Computational Exploration of Polymer Mechanochemistry: Quantitation of Activation Force and Systematic Discovery of Reaction Sites by the Extended Artificial Force-Induced Reaction Method

Abstract

A collection of mechanophores was computationally studied using the extended artificial force induced reaction (EX-AFIR) method, which utilizes two different sets of forces to determine the activation force level (F _act) practically and efficiently. Identifying a mechanophore’s F _act is a focus of mechanochemistry. As have been done in an existing framework, we generated the ΔG _τ ^‡–F _τ curve, where F _τ is the value of external force and ΔG _τ ^‡ denotes the force-coupled free energy barrier of a specific reaction under F _τ. Such a curve is then used to determine the F _act of a certain mechanophore when combined with the Eyring equation. Although generating such a ΔG _τ ^‡–F _τ curve was tough because locating the force-coupled transition states is time-consuming, the extended AFIR method allowed an efficient exploration of all the relevant transition states on the force-modified potential energy surface (FMPES). The EX-AFIR method was later applied to study the problems encountered in current polymer mechanochemistry research, deriving a concept of “node” which could be used for the design of thermostable mechanophores. The first-ever case study of cubane using EX-AFIR is a vivid example of how the fully automated search of possible reaction pathways on the FMPES is facilitated. Furthermore, it also provided insights into the further design and application of unconventional mechanophores.

1. Introduction

Mechanochemistry, an emerging field in modern chemistry, involves the use of mechanical forces to trigger chemical reactions. Occasionally, the external mechanical force can prompt the reactions that cannot occur under normal circumstances, or effectively redirect the reaction to a less favored pathway under thermal conditions. − More interestingly, unlike thermal and photochemical reactions, the mechanical force is believed to distort the original potential energy surface and may therefore result in new reaction patterns. − Polymer mechanochemistry focuses on the chemical reactions of polymers under force. − Early polymer mechanochemistry studies involved applying mechanical force to polyalkenes, leading to the homolysis of the backbone and the generation of fragments with lower molecular weight. −

Recently, despite the reaction of bulk polymeric material under force, the concept of polymer mechanochemistry has considerably expanded. Specifically, versatile reaction patterns have been realized by applying forces to specific molecules. These specially designed molecules, called mechanophores, are a group of compounds that can easily undergo chemical reactions when loaded with a mechanical force. − Notably, mechanophores are usually embedded into long polymer chains to facilitate the transmission of mechanical force, and this also becomes part of polymer mechanochemistry. Our work presented here mainly tackles this issue. In pioneering works, a series of mechanophores were synthesized and subsequently investigated experimentally, such as gem-dihalocyclopropanes (gDHC), − benzocyclobutenes (BCB), − spiropyrans (SP), − and so forth. The properties of mechanophores, such as the activation force level, F _act, (i.e., the level of force at which the mechanophore gets activated, also known as the characteristic plateau force in experiments), can be quantitatively characterized by single molecule force spectroscopy (SMFS).

Computational methods have also been developed to study mechanophores. − The constrained geometry to simulate external force (CoGEF) method, which derives the force from the energy of a constrained geometry, is widely used nowadays to study the bond rupture under mechanical forces. , Despite its simple algorithm repeating constrained geometry optimizations, CoGEF is generally useful in qualitative discussions of mechanophore reactivity, except for a few cases reported previously. It is also used to calculate the mechanochemical response index proposed by Geerlings, De Proft, and co-workers, which is used for the description and monitoring of responses of molecules to external force. , Owing to its simple algorithm, the F _max derived from the CoGEF calculation, which represents the force associated with the mechanochemical transformation, is usually several thousand piconewton higher than the experimentally reported data. This is because practically the CoGEF is carried out as a relaxed scan with respect to an increasing head-to-tail distance, and the force-coupled transition states cannot be located in this procedure.

Therefore, quantitative approaches in which the forces are explicitly described have also been developed. Pioneering works include the ab initio steered molecular dynamics (AISMD) method developed by Martínez and co-workers , and the external force is explicitly included (EFEI) method raised by the Marx group. , Compared to the CoGEF, the development of EFEI is a significant progress, since it allows the optimization of transition state with the constraint of force. With the external tensile force properly simulated, another question then arises: how can we quantify the effect of external force on the mechanochemical reactions? In addition, because F _act is the key factor in evaluating the reactivity and performance of a mechanophore, quantitative approaches for the prediction of F _act are thus desired. In fact, to answer the first question, it is not rare to see research using an E _a–F figure to quantify the force effect for a specific chemical reaction. Such an approach can be traced back to 2009, when the EFEI method was developed. ,, More recently, Zimmerman and co-workers generated E _a–F figures by systematically locating transition states on the force-modified potential energy surface (FMPES) using a transition state search method called single-ended growing string method (SE-GSM). , Hasegawa and co-workers reported a Python-written module for mechanochemical simulations and studied the ball-milling-promoted depolymerization of chitin materials by using the E _a–F curve to describe the force effect in a quantitative discussion. ,

The well-known Eyring equation provides a relation between ΔG ^‡ and the kinetics of a reaction. Given this, the time scale (e.g., half-life, t _1/2) can be readily converted to the free energy barrier ΔG ^‡. Therefore, if the ΔG _τ ^‡–F _τ relationship is unveiled (where F _τ is the value of external force and ΔG _τ ^‡ denotes the force-coupled free energy barrier of a specific reaction under F _τ), the force level needed to trigger the mechanochemical reaction on an experimental time scale can be readily solved. Martinez, Craig and co-workers applied this concept to study the force-promoted ring-opening reaction of cyclobutene, successfully rendering the computationally predicted value of F _act using a ΔG _τ ^‡–F _τ graph combined with the Eyring equation. Shortly thereafter, we also used a ΔG _τ ^‡–F _τ relationship to predict the F _act value for the ball-milling degradation of generic polymers on a given time scale. This approach using a ΔG _τ ^‡–F _τ curve was further applied to study mechanochemical polymer degradation of double-network hydrogels. − Similar work using a ΔG _τ ^‡–F _τ relationship to quantify the force effect on the retro-Diels–Alder reaction was also reported by the Remacle group recently.

