Diffusion-induced dissipation and mode coupling in nanomechanical resonators

We study a system consisting of a particle adsorbed on a carbon nanotube resonator. The particle is allowed to diffuse along the resonator, in order to enable study of, e.g., room-temperature mass sensing devices. The system is initialized in a state where only the fundamental vibration mode is excited, and the ring-down of the system is studied by numerically and analytically solving the stochastic equations of motion. We find two mechanisms of dissipation, induced by the diffusing adsorbate. First, short-time correlations between particle and resonator motions means that the net effect of the former on the latter does not average out, but instead causes nonexponential dissipation of vibrational energy. For vibrational amplitudes that are much larger than the thermal energy this dissipation is linear; for small amplitudes the decay takes the same form as that of a nonlinearly damped oscillator. Second, the particle diffusion mediates a coupling between vibration modes that opens a new dissipation channel by enabling energy transfer from the fundamental mode to the excited modes, which rapidly reach thermal equilibrium.

Figure 1

A particle of mass $m$ is adsorbed on a one-dimensional resonator of mass $M$ and length $L$. The transverse displacement of the resonator at coordinate $X$ along the nanotube axis is $w(X,t)$; $x\left(t\right)$ is the position of the adsorbed particle. The resonator is initialized in its fundamental vibration mode, and the effect of the stochastically diffusing particle on the ring-down of the resonator is studied.

Figure 2

Dimensionless mode energies as function of time during ring-down of the fundamental mode. Here $\epsilon \simeq {10}^{-2}$ and $T=500$ K. For clarity, only the lowest lying flexural modes are shown; higher modes behave similarily. The dashed line indicates the thermal energy in dimensionless units. The system is initialized in a state where all energy (${\mathcal{E}}_0\left(0\right)\simeq {10}^4{k}_{\mathrm{B}}T$) is in the fundamental mode, and then allowed to evolve freely. As can be seen, the effect of the particle diffusion is to damp out the fundamental mode and establish equilibrium with higher-lying modes.

Figure 3

Top: fundamental mode energy as a function of time, together with a linear fit to the initial decay. Center: the parameter ${\omega }_0\mathcal{E}/\gamma \mathcal{D}$, shown below to govern the qualitative behavior of the system, as a function of time. The horizontal dashed line indicates where ${\omega }_0\mathcal{E}=\gamma \mathcal{D}$. Bottom: evolution of the probability distribution $p(\xi ,t)$, where $\xi =x/L$, for the particle to be at a certain position $0<\xi <1$ along the resonator as a function of time. For large initial amplitudes, the particle remains inertially trapped at the antinode of the vibration around $\xi \approx 0.5$. The corresponding energy decay is linear in time. As the thermal fluctuations overcome the inertial trapping potential, the particle diffuses freely, and the energy decays algebraically towards equilibrium.

Figure 4

Damping parameter $\alpha$, as calculated by Eq. (14) (solid line) and by a numerical fit (dots). The same simulation parameters as in Fig. 2 were used, with the exception of the initial amplitude, here ${\mathcal{E}}_0\left(0\right)={10}^{-4}$. The ratio ${\omega }_0{\mathcal{E}}_0/\gamma \mathcal{D}$ was varied by changing the simulation temperature. We see that the perturbative approach is indeed valid for ${\omega }_0{\mathcal{E}}_0/\gamma \mathcal{D}\ll 1$. In the intermediate region ${\omega }_0{\mathcal{E}}_0/\gamma \mathcal{D}\lesssim 1$, Eq. (14) overestimates the magnitude of the damping, but captures the overall shape of the curve and the location of the maximal damping.

Figure 5

Dependence of the thermalization time (the time at which ${\mathcal{E}}_n$ exceeds $k_{\mathrm{B}}T$ for some $n$) on $\epsilon$ and ${\mathcal{E}}_0\left(0\right)$. The black line is a least-squares fit to the data. The numerical simulation becomes more sensitive the more energy is put into the system, which explains the increased spread of the data points as ${\mathcal{E}}_0\left(0\right)$ increases.

Figure 6

Slope of the decay of the fundamental mode during the initial, linear regime, as a function of the number of modes $N$ included in the simulation; black circles are data points while the red dashed line is a linear fit. The values have been divided by $-\epsilon \mathcal{D}\gamma /{\omega }_0$ to show the agreement with the theoretical case (10). As more excited modes are included, more decay channels are opened and the rate of decay increases.

Figure 7

Equilibrium distribution $p_{\mathrm{st}.}\left(\mathcal{E}\right)$. (Black triangles) Distribution obtained from numerical simulation of Equations. (7, 8, 9) after relaxation. (Red line) The distribution $p\left(\mathcal{E}\right)=(\epsilon \mathcal{D}\gamma /{\omega }_0)exp(-\mathcal{E}(\epsilon \mathcal{D}\gamma /{\omega }_0)).$