Force Dependent Barriers from Analytic Potentials within Elastic Environments

Abstract

Bond rupture under the action of external forces is usually induced by temperature fluctuations, where the key quantity is the force dependent barrier that needs to be overcome. Using analytic potentials we find that these barriers are fully determined by the dissociation energy and the maximal force the potential can withstand. The barrier shows a simple dependence on these two quantities that allows for a re‐interpretation of the Eyring‐Zhurkov‐Bell length $\Delta x^{‡}$ and the expressions in theories going beyond that. It is shown that solely elastic environments do not change this barrier in contrast to the predictions of constraint geometry simulate external force (COGEF) strategies. The findings are confirmed by explicit calculations of bond rupture in a polydimethylsiloxane model.

Force dependent barriers for breaking of Si−C bonds and the pulling positions indicated.

Introduction

Theoretical modelling of the behavior of materials under the influence of external forces can give important insights into mechanochemical processes.[ ¹ , ² ] These are extremely hard or impossible to observe in the experiment directly.[ ³ , ⁴ , ⁵ , ⁶ , ⁷ , ⁸ ] Mechanochemical modelling is faced with an intrinsic multi‐scale problem ^[9] as forces are often applied at macroscopic scales, but bonds on atomic scales are broken. ^[10] Even in single molecule stretching experiments[ ⁶ , ¹¹ , ¹² ] polymers of hundreds of monomers are stretched and a single bond eventually breaks. The modelling problem is further complicated by the fact that finite temperature is usually responsible for bond breaking.[ ⁷ , ¹³ , ¹⁴ ] This causes the observed forces to depend crucially on the speed of force increase, where thermally activated bond rupture dominates for slow pulling, but extremely fast force increase may trigger activationless bond breaking. ^[10]

A theoretic description has to reduce the complexity of an experimental situation, where it was shown that in some cases the properties of a single monomer are sufficient to describe the mechanochemical properties of the full polymer. ^[15] Changing side groups ^[16] or also cis‐trans configurations ^[17] can have a large impact on the observed rupture force, however. Molecular junctions[ ¹⁸ , ¹⁹ ] or knots ^[20] need the description of larger portions of the polymer. This leaves the question about the role of the environment coupled to the bond on its vulnerability under the action of the external force.

The nature of force induced reactions can be theoretically investigated by various methods,[ ¹ , ²¹ ] where the constrained geometry simulates forces (COGEF) method introduced by Beyer ^[22] is popular.[ ¹⁴ , ²³ , ²⁴ ] While COGEF leads to qualitatively correct chemistry for many reactions (with exceptions), ^[24] the rupture forces obtained are usually to too large as compared to experiment. What is missing is the inclusion of finite temperature effects as COGEF simulations operate at zero temperature.

Finite temperature can be considered through transition rate theory,[ ¹⁴ , ²² ] where the force‐free potential energy surface (PES) governing the molecular structure is modified by the external force.[ ¹ , ²⁵ ] This also modifies the barrier for bond opening as was clear already in the first COGEF application. ^[22] Force modified activation energies ^[21] result in force dependent transition rates that determine the forces observed in the experiment.

The key for understanding of mechanochemistry of bond breaking is thus the understanding of the force dependence of the corresponding barriers.[ ¹ , ¹⁴ ] Often analytical expressions are used for the force dependence of the barrier, ^[26] where the parameters entering are determined by fitting to experimental data or by explicit numerical simulations in various levels of sophistication.[ ⁷ , ²⁷ ]

Here we study the force dependent barriers resulting from bonds described by analytic potentials, where we find intriguing similarities in the barriers for rather different potentials. ^[28] A simple, solely elastic environment is shown to have no influence on the form of the barrier at all. We also present density functional theory calculations of the barriers for bond rupture in propane and a model polydimethylsiloxane (PDMS) molecule where the force dependent barriers closely follow the behavior seen in the analytical potentials.

