Industrial Mathematics Group

This project is in collaboration with Dominic Vella, and Janine Nunes who is based in the Complex Fluids Group at Princeton University.

Micro- and nanoscale coiling and wrapping phenomena find themselves in many natural scenarios, from the functionality of a spider web to the packing of RNA within virus capsids. Similar behaviour is at play in manufacturing technologies including soft robotics, flexible electronics devices like bendable smartphones, and in multifunctional soft materials for bioengineering and healthcare applications, such as materials for tissue engineering. The successful manufacture of such soft devices, however, depend on our understanding of the wrapping phenomena and our ability to control it to engineer such products at small length scales.

In this project we will study the mathematics of microscale wrapping using combinations of deformability, fluid flow, electrostatics and surface tension. Our work will be underpinned by two example cases: one inspired by nature and one in technology. The first example case we will study concerns the internal wrapping of fibres in droplets in spider webs. Some spider's webs consist of long threads that are held taught by small scale droplets on them: the surface tension of these droplets is strong enough to 'spool' additional thread within them. This ensures that excess length is reeled into the droplets and the thread remains taut but may also be unravelled if the fibre in the web is stretched (for example when an insect flies into the web). It is thought the presence of droplets also helps to dissipate the fly's kinetic energy without too much stretching.

In this project, we will study the unravelling and reel-back process in spider-web droplets. The second case concerns the external wrapping of microscale charged fibres around oppositely charged soft particles through mutual electrostatic attraction. As they attract one another, the fibre will wrap around the particle, squeezing it and deforming its shape as it does so. The initial configuration and subsequent rate of wrapping have been observed to dictate the shape of the particle and the configuration when the fibre ceases to wrap. In this project, we will investigate the wrapping process and dependence on the deformability of the particle and the fibre, the strength of the electrostatic interaction, and its dependence on the initial configuration. The mathematical models we derive in both of these example cases will be used to compare and contrast the wrapping dynamics and dependence of the behaviour and resulting configurations on the forces that are involved.



(left) A fibre is spooled by a droplet; (right) A fibre wraps around a microscale particle via electrostatic interaction.

This project is in collaboration with Sourav Mondal, Chemical Engineering, Indian Institute of Technology Kharagpur, India and Apala Majumdar, Mathematics and Statistics, University of Strathclyde, Scotland.

Liquid-crystal (LC) thin films may be used to detect biomolecules by studying the change in their birefringence pattern, which is captured using a polarised micrograph. The micrograph is unique for a particular biomolecule and also follows a trend with varying concentration and so may be used as a signature for their detection. The key challenge lies in the stability of the LC layer and the selectivity of detection, which can be improved using external field perturbations (electric, thermal, etc.). In this project, we will develop a mathematical model that relates the birefringence pattern to a particular biomolecule and the concentration under a given electric field. We will work with experimental data to validate the model. We will use the mathematical model to guide the experimental protocol to generate a stable LC configuration to ensure reliable predictions for the biomolecules that are identified and their concentrations. We will then use the model to propose an electric-field distribution in the thin film to enhance the selectivity and detection limit. We also need to understand the impact of the internal domain motion due to the electrical field perturbations on the dynamics of the LC molecules in the film, leading to fluctuations in the final micrograph.



Schematic of the LC-aqueous sensor fabricated in a transmission electron microscopy (TEM) grid characterized with a microscope showing the LC texture.

If you are interested in working on a different project to those mentioned above then you can apply for a DPhil through the Mathematical Institute. More details may be found on the application process and deadlines here.