Operator asymptotics, a new approach to length-scale interactions in metamaterials.

Many real-world phenomena, such as the spread of disease or the fluctuations of stock prices, can be understood and predicted by mathematics. This is achieved by building a model, written in mathematical language, that is simple to use yet contains all the essential information required to describe the system in question. For example, imagine we wanted to model the formation of sand dunes in a desert by using the information about each individual grain of sand. This would produce a model too complicated to understand and impossible to use. Yet, to build a successful model it suffices to exploit the fact that on the scale of a dune length that is much larger than an individual grain, sand appears to behave like a continuous medium. In this example, the mathematical model addresses the phenomenon in question by focusing on a single scale, the length scale of the individual dunes. To put it another way, the essential behaviour of the system on the scale in question (length of a dune) is not dictated by the behaviour on finer scale (size of a grain). This however is not the case for many phenomena of great significance today, for example protein folding, catalytic processes or corrosion of metals, where the essential underlying behaviour spans a wide range of length and time scales.

As an illustrative example let us consider a typical Long bone in the body, whose primary function is to provide structural support under mechanical load. A particular phenomenon within Long bone is that its composition can change due to mechanical stresses applied over long time scales. This is achieved by the remodelling of bone to form patterns that are the most resistive and supportive to the changes in mechanical stress. This process begins with the mechanical load stimulating protein translation; the formed proteins then aid in the formation of vesicles and transporter molecules which finally deposit the minerals that form the bone. The complexity of the above explanation shows that for a mathematical model to successfully address the dynamical properties of bone it must include hierarchical behaviour that spans many time and length scales. This illustrates the fact that the simplest models of multi-scale phenomena are incredibly complicated and too difficult to use. The objective we face is to derive effective macroscopic models from complicated multi-scale models. This is where 'multi-scale analysis' and in particular the proposal comes in.

The project will develop new tools in the study of multi-scale models in material sciences, with a focus on a new class of artificial materials called "metamaterials". The role of these tools will be to extract macroscopic properties from multi-scale models of such materials while preserving the key information about the microscale in order to produce simple and accurate models for metamaterials. To achieve this goal I will cast the multi-scale models in question in the mathematical language of "operators" and by using recent advances in operator theory and multi-scale analysis provide a new analytical approach to describing the dominant behaviour of these operators (and, in turn, their model counterparts) resulting from the interaction of the different scales. The proposal will also make advances in the theory of homogenisation, which is a mathematical method used to find effective models of multi-scale systems. The proposal will exploit new advances in homogenisation theory to analyse a broad class of differential equations that model metamaterials.

The proposed work will make advances in the context of multi-scale analysis, an area of great importance in many aspects of modern science, medicine and industry, such as in the understanding and predicting the processes in metal corrosion, catalytic mechanisms and organ function. The fundamental mathematical component of this work is to understand the relationship between microscopic and macroscopic processes that produce such multi-scale phenomena. The mathematical tools developed from this study are likely to be of interest in multi-scale modelling, as they will be able to strictly identify the dominant small-scale behaviour in a given multi-scale process and provide a simplified large-scale model with stharp information about its accuracy.

An equally crucial component of this proposal is the application of these novel tools to a class of emergent materials called "metamaterials". These man-made composites are created to have desirable macroscopic properties by specifying its microscopic structure. The tunable macroscopic properties of metamaterials has lead to metamaterial-based products which are compact, light and more energy efficient than the existing counterpart devices in the commercial, civil and defence sectors. The importance of this emerging industry is further highlighted by noting that the UK government recently began to focus on maintaining the high-level of UK-based research in metamaterials (with increased funding in the "Photonic Materials and Metamaterials" research theme of the EPSRC portfolio).

This proposal is poised to provide long-term benefits by deepening the understanding of the principles that underpin the development of metamaterials. This will be achieved by producing novel mathematical tools that accurately predict the macroscopic behaviour of a metamaterial with prescribed microstructure. Such studies will not only broaden the horizons of metamaterial applications but also provide the blueprints needed for their future synthesis.