Journal of Nuclear Materials 418 (2011) 98–105

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A variable-gap model for calculating free energies of helium bubbles in metals T. Jourdan ⇑, J.-P. Crocombette CEA, DEN, Service de Recherches de Métallurgie Physique, F-91191 Gif-sur-Yvette, France

a r t i c l e

i n f o

Article history: Received 8 April 2011 Accepted 15 July 2011 Available online 31 July 2011

a b s t r a c t We propose a variable-gap energy model for helium bubbles in metals, based on molecular dynamics (MD) calculations. The emphasis is put on the appropriate description of the helium-metal repulsion, which can be modelled as a variable-size gap between regions occupied by helium and metal atoms. Each contribution to the bubble energy is parametrized on MD calculations performed in iron. The model is shown to reproduce accurately the dissociation energies obtained by MD over a large range of helium-to-vacancy ratios. Improvements over previous models are shown on a few equilibrium properties: binding energies, solid to fluid transition, helium density in bubbles and validity of Laplace law. Beyond the iron case, such a model should be valid in other metals where helium behavior is similar. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Neutron irradiation of metals in fission and fusion reactors creates atomic displacements, which result in vacancies and self-interstitials atoms (SIA), and produces foreign atoms through nuclear transmutations. Among these atoms, helium is particularly detrimental to the mechanical properties of the material, since it has been shown to increase void swelling and cause intergranular embrittlement [1]. As a noble gas, helium does not interact chemically with atoms of the metallic matrix and tends to cluster with vacancies, thus creating bubbles. The pressure inside these bubbles reduces the vacancy emission rate and favors the bubble nucleation; these bubbles can then grow to large sizes and lead to significant void swelling [2]. It has been shown that under certain conditions, helium pressure can be high enough to trigger the emission of self-interstitial loops [3]. A key parameter governing these phenomena is the heliumto-displacement per atom (He/dpa) ratio, which typically varies from a few tenths to a few tens of appm per dpa in fission [4] and fusion [5] devices respectively. This ratio can span an even broader range in experiments where helium is directly injected into the samples. This is the case in dual beam irradiations, where one beam is used to create atomic displacements, while the other one injects helium into the sample [6,7]. Finally, in thermal desorption spectroscopy (TDS), which is a useful way to obtain information on small He–V clusters [8–10], atomic displacements are created only by helium implantation, so the He/ dpa ratio is generally very high compared to other irradiation conditions. ⇑ Corresponding author. E-mail address: [email protected] (T. Jourdan). 0022-3115/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jnucmat.2011.07.019

The interpretation of experiments can benefit strongly from the modelling of bubble nucleation and growth. To reach irradiation doses comparable to those obtained experimentally, mesoscopicscale methods are needed, such as object kinetic Monte Carlo [11] (OKMC) and cluster dynamics [12,13] (CD). In such methods, bubbles are considered as single objects which migrate as a whole and which can absorb and emit mobile species (interstitial helium atoms, vacancies, etc.). The lack of atomistic description of the bubbles allows one to considerably speed up the computations with respect to atomic scale methods. This approximation requires the use of an energy model for bubbles, in order to predict dissociation energies and thus emission rates [14–16]. Their validation is based on the comparison of binding energies of vacancies and helium atoms, deduced from the model on one hand, and calculated by molecular dynamics (MD) on the other hand, for bubbles containing less than 20 vacancies [16]. Binding energies prove to vary with the helium-to-vacancy (He/V) ratio with discrepancies between the models and the MD results. First, at low He/V ratio, the models predict a rather flat profile for the variation of the binding energy with the He/V ratio. Conversely MD results exhibit a linear variation. Second, models predict a much sharper variation than MD when the He/V ratio is larger than one. Since the He/V ratio is expected to vary a lot, depending on the experimental conditions, it is important to build a model able to describe accurately both low and large ratios. In the following we present a model for helium bubbles in iron, where each energy term (He–He, Fe–He and Fe–Fe) is parametrized separately, and which is shown to be in good agreement with MD simulations for He/V ratios between 0 and around 3. Implications on the thermodynamics of bubbles, i.e. binding energies, fluid to solid transition and helium content at equilibrium, are then discussed. Since the behavior of helium in other metals is essentially

99

T. Jourdan, J.-P. Crocombette / Journal of Nuclear Materials 418 (2011) 98–105

the same, this model is expected to be extendable to other cases of interest.

