Behaviour of hydrogen inzirconium intermetallics

Patrick A. Burr

Sunday 8th May, 2011

1Introduction

1.1 zirconium in industry

Zirconium (Zr) is seldom used in industry in its metallic form, and even then it is mostlyan alloying agent due to its good corrosion resistance. In other words, it is not a veryimportant metal for most applications. However, there is one niche market in which itcovers a fundamental role: the nuclear power generation industry. In fact, over 90% ofthe Zr that is produced every year is for that market1. Zr is the cladding material ofchoice for water cooled reactors (of all varieties: PWR, BWR, CANDU, VVER, RBMK, etc),and it is also used as a structural material in some designs. Water cooled reactors areby far the most widespread type of nuclear reactors in the world, with the PressurisedWater Reactor (PWR) being the most common subtype. They operate with sintered fuelpellets of Uranium dioxide — usually enriched up to ∼ 5.5% of the fissile isotope — withdimensions of 10–15 mm nominal diameter and 2–3 cm in height. The power produced bythe nuclear reaction is extracted in the form of heat by a flow of pressurised water aroundthe fuel. The role of the zirconium cladding (or sheath) is to mutually protect the fuel andwater from each other. Contamination of fission products into the water stream wouldinevitably increase the amount of radioactivity released into the biosphere, and the waterwould corrode and erode the fuel pellets reducing their integrity, which in turn wouldincrease the chance of dispersion of both fissile elements and radionuclides. In additionto encapsulating and protecting the fuel, the cladding also provides an easy means ofhandling the many thousands fuel pellets by stacking them into long (∼ 4m) tubes, which

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are then grouped into assemblies which are easy to deal with. In some reactors — such asthe CANDU and the RBMK designs — Zr alloys are adopted also for the pressure tubes(and the calandria in the case of the Canadian model). The mechanical properties requiredfor such applications are radically different from that of the cladding, therefore differentalloys are engineered for this purpose.

The preferential choice of Zr as a cladding material over other, possibly cheaper, corrosion-resistant structural materials such as steel, is a result of its exceptionally low cross-sectionfor neutron absorption (σa = 0.185b, Σa = 0.00794cm−1 for thermal neutrons)2, and, moreimportantly, the excellent cross-section to strength ratio. This implies that Zr acts as aneffective cladding material in that it retains its structural properties without poisoningthe neutron flux, which would otherwise have to be compensated by either higher fuelenrichment, or better moderators; both of which would aggravate the economics of powergeneration. When comparing the strength to cross-section ratio of the candidate materialsfor cladding, it transpires that best options are Zr, Be and Mg (followed by a large margin byFe, Al and Ni). Magnesium, used in Magnox reactors, was discarded for water reactors dueto the low melting temperature and — more importantly — high reactivity with oxygen.Beryllium was abandoned due to high handling complexity and increased costs thta arise asa consequence of its toxicity in the powder form (though in more recent times the researchinto Gen IV reactors has invigorated the attention to Be, together with more exotic claddingmaterials, in order to overcome some of the intrinsic limitations of Zr).

Since the birth of water cooled reactors, there has been a large volume of research intoimproving the mechanical and chemical properties of zirconium. There have been twomain paths for the engineering of zirconium alloys: one based on tin (Sn) as the mainalloying component, and the other based on niobium (Nb) additions. As we can see fromtable 1.1, all Zircaloy claddings (developed by Westinghouse) fall under the former class.Zircaloy 2 had much improved corrosion resistance compared to the initial alloy, howevernickel was later removed to form Zircaloy 4 as it is believed to contribute to the formationof hydrides in the zirconium matrix3,4. At the same time many Nb-based zirconium alloyshave been developed all around the world; for example: the Russian Zr-1%Nb, E110, andE635; the Canadian Zr-2.5%Nb; and the European Scanuk. Moreover, newer alloys suchas Zirlo and M5 have evolved taking advantage of Nb additions together with Sn or Fe,however their exact composition is not available in the public domain.

1.2 hydrogen and Hydrides in zirconium

One of the main problems with zirconium cladding is hydrogen embrittlement. H in solidsolution sits in the many interstitial sites of the α phase, described in Fig. ??, however it isnow widely accepted that H in solution does not have a great impact on the mechanical

2

Table 1.1: chemical compositions of commercial zirconium alloys. Values inparentheses express electro-negativity (in Pauling scale) of each of the elementstaken in the pure state (from the CRC Handobook5). All other units are in wt%unless otherwise stated.

Zr-Alloy (1.33) Sn (1.96) Fe (1.83) Cr (1.66) Ni (1.91) Nb (1.60) O (3.44)

Zircaloy 16 2.50 – – – – 0.14 max

Zircaloy 24,6–10 0.10±0.05 1.45±0.25 0.135±0.07 0.055±0.025 – 0.14 max

Zircaloy 3 0.25 0.25 – – – 0.14 max

Zircaloy 44,6–11 1.45±0.25 0.21±0.03 0.1±0.03 – – 0.14 max

Zr-1%Nb – – – – 0.6–1.0 0.09–0.13

Zr-2.5%Nb? – < 0.13 – – 2.4–2.8 0.09–0.13

Zirlo6,12 0.7–1.5 0.07–0.14 { 0.03 – 0.14 } 0.5–2.0 –

M513? – 0.014-0.037 – – 1.00 0.12-0.15

properties of cladding, but rather that:

i. the mechanism through which hydrogen embrittles zirconium is by formation ofhydride platelets, which cause Delayed Hydride Cracking (DHC),14

ii. the degree of embrittlement is strongly dependent on morphology and orientation ofthe hydrides.15,16

The main process for hydride formation is thermal cycling: at high temperatures(T > 550◦C) the solubility of H in Zr is very high (up to 5.9 at% in the α phase17 andup to ∼50 at% in the β), and at operating temperature of roughly 300◦C it is still between0.4–0.7 at% (∼60 ppm in mass). On the other hand, the room temperature solubility iscompletely negligible (see table 1.2 and Fig. 1.1 for further details).

Table 1.2: H solubility in α-Zr at T=300◦C

Paper at %

Kearns (1967)9,15 ∼ 0.7

Zuzek (1990)17 < 1.0

Wang (1995)21 < 1.0

Setoyama (2004)20 ∼ 0.47(

52 ± 2wt ppm)

Okamoto (2006)18 0.4 – 0.5

The direct implication is that during operation the hydrogen absorbed goes into solidsolution in the zirconium matrix, but if the cladding is allowed to cool down the hydrogen

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Figure 1.1: The Zr-H pseudo-phase diagram from Okamoto18, with acomparison of the data from Zuzek17, Dupin19 and Setoyama20 (left) and adetail of the α-Zr solubility of H (right).

content will be above the solubility limit and therefore will segregate in one of the δ, ǫ,γ (metastable) hydride forms. In normal operating conditions the thermal cycling is keptat minimal levels (though unavoidable during re-fuelling and shuffling of assemblies).However, this becomes of particular interest in Reactivity Initiated Accidents (RIA), andin dry storage environment, where a thermal spike is observed coupled with a potentialincrease in hoop stress. The three hydride phases are described in the Fig. 1.2.

The hydrides grow in long chains of small, hard and very sharp lenticular particles. Forthis reason the orientation of hydrides is crucial in the structural integrity of the claddingtubes. As extensively reviewed by Ells15, the hydrides grow preferentially along the basalplane of the HCP structure of α-Zr. This has been exploited by texture engineering the alloyduring manufacturing of tubular cladding in such a way that the normal to basal planesof the zirconium grains (the < 0002 > direction) aligns along the radial axis of the tube6,34.The end result is a series of hydrides aligned circumferentially (see fig. 1.3), and thereforehindering, to some extent, the propagation of radial cracks.

Growth direction of the hydrides will also be heavily influenced by stress gradients.There are currently two conflicting models of how Delayed Hydride Cracking occurs:the Diffusion First Model (DFM) also known as the Dutton-Pulls model23–26; and thePrecipitation First Model (PFM), strongly supported by Kim27–30. Even though at the heartof the debate is the diffusion mechanism of hydrogen, both models infer that it diffusesup the stress gradient from compressive to tensile stress regions. This could be drivenby chemical potential difference (DFM), which is influenced by the stress field, or by Hconcentration (PFM). This means that at a stress concentration point (which could be a

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(a) γ (b) δ (c) ǫ

Figure 1.2: The 3 hydride phases, courtesy of S. Lumley22. a) The metastable γphase has a fct Zr sublatice with c/a > 1 and only up to 0.5 of the interstitial sitesoccupied. b) The hypo-stoichiometric δ phase has a fcc sublatice and it is mostlyfound with a composition of ZrH1.66, therefore not all the tetragonal intersticesare occupied, up to the extreme case of a diamond-like structure. c) The ǫ phasehas an fct (c/a < 1) structure and occupancy of most of the tetragonal sites, up toall in the case of ZrH2.

pre-existing crack tip, a surface defect, an impurity or another hydride) the hydrogen willprecipitate to form a hydride aligned in the direction of the crack. This step is followed byrapid growth of the crack through the hard and brittle hydride. This leads to a situationwith the same conditions as the starting point, except effective crack size is increased bythe size of the hydride. DHC is a stable crack growth mechanism (constant crack growthrate), until the critical stress concentration factor KIC is achieved for which the crack canpropagate through the ductile metal matrix, causing failure of the cladding.

