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My PhD in 5 Pictures: Eleanor Russell

12/14/2021

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In this series our group members pick five images that each illustrate an aspect of their PhD.

Eleanor Russell, 3rd Year PhD student in Applied Mathematics.

My thesis title: Thermal metamaterials and heat spreaders

What does this mean?
  
Heat spreaders: Our funnel-shaped heat spreader designs act as point-to-plane heat source converters. The material properties of each spreader guide the flow of heat in a spatially uniform manner such that, when forced by a small thermal source at the base, a uniform temperature is achieved across the larger top surface. Anisotropy (directional dependence) and inhomogeneity (spatial dependence) are typically needed to achieve these desired effects. We design and employ thermal metamaterials together with other techniques to obtain the necessary anisotropic and inhomogeneous properties.
  
Metamaterials: Meta is a Greek word meaning beyond, and so the concept of metamaterials refers to materials that are beyond that of the natural world so to speak. The unique, engineered properties of metamaterials means that we can manipulate and control physical fields (such as electromagnetic, acoustic and elastic wave fields) in new and exciting ways - the results of which are unachievable with natural materials alone! In my PhD we focus on the manipulation of thermal fields with thermal metamaterials, the properties of which are determined through transformation theory.

Transformation theory:  In the context of my research, transformation theory is based on the heat equation being form invariant after a spatial transformation. To apply the theory for a given problem we:
  • Consider what is needed in terms of manipulation.
  • Apply an appropriate transformation between a virtual and physical space.
  • Measure how much this transformation stretches (or compresses) the virtual space using the Jacobian matrix. (See A-to-Z)
  • Use the Jacobian matrix to determine the properties of the thermal metamaterial that achieves the desired manipulation.
  
These properties are often anisotropic and inhomogeneous. We approximate this behaviour with layered, composite materials whilst also considering interfacial effects.
1
Illustration of the transformation process used to design metamaterials. Here we show an example of a ground cloak transformation where a virtual space with coordinates \(\mathbf{x}’=[x’,y’]\) is compressed upwards to form the physical space with coordinates \(\mathbf{x}=[x,y]\). This mapping from \(\mathbf{x}'\) to \(\mathbf{x}\) creates a triangular void in the physical space. We measure how much this transformation compresses each point in the virtual space using the Jacobian, denoted by \(\mathbf{F}\). From this we can determine the necessary properties for the metamaterial regions, for example, in my PhD we focus on steady-state thermal conduction and so we only need to engineer the thermal conductivity, which is given by
\begin{equation}
\mathbf{k} = \dfrac{\mathbf{F}k'\mathbf{F}^T}{\det\mathbf{F}}\tag{1}         ,
\end{equation}
where \(k’\) is the conductivity in the virtual domain. When the conductivity in the transformed, metamaterial regions satisfy (1), any object placed within the void is undetectable from the external field – making it invisible! The effects of a traditional ground cloak are simulated in image 2. In my PhD I use a modified version of this mapping to utilize it in a new, heat spreading context. A simulation of this application is provided in the image 3.
2
Simulation of a traditional ground cloak application:
a) The temperature field for a given material and set of boundary conditions. Black lines represent isotherms.

b) Same material and boundary conditions, but with part of the domain removed. This triangular void causes disturbances, referred to as perturbations, to the temperature field. We can use these perturbations to detect the apple.
​
c) Same material and boundary conditions, but with triangular void surrounded by an anisotropic ground cloak. The unique properties inside the ground cloak counteract the perturbations caused by the void. As a result, we can no longer detect the apple – making it invisible! 
3
Simulation of a modified ground cloak mapping being used in a new, heat spreading context.
​a) The temperature field through a funnel-shaped design when a natural, isotropic material is used. White lines represent isotherms and show that a uniform temperature is not naturally achieved across the top surface when the funnel is forced by a heat source at its base.
​  
b) The temperature field through a heat spreader designed using a modified ground cloak. The design has three components: an upper isotropic component that is unaffected by the mapping; and two lower metamaterial layers that counteract the perturbations caused by the funnel-shaped geometry. The unique properties in the metamaterial layers guide the flow of heat through the spreader in spatially uniform manner, leading to uniform temperature across the top surface, as desired.
​Ground cloaks are a particularly nice example of thermal metamaterials as their properties are anisotropic, but not homogeneous. Therefore we can accurately approximate them with a laminated design, which can be seen in image 4. 
4
The realisation process for the metamaterial in image 3(b), which can be approximated with a rotated laminate. 
​a) Illustration of the composite design consisting of alternating layers aligned with the principal axes of the system. The thickness and conductivity of each layer are determined through effective medium theory.  