Results and Discussion

Action of an External Force

The COGEF strategy constrains the distance d between two chosen atoms of a molecule (usually a pair of atoms the are thought to be clamped to the external force) while all other degrees of freedom are allowed to relax. Recording the energy of the constrained configurations for different d leads to the corresponding inner energy U(d). The force F needed to fix the molecule at the given d is then obtained by the derivative of the potential, i.e.

$${\displaystyle F\left(d\right)=\frac{\partial U\left(d\right)}{\partial d}.}$$ (1)

We define the force as the positive derivative here which is not the force exerted by the atoms in the potential, but rather the external force needed to stretch the potential to the length d. The force is denoted as scalar quantity. Its direction is along the connection between the two atoms and in opposite directions through the actio est reactio principle. ^[21] The bond spontaneously breaks in COGEF simulations if F(d) exceeds the maximal force $F_{\mathrm{m}ax}$ and this force corresponds to the zero temperature limit for bond breaking. At the corresponding bond breaking point[ ²⁹ , ³⁰ ] the reactant minimum and transition state maximum collapse.[ ¹⁴ , ³⁰ ]

The situation of a bond in a material under strain is often better described by the constraint of a constant external force F than by an outer constraint length d.[ ³¹ , ³² ] Force F and length d are conjugated variables, ^[1] where we may always define F(d), but d(F) is generally not unique. Under the constraint of a given external force the relevant quantity is not anymore the internal energy, but the enthalpy that includes the work done by the force[ ³³ , ³⁴ ]

$${\displaystyle H(F,d)=U\left(d\right)-Fd}$$ (2)

with $d=d\left(F\right)$ at stationary states. Note, that only scalar quantities appear here as the force is thought to be always pointing into the direction of the connection vector between the two atoms chosen. Rotation of the molecule under a change of d is therefore assumed to have no energy contribution. The similarity of the enthalpy with the usual definition through pressure and volume and the connection of Eq. (2) as Legendre transformation ^[1] of U(d) is further elucidated in Supporting Information (SI). It is useful to keep the variable d as argument in H despite that the independent variable in the enthalpy is F and $d=d\left(F\right)$ holds.

Isolated Bonds

In the first step we consider a single bond[ ²² , ³⁵ ] (or a more general reaction coordinate of similar behavior) and its corresponding potential V. We model V by three different analytic potentials: a Morse potential as is often used in mechanochemistry simulations,[ ¹³ , ²⁸ , ³⁶ ] a Lennard‐Jones (12‐6) (LJ) potential ^[37] and a cut harmonic (cH) potential[ ³⁸ , ³⁹ ] as detailed in SI. The cH potential is “cut” (i. e. set to a constant when the dissociation energy is reached) to enable the description of bond breaking. The simplicity of the cH potential allows for straightforward analytic considerations below.

A bond between two atoms is usually characterized by its properties in near equilibrium situations. These are the equilibrium bond length r ₀, the stiffness described by the corresponding spring constant k_b (or the compliance $k_b^{-1}$ ) and the dissociation energy D_e . Morse and cH potentials allow for an arbitrary choice of r ₀, but this quantity is connected to k_b and D_e in the more restricted LJ potential (there is only one unique LJ potential, all other LJ potentials can be obtained by scaling energy and bond lengths ^[40] ). The actual value of the equilibrium bond‐length r ₀ is not important for a study of the action of forces, such that we characterize the potentials by the elongation $b=r-r_0$ in what follows. We will furthermore see, that the maximal force, i. e. the maximal derivative of the potential, $F_{\mathrm{m}ax}$ , plays a central role: potentials can be equally well characterized by D_e and $F_{\mathrm{m}ax}$ instead of D_e and k_b as is shown in Table S1 in SI, a choice more suitable in the realm of mechanochemistry.[ ²² , ²³ ]

The action of an external force on the three potentials is compared in Figure 1a), where we have chosen the parameters $D_e=1$ eV and $F_{\mathrm{m}ax}=3$ eV/$\AA \approx$ 4.8 nN for all potentials consistently (the resulting k_b are given in Table S1). The shape of the force transformed potentials $H(F,b)=V\left(b\right)-Fb$ varies substantially despite the common choice of D_e and $F_{\mathrm{m}ax}$ . The tilt from the $-Fb$ contribution in $H(F,b)$ creates a local minimum at small b and local maximum at somewhat larger b (the two solutions of $\partial V\left(b\right)/\partial b=F$ , c.f. Figure S1b in SI). The enthalpy difference of these states defines the barrier $\Delta H^{‡}$ that has to be overcome by temperature ^[10] to break the bond. This barrier is D_e in the limit of zero external force and vanishes when the external force reaches $F_{\mathrm{m}ax}$ . The value of this barrier on forces between zero and $F_{\mathrm{m}ax}$ can be evaluated analytically for Morse ^[28] and cH potentials (see SI) and numerically for the LJ potential. In case of the cH potential the force dependent barrier $\Delta H^{‡}$ takes the particularly simple form

$${\displaystyle \Delta H^{‡}=D_e{(1-f)}^2}$$ (3)

