Available online at www.sciencedirect.com

www.elsevier.com/locate/actamat

ScienceDirect

Acta Materialia 79 (2014) 248–257

Equilibrium thermodynamics of helium in d-phase Pu–Ga alloys

A.V. Karavaev a,⇑, V.V. Dremov a, G.V. Ionov a, B.W. Chung b

a Russian Federal Nuclear Center – Zababakhin Institute of Technical Physics (RFNC-VNIITF), 13, Vasiliev st., Snezhinsk, Chelyabinsk reg. 456770, Russiab Lawrence Livermore National Laboratory, Livermore, CA 94551, USA

Received 16 January 2014; received in revised form 27 May 2014; accepted 8 July 2014

Abstract

The paper presents a theoretical study of the response of d-phase Pu–Ga alloys to self-irradiation. We investigate the long-termbehaviour of He atoms in the face-centred cubic (fcc) lattice under ambient conditions by means of classical molecular dynamics. Becauseof the relatively low diffusive mobility of He atoms in the lattice their dynamics cannot be tracked directly in molecular dynamicssimulations. Instead, we use the Helmholtz free energy function to investigate the equilibrium thermodynamics of metastable microcon-figurations of perfect crystals with artificially introduced He atoms in different configurations. The Helmholtz free energy of themicroconfigurations are calculated using the thermodynamic integration method. Based on the free energy evaluation, inferences aboutthe relative thermodynamic stability of various microconfigurations under ambient conditions are drawn. Estimates of the equilibriumparameters of He bubbles in the lattice and the distribution of He bubble radii are calculated and compared to those observedexperimentally.� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: d-phase Pu–Ga alloys; Radioactive aging; He bubbles; Classical molecular dynamics; Thermodynamic integration

1. Introduction

For the past few decades one of the most extensivelystudied material science problems has been the modifica-tion of structural and radioactive materials’ propertiescaused by irradiation and/or self-irradiation, which isknown as radioactive aging. The self-irradiation of actini-des and their alloys and compounds results in the produc-tion of radiogenic helium and primary radiation defects ofthe crystal structure, their accumulation, diffusive migra-tion, clustering, etc. The morphology of the defect clusters(i.e. equilibrium size and shape, the fraction of free andclustered defects) is a subject of great importance, since ithas a direct impact on the elastic–plastic properties, staticand dynamic strength, transport coefficients, etc. The pres-ent study investigates the behaviour of the He atoms

http://dx.doi.org/10.1016/j.actamat.2014.07.022

1359-6454/� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights r

⇑ Corresponding author.E-mail address: a.v.karavayev@vniitf.ru (A.V. Karavaev).

accumulated in the face-centred cubic (fcc) lattice of d-phase Pu–Ga alloys during long-term storage under ambi-ent conditions.

The accumulation of He atoms with a rate � 41 ppm peryear [1] could affect the properties of the material [1–4].Indeed, according to one estimate [1], the He concentrationreaches 2000 ppm after 50 years of storage, while it is wellknown that a considerably smaller amount of helium(P 100 ppm) in stainless steel causes swelling and signifi-cant embrittlement.

Produced in the decay, a-particles lose most (� 99:9%)of their initial kinetic energy (� 5 MeV) through electronicexcitations, pick up two electrons and become neutral Heatoms. They occupy the nearest vacancies in the lattice orcreate and occupy vacant sites by knocking out Pu or Galattice atoms, thus generating interstitial atoms [5–8].Diffusive migration of secluded He atoms in the fcc latticeoccurs as a motion of He atom–vacancy complexes with arather high (compared to the ambient temperature)

eserved.

A.V. Karavaev et al. / Acta Materialia 79 (2014) 248–257 249

diffusive activation energy � 0:7 eV [6]. If two diffusing Heatoms meet, then the nucleus of an He bubble is formed.

The first observation of helium bubbles in naturallyaged material obtained with a transmission electron micro-scope (TEM) was reported in Refs. [9,10] for a 21-year-oldd-phase Pu–Ga alloy annealed at 400 �C for 1 h. Then, inRefs. [11–15], d-phase Pu–Ga alloy samples ranging inage from 6 months to 42 years were studied using a Fresnelfringe-imaging technique on an LLNL 300 keV TEM incombination with positron annihilation spectroscopy. Inthe samples aged under ambient conditions, the authorsobserved that the dominant defects are nanometer-sizedhelium bubbles, which are distributed homogeneously inthe bulk of the material. For 16- to 42-year-old samples,the observed number density of helium bubbles is in therange 0:6� 2:0� 1017 cm3, and increases with the age ofthe samples at a nearly steady rate. The observed averagesize of helium bubbles is � 1:4 nm, and this value, alongwith the helium density inside the bubbles (2–3 He atomsper vacant site), remains practically the same as the mate-rial ages.

