Isomerization dynamics of a buckled nanobeam

We analyze the dynamics of a model of a nanobeam under compression. The model is a two-mode truncation of the Euler-Bernoulli beam equation subject to compressive stress applied at both ends. We consider parameter regimes where the first mode is unstable and the second mode can be either stable or unstable, and the remaining modes (neglected) are always stable. Material parameters used correspond to a silicon nanobeam. The two-mode model Hamiltonian is the sum of a (diagonal) kinetic energy term and a potential energy term. The form of the potential energy function suggests an analogy with isomerization reactions in chemistry, where "isomerization" here corresponds to a transition between two stable beam configurations. We therefore study the dynamics of the buckled beam using the conceptual framework established for the theory of isomerization reactions. When the second mode is stable the potential energy surface has an index one saddle, and when the second mode is unstable the potential energy surface has an index two saddle and two index one saddles. Symmetry of the system allows us to readily construct a phase space dividing surface between the two "isomers" (buckled states); we rigorously prove that, in a specific energy range, it is a normally hyperbolic invariant manifold. The energy range is sufficiently wide that we can treat the effects of the index one and index two saddles on the isomerization dynamics in a unified fashion. We have computed reactive fluxes, mean gap times, and reactant phase space volumes for three stress values at several different energies. In all cases the phase space volume swept out by isomerizing trajectories is considerably less than the reactant density of states, proving that the dynamics is highly nonergodic. The associated gap time distributions consist of one or more "pulses" of trajectories. Computation of the reactive flux correlation function shows no sign of a plateau region; rather, the flux exhibits oscillatory decay, indicating that, for the two-mode model in the physical regime considered, a rate constant for isomerization does not exist.