Mathematical modelling of unsteady tube stretching with internal channel pressurisation for fabricating electrospray ionisation emitters

Abstract

Nanoelectrospray ionisation (nESI) is a useful technology for assessing the chemical composition of various liquid samples using mass spectrometry (MS). Signi cant e orts have been made in the design of nESI emitters, as their shape and geometry are critical to the electrospray performance and subsequent MS detection. In the actual manufacturing of these emitters through the heat and draw process, the desired geometry cannot, at present, be achieved. In particular, the inner channel reduces in size, which is not desirable. To improve the sensitivity of biological and chemical mass spectrometry and avoid clogging of the tip, a small near-uniform bore of 10 - 20 m is desirable with the external wall tapering over a length of around 5mm from 75 - 150 m in radius to a sharp end with a radius around 8 - 15 m. Through mathematical modelling, we demonstrate, for the rst time, the feasibility of producing such emitters using the heat and draw process with the addition of pressure in the channel to prevent any reduction in size. In this thesis, we consider the unsteady problem of heating and pulling of an axisymmetric cylindrical glass tube, using asymptotic methods to exploit the slenderness of the tube and over-pressure applied within the inner channel, to form tapers with a near uniform bore and small wall thickness at the tip. This is an unsteady extensional ow problem. As the glass temperature increases, the viscosity reduces until the central heated region extends and thins rapidly to yield an hour-glass shape. During stretching, the cross-sectional geometry will also deform under the e ects of surface tension and applied pressure, with the pressure counteracting the closure of the channel by surface tension and, perhaps, further expanding it. When cooled and cut transversely at the centre, two identical tapered capillaries are obtained. In this thesis, we assume molten glass is a Newtonian uid, and develop coupled ow and energy models to examine in detail the in uence of the process parameters on the geometry, namely the pulling force, pressure, temperature, and surface tension. The use of an over-pressure in the channel, to counteract the reduction in its size as the crosssectional area decreases due to pulling and the channel closes due to surface tension, is of particular interest. The model and solution method described in this thesis enable determination of a pulling force, channel over-pressure, and draw time to achieve tapers with the desired internal diameter and wall thickness at the very tip from a given tubular bre for a temperature dependent viscosity.