More recently, Jeffries et al. [16] carried out a TEMstudy of aged (20- and 42-year-old) samples of Pu–3 at.%Ga alloy using the same methods in order to evaluate themechanism of helium bubble coarsening. After the initialcharacterisation, the samples were annealed in vacuum attemperatures from 250 to 425 �C for periods of 2–90 h.The samples were then cooled down to room temperatureand examined in the TEM. In addition, time-resolvedin situ TEM microphotographs of 42-year-old sampleswere taken during annealing at 450 �C using a Gatan hotstage. Two mechanisms of helium bubble coarsening aregenerally thought to occur [17]: so-called Ostwald ripening,when the bubbles are thought to grow due to the diffusivemotion of individual He atoms through the lattice from smal-ler bubbles to larger ones [18]; and the bubble migration andcoalescence (MC) mechanism, whereby helium migration inthe lattice occurs as the motion of bubbles as a whole, andthe bubbles grow by coalescing when they meet one another[19]. In Ref. [16] the authors concluded that in naturally agedd-phase Pu–Ga alloys the annealing causes the coarsening ofhelium bubbles, with a decrease in the overall number densityof the bubbles in the material and an increase in their averagesize. The time-resolved in situ TEM microphotographs showbubble migration and coalescence, strongly suggesting thatMC is the dominant mechanism.

The thermally activated time-dependent growth ofhelium bubbles at elevated temperatures, the migration ofthe bubbles as a whole, the strongly peaked distributionof bubble sizes and the nearly constant He atom to vacancyratio in naturally aged material suggest that there are fac-tors limiting the growth and defining the equilibriumparameters of the bubbles. In the following sections weinvestigate the solubility of helium in the fcc lattice of d-phase Pu–Ga alloys, the nature of the attractive/repulsiveinteraction of He atoms with the bubbles, the possiblemechanism limiting the size of the bubbles and the

equilibrium bubble parameters (size and He concentration –number of He atoms per vacant site) under ambient condi-tions by means of classical molecular dynamics (CMD).

While the diffusive dynamics of interstitial atoms in thefcc lattice of Pu–Ga alloys is accessible, due to their highmobility, for direct CMD simulations even below ambienttemperature [20,21], characteristic activation energies forthe motion of vacancies and He atoms are rather high[6,20], and consequently diffusive migration times of vacan-cies and He atoms are in the range from seconds to hoursat room temperature. It makes the direct modelling of theirdynamics impossible even using the accelerated (parallelreplica) CMD. Instead, in the present study, we evaluatethe relative thermodynamic stability of metastable micro-configurations of an ideal crystal with artificially intro-duced systems of defects of various morphologies usingthe Helmholtz free energy.

2. Free energy calculations in the CMD

To investigate the relative thermodynamic stability of asystem at finite temperatures, it is not sufficient to calculateand compare only the internal energies of the differentstates of the system. At finite temperatures the entropyterm of the Helmholtz free energy plays an important role.If one could calculate the “absolute” values of free energyfor the different metastable microconfigurations of asystem for the same external conditions, it would becomepossible to draw unambiguous conclusions about the rela-tive thermodynamic stability of those states. Unfortu-nately, the absolute values of thermodynamic potentialscannot be calculated in a simple CMD simulation as anaverage of some quantity or expressed as a function ofsome averages (nor can it be determined in real experi-ments). However, it is possible to evaluate absolute valuesof free energy in the frames of CMD using the so-calledthermodynamic integration method (TIM) [22–24].

The TIM is based on a reversible quasi-equilibriumtransition of a system from the state of interest to a refer-ence state for which one can somehow calculate the freeenergy. Following the formalism by Frenkel and Laad[22–24], let us consider a system of N particles in a volumeV interacting through the interatomic potential U, whichdepends linearly on the parameter k changing in the rangefrom 0 to 1. At k ¼ 0 the potential U becomes a potentialUI for the reference state, while at k ¼ 1 the potential U

becomes UII – the potential in the state of interest:U ¼ ð1� kÞU I þ kU II . If one uses the model potential U

to calculate the free energy F ðV ; T Þ expressed through thestatistical integral and take the derivative of the free energywith respect to the parameter k, one obtains

@F ðkÞ@k

� �N ;V ;T

¼ @U kð Þ@k

� �k

¼ hU II � U Iik ð1Þ

where h. . . ik is canonical average value at a fixed k. Inte-grating Eq. (1), one obtains the free energy of the state ofinterest (k ¼ 1) as

250 A.V. Karavaev et al. / Acta Materialia 79 (2014) 248–257

F k ¼ 1ð Þ ¼ F k ¼ 0ð Þ þZ 1

0

dkhUII � U Iik ð2Þ

Thus, the free energy of the state of interest is expressedas a sum of known free energy of the reference state and thefunction of canonical averages, which can be calculated inthe CMD.

