Integrated Computational Modeling of Water Side Corrosion

Integrated Computational Modeling of Water Side Corrosionin Zirconium Metal Clad Under Nominal LWR OperatingConditions

ASGHAR ARYANFAR,1 JOHN THOMAS,2 ANTON VAN DER VEN,2

DONGHUA XU,3 MOSTAFA YOUSSEF,4 JING YANG,4 BILGE YILDIZ,4

and JAIME MARIAN1,5

1.—University of California, Los Angeles, Los Angeles, CA, USA. 2.—University of California,Santa Barbara, Santa Barbara, CA, USA. 3.—University of Tennessee-Knoxville, Knoxville, TN,USA. 4.—Massachussetts Institute of Technology, Cambridge, MA, USA. 5.—e-mail:[email protected]

A mesoscopic chemical reaction kinetics model to predict the formation of zir-conium oxide and hydride accumulation light-water reactor (LWR) fuel clad ispresented. The model is designed to include thermodynamic information fromab initio electronic structure methods as well as parametric information interms of diffusion coefficients, thermal conductivities and reaction constants. Incontrast to approaches where the experimentally observed time exponents arecaptured by the models by design, our approach is designed to be predictive andto provide an improved understanding of the corrosion process. We calculate thetime evolution of the oxide/metal interface and evaluate the order of the chem-ical reactions that are conducive to a t1/3 dependence. We also show calculationsof hydrogen cluster accumulation as a function of temperature and depth usingspatially dependent cluster dynamics. Strategies to further cohesively integratethe different elements of the model are provided.

INTRODUCTION

Oxidation and hydriding performance of zirco-nium fuel components in light-water reactors(LWRs) may limit the maximum fuel dischargeburn-up, which makes corrosion a critical aspect ofZr materials response in nuclear environments.Although out-of-pile autoclave corrosion tests devel-oped over the years can predict some aspects of in-pile corrosion performance of zirconium alloys,these autoclave tests normally underestimate cor-rosion rates and do not account for irradiationeffects. To help bridge the gap between autoclavemeasurements and in-reactor Zr corrosion, severalmodels of zirconium alloy corrosion and hydridingprocesses have been developed over the years. Thesemodels focus only on certain elements of thereported observations of corrosion and hydriding,and are generally formulated ‘reactively’, i.e. toreproduce observed experimental time evolutionswithout delving into the underlying causes behindthe experimental observations. This lack of

predictiveness prevents these models from beingused for materials evaluation and design. In thispaper, we present a corrosion model that provides afirst-principles approach to understand corrosionand hydriding performance of Zircaloys in a generalmanner, based upon the material microstructure.

Oxide Layer Formation and Growth

Corrosion occurs by diffusion of oxygen andhydrogen into the Zr metal and chemical reactionwith the host atoms according to:1

Zr sð Þ þ xH2O gð Þ ! ZrOx sð Þ þ xH2 gð Þ ð1Þ

where x is a mass balance factor set by the thermo-dynamics of the Zr-O system. The corrosion kineticsis generally interpreted as diffusion controlled,2 witha one-third power law dependency with time.2–5 At acertain thickness, the oxide scale cracks,6,7 whichopens up pathways for the water to reach the metallicsubstrate, leading to a loss of the protective layer anda sudden jump in corrosion rates. Fracture is

JOM, Vol. 68, No. 11, 2016

DOI: 10.1007/s11837-016-2129-1� 2016 The Minerals, Metals & Materials Society

2900 (Published online October 7, 2016)

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attributed to the contribution of several factors, suchas (1) compressive stresses in the oxide due tovolumetric expansion during oxidation (Pillar-Bed-worth ratio of 1.56),2 (2) tensile stresses in the oxidelayer due to thermal and irradiation induced expan-sion on the fuel side, and (3) dynamic embrittlementcaused by hydride precipitation.8–12 Due to the highlycomplex and synergistic nature of corrosion kinetics,with coupled oxygen/hydrogen transport, diffusion/reaction kinetics, temperature effects, and complexoxide microstructures, to name but a few, a multi-pronged approach that includes modeling as anessential element must be adopted to improve ourunderstanding of the process.

In this paper, we present an integrated model ofZr oxidation and hydriding kinetics starting fromfirst-principles reaction kinetics, fundamental ther-modynamics, and parameterization from electronicstructure calculations and experiments. First, wedescribe the thermodynamic modeling that fur-nishes phase diagrams and structural details. Thisis followed by a formulation of the chemical reactionmodel for oxide layer growth, in which the non-stoichiometric oxidation of Zr to ZrOx is computed inreal time, considering the simultaneous spatialvariation in the effective parameters such as diffu-sivities, thermal conductivities, and reaction kinet-ics. Electronic structure calculations of the modelparameters are then discussed, and the paperconcludes with a rate theory study of hydrideaccumulation in the Zr metal layer.