F bV ðm; nÞ ¼ F bV ðm; 0Þ þ DF bV ðm; nÞ

ð8Þ

F bHe ðm; nÞ

ð9Þ

2. Energy model for helium bubbles in iron

where

2.1. Previous models and definitions

DF bV ðm; nÞ ¼ DF f ðm  1; nÞ  DF f ðm; nÞ;

ð10Þ

DF bHe ðm; nÞ

ð11Þ

Models of helium bubbles [14,15,17,16] are essentially based on a bulk helium energy deduced from a helium equation of state (EOS), and include corrections to account for the confinement of helium in a small cavity. In the description of these models, the need for a high density helium EOS has been extensively discussed, since pressure in bubbles can amount to several GPa [18,19]. In the case of helium bubbles in aluminum, fluid to solid phase transition has even been suggested by some experiments, coupled to theoretical calculations [20]. Such a transition has been shown by MD calculations in iron [21]. The free energy of a bubble containing m vacancies and n helium atoms can be decomposed into three terms, due to the Fe–Fe, Fe–He and He–He interactions:

F f ðm; nÞ ¼ F He—He ðm; nÞ þ F FeHe ðm; nÞ þ F FeFe ðm; nÞ:

ð1Þ

This decomposition is particularly convenient as it can be directly mapped on the different energy contributions obtained by MD calculations. Following the model by Trinkaus [14], the He– He contribution should be equal to some bulk energy deduced from an appropriate EOS, corrected by a curvature effect. The Fe– He contribution is due to the repulsive Fe–He interaction. In the existing models, the effect of this repulsion is taken into account by considering that the volume available for helium atoms is not the full cavity volume VV, corresponding to a radius rV, but a smaller volume VHe, associated with a radius rHe. The difference rV  rHe = dHe was estimated by Trinkaus to be a constant equal to around 0.1 nm in nickel. Finally, the Fe–Fe contribution contains a surface free energy, independent of the presence of helium, and a term due to the lattice relaxation around the bubble, which is treated by elastic theory. Since the first part of the Fe–Fe term does not depend on the presence of helium, the formation free energy of a bubble can be rewritten as f

f

f

F ðm; nÞ ¼ F ðm; 0Þ þ DF ðm; nÞ;

ð2Þ

f

where F (m, 0) is the surface free energy, and

DF f ðm; nÞ ¼ F He—He ðm; nÞ þ F Fe—He ðm; nÞ þ DF Fe—Fe ðm; nÞ;

ð3Þ

where DFFe–Fe(m, n) is due to the elastic relaxation. The surface energy will be omitted from the parametrization, since we are only interested in the effect of helium. We may thus simply call DFf(m, n) the formation free energy of the bubble. The binding free energies of vacancies (V), helium atoms (He) and SIA (I) are deduced from the formation free energies, using the following expressions:

F bV ðm; nÞ ¼ F fV þ F f ðm  1; nÞ  F f ðm; nÞ F bHe ðm; nÞ F bI ðm; nÞ

¼

¼

F fHe

F fI

f

f

þ F ðm; n  1Þ  F ðm; nÞ f

f

þ F ðm þ 1; nÞ  F ðm; nÞ;

ð4Þ ð5Þ

¼

F fHe

þ

DF bHe ðm; nÞ;

f

f

¼ DF ðm; n  1Þ  DF ðm; nÞ:

The energies in Eqs. (10) and (11) contain all the dependency of the binding energies on the He/V ratio; these are the quantities that should be reproduced by the model. 2.2. Variable-gap model Recently, MD calculations on large bubbles [21] (rV = 1 nm) have confirmed the reduction of the volume available for helium atoms, due to the Fe–He strong repulsion. However, as the He/V ratio increases from 1 to 2, a clear decrease of the gap between helium and iron atoms has been highlighted. Although this effect may seem marginal for the thermodynamics of large bubbles, where bulk effects due to the helium pressure are likely to dominate, the reduction of the gap could be of importance when its size is not negligible compared to the bubble size. To include such a variable gap (Fig. 1), we look for a function DUf(m, n, rHe, rV), whose minimum with respect to rHe and rV, under the constraint rHe < rV, would be equal to DFf(m, n). We assume that this function can be written as

DUf ðm; n; r He ; r V Þ ¼ UHe—He ðn; r He Þ þ UFe—He ðn; r V  r He Þ þ DUFe—Fe ðm; r V Þ:

ð12Þ

The expression for the terms in Eq. (12) and their parametrization are given in the following sections. The parametrization is based on MD simulations; unless specified, the structures are relaxed using energy minimization. As in previous models, the effect of temperature is introduced only in the He–He term, through the helium EOS. Therefore, in the following the formation energy at 0 K, Ef(DEf), is used instead of the formation free energy Ff(DFf). 2.3. Parametrization 2.3.1. Methods The computation of formation energies of helium bubbles in iron by MD, using energy minimization, has been the subject of several studies [22,23]. However, to our knowledge, only small bubbles, containing less than 20 vacancies, have been considered so far. For such small sizes, Fe–He interface effects are expected to be a significant part of the bubble energy even for low helium content, which is probably not the case for larger bubbles where helium atoms should lie somewhere in the bubble far from iron atoms, due to the strongly repulsive Fe–He interaction. To parametrize the model, it is thus interesting to determine the formation energy of larger bubbles. We have performed MD calculations for bubbles containing up to 200 vacancies, with a maximum He/V ratio equal to 3.