DHC is a complex mechanism, with many influencing factors, including minimum stressconcentration factor KIH, H solubility hysteresis and temperature dependence of crackpropagation. Owing to its complexity and the debate on the process involved, there isno comprehensive review to date. For further details on the subject refer to the followingdocuments:23–33

1.3 zirconium Intermetallics

In the previous section, the effects of zirconium hydrides were discussed; however, oneshould also consider the origin of hydrogen in the cladding. There are three distinct, butnonetheless related, sources of H:

i The acidic environment in which the reactor operates. The coolant of water reactors iseffectively a slightly acidic solution as a result of a number of chemical additions. The

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Figure 1.3: Typical Hydrides orientation in Zircalloy cladding. From LingaMurty6.

most relevant are: boric acid, to control the neutronics of operation in a more uniformfashion, as opposed to discrete control rods, which may distort the neutron flux in theupper region of the core more than the bottom (or the other way round in the caseof BWRs); lithium hydroxide, mostly to counterbalance the decrease in pH caused bythe boric acid, but limited by the increase of corrosion rates of cladding material inpresence of Li+; and H2 gas, added in the cold end of the primary circuit to scavengeexcess oxygen and help maintain the reducing environment.

ii Radiolysis of water molecules into aggressive free radicals. The products of radiolysismay react with other water molecules or impurities (including organic molecules) inthe water stream, or may be subjected to further slicing from radiation. Examples ofsuch species are: OH·, H·, HO2·, H3O+, O−2 , OH−, e−aq, H2O2, H2, O2

35.

iii Oxidisation of the cladding surface. When, for example, a surface atom of Zr is oxidisedinto ZrO2 by two molecules of H2O, four atoms of H are released. These may dissolveback into the water solution, recombine to form H2 gas molecules, or be adsorbed by theZr metal and diffuse into the bulk of the cladding. Equation 1.1 describe this behaviour(from Lelievre36).

(

1 +4y

x

)

Zr + 2H2O→ ZrO24y

xZrHx + 2

(

1 − y)

H↑2 (1.1)

where y tends to be very small (i.e. most of the hydrogen recombines).

As the oxide grows inward, mostly mediated by V••O

diffusion37, the hydrogen has anoxide layer of increasing thickness to diffuse through before reaching the zirconium metal.Whilst it is well known that the hydrogen diffusivity in metals is very high, diffusionbehaviour in Zirconia is less well understood despite having been the subject of manystudies38,39. It is generally accepted that the measured and calculated diffusivity of H inZrO2 is relatively low and does not predict the rate of H uptake in zirconium alloys. Onemodel — currently the most agreed upon — reviews the role of the alloying elements in

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H uptake. As shown in table 1.1, all the common alloying agents have higher electro-negativity compared to zirconium (1.33 on the Pauling scale). This implies that if someintermetallic phases form within the zirconium matrix (in the form of Second Phase Particlesor SPP) they will be less reactive to oxygen, and therefore oxidise at a lower rate, so that asthe oxide layer grows around them they retain a partially metallic form and act as a bridgeto H diffusion36,40. Fig. 1.4 shows a schematic of this process (from Hatano40). From thediagram one can also observe, according to this model, that larger SPPs exacerbate thiseffect and lead to the mechanical failure of the cladding. This was subsequently confirmedexperimentally11. Furthermore, oxidation temperature will influence the stability of theintermetallic precipitates. At lower temperatures the SPPs will retain their metallic formfor longer, allowing increased absorption of H.

Various intermetallic phases have been identified in zirconium alloys, particularly forZircaloys as it contains a greater variety of alloying agents (see table 3.2 in the Resultssection for a comprehensive list. However, it is worth pointing out that the presence andcomposition of most intermetallic phases has been — and still is — subject to a great deal ofdispute. In the literature there are reports of conflicting results regarding solid solubility ofFe and Cr alloying additions in α-Zr. Some papers report that all of the Fe and Cr content issegregated in SPPs41, whilst others have shown that a certain degree of solubility in the Zrmatrix is retained42,43, and even that a supersaturated solution is obtained when quenchedfrom the β phase43. Furthermore Fe-rich intermetallic precipitates have been observedin some cases42,44, but not others41,45. Similar controversy surrounds Sn solubility. It iswell known that Sn has very large solid solubility in α-Zr — it is indeed an α stabiliser— but there have been numerous reports of Sn-based SPPs with a very wide range ofcompositions42,46,47. Note that Yang45 has shown that redeposition of Sn onto precipitatesand grain boundaries during TEM sample preparation is a possible cause of misinterpretedSn segregation.

The issue of intermetallic precipitates in zirconium is obviously a complex one, and it ismade even more intricate by the different morphologies that each SPP may assume, andby their subsequent relation and influential growth on one-another41. For this reasonit is important to extend our knowledge of zirconium intermetallics, particularly theirinteraction with H. Indeed hydrogen uptake might not be the only factor that is affectedby SPPs: hydride nucleation might be influenced by the presence and morphology of theintermetallics, as observed by Vitikainen48.

The necessity for further work in this area is evident. Clarification on the role of thedifferent alloying elements with regards to H uptake could lead to the development ofcladding alloys with greater reliability, and potentially also longer life expectancy, whichwould accommodate fuel with higher burn-up. These are the foundations for the currentproject.

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Figure 1.4: Proposed mechanism for H diffusion through the Zircaloy oxidelayer. (A) represents a zircaloy specimen that has been quenched from theβ phase and annealed to produce uniformly distributed small intermetallicparticles, whilst (B) is slow cooled from the β phase to produce fewer largeparticles of similar composition

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2Methodology

In the past half-century the steady increase in computational power coupled with Moore’spredictions49 for the future of computation, which still hold true nowadays, haveencouraged the development of many algorithms and codes that investigate the behaviourof materials at an atomic scale. Atomistic modelling is a field in ever greater growth, thatis now also being largely adopted by industry. Its growing popularity is not only due tothe increase in the accessibility of computational power, but also because of its inherentcost effectiveness (compared to large scale experiments), time saving nature (acceleratedmodelling of processes), as well as the excellent rate of success.

Atomistic modelling is a field that has been in continuous change since it’s birth, alwaysadopting newer and more accurate techniques and rejecting those that have becomeobsolete. These transitions have always been driven by the availability of greatercomputational power, and the present situation is no different. We are currently observingthe shift from classical potential models to quantum mechanical ones. Because we are but atthe beginning of this process, and each model has its limitations, the two methods need tocoexist and supplement each-other in order to achieve a better understanding of the greaterpicture. The current work has been carried out using quantum mechanical simulations,as it is more suitable for the study of zirconium and zirconium intermetallic systems aspointed out in the comprehensive reviews by Domain50–52.

The main difference between Quantum Mechanical (QM) simulations and empirical pairpotential simulations is that the latter represents ions as a whole entity (commonly as a hard

9

sphere shell and a spring), whilst the former simulates the behaviour of the electron cloudthat form the atom and defines its interactions with its surroundings. Quantum mechanicsis universally considered as the model that provides the most complete description ofnature, and it is the closest we have yet attained in answering the ”ultimate question ofLife the Universe and Everything”. But in order to do so it inevitably carries with it a largeload of detail, and it is this complexity of information that prohibits the use of QM in therepresentation of large systems. In fact the state of the art in QM simulation is limited tocomplex molecules, or crystals of the order of 100 atoms (see convergence section for detailson the size limits of the current work).

2.1 Quantum Mechanics

At the root of quantum mechanics is the non-relativistic many body Schrodinger equation:

HΨ (RI, ri) = EΨ (RI, ri) (2.1)

WhereΨ is the wavefunction of the system as a function of ion positions (RI) and electronpositions (ri), E is the total energy of the system and H is the Hamiltonian operator, whichcontains the kinetic (T) and potential (V) energy contributions of all the ions and electronsin the system:

H = TN + Te + VNe + VNN + Vee (2.2)

where TN and Te represent the kinetic energy components of the electrons and nucleirespectively, and the remaining terms take into account the potential that arises due tothe ion-electron

(

VNe

)

, ion-ion(

VNN

)

and electron-electron(

Vee

)

interactions. These can bemathematically expressed as follows (in the same order as in equation 2.2):

HT = −∑

I

~2

2MI∇2

RI−

∑

i

~2

2mi∇2

ri(2.3a)

HV = −∑

I,i

ZIe2

|RI − ri|+

∑

I,J

ZIZJe2

2∣

∣

∣RI − RJ

∣

∣

∣

+∑

i, j

e2

2∣

∣

∣ri − rj

∣

∣

∣

(2.3b)

where RI, MI and ri and mi are the positions and masses of the Ith ion and ith electronrespectively, ~ is the Planks constant divided by 2π (also known as the Dirac constant), ZI

and ZJ are the atomic numbers of the Ith and Jth ion and e is one electric charge unit.

By minimising the LHS of equation 2.1 with respect to R and r it is possible to find theground state of a system. Note, however, how the number of components within theHamiltonian operator may escalate quite rapidly for a relatively large system (10 for H2,127 for methane, 279 for ethane), therefore making the computation of such systems quitearduous.

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2.2 Born-Oppenheimer Approximation

Due to the complexity of the full Hamiltonian operator, it is necessary to adopt a seriesof approximations in order to compute the ground state of large systems. The first ofsuch operations is the Born-Oppenheimer approximation: because the difference in massbetween electron and ions is very large (me/MI ≪ 1), but the momenta are of the same order ofmagnitude (pe/pI ∼ 1), the ions are effectively immobile with respect to the electrons and theelectronic response to ionic motion is effectively instantaneous. Therefore the wavefunctioncan be separated into two independent component, an electronic one and an ionic one, tobe solved in distinct steps.

Ψ (r,R) = Ψe (r; R)χ (R) (2.4)

whereΨe is a function of r only, with the R dependence parameterized.