THERMODYNAMICS OF Zr OXIDES ANDHYDRIDES

Clad corrosion is a non-equilibrium, multi-scaleprocess that can be viewed as a sequence of phasetransformations of the original hcp Zr-based alloyinto its various oxides and hydrides due to theingress of oxygen and hydrogen. Relevant kineticprocesses include oxygen and hydrogen diffusiondown chemical potential gradients as well as themigration of interfaces separating the alloy from thevarious oxide and hydride phases. These thermallyactivated, non-equilibrium processes can bedescribed at a phenomenological level with kineticlaws based on irreversible thermodynamics and/orCahn-Hilliard-Allen-type descriptions.6,13

Predicting the dynamical evolution at a phe-nomenological level, while very well suited forcorrosion processes, nevertheless requires accuratethermodynamic and kinetic information, which isoften difficult if not impossible to measure experi-mentally, especially in isolation. It is here wherefirst-principles electronic structure methods canplay a crucial role. However, first-principles elec-tronic structure methods only predict properties atzero temperature, while corrosion occurs at elevatedtemperature where entropy and thermal activationare important. Statistical mechanics is the key linkconnecting the 0 K electronic structure properties of

a particular phase to its high-temperature thermo-dynamic and kinetic properties needed in phe-nomenological theories. The development ofpowerful statistical mechanics methods and compu-tational tools to predict free energies14,15 and diffu-sion coefficients16,17 in complex alloys and multi-component ionic and semiconducting compounds isrelatively recent, but it has opened the door tounderstanding the rich thermodynamics of the Zr-O-H system, as well as to new interpretations of thecomplex corrosion kinetics under service.

The phases that form during the corrosion ofzirconium and its alloys are especially challengingto describe with first-principles statistical mechan-ics methods. Zirconium forms a rich variety ofoxides and hydrides having a range of crystalstructures and compositions. ZrO2, for example, isstable in one of three polymorphs at ambientpressure depending on temperature. Zirconium alsoforms a series of sub-stoichiometric oxides at verylow oxygen partial pressures. In fact, a-zirconium,which exists in the hcp crystal structure, is able todissolve oxygen interstitially up to concentrations of33%, with the dissolved oxygen exhibiting substan-tial disorder at high temperature, but formingordered sub-stoichiometric oxides at low tempera-ture. The hydrides are equally complex. ZrH2, forexample, undergoes a cubic to tetragonal marten-sitic transformation upon cooling from high tem-perature and can tolerate very high vacancyconcentrations over the hydrogen sub-lattice atlow hydrogen partial pressures. Metastable hydridesare also observed and are likely stabilized as theirformation in the early stages of precipitation isaccompanied by less coherency strain energy thanthe hydrides that are more stable under ambienthydrostatic pressure conditions.

The elevated temperatures at which corrosionoccurs requires an explicit treatment of entropiccontributions arising from thermal excitations overthe many available configurational and vibrationaldegrees of freedom. The oxygen disorder at hightemperature in the ZrOx sub-oxides, for example, isresponsible for configurational entropy. CubicZrH2�x will also have a sizable configurationalentropic contribution to the free energy due to themany possible ways of arranging hydrogen vacan-cies over the interstitial sites of an FCC Zr sub-lattice. Even more challenging to predict are con-tributions from vibrational excitations, which areresponsible for the structural transformationsamong the various polymorphs of ZrO2 and ZrH2.

A powerful method to describe configurationaland vibrational excitations in multi-componentcrystalline solids is with the use of effective Hamil-tonians. These are mathematical expressions thatcan be parameterized with first-principles calcula-tions and that can then be used to predict theenergies of arbitrary excitations, often with close tofirst-principles accuracy. This makes effectiveHamiltonians invaluable within Monte Carlo

Integrated Computational Modeling of Water Side Corrosion in Zirconium Metal Clad UnderNominal LWR Operating Conditions

2901

simulations where thermodynamic averages areestimated. A cluster expansion effective Hamilto-nian,18,19 for example, is expressed in terms ofpolynomials of site occupation variables and enablesthe rapid calculation of the energy of a multi-component solid as a function of the degree of orderamong its various constituents. A harmonic phononHamiltonian is a similar expansion, expressed interms of polynomials of the components of atomicdisplacement vectors belonging to pair clusters.Phonon Hamiltonians are restricted to small per-turbations and are accurate for stiff bonds or at lowtemperatures where vibrational amplitudes aresmall. Many important high temperature phases,however, exhibit large anharmonic vibrational exci-tations requiring Hamiltonians that go beyond theharmonic approximation. In this context, a varietyof anharmonic lattice dynamical effective Hamilto-nians have been developed that are expressed eitheras a function of lattice Wannier functions,20,21 or asa function of symmetrized collective displacementmodes of clusters of sites.22 These effective Hamil-tonians have been especially invaluable in predict-ing properties of high-temperature phases thatexhibit dynamical instabilities at low temperatures.