ð6Þ

where F fV ¼ F f ð1; 0Þ; F fI and F fHe ¼ F f ð0; 1Þ are the formation free energies of a vacancy, a SIA and an interstitial helium atom respectively. It should be noted that the interstitial binding free energy can be readily deduced from the vacancy binding energy:

F bI ðm; nÞ ¼ F fI þ F fV  F bV ðm þ 1; nÞ:

ð7Þ

Setting the surface energy contribution apart, the binding free energies can be rewritten as

Fig. 1. Schematic layout of a bubble within the variable-gap model, for two helium contents n0 > n, resulting in different radii r 0He > r He and r 0V > r V .

T. Jourdan, J.-P. Crocombette / Journal of Nuclear Materials 418 (2011) 98–105

Ef ðm; nÞ ¼ EððN  mÞFe;nHeÞ  ðN  mÞEðFeÞ  nEðHeÞ:

ð13Þ

In this equation, E((N  m)Fe, nHe) is the energy of the bubble configuration obtained by the MD calculation, and E(Fe) and E(He) are energies of iron and helium atoms in a reference state (bcc for iron). As it can be shown by substituting Eq. (13) into Eqs. (4) and (5), the helium reference energy E(He) is not required to compute binding energies: only the difference E(NFe, 1He)  NE(Fe) is needed, where E(NFe, 1He) refers to a helium atom in interstitial tetrahedral position. This difference is equal to 4.35 eV with the set of potentials used. 2.3.2. He–He interactions As suggested by Trinkaus [14], a convenient way to parametrize the He–He interaction is to use an expression of the energy valid in the bulk, and to add a surface correction, so the energy appearing in Eq. (1) reads surf EHe—He ðm; nÞ ¼ nb ebulk He—He ðn; mÞ þ ns eHe—He ðn; mÞ

¼ neHe—He ðn; mÞ;

ð14Þ

where nb and ns are the number of bulk and surface sites and surf ebulk He—He ðn; mÞ and eHe—He ðn; mÞ are the associated energies. We have parametrized both contributions on our MD results. In practice, bulk helium sites are distinguished from surface helium sites by comparing two Voronoi decompositions: in the first one iron atoms are included, while in the second only helium sites are considered. We define bulk sites as atoms whose Voronoi volume remains unchanged from one decomposition to another. Moreover, this

(a)

0.30

m 5 10 15 20 30 50 100 200

eHe− He (eV)

0.25 0.20 0.15 0.10 0.05 0.00 0

5

10

15

20

15

20

vHe (× 10− 3 nm3)

(b)

0.5 bulk eHe− He (eV)

The potentials due to Ackland et al. [24] for Fe–Fe interactions, Juslin and Nordlund for Fe–He [25], and Beck for He–He [26] have been used in the present work. It has been shown recently that this combination of potentials gives similar results to density functional theory (DFT) calculations [27] for small cluster sizes, and in particular allows one to reproduce the values of binding energies of vacancies and interstitial helium atoms to vacancy-helium clusters as a function of the He/V ratio [23]. Simulations were performed at constant volume in bcc iron, using a box size ranging from 16a  16a  16a for small bubbles to 30a  30a  30a for large bubbles, where a = 0.287 nm is the lattice parameter at 0 K obtained for pure iron with the potential used. Periodic boundary conditions were used in each direction. The bubbles were first created without helium, using two different procedures. Starting from a single vacancy, larger voids were created iteratively by removing at each step either the atom of highest potential energy after atomic relaxation, or the atom closest to the center of the void. The first method, which was used in Ref. [22], is well adapted to small sizes since it gives the correct stable configuration of vacancy clusters, but is more dubious for larger voids. We have used this method for voids containing less than 20 vacancies. For larger voids, we considered the second option, which leads to spherical voids. Atomic structure was relaxed at 0 K using a conjugate gradient (CG) algorithm. To determine the formation energies at 0 K with helium, helium atoms were introduced at random positions in the void and the structure was relaxed with a CG calculation. Then, to allow the system to overcome the energy barriers imposed by the initial position of helium atoms, 10,000 MD steps of 0.2 fs were performed in the NVE ensemble. We imposed a target temperature of 300 K by an appropriate scaling of the initial velocity distribution. Every 100 time steps, the configuration was retrieved and quenched. The lowest energy configuration over the 100 relaxed configurations was then used to compute the formation energy at 0 K. For a simulation box containing N  m iron atoms (m vacancies) and n helium atoms, the formation energy is defined as

0.4 0.3 0.2 0.1 0.0

0

5

10

vHe (× 10− 3 nm3) 1.0

(c)

0.9 bulk eHe− He/ eHe− He

100

0.8 0.7 0.6 0.5 0.4 0.3 0.2

0

100

200

300

400

500

600

n Fig. 2. Energy per helium atom due to He–He interactions, as a function of the mean volume of helium atoms, for different numbers of vacancies m, considering   (a) all helium sites (eHe–He) and (b) only bulk sites ebulk He—He . The line is the energy– volume curve for a helium fcc crystal. (c) Ratio eHe—He =ebulk He—He as a function of the number of helium atoms n in the bubble, which highlights the surface contribution. The line is a fit of the data set (see text).

