Supplementary Information Thermophoretically Driven Carbon Nanotube Oscillators V. R. Coluci, V. S. Tim´ oteo, and D. S. Galv˜ ao EQUATION OF MOTION 11 Thus, the term αdT/dx represents the thermal force due to thermal vibrations of the outer tube caused by the externally imposed thermal gradient (Fig. S1). The regime of a constant thermal gradient is obtained after a transient period once the electrical current pulse is applied. We have analyzed the dynamics of the transient period using molecular dynamics simulations (see below). The thermal gradient was chosen to be practically constant on the outer tube length based on molecular dynamics simulations (see below) according to the following expression where F$_{vdW}$ is the magnitude of the van der Waals restoring force, L and l are the outer and inner tube lengths, respectively. The form of F ( x ) is shown in Fig. S1, for L = 100 nm and l = 5 nm. where the restoring force is written as with The motion of the inner tube of the CNT oscillator considering only translational motions (rigid tubes) is given by m ¨ x − F ( x ) + γ sgn( ˙ x ) | ˙ x | 2 = α dT dx p ( t ) (2) F ( x ) = F$_{vdW}$ { Θ( x + r )[1 − Θ( x − r )][Θ( − x − Δ) − Θ( x − Δ)] } (3) Θ( x ) = 1 2 + x 2 √ x 2 + δ 2 (4) r ≡ L + l 2 Δ ≡ L − l 2 , (5) ( δ = 0 . 1 ˚ A) and dT dx ( x ) ≡ ∣ ∣ dT dx$^{∣}$ ∣ ∣ [ 2 1 + exp( − 30 x ) − 1$^{]}$ . (6)