Figure 1 shows a Zr-O binary phase diagramcalculated from first-principles calculationsaccounting for configurational and vibrational exci-tations. The configurational degrees of freedomwere captured by combining a first-principlesparameterized cluster expansion with Monte Carlosimulations. Vibrational excitations were accountedfor within the quasi-harmonic approximation, with

vibrational free energies explicitly calculated for thestoichiometric oxides and linearly interpolated forthe ZrOx solid solution between x = 0 and ½. Thephase diagram shows the remarkably high oxygensolubility of oxygen in hcp-based Zr. At high tem-peratures, the oxygen forms a solid solution over theoctahedral interstitial sites of hcp Zr. At low tem-peratures, the dissolved oxygen orders at stoichio-metric compositions, including ZrO1/6, ZrO1/3 andZrO1/2. The ZrO1/6 sub-oxide can be viewed asstaged phase in that every other layer of interstitialsites is empty while the oxygen in the filledalternating layers are well ordered, forming affiffiffi

3p

a�ffiffiffi

3p

a supercell. In ZrO1/3, every layer fillsup, but the in-plane ordering is the same as in thefilled layers of ZrO1/6. The ZrO1/2 suboxide main-

tains affiffiffi

3p

a�ffiffiffi

3p

a supercell ordering perpendicularto the c-axis, but achieves half-filling over theoctahedral sites by having layers with 1/3 fillingalternated by layers having 2/3 filling. However, asis clear from the calculated phase diagram, thisphase is able to tolerate a large degree of off-stoichiometry and disorders through a second-ordermechanism. The phase diagram also shows a tran-sition temperature for ZrO2 from its monoclinicform a to its tetragonal form b. This transitiontemperature was predicted within the quasi-har-monic approximation.

While the Zr-O binary has been studied fordecades, it was only recently that a new monoxidephase having ZrO stoichiometry was discovered.Experiments showed the existence of a ZrO

Fig. 1. The calculated Zr-O binary phase diagram.25,26

Aryanfar, Thomas, Ven, Xu, Youssef, Yang, Yildiz, and Marian2902

monoxide in oxidation microstructures.23,24 How-ever, it was with systematic first-principles enu-meration and structure search algorithms that thecrystal structure of this sub-oxide was unraveled.25

This new phase, labeled d’, consists of an omega Zrsub-lattice (made of alternating honeycomb Zrlayers interleaved with triangular Zr layers) withthe oxygen occupying square pyramidal interstitialsites of the Zr sub-lattice. The vibrational entropy ofthis phase was predicted to be lower than that ofcompeting oxides due to the tight packing; however,the phase was found to remain stable until at leastabout 1500 K. It appears as a line compound inFig. 1 as it does not tolerate off-stoichiometrythrough the introduction of oxygen vacancies orinterstitials.25

The hydrides of Zr pose even greater challenges tofirst-principles statistical mechanics, in large partbecause the important cubic ZrH2�x form is pre-dicted to be dynamically unstable at low tempera-tures. In addition to tolerating high vacancyconcentrations over the hydrogen sub-lattice, lead-ing to sizable configurational entropic contributionsto the free energy, cubic ZrH2�x also exhibits largeanharmonic vibrational excitations that dynami-cally stabilize its cubic symmetry. Recently, acluster expansion Hamiltonian describing vibra-tional degrees of freedom was developed to accountfor the large anharmonic fluctuations that stabilizehigh symmetry phases such as cubic ZrH2.22 Com-bined with Monte Carlo simulations, it was possibleto predict the experimentally observed tetragonal tocubic phase transformation upon heating. Figure 2ashows the predicted dependence of the unit celllattice parameters on temperature, clearly showinga second order phase transition from the tetragonalform of ZrH2 to the cubic form. Figure 2b showscalculated free energy curves as a function of the c/a ratio. At zero Kelvin, the cubic phase, correspond-ing to c/a = 1, is clearly dynamically unstable dueto the negative curvature of the free energy. Withincreasing temperature though, anharmonic

vibrational excitations reduce the difference in freeenergy between the tetragonal phase, correspondingto the minimum at c/a< 1, and the cubic phase atc/a = 1. At the second-order transition tempera-ture, the minimum of the free energy shifts to c/a = 1, rendering the cubic phase thermodynamicallystable. It was only upon the introduction of newstatistical mechanics tools accounting for anhar-monic vibrational excitations that important ther-modynamic properties such as the elastic modulicould be calculated for phases such as cubic ZrH2,27

which are dynamically unstable at low temperature,but become dynamically stabilized through largevibrational excitations at high temperature.

MESOSCOPIC KINETIC MODEL OF OXIDELAYER FORMATION AND GROWTH

The oxygen ions from the electrolysis of the waterreach the clad outer surface and diffuse in the Zrmatrix. Free oxygen diffusion occurs preferentiallyalong the grain boundaries, accelerating ion trans-port with respect to bulk diffusion. As the oxygenconcentration builds up, the resulting Zr and Omixtures evolve, giving rise first to stable solidsolutions, followed by intermetallic structures aswell as additional sub-oxide phases such as the(metallic) d’ ZrO phase. When the atomic fraction ofO approaches the oxide stoichiometric phase bound-ary in the phase diagram,25 the system begins totransform into a substoichiometric form of tetrago-nal zirconia, ZrO2�x (with x < 0:02). It is believedthat this layer consists of arrays of roughly equiaxednanocrystals with various stress-minimizing orien-tations relative to the metallic substrate, stabilizedby the relatively large internal surface-to-volumeratio. This metastable phase is soon replaced by astable monoclinc saturated (stoichiometric) zirconiaphase, as more free oxygen is picked up andimmobilized into the oxide phase. Multilayer evolu-tion systems are typically treated as interface-evolving Stefan problems. However, this naturally