In a crystalline solid the atoms spend almost all of thetime near their equilibrium positions in the lattice nodes.The root mean square (rms) deviations of atoms from theirequilibrium positions are defined by the interatomic inter-actions and the state of the crystal (stresses and its temper-ature). Since we are considering a crystal in the classicalstatistics approximation, it is quite natural to use as thereference state the Einstein harmonic crystal in the high-temperature limit, i.e.

UI rið Þ ¼XN

i¼1

ai

2ri � r

ð0Þi

� �2

þ U ð0ÞII ; ð3Þ

where rð0Þi are the equilibrium positions of atoms at

k ¼ 1; ai are effective harmonic force constants, whichare free parameters of the method and may take almostarbitrary values, with the only requirement that atomsare effectively held near their equilibrium positions duringthe entire transition from k ¼ 0 to k ¼ 1; and U ð0ÞII is thevalue of the potential for equilibrium atomic positions r

ð0Þi .

Using Eqs. (2) and (3), one can calculate the free energyof a crystalline solid as

F k ¼ 1ð Þ ¼ 3kBTX

i

lnh

pkBT

ffiffiffiffiffiai

mi

r� �þ U ð0ÞII

þZ 1

0

dkhU II � U Iik ð4Þ

The only requirement for Eq. (4) to be valid is the meta-stability of the modelled system in the sense that particlesoscillate around their equilibrium positions, which aresteady in time during the rather long time period in theframes of the CMD simulation. Eq. (4) can be used for cal-culations of the free energy of multicomponent crystalalloys containing defects if the microstructure is stablewithin the timescale of the CMD (up to ns). This require-ment is satisfied for He atoms under ambient conditionsdue to the high activation energy of the diffusive migration[6].

Here we present the results of free energy calculationunder ambient conditions for the fcc d-phase Pu–Ga alloycontaining 3 at.% Ga and He atoms as a disordered substi-tutional solution. The CMD simulations were performedwith the modified embedded atom model (MEAM) forPu–Pu, Pu–Ga and Ga–Ga interactions (see Ref. [7]) withthe parameterization for binary fcc Pu–Ga alloys [25].For He–He, Pu–He and Ga–He interactions we used exp-6 potential with the parameterization from Ref. [8].

In the calculations we used Pu–Ga samples of21� 21� 21 unit cells (37,044 atoms) with periodic bound-ary conditions. Helium (200 atoms) was added to the

sample by the substitution of 200 randomly chosen atoms.One first needs to calculate the equilibrium density q of thecrystal under the conditions of interest (or, rather, theequilibrium sizes and shape of the sample in order toavoid tangential stresses), which will be fixed in the follow-ing calculations for the canonical NVT ensemble. Tothis end, we simulated an NPT ensemble at P ¼ 0 andT ¼ 300 K. In the simulations a Nose–Hoover thermostat[26,27] was used.

The simulation of the NPT ensemble is followed by thesimulation of the canonical NVT ensemble. In this simula-tion we determined the equilibrium positions of atomswhich differ from the nodes of the ideal crystal lattice dueto the presence of Ga and He atoms, and calculated therms deviations of atoms from their equilibrium positions.The calculated equilibrium positions of atoms r

ð0Þi were

used to construct the potential U I (Eq. (3)), and the rmsdeviations were used to fix the force constants ai. If atomsdiffer in type and/or the atomic positions in the crystallattice are not equivalent, it is necessary to calculate therms deviation for each type/position of atoms and, accord-ingly, the appropriate values of ai.

The next step is to perform a reversible transition fromthe state of interest to the harmonic crystal, and then to dothe integration in Eq. (4). To do that we discretely varied kin a chain of successive simulations to thermalize the sys-tem and then calculated the time average of huII � uIik atfixed k. We verified the correctness of the result by checkingif the transition was reversible. That is, after the transitionfrom k ¼ 1 to k ¼ 0 , we did the reverse transition fromk ¼ 0 to k ¼ 1 using the same values of k in reverse order(see Fig. 1). The values obtained in the direct and reversetransitions are seen to agree well for all the values of k,i.e. the transition is reversible and in quasi-equilibrium.

Note here that there is no need to add the configura-tional entropy term to Eq. (4) since we consider He atomsthat are placed at the lattice sites in the same sense as Puand Ga atoms are constituent of the lattice. This can bejustified by the fact that the He atoms do not diffuse intothe lattice (i.e. they do not leave their lattice sites) in thetimescale of the CMD simulations.

The helium in the He bubbles is in the fluid state, but theatoms are restricted by the surrounding lattice from leavingthe bubbles and do not dissolve into the lattice under ambi-ent conditions [8]. Thus, the application of the TIM for thesystems with He bubbles requires a technique which allowsthe simultaneous calculation of the free energy of a crystaland a liquid restricted in a certain volume.