In short, to computationally derive F _act, one needs to obtain the ΔG _act ^‡ and the ΔG _τ ^‡–F _τ curve. The ΔG _act ^‡ threshold required to cause the target reaction on a given time scale can be identified using the Eyring equation, and the ΔG _τ ^‡–F _τ relationship can be obtained by locating a set of transition states for a target reaction on the FMPES with different F _τ values. ,− The most time- and labor-intensive part of this approach is the latter. This includes identifying the bonds to be rearranged when the system does not include weak bonds. Moreover, in complex mechanophores, the bond rearrangement pattern may change depending on the F _τ, as presented in this paper (see the application to cubane in Figure ), and systematic explorations of all possible reactions under F _τ are required in such cases. This paper addresses this problem in detail.

13.

Model structure of 1,3-disubstituted cubane (C13, left top), the reaction path network of C13 under a tensile force of 1800 pN (left bottom), and energy profiles for the paths starting from the initial structure (right). The reaction path network illustrates connections among local minimum structures with different SMILES representations and colored with their stability as indicated by the color code besides the network.

The artificial force induced reaction (AFIR) method developed by our group involves explicitly adding an artificial force between certain fragments inside a molecule or a complex to trigger reactions. − Normally, the AFIR method employs an artificial force to suppress the potentially interactive atoms or to dissociate a bond so that a reaction can occur, subsequently generating a force-free reaction pathway based on the force-induced reaction path. With the assistance of the artificial force, the AFIR has been proved to be an efficient method to locate the approximate transition state or to extensively explore the reaction potential energy surface in an automated manner. − Here, an extended AFIR (EX-AFIR) method was proposed and then employed for the computational exploration of polymer mechanochemistry. Indeed, as required to quickly locate transition states, an artificial force similar to that used in the conventional AFIR calculations is necessary. Moreover, to properly simulate the genuine tensile force that stretches the polymer chain, a repulsive force, F _τ, is added to the terminal groups of a polymer chain. This is the first time that two fundamentally different sets of forces have been utilized simultaneously for the rapid search of various transition states on the FMPES, making our method efficient in practice. Therefore, one can readily locate the force-coupled transition states at any given level of force, eventually rendering the ΔG _τ ^‡–F _τ relationship. Despite the mechanophores with evident scissile bonds, the EX-AFIR method is also applicable to those without obvious weak bonds (e.g., highly symmetric molecules such as cubane).

This paper first presents the theoretical background of the EX-AFIR method and subsequently introduced its application in polymer mechanochemistry. First, a series of mechanophores having weak bonds (species A, B, C and D shown in Figure ) were used to demonstrate the accuracy of our approach for rendering F _act. Notably, for the generation of mechanoradicals from polymeric materials, − our EX-AFIR method has already been employed to rationalize the experimental results observed by our groups. − The current study provides the first systematic assessment of this approach through applications to a large set of well-characterized mechanophores. A comparison between our calculation results and the experimental data confirmed that the EX-AFIR method is a simple yet reliable computational tool. Second, this method was employed to study a recently reported mechanodegradation reaction of cubane derivatives (species E in Figure ). Owing to the existence of multiple bonds that can potentially be broken, such a computational study remains a challenging task, as the existing tools require manual input of the approximate force-coupled transition states. Moreover, for one of these derivatives, the product changes between mechanochemical and thermal conditions, indicating that multiple reaction channels compete for a given force level. The EX-AFIR method, which uses two sets of forces simultaneously, allows a fully automated and comprehensive search on the FMPES, thus solving the problem. Moreover, the mechanistic insights obtained from our computational study can be used for the further design and improvement of mechanoresponsive molecules.

1.

Mechanophores computationally studied in this article (dihalocyclopropane, benzocyclobutane, spiropyran, furan-maleimide adduct, cubane).

Please note that, in this computational study, the force level required to cause a bond activation under a given condition (temperature T in Kelvin and time scale t in seconds) is denoted as F _act[T,t], together with an indication of whether the value was obtained experimentally (i.e., F _act[T,t],exp) or computationally (i.e., F _{act[T,t],calc}). The computational time scale corresponds to the half-life estimated by the Eyring equation. Common assumptions such as the standard ideal gas, rigid rotor, and harmonic approximations were also employed. Similarly, the (overall) Gibbs energy barrier, ΔG _act ^‡, and rate constant, k _act, for the bond activation under a given condition (temperature T in Kelvin and tensile force τ in piconewton) are denoted as ΔG _act[T,τ] ^‡ and k _act[T,τ], respectively, with either “exp” or “calc”. The ΔG _τ ^‡–F _τ relationship was fitted by the second-order polynomials, based on its harmonic potential model as described in Section V of the Supporting Information.

2. Theory

In this study, the EX-AFIR method was employed to facilitate an efficient exploration of the FMPES. Notably, similar to other reaction pathway searching strategies, , such as the ab initio nanoreactor developed by Martinez and co-workers, the conventional AFIR method has been extensively used to search for possible reaction pathways in organic and organocatalytic reactions. − The original AFIR concept is simple: pushing or pulling two specific intramolecular (or intermolecular) fragments using an artificial force can eventually lead to reactions. The force is introduced to the system using the AFIR function F(Q), which has the following general form

$$F(\mathbf{Q})=E(\mathbf{Q})+\rho \alpha \frac{\sum _{i\in A}\sum _{j\in B}{\omega }_{\mathit{ij}}{r}_{\mathit{ij}}}{\sum _{i\in A}\sum _{j\in B}{\omega }_{\mathit{ij}}}$$ 1

In eq , F(Q) is the AFIR function, E(Q) is the Born–Oppenheimer potential energy surface (PES) of the given geometrical parameter Q, and the second term applies the force to the system of interest. Specifically, the term α controls the magnitude of the added force. The parameter ρ can be set to either 1 or −1, where the former indicates an attractive force, and the latter indicates a repulsive force. The force in eq is given as a weighted sum of the interatomic distance r _ij between atom i in fragment A and atom j in fragment B, and the weight ω _ij is given as follows

$${\omega }_{\mathit{ij}}={\left(\frac{R_i+R_j}{r_{\mathit{ij}}}\right)}^6$$ 2

where R _i and R _j are the covalent radii of atoms i and j, respectively. The force-induced path was then optimized to a minimum energy path without mechanical force. This is realized by using a chain-of-states method, such as the locally updated planes (LUP) method, to find an actual force-free reaction path. In our study, an in-house-modified LUP method was used in the implementation. −

