Figure 1

Schematic representation of a single-well free-energy surface. (a) Intrinsic free-energy surface $U_0(x)$ with minimum-to-barrier distance $x^{‡}$, activation free energy $\Delta G^{‡}$, and inflection point $x^{*}$; (b) decrease in the free-energy barrier $\Delta U(F)$ of the combined free-energy surface, $U(x)=U_0(x)-Fx$, with increasing external force $F$.Reuse & Permissions

Figure 2

Comparison of analytical results [solid lines; $\nu =2/3$ in Eqs. (5, 6)] and Brownian dynamics simulations (symbols) for the mean (black squares) and variance (red circles) of the rupture force for the linear-cubic free-energy surface with $x^{‡}=0.4\mathrm{nm}$, $\Delta G^{‡}=20k_{B}T$, and the diffusion coefficient $D$ chosen so that the Kramers rate $k_0=3{\pi }^{-1}\Delta G^{‡}(x^{‡}{)}^{-2}D\exp (-\Delta G^{‡})={10}^{-4}{\mathrm{s}}^{-1}$. Short-dashed (green) lines are the predictions of the phenomenological theory with the above parameters. Long-dashed (blue) lines are the fit of the phenomenological model ($\nu =1$) to $⟨F⟩$ for loading rates from 0.4 to $100\mathrm{pN}/\mathrm{ms}$. Even though the fit to $⟨F⟩$ is quite good, $k_0$ is too large (by a factor of 33). The inset compares the distributions of the rupture force from theory Eq. (4) (lines) and simulations (shaded bars) for loading rates of 10 and $500\mathrm{pN}/\mathrm{ms}$. Although Eq. (4) already fails at $500\mathrm{pN}/\mathrm{ms}$, the range of validity of Eqs. (5, 6) for $⟨F⟩$ and ${\sigma }_F^2$ extends further.Reuse & Permissions

Figure 3

Constant-force rupture rate $k(F)$ from collapse of constant-speed rupture-force histograms using Eq. (7). Simulation data (symbols) cover pulling speeds $V$ from ${10}^{-4}$ to ${10}^3\mathrm{pN}/\mathrm{ms}$, as indicated (see Fig. 2 for details of simulations), with each $V$ probing a different range of $F$. The solid line is the analytical prediction for $k(F)$, Eq. (3), with $\nu =2/3$. Deviations at the highest pulling speeds (and rupture forces) are caused by a breakdown of the Kramers high-barrier approximation. The inset shows the force histograms (lowest to highest ramp speed from left to right) that were used in Eq. (7).Reuse & Permissions