In the first step the ions are considered stationary (or ”clamped”) which allows to the reducethe Hamiltonian operator by ignoring the kinetic energy of ions TN (being smaller than Te

by a factor of ∼ MI/me) and the ion-ion interaction potential VNN (which is now constant and,therefore, only provides an external potential that is felt by all electrons). This yields what isknown as the ”clamped nuclei” approximation:

He (r; R) = Te (r) + VNe (r; R) + Vee (r)

= −∑

i

~2

2mi∇2

ri−

∑

I,i

ZIe2

|RI − ri|+

∑

i, j

e2

2∣

∣

∣ri − r j

∣

∣

∣

(2.5)

and the respective Schrodinger equation to be solved becomes:

He (r; R)Ψe (r; R) = EeΨe (r; R) (2.6)

In the second stage, the nuclear kinetic energy term TN, and the ion-ion interactions VNN

are reintroduced together with the electronic contribution to the total energy Ee to solve thenuclear part of the Schrodinger equation:

HN (R)χ (R) =Etotχ (R){

TN + VNN (R) + Ee

}

χ (R) =Etotχ (R) (2.7)

2.3 Hartree-Fock Method

From the section above it is clear that the troublesome stage in the minimisation process isto solve for the electronic contribution to the system’s total energy. The starting point forsuch calculation is to consider the electrons as non-interacting fermions. This is obviously

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a very bold assumption, and the consequences of which are discussed below. For now let’sconsider the case of a system with non interacting electrons, the electronic wavefunctioncan be re-written as a ”Hartree product” (after Douglas Hartree? ):

ΨHP ({ri}) =n

∏

i=1

φ (ri) (2.8)

where φ are the normalised wavefunctions of the individual electrons. The above equationonly considers the spacial variation of the electrons, neglecting the 4th degree of freedom:the spin coordinate ω. It is easy to add the spin dependance by defining x = {r, ω} (usuallyjust as x = rα ∨ x = rβ) so that equation 2.8 becomes:

ΨHP ({xi}) =n

∏

i=1

χ (xi) (2.9)

Once the Hartree wavefunction is produced, it is possible to calculate the expectation valueof the electronic contribution to the energy of the system, using the usual probabilisticinterpretation of quantum mechanics. Exploiting the variational theorem, the expectationvalue of the Hamiltonian, generated with any trial wavefunction, must be greater or equalthen the ground state of the system. Mathematically:

EHP =⟨

Ψ∗HP

He

ΨHP

⟩

≥ Ee (2.10)

In other words we are seeking for the minimum value of EHP. This means that we can startwith a trial set of wavefunctions φi, which will output a corresponding set of φ j and theniterate the process with the output as the new input until the two quantities are the selfconsistent.

Even though the Hartree product is a very convenient form of the wavefunction, it hasone shortcoming : it does not generally satisfy the antisymmetry principle, for which the”wavefunction [must be] antisymmetric with respect to interchange of the coordinates(including spin) of a pair of electrons”53, or mathematically:

Ψe (x1, x2, . . . , xn−1, xn) = −Ψe (xn, xn−1, . . . , x2, x1) (2.11)

This is a very important principle in quantum mechanics (in fact the Pauli exclusionprinciple is a corollary of it) and it is crucial to overcome this limitation. This can be achievedif the anti-symmentry condition is imposed by expressing the Hartree wavefunction as a

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Slater determinants (after John C. Slater54):

ΨHF ({ri}) =1√

N!

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

χ1 (x1) χ1 (x2) · · · χ1 (xN)

χ2 (x1) χ2 (x2) · · · χ2 (xN)...

.... . .

...

χN (x1) χN (x2) · · · χN (xN)

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(2.12)

which may be compressed using bra-ket (or Dirac) notation into

χiχ j · · ·χk

⟩

, since it isalways possible to construct a determinant from the vector components.

The new equation for the wavefunction is known as Hartree-Fock (HF), and the samevariational argument as before can be used to solve for the energy of the system, whichyields a similar solution but with an extra term: the ’exchange’ term. Essentially this meansthat if the positions of two electrons are swapped, the sign of the determinant is changedleaving

∣

∣

∣Ψ2∣

∣

∣ unaltered.

As mentioned above, the underlying assumption upon which the HF method was foundis that electrons are non-interacting entities in the system. This is very convenientmathematically — because it transforms the many-electron problem into a one electronin a mean filed problem — but even with a good field of choice it represents only verypoorly a many electron system. The exchange functional has introduced a relation betweenparallel-spin electrons, but a more generic relationship between electrons was needed todescribe the Coulomb ’correlation’ effect.

2.4 Kohn-Sham and Density Functional Theory (DFT)

In 1964 Hohenberg and Kohn55 suggested that the main problem of the HF approach is thechoice of fundamental variable: they believed that the many-electron wavefunction wasfar too complex to be suitably approximated with only a few terms. The new starting pointproposed was the electron density, and for the validation of a new Hamiltonian equationin terms of this variable they proved

1. by reductio ad absurdum that the external potential, generated by the stationary ions(BO approximation), is a unique functional of electron density Vext = Vext [n (r)]; andbecause the electron wavefunction depends on the external potential, it too must bea functional of electron densityΨ = Ψ [n (r)];

2. that the groundstate energy, and all the following properties, can be foundvariationally as a function of density.

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From the above principles, Kohn and Sham, formulated a new Hamiltonian equation thatresembles Schrodinger equation but with ”fictitious” non-interacting particles that behavelike (i.e. have the same density as) real interacting electrons:

E [n (r)] =⟨

Ψ∗

HKS

Ψ

⟩

(2.13)

Where the Hamiltonian operator is composed of the electronic kinetic energy term Te, theclassic (Hartree) electron interaction potential Ve — both of which can be expressed directlyas functions of electron density — and the external potential Ve f f [n (r)]. By expandingequation 2.13, the following is equations are obtained:

E [n (r)] =⟨

Ψ∗

Te + Ve

Ψ

⟩

+

∫

V (r) n (r) dr

= F [n (r)] +∫

V (r) n (r) dr (2.14)

From this point onwards, DFT proceeds in a similar fashion to the Hartree-Fock approach togenerate the self-consistent method for the minimisation of the energy’s expectation value.The single-particle equation becomes:

(

− ~2

2me∇2

r + Ve f f

(

r, n (r))

)

φi (r) = ǫiφi (r) (2.15)

where Ve f f is the effective potential that acts upon the fictitious electron-like particles.

Ve f f = Ve + VXC + Vext (2.16)

and VXC is the exchange-correlation function. This term is the only one that cannot bederived exactly ab-initio so further approximations must be made.

2.5 Approximations

The most common methods to estimate the XC function are the Linear Density Approxi-mation (LDA) and the Generalised Gradient Approximation (GGA). The former simplycalculates the exchange-correlation potential from a uniform electron gas distribution.Such a simple approximation works remarkably well in many systems, especially weaklycorrelated ones, however, previous work from Domain50 shows that for the zirconiumsystem the GGA works best. This takes into account the first derivative of the electrondensity variation in the system as a weighting factor for the exchange correlation effect.

Another very powerful approximation that allows DFT calculations of large systems — andin particular systems with heavy atoms — is the pseudo-potential method, first developed

14

by Phillips56 and later adapted by Heine and Cohen57. When considering the chemicaland physical properties of materials, the interest is usually limited to the valence electronsof the atoms, since the core electrons are so tightly bound by the nucleus that are effectivelyinvisible to the other atoms. Even though the contribution of the core electrons is important— the atom would behave differently if those weren’t there — it is only the effect that theyhave upon the valence electrons that is relevant.

The pseudo-potential method allows to compute such approximation by curve-fitting asofter potential to the real potential of the electron to match only the outer part of thewave function (which, as mentioned in the previous section, is a unique functional of theexternal potential, which in turn depends on the ionic potential). See fig. 2.1 for a graphicalrepresentation.

Figure 2.1: The pseudopotential (red) is made to fit the real potential (blue)in such a way that the respective wavefunction is matched from infinity to acritical radius rc. Modified by A. Patel58, originally from Payne59.

Lastly, As we are interested in the bulk properties of materials, we employ periodicboundary conditions. This approach consists in creating a unit-cell of the crystal of interest,and then repeating this cell infinitely in each direction. This procedure is possible thanksto Bloch’s theorem, which allows to rewrite any wavefunction ψi (r) as the product of acell-periodic components fi (r) and a wavelike component eik·r:

ψi (r) = fi (r) · eik·r (2.17)

As a consequence the problem of solving infinite wavefunctions is reduced to the problemof solving a finite number of wavefunctions at an infinite number of points in k-space.However, because the electronic wavefunction at two k-points very close together are

15

almost identical, it is therefore possible to represent the wavefunction over a region ink-space by that at a single k-point59. Therefore only a finite number of k-points is requiredto calculate the ground state properties of an infinitely large bulk of periodic crystal.

The great advantage of this approach is also it’s main limitation: it will periodically repeateach and every point within the box boundaries. This implies that it approximates a perfectcrystal to near perfection, but for point defect calculations it will effectively produce adefect concentration. To minimise the effect of defect-defect interaction one should usethe largest box possible, but this necessarily means having more atoms, therefore greatercomputational power is required — see fig. 2.2.

0 20 40 60 80 100 120 140 160

0

2

4

6

8

10

12

14

Number of atoms in SC

Defect interaction distance (Å)

Figure 2.2: The blue points are simulations ofα-Zr with an interstitial hydrogen.The size of the cell of Zr was increased from a single unit-cell (2 atoms) to a5×5×3 supercell. The red line is not a best-fit line, but the predicted behaviourof y = ax3, where a is a geometrical factor. As the size of the supercell increases,and therefore the linear defect-defect distance increases, the number of atomsto compute increase according to a cube law. The reason why the pointsare slightly off the predicted trend is mostly due to the anisotropy of the α-zirconium unit-cell

16

2.6 Defect Energy

In the previous sections it was explained how DFT allows calculation of the ground stateenergy of a system. This section will briefly discuss how the ground state energy is used todetermine the energy of point defects.