Experimentally, forces are added to terminal groups rather than single atoms. Here in our EX-AFIR approach, to reproduce the experimental conditions, the stretching force is added to the two terminal groups (rather than two atoms) using the AFIR function-like formula, where the force term is given as a weighted sum of the distances r _ij between atom i in the fragment L and atom j in the fragment R using the weight function ω _ij . In the EX-AFIR method, as shown in Figure , a repulsive force F _τ with a magnitude of τ is first added to the termini of the molecule of interest (as L and R), which turns into a stretching force and helps to construct the FMPES E ^FM(Q).

$$E^{\mathrm{FM}}(\mathbf{Q})=E(\mathbf{Q})-\tau \frac{\sum _{i\in \mathrm{L}}\sum _{j\in \mathrm{R}}{\omega }_{\mathit{ij}}{r}_{\mathit{ij}}}{\sum _{i\in \mathrm{L}}\sum _{j\in \mathrm{R}}{\omega }_{\mathit{ij}}}$$ 3

It is noteworthy that the computation of the force-coupled barrier ΔG _τ ^‡ is based on this FMPES, E ^FM(Q). More specifically, the parameter τ has the following formula.

$$\tau =\frac{\gamma }{[2^{-(1/6)}-{(1+\sqrt{1+\frac{\gamma }{\epsilon }})}^{-(1/6)}]·R_0}$$ 4

R ₀ and ε, represent the Ar–Ar Lennard–Jones parameters, that is, 3.8164 Å and 1.0061 kJ·mol^–1, respectively. In practice, we can specify the τ directly, and it is actually realized by adjusting the value of γ accordingly. Meanwhile, as expressed in eq , the second set of artificial force, F ₂, in which the direction and magnitude are controlled by ρ and α, is then added to the system.

$$F(\mathbf{Q})=E^{\mathrm{FM}}(\mathbf{Q})+\rho \alpha \frac{\sum _{i\in \mathrm{A}}\sum _{j\in \mathrm{B}}{\omega }_{\mathit{ij}}{r}_{\mathit{ij}}}{\sum _{i\in \mathrm{A}}\sum _{j\in \mathrm{B}}{\omega }_{\mathit{ij}}}$$ 5

The latter force, F ₂, is a purely artificial force that triggers chemical reactions, and the reaction should be discussed in terms of E ^FM(Q) rather than F(Q). This is the first time that two types of forces have been used simultaneously in AFIR calculations. Compared to the existing methods, the use of the second set of artificial forces greatly facilitates the process of locating the transition states on the FMPES. Furthermore, it also realizes the fully automated exploration of various reaction pathways on the FMPES, which has not been done previously.

2.

Two types of forces used in our EX-AFIR calculations, where F _τ represents the actual force felt by the polymer; L is the left-hand-side terminal group; R is the right-hand-side terminal group; and A and B are fragments that are pushed or pulled in the AFIR procedure to induce a reaction.

3. Results and Discussion

3.1. gem-Dihalocyclopropane System

Our study first focused on a variety of gem-DHC (gem-dihalocyclopropane) molecules that have been well-studied. − , Here, with the assistance of EX-AFIR, we can readily locate the transition state at any given F _τ. The collected data can be subsequently fitted to a curve that correlates the force-coupled free energy barrier ΔG _τ ^‡ and the force level F _τ. With such a curve, we can computationally predict the activation force level (F _{act[T,t],calc}) given the time scale of experiment.

Shown in Figure is the ΔG _τ ^‡–F _τ graph for a series of gDHCs. Each dot in the figure was derived from a full TS optimization and the following intrinsic reaction coordinate (IRC) calculation on the FMPES. It is clear from Figure that the force-coupled free energy barrier ΔG _τ ^‡ of the C–C cleavage process decreases monotonically with an increasing F _τ. More specifically, a simple model derived from the Marcus theory − suggests that the barrier decreases as a quadratic function of the force level F _τ (see Section V in SI for the derivation process). Although a linear function can also be fitted at a moderate force level, relatively large errors are expected at both low and high force levels. An inspection of the four curves shows the reactivity of these four gDHCs under the same F _τ, where gDBC_PNB > gDCC_PNB > gDBC_PB > gDCC_PB. This result is consistent with the experimental observations. Moreover, since the SMFS experiments that derived the F _act were conducted on a time scale of 0.1 s, the force-coupled free energy barrier ΔG _{act[T=298.15K,τ],calc} ^‡ should be around +68.2 kJ·mol^–1, according to the Eyring equation (i.e., corresponding to a half-life of 0.1 s at 298.15 K). Given this, the computed activation forces F _{act[T=298.15K,t=0.1s],calc} for gDCC_PB, gDBC_PB, gDCC_PNB and gDBC_PNB are 1180, 1000, 820, and 660 pN, respectively. Notably, the calculated F _act values are close to those reported experimentally (with a root-mean-squared deviation of 180 pN), as shown in Table .

3.

Fitted curves that depict the relationship between the force-coupled Gibbs energy barrier ΔG _τ ^‡ and the level of tensile force F _τ for gDCC_PB, gDBC_PB, gDCC_PNB and gDBC_PNB. Here PB means polybutadiene backbone and PNB represents polynorbornene backbone.

1. Experimental and Computed Activation Forces F _{act[T=298.15K,t=0.1s]} of Various gem-DHCs.

species	F act[T=298.15K,t=0.1s],exp (pN)	F act[T=298.15K,t=0.1s],calc (pN)
gDCC_PB	1330	1180
gDBC_PB	1200	1000
gDCC_PNB	900	820
gDBC_PNB	740	660
trans-gDCC_PB	2290	2230
gDCC_DBE	800	550
gDCC_DBZ	1160	870

In addition to the calculations concerning the halogen and backbone effects, the stereochemical effect was also investigated using the EX-AFIR method. trans-gDCC has been reported to have a much higher F _act than that of the cis-isomer. The experimental data (F _{act[T=298.15K,t=0.1s],exp}) and our computed result (F _{act[T=298.15K,t=0.1s],calc}) are 2290 and 2230 pN, respectively, which are considerably close (see Figure ).

4.

