טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentHalioua Guy
SubjectWaves of Collapse in Nondegenerate Chain Arrays
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Oleg Gendelman
Full Thesis textFull thesis text - English Version


Abstract

The thesis is devoted to the study of front propagation in non-degenerate bistable nonlinear chains and lattices. Models of this type turn out to be relevant for simulation of collapse in carbon nanotubes, as well as for many other processes: martensite transitions, chemical reactions in solid state etc.

Analysis of this process is rather challenging. The velocity of the front may be close or even can exceed a sound velocity in the system; therefore, traditional approaches based on continuum approximation cannot be used. Besides, the process of the front propagation cannot be stationary. The energy is released as the transition occurs at every site, and should be removed from the front zone. In current study, we reveal a formation of oscillatory tail with varying length, which absorbs the energy of the oscillations and removes it from the front propagation zone. Certain analytic approximations are available only for the simplest case of linear chain subject to double-parabolic on-site potential. For this reason we primarily rely on numeric methods.

As for one-dimensional chains, we extend known simplified models by addition of gradient nonlinearity into the chain. It was demonstrated that even small nonlinearity drastically modifies the process and increases the velocity of the front propagation. The oscillatory tail becomes smooth (corresponds to lower wave-vectors) and exhibits instabilities related to excitations of discrete breathers. Such instabilities are not possible in the linear model.

Further research efforts were devoted to a two-dimensional model, in which the chains are coupled by Lennard - Jones potential bonds. In this model the front propagation is also observed. The conditions for the front initiation turned out to be somewhat unexpectable and nontrivial. For instance, it is not enough to "ignite" initially only one chain - the front will be arrested by the neighboring chains. Still, if a few neighboring chains are ignited simultaneously; the reaction front gradually arises at all chains in the model crystal.