decomposition also enables us to assign a volume to each helium bulk site, and thus a mean volume for helium sites vHe. The mean energy per helium atom due to He–He interactions as a function of vHe is represented in Fig. 2 for different bubble sizes. All helium sites (eHe–He) are considered in Fig. 2a, whereas only   bulk sites ebulk He—He are taken into account in Fig. 2b. The energy– volume curve corresponding to a fcc phase is also represented. The bulk data do not depend on the bubble size but only on vHe, which validates our approach of bulk site determination. In addition, they are nearly aligned on the fcc curve; the slight shift comes from the fact that in the bubble, the structure is not fcc [21]. To obtain an expression for the helium bulk energy, we use the helium pressure at 0 K p0 which is related to the He–He energy by

ebulk He—He ðv He Þ ¼

Z

1

v He

p0 dv ;

ð15Þ

the integral going from the actual helium volume to infinite dilution. The energy is fitted using a Vinet EOS [28] for the pressure p0:

p0 ¼

3K 0 X2

ð1  XÞ expðg0 ð1  XÞÞ;

ð16Þ

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T. Jourdan, J.-P. Crocombette / Journal of Nuclear Materials 418 (2011) 98–105

where

X¼

v v0

1=3

3 2

g0 ¼ ðK 00  1Þ:

ð18Þ

The fit gives v0 = 1.951  102 nm3, K0 = 1.497 eV/nm3 and K 00 ¼ 8:465. We find that the surface correction is roughly proportional to the bulk helium energy, since it results from missing He–He interactions, and to n2/3, so Eq. (14) becomes 2=3 bulk EHe—He ðm; nÞ ¼ nebulk eHe—He ðv He Þ: He—He ðv He Þ  an

T 0K 300 K 600 K

25

ð17Þ

;

20

p (GPa)



30

15 10 5

ð19Þ

0

The fit of the ratio of the total energy to the bulk contribution,

4

6

8

10

vHe ð20Þ

shown in Fig. 2c, leads to a = 1.354, in close agreement with the value used by Morishita [16]. To include the effect of temperature in the model, we follow the approach adopted by Trinkaus [14]. For solid helium, a temperature-dependent term is added to the 0 K pressure term (Eq. (16)). This is particularly convenient, as this contribution has been derived from calculations performed with the same He–He potential as we used in the present study. For fluid helium, the pressure is deduced from a four term virial expansion of the compressibility factor. The choice of the solid or fluid state for helium is done by comparing the helium volume to the volume upon freezing, as determined semi-empirically by Trinkaus. The bulk free energy per helium atom due to He–He interactions is then bulk fHe—He ðv He Þ ¼

Z v1 v He

ideal p dv þ fHe—He ðv 1 Þ:

ð21Þ

The volume v1 should be sufficiently large, so that the expression of ideal the free energy for an ideal gas fHe—He ðv 1 Þ is valid. The value v1 corresponding to a pressure of 103 Pa in the ideal gas approximation was adopted; it has been checked that the variation of v1 around this value does not modify the free energy of the fluid obtained after integration. The computation of the He–He free energy cannot be performed easily by MD [29]. To validate the model, we simply check that the pressure given by the model is consistent with MD simulations at 300 K and 600 K. The average pressure is calculated using the Lutsko stress tensor [30,31], considering only interactions between helium atoms. The averaging volume is computed as nvHe. This calculation is compared with the bulk pressure given by the model, using the same helium volume and number of helium atoms in the bubble. Although bulk pressure does not depend on the number of helium atoms n, consistency with MD is only achieved by multiplying the bulk pressure by the corrective surface factor given in Eq. (20). The plot of the pressure for a bubble containing m = 200 vacancies shows satisfactory agreement between the two approaches (Fig. 3). In the framework of the model, it is convenient to make expressions depend on rHe instead of the helium volume vHe. We define this radius with the following relation:

nv He ¼

4 3 pr : 3 He

ð22Þ

So finally, the He–He term can be written as



UHe—He ðn; r He Þ ¼ nf bulk He—He ðr He Þ 1 

a  n1=3

:

ð23Þ

16

(× 10− 3

nm3)

18

20

22

Fig. 3. Helium pressure calculated by MD (symbols) and deduced from the model (lines), at different temperatures, for a bubble containing 200 vacancies. The helium and number of helium atoms are the only variables needed to compute the pressure with the model; they are obtained from the MD simulations.