Figure 1.

a) The potentials including the work done by the external force of value $F=0.4F_{\mathrm{m}ax}$ . The corresponding barriers are indicated. b) Force dependent relative barriers for the three potentials considered. Linarization and as proposed in Ref. [41] are shown also.

where $f=F/F_{\mathrm{m}ax}$ . Figure 1b) compares the the force dependent barriers of all potentials. Despite the differences in the potentials in Figure 1a), their force dependent barriers are all very similar ^[28] and decrease smoothly from D_e at $F=0$ to zero at $F=F_{\mathrm{m}ax}$ . Solely in the small force range, LJ and Morse barriers decrease slightly faster than the barrier of the cut harmonic potential. ¹

Elastic Environment

The bond that breaks can not be directly addressed in real situations. Bonded atoms are always connected to other atoms within the molecule, polymer or material from which the force is transferred. ^[9] We want to address the impact of this situation on the barrier for bond‐breaking in a simple setup now. Generally, the probability to break is largest for the most scissile bonds, while other, stronger bonds still answer elastically at a given external force. We therefore approximate the mechanical behavior of the rest of the material as an effective spring with spring constant k_s connected to the chosen bond. This spring represents many bonds within the material, such that rather generally $k_s<k_b$ , despite that each of these single elastic bonds may be stiffer than the scissile bond under observation. The picture of a purely elastic environment that imposes no directional constraint on our bond is an approximation to most real situations, but can be surprisingly accurate as we will see in the example of polydimethylsiloxane below.

The picture adopted is further illustrated in Scheme 1. The corresponding potential depends on the extension b and the total extension d (as in the case of the single bond, these extensions are deviations from the equilibrium positions in the absence of an external force r ₀ and l ₀, respectively) and is given as

$${\displaystyle U(b,d)=V\left(b\right)+\frac{k_s}{2}{(d-b)}^2.}$$ (4)

Scheme 1.

Two spring model with the spring of the bond with spring constant k_b , the spring of the rest of the molecule with spring constant k _s and the external force F. The lengths b and d are deviations from the equilibrium lengths in the absence of force r ₀ and l ₀, respectively.

According to Eq. (1) the external force is $F=k_s(d-b)$ . Fixing d in a COGEF simulation and relaxing b (i. e. requiring $\partial U(b,d)/\partial b=0$ at given d) leads to

$${\displaystyle d=b+\frac{1}{k_s}\frac{\partial V\left(b\right)}{\partial b}.}$$ (5)

This equation is not particularly useful as it can not be inverted directly. It nevertheless shows that the total extension d gets equal to b for very stiff springs ($k_s\to \infty$ ), but $d\gg b$ for small k_s (the usual limit).

Figure 2 shows the result of a COGEF simulation for our model potentials coupled to a soft spring with $k_s=0.2$ (i. e. much softer than the spring constants k_b of the potentials themselves, c.f. Table S1). The use of different optimizers as implemented in scipy ^[42] reveals the surprising effect that the bond‐breaking point and therefore also the maximal energy observed in the COGEF simulation varies dramatically with this choice. The softer cH potential generally breaks at smaller distances than the LJ and Morse potentials for all the optimizers, but shows similar differences between the optimizers themselves. The underlying reason for this remarkable variation with optimizers is, that differences in their numerical settings allow to overcome small barriers as seen in Figure 2d–f). Numerical parameters may therefore act as an effective “temperature” which allows to overcome small barriers. In this case, only a fraction of the maximal force is obtained by taking the derivative at the point of the discontinuity.