Consider the following incorporation reaction for a box containing 64 atoms of zirconium(a 4 × 4 × 2 supercell):

12H2

(

g)

+ Zr64 (s) =⇒ Zr64H (s) (2.18)

The standard enthalpy of reaction is given by:

∆H◦r = ∆H◦f (Zr64H) − ∆H◦f (Zr64) − 12∆H◦f (H) (2.19)

where H◦f

is the standard enthalpy of formation. By definition the enthalpy of solutionEsol is the enthalpy change associated with the dissolution of a substance [H] in a solvent[Zr64] at constant pressure, resulting in infinite dilution5. This means that if the boxof α-Zr were infinitely large the above enthalpy of reaction would equate the enthalpyof solution. And, as shown in the convergence section, it is reasonable to state that thehydrogen interstitials are dilute non-interacting defects. In practical terms, this means thatthe enthalpy of solution of a hydrogen interstitial may be calculated ab-initio by computingthe ground state energy for three systems, as shown in fig. 2.3.

H H+H+ H

Figure 2.3: graphical representation of the incorporation reaction of a hydrogenatom into a zirconium (left) or zirconium intermetallic (right) box as a series of3 distinct simulations in each case.

The current work reports the solution energy as a comparison tool; what is of crucialinterest is the difference in the enthalpy of solution between the pure zirconium and eachof the zirconium intermetallics. When subtracting one term to the other, the hydrogen’senthalpy of formation term cancels out; taking for example a box of 96 atoms of Zr2Fe asthe intermetallic phase:

∆EHsol = EH

sol(IM) − EH

sol(Zr) (2.20a)

= ∆H◦f (Zr64Fe32H) − ∆H◦f (Zr64Fe32) − 12∆H◦f (H)

−(

∆H◦f (Zr64H) − ∆H◦f (Zr64) − 12∆H◦f (H)

) (2.20b)

= ∆H◦f (Zr64Fe32H) − ∆H◦f (Zr64H) − ∆H◦f (Zr64Fe32) + ∆H◦f (Zr64) (2.20c)

17

or, schematically:

+ +⇇H H

Zr ZrIM IM

Figure 2.4: Graphical representation of the series of simulations required tocalculate the difference in solution energy ∆Esol between two different solvents(Zr and IM).

2.7 Nudged Elastic Band (NEB)

In the latter part of the project, the interest of the investigation moved towards the kineticsof hydrogen diffusion in zirconium. A useful parameter to quantify the diffusivity ofhydrogen in zirconium is the activation energy for diffusion, which corresponds to thelargest energy barrier between two stable minima. To extract such parameter, the NudgedElastic Band (NEB) method was adopted. This technique consists of a number of parallelreplicas (images) of the same system, with the defect in a different position between twoknown minima. Each image is minimised independently, but during the ionic step the onlymotion allowed is that along the directions orthonormal to the hyper-tangent, which iscalculated as the normal vector between two neighbouring images60. As a result, when thesimulation is completed, the chain of images form what is known as the Minimum EnergyPath (MEP), see fig. 2.5.

Figure 2.5: An energy map around two local minima (red points). Theblack dashed line represents the initial guess for the position of the images(equidistant linear interpolation), and the blue solid line is the MEP. Modifiedfrom QuantumWise©61

18

2.8 One last word about simulation vs experimental

Even though computer modelling of materials may be considered more predictive thenexperimental works, it must be stressed that the results of modelling can only be validatedby direct (experimental) observations. For this reason a constant collaboration betweenexperimentalists and modellers should be sought.

2.9 general references for the methodology

...

19

3Results

3.1 Convalidation of results

The first part of the project involved validating the results of the project by proving theconvergence of the variable that have a significant impact on the accuracy. These wereidentified (in order of importance) as

1. the nature of the potential used,

2. the supercell size,

3. the cutoff potential,

4. the k-space sampling density,

5. and the σ-smearing of the partial occupancies.

In the following chapter a brief outline of the convergence of each parameter is presented,followed by a summary of the parameters used in the calculations

3.1.1 Choice of functional

Previous work from C. Domain50 has showed that GGA potentials are more suitable thanLDA ones to investigate bulk properties of the Zirconium system. Furthermore, because

20

Fe Nb Sn Y (a,c) Zr (a,c)

Experimental62–65 (Å) 2.865 3.293 6.4892 3.6474 5.7306 3.23 5.145

US-PPLDA -11.50 -6.80 -3.32 -13.24 -25.26 8.50 -11.70

GGA 2.80 -0.10 14.38 -2.04 -11.26 -1.80 -2.10

PAW

LDA -13.10 -3.00 -1.12 -10.94 -20.06 -6.80 -6.80

PW91 -3.80 2.90 16.08 0.06 -6.56 0.40 1.50

PBE -3.30 3.30 16.58 1.06 -5.36 0.30 2.80

Table 3.1: Lattice constant calculation compared to experimental values. Thevalues for the 3rd row onwards are the percentage deviations from the referencevalue (1st row). Modified from, and all calculations carried out by Lumley22.Parameters used in calculations unknown.

we are interested in the bulk properties, it is intuitive to use Projector Augmented Wavepotentials rather than localised ones. This reduced the choice to only two options, bothPAW-GGA: Perdew-Wang 1991 (PW91) and the Perdew-Burke-Ernzerhof 2002 (PBE).

However, as mentioned in the 1, the current project is closely related to the work carriedout by S. Lumley in his PhD, and it was therefore important to have results that werecomparable.

In his work, Lumley produced a comparison of predicted versus experimental latticeparameters for different functionals, see Table 3.1. From this comparison it was evincedthat the PW91 was the most appropriate potential.

3.1.2 Planewave cutoff energy

When dealing with a problem of infinite points, in practical terms one must resort to alarge but finite number of points to be able to solve it. This is the case with the basis set forelectrons in DFT (or any other QM computation method).

Even at a single k-point the solution to the Kohn-Sham equaiton is given by an infinitesummation. Fortunately not all terms have the same weight, in fact each term representsan energy level with a kinetic energy associated with it, and the higher the level the largerthe kinetic energy contribution. It is clear that the terms with the lowest kinetic energy havethe greatest influence in the ground state of the system, it is therefore possible to define anarbitrary cut-off value for which any higher order solution will be neglected.

A suitable cut-off potential is one that minimises computational power — associated withthe number of bands to compute — whilst also providing a sufficiently large number ofempty levels to avoid artefacts in the results.

21

The graph in fig. 3.1 shows the convergence for the planewave cut-off potential, with thesystem’s total energy. It is clear that if a precision of 10−3 eV/atom is sought than a potential

300 400 500 600 700

-6

-5

-4

-3

-2

-1

0

α-Zr

β-Zr

γ-ZrCr2

Zr3Fe

Zr2Fe

ZrFe2

Zr3Sn

Zr5Sn3

Cut-off potential (eV)

log(E

0/a

tom

)

Figure 3.1: The abscissae axis represents the cut-off value (in eV) and theordinate axis shows the difference in energy per atom between each point andthe previous one, logarithmically. All the calculations were carried using unitcells with a very large (and constant) BZ sampling.

in the region of 450 eV should be used. However, because other parameters (e.g. thesupercell size) would be limited to a precision below 10−2 if such cut-off was adopted, itwas necessary to trade-off one order of magnitude of uncertainty here. It was thereforechosen to use a potential cut-off of 350 eV, which produced a gain in computational powerof roughly 50%. Further analyses showed that this cut-off value limited the precision onlyto ∼ 5 × 10−3 eV/atom, therefore still below the uncertainty obtained from defect-defectinteractions (see section 3.1.4). The self consistent loops were minimised to 10−4eV, whichis a relatively inaccurate value, and this explains the large scatter at lower values. Followingthis, further results were calculated using more accurate energy stopping criteria for theelectronic relaxation. It is worth noticing that the speed of convergence is very element-specific: comparing the curves for Zr3Fe, Zr2Fe and ZrFe2 one can observe a simple shiftupwards (slower conversion) with increasing Fe content. Similarly the Zr-Sn intermetallicshave similar curves, and so do the pure Zr phases. It is perhaps not very surprising thatFe and Cr are the most ”problematic” species, given that they have the largest number ofelectrons in the outer shell.

22

3.1.3 Brillouin zone sampling

As mentioned in the methodology section (), it is necessary to approximate the (smooth)variation of the wavefunctions in k-space as a series of discrete solutions around distinctpoints. The approximation works best for infinitely close points, and degrades as thedistance between sampling points increases. However, denser sampling requires greatercomputational power, making it prohibitive to use very large number of k-points.

The graph in fig. 3.2 illustrates the increase in precision as the distance between k-pointsdecreases. It is clear that a precision of 10−3eV/atom requires a sampling density of roughly

0 20 40 60-5

-4

-3

-2

-1

α-Zrβ-Zrα-ZrCr2γ-ZrCr2Zr3FeZr2FeZrFe2Zr3SnZr5Sn3

linnear K-point density (Å)

log(

ΔE

0/at

om)

Figure 3.2: The ordinate-axis represents the difference in groundstate energyper atom between each point an the one that precedes it (i.e. with fewer k-points), logarithmically. Owing to the large scatter of the plot, it was decidednot to represent the individual convergence curve of each series, but rather thecollective behaviour of all phases. The shade was drown as a guide to the eyeonly. Data obtained with unit cells of each phase, fixed cut-off potential of 450eV, and altering the Monkhorst-Pack grid by increasing the least dense vectorfirst. Γ-centering used for hexagonal, rhombohedral and triclinic cells.

60 k-points Å, equivalent to k-point distance of 0.016 Å−1. However, a precision inferior byone order of magnitude will be easily achieved around 25 k-points Å(0.04 Å−1 in reciprocalspacing). As a consequence of this, and considering the grids that would have been optimal

23

for the relevant supercells, a k-point spacing of 0.03 Å−1 was opted for (linear densityequivalent is ∼ 33 Å). The decision was done in light of the fact some Monkhorst-Packgrids are more efficient than other. For instance, for cubic, tetragonal and orthorhombicsystems, non-Γ-centred even number grids are more efficient than odd number or Γ-centredones. This is because when the Brillouin Zone is reduced to its irreducible constituent, manyof the k-points (up to 1⁄2) will become redundant by symmetry, thus reducing computationalpower needed for the same degree of accuracy. However, if the k-points are not positionedsymmetrically to the symmetry axis (as is the case for Γ-centred grids in a square system),then the total number of k-points in the IBZ is only a little less than that of the full BZ. Asa direct consequence, decreasing the number of k-points (and therefore accuracy) does notnecessarily mean a decrease in computational power: a 3 × 3 × 3 grid will contain morepoints in the IBZ than a 4 × 4 × 4 grid for a square system.

