A simple transcendental travelling wave solution and stability study for the thermophoretic motion with variable heat transmission factors on substrate-supported grapheme sheet

TL;DR

This paper introduces a simple transcendental wave solution for thermophoretic motion in graphene sheets, using an energy method to transform and solve PDEs, providing insights into wrinkle formation and control via heat sources.

Contribution

It presents a novel, easily solvable transcendental solution for thermophoretic motion, enabling better understanding and control of wrinkle patterns in graphene sheets.

Findings

Transformed third-order PDE into first-order PDE for easier analysis.

Derived semi-group and transcendental solutions for the motion.

Linked wrinkle formation to equilibrium point evolution.

Abstract

Manually tailored wrinkled graphene sheets hold great promise in fabricating smart solid-state devices. In this paper, we employ an energy method to transform the original third-order partial differential equation (pde), i.e. Eq. (1) into the first-order pde, i.e. Eq. (8) for the thermophoretic motion of substrate-supported graphene sheets, which can be solved in terms of semi-group and transcendental solutions. Unlike soliton solutions derived using other more sophisticated techniques [9, 23], the present transcendental solution can be easily solved numerically and provides physical insights. Most importantly, we verify that the formation of various forms for wrinkling wave solutions can be determined by the evolution of equilibrium points for Eq. (1). This sheds a light on modifying the heat sources in order to control the configuration of wrinkle waves that has not been previously addressed.