Figure 2.

a–c) COGEF simulation for $k_s=0.2$ with all potentials for different optimizers. ^[42] The optimizers were used with default settings and an initial guess value of $b=0$ . The maximal derivative is given as fraction of F_max at the discontinuity. d–f) Potentials for the given values of d. The corresponding minima found by the conjugate gradient optimizer are indicated red dots.

The sudden energy drop in Figure 2a–c) is a very common phenomenon in COGEF simulations[ ¹⁴ , ²⁴ ] and is seen even in stretching small molecules like ethane. ^[43] The cH potential allows for further analytic considerations to shine light on this effect. The stationary state describing the initial minimum at a given d is found at

$${\displaystyle b=\left\{\begin{array}{c}\frac{k_s}{k_s+k_b}d\mathrm{for}b\le b_{max}=2D_e/F_{max}\\ d\mathrm{else}\end{array}\right.}$$ (6)

in the cH potential (c.f. Eq. (5)). The elongation b is a fraction of the total elongation d as long as the bond is still intact and this fraction decreases with decreasing k_s (softer springs). Accordingly, the softer of the two springs stores a larger fraction of the elastic energy (c.f. Eq. (S13) in SI). Following this potential minimum, we would have to elongate the full structure up to

$${\displaystyle d_{\mathrm{m}ax}=\frac{k_s+k_b}{k_s}{b}_{\mathrm{m}ax}}$$ (7)

until the bond breaks spontaneously, i. e. without barrier. This corresponds to an energy of

$${\displaystyle U(b_{\mathrm{m}ax},d_{\mathrm{m}ax})=\frac{k_s+k_b}{k_s}{D}_e}$$ (8)

which is always larger than D_e and much larger for the usual case of $k_s\ll k_b$ . The extra energy is consumed by elastic deformation of the environment. Figure 2 shows that even the “downhill symplex” optimizer, which shows the largest elongation before bond breaking, does not reach the theoretical value of $d_{\mathrm{m}ax}\approx 15.67$ $\AA$ for our chosen parameters.

The underlying reason for the large accumulation of elastic energy seen in Figure 2 is the lack of consideration of finite temperature where force dependent barriers come into play again. Taking the external force F as constraint, we have to switch to the enthalpy, Eq. 4

$${\displaystyle H(b,F,d)=V\left(b\right)+\frac{k_s}{2}{(d-b)}^2-Fd}$$ (9)

where again $d=d\left(F\right)$ at stationary states. $\partial H/\partial d=0$ gives

$${\displaystyle d=\frac{F}{k_s}+b}$$ (10)

as can be expected for the spring in Figure 1. The requirement of the enthalpy to be stationary in b ($\partial H/\partial b=0$ ) and inserting d according to Eq. (10) immediately leads to

$${\displaystyle F=\frac{\partial V\left(b\right)}{\partial b}.}$$ (11)

This is the same requirement as for the potential without the additional spring and is therefore independent of the value of k_s . We hence arrive at exactly the same picture for breaking the bond via the barrier $\Delta H^{‡}$ as for the bond without coupling to the environment. This means that the barrier for bond breaking is completely determined by the properties of V(b) alone, which is not what the COGEF simulations in Figure 2 predict.

How can this problem be solved in simulations? Correct barriers for a given external force are obtained by the consideration of the full force transformed potential, e. g. by the external force is explicitly included (the EFEI method[ ¹ , ⁴⁴ ]) or related variants.[ ⁴⁵ , ⁴⁶ ] This leads to stationary states (called Newton trajectories by some authors[ ²⁹ , ³⁰ ]) that include reactant minima as well as the transition states. ^[14] These states can e. g. be found by nudged‐elastic‐band calculations[ ⁴⁵ , ⁴⁷ ] if the product state is still bound. ^[48] Some of us have developed the 3S‐COGEF method ^[14] to obtain such force dependent barriers effectively also for unbound final states. In brief, a usual COGEF simulation is performed that provides force dependent reactant minima. These are used as starting points for the calculation of the barrier for breaking of a specific bond at the corresponding external force. The barrier is obtained by elongation of this bond‐length under the constraint of the given external force in an EFEI approach. Numerical evaluation of the barriers for bond breaking in Morse and LJ potentials using the 3S‐COGEF method coupled to springs for a wide variety of k_s confirm the independence of the barrier from the environment numerically (see Figure S5 in SI). Accordingly, also in these potentials the force dependent barrier is completely independent of the value of the spring potential.

