2rad) and cos1 ðrcad =rad Þ, respectively, where raa is the GB energy of Zr, rad is the disordered interfacial energy and rcad is the coherent interfacial energy. 2.3. Influences of GB structure and grain orientation on hydride nucleation For a quantitative analysis we take rcad =rad ¼ 1=10, which is a reasonable estimation for the ratio of a coherent interfacial energy to a disordered interfacial energy [29,30]. Changes in fi (i = a, b or c) with raa/rad are calculated and shown in Fig. 3. It is seen that fi decreases with increasing raa/rad, indicating decreases in DGa , DGb and DGc . The value of rad is uncertain. The GB energy (raa) is dependent

on the misorientation between adjacent grains (h), as expressed by the well-known Read–Shockley equation [31]. The GB with higher misorientations have higher energy and therefore are preferable sites for hydride precipitation. The coincident site lattice (CSL) notation is usually used to describe the GB structure. GB with high R values of CSL may be expected to have a higher energy than ones with low R (http://en.wikipedia.org/wiki/Grain_boundary). Recently Mani Krishna et al. [10] reported that GB with smaller R have higher resistance to hydride formation, despite the fact that there is wide scatter in the correlation between hydride preference and R value in their work. In addition, Fig. 3 shows that Zr grain orientation also affects hydride nucleation. For a given raa/rad, fa > fb > fc, indicating that an orientation parallel to the habit plane and the GB plane can reduce the critical nucleation energy of intergranular hydrides, i.e. the formation of intergranular hydrides depends not only on GB structure, but also on Zr grain orientation. This might be one reason why a wide scatter in the relation between hydride preference and R value was observed by Mani Krishna et al. [10]. 3. EBSD studies Theoretical results show that the combined effects of GB structure and grain orientation simultaneously govern hydride formation at GB. In order to verify this viewpoint Zircaloy-4 with the composition given in Table 1 was used for experimental observations. Specimens were hydrided at 648 K to a hydrogen content of 240 p.p.m. and then furnace cooled to room temperature. EBSD was used for microstructural characterization, because this technique can statistically collect data on crystal orientation over a large area (http://en.wikipedia.org/wiki/Electron_backscatter_diffraction). For observations the hydrided specimen was polished using colloidal silica and etched in a solution of 45% HNO3, 45% H2O and 10% HF. EBSD measurements were carried out with an orientation imaging microscopy (OIM) system installed on a FEI XL30 environmental scanning electron microscope. The microscopy settings for the measurements were as follows: 30 keV acceleration voltage, 15 mm working distance, 70° tilt and 0.06 lm step size. EBSD patterns were analyzed using the TSL OIM software. Representative Kikuchi patterns of the Zr matrix and hydrides are given in Fig. 4. They can be indexed as a-Zr and d-ZrH1.66, respectively. Fig. 5a is a typical EBSD pattern of a hydrided specimen, in which the red colored region is d-ZrH1.66 and the matrix is a-Zr. Both intragranular and the intergranular hydrides are found. The crystallographic orientation relationships between hydrides and Zr matrix were determined Table 1 Chemical composition of Zircaloy-4 (wt.%).

Fig. 3. Changes in fi (i = a, b or c) with the GB energy and the grain orientation showing that the GB having high energy and lying on the habit plane substantially lower the critical nucleation energy.

Sn

Fe

Cr

Ni

O

Zr

1.52

0.21

0.11

<35 ppm

0.125

Balance

W. Qin et al. / Acta Materialia 59 (2011) 7010–7021

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Fig. 4. Typical Kikuchi patterns. (a) Kikuchi pattern of a matrix (left) indexed as a-Zr (right); (b) Kikuchi pattern of a precipitate (left) indexed as dZrH1.66 (right).

