Time Responses of Casimir Forces Induced on a Flake in a Liquid Controlled by a Tunable Graphene Layer

Casimir forces arise between two bodies nearby due to electromagnetic quantum fluctuations. In thermal equilibrium, Casimir forces are attractive in vacuum except for forces on nonreciprocal materials. On the other hand, repulsive Casimir forces can be generated in a liquid environment, which enhances the controllability of Casimir forces and expands their potential for applications. One attractive example of such applications is the stable levitation of a nanoflake in a liquid. Recently, it has been experimentally demonstrated that a gold nanoflake in ethanol can stay at a distance away from the substrate by manipulating the Casimir force with a proper choice of materials. For further development of this technique, the dynamic control of the levitation by Casimir forces is expected.<br/>In this research, we focus on graphene due to its tunable optical response by adjusting the chemical potential. By numerical calculations, we show that the floating position of a nanoflake in a liquid can be dynamically modulated by tuning the chemical potential of graphene inserted in the substrate. For instance, the zero-force position of a graphene nanoflake in ethanol can be varied from 38.6 to 43.0 nm above the Teflon/graphene/SiO₂ substrate when the chemical potential is adjusted from 1 to 0 eV. The stable levitation by zero Casimir forces is induced by the mechanism as follows: The total Casimir force on the nanoflake can be decomposed into attractive and repulsive force components, and the magnitude of each component increases as the distance between two bodies decreases. The nanoflake stably levitates above the substrate where attractive and repulsive force components are balanced. By increasing (decreasing) the chemical potential of graphene in the substrate, the attractive force component becomes larger (smaller), and the stable zero-force position of the nanoflake approaches (departs from) the substrate.<br/>The dynamical behavior of the floating nanoflake is investigated by calculating the equation of motion considering the Casimir force, gravity, and buoyancy. By assuming that the Casimir force is proportional to the distance variation around the zero-force position, we develop a harmonic oscillator model for the system. The validity of this analytical model is confirmed by showing that the time constant based on the harmonic oscillator model is in good agreement with that obtained by the numerical calculation results. This allows us to discuss the tractability of the levitated nanoflake by the spring constant which indicates how strong the Casimir force is, and by the fluid resistance coefficient. As a result, we find that the thickness of the topmost film of the substrate can largely change the zero-force position and the time constant.<br/>In addition, proper choice of materials allows high tunability of the zero-force position and the tractability in our control scheme. For example, the time constant in the dynamics for a graphene nanoflake is 3 to 6 times longer than that for a gold nanoflake. Furthermore, the upper limit of the zero-force position of the nanoflake is discussed. Our scheme significantly broadens the degrees of freedom for dynamic control of nanoflakes in a liquid.<br/>Our system, based on tunable levitation by the zero Casimir force, is applicable to various applications. One such application is a new concept of a graphene chemical sensor device. Since the Casimir force can be modulated by altering the chemical potential of graphene, the absorption of chemical materials on the surface of graphene results in the position shift of the nanoflake in a liquid.