200 400 600 800 1000 12004.4

4.5

4.6

4.7

4.8

4.9

5

5.1

Temperature (K)

Latti

ce P

aram

eter

(A

ngst

r.)

c

a & b

13824-Zr MC

64000-Zr MC

1400

Experimental Extrapolation

0.8 0.85 0.9 0.95 1 1.05 1.1 1.151.2−25

−20

−15

−10

−5

0

5

10

c/a

Hel

mho

ltz F

ree

Ene

rgy

(meV

/Zr)

400 K800 K

1200 K1600 K

0 K

(b)(a)

Fig. 2. (a) Calculated lattice parameters as a function of temperature for ZrH2. (b) Calculated free energies of ZrH2 as a function of c/a ratio atseveral temperature below and above the second order tetragonal to cubic phase transition.


2903

leads to a parabolic growth scaling law which isknown to not hold for Zircalloy corrosion. Instead,here, we develop a one-dimensional chemical reac-tion kinetics model.

Throughout the process, the atomic fraction ofoxygen (related to the partial pressure) in Zr isdefined as:

X ¼ ½Oi�½Oi� þ ½Zr� ð2Þ

where [Oi] is the concentration of chemically immo-bilized oxygen (i.e. not available for diffusion). Thevalue of X for the substoichiometric tetragonalphase denoted by ZrO2�x that forms first is thengiven by:

X ¼ 2 � x

3 � xð3Þ

which corresponds to non-stoichiometric ratio ofX � 0.6644 when x = 0.02. The value for monocliniczirconia is a constant at X = 2/3. Therefore, once theoxide layer emerges, oxygen diffusion proceeds on twodifferent media with different physical properties.28

Thispicture including a three-layer systemis schemat-ically depicted as a one-dimensional process in Fig. 3.

A temperature gradient exists within the clad,such that the temperature depends on depth, whichwe denote by the spatial variable y. A simple orderof magnitude analysis of the values of the thermaldiffusivity a and the diffusion coefficient D yieldsapproximately:

a ¼ jqcp

� 10�5 m2

s

D � 10�14 m2

s! @T

@t� @½O�

@t

(

ð4Þ

where q is the mass density, cp is the heat capacity,and T is the absolute temperature. This means thatthe thermal propagation rate is significantly fasterthan mass transport of O in Zr. Hence, we canassume that, at any stage in the zirconium oxidegrowth, the temperature distribution equilibratesvery rapidly on the scale of atom diffusivity. Thisallows us to discard the time dependence of thethermal profile an any time (T = T(y)). With this,the set of parabolic differential equation thatexpresses the spatio-temporal variation of freeoxygen in the system is:

d½O�dt ¼ DðTÞ @2½O�

@y2 þ QKBT2

@T@y

@½O�@y

� �

� krxn½O�bd½Oi�

dt ¼ krxn½O�b

(

ð5Þ

where b is the reaction order. This equation resultsfrom a first-order Taylor series expansion of Fick’sLaw with a temperature gradient. The first term inthe right-hand side of the equation representsconcentration-driven diffusion, the second termrepresents thermomigration, and the third is oxy-gen consumption due to chemical immobilization(i.e. absorption of oxygen to form chemical com-pounds). The oxygen diffusivity follows a standardArrhenius law with temperature T, which itselfdepends on the spatial coordinate y via the temper-ature gradient.

DðTÞ ¼ D0 exp�QD

kBT

� �

ð6Þ

Suitable initial conditions to solve the differentialequation system (5) by using finite differences canbe defined at t = 0 for stage I:

Fig. 3. Schematic representation of model parameters and evolved layers. The chemistry defining each layer is provided in simplified form in theleft hand side, while the global layer designation is written in the right-hand side.


I:C :½O�ðy; 0Þ ¼ 0½Zr�ðy; 0Þ ¼ ½Zr�0

�

ð7Þ

where [Zr]0 is the (constant) initial zirconiumconcentration, obtained from its atomic density.

B:C :½O�ð0; tÞ ¼ C0@½O�ðL;tÞ

@y ¼ 0

�

ð8Þ

where C0 is the (constant) oxygen concentration,available from electrolysis on the water side, whichis set by water chemistry, temperature, pressure,etc. The second equation reflects that no free oxygencan escape the clad into the fuel side. (i.e.JOðL; tÞ ¼ 0). Our critical assumption for post-crackgrowth (stage II) regime is that the free oxygen fromthe environment side becomes immediately avail-able on the oxide/metal interface via transportthrough the microcrack network and remains con-stant at C0 as the crack grow. Therefore, theupdated boundary condition becomes:

½O�ðs; tÞ ¼ C0 ð9Þ

A further simplification is to assume a single oxygenpopulation that penetrates the clad by diffusion and isconsumed due to the oxide reaction process. Oxygendiffusion results in a spatial depth distribution fromwhich we obtain the concentration fraction X. Hence,we can obtain the spatial distribution of the emergingzirconium oxide phases based on the time evolution ofX. The thickness of the oxide layer, s, is then self-consistently obtained as the depth at which theoxygen in the clad is dissolved by the Zr.

