Out-of-plane buckling in two-dimensional glass drawing

We derive a mathematical model for the drawing of a two-dimensional thin sheet of viscous fluid in the direction of gravity. If the gravitational field is sufficiently strong, then a portion of the sheet experiences a compressive stress and is thus unstable to transverse buckling. We analyse the dependence of the instability and the subsequent evolution on the process parameters, and the mutual coupling between the weakly nonlinear buckling and the stress profile in the sheet. Over long time scales, the sheet centreline ultimately adopts a universal profile, with the bulk of the sheet under tension and a single large bulge caused by a small compressive region near the bottom, and we derive a canonical inner problem that describes this behaviour. The large-time analysis involves a logarithmic asymptotic expansion, and we devise a hybrid asymptotic–numerical scheme that effectively sums the logarithmic series.

Fabrication and Use of High-Speed, Concentric H+- and Ca2+-Selective Microelectrodes Suitable for In Vitro Extracellular Recording

Ion-selective microelectrodes (ISMs) have been used extensively in neurophysiological studies. ISMs selective for H+ and Ca2+ are notable for their sensitivity and selectivity, but suffer from a slow response time, and susceptibility to noise because of the high electrical resistance of the respective ion exchange cocktails. These drawbacks can be overcome by using a “coaxial” or “concentric” inner micropipette to shunt the bulk of the ion exchanger resistance. This approach was used decades ago to record extracellular [Ca2+] transients in cat cortex, but has not been subsequently used. Here, we describe a method for the rapid fabrication of concentric pH- and Ca2+-selective microelectrodes useful for extracellular studies in brain slices or other work in vitro. Construction was simplified compared with previous implementations, by using commercially available, thin-walled borosilicate glass, drawing an outer barrel with a rapid taper (similar to a patch pipette), and by use of a quick and reliable silanization procedure. Using a piezoelectric stepper to effect a rapid solution change, the response time constants of the concentric pH and Ca2+-electrodes were 14.9 ± 1.3 and 5.3 ± 0.90 ms, respectively. Use of these concentric ISMs is demonstrated in rat hippocampal slices. Activity-dependent, extracellular pH, and [Ca2+] transients are shown to arise two- to threefold faster, and attain amplitudes two- to fourfold greater, when recorded by concentric versus conventional ISMs. The advantage of concentric ISMs for studies of ion transport and ion diffusion is discussed.