The form of the barrier from Eq. (3) determined by the values of D_e and $F_{\mathrm{m}ax}$ from the most scissile bond should therefore be a good approximation for the force dependent barriers in extended materials. Similar forms were proposed in the literature as ^[3] $\Delta H^{‡}=D_e{(1-f)}^n$ with exponents $n=1$ or $n=3/2$ in tribology. ^[41] The appearance of exponents n smaller than two is not supported by Morse and LJ potentials that show a rather slightly steeper decrease of the barrier with increasing f, however.

Connection to Bell Theories

Eq. (3) furthermore allows an alternative interpretation of the meaning of $\Delta x^{‡}$ from the Eyring‐Zhurkov‐Bell (EZB) theory. The usual interpretation of $\Delta x^{‡}$ is the distance between reactant and transition state on the reaction coordinate ^[1] (b here). This is the case if the potential is harmonic in both states in the absence of an external force (see also SI). The distance itself changes strongly by the force applied, ^[49] however. The variation of this distance with f is particularly apparent for Morse and LJ‐potentials, where small forces produce small tilts in the potential $H(F,b)$ (c.f. Figure S1) such that the position of the barrier b_TS tends to infinity while the position of the reactant minimum b_R tends to zero in the limit of $f\to 0$ (this effect is directly seen in the crossing points of the external force F with force vs elongation in Figure S1). It is straightforward to show that $b_{TS}-b_R=b_{\mathrm{m}ax}{(1-f)}^2$ for the cH potential. The force dependent distance between initial minimum and barrier for the three potentials depicted in Figure 3 shows that the assignment of a fixed value for $b_{TS}-b_R$ would be misleading. An alternative picture for $\Delta x^{‡}$ is revealed by linearization of Eq. (3) leading to

$${\displaystyle \Delta H^{‡}=D_e-F\frac{2D_e}{F_{\mathrm{m}ax}}+D_e{f}^2}$$ (12)

Figure 3.

Dependence of the distance between the transition state and the reactant minimum on f. The horizontal dotted line indicates $2D_e/F_{\mathrm{m}ax}$ .

from which we identify

$${\displaystyle \Delta x^{‡}=\frac{2D_e}{F_{\mathrm{m}ax}}.}$$ (13)

This quantity has the unit of length, but describes the ratio of the dissociation energy D_e to the maximal steepness of the potential $F_{\mathrm{m}ax}$ rather than a distance. It agrees with the distance between reactant minimum and transition state ^[50] in the cH‐potential at zero force only.

In the same spirit, Eq. (3) also gives a connection to extended Bell theory (EBT),[ ²¹ , ⁴⁹ , ⁵¹ , ⁵² ] where the barrier is expressed as

$${\displaystyle \Delta H^{‡}=D_e-F\frac{2D_e}{F_{\mathrm{m}ax}}+\Delta \chi \frac{F^2}{2}}$$ (14)

with Δχ having the unit of a compliance[ ⁴⁹ , ⁵³ ] and is given by the difference between compliances of reactant and transition states in the harmonic approximation[ ⁵⁰ , ⁵² ] (see also SI). Δχ is equal to the inverse spring constant in the cH potential in the reactant state, but this is not the case for Morse and LJ (c.f. Table S1). We identify $\Delta \chi =2D_e/F_{\mathrm{m}ax}^2$ from Eq. (14) which is completely defined from D_e and F_max and applies for all three potentials to good approximation.

The interpretation of D_e and $F_{\mathrm{m}ax}$ as central quantities allows for further simplifications if a constant force increase can be assumed as is often the case in the experiment.[ ⁵ , ⁷ , ¹⁰ , ³² ] Bond breaking is a stochastical process induced by finite temperature with its probability determined by the force dependent barrier. The intrinsic randomness of temperature fluctuations leads to the measurement of force distributions, but the most probable force for bond breaking $F^{*}$ can be predicted. The simplicity of the force dependent barrier from Eq. (3) implies that $F^{*}$ solely depends on a restricted set of parameters. These are firstly two time‐scales, where the bond's dissociation energy D_e specifies a theoretical life‐time ² of the bond ${\tau }^{-1}=A\exp (-\beta D_e)$ . Here A is the pre‐exponential factor ^[54] and β ⁻¹ is thermal energy. Another timescale is defined by the constant force increase $\alpha =dF/dt$ (the loading rate).[ ¹⁰ , ¹⁴ ] The loading rate constitutes the time needed to reach $F_{\mathrm{m}ax}$ from $F=0$ as $t_{\mathrm{m}ax}=F_{\mathrm{m}ax}/\alpha$ (there is no barrier anymore for $F=F_{\mathrm{m}ax}$ and the bond breaks spontaneously). Hanke and Kreuzer ^[28] derived using the barrier of Eq. (3) that the most probable force measured in the experiment follows

