Effect of flexible supports on the frequencies of nanobeams with tip mass and axial load for applications in atomic force microscopy (AFM)

Doctoral Degree. University of KwaZulu-Natal, Durban.This aim of this investigation is to describe the mechanical performance of a beam (probe) used in dynamic atomic force microscopy (dAFM) which can be utilized in scanning the topographical features of biological samples or "pliable" samples in general. These nanobeams can also be used to modify samples by using high frequency oscillating contact forces to remove material or shape nano structures. A nanobeam with arbitrary boundary conditions is studied to investigate different configurations and the effects of the relevant parameters on the natural frequencies. The nano structure is modelled using the Euler-Bernoulli theory and Eringen's theory of nonlocal continuum or first order stress-gradient theory is incorporated to simulate the dynamics of the system. This theory is effective at nanoscale because it considers the small-scale effects on the mechanical properties of the material. The theory of Nonlocal continuum is based on the assumption that the stress at a single point in the material is influenced by the strains at all the points in the material. This theory is widely applied to the vibration modelling of carbon nanotubes in several studies. The system is modelled as a beam with a torsional spring boundary condition that is rigidly restrained in the transverse direction at one end. The torsional boundary condition can be tuned, by changing the torsional spring stiffness, such that the compliance of the system matches that of the sample to prevent mechanical damage of both the probe tip and the sample. When the torsional spring stiffness is zero, the beam is pinned and when the stiffness is infinite, the beam is a cantilever. In the first case, a mass is attached to the tip and a linear transverse spring is attached to the nanobeam. The mass and spring model the probe tip and contact force, respectively. In the second case, at the free end is a transverse linear spring attached to the tip. The other end of the spring is attached to a mass, resulting in a single degree of freedom spring-mass system. When the linear spring constant is infinite, the free end behaves as a beam with a concentrated tip mass. When the mass is infinite, the boundary condition is that of a linear spring. When the tip mass is zero, the configuration is that of a torsionally restrained cantilever beam. When tip of the nanobeam vibrates, the system behaves like a hammer and chisel. The motion of the tip of the beam and tip mass can be investigated to observe the tip frequency response, force, acceleration, velocity and displacement. The combined frequencies of the beam and spring-mass systems contain information about the maximum displacement amplitude and therefore the sample penetration depth

Buckling of elastically restrained nonlocal carbon nanotubes under concentrated and uniformly distributed axial loads

Buckling of elastically restrained carbon nanotubes is studied subject to a combination of uniformly distributed and concentrated compressive loads. Governing equations are based on the nonlocal model of carbon nanotubes. Weak formulation of the problem is formulated and the Rayleigh quotients are obtained for distributed and concentrated axial loads. Numerical solutions are obtained by Rayleigh–Ritz method using orthogonal Chebyshev polynomials. The method of solution is verified by checking against results available in the literature. The effect of the elastic restraints on the buckling load is studied by counter plots in term of small-scale parameter and the spring constants.</p

Nonlinear Dynamics of Carbon Nanotubes Under Soft Alternating Current Actuation

This thesis work deals with electrostatically actuated Carbon Nanotubes (CNTs) cantilevers. Four forces are acting on the CNTs cantilever, namely damping, elastic, electrostatic and van der Waals forces. The van der Waals force is significant for values of 50 nm or lower of the gap between the CNTs and the ground substrate. As both electrostatic and van der Waals forces are nonlinear, and the CNTs electrostatic actuation is given by alternating current (AC) voltage, the CNTs undergo nonlinear parametric dynamics. The Method of Multiple Scales (MMS) and Reduced Order Model (ROM) are employed to investigate the system under soft excitation and weak nonlinearities. The frequency-amplitude and voltage-amplitude responses are reported in the cases of AC near half natural frequency and AC near primary natural frequency

A Review of Double-Walled and Triple-Walled Carbon Nanotube Synthesis and Applications