using pole figures. For the intragranular hydrides the (0 0 0 1)a-Zr//{1 1 1}d-ZrH1.66 relationship is found at all nucleation sites (Fig. 5b). This is consistent with a number of previous studies using TEM [14,15,24–26] and a very recent one using a synchrotron technique [16]. For the intergranular hydrides the (0 0 0 1)a-Zr//{1 1 1}d-ZrH1.66 relationship is retained, but only exists between hydrides and one bounding Zr grain, implying that Fig. 2b is the main type of nucleation of intergranular d-hydrides. The preference of intergranular hydrides for the basal plane of the Zr grain has also been reported by another EBSD study [8] and TEM observations [14,24]. (0 0 0 1) and {1 1 1} are the close-packed planes of a-Zr and d-hydride, respectively. When they are parallel to each other a good match of atom patterns and spacings across the d-hydride/a-Zr interfaces favors the precipitation of hydrides. A detailed discussion can be seen in Carpenter [32]. In recent decades it has been confirmed that the (0 0 0 1)a-Zr//{1 1 1}d-ZrH1.66 relationship is most common for the precipitation of hydrides in Zr alloys, although some occasional exceptions have been reported [8,14– 17,24–26]. The reasons that the case in Fig. 2a is not observed might be due to the fact that the critical nucleation energy in the case of Fig. 2a is higher than that in the case of Fig. 2b, as shown in Fig. 3, and therefore hydride nucleation via the mechanism in Fig. 2a is energetically unfavorable. The low probability of hydride nucleation by the mechanism shown in Fig. 2c may be

attributed to two factors: (i) the GB in Fig. 2c is a mirror image symmetrical structure with respect to the habit plane (i.e. a twin structure), and such boundaries generally have low energy and are unfavorable for hydride precipitation; (ii) a-Zr has a much higher stacking fault energy on the (0 0 0 1) basal plane than on the prismatic planes [33,34], and therefore the appearance of a (0 0 0 1) basal twin boundary in a-Zr is energetically more difficult, which has been confirmed by experimental results for pure Zr and Zircaloy-4 [35]. The lack of nucleation sites might also be one reason why hydride formation by the mechanism shown in Fig. 2c is not observed in the present experiment. Further, the information of GB misorientation obtained by the analytic technique EBSD shows that the GB at which d-hydrides form have large misorientation angles, i.e. larger than the general misorientation limit for low angle boundaries (h > h = 15°) in the Read–Shockley model [31] (Fig. 5b). These results imply that the most probable sites for the nucleation of intergranular hydrides are GB consisting of facets lying on the basal plane of one bounding Zr grain, as depicted in Fig. 6a and having a high energy, as indicated in Fig. 6b. 4. Lattice matching between the intergranular hydride and Zr matrix The presence of a hydride–matrix orientation correlation indicates that lattice matching is still important in

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(a) hα1

cδ cα

hδ hα2

dα

dδ

gα2 gα1 gδ iδ iα2 iα1 jα1 fδ fα2 f α1

jδ jα2

bα

kδ kα1 kα2

bδ

TD

eδ eα1

lα1

aα aδ

lα2 lδ

eα2

RD

(b)

RD

RD

RD

RD

TD

TD

(aα)

(aδ)

(bα)

(bδ)

TD

TD

(cα)

(cδ)

RD

θe = 44°

(dα)

(dδ)

RD

θf = 76°

TD

(eα1)

(eδ)

(eα 2)

TD

( fα1)

( fδ)

θg = 83°

( fα 2)

TD

(gα1)

(gδ)

(gα 2)

θi = 70°

TD

(hα 1)

(hδ)

(hα 2)

TD

(iα1)

(iδ)

( jδ)

( jα 2)

θl = 27°

θk = 30°

TD

TD

(kα1)

(kδ)

(kα 2)

θj = 29° TD

( jα 1)

(iα 2)

θh = 84°

(lα 1)

(lδ)

(lα 2)

Fig. 5. EBSD experimental results. (a) EBSD pattern of the hydrided Zircaloy-4 showing the intergranular hydrides (yellow boxed region) and the intragranular hydrides (pink boxed region). (b) The crystallographic orientation relationship between a-Zr and d-ZrH1.66 and the GB misorientation angles. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

W. Qin et al. / Acta Materialia 59 (2011) 7010–7021

(a)

(b)

Fig. 6. The nucleation sites of the intergranular d-hydrides. (a) The intergranular hydrides prefer GB parallel to the basal plane of one bounding Zr grain. (b) The GB energy calculated using the Read– Shockley equation showing that GB lying on the basal plane and having high energy are the most probable sites for intergranular hydride nucleation.

the precipitation of intergranular hydrides. The strain energy ðDGGB strain Þ associated with the hydride nucleus can be estimated as [36,37] DGGB strain ¼

