Mathematical model of laser modification of nanomaterials within the framework of catastrophe theory

Abstract

A nonlinear mathematical model of the interaction of laser radiation with nanoscale objects, based on the formalism of catastrophe theory, is proposed. It is shown that the modification of matter under ultrashort pulses is described by rearrangements of the potential function in the space of control parameters: energy density, pulse duration, and nanostructure size. Catastrophes of the cusp (A₃) and swallowtail (A₅) types are identified, which are responsible for abrupt order – disorder and melting-fragmentation transitions. Bifurcation sets that determine the stability boundaries of structural states are constructed. Special attention is paid to the size-dependent parameter associated with surface energy and Laplace pressure: as the particle size decreases, the system enters a regime that allows direct fragmentation without a melting stage. Numerical simulation of the dynamics of femtosecond pulses acting on thin vanadium dioxide films and gold nanoparticles is carried out, and the obtained regime maps are compared with known experimental data. It is established that the pulse duration acts as an additional control parameter of dynamic bifurcation. The possibility of predicting the morphology of the final state of a nanosystem along the trajectory of control parameters, without resorting to resource-intensive atomistic modeling, is demonstrated. The proposed approach can serve as a basis for technological maps of laser processing of nanomaterials.