Nuclear fusion promises to deliver an unlimited supply of clean, green energy, of paramount importance to help address the climate emergency. There have been recent successes in generating fusion energy – notably at the UK Atomic Energy Agency (UKAEA) reactor at Culham where record-breaking fusion energy production was demonstrated earlier this year. However, a major barrier to the wider adoption of fusion remains: the materials used to build a fusion reactor need to withstand bombardment from high-energy radiation. In the case of metals, irradiation damage can be pictured in terms of the accumulation of dislocation loops which self-organise into complex microstructures, changing the mechanical properties of the material. To predict this phenomenon accurately, new models are needed. This project will therefore focus on developing a new mathematical framework to connect discrete atomistic models of dislocation loops to continuum differential equations. This project will need a strong background in either applied mathematics or theoretical physics, and a willingness to collaborate across disciplines, both within HetSys and with our project partners from the UKAEA, based at the Culham Centre for Fusion Energy.

Click here to read more about Tom Hudson (Project Supervisor) and his enthusiasm for this project!

Summary: The materials used to build a fusion reactor undergo bombardment from high-energy radiation. In the case of metals, irradiation causes the accumulation of dislocation loops which self-organise into complex microstructures, changing the mechanical properties of the material. To predict this phenomenon accurately, new models are needed. This project will therefore focus on developing a new mathematical framework to connect discrete atomistic models of dislocation loops to continuum differential equations. The resulting modelling hierarchy will be applied computationally to predict the evolution of dislocation loop microstructures, providing an assessment of tungsten's suitability for fusion applications.

Background: Dislocations are topological line defects found in crystals, and are the carriers of plasticity in metals i.e. irreversible deformation. The discovery of dislocations was a key scientific achievement of the 20th century, providing an explanation of the mechanism by which humanity has been able to work metals for thousands of years. As such, understanding how dislocations behave in a metal sample is crucial to understanding how it may deform, crack and fail. A key complexity in studying dislocations is their number: a single cubic centimetre of tungsten may contain on the order of 10,000km of dislocation line. Moreover, dislocations interact in a complex non-local fashion through stress fields in the material. Mathematical theories to describe this have been developed over the last 60 years, and this project seeks to exploit some of these advances in a new setting (see references below).

In fusion material in particular, materials are exposed to bombardment by high-energy radiation, taking them out of the usual equilibrium setting. This is a completely new frontier in materials engineering, and so new mathematical and computational models are in development in this setting. Using the computational and mathematical methodologies which are developed in HetSys training, the student working on this project will study the physical properties of Tungsten, and use this along with experimental data to develop a new modelling hierarchy for dislocation loop microstructures. A possible direction we will explore in the project is to take a discrete model of dislocation loop interaction and pass to a continuum limit for a density of loops, allowing efficient computation of statistics for comparison with experimental data.

References

Those references below marked with an asterisk (*) are particularly good starting points to get some background.

Books:

• Micromechanics of Defects in Solids, Mura. Springer, 1987.
• Introduction to Dislocations, Hull and Bacon. Butterworth-Heinemann, 2011. (*)
• Introduction to Elasticity Theory for Crystal Defects, Balluffi. World Scientific, 2017.
• Computer Simulations of Dislocations, Bulatov and Cai. Oxford University Press, 2006.

Research Articles:

• The Forces Exerted on Dislocations and the Stress Fields Produced by Them, Peach and Koehler. Physical Review, vol 80, 436, (1950). (*)
• Anisotropic Continuum Theory of Lattice Defects, Bacon, Barnett and Scattergood. Progress in Materials Sciences, vol 23, pp51-262 (1979). (*)
• Motion and Rotation of Small Glissile Dislocation Loops in Stress Fields, Wolfer, Okita and Barnett. Physical Review Letters, vol 92, no 8, 085507 (2004).
• Statistical model for diffusion-mediated recovery of dislocation and point-defect microstructure, Rovelli, Dudarev and Sutton. Physical Review E, vol 98, 043002 (2018).
• An Existence Result for Discrete Dislocation Dynamics in Three Dimensions, Hudson. arXiv preprint: https://arxiv.org/abs/1806.00304.
• Diffusion and interaction of prismatic dislocation loops simulated by stochastic discrete dislocation dynamics, Li et al. Physical Review Materials, vol 3, 073805 (2019).
• Atomistic origins of continuum dislocation dynamics, Hudson, van Meurs and Peletier. Mathematical Models and Methods in Applied Sciences, vol 30 no 13, pp2557-2618 (2020).
• Asymptotic Expansion of the Far-field of a Crystalline Defect, Braun, Hudson and Ortner. arXiv preprint: https://arxiv.org/abs/2108.04765