# Materials: Models and Simulation

Molecular dynamics, stochastic models, optics

### Optimal Control of PDEs for Multiscale Particle Dynamics

The exciting field of optimal control problems involving partial differential equation constraints enables us to gain insight into a wide range of scientific applications, such as biological and chemical mechanisms, opinion formation, fluid flow control, as well as a number of cutting-edge industrial processes. In each case a practitioner has a desired objective, or “target”, in mind, for instance maximising the production of a chemical, or ensuring that a fluid flow behaves in a specific way. In order to achieve this objective, one must first build an accurate mathematical model describing the process, and then devise a potent numerical scheme to solve the resulting system of equations on the discrete level (i.e., on a computer).

Recently, the supervisors have made progress in applying this methodology to industrially-relevant, multiscale particle-dynamics problems. Current application areas include: (i) the control of biophysical processes, such as the brewing of beer or production of biogas in bacterial reactors; (ii) micro/nanofluidics, such as the design of microfiltration devices or the control of nanoparticles for targeted therapeutic drug delivery. There are existing links with a number of local companies, who are keen to collaborate on these topics, if they are of interest to the student. In addition, there are potential collaborators in Biology and Engineering, who are able to provide experimental data. Informal enquiries can be made to Ben Goddard (b.goddard@ed.ac.uk) or John Pearson (j.pearson@ed.ac.uk).

### Mathematics of ultra-fast chemical reactions

Recent experimental advances in light sources and ultrafast lasers have driven the need for the development of theoretical and computational tools to study and understand ultra-fast chemical reactions. In particular, tools are required to bridge the gap between traditional quantum chemistry (describing electrons) and molecular dynamics (describing nuclei). Current computational approaches require vast resources and can describe the dynamics over only very short times. This project will start from existing mathematical models for low-dimensional systems and extend them to higher dimensions, where far more interesting examples can be studied. There are also many practical chemistry questions that can be investigated, and the methodology can also be incorporated into more widely-used techniques. The project combines asymptotics, PDEs and some light numerical components. No knowledge of chemistry is required, but an interest in it, and a desire to work in a truly interdisciplinary field would be advantageous. In particular there are strong links to work by members of the School of Chemistry. Informal enquiries can be made to Ben Goddard (b.goddard@ed.ac.uk).

### Statistical mechanics for reaction-diffusion systems

Many systems in chemistry, biology and engineering can be modelled by reaction-diffusion equations in which the populations of a number of different species move and interact or react, leading to the interchange of population or mass between the species. When the populations are large and/or reside in a complex environment (such as particles suspended in a turbulent flow), a powerful approach is to use techniques from statistical mechanics, which describes the 'average' behaviour of such systems. Dynamical density functional theory is one such approach that has met with great success over the past decade or so. This project will extend existing models, which generally describe only the dynamics, to include the reaction terms. The topics covered can be tailored to the interest of the student, covering both rigorous analysis and numerics. Techniques include statistical mechanics, stochastic dynamics, mathematical modelling, homogenisation theory of PDEs, and computational methods such as pseudo-spectral methods and finite elements. This project also has strong links to the work of members of the School of Engineering. Informal enquiries can be made to Ben Goddard (b.goddard@ed.ac.uk).

### Numerical methods for Brownian dynamics simulations with applications in physics and biology

Brownian dynamics is the method of choice in the simulation of polymers, complex fluids and biomolecules at intermediate scales. In the standard model, a stochastic differential equation is formulated to model the interaction forces between bodies and the effective stochastic interactions with a solvent bath. To model hydrodynamic interactions, the equations of motion must incorporate multiplicative stochastic noise, which just means that noise enters into the equations multiplied by a function of positions. (In difficult cases this function may need to be computed on-the-fly, from atomistic simulations.) The incorporation of multiplicative noise complicates the design of numerical methods and the analysis of numerical error. In this project, the existing schemes for performing Brownian dynamics will first be exhaustively compared and assessed in terms of different quantitative measures of error. Model problems will be used to evaluate the performance of different schemes. Analytical methods will be used to estimate the errors in different regimes and to suggest new numerical methods which can improve the computational performance and accuracy obtained in simulation. Examples for demonstrating the performance of methods will be found in applications in polymer physics and biological systems. Informal enquiries can be made to Ben Leimkuhler (b.leimkuhler@ed.ac.uk).

### Electromagnetic surface waves: optical sensing and solar cell applications

Electromagnetic surface waves may be excited at the interface of two dissimilar materials. Typically, the excitation of these waves is extremely sensitive to the morphology of the interface. Consequently, electromagnetic surface waves represent an attractive proposition for optical sensing applications. Indeed, surface-plasmon-polariton waves - which are electromagnetic surface waves associated with the interface of metals and dielectric materials - are harnessed in highly sensitive optical sensing devices currently used in analytical chemistry and biology. In addition, the excitation of electromagnetic surface waves is associated with a sharp increase in absorption of incident light. For this reason, in recent years surface-plasmon-polariton waves have been considered for possible solar cell applications. Beyond simple metals and dielectric materials, electromagnetically-complex materials offer greater scope for the excitation of electromagnetic surface waves. For example, certain complex materials can support multiple modes of electromagnetic surface waves which are promising for both optical sensing and solar cell applications.Projects are available which involve the theory underpinning electromagnetic surface waves at the interfaces of dissimilar complex materials, with a view to optical sensing and solar cell applications. For further details, please contact Tom Mackay (T.Mackay@ed.ac.uk).

### Stochastic hybrid modelling of chemical systems

It is well known that stochasticity can play a fundamental role in various biochemical processes, such as cell regulatory networks and enzyme cascades. Isothermal, well-mixed systems can be adequately modelled by Markov processes and, for such systems, methods such as Gillespies algorithm are typically employed. While such schemes are easy to implement and are exact, the computational cost of simulating such systems can become prohibitive as the frequency of the reaction events increases. This has motivated numerous coarse grained schemes, where the fast reactions are approximated either using Langevin dynamics or deterministically. While such approaches provide a good approximation for systems where all reactants are present in large concentrations, the approximation breaks down when the fast chemical species exist in small concentrations, giving rise to signicant errors in the simulation. This project will be concerned with developing further and analysing a new hybrid approach (a stochastic dierential equation with jumps) capable of dealing with more general systems. Potential applications in the eld of parameter estimation for chemical systems would also be investigated. Informal enquiries can be made to Konstantinos Zygalakis (kzygalakis@ed.ac.uk).