2.0

P(r) ϕ (r) ( × 103 eV/ nm)

eHe—He a ¼ 1  1=3 ; n ebulk He—He

14

12

1.5 1.0 0.5 n = 440

0

n = 340

0

n = 240

0

n = 140

0 0.0

0.1

0.2

0.3

0.4

0.5

r (nm) Fig. 4. Density of Fe–He bonds, weighted by the Juslin–Nordlund potential, as a function of the Fe–He distance, for a bubble containing 200 vacancies. Results are shown for several helium contents. The vertical dotted lines correspond to rFe–He, as defined in Eq. (26).

2.3.3. Fe–He interactions As mentioned in Eq. (12), in our model we try to make the Fe– He term depend on the difference between the position of sharp interfaces rV  rHe. The Fe–He energy computed by MD can be written as

EFeHe ðm; nÞ ¼

X i

uðri Þ 

Z

1

PðrÞuðrÞ dr;

ð24Þ

0

where the summation is performed on all Fe–He pairs, u is the Fe–He potential and P(r) represents the density of Fe–He bonds at distance r. The quantity to integrate P(r)u(r) is shown in Fig. 4 for a bubble containing 200 vacancies, and for several contents of helium. Note first that this figure clearly exhibits a closure of the gap between helium atoms and the iron surface upon accumulation of helium in the bubble, thus justifying our variable-gap approach. We want to express EFe–He(m, n) with a characteristic Fe–He distance (rFe–He) and the number of helium atoms n in the bubble. Naturally, Fe–He interactions occur mainly at the Fe–He interface, so that EFe–He(m, n) is proportional to n2/3. Although the distribution presents some spread, one can express P(r) as

102

T. Jourdan, J.-P. Crocombette / Journal of Nuclear Materials 418 (2011) 98–105

EFe− He (eV)

120

(a)

120

m 5 10 15 20 30 50 100 200

100 80 60 40 20

EFe− He/ n2/ 3 (eV)

0 0.24

0.26

0.30

0.32

0.34

0.36

0.38

(b)

4

80 60 40 20

3

0 0.00

0.02

0.06

0.08

0.10

ΔrV (nm)

1

0.26

0.28

18 16 14 (c) 12 10 8 6 4 2 0 0.24 0.26

0.30

0.32

0.34

0.36

0.38

Fig. 6. Relaxation energy DEFe–Fe of the iron lattice for a bubble containing 200 vacancies, as a function of the displacement of the bubble surface DrV. The crosses are the MD results and the line is the energy given by the isotropic elastic theory (Eq. (34)).

One point to emphasize is that in order to relate rFe–He to the smallest distance between helium and iron atoms, we can use the standard deviation of the distribution P(r)u(r) around the mean value rFe–He. A fit of this standard deviation leads to

rðrFe—He Þ ¼ r1 rFe—He þ r0 ; 0.30

0.32

0.34

0.36

0.38

Fig. 5. (a) Fe–He total energy for different numbers of vacancies m as a function of rFe–He (Eq. (26)), (b) divided by n2/3 and (c) divided by n2/3u(rFe–He). The line is a fit, using Eq. (28).

PðrÞ ¼ n2=3 gðrÞdðr  r Fe—He Þ;

ð25Þ

P

r i uðri Þ i r Fe—He ¼ P : uðri Þ

2.3.4. Fe–Fe interactions As mentioned in Section 2.2, only the elastic relaxation of the iron lattice due to helium is parametrized on MD calculations. The binding energy of a vacancy to a void EbV ðm; 0Þ is deduced from the formation energies determined by DFT calculations [32] for small sizes (m 6 5), and from the usual capillary law [33,34], parametrized on the same DFT calculations:

EbV ðm; 0Þ ¼ Ef ð1; 0Þ þ

where g(r) is to be specified and rFe–He is defined by

ð26Þ

EbV ð2; 0Þ  Ef ð1; 0Þ

Ef ð1; 0Þ  EbV ð2; 0Þ 2

2=3

1

This distance is represented in Fig. 4 with vertical dashed lines. The Fe–He energy given by the model is thus

where r 0V is defined by

UFe—He ðn; r Fe—He Þ ¼ n2=3 gðr Fe—He Þuðr Fe—He Þ:

4  0 3 a3 p rV ¼ m ; 2 3

ð27Þ

Although the use of u(rFe–He) does not account for the spread of the distribution, we have checked that including this spread in P(r) only improves marginally the results with respect to the simple expression (25). Taking into account the shift of the distribution, and thus the reduction of the gap, through the variation of rFe–He, is sufficient for our model. To ensure that UFe–He(n, rFe–He) = EFe–He(m, n), we fit the quantity EFe–He(m, n)/(n2/3u(rFe–He)) (Fig. 5c). This is done with the following function

g rr00  Dr

þ1

;

ð28Þ

with g0 = 19.17, r0 = 0.3004 nm and Dr = 0.0247 nm. This function accounts, among other things, for the increase of the number of Fe–He bonds for a given helium atom close to the iron surface when rFe–He decreases. To make expression (27) complete, we need a link between rFe–He and rV  rHe. This expression is given in the following section.

