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Phys. Rev. E 86 , 056218 PHYSICAL REVIEW E 86, 056218 (2012) Isomerization dynamics of a buckled nanobeam Peter Collins School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom Gregory S. Ezra* Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York 14853, USA Stephen Wiggins† School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom (Received 14 June 2012; published 29 November 2012) 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. DOI: 10.1103/PhysRevE.86.056218 PACS number(s): 05.45.−a, 82.20.−w, 62.25.Fg I. INTRODUCTION the modes is always unstable, while the second mode can be either stable or unstable, and all remaining modes (which are There is currently much interest in the mechanical proper- not considered explicitly in our model) are stable. ties of nanoscale objects such as rods and cantilevers [1–4]. For Of course, it is natural to ask how well such a two-mode example, changes in resonant frequencies with mass loading Galerkin truncation of the solution of the Euler-Bernoulli beam can enable sensitive detection of molecular species with given equation models the solution of the full partial differential mass. The possible manifestation of quantum effects are also of great interest (see Ref. [5] for a recent review). The effects equation describing the Euler-Bernoulli beam. If one considers of mechanical stress on nanostructures, such as the buckling the eigenfunctions (“modes”) for the linearized the Euler- of nanobeams, has been studied both experimentally and Bernoulli equation (linearized about some equilibrium solu- theoretically [6–8]. This latter work is directly related to the tion, which in this case, is the straight, uncompressed beam) work in this paper. then there are parameter regimes in which only a “few” modes The system studied in the present paper is a nanobeam are unstable, with the remaining (infinite) modes being stable. subject to compressive stress. According to standard contin- In particular, the number of unstable modes can be chosen uum mechanics, the beam will buckle as the magnitude of according the the magnitude of the compressive stress, as we the associated compressive strain increases. More specifically, describe in Sec. II. It is a standard engineering approach to following previous work [9–11], we consider physical parame- approximate the full solution of a partial differential equation ters for which the dynamics of the buckled beam is accurately by considering an expansion of the full solution in terms of described by a two-mode model obtained by Galerkin trun- these “linearized” modes and then to only retain the terms in the cation of the full dynamics described by the Euler-Bernoulli expansion corresponding to the unstable modes. The physical beam equation. In the parameter regime considered, one of reasoning underlying this approach is that the evolution near the equilibrium solution is dominated by the evolution of the unstable modes. *[email protected] Over the past 30 years, the accuracy of such approx- †[email protected] imations, in the appropriate parameter regimes, has been 1539-3755/2012/86(5)/056218(16)056218-1 ©2012 American Physical Society PETER COLLINS, GREGORY S. EZRA, AND STEPHEN WIGGINS PHYSICAL REVIEW E 86, 056218 (2012) rigorously proven using center manifold or inertial manifold reaction rate theory. The potential function that emerges from a techniques (see, e.g., Refs. [31,32]) in a number of physically standard modal analysis of the transverse beam displacements interested cases. A seminal example of this approach that has a high degree of symmetry, and the appropriate dividing is directly applicable to our work is the work of Holmes, surface for computation of reactive flux is then, in fact, Marsden, and Moon on the dynamics of a buckled beam completely determined by symmetry. Moreover, we are able subject to periodic (temporal) forcing [33–35]. Initially, the to rigorously prove that the dividing surface is a normally work was concerned with a single mode truncation subject hyperbolic invariant manifold in a specific energy range. The to periodic forcing (the classical “forced Duffing oscillator”) energy range is sufficient for treating the effects of the index in a parameter regime where there was only one unstable one and index two saddles on the isomerization dynamics in a mode (with the infinite number of remaining modes being unified fashion. stable). In this setting a combination of experimental and We compute both the distribution of gap times and the theoretical work showed that the experimentally observed reactive flux at a number of energies in a physically relevant chaotic behavior was indeed captured by the evolution of range. Our results show that, in this regime for the Si nanobeam the one unstable mode, subject to forcing [33,34]. Later, it considered: (i) the isomerization dynamics is extremely regular was rigorously shown [35] that this single mode truncation and nonergodic, and (ii) a rate constant for isomerization does captured the dynamics of the full partial differential equation not exist. Rather than exhibiting a rapid drop to a “plateau” governing the beam in the parameter regime with only one value followed by slow exponential decay [14,16,19], the unstable mode. This work is mathematically very technical reactive flux shows damped oscillatory decay. The system and uses partial differential equation and functional analytic considered is in a regime (T 100 K) where quantum effects techniques in an infinite dimensional geometrical setting. A are likely to be negligible. mathematical study along these lines is not the goal of this The structure of the paper is as follows. In Sec. II we derive paper. Rather, we will use the intuition developed from this the equations of motion for the nanobeam. In Sec. III we approach and take as the starting point of our analysis the discuss in further detail the 2-degree-of-freedom Hamiltonian 2-degree-of-freedom (DOF) Hamiltonian system governing system obtained for the two-mode truncation of the beam the two-mode Galerkin truncation of the Euler-Bernoulli beam dynamics. Particular attention is given to the phase space equation in a parameter regime where there are (at most) two geometry, specifically, the existence of a normally hyperbolic unstable modes, with the remaining modes being stable. The invariant manifold (NHIM) [28] in the system phase space. In 2-DOF Hamiltonian obtained in this way describes a bistable Sec. IV we review some concepts from reaction rate theory: (reactive) mode coupled to a transverse degree of freedom; phase space dividing surfaces and volumes, gap times, and the dynamical system thereby obtained has precisely the form reactive fluxes. In Sec. V we discuss the physical parameter of simple model potentials that have been used to describe values and energy scales appropriate to our calculations. isomerization reactions in chemistry [12–20]. However, for Section VI presents the results of our numerical calculations, the nanobeam problem, the simple form of the potential and Sec. VII presents our conclusions. In Appendix we apply is a rigorous consequence of the two-mode truncation of the concept of exponential dichotomies [29] to provide a proof the dynamics rather than an approximation to an unknown of the existence of a NHIM in the phase space of the truncated molecular potential energy function. nanobeam problem. We can therefore investigate the dynamics of the buckled nanobeam using the conceptual framework established for the II. EQUATIONS OF MOTION FOR THE NANOBEAM theory of isomerization reactions. A number of approaches to the study of isomerization dynamics have been developed in In this section we derive a two-mode (classical) model for the chemical literature (see, for example, Refs. [16–18,20,21]). an Euler-Bernoulli beam subject to compressive stress applied Some basic relevant questions are the following: can “iso- at both ends.
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