6lgcv2 gðc  1Þ þ 1

ð4Þ

where l is the shear modulus, c is the ratio of the bulk modulus of the precipitate to that of the matrix, g = (1 + m)/3(1  m), with m being the Poisson ratio, and v is the Zr matrix–hydride misfit strain, which is, to a large extent, dependent on the coherent lattice misfit (e) in the habit plane. The d-hydride (ZrH1.66) with lattice parameter a = 0.4778 nm has a minimum lattice misfit with pure a-Zr with lattice parameters a = 0.3231 nm and c = 0.5146 nm when the relationship (0 0 0 1)a-Zr//(1 1 1)d-ZrH1.66 holds [32]. For this relationship e = 0.0458, obtained from e = (dhyd  dzir)/dhyr, where dhyd and dzir are the interplanar spacings of the corresponding planes in the hydride and Zr matrix, respectively [32]. This means that hydrides formed at GB lying in the basal plane have the optimum lattice matching with the Zr grains, which favors hydride

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precipitation. This is consistent with the present experimental results. For intragranular hydrides the matrix–hydride misfit strain mainly results from coherent lattice matching on both sides of the thin plate-shaped d-hydride nucleus. However, for intergranular hydrides the case becomes more complicated. Firstly, due to the different orientations of the two adjacent grains coherent lattice matching generally exists on one side of the thin plate-shaped d-hydrides, whereas on the other side the degree of matching is uncertain. Secondly, some solutes in alloys tend to diffuse towards the GB and affect the interplanar spacing in the region near the GB. Thirdly, the compatibility stresses produced by the different orientations of the adjacent grains can lead to local lattice strain and change the interplanar spacing in the vicinity of the GB. These factors more or less affect the hydride–matrix accommodation strain. For pure a-Zr DGchem = 4.02  108 J m3, l = 33.82 GPa, m = 1/3, and c = 12/13.5 [37]. We take rcad =rad ¼ 1=10 (as in Section 2). The changes in the critical nucleation energy of intergranular hydrides with DGGB strain are calculated using Eq. (2) and are given in Fig. 7a. The change in DGGB strain with v is calculated using Eq. (4) and is given in Fig. 7b. When 8 3 v > vc = 0.0541, DGGB and DGb strain > 3:8  10 J m , increases rapidly with a slight increase in v, meaning that the precipitation of d-hydrides becomes difficult. Since lattice matching between intergranular hydrides and the Zr matrix is restricted by some uncertain factors, the formation of intergranular hydrides has a larger uncertainty relative to intragranular hydrides. In addition, the GB hardening effect also affects hydride formation. Due to solute segregation at the GB and the strain incompatibility across the GB the hardness in the vicinity of the GB is generally higher than that in the interior of grains [38,39]. Fig. 7b shows that vc decreases from 0.0541 to 0.0496 when the shear modulus of the Zr matrix increases from 33.82 to 40 GPa, meaning that when the GB are surrounded by an elastically harder region, hydride formation requires better lattice matching, i.e. precipitation becomes more difficult. Recent EBSD studies of Zircaloy-2 have provided some evidence supporting this conclusion [10]. Our results show that intergranular hydrides energetically prefer GB lying in the basal plane. However, some exceptions have also been reported. For example, Une and Ishimoto observed that intergranular hydrides do not prefer any special GB [17]. The reason for this might be the presence of high local strain at the GB in the specimens investigated by them, which may destroy hydride–matrix lattice matching and result in loss of the orientation correlation between the two. In addition, Arunachalam et al. found that intergranular hydrides prefer prism tilt boundaries, rather than the basal boundary in Zrcaloy-2 with a hydrogen concentration of 550 p.p.m. [18]. The different habit planes observed might be attributed to the different alloy compositions and hydrogen concentrations used in the different experiments, which changes the matching of

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behavior. For simplicity the influence of dislocations on the formation of intragranular hydrides is not considered in this work. From the point of view of classical nucleation theory the nucleation rate J is expressed as [41] J  ¼ Zb N expðDG =kT Þ expðs=tÞ

ð5Þ

where Z is the Zeldovich non-equilibrium factor, b is the rate of single solute atoms being added to the nucleus, which is proportional to the diffusivity of solute (D) [41], N is the number of nucleation sites, kT has its usual meaning, s is the incubation time and t is the isothermal reaction time. For steady-state nucleation s  t and thus exp(s/ t) ! 1. The present work shows that the nucleation of intergranular hydrides by the mechanism in Fig. 2b is dominant, and therefore here we will compare the preference for hydride formation in the grains with that at GB in Fig. 2b. According to Eq. (5) the ratio of nucleation rate at GB (J GB;b ) to that in grains (J intra ) is given by J GB;b Z b Db N b expðDGb =kT Þ ¼  J intra Z intra Dintra N intra expðDGintra =kT Þ