Double- and triple-walled carbon nanotubes (DWNTs and TWNTs) consist of coaxially-nested two and three single-walled carbon nanotubes (SWNTs). They act as the geometrical bridge between SWNTs and multi-walled carbon nanotubes (MWNTs), providing an ideal model for studying the coupling interactions between different shells in MWNTs. Within this context, this article comprehensively reviews various synthetic routes of DWNTs’ and TWNTs’ production, such as arc discharge, catalytic chemical vapor deposition and thermal annealing of pea pods (i.e., SWNTs encapsulating fullerenes). Their structural features, as well as promising applications and future perspectives are also discussed. Keywords: carbon nanotubes; double-walled carbon nanotubes; triple-walled carbon nanotubes; synthesis; catalytic chemical vapor deposition; arc discharge; fullerenes; pea pod

Multiscale Mathematical Modelling of Nonlinear Nanowire Resonators for Biological Applications

Nanoscale systems fabricated with low-dimensional nanostructures such as carbon nanotubes, nanowires, quantum dots, and more recently graphene sheets, have fascinated researchers from different fields due to their extraordinary and unique physical properties. For example, the remarkable mechanical properties of nanoresonators empower them to have a very high resonant frequency up to the order of giga to terahertz. The ultra-high frequency of these systems attracted the attention of researchers in the area of bio-sensing with the idea to implement them for detection of tiny bio-objects. In this thesis, we originally propose and analyze a mathematical model for nonlinear vibrations of nanowire resonators with their applications to tiny mass sensing, taking into account thermal, piezoelectric, electromagnetic, surface, and external excitations.~The mathematical models for such nanowires are formulated using the Euler-Bernoulli beam theory in conjunction with the nonlocal differential constitutive relations of Eringen type. In order to analyze the obtained nonlinear partial differential equation (PDE), we first use the Galerkin method in combination with a perturbation technique to obtain the primary resonance.~After finding the primary resonance, a parametric sensitivity analysis is carried out to investigate the effects of key parameters on the sensitivity of the nanowire resonators in mass sensing.~Our main hypothesis is that bio-particles attached to the surface of the nanowire resonator would result in a detectable shift in the value of the jump frequency.~Therefore, a mathematical formula is developed based on the jump frequency to scrutinize the sensitivity of the considered nanowire resonators. Our mass sensitivity analysis aims at the improved capability of the nanowire resonators in detection of tiny bio-particles such as DNA, RNA, proteins, viruses, and bacteria.~Numerical solutions, obtained for the general nonlinear mathematical model of nanowire resonators, using the Finite Difference Method, are compared with the results obtained with a simplified approach described above. Finally, we investigate the sensitivity of the nanowire resonator for mass sensing using molecular dynamics simulations to provide a validation for our results from the obtained continuum models. It is expected that the results of this research may assist in our better understanding of key characteristics of nanowire resonators for their applications in detection of bio-particles, ultimately impacting the development of advanced approaches to disease diagnostics and treatments

Nanomechanical Resonators: Toward Atomic Scale

The quest for realizing and manipulating ever smaller man-made movable structures and dynamical machines has spurred tremendous endeavors, led to important discoveries, and inspired researchers to venture to new grounds. Scientific feats and technological milestones of miniaturization of mechanical structures have been widely accomplished by advances in machining and sculpturing ever shrinking features out of bulk materials such as silicon. With the flourishing multidisciplinary field of low-dimensional nanomaterials, including one-dimensional (1D) nanowires/nanotubes, and two-dimensional (2D) atomic layers such as graphene/phosphorene, growing interests and sustained efforts have been devoted to creating mechanical devices toward the ultimate limit of miniaturization— genuinely down to the molecular or even atomic scale. These ultrasmall movable structures, particularly nanomechanical resonators that exploit the vibratory motion in these 1D and 2D nano-to-atomic-scale structures, offer exceptional device-level attributes, such as ultralow mass, ultrawide frequency tuning range, broad dynamic range, and ultralow power consumption, thus holding strong promises for both fundamental studies and engineering applications. In this Review, we offer a comprehensive overview and summary of this vibrant field, present the state-of-the-art devices and evaluate their specifications and performance, outline important achievements, and postulate future directions for studying these miniscule yet intriguing molecular-scale machines

