Thermally Activated Snap-through Transitions Controlled by Tunable Metastability

Abstract

The effects of thermal fluctuations on the morphology of two-dimensional materials are hard to harness. We propose that a geometrically constrained graphene nanoribbon (GNR) can exhibit thermally activated snap-through transitions with a predictable and controllable transition rate constant. The energetics and kinetics of such transitions can be fully captured by combining enhanced sampling methods and generalized transition state theory. Using well-tempered metadynamics, we determine the free energy landscape and a pair of asymmetric transition pathways of the GNR system. Notably, generalized transition state theory accurately captures how the transition rate constant responds to temperature and the tunable free energy landscape of our system. This work offers a theoretical framework for elastic metastability, introduces rare event methods into thermalized nanomechanical systems, and provides potential applications in designing nanoscale thermal switches and thermal actuators.

Figures (5)

• Figure 1: (a) Representative configurations of the GNR corresponding to the natural state (top), the asymmetric unstable equilibrium ($\mathcal{A}$ state, middle), and the inverted state (bottom). The rigidly constrained atoms at the two ends impose a fixed end-shortening $\Delta L$ and a symmetric tangent angle $\alpha$ at the boundaries. (b) The bifurcation diagram for the GNR system under consideration. Based on the equilibrium solutions of the Euler–beam equation, the midpoint displacement $w_0$ is plotted as functions of the geometric control parameter $\mu$, with solid lines indicating the natural state (violet upper branch) and the inverted state (blue lower branch), and the orange dashed line indicating the $\mathcal{A}$ state. Diamond markers represent the equilibrium averages of $w_0$ computed from MD simulations.
• Figure 2: (a) Midpoint displacement $w_0$ of the GNR (left y-axis, orange) and the corresponding atomistic potential energy (APE) (right y-axis, light blue) in a representative trajectory of the thermally activated snap-through transition. The zero of APE has been shifted for clarity. (b) Normalized time-correlation function of $w_0$ within the local equilibrium of the inverted state. (c) Cumulative distribution function of the escape times from the inverted state (blue solid line), compared with the maximum likelihood estimation fit (red dashed line).
• Figure 3: (a) Free energy surface (FES) computed from 2D WT-MetaD in the ($w_0$, $\Delta w_{\text{qrt}}$) space, and (b) unreweighted histogram from a representative 1D WT-MetaD simulation biasing only $w_0$, as projected into the ($w_0$, $\Delta w_{\text{qrt}}$) space. The collective variable $\Delta w_{\text{qrt}}$ quantifies the degree of asymmetry of the GNR. The locations of the natural state (red circle), inverted state (blue diamond), and the $\mathcal{A}$ state (yellow triangle) are computed from the Euler-beam equation. The minimum free energy paths (MFEPs) on the FES in (a) are consistent with the transition pathways revealed by 1D WT-MetaD in (b). The results shown correspond to $T$ = 300 K and $\alpha = 16^\circ$ ($\mu \approx 0.9873$), exhibiting qualitative features common across the computed range $0 \leq \mu \lesssim 1.789$.
• Figure 4: (a) Free energy profile computed from WT-MetaD (five 500 ns production runs) along the order parameter $w_0$ for the GNR constrained at $\alpha = 29^{\circ}$ ($\mu \approx 1.789$) under temperature $T$ = 300 K. Inset: Magnified view of the free energy profile highlighting the forward transition barrier $F^+$. (b) Forward and backward transition barriers, $F^+$ and $F^-$, as functions of the dimensionless geometric control parameter $\mu$.
• Figure 5: (a) For the GNR system constrained at $\alpha = 29^{\circ}$, $\ln{(k\,T^{-\frac{1}{2}})}$ is plotted versus $1/T$ to reveal the temperature dependence of the transition rate constant $k$ computed directly from unbiased simulations. Based on Eq. (\ref{['eq:rate_const2']}), $\Delta F^\ddagger$ can be inferred from the slope of the linear fit. (b) Transition rate constant $k$ for the forward transition as a function of temperature, calculated using Eq. (\ref{['eq:rate_const1']}) and WT-MetaD for the system constrained at $\alpha = 24^{\circ}$.