ð6Þ

where the subscripts b and intra represent the cases of hydride nucleation at GB and in grains, respectively. The Zeldovich non-equilibrium factor accounts for the probability that a nucleus that has reached the critical size dissolves back into solution. Its range is between 1/20 and 1/40 [42]. In this paper we take Zb = Zintra as an approximation. The critical nucleation energy for the intragranular d-hydride (DGintra ) is calculated using Puls’ model [19] DGintra ¼

18prcad r2ad ðDGchem þ DGintra strain Þ

2

ð7Þ

5. The relation between the intergranular and the intragranular hydride formation

where DGintra strain is the strain energy per unit volume for hydride formation inside grains. Eq. (6) shows that in addition to the critical nucleation energy the hydrogen atomic diffusivity and the number of nucleation sites also affect the site preference of hydride precipitation. According to the semi-empirical relation proposed by Borisov et al. [43] the atomic diffusivity at the GB (Db) is related to the GB energy      kT qDb raa ¼ m ln  ln m ð8Þ br20 r0 k2 Dintra

Although both intergranular and intragranular hydrides have been reported in (nearly) all the relevant literature there is no theoretical model to allow an understanding of the mechanism whereby some hydrides form at GB and some intragranularly. In this section the inherent relation between intergranular and intragranular hydride formation is analyzed. Bailey’s experiment confirmed that hydride nucleation inside Zr grains does not necessarily require the presence of dislocations, and hydride nucleation at GB and in grains is intimately related [40]. The simultaneous precipitation of hydrides inside Zr grains and at GB seems to be an intrinsic

where m is the number of atomic layers forming the boundary, r0 is the mean inter-atomic distance, q is the GB width, and k and b are constants. Borisov et al. suggested m, k and b to be 1, 1 and 2, respectively [43]. Assuming q = mr0, we have Db =Dintra ¼ expð2r20 raa =kT Þ. To determine the number of nucleation sites a simple expression (3q/d) is used to estimate the ratio of the GB volume to the grain volume, where d is the grain size [44]. The ratio of the number of nucleation sites at GB to that in grains may be approximated by Nb/Nintra = 3qK/d, where K is the probability of the GB lying on the habit plane, which is dependent on the texture of the investigated material. Substituting Db/Dintra and N b =N intra into Eq. (6) leads to

Fig. 7. Effects of lattice matching between the hydrides and Zr matrix on the nucleation of intergranular hydrides. (a) Dependence of the critical nucleation energy of the intergranular hydrides on the strain energy. (b) Dependences of the strain energy on the matrix–hydride misfit strain and the strength of the Zr matrix.

atom patterns and spacings at the hydride–matrix interfaces and leads to different orientation relationships.

W. Qin et al. / Acta Materialia 59 (2011) 7010–7021

J GB;b ¼ J intra

  3qK exp½ðDGb  2r20 raa  DGintra Þ=kT  d

ð9Þ

when J GB;b =J intra > 1 (or lnðJ GB;b =J intra Þ > 0) intergranular precipitation is dominant; in contrast, if it is less than 1 (or lnðJ GB;b =J intra Þ < 0) intragranular precipitation is dominant. Therefore, Eq. (9) may be taken as a criterion to determine the hydride precipitation sites. For a quantitative analysis we take d = 10 lm, q = 0.5146 nm, r0 = 0.3231 nm and 8 GB 3 DGintra strain ¼ DGstrain ¼ 2:724  10 J m , and assume K = 1/ 2 c 100, rad = 0.5 J m and rad ¼ 0:05 J m2 , the change in J GB;b =J intra with raa and T is calculated using Eq. (9) and is shown in Fig. 8a. The Zr grain size (d) in currently available Zircaloy-4 is generally in the range 5–20 lm. In this paper we use d = 10 lm for the theoretical calculations. The lattice parameters of a-Zr are a = 0.3231 nm and c = 0.5146 nm. d-hydrides prefer a (0 0 0 1) basal boundary. According to the treatment of Borisov et al. [43] the number of atomic layers forming the boundaries is taken to be 1, and consequently the width of the basal boundary is q = c = 0.5146 nm. Since hydrogen atoms mainly diffuse along the GB plane, the interatomic distance along the basal GB plane is r0 = a = 0.3231 nm. The strain energy associated with the formation of an intergranular hydride nucleus is governed

