I am a theoretical physicist working on the effects of topology and geometry in condensed matter physics. In particular, I am interested in packing problems, foams and the role played by conformal geometries in these systems.
How bees and foams respond to curved confinement: level set boundary representations in the Surface Evolver
A. Mughal, T. Libertiny, G. E. Schroeder-Turk arXiv:1611.10055 (2016).
We present a Surface Evolver framework for simulating single bubbles and multicellular foams trapped between curved parallel surfaces. We are able to explore a range of geometries using level set constraints to model the bounding surfaces. Unlike previous work, in which the bounding surfaces are flat (the so called Hele-Shaw geometry), we consider surfaces with non- vanishing Gaussian curvature, specifically the sphere, the torus and the Schwarz Primitive-surface. In the case of multi-cellular foams - our method is to first distribute a set of N points evenly over the surface (using an en- ergy minimisation approach), these seed points are then used to generate a Voronoi partition, that is clipped to the confining space, which in turn forms the basis of a Surface Evolver simulation. In addition we describe our ex- perimental attempt to generate a honeycomb on a negatively curved surface, by trapping bees between two Schwarz Primitive-surfaces. Our aim is to understand how bees adapt the usual hexagonal motif of the honeycomb to cope with a curved surface. To our knowledge this is the first time that an attempt has been made to realise a biological cellular structure of this type.
Attempt to build a bee honeycomb on a gyroid.
Curvature driven motion of soap cells in toroidal Hele-Shaw cells
A. Mughal, S. J. Cox, G. E. Schroeder-Turk arXiv:1610.09239 (2016).
We investigate the equilibrium properties of a single area-minimising bubble trapped between two narrowly-separated parallel curved plates. We begin with the simple case of a a bubble trapped between concentric spherical plates. We develop a model that shows that the surface tension energy of the bubble is lower when confined between spherical plates as compared to a bubble trapped between flat plates. We confirm our findings by comparing against Surface Evolver simulations. Next, we derive a simple model for a bubble between arbitrarily curved parallel plates. The energy is found to be higher when the local Gaussian curvature of the plates is negative and lower when the curvature is positive. To check the validity of the the model we consider a bubble trapped between concentric tori. In the toroidal case we find that the sensitivity of the bubble's energy to the local curvature acts as a geometric potential capable of driving bubbles from regions with negative to positive curvature.
A geodisic circle in a region of positive curvature
Phyllotaxis, disk packing and Fibonacci numbers
A. Mughal and D. Weaire arXiv:1608.05824 (2016).
We consider the evolution of the packing of disks (representing the position of buds) that are introduced at the top of a surface which has the form of a growing stem. They migrate downwards, while conforming to three principles, applied locally: dense packing, homogeneity and continuity. We show that spiral structures characterised by the widely observed Fibonacci sequence (1,1,2,3,5,8,13...), as well as related structures, occur naturally under such rules. Typical results are presented in a animation.
Projection of cylindrical disk packings on to a stem