To summarise the grid type used:

for(

α, β, γ)

= 90◦ even number MP grids were adopted up to 8 points per reciprocal vector,then odd number ones were employed (starting from 11).

for(

α, β, γ)

< 60◦∨(

α, β, γ)

> 120◦ Γ-centred grids were used, were possible evennumbers below 8 and odd numbers above 11.

3.1.4 Defect concentration limit

As mentioned in the Methodology section 3.1.3, it is very computationally expensive tomodel larger supercells, however because we intend to model isolated point defect energy,it is vital to minimise the defect-defect interaction. The largest supercells that possibleto compute contained 150 atoms, equivalent t a H concentration of 73.6 wt. ppm, whichis just above the solid solubility of H in α-Zr at operating temperature. It is clear thenthat supercell size should be kept as large as possible. In support of this, the variation ingroundstate energy with supercell size is of the order of 0.02eV for Zirconium (see table 3.4and 3.5), which is much larger than the precision limit imposed by the other parametersdescribed in this chapter.

3.1.5 σ-smearing of partial occupancies

Partial occupancies are commonly used in DFT calculation since they reduce the numberof k-points required for the calculation of an accurate calculation band-structure energy60.The smearing method used was the first order Methfessel-Paxton, which produces similarresults to simple Gaussian smoothing but with much lower entropy terms. Ideally,the fictitious temperature smearing parameter σ should be as large as possible whilstmaintaining the entropy value (which can be considered as the magnitude of uncertainty)

24

negligible. From the above analyses it is clear that a source of error can be defined as”negligible” if it is less than 10−3 eV/atom. Fig. 3.3 shows the variation entropy componentof the total energy as a function of σ. It is clear from the above graph that if a precision

0 0.05 0.1 0.15

-6

-5

-4

-3

-2

α-Zr

β-Zr

γ-ZrCr2

Zr3Fe

Zr2Fe

ZrFe2

Zr3Sn

Zr5Sn3

σ (K)

log(Δ

S)

Figure 3.3: The ordinate-axis represents the difference in entropy per atombetween each point an the one that precedes it, logarithmically. The shade wasdrown as a guide to the eye only.

of 10−3 is desired, than σ should be less than 0.04 K. However it was later discovered thatthe increase in computational power between 0.01 and 0.04 K was marginal, and thereforedecided to retain the original value used by S. Lumley in his work.

25

3.2 Summary of intermetallic phases

Various intermetallic phases have been identified from the literature as relevant ones forthe current work, and they are summarised in Table 3.2. When examining the intermetallicphases it is useful to refer to the relevant phase diagram, which are attached in theAppendix ??.

Table 3.2: Summary of the main intermetallic phases found in Zirconium alloys

Alloy Phase Composition Occurrence and Notes

Zr (Fe,Cr)2HCP: ...(

β-ZrCr2)

ZrCr1.1Fe0.941,46

Zr (Fe0.75Cr0.25)266

Zr (Fe0.6Cr0.4)245

Zr (Fe0.4Cr0.6)245

ZrxCr2Fe546,67

Zr0.36Cr.43Fe.2168

Zr0.32Cr.24Fe.4468

Very common in Zy-2 and Zy-4(often with different composi-tions)

FCC: ...(α-ZrCr2)46,48,69

Zr (Fe0.1Cr0.9)244 Found only within grains, whilst

in the same experiments the HCPstructure was found betweengrains. Especially abbundant inScanuk alloys.

Zr2 (Fe,Ni) BCT: ...(Zr2Ni)

Zr2Ni0.4Fe0.641,46,48

Zr2 (Fe0.5Ni0.5)45

Zr2 (Fe0.8Ni0.2)45

Very common in Zy-2

Zr4 (Fe,Cr) FCC: ...(Zr4Fe)

Zr4Fe0.7Cr0.345

Zr4Fe44Metastable form found to dec-orate the boundaries of thelamellae formed in β-quenchedZircaloy

Zr-Sn ZrSn, Zr5Sn3, Zr3Sn,Zr5Sn4, Zr4Sn, BCT Sn42,44,46,47,67

Often with large Fe content. Maybe an artefact of TEM samplepreparation45.

Zr (FexM1−x)2 HCP (x < 0.8),FCC (x > 0.9)

M=V,Cr,Mn70

Zr-Nb β-Nb Nb is a β stabiliser for Zr.

Of the all intermetallic phases, the only ones that are universally accepted to be present inthe relevant alloys are Zr(Fe,Cr)2 and Zr2(Fe,Ni), and their composition has been identifiedto a confident degree of certainty. Because Nickel has been discarded as an alloying agentin Zircalloy-4 onwards (see table 1.1), the current study has focused on the intermetallicphases not containing Ni.

26

3.3 Pure Zirconium

Zirconium metal has two stable phases, α and β, represented in fig. 4.5 and described inTable 3.3.

(a) α-Zr (b) β-Zr

Figure 3.4: Graphical representation of the unit cells of the twozirconium phases with relevant intersticial sites. (a) The intersticials andtheir crystallographic positions of α-Zr are (Clockwise from the bottomleft): hexahedral

(

13 ,

23 , 0

)

, tetrahedral(

13 ,

23 ,

18

)

, non-basal trigonal(

59 ,

19 ,

16

)

,

octahedral(

23 ,

13 ,

14

)

and basal trigonal(

23 ,

13 , 0

)

. Notice that whilst the trigonalsite repeats every {0001} plane, the hexahedral repeats only every {0002} plane,since a Zr atoms occupy the

(

13 ,

23 ,

12

)

site. (b) The intersticial sites for β-

Zr are (from top to bottom): octahedral(

12 , 0,

12

)

, tetrahedral(

12 , 0,

14

)

and

trigonal(

16 ,

12 ,

16

)

Table 3.3 also contains the calculated lattice parameters for both phases compared toexperimental work. It is clear that there is a decisive agreement between predicted andexperiment values, which confidence for subsequent results.

The enthalpy of formation (Esol) and the defect volume (∆V) were calculated for each of thedefect sites described above; they are summarised in Table ??.

27

Table 3.3: Calculated lattice parameters compared to experimental values. Theuncertainty in the predicted lattice parameter derives from the variation of theresults with supercell size.

Phase Lattice type Space group Calculated Reference

(Int. table n.) a0 c0 a0 c0

α-Zr HCP P63/mmc (194) 3.233 ± 0.003 5.162 ± 0.011 3.2362 5.14562

3.241471 5.169671

3.23172 5.14272

β-Zr BBC Im3m (299) 3.570 ± 0.001 3.545373

3.48874

3.616275

Table 3.4: α-Zr: Energy of solution (Esol) and Defect volumes (∆V) for H in allthe stable interstitial sites, for different supercell sizes.

IntersticialSite

SuperCellsize

Atoms in SC([H] wt. ppm)

r(H-H) in Å Esol ∆V

Tet

2 × 2 × 1 8 (1380) 5.23 -0.427 3.36

2 × 2 × 2 16 (690) 6.48 -0.449 3.61

3 × 3 × 2 32 (345) 9.71 -0.463 3.20

4 × 4 × 2 64 (173) 10.34 -0.442 3.44

4 × 4 × 3 96 (115) 12.93 -0.474 2.13

5 × 5 × 3 150 (73.6) 15.47 -0.467 3.07

Oct

2 × 2 × 1 8 (1380) 5.18 -0.436 0.93

2 × 2 × 2 16 (690) 6.47 -0.465 1.13

3 × 3 × 2 32 (340) 9.71 -0.438 0.82

4 × 4 × 2 64 (173) 10.33 -0.404 1.05

4 × 4 × 3 96 (115) 12.92 -0.427 0.26

5 × 5 × 3 150 (73.6) 15.47 -0.432 0.76

Hex4 × 4 × 2 64 (173) 10.31 -0.341 3.77

5 × 5 × 3 150 (73.6) 15.46 -0.356 3.40

Tri4 × 4 × 2 64 (173) 10.34 0.018 4.49

5 × 5 × 3 150 (73.6) 15.48 0.074 5.34

nbT 4 × 4 × 2 64 (173) 10.34 -0.040 3.79

28

Table 3.5: β-Zr: Energy of solution (Esol) and Defect volumes (∆V) for H in allthe stable interstitial sites, for different supercell sizes.

IntersticialSite

SuperCellsize

Atoms in SC([H] wt. ppm)

r(H-H) in Å Esol ∆V

Tet

2 × 2 × 2 16 (690) 7.10 -0.644 7.3

3 × 3 × 3 54 (204) 10.59 -0.865 5.35

4 × 4 × 4 128 (86.3) 14.26 -0.650 3.16

Oct

2 × 2 × 2 16 (690) 6.85 -0.770 3.04

3 × 3 × 3 54 (204) 10.58 -0.699 9.25

4 × 4 × 4 128 (86.3) 14.25 -0.501 4.13

Tri2 × 2 × 2 (204)

3 × 3 × 3 (86.3)

3

29

3.4 Zr-Fe intermetallics

(a) Zr3Fe (b) Zr2Fe

(c) ZrFe2

Figure 3.5: Graphical representation of the unit cells of the three Zr-Feintermetallic phases.

30

In the Zr-Fe phase diagram (see ??), there are three stable intermetallic phases: Zr3Fe, Zr2Feand ZrFe2. Graphical representation of these are presented in fig 3.6. Furthermore, there isalso a metastable form, with a stoichiometry of Zr4Fe, that has been documented in variousZy-4 studies44,45, mostly with some degree of Cr solution in the Fe site. However insufficientcrystallographic data was found regarding the Zr4Fe structure in order to perform reliableDFT simulations.