ΔG _τ ^‡–F _τ graph for trans-gDCC_PB, where the concerted transition state is more energetically favored in the low-force-level region, while the stepwise transition state is predominant in the high-force-level region.

Interestingly, our calculations revealed two different transition states for trans-gDCC: a concerted transition state and a stepwise one, the latter of which first leads to a diradical intermediate. The concerted transition state (shown in gray in Figure ), in which the transfer of chloride and C–C bond breaking occur simultaneously, has a lower barrier in the low-force-level region (i.e., with a small F _τ). However, the stepwise transition state becomes predominant once F _τ is over 1490 pN. Thus, trans-gDCC actually undergoes a stepwise transition state for activation in the SMFS experiments (please refer to the SI for the IRC curves of these two transition states).

The EX-AFIR calculations simulating the ring-opening reaction of gDCC_DBE and gDCC_DBZ were performed and the results are also shown in Figure and Table . The experimental results indicated that a CC double bond, if directly attached to the cyclopropane ring, can lead to a ring-opening reaction at a lower force level. Moreover, the stereochemistry of the adjacent alkene moiety also plays a vital role in lowering the F _act. Compared with the CC double bond in the Z-configuration, the E-alkene moiety can significantly lower the activation force to the level of 800 pN. Despite the semiquantitatively accurate data that reproduced the experimental F _act values, our calculations also showed both gDCC_DBE and gDCC_DBZ have similar reactivities under force-free conditions.

5.

Relationship between ΔG _τ ^‡ and F _τ for the ring-opening reaction of gDCC_DBE and gDCC_DBZ, which have similar reactivities in the low-force-region, and gDCC_DBE is much more sensitive toward external force.

When F _τ = 100.0 pN (see Figure ), the computed Gibbs energy barriers for the ring-opening reaction are +91.0 and +94.4 kJ·mol^–1 for gDCC_DBE and gDCC_DBZ, respectively. The higher reactivity of gDCC_DBE, is due to its higher sensitivity to the external force, which is confirmed by the much steeper slope of its ΔG _τ ^‡–F _τ curve. Therefore, the double bond in the E-configuration, if directly connected to the scissile bond, can be regarded as a structural motif that enhances the force effect.

Epoxides can be treated as an analogue to the gem-DHC system in the force-promoted ring opening reactions, although it requires much higher force to be activated. , Notably, CoGEF fails to give the qualitatively correct results for the bond rupture of epoxide under tension. Although it was difficult to measure the exact F _act in the SMFS experiments because the force required is much higher than that can be applied experimentally, the EX-AFIR calculations can deliver the F _act required to break the C–C bond on the time scale of 0.1 s. Table lists the calculated F _{act[T=298.15K,t=0.1s],calc} data for epoxides embedded in different polymer scaffolds. The calculation results clearly show that a CC bond in the vicinity of epoxide moiety can significantly increase the F _act. Notably, it was experimentally found that under a tensile force of 2500 pN, the force-coupled rate constant of epoxide_DBE or epoxide_DBZ is less than 5 s^–1, indicating a force-coupled barrier above 69.0 kJ·mol^–1. We computed the force-coupled barriers as 80.3 and 84.9 kJ·mol^–1, which are close to the experimental data. Further comparison with the ΔG _τ ^‡–F _τ curve of gDCC reveals that the much higher F _act of epoxide is owed to its intrinsically lower reactivity even under force-free conditions (see SI for details). A further inspection of the C–C bond length in both epoxide and gDCC reveals a much shorter distance in the former (1.47 vs 1.53 Å). The sensitivity toward the external force, at least for epoxide_PB and gDCC_PB, is similar.

2. Computed Activation Forces F _{act[T=298.15K, t=0.1s]} of Various Epoxides.

species	F act[T=298.15K, t=0.1s],calc (pN)
epoxide_PB	3800
epoxide_PNB	3470
epoxide_DBE	2890
epoxide_DBZ	3020

3.2. Spiropyran System

Compared to the gem-DHC system, spiropyrans are far more reactive under tensile forces. The ring-opening reaction can be achieved with an extraordinarily low force (i.e., ≈300 pN). In addition, the sensitivity of spiropyrans toward the external force F _τ can be easily manipulated by changing the position of two polymer arms or switching the substituent on the benzene ring where the C–O bond is located. Spiropyran SP1 was reportedly less reactive than SP2. , Figure shows the transition states of ring-opening processes for both SP1 and SP2 at F _τ = 500 pN.

6.

Optimized geometries of the force-coupled transition states linked to the ring-opening reactions for SP1 and SP2 (F _τ = 500 pN).

The force-coupled free energy barriers ΔG _τ ^‡ associated with these two transition states are +52.4 and +25.6 kJ·mol^–1, respectively. This clearly indicates that SP2 is much more reactive at the same force level. To furnish the ring-opening reaction on the time scale of 0.1 s, the computed F _{act[T=298.15K,t=0.1s],calc} is indeed lower than 500 pN. Our calculations also show that the open-ring product is thermodynamically less stable. An s-cis to s-trans isomerization of the newly formed butadiene moiety occurs immediately after the ring-opening step, affording a more stable merocyanine isomer (see SI for details).

Subsequently, we reproduced the experimentally observed substituent effects. The force-coupled rate constants k _{act[T,τ],exp} for a series of SP2 molecules were carefully measured by Craig and co-workers. Based on the Eyring equation, the rate constant k _{act[T,τ],exp} can be readily transformed into the force-coupled Gibbs energy barrier as shown in Table . We carried out calculations for three different SP2 molecules, namely SP2_H, SP2_Br, and SP2_NO2, to obtain the force-coupled free energy barriers ΔG _τ ^‡ under F _τ = 375 pN. As shown in Table , SP2_NO2 is the most reactive, followed by SP2_Br. Unsubstituted SP2_H, is the least reactive. Note that the process consists of two steps: the C–O cleavage, and the following s-cis to s-trans isomerization. Table presents the overall barriers that correspond to the free energy gap between the highest transition state and the reactant (see SI for more details).