$${\displaystyle 1-f^{*}=\frac{t_{\mathrm{m}ax}}{\tau }exp(-\beta D_e{f}^{*})}$$ (15)

where $f^{*}=F^{*}/F_{\mathrm{m}ax}$ . Neglecting small values of $f^{*}$ on the left side of Eq. (15) leads to the expression for the most probable force as[ ³³ , ⁵⁵ ]

$${\displaystyle f^{*}=\frac{1}{2\beta D_e}log\left(2\beta D_e\frac{\tau }{t_{\mathrm{m}ax}}\right),}$$ (16)

which is very similar to the high velocity result of Hummer and Szabo ^[38] (see also SI). Notably, this equation just involves ratios of clearly defined energies (D_e and β ⁻¹) as well as time‐scales (τ and $t_{\mathrm{m}ax}$ ) that depend only on $D_e,F_{\mathrm{m}ax}$ , loading rate and temperature.

Real Molecules

Force dependent barriers of real molecules studied in the literature were indeed found to roughly follow the form of Eq. (3), such as the outward‐pathway ^[56] or disrotatory pathways of ring‐opening reactions,[ ⁹ , ⁵⁷ , ⁵⁸ ] and β‐rotation in the spiropyran‐merocyanine‐transition. ^[7] As predicted by the spring model described above, variation of the environment via the connecting chain molecules have little influence on the barriers themselves changing only the corresponding value of $F_{\mathrm{m}ax}$ (determined by the force value where the barrier vanishes) slightly. ^[57] The finding is also in line with the effect that a single monomer is sufficient to characterize the force response of the polymer chain. ^[15] We want test Eq. (3) in more realistic situations. To this end, we employ density functional theory (DFT) calculations of real molecules under the influence of external forces.

We first study the influence of spin‐polarization on the force dependent barriers as a broken bond produces radicals. We apply the COGEF strategy to propane as shown in Figure 4a), where we elongate the distance between the outer carbon atoms in the molecule and study the barrier to break one of the equivalent carbon‐carbon (CC) bonds. Bond‐rupture results in two radical species that have unpaired electrons. While these unpaired electrons can still form a singlet state, ^[59] the energy of such states pose severe problems for intrinsically single Slater determinant approaches like Kohn‐Sham DFT.[ ⁶⁰ , ⁶¹ ] Spin polarized treatment (spin unrestricted calculations) sometimes improves energetics, but results in unphysical localized spins ^[62] as visible in Figure 4b).

Figure 4.

Structure of propane a) in the ground state (C: grey, H: white) and b) in the broken form (the colors indicate localized spin contributions). Spin polarized and unpolarized c) COGEF energies and d) force dependent barriers.

The potentials of unpolarized and spin‐polarized calculations are compared in Figure 4c). We have restricted the total spin‐projection $S_z=0$ in the spin polarized case. No discontinuities are found in the potentials of this small molecule. Both potentials are very similar for small elongations, but differ substantially at larger distances. Relaxation in the local spin‐degrees of freedom allows the spin polarized separation energy of $D_e=4.05$  eV to be 1.01 eV lower than the unpolarized value. In contrast, the maximal force $F_{\mathrm{m}ax}=$ 5.76 nN is not affected by spin‐polarization as the electronic state at the maximal derivative of the potential is still spin‐paired (see also Figure S3c).