Temperature Profile Update

As shown in Eq. 5, T(y) and dT=dy are essentialinputs with spatio-temporal distributions. As theoxide builds up, it reduces the heat transfer rate.Hence, heat flux evacuated radially from the fuel is:

q ¼ �jðyÞdT

dy� jZr

Tf � Tw

Lð10Þ

Further, we assume a linear temperature profilethrough the clad with (at most) two separate phases(oxide plus metal):

TðyÞ ¼ Tw þ Ti�Tw

s y y < s

Ti þ Tf�Ti

L�s y y s

(

ð11Þ

where Ti is the interface temperature obtained fromEq. 10 as:

Ti ¼jrðL� sðtÞÞTw þ sðtÞTf

jrðL� sðtÞÞ þ sðtÞ ð12Þ

where jr ¼ jZrO2

�

jZr, L and s(t) are the thermalconductivity ratio, the clad, and oxide layer thick-ness, and Tf and Tw are the temperatures on thefuel and water sides, respectively, which are takento be constant. Hence, the temperature gradients atany given instant are obtained as:

dT

dy

ox

¼ Tf � Tw

jrðL� sðtÞÞ þ sðtÞ ð13Þ

and

dT

dy

Zr

¼ Tf � Tw

L� sðtÞ þ sðtÞjr

ð14Þ

Iterative Procedure

The initial conditions are defined assuming a pureZr layer, which sets the original temperature gra-dient and the rest of the associated temperature-dependent parameters. As the oxide layers grow,this assumption is updated accounting for the newthree-layer metal/oxide system and temperaturedistributions. Hence, the simulation is carried outdynamically by iteration of time and space varia-tions in diffusivity (D), thermal conductivity (j) andreaction rate (k), as shown in Fig. 4.

The experimental kinetics of growth is shown todisplay a third power law relationship with thetime. In this work, we first assume linear kinetics(b = 1). Subsequently, we iteratively vary b until at1/3 dependence is recovered, adjusting the units ofthe rate constant k in each case. The present modelrepresents a simplified view of the oxidation pro-cess, which is nonetheless useful to gain qualitativeinsights into the kinetics of corrosion, and whichwill be augmented in future studies with a morefundamental description of the relevant processes.

MODEL PARAMETERIZATION

Electronic Structure Calculations of OxygenDiffusion in Zr Oxide

By combining density functional theory (DFT)calculations and thermodynamic/kinetic continuummodels, we have conducted a series of studies on thedefect chemistry and transport properties of zirco-nia. The goal of these studies is to deduce physically

Fig. 4. Iterative procedure for the growth of oxide layer s(t).


2905

based kinetic parameters, such as diffusion coeffi-cients and surface reaction rates, and feed them tothe higher-level kinetic models.

Conventional corrosion models are generallybased on the assumption that doubly-charged oxy-gen vacancies and electrons are the two dominanttypes of intrinsic defects, ignoring the fact that thesituation might change with varying temperatureand oxygen chemical potential. In fact, the oxidelayer is constantly exposed to a gradient in thechemical potential of oxygen from the oxygen-poorzone at the metal/oxide interface to the oxygen-richzone at oxide/water interface. We assessed allpossible intrinsic point defects in both tetragonal29

and monoclinic30 phases of ZrO2. Our resultsrevealed that, for the tetragonal phase and underoxygen-poor conditions, the dominant defects aredoubly-charged oxygen vacancies compensated byelectrons. With continuous depletion of oxygen, theneutral oxygen vacancy takes over. For the mono-clinic phase under oxygen-rich conditions and tem-peratures less than 500�C, quadruple-chargedzirconium vacancies prevail compensated by holes.

Oxygen diffusion inside the oxide film consists ofvarious defect-mediated migration processes. Com-bining the formation free energies and migrationbarriers in a random-walk model, we predictedoxygen self-diffusion coefficients in tetragonalZrO2.31 Migration barriers of oxygen defects werecalculated with the nudged elastic band (NEB)method.32 The resulting oxygen self-diffusivity asa function of temperature and off-stoichiometry (x inZrO2�x) is reproduced from Ref. 31 in Fig. 5. Atx = 10�4, the simulation results compare favorableto the experimental data.33 The origin of thedeviation of the slope at this x is the assumptionof a pure ZrO2, which is not the case in theexperiments. Accounting for impurities also resultsin better agreement in the slope, as discussed inRef. 31.

Impurities in ZrO2 arise from the alloying ele-ments that are present in the zirconium alloy.Aiming at quantifying the effect of metal dopants onoxidation and hydrogen pickup kinetics and identi-fying design principles for degradation-resistantzirconium alloys, we performed studies of commonmetal alloy elements dissolved in ZrO2. The exis-tence of aliovalent metal dopants, even on the ppmlevel, could substantially change the defect chem-istry and corrosion kinetics. In tetragonal ZrO2, weexplicitly evaluated the effect of dissolved Nb.34

DFT calculations show that Nb5+ substituting zir-conium is the predominant Nb defect. Positivelycharged Nb dopants are compensated mainly byzirconium vacancies but to some extent also withelectrons. This is also accompanied by a decrease inthe concentration of oxygen vacancies. Kinetically,this is reflected in the enhancement of electrontransport and the suppression of oxygen migration.