The potential UI for the systems with He bubbles wasconstructed in the following way. It is natural to use theapproximation of the Einstein harmonic crystal for theatoms which form the lattice while, for the He atoms in abubble, one needs a potential that would keep them inthe liquid state but would not allow the bubble to disperseduring the transition from k ¼ 1 to k ¼ 0. This is why wepropose to use for the He atoms in the bubble an approx-imation of ideal gas in a potential well with a rather steep

Fig. 1. Typical dependence of average values of hUII � UI ik on coupling parameter k along direct (from k ¼ 1 to k ¼ 0) and reverse (from k ¼ 0 to k ¼ 1)transitions obtained in numerical simulations for Pu–Ga samples with He atoms in the form of single He atom–vacancy complexes (} and �) and Hebubbles (� and +).

A.V. Karavaev et al. / Acta Materialia 79 (2014) 248–257 251

wall (to effectively keep the He atoms in the bubble). Thewell has a width corresponding to the equilibrium size ofthe bubble and is centred at the centre of the bubble. So,we calculated the free energy of systems with He bubblesusing the model potential UI in the form

U IðriÞ ¼X

crystal

ai

2ri � r

ð0Þi

� �2

þ U ð0ÞII crystal

( )

þX

bubble

U 0

ri � rcj jR

� �n( )

ð5Þ

where rc is the bubble centre, R is its equilibrium radius, n

is an arbitrary power and U 0 is the energy constant of thepotential. The values of n and U 0 should be chosen suchthat the potential well would be quite flat within the bubble(for r� rcj j < R) and would have a steep wall atr� rcj j > R, restricting the He atoms from escaping the

bubble. In our CMD simulations we used n ¼ 10 andU 0 ¼ 0:1 eV . For this potential one can analytically calcu-late the free energy as

F bubbleðk ¼ 0Þ ¼ kBT lnðCðNb þ 1ÞÞ

� 3NbkBT ln2R

nKTC

1

n

� �kBTU 0

� �1n

!ð6Þ

where N b is the number of He atoms in the bubble and fac-tor CðNb þ 1Þ ¼ N b! is due to the equivalence of the parti-cles in the bubble. Using Eqs. (4) and (6), one can calculatethe free energy of the reference state for the Pu–Ga systemwith the He bubble.

A further peculiarity of the free energy calculation forPu–Ga systems with an He bubble comes from the fact that

the exp 6 potential that models the interaction of He atomswith other components is singular at r ¼ 0; moreover, it isnegative as r approaches 0. This means that, in the transi-tion from the state k ¼ 1 to the state k ¼ 0, the He atomswhich overcome the repulsion barrier will attract oneanother. This will give a non-physical segregation of theHe atoms in the bubble which would not dissolve inthe reverse transition. To avoid this, we replaced theoriginal exp 6 potential by a potential which equals exp 6at r P r0 ¼ 0:15 nm and is the straight line UðrÞ ¼dU exp 6=dr

r¼r0ðr � r0Þ þ U exp 6ðr0Þ at r < r0. We thus we

obtained a smooth, continuous potential.In Fig. 1 one can see the dependence of the average

potential energy hUII � UIik on the parameter k in the suc-cessive quasi-equilibrium transition from k ¼ 1 to k ¼ 0.The rapid increase as k approaches 0 corresponds to thearbitrary approach of He atoms inside the potential wellwhen the repulsion of the exp 6 potential is “turned off”

by decreasing k. The reverse transition from k ¼ 0 tok ¼ 1 shows the resulting values of hU II � U Iik (shown by+ in Fig. 1) to agree reasonably well with the valuesobtained in the direct transition for all ks. Thus, wecan assert that the transition is reversible and at quasi-equilibrium, and can be used to calculate the free energyusing Eq. (2).

3. Solubility of He in Pu–Ga alloys

Here we present CMD calculations of free energies usingthe TIM performed to verify helium solubility in the fcc lat-tice of d-phase fcc Pu–Ga alloys under ambient conditions.For this, we calculated the free energies for two kinds of

252 A.V. Karavaev et al. / Acta Materialia 79 (2014) 248–257

systems which were identical in the composition and num-ber of atoms but varied in their microconfigurations. Wegenerated a set of fcc Pu–Ga samples (box size is� 10 nm – 37,044 atoms) and added 50 He atoms in twodifferent ways. First, we simply replaced 50 randomly cho-sen lattice atoms by He atoms. Second, we replaced 50 lat-tice atoms in the centre of the sample by He atoms to forma helium bubble, with one helium atom per vacant latticesite. Thus, the systems with distributed helium and heliumin the form of a bubble consist of identical numbers of par-ticles and can be interpreted as rearrangements of Pu, Gaand He atoms inside the box. Their internal and freeenergies can thus be compared directly in order to drawconclusions about the energetic and thermodynamic prefer-ences of those microconfigurations. For better statistics, we

Fig. 2. (a) Internal and (b) free energy differences of the systems with all He aaway from the bubbles He atoms as functions of helium concentration (numb

generated 10 samples of each kind, varying the distributionof Ga and He atoms inside the samples.