3. Substituent Effect on the Ring-Opening Reaction of Spiropyran SP2 at F _τ = 375 pN.

species	ΔG act[T=298.15K,τ=375pN],exp ‡ (kJ·mol–1)	ΔG act[T=298.15K,375pN],calc ‡ (kJ·mol–1)
SP2_H	+67.5	+51.8
SP2_Br	+64.6	+49.2
SP2_NO2	+54.7	+43.1

^aObtained by transforming k _{act[T=298.15 K, τ=375 pN],exp} (= 9.0, 32.0, and 1600.0 s^–1, respectively, for SP2_H, SP2_Br, and SP2_NO2) via the Eyring equation.

3.3. Benzocyclobutane System

Another well-known mechanophore, the benzocyclobutane system (BCB), was also revisited using the EX-AFIR method. Notably, the ring-opening of cis-BCB under mechanical force was confirmed to violate the Woodward–Hoffmann rules. According to the experimental data, the thermally inert cis-isomer was found to have a lower F _{act[T=298.15K,t=0.1s],exp} (i.e., being more reactive) than the trans-isomer. Consistent with the experimental data, our calculations also suggest that the cis-isomer (cis-BCB) is more reactive than the trans-isomer (trans-BCB), whose F _{act[T=298.15K,t=0.1s],calc} were computed to be 1430 pN and 1620 pN, respectively. As shown in Table , our calculation results successfully reproduced the quantitatively consistent F _act with an error of less than ∼ 120 pN. Notably, we also found that the disrotatory ring-opening reaction was not favored for the cis-BCB in the low-force region, as the slope of the fitting curve of cis-BCB was much steeper than that of trans-BCB (see Figure S11 for more details). This indicates that the ring-opening reaction should follow the Woodward–Hoffmann rules under force-free conditions.

4. Experimental and Computed Activation Forces F _{act[T=298.15K,t=0.1s]} for Two Benzocyclobutanes.

species	F act[T=298.15K,t=0.1s],exp (pN)	F act[T=298.15K,t=0.1s],calc (pN)
cis-BCB	1370	1430
trans-BCB	1500	1620

3.4. Force-Triggered Retro-Diels–Alder Reactions

The Diels–Alder reaction is one of the most famous pericyclic reactions leading to the formation of a stereospecific six-membered ring from a diene and a dienophile. Due to the simplicity of the reactants and the complexity of the products, it therefore serves as a powerful and reliable tool in organic synthesis. − Notably, the Diels–Alder reaction is less reversible under low temperatures. Nevertheless, recently, De Bo and co-workers revealed that the retro-Diels–Alder reaction, can be realized with the assistance of mechanical force at low temperatures (5–10 °C). It was also found that the geometry of the Diels–Alder adduct strongly affects its reactivity. , Shown in Figure are the four furan-maleimide Diels–Alder adduct compounds subjected to the force-triggered retro-Diels–Alder reaction.

7.

Furan-maleimide Diels–Alder adducts subjected to tensile forces. Note that the positions of the two adjacent polymer chains and the stereochemistry of the furan-maleimide adduct are different from one to another.

We noted that De Bo and co-workers have already studied the mechanism of retro-Diels–Alder reaction using the EFEI method. They highlighted that the external force F _τ helps to transform the concerted transition state to a stepwise one. Here, we also would like to explore this force-promoted retro-Diels–Alder reaction using the EX-AFIR method, aiming at providing a prediction of the activation force level, which are currently unavailable experimentally. In our case, a repulsive force F _τ is directly added to the two terminal −CH₃ groups (see Figure ). The optimization to find minima and transition states was with this force constraint. A set of force levels (F _τ) were eventually tested, and the force-coupled free energy barriers (ΔG _τ ^‡) were successfully obtained for all four isomers (see Figure for the details).

8.

Effect of stretching force F _τ on the force-coupled free energy barrier ΔG _τ ^‡ of retro-Diels–Alder reaction for furan-maleimide.

In our EX-AFIR exploration, Distal-Exo (4c) was also found as the most force-resistant compound. Even at F _τ = 3000 pN, the barrier ΔG _τ ^‡ is still up to +72.2 kJ·mol^–1. However, the retro-Diels–Alder reaction for the other three isomers (1c, 2c and 3c) are almost barrierless with a tensile force F _τ of 3000 pN, being +10.5, +14.6 and 24.1 kJ·mol^–1, respectively. Though the SMFS data for this set of compounds is not available until now, based on the ΔG _τ ^‡ – F _τ relationship concluded in Figure , we can predict the activation force F _{act[T=298.15K,t=0.1s],calc} for 1c (Proximal-Endo), 2c (Proximal-Exo), 3c (Distal-Endo) and 4c (Distal-Exo) as 900, 1370, 1330 and 3220 pN. From the generated ΔG _τ ^‡–F _τ graph, it is also noteworthy that compound 2c is much more sensitive to the external force if compared to 3c. At high force region, 2c is more reactive than 3c, while it is more inert than 3c if F _τ is lower than 1550 pN. It is consistent to the experimental data that 3c is more reactive than 2c under thermal conditions.

Shown in Figure are the transition states of all four isomers at F _τ = 2000 pN. Similar to what De Bo and co-workers have indicated, the external force greatly affects the geometry of transition states, resulting in unequally long C–C bonds which are prone to break. For example, in 1c (Proximal-Endo), these two C–C bonds are 2.27 and 1.64 Å long, respectively. The C–C bond, which has the tensile force directly passing through, extends much longer and breaks first. The stepwise transition state is defined based on the bonding rather than the energy. According to the IRC calculations, in fact, there is no intermediate after the first C–C bond breaks on the FMPES E ^FM(Q). Based on our EX-AFIR exploration, we also confirmed there is a certain threshold for the special stepwise transition state switching to the conventional concerted transition state with equally long C–C bonds. For 1c and 2c, the force level is around 500 pN, while it is around 200 pN for 3c. For compound 4c, the switching point comes much earlier as the critical force is around 1000 pN.

9.

Optimized transition state geometries for all four isomers at F _τ = 2000 pN.