22=3  1

ðm2=3  ðm  1Þ2=3 Þ:

ð30Þ

The corresponding formation energy is

Ef ðm; 0Þ ¼

i

exp

ð29Þ

with r1 = 0.3 and r0 = 0.122 nm. 0.28

rFe− He (nm)

gðrÞ ¼

0.04

2

0 0.24

EFe− He/ n2/ 3/ ϕ (rFe− He)

0.28

100

ΔEFe− Fe (eV)

140

 2 m2=3 ¼ 4p r0V c;

ð31Þ

ð32Þ

and c is the surface energy. This energy reads

c¼

1 ð9p

Þ1=3 a2

Ef ð1; 0Þ  EbV ð2; 0Þ 22=3  1

:

ð33Þ

Using the DFT values for the formation and binding energies, we obtain c = 1.98 J m2, in reasonable agreement with theoretical [35] and experimental [36] values. We think a more involved parametrization of this energy term on our MD calculations would not make sense, since the vacancy formation energy obtained with the present Fe–Fe potential is significantly lower than the reference value given by DFT calculations. The parametrization of the relaxation energy is based on the following expression, given by the isotropic elastic theory [37]: 2 2 0 DEFeFe ðm; nÞ ¼ 8plr cav V ðDr V Þ  8plr V ðDr V Þ ;

ð34Þ

where r cav V is the true radius of the cavity without helium, DrV is the displacement of the iron surface under the helium pressure and l is

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T. Jourdan, J.-P. Crocombette / Journal of Nuclear Materials 418 (2011) 98–105

 2 DUFeFe ðm; r V Þ ¼ 8plr 0V r V  r 0V :

ð35Þ

6 5

E d (m, n) ( eV)

the shear modulus. Approximating r cav with r0V as in Eq. (34) still V leads to a satisfactory agreement with MD calculations, as soon as DrV is directly measured by MD (Fig. 6), even for large displacements of around 0.1 nm. To compute the shear modulus we use the Voigt-Reuss-Hill average [38] of the elastic constants obtained with the present potential [24], which leads to l = 82.2 GPa, in close agreement with the experimental value [39]. We thus define the Fe–Fe term in the model as:

4

2 m

The bubble radius rV must be chosen so that r V  r 0V is equal to DrV. We define it as

1

r V ¼ r He þ r FeHe  brðr FeHe Þ  r s ;

0 0.0

ð36Þ

3. Thermodynamic properties of bubbles

0.5

1.0

1.5

2.0

2.5

3.0

Fig. 7. Dissociation energy of a vacancy (squares) and of a helium atom (triangles), obtained by MD (symbols) and by the bubble model (lines), as a function of the He/ V ratio n/m.

5.5 (a)

T 0K 300 K 600 K

5.0 4.5 4.0 3.5 3.0 2.5

We now apply our model to a few interesting properties of helium bubbles in iron. Firstly the problem of inconsistency between results obtained with previous analytical models and MD simulations is addressed. Secondly we look for the effect of the variable-size gap on the solid–fluid transition, and finally we discuss the validity of the Laplace law to determine the equilibrium density of helium when such a gap exists.

5 31 101 201

He/ V ratio (n/ m)

F b (m, n) ( eV)

with b = 2.5 and rs = 0.09 nm. The two last correcting terms in Eq. (36) can be explained as follows. Since rFe–He was computed using Eq. (26) and the distribution of distances exhibits some spread around this value, rFe–He should be reduced to determine the position of iron atoms when adding rHe to rFe–He. The value b = 2.5 can be justified in the case of a gaussian distribution; it means we are interested in the lowest populated values of the distribution. In addition, the values rHe and r 0V determined by Eqs. (22) and (32) are slightly shifted off the atomic position of helium and iron surface atoms, which justifies the addition of the last correcting term. This way the minimum of the function DUf (Eq. (12)), written as the sum of contributions given by Eqs. (23), (27) and (35), is shown to give a contribution of the different energy terms in reasonable agreement with those obtained by MD.

EVf + EVm

3

(b) EVf

2.0 1.5 0.0

0.5

1.0

1.5

2.0

2.5

3.0

He/ V ratio (n/m) Fig. 8. Binding free energy of (a) a vacancy and (b) a helium atom, as given by the model, for a bubble containing m = 200 vacancies. The horizontal dashed line is used in Section 3.3 to determine the equilibrium density of helium in bubbles.