Figure 4d) shows the force‐dependent barriers for CC bond breaking evaluated with different methods. The symbols were obtained by a 3S‐COGEF ^[14] investigation while the solid lines are barriers calculated from the minima and maxima of $U_F\left(d\right)=U\left(d\right)-Fd$ using U(d) from Figure 4c). Due to the absence of discontinuities in U(d), the two methods lead to the same result. Broken and dotted lines are the analytic functions $\Delta H^{‡}=D_e{(1-F/F_{\mathrm{m}ax})}^2$ [Eq. (3)] with the two differing D_e values for spin‐polarized and spin‐unpolarized calculations, respectively. Interestingly, the more correct spin polarized barrier follows its own analytic expression closely for small forces, but goes over into the analytic expression of the spin‐unpolarized case for larger forces. This shows the somewhat counter‐intuitive effect, that the nature of the transition state for small forces is determined by the property of the potential at large distances (i. e. the radical solution), while the nature of the transition state at large forces is more similar to the spin‐paired ground state.

We finally apply the description developed to stretching of a small polydimethylsiloxane (PDMS) model molecule, which is a common substance and is extensively studied in triboelectric nanogenerators. ^[63] It is not clear how charge transfer in this process is generated[ ⁶⁴ , ⁶⁵ , ⁶⁶ ] and there is a conjecture that mechanochemical bond‐breaking might be a key issue.[ ⁶⁴ , ⁶⁷ ]

We model PDMS as CH₃[Si(CH₃)₂O]₃Si(CH₃)₃ in the structure shown in Figure 5a). The Si−C bond is the energetically weakest bond in the system with $D_e=$ 3.95–4.0 eV in a spin‐unrestricted calculation as compared to 4.57–4.96 eV for C−H and 5.32–5.49 eV for Si−O, respectively (see SI for details). The small variation in bond energies when removing methyl groups on different Si atoms evinces the locality of this bond. In contrast to propane, it was not possible to find the spin polarized broken radical state for the Si−C bond in the COGEF calculations, such that we report spin‐paired results in Figure 5b) only. The spin‐paired dissociation energy is 4.85–4.95 eV and thus again roughly 1 eV higher.

Figure 5.

a) Structure of the PDMS model (C: grey, H:white, Si: brown, O: red). The pulling positions in the COGEF simulations are indicated by the carbon atom indices. b) Energies and c) forces vs. total elongation d of the molecule.

We fix the COGEF anchor point on one selected carbon atom and vary the other carbon atom with increasing distance between these points. Eventually one of the terminal Si−C bonds breaks (see Figure S6 in SI). Despite the chemically similar pulling positions some of the resulting COGEF energies show a discontinuity, but others do not. The height of the discontinuity peak clearly correlates with the total elongation of the molecule, while the final energies are very similar. Here we see the effect of the increasing contribution of the environment (the rest of the molecule between the anchor points except of the bond that breaks) as predicted by the consideration of the spring‐model above.

The forces shown in 5b) have very similar maxima of $F_{\mathrm{m}ax}$ =4.21–4.23 nN as expected due to the similarity of the Si−C bonds. While the force distribution is smooth at near pulling positions (small environmental effects), discontinuities appear for larger influences of the environment matching the discontinuities seen in the energies. These are a result of the sudden relaxation of the environment that is possible when the bond actually breaks.

Pulling the silicon atoms as detailed in SI reveals that breaking of the Si−O does not only require more energy, but also larger forces of at least $F_{\mathrm{m}ax}$ of at least 5.85 nN for SiO bond dissociation. There is spontaneous hydrogen transfer in our COGEF simulations which might be a channel lower energey PDMS Si−O backbone rupture (see SI fore details). This investigation is out of scope of the present study, however.

We now investigate the force dependent barriers for breaking of the terminal Si−C bond when pulling on the carbon atom pairs indicated in Figure 5. The 3S‐COGEF simulations were performed spin‐paired for the same reason as as in the COGEF simulations above. As predicted by Eq. (11), the 3S‐COGEF barriers are independent from the COGEF anchor positions and show all the same behavior as seen in Figure 6. The force dependence of the barrier is remarkably similar to the expectation of Eq. (3) for the spin‐paired D_e . The lower spin‐polarized D_e leads to lower barriers and the analysis of Figure 4 strongly suggests that these are relevant for smaller forces, i.e forces up to 1 nN. This poses us into a dilemma as it is often the low force range that is most important for many experimental situations.[ ⁵ , ⁶ , ⁷ , ⁸ , ¹¹ , ¹⁴ ] The analytical expression of the barrier using the spin‐polarized D_e may therefore give a better description than the 3S‐COGEF calculation itself. This observation on the other hand releases us of the burden to evaluate all the barriers numerically as these are determined to good accuracy by D_e and $F_{\mathrm{m}ax}$ through Eq. (3).