The results are as follows: the configuration with the Hebubble was found to be about 19 eV (0.38 eV per atom) ofinternal energy and about 24 eV (0.48 eV per atom) of freeenergy more favorable than the configuration with distrib-uted He. The conclusion is that, under ambient conditions(kBT ¼ 0:026 eV), He atoms do not dissolve in Pu–Gaalloys and do not form a disordered substitution solution.

The next series of calculations was performed to investi-gate the interaction of single He atoms with He bubbles ofvarious size and He concentration. Again, the CMD calcu-lations of free energies using the TIM were done for twokinds of systems identical in the composition and numberof atoms but varying by microconfiguration. A set of fcc

toms placed in the bubbles and systems with He bubbles and distributeder of He atoms per vacant site) and the size of initial bubbles.

A.V. Karavaev et al. / Acta Materialia 79 (2014) 248–257 253

Pu–Ga samples was generated by taking away atoms insidespheres of various radii, from 0.8 to 1.5 nm, in the centre ofthe samples and filling the voids with He atoms with vari-ous concentrations 1, 2, 2.5, 3, 4 of He atoms per vacantsite. Again, for better statistics, 10 samples of each typewere generated.

The long-term CMD simulation of the equilibration ofHe bubbles in the fcc lattice of Pu–Ga alloys is crucialsince, as was demonstrated in Ref. [8], if we stop the simu-lation at t � 100 ps we can come to an erroneous conclu-sion that He bubbles are unstable. Note, that effectivepressure in He bubbles is estimated to be rather high(� 3 GPa) [28], as manifested by a slight expansion of thebubbles.

To the thermalized systems with He bubbles of varioussizes and He concentrations we added 50 He atoms intwo different ways. First, the He atoms were added in theform of He atom–vacancy complexes located outside the5 nm sphere located at the centre of the He bubble. Second,the He atoms were added directly to the bubble by replac-ing 50 Pu or Ga atoms at the edge of the bubble. The resul-tant microconfigurations were again thermalized for� 1 ns.

For the systems with the He bubbles that were increasedin size and with the He bubbles with the 50 additional Heatoms distributed in the bulk of the samples, we calculatedthe Helmholtz free energy and the internal energy and com-pared them. the results of this comparison are presented inFig. 2. The first thing that one notices is that both the inter-nal energy difference and the free energy difference are neg-ative across the entire ranges of He concentrations andbubble sizes. That is, for all sizes of bubbles and all He con-centrations, both the internal and free energies of the sys-tems gain from the joining of 50 He atoms to any of thebubbles. Thus, the configurations with the He bubble ofincreased size are more preferable in the thermodynamicsense. This means that if a He atom were able to migratein the lattice under ambient conditions it would tend to joinone of the He bubbles rather than to remain a singlesubstitutional atom in the lattice. In other words, underambient conditions, He atoms do not dissolve in Pu–Gaalloys.

The second important feature in Fig. 2 is the dramaticdifference in the characters of the internal energy surfaceand of the free energy. The surface of the free energy differ-ence (the free energy profit from the joining of 50 atoms tothe initial He bubble) has a minimum at the point C � 2:7and R � 1:1 nm. This means that if 50 single He atomswere to join to He bubble of such size and He concentra-tion it would result in the greatest decrease in the Helm-holtz energy of the system. Thus the values of C ¼ 2:7and R ¼ 1:1 nm can be treated as estimates for the thermo-dynamic equilibrium parameters of He bubbles in thePu–3 at.% Ga fcc alloy under ambient conditions. Thecalculated equilibrium size is slightly higher than the exper-imentally observed average radius, which is of the order0.7–0.8 nm [14]. The obtained equilibrium He concentration

value, C ¼ 2:7, agrees well with the experimentallyobserved values of 2–3 He atoms per vacant site [14].