b) Simulation of this design using 5 bilayers. The temperature field here approximates that of the metamaterial simulation in image 3(b). This design better represents the true metamaterial as we increase the number of bilayers. A sufficiently large number of layers would eliminate temperature variation across the top surface completely.
​Although not shown here, we have extended this model (and the other models in our research) to include interfacial effects. These effects can dominate in multi-layered designs such as laminates.
5
Although PhD students obviously spend most of their time researching, this is not the only thing we do! A PhD can involve many responsibilities, for example, during my PhD I have spent time:
  • Being a teaching assistant for undergraduate courses.
  • Attending conferences to present work and network with other researchers in my field.
  • Creating online material for outreach and widening participation – like writing blog posts and A-to-Z entries for this website!
  • Helping at science festivals to promote the research focuses of the Mathematics of Waves and Materials group.
In the image below there is a photo of me helping at Bradford Science Festival earlier this year. The main aim of this weekend was to get primary school children excited about maths and science by explaining some simple concepts about sound. The next image shows a simple maze game that I designed on Scratch to help demonstrate the effects of different thermal metamaterials. The final image shows a photo I took of the Utah State Captiol when I attended a conference in Salt Lake City back in 2019.
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Sophie Germain

3/8/2020

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Picture
On International Women's Day in the International Year of Sound we want to acknowledge the extraordinary life and work of mathematician and philosopher Marie-Sophie (Sophie) Germain (1776-1831).

The daughter of a bourgeois Parisian silk merchant, Sophie’s early interest in mathematics grew during the reign of terror, which kept her confined to the family home.  Sophie found intellectual stimulation in her father’s library, and spent long hours studying mathematics, Greek and Latin; teaching herself the latter in order to understand texts by Newton and Euler. This interest in intellectual pursuits was initially strongly disapproved of by her parents, who removed her candles and her fire in a vain attempt to discourage her studies.  They eventually relented, however, realising that their daughter was serious in her work.
​

Image source: http://mathshistory.st-andrews.ac.uk/PictDisplay/Germain.html

As a young woman, Sophie was barred by her sex from attending the newly established Ecole Polytechnique, but was able to obtain the lecture notes and submit work under the borrowed name of Monsieur Leblanc, a former student who had left the city.  She was to use this same pseudonym in her initial correspondence with mathematicians including Legendre, Lagrange and Gauss, fearing that as a woman, she would not be taken seriously. On discovering Germain’s true identity, however, Gauss responded with admiration for her tenacity.

Germain was particularly interested in number theory, but a visit to Paris by Ernst Chladni in 1808 turned her attention in another direction.  Chladni reproduced his classic experiment, producing nodal figures in sand on vibrating plates. In response, in 1809 the Institut de France offered a kilo of gold to any person who could formulate a mathematical theory of elastic surfaces.

Picture
Germain submitted her first anonymous entry to the competition in 1811 (the only submission received), but her work, while radical in approach, was flawed, and the competition extended a further two years.  Her second entry was given an honourable mention, but contained errors and was deemed insufficient for the prize. Finally, in 1816, Germain submitted an entry in her own name which, although still imperfect, was awarded the prize.  The Institut did not publish her paper, however, and Germain later published it herself, pointing out the errors in her work (Recherche sur la théorie des surfaces élastiques,  1821). 

It is likely that, being self taught, and lacking in formal education and guidance, there were gaps in Germain’s knowledge that affected her work in mathematical physics more than her work in number theory.  However, this makes her achievements in the field all the more extraordinary.  Germain took a different approach to that of her rival in the early days of the challenge, Siméon-Denis Poisson, whose initial work on the problem was based on molecular theory.  Poisson was admitted to the institute early in the competition (something not permitted to Germain) and became a judge rather than a competitor.  As such, he had access to Germain’s work, and she consulted with him on the subject. However Poisson did not regard Germain as a serious scholar, and did not acknowledge her in his own 1814 work on elasticity.
​
In later life, Germain continued her work in both number theory and elasticity.  She contributed to the body of work on Fermat’s Last Theorem with what became known as “Sophie Germain’s Theorem”, published in a supplement to the second edition of Legendre’s Théorie des Nombres.  Before her death from breast cancer in 1831, Germain published her work on surface curvature, Mémoire sur la courbure des surfaces.  After the intervention of Gauss, Sophie Germain was eventually awarded a posthumous honorary degree from the University of Göttingen.

References and further reading
  • Del Centina, A., Fiocca, A., (2012), ‘The correspondence between Sophie Germain and Carl Friedrich Gauss’, Arch. Hist. Exact Sci. 66:585–700
  • Laubenbacher, R., Pengelley, D.,  (2010), ‘Voici ce que j’ai trouvé: Sophie Germain’s grand plan to prove Fermat’s Last Theorem’, Historia Mathematica 37 (2010) 641–692
  • Frize, M. (2010). The Bold and the Brave: A History of Women in Science and Engineering. Ottawa: University of Ottawa Press.
  • GRAY, M. (2005). Sophie Germain. In CASE B. & LEGGETT A. (Eds.), Complexities: Women in Mathematics (pp. 68-74). Princeton; Oxford: Princeton University Press.
  • Bucciarelli L.L., Dworsky N. (1980) An Award with Reservations. In: Sophie Germain. Studies in the History of Modern Science, vol 6. Springer, Dordrecht.
  • https://scientificwomen.net/women/germain-sophie-39
  • http://mathshistory.st-andrews.ac.uk/Biographies/Germain.html
  • https://simonsingh.net/books/fermats-last-theorem/sophie-germain/
  • https://www.thoughtco.com/sophie-germain-biography-3530360

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    The Mathematics of Waves and Materials group are are a research group in the Department of Mathematics at the University of Manchester.  We work on the theoretical, numerical and experimental aspects of both materials and waves. See our research page for more information and details.

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