Gigahertz nanomechanical oscillators based on ions inside cyclic peptide nanotubes: a continuum study

The present work aims to investigate the mechanical oscillatory behavior of ions, and in particular Li+,Na+,Rb+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Li}^{+}, {\rm Na}^{+}, {\rm Rb}^{+}}$$\end{document} and Cl-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Cl}^{-}}$$\end{document} ions, inside a cyclo[(–d-Ala–l-Ala)4–] peptide nanotube using the continuum approximation along with the 6–12 Lennard–Jones (LJ) potential function. Assuming that each peptide unit is comprised of an inner and an outer tube, the van der Waals (vdW) potential energy and interaction force between an ion and a cyclic peptide nanotube (CPN) are determined analytically. With respect to the present formulations, a detailed parametric study is conducted on the vdW potential energy and interaction force distributions by varying the number of peptide units. Employing the conservation of mechanical energy principle, a novel expression for precise evaluation of oscillation frequency is introduced. To verify the accuracy of the proposed frequency expression, the results obtained from energy equation are compared with the ones predicted through solving the equation of motion numerically. The effects of number of peptide units and initial conditions including initial separation distance and velocity on the oscillatory behavior of various ions inside CPNs are explored. Among the considered ions, Cl-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Cl}^{-}}$$\end{document} ion is found to generate the highest frequency. According to the potential energy profile, one oscillatory zone for one peptide unit and different oscillatory zones for more than one peptide unit are observed. Numerical results indicate that optimal frequency decreases with increasing the number of peptide units and almost remains unchanged when the number of peptide units exceeds four.