Publications

Computing the multimodal stochastic dynamics of a nanobeam in a viscous fluid

Abstract: The stochastic dynamics of small elastic objects in fluid are central to many important and emerging technologies. It is now possible to measure and use the higher modes of motion of elastic structures when driven by Brownian motion alone. Although theoretical descriptions exist for idealized conditions, computing the stochastic multimodal dynamics for the complex conditions of experiment is very challenging. We show that this is possible using deterministic finite element calculations with the fluctuation dissipation theorem by exploring the multimodal stochastic dynamics of a doubly-clamped nanobeam. We use a very general, and flexible, finite-element computational approach to quantify the stochastic dynamics of multiple modes simultaneously using only a single deterministic simulation. We include the experimentally relevant features of an intrinsic tension in the beam and the influence of a nearby rigid boundary on the dynamics through viscous fluid interactions. We quantify the stochastic dynamics of the first eleven flexural modes of the beam when immersed in air or water. We compare the numerical results with theory, where possible, and find excellent agreement. We quantify the limitations of the computational approach and describe its range of applicability. These results pave the way for computational studies of the stochastic dynamics of complex 3D elastic structures in a viscous fluid where theoretical descriptions are not available.

Citation: J. Barbish and M. R. Paul, "Computing the multimodal stochastic dynamics of a nanobeam in a viscous fluid" Journal of Applied Physics, vol. 136, no. 23, p. 234502, Dec. 2024.

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Using Covariant Lyapunov Vectors to Quantify High Dimensional Chaos with a Conservation Law

Abstract: We explore the high-dimensional chaos of a one-dimensional lattice of diffusively coupled tent maps using the covariant Lyapunov vectors (CLVs). We investigate the connection between the dynamics of the maps in the physical space and the dynamics of the covariant Lyapunov vectors and covariant Lyapunov exponents that describe the direction and growth (or decay) of small perturbations in the tangent space. We explore the tangent space splitting into physical and transient modes and find that the splitting persists for all of the conditions we explore. In general, the leading CLVs are highly localized in space and the CLVs become less localized with increasing Lyapunov index. We consider the dynamics with a conservation law whose strength is controlled by a parameter that can be continuously varied. Our results indicate that a conservation law delocalizes the spatial variation of the CLVs. We find that when a conservation law is present, the leading CLVs are entangled with fewer of their neighboring CLVs than in the absence of a conservation law.

Citation: J. Barbish, and M. R. Paul, “Using Covariant Lyapunov Vectors to Quantify High Dimensional Chaos with a Conservation Law," Physical Review E, vol. 108, p. 054202 Nov. 2023.

Download here: here. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in J. Barbish, and M. R. Paul, “Using Covariant Lyapunov Vectors to Quantify High Dimensional Chaos with a Conservation Law," Physical Review E, vol. 108, p. 054202 Nov. 2023. and may be found at https://doi.org/10.1103/physreve.108.054202.

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Multi-Mode Brownian Dynamics of a Nanomechanical Resonator in a Viscous Fluid

Abstract: Brownian motion imposes a hard limit on the overall precision of a nanomechanical measurement. Here, we present a combined experimental and theoretical study of the Brownian dynamics of a quintessential nanomechanical system, a doubly clamped nanomechanical beam resonator, in a viscous fluid. Our theoretical approach is based on the fluctuation-dissipation theorem of statistical mechanics: we determine the dissipation from fluid dynamics; we incorporate this dissipation into the proper elastic equation to obtain the equation of motion; and the fluctuation-dissipation theorem then directly provides an analytical expression for the position-dependent power spectral density (PSD) of the displacement fluctuations of the beam. We compare our theory to experiments on nanomechanical beams immersed in air and water and obtain excellent agreement. Within our experimental parameter range, the Brownian-force noise driving the nanomechanical beam has a colored PSD due to the “memory” of the fluid; the force noise remains mode independent and uncorrelated in space. These conclusions are not only of interest for nanomechanical sensing but also provide insight into the fluctuations of elastic systems at any length scale.

Citation: H. Gress, J. Barbish, C. Yanik, I. I. Kaya, R. T. Erdogan, M. S. Hanay, M. Gonzalez, O. Svitelskiy, M. R. Paul, and K. L. Ekinci, “Multi-mode Brownian Dynamics of a Nanomechanical Resonator in a Viscous Fluid," Physical Review Applied, vol. 20, no. 4, p. 044061, Oct. 2023.

Download here: here. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in H. Gress, J. Barbish, C. Yanik, I. I. Kaya, R. T. Erdogan, M. S. Hanay, M. Gonzalez, O. Svitelskiy, M. R. Paul, and K. L. Ekinci, “Multi-mode Brownian Dynamics of a Nanomechanical Resonator in a Viscous Fluid," Physical Review Applied, vol. 20, no. 4, p. 044061, Oct. 2023. and may be found at https://doi.org/10.1103/physrevapplied.20.044061.

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The Dynamics of an Externally Driven Nanoscale Beam that is Under High Tension and Immersed in a Viscous Fluid

Abstract: We explore the dynamics of a nanoscale doubly clamped beam that is under high tension, immersed in a viscous fluid, and driven externally by a spatially varying drive force. We develop a theoretical description that is valid for all possible values of tension, includes the motion of the higher modes of the beam, and accounts for a harmonic force that is applied over a limited spatial region of the beam near its ends. We compare our theoretical predictions with experimental measurements for a nanoscale beam that is driven electrothermally and immersed in air and water. The theoretical predictions show good agreement with experiments, and the validity of a simplified string approximation is demonstrated.

Citation: J. Barbish, C. Ti, K. L. Ekinci, and M. R. Paul, "The Dynamics of an Externally Driven Nanoscale Beam that is Under High Tension and Immersed in a Viscous Fluid" Journal of Applied Physics, vol. 132, no. 3, p. 034501, Jul. 2022.

Download here: here. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in J. Barbish, C. Ti, K. L. Ekinci, and M. R. Paul, "The Dynamics of an Externally Driven Nanoscale Beam that is Under High Tension and Immersed in a Viscous Fluid" Journal of Applied Physics, vol. 132, no. 3, p. 034501, Jul. 2022. and may be found at https://doi.org/10.1063/5.0100462.

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