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by some uncertain factors. For a quantitative calculation GB we approximately take DGintra strain ¼ DGstrain . If the relationship of (0 0 0 1)a-Zr//(1 1 1)d-ZrH1.66 is strictly followed during nucleation, v = e = 0.0458. Based on Eq. (4) and using the parameters given in Section 4, we have 8 GB 3 DGintra strain ¼ DGstrain ¼ 2:724  10 J m . It can be seen that intergranular hydrides are prevalent when raa > 0.16 J m2. Although this critical value (0.16 J m2) has uncertainty due to some uncertain parameters used in the calculations, the results clearly show that the GB energy is crucial in determining hydride precipitation sites. The GB energy distribution in a real polycrystalline material is dependent on the preparation process and method. When the GB energy distribution changes from case I to II in Fig. 8b the fraction of high energy GB increases and hydrides are more likely to form at GB. This result is in agreement with comparative studies of stress-relieved and recrystallized Zr alloys, with intergranular hydrides being more frequently observed in recrystallized Zr alloys than in stress-relieved ones [5–7,45]. The reason for this may be the fact that stress relief increases the fraction of low energy boundaries, while recrystallization has the reverse effect. In addition, Fig. 8a indicates that the choice of hydride precipitation site is also related to temperature. However, a much stronger dependence of lnðJ b;GB =J intra Þ on GB energy than on temperature, as shown in Fig. 8a, implies that GB energy plays a more important role in determining the precipitation sites of hydrides. It should be pointed out that the present calculation does not consider the influence of temperature on GB energy. For a pure system the GB energy usually decreases with increasing temperature [46]. However, for impure systems or alloys the GB energy may increase as the temperature increases due to the reduction in GB segregation [47,48], and as a result the preference for hydride formation at GB may increase at elevated temperatures. 6. Stress-induced reorientation of intergranular hydrides It is a general behavior that hoop (tensile) stress induces hydride reorientation from the circumferential to the radial direction in Zr alloy tubes. Although this problem has been theoretically studied by Puls [19] and Ells [20], they did not consider the influence of GB. In this section attention is restricted to intergranular hydrides. This work may be used as a supplement to their previous classical work. 6.1. Effect of tensile stress on critical nucleation energy of intergranular d-hydrides

Fig. 8. Choice of the hydride precipitation sites. (a) Effects of the GB energy and the temperature on the ratio of nucleation rates at the GB to that in the grains showing that the GB structure is the decisive factor in determining precipitation sites. (b) Schematic diagram showing the influence of the different GB energy distributions on the hydride precipitation sites.

A large body of work has shown that nucleation is the most important stage in determining hydride orientation [19,20,25]. Hydrides always tend to precipitate with the smallest nucleation barrier. The change in critical nucleation energy of differently oriented hydrides with applied stress decides the hydride orientation. Under tensile stress (-) there is stress interaction energy with hydrides [19] DGint = (-consu)v, where u is the angle between the tensile

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stress axis and the normal to the hydride plane. Tensile stress can change the chemical free energy of nucleation [2,3,19], DGstress thyd Þ½ð-consuÞ=3tH , where thyd is chem ¼ DGchem  ðx= the molar volume of hydrides of composition ZrHx, with x = 1.66 for d phase, and tH is the molar volume of hydrogen in Zr. The critical nucleation energies of d-hydrides at GB (DGtens;i ) under tensile stress are thus (Appendix B) DGtens;i ¼

16r3ad 2 3½DGchem  ðx=thyd Þ½ð-consuÞ=3tH  ð-consuÞv þ DGGB strain 

fi

ð10Þ

with i = a, b or c for the cases in Fig. 2a–c. The increase in - increases the value of [. . .] in Eq. (10) and thus DGtens;i decreases. 6.2. Effect of tensile stress on the GB energy The analyses in Sections 2 and 5 showed that the critical nucleation energy and the nucleation rate of intergranular hydrides are related to the GB energy. Tensile stress can give rise to local strain in the GB and therefore change the GB energy. The GB energy (raa,ten) under tensile stress is given by (Appendix C) raa;ten ¼ raa þ ð-cons#Þjq

