Batteries and Solar Cells

Crystalline silicon photovoltaics

The dramatic improvement in the energy output of crystalline silicon photovoltaics has moved this technology from novelty to a key ingredient having a tangible impact on renewable energy sources. This project is an investigation of the electric contact between n-type silicon and the silver electrode in a p-base crystalline photovoltaic cell. Currently, models indicate that the electron flow path is through a thin interfacial glass layer existing between the bulk silicon and silver conductor. A mathematical model is under development, based on drift diffusion equations, for the electron transport through this glassy layer to determine whether "crystalline" or "colloid assisted tunneling" theories best describe the situation.

The first step is a one-dimensional model describing the flow of electrons through a homogeneous glassy layer. This model has been solved and the solutions analysed using a number of asymptotic and numerical techniques. The model predicts that the effective resistance across a glass contact may be a nonmonotonic function of the current.

In the future the model will be developed further by first extending to two dimensions and then considering the effect on the contact resistance of silver precipitates present in the glassy layer. The industrial partner for this project is DuPont (UK) Ltd.

For further information contact Jonathan Black, Chris Breward or Peter Howell.

Key references in this area

  • C Ballif, D Huljic, G Willeke and A Hessler-Wyser (2003). Contact resistance scanning for process optimization: the corescanner method. Applied Phys Letters 82(12): 1878-1880.
  • Z Li, L Liang, A Ionkin, B Fish, L Cheng, K Mikeska (2011). Microstructural comparison of silicon solar cells' front-side Ag contact and the evolution of current conduction mechanisms. Journal of Applied Physics 110(7), 074304.

Mathematical modelling of lithium ion batteries

The lithium-ion battery has long been recognised as a strong candidate for the next generation of clean energy storage. However, its wide application is currently limited by technology barriers including energy storage density, charging/discharging speed, long-term stability, and safety issues. Experimental efforts have been made to search for new materials as well as new structures, but a clear understanding of some basic mechanisms is still lacking. In this project we aim to derive and solve mathematical models of some key aspects of lithium ion battery operation.

Currently we are working on the traditional anode material of graphite. Graphite is composed of sheets of carbon atoms; the lithium intercalates between these sheets. However, this intercalation is non-uniform, as lithium ions in one layer inhibit lithium ions intercalating in neighbouring layers. Our model equation comprises a set of coupled Cahn-Hilliard equations for the phase change as lithium ions enter a given layer, taking into account the interaction between adjacent layers. This model problem has rich dynamics, which we are exploring through numerical and asymptotic methods.

Although simple the model is able to reproduce experimental data concerning the transformation from stage-2 lithium-graphite to stage-1 graphite. In the future we will extend the model to include longer range interactions between layers, a more detailed description of mechanical effects, and perhaps to a continuous description instead of modelling individual discrete layers. In doing so we wish to address performance-limiting aspects of the anode design. Further issues including percolation, temperature effects, material expansion and deformation will also be modelled. The ultimate goal is to relate the micro-scale phenomena to properties that are observable at a macroscopic scale.

For more information please contact Peter Howell or Chang Wang.

Key references in this area

  • Damian Burch, Gogi Singh, Gerbrand Ceder and Martin Z. Bazant (2008). Phase-Transformation Wave Dynamics in LiFePO4. Solid State Phenomena 139: 95-100.
  • Tsutomu Ohzuku, Yasunobu Iwakoshi and Keijiro Sawai (1993). Formation of Lithium-Graphite Intercalation Compounds in Nonaqueous Electrolytes and Their Application as a Negative Electrode for a Lithium Ion (Shuttlecock) Cell. J. Electrochem. Soc. 140(9): 2490-2498.

 

Dynamics of collisonless plasmas

A plasma contains a mixture of positively and negatively charged particles (typically positive ions and electrons) whose net charges almost balance so that the plasma as a whole is approximately electrically neutral. Plasma is sometimes known as the fourth state of matter, after solid, liquid and gas, and is thought to comprise more than 99% of the material in the universe. Plasmas are used for example in fluorescent lightbulbs and in industrial processes such as coating and etching, and highly energetic plasmas are also central to current proposals for generating electricity using nuclear fusion. 

In a highly ionised plasma, the particles interact predominantly through electrostatic forces, and the effects of inter-particle collisions are often negligible. The behaviour of a collisionless plasma may be modelled using a relatively simple kinetic theory, with the distribution of ion velocity v at a position x at time t described by a density function f(x,v,t) which satisfies a partial differential equation known as the Vlasov equation.

As the ion temperature tends to zero, the Vlasov equation apparently reduces to a cold-ion model which resembles the standard equations of gas dynamics. However, in contrast with classical gas dynamics, in one dimension the cold-ion model permits the formation of “spikes” where the ion density tends to infinity in finite time. Through a combination of asymptotic analysis and scientific computation, we have shown that, once a spike has formed, the cold-ion model ceases to provide valid approximations to solutions of the Vlasov equation, which exhibit unexpected multi-modal behaviour, as shown in the figure. This corresponds to fast ions overtaking slower ones, which is impossible in the cold-ion model.

Many open problems remain to be analysed, for example:-

  • What are the conditions for a spike to form?
  • In such cases, when the standard cold-ion model fails, what is the correct limiting model as the ion temperature tends to zero?
  • How can the model be extended into 2 or 3 dimensions?

For more information please contact Peter Howell, John Allen, or John Ockendon.

Key references in this area

  • Mora, P. Plasma expansion into a vacuum. Phys. Rev. Lett. 90 (2003), 185002.
  • Grismayer, T., Mora, P. Influence of a finite initial ion density gradient on plasma expansion into a vacuum. Phys. Plasmas 13 (2006), 032103.
  • Perego, M., Howell, P. D., Gunzburger, M. D., Ockendon, J. R., Allen, J. E. The expansion of a collisionless plasma into a plasma of lower density. Phys. Plasmas 20 (2013), 052101.