Buckled Nano Rod—A Two State System and Its Dynamics

Abstract

We consider a suspended elastic rod under longitudinal compression. The compression can be used to adjust potential energy for transverse displacements from harmonic to double well regime. As compressional strain is increased to the buckling instability, the frequency of fundamental vibrational mode drops continuously to zero (first buckling instability). As one tunes the separation between ends of a rod, the system remains stable beyond the instability by developing a double well potential for transverse motion. The two minima in the potential energy curve describe two possible buckled states at a particular strain. From one buckled state it can go over to the other by thermal fluctuations. Using a continuum approach and transition state theory (TST) we have calculated the rate of conversion from one state to other. If the strain is between $\epsilon_c$ and $4\epsilon_c$, the saddle point is the straight rod. Using TST we find exact expressions for the rate under the harmonic approximation. The expression, however, diverges at the second buckling instability $(\epsilon_c = 4\epsilon_c)$. At this point, the straight rod configuration, which was a saddle till then, becomes hill top, and two new saddles are generated. The new saddles have bent configurations (S-shaped) and as rod goes through further instabilities, they remain stable and the rate calculated under the harmonic approximation around saddle point remains finite. However, the rate diverges near the second buckling instability. We suggest methods to correct this divergence. Using these expressions, we calculate rate of passage from one buckled state to the other, for Si rods of nano dimensions. A calculation including friction has been carried out, by assuming that each segment of the rod is coupled to its own collection of harmonic oscillators. We find that friction lowers the rate of conversion.