Further inspections on the geometries suggest that all the reactive isomers have a “node” at the breaking C–C bond. Regarding the definition of node, careful inspections on the structural motif shown in Figure suggest that, if we take the Newman projections concerning the breaking C–C bond, we will find the angle between two stretching forces is less than or around 90°. It can be seen in the cases of 1c (Proximal-Endo), 2c (Proximal-Exo) and 3c (Distal-Endo). However, the angle between two tensile forces is around 180° in 4c (Distal-Exo). We believe such a “node” inhibits the transduction of tensile force and is supposed to generate a torque on the bond prone to break. In the most force-resistant isomer 4c (Distal-Exo), such a node does not exist (as the dihedral angle is 177.9°), and the tensile force can pass freely through the breaking C–C bond without any hindrance. The computational study on the retro-Diels–Alder reaction shows a vivid example that how a specially designed geometry can significantly amplify the effect of mechanical force. The structural information (i.e., “nodes” on the transduction direction of force) revealed by the EX-AFIR calculations can be used for the further enhancement of force-responsive ability, even for those less reactive compounds. In fact, the concept of “node” proposed in the initial version of this paper, which was made available on ChemRxiv in 2022, led to the successful design and discovery of a mechano-reactive yet thermally and photostable mechanophore. Moreover, a similar concept was subsequently suggested in 2024 by Moore and co-workers. , Consequently, this concept is likely to be widely applicable and highly beneficial in the future design of mechanophores.

10.

Illustration of “node” along the transduction path of force F _τ.

3.5. Force-Triggered Ring-Opening Reactions of Cubanes

Cubane, a unique molecule with tremendous strain yet remarkable kinetic stability, has attracted broad attention since it was first synthesized almost 60 years ago. − Recently, Craig and co-workers have reported the first-ever case of cubane mechanodegradation at ambient temperature. Compared with the reactions carried out under thermal conditions, the external stretching force led to intriguing results. For example, instead of forming cyclooctatetraene, the force led to the formation of tricyclooctadiene, which was later detected by nuclear magnetic resonance (NMR). Moreover, the experimental outcome is highly dependent on the positions of the two polymer chains through which the tensile force is transmitted. The 1,2-disubsituted cubane was the only isomer found reactive under sonication at room temperature. A further SMFS experiment of 1,2-disubstituted cubane revealed a characteristic plateau at around 1550 ± 80 pN.

In this study, the EX-AFIR method was applied to computationally analyze the mechanodegradation reactions of cubane. In contrast to conventional mechanophores, for which we can predict vulnerable bonds under tension, the cubane derivatives, particularly 1,3- and 1,4-disubstituted cubanes, have a highly symmetric structure, and identifying the bond prone to breaking under tension is difficult. In addition, because the ring-opening reaction is expected to generate a pair of radical species, hydrogen atom transfer (HAT) reactions might also occur afterward. Consequently, it was not easy to computationally study this “complex” molecule, as the existing methods require a pre-evaluation of the scissile bonds to suggest the approximate force-coupled transition states for further optimization. In this case, the introduction of the second set of forces ideally solves this problem, as they are automatically added to the various bonds in the molecule under stress, helping to generate a reaction path network of the possible products and the approximate force-coupled transition states connecting them. The analysis of such a network can subsequently deliver the most probable reaction pattern and the F _act required. To the best of our knowledge, such a fully automated, systematic search on the FMPES has not been previously reported. Moreover, this method enables the computer-guided design of new mechanophores, as chemical bonds in the potentially reactive molecules can be systematically screened in this manner.

The left-hand-side of Figure shows the computed mechanodegradation path network of 1,2-disubstituted cubane (C12). A simulated tensile force F _τ of 1800 pN is applied to the two methylene (CH₂) groups of C12 molecule (see the red-highlighted part for how the force F _τ is added). A second set of force with α = 6000 pN, which is artificial, was then employed on the eight carbon atoms and six hydrogen atoms on the cubane moiety to trigger different reactions under F _τ. In other words, various possible products and their formation pathways were explored systematically using the extended SC-AFIR method with τ = 1800 pN (in eq ) and α = 6000 pN (in eq ). The obtained products (i.e., EQs, shown as the dots in Figure ) were further analyzed and categorized into 150 different groups based on their bonding connectivity (SMILES representation). The first step in this mechanodegradation is the collapse of the C₈ cubic structure, for which we located four different reaction patterns. The approximate force-coupled transition states obtained were further optimized and the details of the energy profiles are shown on the right side of Figure . Note that the C–H activation and the following HAT reaction can also tear down the cubic structure, but this is accompanied by an inaccessibly high barrier.

11.

Model structure of 1,2-disubstituted cubane (C12, left top), the reaction path network of C12 under a tensile force of 1800 pN (left bottom), and energy profiles for the paths starting from the initial structure (right). The reaction path network illustrates connections among local minimum structures with different SMILES representations and colored with their stability as indicated by the color code besides the network.

Our calculations successfully predicted the correct product of C12 under a stretching force of 1800 pN, as the force-coupled barrier of C _a -C _b bond homolysis was only +70.6 kJ·mol^–1, which is the lowest of all four possible pathways. Figure also shows that the covalent C _a –C _e bond is vertical to the C _a –C _b bond and directly crosses it at point a. The force-coupled barrier for breaking such a bond is +144.0 kJ·mol^–1, while the C _g –C _h bond, which is parallel to the C _a –C _b bond, has an even higher barrier of +167.9 kJ·mol^–1. Moreover, rather than stepwise homolysis, a concerted mechanism exists for the collapse of the cubic motif. The structure C12-PT92_TS shown in Figure represents such a way to break the cubic structure: however, the barrier was found as high as +289.5 kJ·mol^–1. Based on the calculation results, we believe that the ring-opening proceeds via a stepwise homolysis mechanism rather than a closed-shell concerted mechanism. Once the ring-opening transition state is located, investigating how the force-coupled barrier varies with the level of force becomes possible, allowing F _act to be computationally predicted. Based on the ΔG _τ ^‡–F _τ graph generated by our calculations (see SI for details), the F _act should reach 1910 pN for a force-coupled rate constant of 33 s^–1. This result is semiquantitatively consistent with the experimental data (i.e., 1550 pN).