The values for oxygen diffusion (D0 and QD)shown in Fig. 5 correspond to very small values of x.As a first-order approximation, we extrapolate thevalues of D0 and QD to higher values of x using thefollowing expressions:

Qox ¼ 1:182 � 0:152 logðxÞ � 0:036 logðxÞð Þ2 eV½ �ð15Þ

logðD0Þ ¼ �1:95 þ 0:0304 logðxÞ � 0:1161 logðxÞð Þ2

� Dox in cm2 s�1 �

ð16Þ

These expressions allow us to expand the nonsto-ichiometry range of oxygen diffusion in ZrO2�x for xvalues up to �0.02.

Mechanisms of Hydrogen Pickup andDiffusion on Zr

Hydrogen pickup and hydride formation are alsoimportant degradation processes in zirconium alloysthat could potentially lead to embrittlement. Here,we focus on hydrogen pickup mechanisms, whilehydride formation is addressed in more detail in‘‘Summary’’ section. In order to understand howhydrogen is incorporated into the oxide passive filmand how it enters the zirconium alloy, we havestudied hydrogen-related defects in tetragonal (t-)ZrO2

35 and monoclinic (m-) ZrO2.30 Both in t-ZrO2

and under oxygen-poor conditions, hydrogen isfound to be more energetically favorable as asubstitutional H+ occupying an oxygen site. As

Fig. 5. Oxygen self-diffusion coefficient at constant off-stoichiome-try, x, in tetragonal ZrO2�x as a function of temperature. Theexperimental data are tracer diffusion coefficients from Ref. 33scaled by a correlation factor of 0.65.


such, the concentration of substitutional hydrogenincreases as one approaches the metal/oxide inter-face. Our results have also demonstrated that thereis a tendency for hydrogen species to cluster at azirconium vacancy site in oxygen-rich conditions,which could be potentially detrimental to themechanical integrity of the oxide film, analogousto hydrogen embrittlement in the Zr metal.

By analyzing the effects of different alloyingelements in m-ZrO2, we have seen that the hydro-gen solubility in the oxide exhibits a minimum withrespect to the electron chemical potential, le.

30 Weidentified two doping strategies that could poten-tially suppress hydrogen uptake in the oxide.Dopants such as Cr could reduce the hydrogensolubility, while dopants such as Nb, Ta, Mo, W or Pcould maximize le and accelerate the H2 evolutionkinetics at the surface. However, increasing le alsoresults in enhanced hydrogen trapping at zirconiumsites in m-ZrO2, thus hindering its diffusion towardthe oxide/metal interface. While Nb and Cr arealready widely used in current zirconium alloys, Ta,Mo, W and P have been predicted in our work toaccelerate the H2 evolution kinetics at the surface.So far, our studies have comprehensively covereddefect chemistry related to oxidation kinetics andhydrogen incorporation in bulk t-ZrO2 and m-ZrO2.The computed formation and migration energies aresufficient for kinetic models where the passive filmis treated as a homogeneous medium. However,extended defects, such as oxide/metal, oxide/waterand oxide/secondary phase particles interfaces, alsoexist in the system and alter the energetics andkinetics nearby. Extending the current frameworkto account for structural variation and space-chargeeffects in the vicinity of these extended defects is thenext step, and will provide more accurate quantifi-cations of reaction rates and transport processesclose to these critical interfaces.

Simulation Parameters

The values for simulation are listed in Table I.

The above parameters are taken from differentsources including experiments dating back severaldecades, DFT calculations, etc., and have not beenchecked for self-consistency. For this reason, theresults presented below are more a qualitativereflection of the oxidation process, rather than arigorous quantitative analysis.

RESULTS

Oxide Layer Growth Simulations

Figure 6 shows the spatio-temporal evolution ofthe oxygen and atomic fraction according to theoxidation model. The thickness of the substoichio-metric ZrO2�x (tetragonal layer) is defined by theiso-contour X = 0.6644. Figure 7 shows the evolu-tion of the monoclinic oxide scale s(t) for b = 1. Thecurve is compared with a t1/3 power law, with verygood agreement achieved. Further analysis showsthat the closest agreement with a t1/3 dependence isachieved for b = 0.7. Figure 8 shows the oxygenconcentration profiles during stage I before andafter the oxide layer emerges.

According to experimental observations, stage IIis seen to approximately begin at an oxide layerthickness of approximately s � 80 lm. As men-tioned earlier, this is brought about by fragmenta-tion and cracking of the oxide scale, which occurswhen pre-exisiting cracks propagate inward andreach the oxide/metal interface. At that point,oxygen and hydrogen ions from the water sidebecome immediately available at the interface, sothat diffusion through the oxide ceases to be therate-limiting step for oxygen availability and oxidelayer growth. During stage II, s starts to scalequasilinearly with time, which is captured qualita-tively by the model, as shown in the inset to Fig. 7.It is worth noting that the model at the presentstage captures the qualitative features of thisbreakaway growth regime, which is seen to stronglydepend on coolant chemistry and irradiation condi-tions, among other factors.

