[V] RESULTS •  Both expressions for the elastic interaction energy, (i) and (ii) are shown and compared. The

defect forces used are that of a tetrahedral site where all points can be related by symmetry.

•  The elastic interaction energy profile for a H tetrahedral interstitial and a prismatic edge dislocation within Zr is shown to be asymmetric, however with a profile close to a Cottrell atmosphere.

[I] OVERVIEW Pressurized water reactors (PWRs) are the most common reactor design in the world. PWRS use water as both coolant and moderator. Zirconium (Zr) alloys are used as the fuel cladding material due to their corrosion resistance and low thermal neutron capture cross section. However, under the harsh reactor environment and high temperatures, the alloy does corrode. The corrosion reaction produces Hydrogen (H) atoms that can enter the metal matrix. H has a low solubility limit (~60 ppm) in Zr. It also tends to embrittle the zircaloy and can cause delayed hydride cracking (DHC), which is a result of a H assisted phase transformation within the metal. Fuel cladding failures impact on both the safety and economics of the fuel cycle of nuclear reactors. This project is focused on modelling DHC with input from first principles calculations.

MULTISCALE MODELLING OF DELAYED HYDRIDE CRACKING J. Majevadia1, R. Nazarov2, D. S. Balint1, M.R. Wenman1, A.P. Sutton1 1 Imperial College London, 2 Max Planck Institut Für Eisenforschung

[II] RESEARCH SUMMARY •  The purpose of this work is to develop a model for stress driven diffusion of H to crack tips. This is

significant to the nuclear industry particularly with regard to improving the safety and lifetime of the fuel pin.

•  A first principles approach has been used to calculate the relaxation of atomic H in tetrahedral interstitial in Zr.

•  Using this first principles data, we have calculated the elastic interaction energy between atomic H and edge dislocations in a way that respects the symmetry of the local atomic environment. This is in contrast to a misfitting sphere model of the elastic interaction.

•  In this work we have shown that the two methods for calculating interaction energies are equivalent, only when the relaxed atomic environment has spherical symmetry. This approach is significant in its multiscale nature, combining atomic-scale data with linear elasticity theory.

•  A stress driven diffusion equation has been derived, which incorporates the new expression for the elastic interaction energy. The diffusion equation has been implemented to determine the time evolution of H in the presence of a single dislocation.

A Transmission Electron Micrograph (TEM) image of a notch on the surface of Zr fuel cladding. A crack has propagated through the material, assisted by the formation of a brittle hydride at the crack tip, shown by the clouded region around the crack (reproduced from Sagat 2001 in a review by C.E. Coleman [1]).

ACKNOWLEDGEMENTS Funding for this project is provided by EPSRC (EP/G036888/1) through the Centre for Doctoral Training in the Theory and Simulation of Materials at Imperial College London.

@cH(r, t)

@t=

D

kBT[r · cH(r, t)rEint(r)

+cH(r, t)r2Eint(r) + kBTr2cH(r, t)]

[IV] METHOD

Density Functional Theory The energetics and equilibrium structure of point defects in Zr were calculated using Density Functional Theory (DFT) with the Vienna Ab Initio Simulation Package (VASP).

•  H solution enthalpies and H-vacancy binding energies were calculated to ensure that results were in agreement with previous work [2]. The most stable site for H is the tetrahedral interstitial.

Interstitial (y) (eV)

- present work (eV)

- Domain 2002

T -0.423 -0.604[-0.67]

O -0.374 -0.532

�HyH

�HyH

Displacement field for a prismatic edge dislocation in anisotropic Zr [3]. The primary slip system for Zr is the prismatic plane.

y (eV)Substitutional 1.110604

Local T site -0.239119

Local O site -0.210476

Local Hex site -0.122295

Local BT site 0.427108

EH�vacbinding(y)

Elastic interaction energy between H and an edge dislocation as calculated using expression (i) however with both forces and distance between the interstitial and the surrounding atoms fully symmetric.

[VI] CONCLUSIONS •  The defect forces of H on Zr atoms in a tetrahedral

s i te have been determined through DFT calculations.

•  When the defect forces are fully symmetric, the resulting elastic interaction profile agrees with that calculated using the misfitting sphere model.

•  Using our novel approach to defect force calculations, we have shown the elastic interaction energy profile for atomic H in its most stable tetrahedral site in Zr. Our result captures the full symmetry of the local atomic environment of Zr, which has a non-ideal c/a ratio.

Wint(r, �) = �FH ·UD(r, �)

Defect Forces •  The defect forces are a set of fictional forces which characterize the defect. With this approach it is

possible to determine from ab initio DFT simulations, the force exerted by H on surrounding atoms in the cell.

•  The forces exerted by H on surrounding Zr atoms was calculated by removing a H atom from a fully relaxed Zr96H system and performing a single point calculation. The forces calculated are that of Zr on a vacancy left by H, which is the negative of the required defect forces.

Forces exerted on Zr by interstitial H.

Wint(r, ✓) = PD4VH

Elastic Interaction Energy •  The elastic interaction energy between a H atom and a defect such

as a dislocation can be calculated in two ways:

•  The defect forces capture the symmetry of the local environment of Zr.

•  It is shown in the calculation of Wint that expression (i) captures the full symmetry of the local environment, whereas expression (ii) approximates the effect of H on the surrounding environment as that of a misfitting sphere.