It should be noted that, unlike the sonication reaction in which the tensile force is not persistent, in the SMFS experiments, the tensile force is always applied on the cubane molecule, even after the first ring-opening reaction. Although cyclooctadiene is generally kinetically stable without UV irradiation, − the tensile force promotes subsequent reactions. Figure depicts the subsequent reactions of C12-PT77_P under F _τ = 1800 pN, from which it is clear that the C _c –C _d bond along the force transmission direction is highly vulnerable. The homolytic breaking of the C _c –C _d bond under F _τ = 1800 pN is only +33.8 kJ·mol^–1, even lower than that of the first C _a –C _b homolysis. Such a lower barrier can also explain why there is only one plateau on the SMFS curve rather than two. Under SMFS conditions, the formed intermediate C12-PT77_P cannot survive, as it immediately experiences another C–C homolytic cleavage to generate the diradical species C12-PT519_P. Interestingly, further C _g –C _h bond breaking in C12-PT519_P has a force-coupled barrier up to +125.9 kJ·mol^–1. Our further calculations revealed that the barrier is still at the level of 103.6 kJ·mol^–1, even when the force increases to 2500 pN. This pathway is therefore energetically unfavored, probably owing to the formation of two unstable antiaromatic cyclobutadiene molecules. Therefore, the formation of two cyclobutadiene molecules was not observed on the given time scale of the SMFS experiments. No further rupture was detected even at F _τ = 2500 pN, which is also consistent with the experimental SMFS curve. Furthermore, our calculation results suggested that if the force is retracted, this diradical species promptly undergoes a radical–radical coupling reaction to regenerate tricyclooctadiene.

12.

Following reactions of PT77_P (tricyclooctadiene) derived from 1,2-disubstituted cubane (C12).

Similar SC-AFIR calculations were conducted for both 1,3- and 1,4-disubstituted cubane to obtain the reaction path networks on the FMPES. Figure shows the reaction path network of 1,3-disubstituted cubane (C13) under a constant force of 1800 pN.

Owing to the more symmetric structure of C13, manually identifying the scissile bond under force was challenging. This structural symmetry forces EFEI and other existing methods to iteratively optimize manually generated guesses for all possible bond cleavages individually, which can be a highly demanding task. However, in the EX-AFIR method, the second set of forces significantly facilitates and accelerates the study. The right-hand-side of Figure shows the energy profile of the first ring-opening reaction of the 1,3-disubstituted cubane, in which we observed four different transition states. The transition state possessing the lowest energy is C13_PT98_TS, and the force-coupled barrier for this cleavage pattern is +127.7 kJ·mol^–1 under F _τ = 1800 pN. Notably, two possible cleavage sites exist after the formation of the diradical species. Further calculations strongly suggest that there is a post-TS bifurcation point on the FMPES, and the formation of either product is possible. Additional calculations revealed the ΔG _τ ^‡–F _τ relationship of the C _a –C _b bond homolysis, which is actually highly insensitive to the external force applied. Even at a tensile force of 3000 pN, the force-coupled barrier is still as high as +105.7 kJ·mol^–1, strongly suggesting that such a reaction cannot occur under ambient conditions. Furthermore, based on whether the atoms can directly feel the tensile force, we can divide the cubic into Plane α and Plane β. Plane α, which consists of carbon atoms C _a , C _b , C _c and C _d , is the one having explicit force passing through. Compared with Plane α, the calculations showed that the breaking of any bond on Plane β is energetically unfavored. Similar results were found when breaking the bond connecting Plane α and Plane β (e.g., C _b –C _f bond, C13_PT34_TS), suggesting that the transmission of force is restricted within Plane α.

There are only two possible patterns for the C–C cleavage in the 1,4-disubstituted cubane, as it possesses the most symmetric structure (see Figure ). In C14_PT48_TS, C _a –C _b bond breaks at the node where the force is applied. While in C14_PT72_TS, the breaking C–C bond is not connected to any of the nodes where the force is directly applied. The calculation indicated that the former C–C cleavage pattern has a relatively lower barrier, being +126.9 kJ·mol^–1 under F _τ = 1800 pN. Furthermore, the following calculations revealed that the C–C homolytic cleavage in the most symmetric C14 is even more insensitive to external forces. Even at F _τ = 3000 pN, the C–C homolytic cleavage still possesses a barrier of +120.6 kJ·mol^–1. If the ΔG _τ ^‡–F _τ curve is fitted with a linear equation, the slope of C12 is 4.22-fold if compared with C14 (see Figure , 1000 pN < F _τ < 2000 pN). From this perspective, we can conclude that the greater the number of branches between two nodes where the force is applied, the less the branched bonds will be affected. Overall, our calculation results are consistent with the experimental observation that only 1,2-disubstituted cubanes can undergo intramolecular reactions, whereas 1,3- and 1,4-disubsititued cubanes are chemically inert under ambient conditions, even when a stretching force is applied. In addition, from the ΔG _τ ^‡–F _τ curve, we can see that the C _a –C _b bond in 1,2-disubstituted cubane is intrinsically weaker (probably due to the electronic effect of two carboxylic groups), as the barrier for homolytic cleavage at F _τ = 0 pN is +123.1 kJ·mol^–1, which is lower than that of the 1,3- and 1,4-disubstituted cubanes. Such a difference in reactivity under force-free conditions was also observed by Craig et al. using in situ NMR.

14.

Model structure of 1,4-disubstituted cubane (C14, left top), the reaction path network of C14 under a tensile force of 1800 pN (left bottom), and energy profiles for the paths starting from the initial structure (right). The reaction path network illustrates connections among local minimum structures with different SMILES representations and colored with their stability as indicated by the color code besides the network.

15.

ΔG _τ ^‡–F _τ for three different disubstituted cubanes: curves fitted to a quadratic function up to 3000 pico-newton.

Cyclooctatetraene was the only product obtained under thermal degradation conditions. Given this, we also explored the possible pathways for the generation of cyclooctatetraenes on the FMPES of all three cubane isomers. According to our calculation results, for the 1,3- and 1,4-disubstituted cubanes, although the following pathways were all found to have easily accessible barriers, the first step in breaking down the cubic structure was not energetically favorable (see SI for details). Therefore, the formation of cyclooctatetraene is not possible, because the cubane is kinetically stable under the given conditions (e.g., at room temperature). For 1,2-disubstituted cubane (C12), the existence of force diverts the reaction pathway and prevents the formation of cyclooctatetraene. More specifically, once the tricyclooctadiene species is generated, the tensile force induces the C–C bond along the force-transmission direction (i.e., C _d –C _c bond), rather than the C–C bond vertical to the force direction (i.e., C _c –C _g bond), to be immediately cleaved. Figure shows the ΔG _τ ^‡–F _τ curves for each C–C bond cleavage pattern in syn-tricyclooctadiene, from which we can clearly see a force whose level is above a certain threshold (i.e., F _τ > 440 pN) can divert the reaction pathway to a different direction. However, in the low-force-region (F _τ < 440 pN), although the C _c –C _g bond cleavage has a lower barrier compared with that of C _c –C _d bond, the barrier of the former is now greater than 100 kJ·mol^–1, which means the reaction is kinetically blocked (i.e., extremely slow) under ambient conditions. Therefore, cyclooctatetraene cannot be formed in the presence or absence of an external tensile force. Further calculations concerning anti-tricyclooctadiene species are also incorporated into the SI for reference.

