Temperature and loading rate dependent rupture forces from universal paths in mechanochemistry

Mechanical breaking of covalent bonds is induced by thermal fluctuations which leads to temperature and loading rate dependent rupture forces. These forces are much smaller than the maximum slope of the bond potential at practically accessible loading rates. Calculation of force-dependent activation energies obtained with a one-dimensional constrained geometry method is shown to work for isolated weak bonds or very simple molecules only. The situation is analyzed in a two-dimensional picture taking the broken bond explicitly into account. A single force-independent universal path appears that contains full information about the force dependence of the minima and transition states. The activation energies obtained from this path are of similar accuracy as those from force-dependent nudged elastic band calculations. Strong variations of rupture forces with loading rate and temperature are found for mechanochromophores, where the rupture force is more than one order of magnitude smaller than the maximum slope in agreement with experiment.

Figure 1

Electronic energy $U_F\left(d\right)$ of the given molecules at fixed external force $F=3$ nN. The equilibrium distance of the reactant state $d_R$ and the distance of the transition state $d_{\text{TS}}$ are indicated, respectively.

Figure 2

(a) Energy landscape $U_F(d,b)$ for a linear molecule with four atoms X interacting via equal Morse potentials at the given external force $F=0.3$. Extrema of $U_F(d,b)$ with respect to $b$ are indicated by solid lines (from thick to thin): reactant minimum (green), reactant maximum (black), transition state maximum (red), and product minimum (magenta). (b) Zoom into $U_F(d,b)$ in the region where the extrema vary most. The insets show cuts for fixed $d$ as indicated. The arrows illustrate the variation of the force-dependent reactant (R) and transition states (TS) with increasing external force. (c) $U_F(d,b)$ for ${\mathrm{C}}_6{\mathrm{H}}_{14}$ obtained with DFT at $F=3.5$ nN.

Figure 3

3S-COGEF path (from thick to thin: green, red, and magenta lines in analogy to Fig. 2) for rotation of the dihedral angle $\beta$ in spiropyran (derivative SP-H as shown in the insets). Positions of reactant (green triangle), transition (red circle), and product (magenta square) structures for the external force $F=0.4$ nN are highlighted. Energies are given relative to the energy of the reactant state.

Figure 4

Energy landscape for the $b$-breaking reaction of spiropyran from Fig. 3 at $F=0.7$ nN. Calculated configurations to define the 1S-COGEF path are indicated by green dots and the blue squares show the configurations calculated to obtain the full 3S-COGEF path by fixing $d$ and $b$. Yellow and white lines indicate configurations calculated by following the transition state maximum and product minimum curves directly.

Figure 5

Rupture force $F_{\mathrm{rup}}$ with uncertainty of all molecules investigated for various loading rates (left, for $T=298$ K) and temperatures (right, for $\alpha =10$ nN/s).