The mathematics of glass sheets - how to make their thickness uniform

Oxford Mathematician Doireann O'Kiely was recently awarded the IMA's biennial Lighthill-Thwaites Prize for her work on the production of thin glass sheets. Here Doireann describes her work which was conducted in collaboration with Schott AG.

"Thin glass sheets have many modern applications, including touch-screens, cameras and thumbprint sensors for smartphones. Glass sheets with thicknesses in the range 50–100µm are flexible, and may be used in bendable devices.

In the glass sheet redraw process, a prefabricated glass sheet is fed through a heater and stretched. When the glass is hot it behaves as a viscous fluid. As the glass is stretched, it gets thinner and the edges of the sheet are pulled in. This combined response means that both the thickness and the width of the sheet decrease, and the cross-section of the sheet can change shape so that the final product may not have uniform thickness.

In industrial processes, the heater is typically short compared to the sheet width to minimize width reduction and yield desirable thin, wide glass sheets. However, sheets produced in this way are typically thicker at the edge than elsewhere (see image). Asymptotic analysis of the process in this limit indicates that the behaviour in the main part of the sheet is one-dimensional – it varies only in the direction of motion – and there is a two-dimensional boundary layer near the sheet edge.

Numerical solution of the boundary-layer problem illustrates that the glass in the path of the inward-moving edge accumulates, leading to the observed thick edges. The same numerical scheme can also be used to determine the modified input shape required for the manufacture of a uniformly thin sheet. Physically, a small region at the edge of the sheet is tapered, making it thinner to compensate for the accumulation of glass during redraw."

The image above shows that the redrawn glass sheet is extremely thin, but is also relatively thick in a localised region near the sheet edge. Photo by Dominic Vella.