Table I. Parameter values used in the kinetic model for oxide layer growth

Parameter Value Units

L 600 lmTf 660 KTw 600 KQrxn 47.236 kcal mol�1

[O]0 3.1 9 10�36 a mol l�1

[Zr]0 7.17 9 10�5 b Mol l�1

QD 4.4537 kcal mol�1

k0;rxn 6.1 9 10438 kcal mol�1

jZrO2239 c W m�1 K�1

jZr 8.8 + 0.007T + 0.00003T2 + 0.3T�140 W m�1 K�1

a[O]0 = 0.05 ppb = 0.05 g/l = 3.1 9 10�3 mol/l.bAssuming all Zr is available for oxidation reaction.cAlmost insensitive to temperature.


2907

A Microscopic Model for the HydrogenKinetics in Zr

A critical aspect of the overall corrosion process isthe coupling of hydrogen and oxygen transportthrough the oxide scale towards the metal sub-strate, where hydrogen further diffuses and formshydride platelets. These platelets are preferentiallyoriented along the normal and tangential directions,and have a profound impact on the mechanicalbehavior and loss of ductility of the zircalloy. At thisstage, these synergisms have not been fully estab-lished and hydrogen transport and precipitation aretreated as independent processes. In this section, arate theory model of hydride formation in Zr metalis proposed, without any coupling to the oxide scalekinetics for the moment.

Existing models41 for the hydrogen behavior(diffusion and hydride precipitation) in Zr usesimplified expressions for reaction (precipitationand dissolution) rates to solve for the hydrogenconcentration in the metal matrix (solid solutionform) and the total concentration of hydrogen in all

hydrides without regard to their sizes. While thepartition of hydrogen into the matrix and thehydrides is important, the existing models do notprovide more detailed information, such as thenumber density and the size distribution of thehydrides, which is valuable to mechanics modelingand failure probability analysis of the fuel-claddingsystem. Motivated by this, we have initiated aneffort to develop a microscopic model for the hydro-gen kinetics in Zr. Specifically, we use a clusterdynamics (CD) approach that establishes one equa-tion for each size of the cluster (here hydride). TheCD approach has been extensively used in modelingkinetic processes, e.g., defect clustering and heliumbubble formation, in irradiated materials.42,43 Withthis approach, the hydrogen kinetics is described bythe following system of equations:

@C1ðx;tÞ@t ¼ @

@x J þ 2 � k�2 C2 þP

n>2

k�nCn � 2 � kþ1 C21

�P

n>1

kþnC1Cn

@Cnðx;tÞ@t

n> 1¼ kþn�1CnCn�1 þ k�nþ1Cnþ1 � kþnC1Cn � k�nCn

8

>

>

>

>

>

<

>

>

>

>

>

:

9

>

>

>

>

>

=

>

>

>

>

>

;

ð17Þ

where C stands for the concentration (nm�3), kþn andk�n are the reaction constants for a cluster contain-ing n-H atoms to capture and emit a H-atom,respectively. Here, we assume that the newlyformed hydride has a fixed stoichiometry and hencethe H-number is sufficient to represent a hydridecluster and that a hydride cluster contains at leasttwo H-atoms (C1 refers to the concentration ofhydrogen in the metal matrix.) At present, ourmodel uses the following expressions for the reac-tion rate constants: kþn ¼ 4p r1 þ rnð Þ D1 þDnð Þ and

k�n ¼ 4p r1 þ rn�1ð Þ D1 þDn�1ð ÞC0 exp � EBn

kBT

� �

, respec-

tively, where r is the radius, D is the diffusivity(Dn ¼ D1 (for n ¼ 1Þ or 0 (for n> 1Þ), C0 is thematrix atomic number density and EB

n is the bindingenergy of a H-atom to the cluster. These expressionsare most suitable for 3D diffusion-reaction systems

Fig. 6. (a) The time dependent evolution of oxygen concentration [O] and (b) non-stoichiometric ratio X (right). The tetragonal zirconia evolutionis given by the 0.6644 iso-contour.

Fig. 7. The correlation of simulation growth with third power law.Inset Sudden jump in growth regime assuming tCrack ¼ 0:9tSimulation.


with near-spherical clusters, which may be a rea-sonable approximation to the early stage of hydrideprecipitation (or, very small precipitates). Alterna-tive rate expressions will be used to account for theplate shape of large hydrides that has been reportedin experiments. [e.g., Ref. 44]

Two unique features of the problem of the hydro-gen kinetics in Zr cladding are the presence of apronounced temperature gradient across the clad-ding thickness and an extra diffusion flux inducedby the temperature gradient, the latter known asthe Soret effect. The temperature gradient stipu-lates a spatial dependence of the hydrogen diffusiv-ity and the reaction rate constants. The Soret effectmodifies the overall hydrogen diffusion flux in themetal matrix as

J ¼ �D@C

@xþQ

C

T2

dT

dx

� �

ð18Þ

where Q is the Soret constant. After converting thepartial differential equations intoordinary differentialequations at different spatial positions and applyingan all-zero initial condition, a constant (in-)flux, Jin, ofhydrogen from the water side and a zero flux at thefuel-clad interface, we have performed a first roundsimulation of the hydrogen kinetics using the clusterdynamics model and the parameter values listed inTable I. Note that all the parameter values in Table II

are known to be suitable for the hydride formation in a-Zr according to,45 except those for r(n) and EB

n whichneed to be finalized in the near future based onrelevant experiments and/or atomistic studies.