4. Equilibrium size of He bubbles

In this section we evaluated the thermodynamic equilib-rium size of He bubble with fixed He concentration startingfrom another concept. In the following, we fixed the Heconcentration at C ¼ 2:7, which agrees with the experimen-tally observed value of 2–3 He atoms per vacant site [14].Since the Helmholtz free energy is an additive function,meaning that the free energy of the entire system is equalto the sum of free energies of its parts, it is possible to carryout the calculations as follows. First, we calculated the freeenergy of the Pu–3 at.% Ga samples without He and thesamples containing one He bubble of 54, 108, 162, 216,270, 324, 378 and 432 atoms with a concentration ofC ¼ 2:7. For statistical purposes we used 20 samples iden-tical in composition and number of atoms, but varied bythe distribution of Ga atoms in the lattice. Second, let usimagine a system consisting of K subsystems with one Hebubble of 54 atoms in each. Let us label the free energyof such subsystem as F 54. Then the free energy of the wholesystem is F ¼ K � F 54. Now let us replace one of the subsys-tems with a subsystem without helium (the free energy ofsuch a subsystem is F 0). In order to keep the number ofHe atoms in the entire system constant, we increase thenumber of He atoms in one of the bubbles by 54 atoms(the free energy of such a subsystem is F 108). Then the freeenergy gain from the coalescence of two 54-atom He bub-bles is DF 108 ¼ F 108 þ F 0 � 2F 54. We next increase the sizeof the “main” bubble by 54 He atoms to 162 and, in accor-dance, replace one of the subsystems with a 54 atoms Hebubble by the non-defective crystal. Then the free energygain from the coalescence of three of the 54 atoms He bub-bles is DF 162 ¼ F 162 þ 2F 0 � 3F 54, and so on.

The results presented in Fig. 3a are the internal and freeenergy gains resulting from the coalescence of 54 atoms Hebubbles with fixed concentration C ¼ 2:7. The negative val-ues of the internal and free energy changes mean that thegrowth of the bubble through the mechanism of coales-cence of smaller (54 He atom) bubbles is energeticallyand thermodynamically profitable. The solid red anddashed blue lines in Fig. 3a are polynomial regressions ofcorresponding data.

If we consider the He bubble growing in our CMD sim-ulations as a system at constant pressure and temperaturethat can exchange particles (He atoms) with the surround-ings, then the condition for the thermodynamic equilibriumof such a system is the minimum of the chemical potentiall. The chemical potential l can be calculated as the deriv-ative of the free energy with respect to the number of par-ticles. Differentiating the polynomial approximations inFig. 3a with respect to N, one gets the dependence pre-sented in Fig. 3b. The dashed blue line in Fig. 3b representsthe chemical potential of the He atoms in the bubble withHe concentration C ¼ 2:7 which grows by the mechanism

Fig. 3. (a) Relative internal and free energies as functions of the number of particles N in the He bubble with He concentration C ¼ 2:7; (b) derivatives ofthe internal and free energies with respect to N. (For interpretation of the references to color in this figure legend, the reader is referred to the web versionof this article.)

254 A.V. Karavaev et al. / Acta Materialia 79 (2014) 248–257

of coalescence of 54 atoms He bubbles of the same concen-tration. The chemical potential dependence on N has aminimum at N � 205 and the minimum chemical potentialvalue l0 ¼ �0:315 eV. This means that there is the thermo-dynamic equilibrium size of He bubble is N ¼ 205 if onetakes into account only one mechanism of growth of Hebubbles (coalescence of smaller bubbles) with a fixed con-centration C ¼ 2:7 under ambient conditions.

Combining the dependence of the chemical potential lon the number of particles N (see Fig. 3b) with the depen-dence of the equilibrium He bubble radius R on the num-ber of particles N obtained in the simulations, one gets

lðRÞ and can construct the distribution function f ðRÞ ofHe bubbles with the fixed concentration C ¼ 2:7. Accord-ing to classical statistics, the distribution function isproportional to the Boltzmann factor. Thus, up to a nor-malisation constant,

f ðRÞ � exp � lðRÞ � l0

kBT

� �ð7Þ

The He bubble radius distribution function given byEq. (7) is presented in Fig. 4 by a dashed blue line in com-parison with experimentally observed one [14] shown bybars. The bar plot in Fig. 4 is a combined histogram

Fig. 4. The calculated helium bubble distribution function with a fixed He concentration C ¼ 2:7 (dashed blue line) in comparison with a previouslyexperimentally observed distribution function [14]. (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article.)

A.V. Karavaev et al. / Acta Materialia 79 (2014) 248–257 255

presented in Fig. 4 of Ref. [14] obtained for 26-, 35-, 36-and 42 year-old material and then divided by the maximalnumber of counts.

As can be seen in Fig. 4, the overall characters of the dis-tribution functions calculated and obtained by TEM [14] arequite similar. There are, however, two major discrepancies:namely, the location of the maximum (� 1:6 nm in CMDdependence and � 1:4 nm in the experiment) and the widthof the distributions – the experimental distribution has agreater width than the calculated one. We propose a varietyof plausible explanations for these discrepancies.

