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Published on November 29th, 2013 | by Daphane Ng

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Understanding fibre extrusion using mathematics

Written by Hayden Tronnolone

Optical fibres are thin threads of material capable of transmitting light through their interior, making them invaluable to modern communications due to their ability to transmit data at high speeds. Conventional optical fibres direct light using layers of materials with different refractive indices; however, this design limits the range of optical properties that can be achieved.

A microstructured optical fibre (MOF) is a relatively new fibre design that uses only a single material (often glass), instead directing light using air channels that run along the length of the fibre. Flexibility in the pattern of air channels allows the construction of fibres with highly adaptable and previously unrealised optical properties, allowing improved data transmission and the creation of highly sensitive bio or chemical sensors.

MOFs are typically constructed in two stages: (1) the production of a large-scale version of the fibre, called a preform; and (2) the pulling of the preform into a long fibre. The preform is typically around 30 centimetres in length with a radius of a few millimetres, and is pulled into a fibre around one kilometre long with a radius of only a few micrometres. A particularly flexible technique for constructing preforms is the extrusion method, in which a block of glass is heated until molten and then forced through a die, giving rise to the pattern of air channels within the preform.

In theory, the extrusion method allows the production of almost any specified design; however, in practice the process presents considerable challenges. The molten glass moves like a very thick fluid, similar to honey, resulting in distortions within the air channels that arise between the time the preform leaves the die and its solidification. These include changes in size, shape and position, or the closure of air channels altogether, all of which can render the fibre useless.

At present MOFs are created through a laborious trial-and-error process. As part of my PhD I have been attempting to create a mathematical model of the fluid flow that arises during the extrusion process in order to provide a better understanding of the observed deformations. This involves writing down and solving equations that represent the physical processes that may arise during extrusion. The results are then be compared with experiments to determine how well the models capture the true physics. Eventually, these models will be able to guide the construction of any fibre design.

The process of creating a good mathematical model requires much careful thought. If the the model includes too many features the resulting equations can be impossible to solve. On the other hand, if the model is too simple it will not provide an accurate description of the process being studied. Counter to the conventional view of mathematical problems as having simple unequivocal answers, creating a good model requires much creative thinking and reasoning. Unfortunately, this side to mathematics is often overlooked.

In order to provide an insight into aspect side of mathematics I created a website that outlines my early attempts at modelling the extrusion process described above. The website describes the modelling process, explains the mathematics used, analyses the results and even features a short quiz at the end to test your understanding. You can have a look today using the link above, see how you score and hopefully gain a new understanding of what maths is used for.

Biography
Hayden is a PhD candidate in the School of Mathematical Sciences at the University of Adelaide under the supervision of Dr Yvonne Stokes (UA) and Professor Darren Crowdy (Imperial College London). Hayden has both a bachelor’s and honours degree in mathematical sciences.

You can visit his website here: http://maths.adelaide.edu.au/hayden.tronnolone/project/[subscribe2]




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