Figure 9 shows the resulting dynamics of hydridecluster evolution as a function of depth for a 0.1-cm-thick Zr metal clad in the presence of a temperaturegradient and incorporating the Soret effect. At thebeginning of the process, H atoms that flow into thematerial from the right (water side) boundary aretransported rapidly to the left (fuel side) boundary.Subsequently, small hydride clusters/embryos con-taining a handful of H-atoms start to form, firstnear the right boundary and then also towards theinner (left) positions, owing to monomer-capturingprocesses. The frame at t = 3,000,000 s (�35 days)in Fig. 9 represents an even later stage where apronounced x-dependence of the size and populationof hydrides is clearly developed, with bigger andlarger population of hydrides occurring near theright (water side) boundary. This preferential for-mation and growth of hydrides is qualitativelyconsistent with the general observations in exper-iments (e.g., Refs. 41 and 45).

We emphasize that the initial results presentedhere are intended to demonstrate the general behav-ior and the new functionalities of this model (e.g.,providing separated information of size and number

Fig. 8. (a) Spatial variation of oxygen concentration [O] in short- and (b) long-range time.

Table II. Parameters used in a first round simulation

L (clad thickness) [nm] 106

x (mesh grids) [nm] [�10.^([6:�.05:5 4.9:�.1:2 1.8:�.2:0]) 0]rT (temperature gradient) [K/nm] �6 9 10�5

T(x) (temperature at x) [K] 325 + rT 9 x + 273D0 (H diffusivity prefactor) [nm2 s�1] 7 9 1011

Em (migration energy of H) [eV] 0.47Jin (H pickup rate) [nm�2 s�1] �0.02r(n) (cluster radius) [nm] n1/3 9 0.14EB

n (cluster binding energy) [eV] 0:435 þ 0:18 � 0:435ð Þ n2=3 � n� 1ð Þ2=3h i.

22=3 � 1�

Q (Soret coefficient) [K] 3000


2909

density of hydride clusters at different spatial posi-tions within the cladding) that are not available fromexisting models, rather than to make detailed com-parisons with any particular experiment. Fullparameterization and thorough experimental vali-dation of the model remain as a future set of goals.

SUMMARY

This work contains modeling results of zirconiumoxide layer growth and the formation of hydridephases in Zr metal as a function of time. The modelspresented here represent fully integrated chemicalreaction kinetics models, coupled to fundamentalthermodynamic and kinetic data obtained by elec-tronic structure calculations. The phase diagrams ofthe Zr oxide and hydride phases have been mappedout using cluster expansion Hamiltonians withstructure identification, and provide the basis to

formulate chemical models that account for stoi-chiometry and solubility behavior. Electronic struc-ture methods also provide physical coefficients andparameters to fit the mesoscopic models.

In contrast with other more semi-empirical, qual-itative models, the time dependence of the oxidelayer thickness is not set a priori, but is an outcomeof the calculations. We have found that the reactionorder (which in general does not coincide with thestoichiometric coefficients of the reaction) sets thetime exponent of the thickness growth dependence.Although a reaction order of unity comes close tofurnishing the one-third power law dependenceobserved experimentally, we have found that avalue of 0.7 provides the best fit. In terms of hydrideformation, we have demonstrated a new clusterdynamics type of model that incorporates thehydrogen diffusion, clustering and thermal dissoci-ation in the presence of a temperature gradient in afundamentally self-consistent way, and provides

Fig. 9. Dynamic evolution of concentrations of dissolved H and hydride clusters (defined by number of H atoms contained) at different positionscalculated with a temperature gradient and Soret effect. The spatial meshes are marked as circles. Note that the temperature linearly drops from385�C on the left boundary to 325�C on the right boundary, and that there is a constant influx of H on the right boundary.


additional information (e.g., size distribution andnumber density of hydride clusters) that are notavailable from existing models. We emphasize thatthere are still many open questions in regards toZircalloy corrosion, such as irradiation effects, theimpact of the alloy chemistry and microstructure,synergistic H/O transport and chemistry, valida-tion, etc. The models presented are an attempt tobring the full power of computational/physical mod-eling to a relatively old problem, using a newmodeling philosophy supported by new numericaltechniques and unprecedented computationalpower. The next phases of our modeling effort willinvolve bridging the oxide growth model with thehydride accumulation calculations.

ACKNOWLEDGEMENTS

This research was supported by the Consortiumfor Advanced Simulation of Light Water Reactors(CASL), an Energy Innovation Hub for Modelingand Simulation of Nuclear Reactors under USDepartment of Energy Contract No. DE-AC05-00OR22725.

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