First, the results obtained in CMD simulations aredependent on the model of interatomic interaction used.The MEAM potential for Pu–Ga alloys was parameterizedfor certain conditions and, despite it having been shown toprovide good results when modelling fcc Pu–Ga alloysunder a variety of conditions, it does not necessarilydescribe the defective Pu–Ga alloys containing He withthe same accuracy. On the other hand, the qualitativeagreement between the calculated distribution functionand the experimental one suggests that the MEAM cap-tures the physics of the interaction of He atoms with thePu–Ga matrix.

The second factor that could plausibly affect the equilib-rium parameters of the He bubbles is the possible influenceof the alloy composition (local Ga concentration). If theequilibrium radius of the He bubbles depends on the Gaconcentration, then it would vary from region to regionwithin the same sample, since the Ga content is not uni-form in crystal grains [29]. This would cause broadeningof the distribution.

Another possible reason for the discrepancies is that inour CMD simulations we used the fixed He atom pervacant site ratio of C ¼ 2:7. In reality, there is a range ofHe concentrations in the He bubbles. According to the esti-mate given in Ref. [14], the average He concentration in theexperiments is about 2–3. The distribution of He concen-trations in He bubbles also results in the broadening ofthe overall distribution.

Finally, in our simulation we considered only one mech-anism of He bubble growth – the coalescence of smallerbubbles with the same fixed He concentration. However,at finite temperature this is not the only mechanism: thereare bubbles with different He concentrations and solitaryatom–vacancy complexes. If these can migrate in the latticeat ambient temperature, this would modify the He bubbleradius distribution, broadening it and possibly shifting itsmaximum.

5. Conclusions

Within the scope of a theoretical study using CMD intothe response of d-phase Pu–Ga alloys to self-irradiation, wehave investigated the behaviour of He atoms in the fcclattice of Pu–Ga alloy under ambient conditions. Becauseof the low mobility of He atoms in the Pu–Ga alloy lattice,their dynamics cannot be tracked in direct CMD simula-tions. Instead, we used the Helmholtz free energy functionto evaluate the relative stability of metastable states (micro-configurations) of Pu–Ga crystals doped with He atoms.

The Helmholtz free energy of the microconfigurationswas calculated through the thermodynamic integration

256 A.V. Karavaev et al. / Acta Materialia 79 (2014) 248–257

method (TIM) [22–24]. The TIM was used to calculate thefree energy of the fcc d-phase Pu–Ga alloy with He atomsdistributed as disordered substitutional atoms and with Hebubbles of different sizes and with various He concentra-tions (number of He atoms per vacant site). Based on theresulting free energy values, we drew inferences about therelative thermodynamic stability of different microconfigu-rations under ambient conditions, specifically:

(1) It was found that, for all of the parameters of Hebubbles used (radius of the bubbles and He atomconcentration), the helium bubbles did not dissolvein the matrix. From the thermodynamics viewpoint,the He atom–vacancy complexes produced as theresult of a-decay of 239Pu during diffusive motioninside the lattice should join together when they meetto form bubbles.

(2) It was demonstrated that the free energy surface is ata minimum for bubbles of radius R ¼ 1:1 nm andwith C ¼ 2:7 He atoms per vacant site, which is closeto the experimentally observed parameters of Hebubbles in naturally aged material: R ¼ 0:7 nm andC ¼ 2� 3 [14].

(3) If we fix the He concentration in the bubbles atC ¼ 2:7, then the estimated thermodynamic equilib-rium size of the He bubbles is � 205 He atoms andthe radius R � 0:8 nm, while the experimentallyobserved result for the average number of He atomsin the bubbles is 180 and the radius R ¼ 0:7 nm.

(4) The calculated He bubble radius distribution functionwas compared to that observed experimentally [14].Although the overall behaviour of the experimentaland calculated distributions are quite similar, thereare some differences. A number of plausible explana-tions for that differences are discussed in Section 4.

At first glance, it looks as if there is the contradictionbetween the estimates of the equilibrium bubble radiusobtained by the two sets of calculations in Section 3 –R ¼ 1:1 nm – and in Section 4 – R ¼ 0:8 nm. The differenceis caused by the fact that in the simulations we investigatedtwo different mechanisms of He bubble growth. The first isbubble coarsening by the diffusive motion of secluded Heatom–vacancy complexes, and the second is growth bythe coalescence of smaller bubbles. Our CMD calculationsand experimental TEM observations [14] demonstratedthat the existence of He atoms in the form of single Heatom–vacancy complexes is thermodynamically unfavor-able. If two diffusing He atoms meet each other or if anHe atom meets a bubble, they would join. Moreover, it isthermodynamically profitable to form bubbles with anHe atom concentration of 2–3 He atoms per vacant site.The second mechanism – migration and coalescence ofsmaller bubbles – has been observed, by in situ TEM at ele-vated temperatures, to be the dominant mechanism of Hebubble growth [16]. Of course, both of these mechanismsmay exist at finite temperatures, but the statistical weight

of the first mechanism and its contribution to the overallradius distribution function is much smaller than that ofthe MC mechanism. Thus, the second estimate of the equi-librium He bubble radius, R ¼ 0:8 nm, is more reasonable.