16.

ΔG _τ ^‡–F _τ graphs for C–C bond cleavages in syn-tricyclooctadiene, illustrating the kinetic stability of syn-tricyclooctadiene under low or force-free conditions (i.e., cannot form cyclooctatetraene at room temperature) while a diradical species will be formed via C _c –C _d cleavage if the force is above 440 pN.

The computational exploration of cubane mechanodegradation presented here is not only a challenging task that is barely possible using the existing computational tools, but also a vivid case demonstrating the performance of the EX-AFIR method. From the calculation results, we gained a better understanding of the reaction mechanism for such a highly symmetric molecule, and we believe that the EX-AFIR method can be used to efficiently screen novel mechanophores without obvious scissile bonds.

4. Conclusions

The extended AFIR method was applied to study the force-assisted ring-opening reaction of a variety of mechanophores and successfully rendered the ΔG _τ ^‡–F _τ relationship for each of them. Combined with the Eyring equation, our method semiquantitatively reproduced the activation forces obtained from the SMFS experiments. Compared with the widely used CoGEF method, which delivers an F _max being several thousand piconewtons deviating from the real F _act, our method can offer a semiquantitatively correct prediction of the F _act. The simultaneous utilization of two sets of forces significantly facilitates the location of force-coupled transition states, transforming the traditionally time-consuming task into a feasible and feasible process. In this regard, the EX-AFIR method was further employed to solve the practical problems encountered in this field, including an automated and comprehensive search of reaction pathways on the FMPES for a recently reported cubane mechanophore. As discussed in the article, our calculations not only reproduced the experimental observations, but also revealed the mechanistic details of this mechanodegradation reaction, including those of the less favored pathways. The reason why the reaction did not form the thermal decomposition product, cyclooctatetraene, was well explained. For the 1,2-disubsituted cubane, it is the tensile force that prevents the formation of cyclooctatetraene. While for 1,3- and 1,4-disubstituted cubanes, the first step of breaking down the C₈ cubic structure is not possible at ambient temperature, even if a tensile force up to 2000 pN is applied. In addition, we provided solid computational evidence that covalent bonds in a highly symmetric geometry are less sensitive to external forces. The case study clearly demonstrated that, instead of proposing the possible cleavage sites, which is time-intensive and less accurate, the EX-AFIR method allows us to systematically search for various possible reactive sites under an external force. We believe that the EX-AFIR method will be helpful for the future design of unconventional mechanophores, even for some challenging cases, where heterolysis happens or flex-activated mechanophores get activated. − It should be noted that the same calculation can be performed using the latest distribution version of our GRRM program, i.e., GRRM23, combining the Gaussian 16 program.

5. Computational Details

All the ΔG _τ ^‡–F _τ relationship calculations were performed using the density functional theory (DFT) with the B3LYP hybrid functional , as implemented in Gaussian 16. To describe the dispersion properly, an explicit dispersion correction term called Grimme-D3, , was also employed in the DFT calculations. The 6–311G­(d,p) basis set , was used for all atoms (except Br) involved in this study during both the geometry optimization and the single-point calculation processes. The Stuttgart/Dresden pseudopotential basis set SDD, − as well as a d-polarization function (ζ = 0.428), was used for the Br atom. To describe the open-shell singlet species involved in the homolysis process, an unrestricted DFT, as well as a procedure to test and optimize the wave function were used in this work. To describe the solvation environment in which the molecule resides, the implicit solvation model, IEF-PCM, was applied to all the calculations involved in this study with toluene (ε = 2.3741) as the solvent. Unless otherwise specified, all the structures shown in this article were fully optimized with the consideration and constraint of external tensile force. The external tensile force, F _τ, was properly simulated through our extended AFIR method. A second set of forces, which were artificial and finally eliminated after calculation completion, was added simultaneously to allow the search for reaction pathways on the constructed FMPES. Frequency calculations were carried out for minima and saddle points on the FMPES using the same level of theory associated with the optimization at 298.15 K and 1 atm. All the geometries shown in this article were visualized using the CYLview software. The calculated activation forces, F _{act[T,t],calc}, were rounded to the nearest tenth.

The SC-AFIR automated pathway explorations on the cubane FMPESs were performed at the DFT level of theory with the BP86 pure functional , as implemented in TURBOMOLE 7.6. To properly describe the dispersion, the Grimme-D3 correction term was also used in the DFT calculations. The Def2-SV­(P) basis set was used for all atoms involved. To describe the open-shell singlet species involved in the homolysis process, the unrestricted DFT was also used during the exploration. These calculations were performed in the gas phase. Upon completion of the automated exploration, kinetically important pathways were extracted from the obtained reaction path networks, and the associated ΔG _τ ^‡–F _τ relationships were further investigated with the computational treatments described in the previous paragraph.

Basis sets and functional dependencies of our method have also been tested, and our results indicate that using larger basis sets is beneficial to promoting the accuracy of the predicted F _act value. Meanwhile, functionals with a higher Hartree–Fock exchange fraction (e.g., M06–2X) intend to overestimate F _act significantly (see SI for more details).

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacs.5c06150.

• ΔG _τ ^‡–F _τ curves of a series of mechanophores; computed free energy profiles of mechanochemical transformations using constraint of tensile forces; derivation of the equation that correlates barrier height ΔE ^‡ and force strength τ; basis sets and functional effect on the ΔG _τ ^‡–F _τ curve (PDF)

• Cartesian coordinates of all force-coupled intermediates and transition states, as well as the associated force-coupled energies (XYZ)

The authors declare no competing financial interest.