ð11Þ

where # is the angle between the tensile stress axis and the normal to GB plane and j is the local strain caused by tensile stress in the GB region. The increase in GB energy reduces fi, as indicated in Fig. 3, and consequently DGtens;i decreases. In order to more clearly understand the relation between hydride orientation, GB orientation and the tensile stress direction, a schematic diagram is given in Fig. 9. For hydrides formed at GB, # = u. When # = u = 0 the stress-induced increases in the denominator in Eq. (10) and the GB energy (raa,ten) reach maximum, and the latter yields the largest possible reduction in fi in Eq. (10). This means that hydrides prefer to precipitate at GB perpendicular to the tensile stress axis, which is consistent with the EBSD observations of Zircaloy-2 [8]. 6.3. Hydride reorientation from the circumferential to the radial GB in Zr alloy tubes Recent EBSD studies of Zr alloy tubes has shown that under hoop (tensile) stress the hydrides prefer to form at

radial GB [8]. In general, a commercial Zr alloy tube has a strong radial alignment of the basal poles. With increasing texture strength the probability of the basal plane being parallel with the GB increases in the circumferential direction, but decreases in the radial direction. It is therefore a reasonable assumption that the situation presented in Fig. 2b is more likely when hydrides form at circumferential GB, and the situation in Fig. 2a is more likely when hydrides form at radial GB. Based on this assumption, the ratio of the intergranular d-hydride nucleation rate at radial GB (J GB;rad ) to that at circumferential GB (J GB;cir ) under hoop stress is given by (Appendix D) J GB;rad S rad expf½DGrad  2r20 ðraa;rad  raa;cir Þ ¼  J GB;cir S cir g  DGcir =kT g

ð12Þ

where DGrad and DGcir are the critical nucleation energies of d-hydride formation at radial and the circumferential GB, respectively, and the meanings of the other terms are given in Appendix D. Eq. (12) can be used to analyze stress-induced susceptibility to hydride reorientation from circumferential to radial GB. For simplicity a rectangular shape of the Zr grains is used, in which there are two types of GB in Zr alloy tubes: radial GB perpendicular to the hoop stress and circumferential GB parallel to the hoop stress. For hydride formation at circumferential GB # = u = 90°. In this situation Eq. (10) is simplified to Eqs. (1)–(3). The effect of hoop stress on hydride formation at circumferential GB is negligible. We thus have DGcir ¼ DGb . For hydride formation at radial GB # = u = 0°. Eq. (10) is rewritten as DGtens;i ¼

16r3ad 3½DGchem  ðx=thyd Þð-=3ÞtH  -v þ DGGB strain 

f 2 i ð13Þ

If the situation in Fig. 2a is dominant for hydride nucleation at radial GB we have DGrad ¼ DGtens;a . In addition, according to Eq. (11), the energy of radial GB (raa,rad) is given by raa;rad ¼ r0aa;rad þ -jq, where r0aa;rad is the energy of radial GB when - = 0. The increase in - gives rise to increases in both the denominator of Eq. (13) and the radial GB energy (raa,rad), and the latter causes a decrease

Fig. 9. Schematic diagram showing the relation between the intergranular hydride orientation, the GB orientation and the tensile stress direction.

W. Qin et al. / Acta Materialia 59 (2011) 7010–7021

in fi in Eq. (13), as a result of which DGrad ðDGtens;a Þ decreases. The value of [. . .]in Eq. (12) decreases, and J GB;rad =J GB;cir increases accordingly. Hoop stress promotes hydride formation at radial GB, consistent with the EBSD observations [8]. Strengthening the radial basal pole texture enhances the possibility of the basal plane being parallel to circumferential GB of tubes and results in a lowering of DGcir ðDGb Þ and an increase in the value of g in Eq. (12). As a result, J GB;rad =J GB;cir decreases and the stress-induced increase in J GB;rad =J GB;cir weakens. This means that the strong radial basal pole texture enhances the resistance to hydride formation at radial GB. 7. Resistance to the intergranular hydride formation The suppression of the hydride precipitation at GB, especially at radial GB of Zr alloy tubes, is important in improving the performance of the material. Intergranular hydrides could be reduced if some modifications are introduced. 7.1. Reduce the GB energy Solute segregation in GB can lower the GB energy. The dependence of GB energy on solute segregation may be written as [49] raa ¼ raa;0  Cb0 ½DH seg þ RT ln X 0 