(i)

(ii)

A tetrahedral interstitial in Zr: the force on the three basal sites are identical, the force exerted along the c-axis is different because the point is not related to the others by any symmetry operation.

REFERENCES [1] C.E. Coleman: Cracking of Hydride-forming metals and alloys, Comprehensive Structural Integrity, Chapter 6.03, 103-161 (2007) [2] C. Domain, R. Besson, and A. Legris, “Atomic-scale ab-initio study of the Zr-H system: I. bulk properties,” Acta Materialia , vol. 50 , no. 13 , pp. 3513 –3526 , 2002 . [3] L.J. Teutonico, Materials Science and Engineering, 1970

Elastic interaction energy calculated using expression (ii), giving the expec ted p ro f i l e o f a Cot t re l l atmosphere. The two profiles agree well for the case where the tetrahedral site has ideal symmetry.

Stress Driven Diffusion The diffusion equation is derived from the chemical potential of H in the bulk, and in the region of the stress concentration. The chemical potential of H in the vicinity of the crack tip is dependent on its elastic interaction energy.

Elastic interaction energy between H and a prismatic edge dislocation in Zr as calculated using expression (i), us ing the de fec t fo rces obtained DFT calculations.

The elastic interaction driving diffusion is captured in the first two terms of the diffusion equation

-60 -40 -20 0 20 40 60-60

-40

-20

0

20

40

60

xHAngstromL

yHAngstromL

The steady state solution to the diffusion equation

Y (Å

)

X (Å) •  We have developed a stress driven diffusion model into which the elastic interaction energy is

incorporated, and have shown that the steady-state solution to the diffusion equation also captures the symmetry of the atomic environment of Zr.

•  The significance of this work is that we have used a multi-scale approach to determine the evolution of H within Zr in the presence of stress concentrations, with the possibility of extending the work to modelling diffusion to crack tips and the first stage of DHC.

Prerequisites to obtaining a model that replicates the primary stages of DHC include understanding the relationship between H and Zr on the atomic scale, incorporating H interactions with line defects and determining H-H interactions. The behaviour of H in Zr is studied using DFT and results for the volume dilatation, and forces exerted on neighbouring atoms by hydrogen are presented. The hydrogen-dislocation interaction can thus be determined analytically. The effect of dislocation screening by hydrogen is determined and the resulting reduction in dislocation spacing is shown. The scope for future work using microscale modelling methods such as Discrete Dislocation Dynamics are presented.

FIGURE 3 Zirconium has a hexagonal close-packed structure, with a c/a ratio of 1.593. There are five interstitial sites in which a solute atom might sit. For hydrogen, the most energetically favourable site is the Tetrahedral.

[4]

c)

REFERENCES [1] D. Olander: Nuclear Fuels- Present and Future, Journal of Nuclear Materials Volume 289, 1-22 (2009)

[2] C.E. Coleman: Cracking of Hydride-forming metals and alloys, Comprehensive Structural Integrity, Chapter 6.03, 103-161 (2007)

[3] H.K. Birnbaum, Mechanisms for Hydrogen Related Fracture of Metals – Technical Report, Office of Naval Research, (1989)

[4] S. C. Middleburgh and R. W. Grimes, “Defects and transport processes in beryllium,” Acta Materialia. Vol. 59, pp. 7095–7103, Oct 2011.

[5] L.J. Teutonico, Materials Science and Engineering, 1970

[6] G. Kresse and J. Furthmuller, “Efficiency of ab-initio total energy calculations for metals and semicon- ductors using a plane-wave basis set,” Computational Materials Science, vol. 6, pp. 15–50, 1996.

FUTURE WORK • Incorporate a stress driven diffusion equation for hydrogen into the DDD model.

• Use DDD simulation technique to model the effects of hydrogen on plasticity and vice versa.

• Assess the role of dislocations as possible solute traps and sites for hydride nucleation

Wint[x,y] (eV)

FIGURE 5 A contour plot of the elastic interaction energy between hydrogen and a dislocation. The centre of the plot is the location of the dislocation, with dislocation line perpendicular to the page.

CONCLUSIONS We have found that there is an elastic interaction between hydrogen and an edge dislocation, and that hydrogen has a tendency to accumulate in the tensile region of a dislocation. As more edge dislocations are placed in close proximity, their elastic fields results in a greater accumulation of hydrogen and so the screening between dislocations can be significant. Low temperatures and high concentrations of hydrogen are required to provide sufficient dislocation screening, and since concentrations in the elastic field of a dislocation can reach the solvus we find that dislocations can act as nucleation sites for hydride phases.

FIGURE 6 A contour plot of the concentration profile around an edge dislocation. Bulk concentration is 100ppm and T=300K. A significant accumulation of hydrogen can be seen in the vicinity of the dislocation. Under the appropriate conditions the solvus concentration can be reached in these regions and lead to hydriding.

RESULTS The interaction energy is calculated by combining the Kanzaki forces with the displacement field of the prismatic edge dislocation. The concentration profile is then determined using a Fermi-Dirac distribution function. The results are shown in Figures 5 and 6.

FIGURE 2

The stages of delayed hydride cracking, beginning with the diffusion of hydrogen atoms to a stress concentration due to the reduced local chemical potential around the crack tip, followed by the nucleation, growth, and fracture of the hydride phase.