3.1. Dissociation energies Dissociation energies Ed(Fd) are the sum of the binding energies, as defined by Eqs. (8) and (9), and of the migration energies (0.67 eV for the vacancy [34] in iron and 0.06 eV for interstitial helium atom [27]). The dissociation energies of a vacancy and a helium atom at 0 K are plotted in Fig. 7 as a function of the He/V ratio. Results obtained by MD and by the model are compared for different bubble sizes, from 5 to 201 vacancies. The vacancy binding energies without helium are identical in MD and in the model and they are deduced from the capillary law, as explained in the previous section. The agreement between the two methods is satisfactory, even for very small bubbles. The vacancy dissociation energy increases with the He/V ratio, since the emission of a vacancy results in an increase of the bubble energy due to the helium pressure. This increase is almost linear, as evidenced by previous MD [23] and DFT [27] studies, except for large bubbles at small helium content. Conversely, the dissociation energy of a helium atom decreases with the He/V ratio. The crossover between helium and vacancy curves varies around a He/V ratio equal to 1, close to the values obtained by previous MD [23] and DFT [27] calculations. The present MD calculations and model enable us to highlight a finite size effect on the vacancy and helium dissociation energies and to bridge the gap between previous results. In former MD calculations, at small He/V ratios a linear variation of

the vacancy and helium dissociation energies was found [22], whereas a zero slope was obtained by the models [16]. We show here that the linear variation only holds for very small bubbles, and that from around 30 vacancies the dissociation energy curve clearly exhibits a zero slope at small helium content. The linear variation for small bubbles is due to the Fe–He surface effects, which are significant even at low helium content. It can be shown that in previous models, since the Fe–He energy was treated as a first order correction of the helium bulk free energy, and since the Fe–He distance was assumed to be constant, the vacancy binding energy was proportional to the bulk pressure, which at 0 K varies very softly at small helium content, and thus did not exhibit a linear slope. Another difference with previous models appears for ratios greater than around 2. For such ratios, these models predict an abrupt decrease of the vacancy binding energy, which does not show up in MD calculations. This decrease was explained by a failure of the elastic theory. However, as it has been shown in Section 2.3.4, elastic theory seems adequate even at such He/V ratios. This spurious drop may come from the way the elastic term was included in the previous models, using a first order expansion of the bubble free energy as a function of the increase of bubble size. No such abrupt decrease is seen in our model for the He/V ratios considered, in agreement with MD results.

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The binding free energies are shown in Fig. 8, for a bubble containing 200 vacancies and for three different temperatures. Providing the effect of temperature is correctly given by the sole contribution of the helium pressure, the change in the binding energy seems rather marginal, notably with respect to the finite size effects that have just been discussed. The non-monotonous variation of the helium binding energy with temperature (in particular the increase at small helium content) can be explained by the entropic contribution. 3.2. Solid–fluid transition

2.5 60

150

400

1.5

300

100

n/m

2.0

0 50

1.0 50

0.5

0.5

1.0

1.5

2.0

rV0

2.5

3.0

3.5

3.3. Helium density at equilibrium The experimental measure of helium density in bubbles can be made by several techniques [19], but only a few of them enables one to probe small bubbles. PAS is a convenient way to characterize small vacancy clusters, because positron lifetime increases with the void radius. Since this lifetime saturates for void radii of around 0.5 nm, the decrease in lifetime observed in presence of helium can be unambiguously correlated to the density of helium inside vacancies [41]. This technique is particularly interesting because helium density can be directly deduced from the lifetime measurements and a simple model deduced from theoretical calculations [42,43]. Electron energy-loss spectroscopy (EELS) has also received much attention because of its ability to probe single bubbles when combined with scanning transmission electron microscopy (STEM) [44,45]. In these experiments, the assumption of bubble thermal equilibrium with respect to vacancies is often made. This is achieved by imposing the equality of chemical potentials of vacancies inside bubbles and in the matrix, so the chemical potential inside bubbles should be zero. It is customary to assume that the free energy of the bubble can be written under the simplified form 0 F f ðm; nÞ ¼ F bulk He—He ðm; nÞ þ 4pr V c:

4.0

(nm)

Fig. 9. Apparent helium density (left), or He/V ratio (right) necessary for helium freezing, at different temperatures (in K), as a function of the unrelaxed bubble radius defined by Eq. (32).

ð37Þ

Then if one derives this expression with respect to m to obtain the chemical potential, or equivalently with respect to the bubble volume VV since elastic effects are neglected, one obtains the usual Laplace law

p¼

2c ; r0V

ð38Þ

which defines the content of helium in every bubble type, through an helium EOS, to ensure global thermal equilibrium. Depending on the type of experiment, this law is used to determine the bubble radius [41] or the content of helium [46], when coupled to an appropriate EOS. In PAS and STEM–EELS experiments, it is possible to have access to both parameters, so the validity of the Laplace law can be checked [45,47,48]. Since the free energy given by our model is more complex than Eq. (37), it is interesting to see how it alters the equilibrium helium density. Writing the vacancy chemical potential inside a bubble containing m vacancies as

lV ¼ F f ðm; nÞ  F f ðm  1; nÞ ¼ F f ð1; 0Þ  F bV ðm; nÞ;

0

200

800 700

apparent helium density (nm− 3)