ð14Þ

where raa,0 is the GB energy without solute segregation, DHseg is the enthalpy change of GB segregation per mole solute, X0 is the bulk concentration and Cb0 is the solute excess available for segregation at saturation. The experience of many years manufacturing Zr alloys has shown that some solute atoms at very small concentrations may enhance the resistance to hydrogen embrittlement. The relevant mechanism remains controversial. The present results imply that a decrease in hydride precipitation at GB, due to a lowering of GB energy, might be one reason. In addition, stress-relief annealing can also reduce GB energy and suppress intergranular hydride formation. 7.2. Strengthen Zr matrix The results in Section 4 show that intergranular dhydrides formation is more difficult when GB are surrounded by an elastically harder Zr matrix. Christensen et al. used the first principles density functional approach to analyze the GB segregation of impurity and alloy elements in a-Zr [50]. Their results show that some atoms, such as C, N and O, prefer to remain in the grains as interstitial atoms. These atoms may strengthen the Zr matrix and affect hydride precipitation. 7.3. Strengthen radial basal pole texture of Zr alloy tubes Radially oriented hydrides are most deleterious to the ductility of Zr alloy tubes. The present results show that

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hydrides prefer to form at GB lying on the basal plane of Zr grains. Strengthening the radial texture of the basal pole can reduce the probability of the basal plane being parallel with radial GB. As a result, hydride formation at radial GB is suppressed. 8. Conclusions The theoretical model presented in this paper provides a general understanding of intergranular d-hydrides. While developed mainly for Zr alloys, the proposed treatment should also be applicable to other metals, such as Ti alloys, where hydrides also lead to a loss of ductility. The principal results obtained are as follows. 1. The GB structure and the grain orientation simultaneously govern hydride precipitation at GB. Hydrides prefer GB lying on the basal plane of one bounding Zr grain and having a high energy. EBSD studies provided direct evidence supporting this conclusion. 2. The strain energy associated with the formation of intergranular hydrides is affected by many factors, such as the degree of lattice matching between hydrides and the Zr matrix, solute segregation in the GB, compatibility stress near the GB and the GB hardening effect. Compared with intragranular hydrides, the formation of intergranular hydrides has larger uncertainty. 3. There is an inherent relation between intergranular and intragranular hydride formation. A theoretical criterion was proposed to quantitatively describe this relation. The GB energy distribution in alloys is the decisive factor in determining hydride precipitation sites. 4. The theoretical model gave a general expression for analyzing the stress-induced susceptibility of hydride formation at radial GB in Zr alloy tubes. The preference of hydrides for radial GB under hoop stress is attributed to a stress-induced increase in the radial GB energy and a decrease in the critical nucleation energy of hydrides at radial GB. 5. In order to reduce the precipitation of intergranular hydrides, especially in the radial direction of the Zr alloy tube, it is necessary to: (i) reduce the GB energy, (ii) increase the elastic modulus of the Zr matrix, and (iii) reduce the possibility of radial GB lying on the basal plane.

Appendix A From a thermodynamic viewpoint the change in Gibbs free energy (DGi) accompanying the precipitation of a nucleus in a GB is given by DGi ¼ DGi;V þ DGi;S  DGi;GB

ðA1Þ

with i = a, b, or c for the cases in Fig. 2a–c, respectively, where DGi,v is the contribution of the bulk free energy of transformation, DGi,s is the contribution of the interphase

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W. Qin et al. / Acta Materialia 59 (2011) 7010–7021

boundary energy between nucleus and matrix, and DGi,GB is the contribution of the GB energy between the parent grains. The negative sign in Eq. (A1) means that a part of the GB is covered by the precipitated second phase. For the case in Fig. 2a we have   DGa;V ¼ ta DGchem þ DGGB strain 2 ¼ pr3 ð2  3 cos w þ cos3 wÞ½DGchem þ DGGB strain  3 ðA2Þ DGa;S ¼ S s rad ¼ 4pr2 ð1  cos wÞrad 2

2

DGa;GB ¼ sGB raa ¼ ðpr sin wÞraa

2

DGb;GB ¼ DGa;GB ¼ ðpr sin wÞraa

For the case in Fig. 2c DGc,V is rewritten as DGc;V ¼

pr3 ½2ð2  3 cos w þ cos3 wÞ  2ð2  3 cos / 3 þ cos3 /Þ½DGchem  ðx=thyd Þ

DGi,s and DGi,GB do not change with applied stress (as in Appendix A). Based on the same derivation as in Appendix A we can get Eq. (10).