Figure 6.

Calculated force‐dependent barriers for breaking of Si−C bonds of the terminal Si atom under pulling on the given C atom pairs (c.f. Figure 5). Expectations from spin‐paired and spin‐polarized dissociation energies according to Eq. (3) are superimposed.

Conclusion

We have shown that the force dependent barrier to break a bond is mainly characterized by just two quantities. These are the dissociation energy and the maximal force the system can withstand until it breaks spontaneously. The barrier takes a particularly simple form which allows for a reinterpretation of the Eyring‐Zhurkov‐Bell $\Delta x^{‡}$ as the ratio between these two quantities and also naturally explains the contribution of the term quadratic in the applied force appearing in extended Bell theories.

We have furthermore shown that an environment modelled as effective spring does not change the barrier at all, but leads to exactly the same result as the barrier of the bond alone. Usual COGEF simulations fail to describe this behavior and a multidimensional approach is required instead. The 3S‐COGEF method, which approximately calculates the stationary trajectories under the influence of external forces, is capable to correctly obtain these barriers.

Density functional theory calculations of a model PDMS molecule revealed the predicted behaviour of the numerically determined barrier that is independent on the pulling positions within the molecule. The resulting independence of the bond from the deformation of the environment might be approximate in general, where the properties of other species connected to the atoms forming a bond will influence the properties of the bond itself. This influence nevertheless can be expected to diminish quickly with distance in most configurations in the absence of long‐range effects.

We may expect deviations from our simple picture for more complex structural conditions such as the influence of sterical hindrance, ^[53] cis‐trans transitions. ^[68] or more complex pathways. ^[56] Long range effects can be expected e. g. through aromatic coupling in spyropyrans, ^[69] carbon chains ^[70] or on the diamond surface. ^[71] Further deviations may be expected by relaxations of may degrees of freedom at the same time that lead to snapping motions at rupture ^[38] resulting in catastrophic events. ^[72] It is an interesting question, when these have to be expected and how this is coupled to the time scales relevant in the rupture process.

Computational Details

All electronic structure calculations are performed within Kohn‐Sham density functional theory (DFT) as implemented in the GPAW package.[ ⁷³ , ⁷⁴ ] The electronic wavefunctions and electron‐density are represented in the projector augmented wave method, ^[75] where the smooth parts are represented on real space grids. The grid spacing was chosen to 0.2 Å for the Kohn‐Sham wavefunctions and 0.1 Å for the density. The exchange‐correlation energy was approximated by the PBE ^[76] functional. Molecular structures were set up using the atomic simulation environment. ^[77] All unconstrained degrees of freedom were considered to be relaxed if the corresponding forces were found to be below 0.05 eV/Å.

The bond dissociation energy D_e for a molecule AB separated into two radicals A and B is calculated as

$${\displaystyle D_e=E_{\mathrm{A}}+E_{\mathrm{B}}-E_{\mathrm{A}B}}$$ (17)

with the DFT energies of the isolated species E _A,B,AB . The spin‐angular moment projection S_z is fixed to ħ/2 in spin‐polarized calculations of the isolated radicals A and B.

Conflict of interest

The authors declare no conflict of interest.

Supporting information

As a service to our authors and readers, this journal provides supporting information supplied by the authors. Such materials are peer reviewed and may be re‐organized for online delivery, but are not copy‐edited or typeset. Technical support issues arising from supporting information (other than missing files) should be addressed to the authors.

Supporting Information

S. Khodayeki, W. Maftuhin, M. Walter, ChemPhysChem 2022, 23, e202200237.

Data Availability Statement

The supporting information contains the following. Details of the model potentials and connection of their parameters with dissociation energy and maximal force. Details of the DFT calculations for PDMS: conformer comparisons and various bond breaking scenarios. Discussion of the connection between the enthalpy depending on pressure and volume with the enthalpy depending on force and elongation.