The helium pressure inside a bubble can be very high, so it is interesting to see if some phase transition from fluid to solid can occur in this confined environment. This subject has been investigated experimentally in aluminum [20]. The authors used positron annihilation spectroscopy (PAS) to determine the helium density inside bubbles, which was then converted into pressure by means of the fluid helium EOS proposed by Trinkaus [14]. Comparing this pressure with the value given by the melting curve, they concluded that helium was indeed solid in some of their experiments. Our model may be used to investigate further the phase transition by including the finite size effects, and notably the Fe–He repulsion. The solid–fluid transition is included in the bulk He–He contribution and the melting curve deduced from this parametrization has been shown to be in good agreement with experimental data [14,40]. So it is possible to determine the helium state by comparing the volume per helium atom obtained after minimization of DUf to the volume upon freezing at a given temperature. The He/V ratio necessary for freezing is shown in Fig. 9, for different bubble sizes and at different temperatures. There is a linear relationship between this ratio and the apparent helium density, taking into account the full volume of the cavity, provided the volume associated with a vacancy is constant with the number of helium atoms. We have represented this apparent density in Fig. 9 to give a rough idea of the densities, although it should be kept in mind that the elastic relaxation of the bubble induces a change in the bubble radius, and thus in the apparent density. As expected, for large bubbles the transition is independent of the bubble radius, but for radii smaller than around 1–2 nm, finite size effects become significant. The Fe–He repulsion reduces the

volume available for helium atoms, so the actual density of helium atoms, calculated with rHe, can become much higher than the density computed by using the whole bubble volume.

ð39Þ

the equilibrium helium density can be read graphically in Fig. 8 at the intersection between the binding energy curves and a horizontal line defined by the vacancy formation energy. The effect of the bubble size on the equilibrium density is highlighted in Fig. 7, where a similar graphical construction has been performed. In this case the horizontal line is defined by the sum of the vacancy formation energy and the vacancy migration energy, because dissociation energies, instead of binding energies, are plotted in this figure; however the migration energy plays no role in the equilibrium density. It can be seen that for small helium content, the change of slope of the binding (dissociation) energy as a function of the bubble size compensates for the variation of the y-intercept, given by the vacancy binding (dissociation) energy without helium. That is why for a range of small sizes, the equilibrium helium density is roughly constant (Fig. 10). So we can conclude that the finite size effects, which result in an increase of the vacancy binding energy, lead to smaller helium contents at equilibrium. For radii larger than

T. Jourdan, J.-P. Crocombette / Journal of Nuclear Materials 418 (2011) 98–105

of bubbles under various experimental conditions, for both low and high He/dpa ratios.

100

Acknowledgments T = 300 K

n/m

apparent helium density (nm− 3)

10 2

105

10 1

T = 600 K

10− 1

Useful discussions with Dr. A. Barbu and Dr. C.C. Fu are gratefully acknowledged. This work was partially funded by the French National Research Agency (ANR) through the HSynThEx Project ANR-10-INTB-0905. Calculations were partially performed at the CCRT computing center. References

10 − 1

10 0

10 1

10 2

0 (nm) rV

Fig. 10. Equilibrium helium density in bubbles of radius r 0V , at 300 K and 600 K. Solid lines: model described in this article; dot-dashed lines: helium density deduced from Eq. (38), with a high density EOS (see also Ref. [14] for this curve); dashed lines: helium density deduced from Eq. (38), considering helium as an ideal gas.

[1] [2] [3] [4] [5] [6] [7] [8] [9]

around 1–2 nm, these effects become negligible and we recover the usual Laplace law. Although this size roughly corresponds to the smallest bubbles analyzed experimentally, it is interesting to note that small bubbles considered as under-pressured, using the Laplace law, may in fact be at equilibrium. 4. Conclusion In this article we have proposed a model for helium bubbles in metals. A bubble is described as a sphere containing helium atoms, separated from metal atoms by a gap whose size varies with the number of helium atoms and vacancies in the bubble. Each energy term has been parametrized separately for helium bubbles in iron, using MD simulations for bubbles containing up to 200 vacancies. We have checked that such a parametrization leads to dissociation energies in good agreement with MD simulations for He/V ratios up to around 3. The following properties, which are due to finite size effects, have been highlighted using the model:  The minimum He/V ratio necessary for the fluid to solid transition to occur, which is constant for large bubbles, is shown to decrease as the bubble radius is reduced below around 1–2 nm. This effect is directly linked to the strong contribution of the Fe–He repulsion to the energy of small bubbles, which reduces the volume available for helium atoms.  At low He/V ratios and for small bubble sizes, the dissociation energies exhibit a linear variation, which progressively flattens out as the bubble size increases. A direct consequence of this result is that under the assumption of thermal equilibrium of bubbles with respect to vacancies, the helium content is shown to be lower than predicted by the usual Laplace law. It should be noted that the hypothesis of thermal equilibrium of vacancies, which may be valid under certain annealing conditions, cannot be used in general under irradiation. When introduced in kinetic simulations, the present bubble model should provide a useful insight into the nucleation, growth and coarsening processes

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48]

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