Values for the three phases that were investigated, with comparison to experimental data,are summarised in Table 3.6.

Table 3.6: Summary of the Fe-Zr intermetallic phases with computed latticeparameters and comparison with experimental data.

Phase Space group Calculated Experimental Reference

(Int. table n.) a0 b0 c0 a0 b0 c0

Zr3Fe Cmcm (63) 3.258 10.901 8.996 3.33 10.95 8.82 [Ardisson91]76

3.324 10.974 8.821 [Yartys99]77

3.32 11.00 8.82 [Stein02]72

Zr2Fe I4/mcm (140) 6.268 5.720 6.41 5.56 [Ardisson91]76

6.379 5.599 [Yartys99]77

6.382 5.602 [Stein02]72

ZrFe2 Fd3m (227) 7.011 7.016 [Stein02]72

7.1 [Gorria04]78

7.064 [Zotov08]79

Defect energy calculations were performed using the following supercell parameters:

Zr3Fe: SC size= 3 × 1 × 2; n. atoms: 69; [H] = 127.4wt. ppm; r (H-H) = 9.78Å.

Zr2Fe: SC size= 2 × 2 × 2; n. atoms: 96; [H] = 132.2wt. ppm; r (H-H) = 11.44Å.

ZrFe2: SC size= 2 × 2 × 2; n. atoms: 48; [H] = 310.4wt. ppm; r (H-H) = 9.91Å.

ZrFe2: SC size= 3 × 3 × 3; n.atoms:162; [H] = 91.98wt. ppm; r (H-H) = 14.89Å.

where SC stands for SuperCell, and r (H-H) is the smallest defect interaction distancebetween cell replicas. The resulting solution energies for hydrogen in the various stableinterstitial sites are presented in Tables 3.7. A negative ∆Esol means that the site is morestable than the lowest energy interstice in α-Zr. Many more sites were investigated than theones presented in table 3.7, however they were found to decay into one of the more stablesites.

31

Table 3.7: List of the stable interstitial sites for Zr3Fe, Zr2Fe and ZrFe2. ∆Esol isthe difference in solution energy of H with respect to a tetrahedral interstitial inα-Zr of similar supercell size.

Phase Wyckoff n. Type Coordinates Esol ∆Esol

Zr3Fe 4b Oct 0, 12 , 0 −0.714 −0.239

8 f Oct 0, 0.035, 58 −0.690 −0.215

4c Tet 0, 0.035, 34 −0.521 −0.046

8 f Tet 0, 0.324, 0.452 −0.521 0.017

8 f Tet 0, 0.279, 0.558 −0.415 0.060

4c Oct 0, 0.614, 14 −0.376 0.098

8 f Tet 0, 0.117, 0.665 −0.295 0.180

16h Tet 0.223, 0.302, 0.388 −0.190 0.284

4c Hex 0, 0.242, 14 −0.128 0.347

Zr2Fe 16l Tet 78 ,

38 ,

38 −0.492 −0.018

32m Tet 0.59, 0.27, 0.18 −0.460 0.014

16 j Tri 14 , 0, 1

4 −0.449 0.025

4b Tet 0, 12 ,

14 −0.399 0.075

16k Tet 0.345, 0.452, 0 −0.258 0.217

4c Lin 0, 0, 12 −0.225 0.249

ZrFe2 96g Tet 0.422, 0.422, 0.220 0.110 0.563

32e Tet 0.385, 0.385, 0.385 0.128 0.580

48 f Tri 0.206, 12 ,

12 0.202 0.654

8b Tet 12 ,

12 ,

12 0.236 0.689

96g Tri 0.112, 0.112, 0.319 0.237 0.691

32e Tri 0.792, 0.792, 0.792 0.378 0.831

16c Lin 38 ,

18 ,

38 1.327 1.780

32

3.5 Zr-Cr intermetallics

(a) α-ZCr2

(b) β-ZCr2(c) γ-ZCr2

Figure 3.6: Graphical representation of the unit cells of the three Zr-Feintermetallic phases.

The Zr-Cr binary system (see ??) has only one stoichiometry for intermetallic compounds(1:2), however, within that stoichimetry there are three phases. They are all layered Laves

33

phase with different stacking sequences (c for cubic and h for hexaagonal): α-ZrCr2 hasthe C15 Laves structure composed of three layers with a ccc stacking sequence; β-ZrCr2

has the C36 Laves structure composed of four layers with a chch stacking sequence; andγ-ZrCr2 has the C14 Laves structure composed of two layers with a hh stacking sequence80.It must be stressed that even though the β-ZrCr2 phase hardly appears in the literaturearound zirconium alloys, XRD studies have shown that the C36 structure is often presentin crystals of C14 as a concentration of staking faults in the layered sequence80.

Table 3.8: Summary of the Cr-Zr intermetallic phases with computed latticeparameters and comparison with experimental data.∗ Values with an asterisk are taken from commercial alloys such as Zircaloy-2,Zircaloy-4 and Scanuk.

Phase Space group Calculated Experimental Reference

(Laves structure) (Int. table n.) a0 c0 a0 c0

α-ZrCr2 (C15) Fd3m (227) 7.118 7.204 [Soubeyroux95]81

7.23 [Gonzalez05]82

7.4∗ [Vitikainen78]48

7.19∗ [Krasevec81]69

7.21∗ [Yang87]44

β-ZrCr2 (C36) P63mmc (194) 5.06 16.29 5.11 16.56 ??

γ-ZrCr2 (C14) P63mmc (194) 5.09 8.04 5.113 8.309 [Soubeyroux95]81

5.103 8.268 [Mestink Filho]80

5.079∗ 8.279∗ [Vander Sande]67

5.01∗ 8.22∗ [Krasevec81]69

5.09∗ 8.27∗ [Yang87]44

Defect energy calculations were performed using the following supercell parameters:

α-ZrCr2 SC size = 2 × 2 × 2; n. atoms = 48; [H] = 322.6wt. ppm; r (H-H) = 10.07Å.

α-ZrCr2 SC size = 3 × 3 × 3; n. atoms = 162; [H] = 95.60wt. ppm; r (H-H) = Å.

β-ZrCr2 SC size = 2 × 2 × 1; n. atoms = 96; [H] = 161.3wt. ppm; r (H-H) = Å.

γ-ZrCr2 SC size = 2 × 2 × 2; n. atoms = 96; [H] = 161.3wt. ppm; r (H-H) = 10.16Å.

where SC stands for SuperCell, and r (H-H) is the smallest defect interaction distancebetween cell replicas. The resulting solution energies for hydrogen in the various stableinterstitial sites are presented in Tables 3.9. As for the Zr-Fe case, more sites wereinvestigated than the ones presented in this report but they were found to be unstable,and, hence, decay into one of the sites described in Table 3.9.

34

Table 3.9: List of the stable interstitial sites for α-ZrCr2, β-ZrCr2 and γ-ZrCr2.∆Esol is the difference in solution energy of H with respect to a tetrahedralinterstitial in α-Zr of similar supercell size.

Phase Wyckoff n. Type Coordinates Esol ∆Esol

α-ZrCr2 96g Tet 0.422, 0.422, 0.220 −0.241 0.211

32e Tet 0.385, 0.385, 0.385 −0.204 0.248

48 f Tri 0.206, 12 ,

12 −0.060 0.392

8b Tet 12 ,

12 ,

12 −0.058 0.394

16c Lin 38 ,

18 ,

38 1.212 1.664

γ-ZrCr2 6h Tet 0.426, 0.573, 14 −0.277 0.191

4 f Tet 13 ,

23 ,

716 −0.234 0.234

12k Tet 0.533, 0.467, 0.409 −0.227 0.241

24l Tet 0.384, 0.351, 0.454 −0.185 0.283

6h Tet 0.245, 0.490, 14 −0.172 0.296

12k Tet 0.161, 0.322, 0.128 −0.157 0.310

4e Tet 0, 0, 18 0.128 0.595

2c Lin 13 ,

23 ,

14 1.262 1.730

35

3.6 Zr-Sn intermetallics

(a) Zr3Sn (b)Zr2Fe

Figure 3.7: Graphical representation of the unit cells of the two stoichiometricZr-Sn intermetallic phases that were analysed in the current work.

The zirconium-tin binary system comprises of various phases of dubious nature. Muchcontroversy surrounds the composition and — more importantly — the structure of theZr-Sn intermetallic phases. The two most predominant intermetallics in the binary systempresent partial occupancy or self-interstitial occupancy of lattice sites: the Zr4Sn phasederives from the Zr3Sn structure with 1⁄4 of the Sn sites being occupied by a Zr atom; theZr5Sn3 phase tends to have self-interstitial occupancy of Sn up to a stoichiometry of Zr5Sn4.As a limitation of DFT, we are unable to compute such structures, however in the currentwork the structures with no excess or defect Sn are investigated to provide a platform forfurther analyses with more suitable tools (such as quasi-random techniques). These aresummarised in table 3.10. Another structure that has been reported in the literature isthe orthorhombic ZrSn83, however insufficient crystallographic data is available to modelits structure reliably with DFT. Furthermore the tin-rich side of the binary system (andtherefore the ZrSn2 intermetallic) has been disregarded since such it appears not to berelevant for zirconium alloys.

Defect energy calculations were performed using the following supercell parameters:

Zr3Sn SC size = 2 × 2 × 2; n. atoms = 64; [H] = 160.5wt. ppm; r (H-H) = 11.27Å.

36

Table 3.10: Summary of the Sn-Zr intermetallic phases with computed latticeparameters and comparison with experimental data.∗,† Values accompanied by the asterisk and dagger symbols were taken from theZr4Sn and Zr5Sn4 stoichiometry respectively.

