## Science and Technology

### Breakthrough in Theory and Modeling of Dimensional Instability of Zr-Based Alloys in Light Water Reactors

**S.I. Golubov, R.E. Stoller** - Oak Ridge National Laboroatory

**A.V. Barashev ** - University of Tennessee, Knoxville

Generalization of radiation growth model developed at ORNL

to describe deformation of fuel cladding materials under applied loads

The Consortium for Advanced Simulation of Light Water Reactors (CASL) at ORNL is addressing a wide range of materials performance issues in support of pressurized water reactor (PWR) power uprates and plant lifetime extension. Among these is the development of a physically based model for dimensional instability of PWR fuel cladding. The materials involved are Zr-based alloys with a hexagonal close-packed (HCP) crystal structure. It is well established that dimensional instability of fuel cladding under neutron irradiation is driven by two qualitatively different mechanisms: (a) radiation growth (RG), which is a generic process that occurs in irradiated HCP metals independent of any applied load or mechanical stress, and (b) radiation creep (RC) which takes place only under conditions of an applied load. Note that RG can only be determined in isolation in single crystal material because in multigrain materials, i.e. in any engineering material, RG is accompanied by a creep driven interaction between grains. Therefore, a proper understanding or description of creep cannot be obtained without an accurate RG model. Although the phenomenon of RG in HCP metals was discovered more than 50 years ago, no satisfactory theory or model has been developed and there has been a corresponding absence of a reliable creep model [1,2].

Previously developed models of radiation damage in Zr and its alloys only considered the impact of the individual point defects produced by neutron irradiation, i.e. single vacancies and self-interstitial atoms. It is now known from molecular dynamic simulations that the primary damage formation also includes interstitial clusters that have unique diffusion properties that lead to a qualitative change in reaction kinetics. These anisotropic diffusion properties are responsible for a unique damage accumulation in irradiated HCP metals. A radiation damage theory that accounts for the true nature of the primary damage was developed for cubic crystals by an international collaboration of scientists from Denmark, England, Germany and USA [3]. The Materials Performance and Optimization (MPO) Focus Area working within CASL has used this framework to develop a new radiation growth model for HCP metals. The wide variety of qualitatively different experimental observations accumulated over the last ~50 years, including those which have never been understood, have been explained by the new model.

The primary contributions of the new model are related to the anisotropy of diffusion as reflected in different distributions of dislocations in the HCP crystal as described by the following equations [4]:

where *p*_{x}, *p*_{y} and *p*_{z} are the densities of dislocations in prismatic and basal directions, respectively; *p* is the total dislocation density and *x* is the fraction of interstitial atoms created in the form of clusters. Atomistic molecular dynamics simulations determine the last parameter. Comparison of calculated results for RG [5] with experimental data for the case of annealed Zr is shown in Fig.1. Comparison of the calculated results for effect of cold work on RG with experiment is shown in Fig. 2.

Fig.1: Comparison of calculated results from RG model with experimental data. |
Fig. 2: Effect of cold work on radiation growth, (a) experimental observations, and (b) predictions of computational model. |

The RG model provides a background for developing a true physically based model of dimensional instability in PWR fuel cladding, and for the first time RG is described quantitavely in a single grain. Implementation of the new RG model in the Visco-Plastic Self-Consistent Code (VPSC) developed by the LANL CASL team provides the opportunity to calculate dimensional instability of multigrain materials. In collaboration with ORNL, the original version of VPSC, which was based on purely phenomenological models for RG and creep, has been modified to adopt the new RG model. Recently we suggested the use of the Gittus climb-assisted glide creep model [6] since the RG model provides the climb velocities, *V*_{a}, for all dislocations, and the creep rate, , can be calculated as follows

where are the elastic strain and shear modulus. Preliminary results of the calculations of deformation of a single grain in unstressed and stressed conditions are shown in Figs. 3. Application of the model to multigrain materials is in progress.

Fig. 3a: RG Strain in unstressed single crystal of Zr. |
Fig. 3b: RG Strain in unstressed single crystal of Zr. |

**References**

- R.A. Holt, In-reactor deformation of cold-worked Zr–2.5Nb pressure tubes, J. Nucl. Mater. 372 (2008) 182.
- R. Adamson, F. Garzarolli, C. Patterson, In-reactor creep of Zirconium Alloys, Advance Nuclear Technology International (2009), Krongjutarvägen 2C, SE-730 50 Skultuna Sweden.
- S.I. Golubov, A.V. Barashev, R.E. Stoller, Radiation damage theory. In: Konings R.J.M., (ed.) Comprehensive Nuclear Materials, volume 1, pp. 357-391, Amsterdam, 2012: Elsevier.
- S.I. Golubov, A.V. Barashev and R.E. Stoller, ORNL/TM-2011/473.
- A.V. Barashev, S.I. Golubov, R.E. Stoller, ORNL Report: ORNL/TM -2012/225 (2012).
- J.H. Gittus, Philos. Mag. 25 (1972) 345.