Bubble Growth From First Principals

We consider the simplest relevant problem in the foaming of molten plastics, the growth of a single bubble in a sea of highly viscous Newtonian fluid, and without interference from other bubbles. This simplest problem has defied accurate solution from first principles. Despite plenty of research on foaming, classical approaches from first principles have neglected the temperature rise in the surrounding fluid, and we find that this oversimplification greatly accelerates bubble growth prediction. We use a transport phenomena approach to analyze the growth of a solitary bubble, expanding under its own pressure. We consider a bubble of ideal gas growing without the accelerating contribution from mass transfer into the bubble. We explore the roles of viscous forces, fluid inertia, and viscous dissipation. We find that bubble growth depends upon the nucleus radius and nucleus pressure. We begin with a detailed examination of the classical approaches (thermodynamics without viscous heating). Our failure to fit experimental data with these classical approaches, sets up the second part of our paper, a novel exploration of the essential decelerating role of viscous heating. We explore both isothermal and adiabatic bubble expansion, and also the decelerating role of surface tension. The adiabatic analysis accounts for the slight deceleration due to the cooling of the expanding gas, which depends on gas polyatomicity. We also explore the pressure profile, and the components of the extra stress tensor, in the fluid surrounding the growing bubble. These stresses can eventually be frozen into foamed plastics. We find that our new theory compares well with measured bubble behavior.

A Mineralogical Study of a Dravitic Tourmaline from the Grenville Province

Tourmaline from a gem-quality deposit in the Grenville province has been studied with X-ray diffraction, visible-near infrared spectroscopy, Fourier transform infrared spectroscopy, scanning electron microscopy, electron microprobe and optical measurements. The tourmaline is found within tremolite-rich calc-silicate pods hosted in marble of the Central Metasedimentary Belt. The crystals are greenish-greyish-brown and have yielded facetable material up to 2.09 carats in size. Using the classification of Henry et al. 2011 the tourmaline is classified as a dravite, with a representative formula shown to be (Na0.73Ca0.2380.032)(Mg2+2.913Fe2+0.057Ti4+0.030) (Al3+5.787Fe3+0.017Mg2+0.14)(Si6.013O18)(BO3)3(OH)3((OH,O)0.907F0.093). Rietveld analysis of powder diffraction data gives a = 15.9436(8) Å, c = 7.2126(7) Å and a unit cell volume of 1587.8 Å3. A polished thin section was cut perpendicular to the c-axis of one tourmaline crystal, which showed zoning from a dark brown core into a lighter rim into a thin darker rim and back into lighter zonation. Through the geochemical data, three key stages of crystal growth can be seen within this thin section. The first is the core stage which occurs from the dark core to the first colourless zone; the second is from this colourless zone increasing in brown colour to the outer limit before a sudden absence of colour is noted; the third is a sharp change from the end of the second and is entirely colourless. These events are the result of metamorphism and hydrothermal fluids resulting from nearby felsic intrusive plutons. Scanning electron microscope, and electron microprobe traverses across this cross-section revealed that the green colour is the result of iron present throughout the system while the brown colour is correlated with titanium content. Crystal inclusions in the tourmaline of chlorapatite, and zircon were identified by petrographic analysis and confirmed using scanning electron microscope data and occur within the third stage of formation.