Phase Space group Calculated Experimental Reference

(Int. table n.) a0 c0 a0 c0

ZrSn3 Pm3n (223) 5.633 5.63 [Roßteutscher65]84

5.625∗ [Kwon90]85

5.631∗ [Luo70]86

Zr3Sn5 P63mcm (193) 8.538 5.799 8.456 5.779 [Kwon90]85

8.462 5.797 [Gunnar60]83

8.7656† 5.937† [Kwon90]85

Zr5Sn3 SC size = 2 × 2 × 2; n. atoms = 128; [H] = 77.56wt. ppm; r (H-H) = 11.58Å.

where SC stands for SuperCell, and r (H-H) is the defect interaction distance between cellreplicas. The resulting solution energies for hydrogen in the various stable interstitial sitesare presented in Tables 3.11. Once again, the sites that were found to be unstable are notpresented in the report.

37

Table 3.11: List of the stable interstitial sites for Zr3Sn and Zr5Sn3. ∆Esol is thedifference in solution energy of H with respect to a tetrahedral interstitial inα-Zr of similar supercell size.‡ This particular site is actually shifted off the trigonal plane towards theoctahedral site 2b, this is apparently more stable than the full octahedral site(multiple simulations with different starting guesses were carried out to confirmthis).

Phase Wyckoff n. Type Coordinates Esol ∆Esol

Zr3Sn 6d Tet 14 ,

12 , 0 −0.662 −0.220

8e Hex 14 ,

14 ,

14 −0.376 0.065

48l Tri 0.273, 0.348, 0.424 −0.100 0.342

6b Lin 0, 12 ,

12 0.735 1.177

Zr5Sn3 2a Tri 0, 0, 14 −0.541 −0.074

12k Tri‡ 0.095, 0.553, 0 −0.437 0.030

2b Oct 0, 0, 0 −0.417 0.050

12i Hex 0.1955, 0.391, 0 −0.018 0.449

12 j Hex 0.210, 0.514, 14 0.051 0.518

24l Hex 0.461, 0.143, 0.402 0.464 0.931

12k Tri 0.430, 0, 0.430 0.909 1.376

12i Tri 0.454, 0.908, 0 1.035 1.502

6 f Lin 12 , 0, 0 1.076 1.543

4c Lin 13 ,

23 ,

14 1.574 2.04

38

3.7 Zr-V intermetallics

(a) ZrV2

Figure 3.8: Graphical representation of the unit cells of the only Zr-Vintermetallic compound.

The vanadium-zirconium system comprises of only one intermetallic compound: ZrV2.This has the same crystal structure as ZrFe2 and α-ZrCr2 described above. The calculatedand experimental values of the intermetallic are presented in Table 3.12.

Table 3.12: Details of the only Zr-V intermetallic phase with computed latticaparameter and comparison with experimental values.

Phase Space group Calculated a0 Experimental a0 Reference

ZrV2 Fd3m (227) 7.324 7.443 [Lototsky05]71

7.44 [Daumer88]87

7.441 [Lawson78]88

Defect energy calculations were performed using the following supercell parameters:

ZrV2 SC size = 2 × 2 × 2; n. atoms = 48; [H] = 326.1wt. ppm; r (H-H) = 10.16Å.

ZrV2 SC size = 3 × 3 × 3; n. atoms = 162; [H] = 96.65wt. ppm; r (H-H) = 15.27Å.

where SC stands for SuperCell, and r (H-H) is the defect interaction distance between cell

39

replicas. The resulting solution energies for hydrogen in the various stable interstitial sitesare presented in Tables 3.13.

Table 3.13: List of the stable interstitial sites for ZrV2. ∆Esol is the difference insolution energy of H with respect to a tetrahedral interstitial in α-Zr of similarsupercell size.

Phase Wyckoff n. Type Coordinates Esol ∆Esol

α-ZrCr2 96g Tet 0.422, 0.422, 0.220 −0.717 −0.249

32e Tet 0.385, 0.385, 0.385 −0.644 −0.176

96h Tri 18 , 0.066, 0.184 −0.602 −0.135

48 f Tri 0.206, 12 ,

12 −0.575 −0.107

8b Tet 12 ,

12 ,

12 −0.360 0.107

32e Tri 0.792, 0.792, 0.792 −0.307 0.161

48 f Lin 0.375, 12 ,

12 0.064 0.532

16c Lin 38 ,

18 ,

38 0.197 0.666

40

3.8 Nudged Elastic Band for α-Zr

In aid of clarity, the interstitial sites, described in the 3.3 section, will identified as follows:

TET for the 4 f tetrahedral site at 13 ,

23 ,

18 ;

HEX for the ?? trigonal site at 13 ,

23 , 0;

OCT for the 2d octahedral site at 23 ,

13 ,

14 ;

TRI for the ?? trigonal site at 23 ,

13 , 0;

nbT for the k? non-basal trigonal site at 59 ,

19 ,

16 .

The Nudged Elastic Band (NEB) method was used to calculate the migration enthalpy forhydrogen in α-Zr. the three resulting curves are shown in fig. 3.9.

-1 -0.5 0 0.5 1

0

0.1

0.2

0.3

0.4

0

2

4

6

8

10

Distance (Å)

ΔH

(eV

)Δ

H (kcal/m

ol)

HEX

TETTETOCT

OCT

OCT

TRI

nbT

TET

Figure 3.9: NEB curves for the three possible migration mechanisms. Thecurves have been centred at the saddle point and enthalpy change values (theordinate axis) were taken with respect to the lowest energy point on the curve.Dashed lines indicate that the data was collected for only half the bell curveexploiting the symmetry property of the migration process.

41

It is worth mentioning that the validity of the symmetry operation was proved by comparingthe above curves to ones obtained from modelling the full migration process, which yieldedsimilar results, but with poorer resolution (same number of points over twice the spread).

The three migration processes occur in different directions and are shown in fig. 3.9 insuch a way that the x-axis represents the distance from the saddle point, regardless ofthe direction. As a consequence of the alignment, the same TET site is found at differentabscissae points. Furthermore, because each curve is normalised with respect to the moststable site within the curve, the graph is not representative of the depth of the wells nor of thedifference in stability between TET and OCT sites. However, this normalisation allows tocompare energy barrier height between different migration processes. The energy barriersare presented in table 3.14.

Table 3.14: Energy barriers for diffusion in the three cases described above,with comparison to experimental data.

Migration process ∆H (eV) ∆H (cal/mol)

TET-HEX-TET 0.0997 2298.5

TET-nbT-OCT 0.4001 9228.6

OCT-TRI-OCT 0.4227 9748.3

Sawatzky1960 0.3634 ± 0.0173 8380 ± 40089

Kearns1972 0.4618 ± 0.0117 1065090

42

4Discussion

The literature contains various results (partially conflicting) regarding the interstitialoccupancy of H in zirconium: for the α phase, there is a general consensus that H ismore stable in the tetrahedral site rather than the octahedral one17,19,20,50? . For the β phasehowever, Zuzek17 reports octahedral occupancy, while Setoyama20, Dupin19 and Domain50

assert a preference for the tetrahedral site. Even though the current work is in agreementfor the α-Zr, it is a very small preference (of the order of 0.04eV) and it is realistic to believethat H might occupy both sites. In fact, as discussed below, it is a requirement for diffusionto occur in α-Zr. As for the β phase, the difference in solution energy becomes greater(∼ 0.1eV), but the scatter of points (with different cell sizes) is too large too conclude whichsite is preferred: for smaller supercell sizes the tetrahedral site is more stable, but for thesupercell containing 128 atoms (which was problematic to converge), the trend reverses.Note that the latter simulation was the least reliable.

The results for pure hydrogen showed that in the α phase, the is no net preference betweenthe two sites.

Due to the large number of different intermetallic systems — each with a different set ofstoichiometries, christallographic structures, relative radius sizes and electro-negativities— the comparison across the various systems is very arduous. Table 4.1 summarises thesolution energy of the most stable site only from each of the phases investigated. The firstsurprising feature of the results is that hydrogen exhibits a lesser preference for the ZrFe2

and all ZrCr2 phases compared to α-Zr. As discussed in the introduction section 1.3, these

43

Table 4.1: The difference in solution energy is taken with respect to the α-ZrTET site (with a similar defect concentration)

Phase Esol ∆Esol ∆V

α-Zr -0.467 – 3.07

β-Zr -0.644 -0.202 2.71

Zr3Fe -0.744 -0.239 0.61

Zr2Fe -0.492 -0.018 2.48

ZrFe2 0.110 0.563 3.09

α-ZrCr2 -0.241 0.211 2.66

β-ZrCr2 -0.336 0.132 3.96

γ-ZrCr2 -0.277 0.191 3.79

ZrSn3 -0.662 -0.220 -0.18

Zr5Sn3 -0.541 -0.074 1.87

ZrV2 -0.717 -0.249 2.94

phases are thought to be the main SPP responsible for H uptake in Zircaloy-4. Because oftheir higher electro-negativity, they retain a partially metallic form during the oxidizationof the zirconium alloy, providing a bridge for H diffusion through the zirconia layer. Thecurrent results provide the reason for which hydrogen would migrate out of the SPP intothe bulk Zr. Furthermore, it implies that it will most likely be retained in the Zr matrix,rather than segregate into the Zr(Fe,Cr)2 particles. These results are particularly relevantfor Zircaloy-4 where such SPP is very common.

4.1 Electro-negativity

Various attempts have been made to correlate the stability of H interstitials to the physicaland chemical properties of the various phases. The first of which was the electro-negativityof the alloying metal compared to zirconium. This seems like an intuitive comparisontool, which should reveal whether the chemical affinity is the predominant process in thesolubility of hydrogen in the metals. However there seem to be little correlation betweensolution enthalpy and electro-negativity of the alloying metals, as one can observe from fig.4.1. The graph only shows the lowest energy sites for each phase, since it is reasonable toassume that the H interstitial will chiefly occupy the most stable configurations.

