Chaotic transitions temperature dependency

Convection in Melt Growth. Convection in the melt is pervasive in all terrestrial melt growth systems. Sources for flows include buoyancy-driven convection caused by the solute and temperature dependence of the density surface tension gradients along melt-fluid menisci forced convection introduced by the motion of solid surfaces, such as crucible and crystal rotation in the CZ and FZ systems and the motion of the melt induced by the solidification of material. These flows are important causes of the convection of heat and species and can have a dominant influence on the temperature field in the system and on solute incorporation into the crystal. Moreover, flow transitions from steady laminar, to time-periodic, chaotic, and turbulent motions cause temporal nonuniformities at the growth interface. These fluctuations in temperature and concentration can cause the melt-crystal interface to melt and resolidify and can lead to solute striations (25) and to the formation of microdefects, which will be described later. 

Much of this Volume deals with the transition phenomena observed in isothermal or temperature dependent reaction sequences, involving appropriate cooperative interactions like autocatalysis, and functioning far from equilibrium. Classical bifurcation phenomena involving the loss of stability of a uniform steady state and the evolution to a limit cycle or a space pattern, abrupt overshoots associated to ignition and explosion, or transition to chaotic dynamics are some characteristic examples. 

Whether the behavior is Mly chaotic or not depends on both the initial conditions and the magnitude of the parameters that appear in the equations of motion. In chemical kinetics these parameters are rate constants and their value can be varied, most simply 1 changing the temperature. En route to chaos the system goes through eye-catching patterns and these are our topic here. For surface reactions such patterns have been extensively observed, particularly in oxidation reactions using O2 or NO as the oxidants on transition metal catalysts. 

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