Note that all of the described CMD calculations weredone for the ambient conditions that correspond to thematerial storage conditions. It would be interesting torepeat the present calculations for elevated and reducedtemperatures to quantify the effect of temperature on theequilibrium parameters of He bubbles, as was done exper-imentally in Ref. [16].

Finally, we have to notice that, while the presentedapproach provides the answer to the question of which ofthe examined microstates is preferable from the thermody-namics viewpoint, it does not predict the thermodynami-cally stable state if that was not included in thecalculation set. Thus, the result of calculations dependson the “imagination” and “physics flair” of the researcher,meaning that one should generate the set of configurationsbased on preliminary understanding and available experi-mental information.

Acknowledgements

The work was performed through the collaborationauthorized under Contract No. B593480 between the Rus-sian Federal Nuclear Center – Zababakhin Institute ofTechnical Physics and the Lawrence Livermore NationalLaboratory.

References

[1] Wolfer WG. Los Alamos Sci 2000;26(2):274–85.[2] Hecker SS, Martz JC. In: Mallinson LG, editor. Aging studies and

lifetime extension of materials. New York: Kluwer Academic; 2001.p. 23–52.

[3] Clark DL, Hecker SS, Jarvinen GD, Neu MP. In: Morss LR,Edelstein N, Fuger J, Katz JJ, editors. The chemistry of the actinideand transactinide elements. New York: Springer; 2006. p. 979–89[chapter 7].

[4] Wolfer WG, Kubota A, Soderlind P, Landa AI, Oudot B, Sadigh B,et al. J Alloys Compd 2007;444–445:72–9.

[5] Howell RH, Sterne PA, Hartley J, Cawan TE. Appl Surf Sci1999;149:103–5.

[6] Wirth BD, Shwartz AJ, Fluss MJ, Cartula MJ, Wall MA, WolferWG. Mater Res Soc Bull 2001;26(9):679–83.

[7] Valone SM, Baskes MI, Martin RL. Phys Rev B 2006;73:214209.[8] Dremov VV, Sapozhnikov FA, Kutepov AL, Anisimov V, Korotin

M, Shorikov A, et al. Phys Rev B 2008;77:224306.[9] Rohr DL, Staudhammer KP. J Nucl Mater 1987;144:202–4.

[10] Zocco TG, Rohr DL. Materials research society symposium pro-ceedings, development in TEM examination of Pu and Pu alloys, vol.11. Pittsburgh, PA: MRS; 1988.

[11] Moore KT, Wall MA, Schwartz AJ. J Nucl Mater 2002;306:213.[12] Martz JC, Schwartz AJ. J Metals 2003;55(9):19–23.[13] Zocco TG, Schwartz AJ. J Metals 2003;55(9):24–7.[14] Schwartz AJ, Wall MA, Zocco TG, Wolfer WG. Philos Mag

2005;85:479–88.[15] Moore KT. Micron 2010;41:336–58.[16] Jeffries JR, Wall MA, Moore KT, Schwartz AJ. J Nucl Mater

2011;410:84–8.[17] Schroeder H, Fichtner PFP. J Nucl Mater 1991;179–181:1007.

A.V. Karavaev et al. / Acta Materialia 79 (2014) 248–257 257

[18] Baldan A. J Mater Sci 2002;37:2171.[19] Gruber EE. J Appl Phys 1967;38:243.[20] Baskes MI, Lawson AC, Valone SM. Phys Rev B 2005;72:014129.[21] Dremov VV, Karavaev AV, Samarin SI, Sapozhnikov FA, Zocher

MA, Preston DL. J Nucl Mater 2009;385:79–82.[22] Frenkel D, Laad AJC. J Chem Phys 1984;81:3188.[23] Frenkel D. Molecular dynamics simulations of statistical mechanics

systems, XCVII. Bologna: Soc. Italiana di Fisica; 1986.

[24] Frenkel D, Smit B. Understanding molecular simulations. Fromalgorithms to applications. San Diego, CA: Academic Press; 2002.

[25] Dremov VV, Karavaev AV, Sapozhnikov FA, Vorobyova MA,Preston DL, Zocher M. J Nucl Mater 2011;414:471–8.

[26] Nose S. J Chem Phys 1984;81:511–9.[27] Hoover WG. Phys Rev A 1985;31:1695–7.[28] Arsenlis A, Wolfer WG, Schwartz AJ. J Nucl Mater 2005;336:31–9.[29] Robbins JL. J Nucl Mater 2004;324:125–33.