For those binary systems that have multiple intermetallic stoichiometries, namely theZr-Fe and the Zr-Sn system, it is possible to identify an uprising trend with increase inelectro-negativity. However, this apparent trend is heavily autocorrelated, since the RHSof the graph not only represents higher electro-negativity, but also a larger content of the

44

1.3 1.4 1.5 1.6

-0.2

0

0.2

0.4

0.6

α-Zr

ZrV2

α-ZrCr2

β-ZrCr2

γ-ZrCr2

Zr3Fe

Zr2Fe

ZrFe2

ZrSn3

Zr5Sn3

H_sol=0

"Average" electronegativity of IM (Pauling)

ΔE

so

l (eV

)

Figure 4.1: ∆Esol is plotted with respect to α-Zr TET as the reference value.The dotted line represents the value for which ∆Esol = 0, i.e. when there is noenthalpy gain in having the hydrogen as an adsorbed intersticial rather than inits molecular (gas) form. The abscissae values are a fictitious form of electro-negativity obtained by smearing the individual electro-negitivities of each atominto a single, uniform, weighted value across the entire phase.

non-zirconium element (since all alloying additions have higher electro-negativity thanZr). Therefore, towards the RHS of the graph the properties of the intermetallic — allthe properties, not only electro-negativity — are bound to be more influenced by theadditive element Furthermore, such trend would require the reference value (α-Zr TET) tobe between the negative and the positive values, rather than to the far left of the graph.

4.2 Volumetric parameters

A second means of comparing defect energy across the different systems is by relating itto the available volume in the crystal structure. Such quantity can be expressed either asthe packing fraction or as the atomic density. The latter is a much simplified quantity thatdoes not take into account the various site geometries, but it has the great advantage ofbeing easy to calculate and reliable. While the packing fraction is heavily dependent onthe radii of the atoms, which is relatively straightforward to measure for pure metals, butrather arduous and arbitrary for intermetallic phases. Even though this was attempted,

45

and graph with a clear correlation was produced (as presented in the symposium1), it isnot reproduced here because sensitivity analysis showed that the values of the graph wereexcessively susceptible to the measured metallic radii (for which the estimated error wastoo large). It is believed that such graph could lead to incorrect interpretation of the data,and was of little scientific relevance.

However, the solution energy vs. atomic density graph is presented in fig. 4.2. Toanalyse this graph it is useful to follow the lowest point for each series. The data show a

30 40 50 60 70

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

α-Zr

β-Zr

β-ZrCr2

α-ZrCr2

γ-ZrCr2

Zr3Fe

Zr2Fe

ZrFe2

Zr3Sn

Zr5Sn3

ZrV2

atoms/Vol (nm-3)

Eso

l (e

V)

Figure 4.2: Solution energy ∆Esol vs. atomic density. Only the most stable siteswere considered in the composition of this graph. The zero-line represents thevalue for which there is no enthalpy gain (or loss) for the hydrogen atom to goin solution.

clear depression with the minimum around 45-55 atoms/nm−3. Note also that the well isasymmetric, with the higher atomic density being a much steeper descend compared to thelow number density side. The presence of a minimum rather than a steady increase, seemsto suggest that H does not prefer the largest site possible, but rather that it prefers to feel(and exert) a localised strain around it. This is perhaps not too surprising considering thathydrogen diffuses up the stress concentration gradient25? ? .

To have a better understanding of the relationship between space and stability of aninterstitial site, it is useful to compare the defect energy to the defect volume ∆V(defined as

1Year-4 research project presentation, nuclear group, 17-03-2011, RSM, Imperial College, London

46

the volume difference of a defective cell compared to the perfect crystal). It is often foundin the literature that volume accommodation comes with an energy associated with it, sothat defects with greater ∆V, which impose a higher strain to neighbouring atoms, havehigher defect energy. The present study has two important limitations when analysingdefect volumes: firstly it consider interstitial and not substitutional defects, therefore thedefect volume is unlikely to be negative; secondly it investigates only one defect specie indifferent sites, rather than different species in the same site. This implies that, as well asdifferent defect neighbours, also different defect geometries will be compared against eachother. Inevitably, this will lead to large scatter in the results, as we can see from fig. 4.3.Even though it is hard to understand the behaviour of each phase individually, it is worthnoticing that most data series tend to higher energy values as at larger defect volumes —this is especially obvious for Zr5Sn3 (dark orange) and α-Zr (light purple). Furthermore,there is a noticeable division around ∼ 4Å3. Before said value most points are negativeand mostly distributed within 0 and −0.7eV, whilst beyond it they tend to be positive andlarge. In order to elucidate the general trend of the data, past the scatter, the graph has beensubdivided along the ordinate into six energy bands, and for each band the distribution ofpoints along the abscissae has been calculated. The six normalised distributions have beenplotted in areal histogram form in fig. 4.4. In such a graph it becomes relatively straight-forward that the most stable sites tend to have∆V values closer to 0 (slightly biased towardsthe positive volume changes). Whilst defect that have a higher energy (lighter-colouredbands) are predominantly those that required a larger volume accommodation.

The phases in which H seems to form defects with the smallest volumetric change areZr3Fe and Zr3Sn. However, these results provide no information about how easily doeseach phase accommodate the change in volume required for the stable interstices, in factthis parameter is most probably one of the main sources of the scatter.

47

-2 0 2 4 6 8

-0.5

0

0.5

1

1.5α-Zr

β-Zr

Zr3Fe

Zr2Fe

ZrFe2

α-ZrCr2

β-ZrCr2

γ-ZrCr2

Zr3Sn

Zr5Sn3

ZrV2

ΔV (Å3)

Eso

l (eV

)

Figure 4.3: Energy of solution Esol vs. defect volume∆V. The individual trendsfor each of the series could not be represented in a clean manner due to the largescatter of the data.

Figure 4.4: The distribution of the lowest energy points is represented by thedarkest shade of blue, and the least favourable sites are represented by thepalest shade. The fraction of data points (y-axis) has been normalised by thetotal number of points in each band.

48

4.3 Defect’s neighbouring atoms

Regardless of the comparative method used (electro-negativity, atomic density or defectvolumes), it is quite clear that the interstitials with the lowest energy are mostly observed inthe phases with the largest Zr content (Zr3X in particular, with X=Sn,Fe). By analysing theseit became clear that the most stable sites in such alloys are the ones in which hydrogen’sneighbours are solely Zr atoms. Furthermore, in the low Zr content alloys, such as ZrM2

(with M=Fe,Cr,V), where by stoichiometry the majority of interstitial sites have more M-type neighbours compared to Zr neighbours, the most stable ones are characterised by aration of 1:1 between Zr and M atoms.

With this premise in mind, the degree of electron sharing near the defect site wasinvestigated. This was achieved by plotting the iso-surfaces of the charge density forthe most stable interstitial configurations in each phase. ??

49

4.4 H diffusion in α-Zr

By observing the crystal structure of α-Zr again, re-displayed below for convenience,one can evince that diffusion between basal planes can occur only by hopping betweentetrahedral and octahedral sites via the non-basal trigonal site (for simplicity we will callthis the T-O process). On the other hand migration along the c-axis can occur in two ways:

T-H-T From TET(

at 13 ,

23 ,

18

)

, passing through HEX(

at 13 ,

23 , 0

)

, to the symmetrically

equivalent TET(

at 13 ,

23 ,

18

)

. However this can occur only for half the height of unitcell, since the HEX site appears only every (0002) plane.

O-t-O Alternatively, migration can occur from OCT(

at 23 ,

13 ,

14

)

, via TRI(

at 23 ,

13 , 0

)

, to the

symmetrically equivalent OCT(

at 23 ,

13 ,

14

)

. This process alone is sufficient to obtainc-axis diffusion since it is not constrained by Zr-occupied sites.

Figure 4.5: Structure of α-Zr, from bottom left the intersticial sites (red crossesare): hexahedral

(

13 ,

23 , 0

)

, tetrahedral(

13 ,

23 ,

18

)

, non-basal trigonal(

59 ,

19 ,

16

)

,

octahedral(

23 ,

13 ,

14

)

and basal trigonal(

23 ,

13 , 0

)

.

Even though the former migration process cannot — by it self — contribute to diffusion,it is not necessarily the case that c-axis diffusion is achieved only by O-t-O. In fact, T-

H-T can be coupled with the diffusion from tetrahedral to octahedral and back to (adifferent) tetrahedral site (via the T-O process), to obtain an alternative route. The calculatedactivation energies for both processes agree very well with experimental values, and therespective energy paths were superimposed in fig. 4.6 by plotting the vertical componentof the MEP on the abscissae.

50

0 0.5 1 1.5 2 2.5

0

0.1

0.2

0.3

0.4

0.5

0

2

4

6

8

10

z (Å)

ΔH

(eV

)

TET TET

HEX

nbT nbT

OCT OCT

ΔH

(kcal/mol)

TRI

Figure 4.6: The enthalpy change is taken with respect to the lowest stableenergy state (the TET site). z represents the distance along the c-vector fromthe starting interstitial site. Dashed lines are generated by mirroring the data.Because the T-O and the T-H-T MEP are along the [0001] direction, this alsorepresents the total distance travelled by the hydrogen. However, for the initialpart of the red curve (and its symmetric counterpart), the migration route isalong the

[

3308]

direction, so that the abscissae value represent the projectionsof such migration along the vertical axis.

The activation energy for the red route is dictated by the TET to OCT migration process,which is also the mechanism trough which non-vertical diffusion occurs. Therefore,isotropic diffusion can occur by this process alone. However the difference of energybarrier of such process compared to the blue route is minimal (0.40 compared to 0.42eV),consequently it is safe to state that hydrogen will probably diffuse using both processes. Itis worth noticing that Domain’s work50 predicted the T-O process for non-vertical diffusion(in agreement with the current work) but the O-t-O one for diffusion in the other directions,even though his values were (0.35 compared to 0.41eV).

further work: NEB on IM, ZrNi system, ZrSn2.

51

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