Nanomechanical Resonators: Toward Atomic Scale

The quest for realizing and manipulating ever smaller man-made movable structures and dynamical machines has spurred tremendous endeavors, led to important discoveries, and inspired researchers to venture to previously unexplored grounds. Scientific feats and technological milestones of miniaturization of mechanical structures have been widely accomplished by advances in machining and sculpturing ever shrinking features out of bulk materials such as silicon. With the flourishing multidisciplinary field of low-dimensional nanomaterials, including one-dimensional (1D) nanowires/nanotubes and two-dimensional (2D) atomic layers such as graphene/ phosphorene, growing interests and sustained effort have been devoted to creating mechanical devices toward the ultimate limit of miniaturization--genuinely down to the molecular or even atomic scale. These ultrasmall movable structures, particularly nanomechanical resonators that exploit the vibratory motion in these 1D and 2D nano-to-atomic-scale structures, offer exceptional device-level attributes, such as ultralow mass, ultrawide frequency tuning range, broad dynamic range, and ultralow power consumption, thus holding strong promises for both fundamental studies and engineering applications. In this Review, we offer a comprehensive overview and summary of this vibrant field, present the state-of-the-art devices and evaluate their specifications and performance, outline important achievements, and postulate future directions for studying these miniscule yet intriguing molecular-scale machines

Vibration analysis of viscoelastic single-walled carbon nanotubes resting on a viscoelastic foundation

Vibration responses were investigated for a viscoelastic Single-walled carbon nanotube (visco-SWCNT) resting on a viscoelastic foundation. Based on the nonlocal Euler-Bernoulli beam model, velocity-dependent external damping and Kelvin viscoelastic foundation model, the governing equations were derived. The Transfer function method (TFM) was then used to compute the natural frequencies for general boundary conditions and foundations. In particular, the exact analytical expressions of both complex natural frequencies and critical viscoelastic parameters were obtained for the Kelvin-Voigt visco-SWCNTs with full foundations and certain boundary conditions, and several physically intuitive special cases were discussed. Substantial nonlocal effects, the influence of geometric and physical parameters of the SWCNT and the viscoelastic foundation were observed for the natural frequencies of the supported SWCNTs. The study demonstrates the efficiency and robustness of the developed model for the vibration of the visco-SWCNT-viscoelastic foundation coupling system

Electrostatically Actuated Double Wall Carbon Nanotubes to Include Intertube van der Waals Forces

This work deals with the amplitude-frequency and amplitude-voltage responses of parametric and primary resonances of electrostatically actuated double-walled carbon nanotubes (DWCNTs). Nonlinear forces acting on the DWCNT are intertube van der Waals and electrostatic forces. Soft AC excitation and small viscous damping are assumed. In coaxial vibration, the outer and inner carbon nanotubes move together (in-phase), maintaining their geometric concentricity; while in noncoaxial vibration, the CNTs move in opposite direction (out-of-phase). Modal coordinate transformation is formulated. The Harmonic Balance Method (HBM) is used to find the free response solutions of the DWCNT. The Reduced Order Model (ROM) method is also used in this investigation. All ROMs using one through five modes of vibration (terms) are developed in terms of modal coordinates. ROM using one term is solved and frequency-amplitude response predicted by using the Method of Multiple Scales (MMS). All models and methods are in agreement at lower amplitudes for coaxial vibration, while in higher amplitudes only ROM with five terms provides reliable results. The effects of voltage, detuning frequency, and damping on the various resonance responses of electrostatically actuated DWCNTs are reported. A discussion of stability and bifurcation analysis is presented