# Enhancing Sensitivity of Double-Walled Carbon Nanotubes with Longitudinal Magnetic Field
## 1 Introduction
They theoretically and experimentally showed that, through tailoring the sensor geometry and mass load position, a mass sensitivity enhancement exceeding 400% was noted in a magnetoelastic sensor with the utilization of novel sensor platform geometries. [PERSON] et al. [12] proposed a novel approach to mass sensing and stiffness spectroscopy, focusing on the phenomenon whereby a nanoresonator enters a superposition state characterized by two orthogonal vibrations with distinct frequencies upon the disruption of symmetry. The measurement of these frequencies enables the determination of the mass, stiffness, and azimuthal arrival direction of the adsorbate. In the current research, we focus on how it is possible to increase the frequency shift--and, consequently, to enhance sensitivity--through the use of a longitudinal magnetic field [10]. The applications of magnetic fields in micro- and nanoelectromechanical systems have been addressed in many studies [24; 25; 26]. In the case of sensors, coupling the magnetic effect with acoustic resonator sensors offers significant advantages and the possibility to use a much wider variety of materials for the transducers, when compared to piezoelectric resonators, allowing more flexibility and a potentially improved performance. This approach was used by [PERSON] et al. [27] to detect fluid volumes, liquid viscosity, and density. Moreover, the influence of magnetic forces on the behavior of micro- and nanomechanical elements has been extensively investigated as an external parameter. [PERSON] et al. [28] demonstrated that the frequencies of a DWCNT increase with an increase in the magnetic field. They also investigated the effect of a longitudinal magnetic field on the transverse vibration of a magnetically sensitive material. [PERSON] et al. [29] conducted a comprehensive parametric study on multi-walled carbon nanotubes in a magneto-thermal environment. Their study considered the nanotube geometry and explored other parameters, such as the number of walls, initial curvature, and environmental factors, including the temperature and magnetic field. [PERSON] et al. [30] showed the effect of the strength of the magnetic field on the vibration of an SWCNT embedded in a viscoelastic medium. They proposed a new analytical expression for the complex natural frequencies for certain typical boundary conditions and the Kelvin-Voigt model and demonstrated the efficiency of their proposed methods in the vibration analysis of embedded SWCNTs under a magnetic field. [PERSON] et al. [31] analyzed the vibrational behavior of a CNT conveying a nanoflow subjected to a longitudinal magnetic field. [PERSON] et al. [32] explored the longitudinal magnetic field's influences on the vibration of viscoelastic nanotubes based on the non-local strain gradient Timoshenko beam model. [PERSON] et al. [33] investigated the effect of an external magnetic field on the nonlinear vibration of a CNT conveying a fluid. Their study revealed that the magnetic term significantly attenuated or dampened the system's vibration by approximately 20%. Moreover, in the case of modeling improvement, [PERSON] and [PERSON] [34] recently focused on both the inertia and stiffness of the detected mass by considering it as an elastic body with stiffness. Their findings showed that including the stiffness of the detected mass introduced an additional frequency to the spectrum of the nano-beam, resulting in a shift in the nano-beam's frequencies to the adjacent higher mode. Furthermore, [PERSON] et al. [35] employed the Timoshenko beam theory and [PERSON]'s non-local theory of elasticity to enhance the modeling accuracy, particularly for short beams. They found that the Euler-Bernoulli beam theory overpredicted the frequency shift, compared to the Timoshenko beam theory. Overall, the sensitivity of resonant mass sensors relies significantly on the effective vibratory mass. This is influenced by both the geometry and configuration of the resonant structure and the properties of the materials that make up the structure [36].