ðA6Þ

ðB3Þ

Appendix C (-cons#)S represents the force acting on the GB with S area in the direction of the GB normal. jq is the change in the GB thickness under stress. (-cons#)Sjq represents the stress-induced increase in GB energy with S area. Taking S = 1, (-cons#)jq represents the increase in GB energy per unit area. The GB energy per unit area under tensile stress (raa,ten) is thus raa,ten = raa + (-cons#)jq. Appendix D

ðA7Þ

pr3 ½2ð2  3 cos w þ cos3 wÞ  2ð2  3 cos / ¼ 3 þ cos3 /Þ½DGchem þ DGGB strain 

DGc;S ¼ 4pr2 rad ðcos /  cos wÞ þ 2rcad pr2 sin2 / 2

ðB2Þ

ðA4Þ

Based on the same derivation path as for Eq. (1) we can get Eq. (2). For the case in Fig. 2c, we have DGc;V

 ½ð-consuÞ=3tH þ DGGB strain  ð-consuÞv

 ½ð- cos uÞ=3tH þ DGGB strain  ð-consuÞv

 pr3  2ð2  3 cos w þ cos3 wÞ  ð2  3 cos / þ cos3 /Þ DGb;V ¼ 3  DGchem þ DGGB ðA5Þ strain

2

pr3 ½2ð2  3 cos w þ cos3 wÞ  ð2  3 cos / 3 þ cos3 /Þ½DGchem  ðx=thyd Þ

ðA3Þ

where ta and ss are the volume and the interface area of the hydride nucleus, respectively, sGB is the GB area covered by the hydride nucleus and r is the radius of the spherical portion. Following the treatment given in Johnson et al. [27] we get Eq. (1). For the case in Fig. 2bwe have

1 DGb;S ¼ 4pr2 ½ð1  cos wÞ  ð1  cos /Þrad 2 þ rcad pr2 sin2 /

DGb;V ¼

2

DGc;GB ¼ DGb;GB ¼ DGa;GB ¼ ðpr sin wÞraa

ðA8Þ ðA9Þ ðA10Þ

Based on the same derivation path as for Eq. (1) we can get Eq. (3). Appendix B Under tensile stress there is stress interaction energy with the hydrides and, moreover, the tensile stress changes the chemical free energy of nucleation. The term for the bulk free energy (DGi,V) in Eq. (A1) therefore changes with applied stress. For the case in Fig. 2a DGa,V is rewritten as 2 DGa;V ¼ pr3 ð2  3 cos w þ cos3 wÞ½DGchem 3  ðx=thyd Þ½ð-consuÞ=3tH þ DGGB strain  ð-consuÞv For the case in Fig. 2b DGb,V is rewritten as

Based on Eq. (5), the ratio of hydride nucleation at the radial GB to that at circumferential GB is written as J GB;rad Z rad Drad N rad expðDGrad =kT Þ ¼ Z cir Dcir N cir expðDGcir =kT Þ J GB;cir

We take Zrad = Zcir as an approximation. According to Eq. (8) and following the same derivation as for Db =Dintra , the ratio of the atomic diffusivity at the radial GB to that at the circumferential GB is given by Drad =Dcir ¼ exp ½2r20 ðraa;rad  raa;cir Þ=kT , where raa,rad and raa,cir are the radial GB energy and the circumferential GB energy, respectively. The ratio of the nucleation sites at the radial GB to those at the circumferential GB (Nrad/Ncir) is related to the volume of the radial GB (Vrad) and the circumferential GB (Vcir). Assuming that the width is the same for the radial and circumferential GB, Vrad/Vcir = Srad/Scir, where Srad and Scir are the areas of the radial and the circumferential GB, respectively. If the form of Fig. 2b is dominant for hydride formation at the circumferential GB, hydride formation at the circumferential GB depends on the possibility (g) of the circumferential GB lying on the basal plane. We thus approximately have Nrad/Ncir = Srad/Scirg. Substituting Drad/Dcir and Nrad/Ncir into Eq. (D1) yields Eq. (12). References

ðB1Þ

ðD1Þ

[1] Northwood DO, Kosasih U. Int Met Rev 1983;28:92. [2] Puls MP. Acta Metall 1981;29:1961. [3] Puls MP. Acta Metall 1984;32:1259.

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