Theoretical Approaches for Understanding the Interplay Between Stress and Chemical Reactivity

  • Gurpaul S. Kochhar
  • Gavin S. Heverly-Coulson
  • Nicholas J. Mosey
Part of the Topics in Current Chemistry book series (TOPCURRCHEM, volume 369)


The use of mechanical stresses to induce chemical reactions has attracted significant interest in recent years. Computational modeling can play a significant role in developing a comprehensive understanding of the interplay between stresses and chemical reactivity. In this review, we discuss techniques for simulating chemical reactions occurring under mechanochemical conditions. The methods described are broadly divided into techniques that are appropriate for studying molecular mechanochemistry and those suited to modeling bulk mechanochemistry. In both cases, several different approaches are described and compared. Methods for examining molecular mechanochemistry are based on exploring the force-modified potential energy surface on which a molecule subjected to an external force moves. Meanwhile, it is suggested that condensed phase simulation methods typically used to study tribochemical reactions, i.e., those occurring in sliding contacts, can be adapted to study bulk mechanochemistry.


Chemical simulation Mechanochemistry Modeling Tribochemistry 



Atomic force microscopy


Attachment point


Complete active space Møller–Plesset 2nd order perturbation theory


Complete active space self-consistent field


Car–Parrinello molecular dynamics


Density functional theory


External force is explicitly included


Force field


Force-modified potential energy surface


Generalized solid-state nudged elastic band


Intrinsic reaction coordinate


Molecular dynamics


Minimum energy path


Nudged elastic band


Potential energy surface


Pulling point


Quantum chemical


Quantum mechanics/molecular mechanics


Registry index


Steered molecular dynamics


Transition state

1 Introduction

For a chemical reaction to occur, the reacting species generally need to acquire sufficient energy to overcome the associated reaction barrier as the system moves along a direction that converts the reactants into products. The energy required to activate these reactions has been provided conventionally in the form of heat (thermochemistry), light (photochemistry), or an electric potential (electrochemistry). Thermochemistry involves the distribution of thermal energy amongst the different nuclear degrees of freedom in a molecule, which allows reactions to occur while the system remains in a particular electronic state. Photochemistry focuses on using light to promote chemical systems to higher energy electronic states, which generally have potential energy surfaces with different shapes than that of the ground state, and can thus lead to different reactions. Meanwhile, electrochemistry uses an applied potential to shift the electronic energy levels of reacting species to induce processes such as electron transfer.

Recent years have seen the development and application of techniques that allow one to activate chemical processes by acting on parts of a molecule or material with external forces or stresses [1, 2, 3, 4, 5, 6]. In this approach, termed mechanochemistry, the forces or stresses acting on a system perform work as the system undergoes changes in structure, such as those that occur during a chemical reaction. The work performed on the system provides energy that can activate chemical reactions, whereas the directional nature of an applied force can be used to guide chemical systems along specific reaction pathways. This approach affects nuclear degrees of freedom, and is thus distinct from photochemical and electrochemical methods. In addition, the directional natures of external forces or stresses render mechanochemistry distinct from thermochemical approaches, where heat is distributed throughout the entire system.

The fact that mechanochemical conditions promote reactions in ways that differ distinctly from thermochemistry, photochemistry, and electrochemistry suggests that mechanochemistry may offer a means of activating reactions that are difficult to achieve through these conventional experimental approaches to induce reactions. This potential has prompted extensive research over the last decade into the ability to subject molecules and materials to external forces and stresses in a controlled manner and to use these conditions to activate chemical reactions [1, 2, 3, 4, 5, 6]. The ability to subject molecules and materials to mechanochemical conditions has been made possible through developments in experimental techniques such as atomic force microscopy (AFM) [7, 8, 9], optical and magnetic tweezers [10, 11, 12, 13], molecular force probes [14, 15], sonication techniques [16, 17, 18, 19], and grinding and milling [3, 20, 21, 22]. These techniques operate on various length scales, with methods such as AFMs, tweezers, and force probes acting at the molecular level, and sonication, grinding, and milling being applicable to bulk systems. Overall, the application of these techniques has illustrated the potential to guide chemical systems along specific reaction pathways using applied stresses [19, 23], permitted the examination of processes such as the unfolding of proteins and DNA [10, 24, 25, 26, 27], and even led to practical applications such as visible stress sensors [6, 28].

A large number of theoretical studies of chemical processes occurring under mechanochemical conditions have been reported to complement experimental efforts, help explain experimental outcomes, and provide new insights into the interplay between applied forces or stresses and chemical reactivity [29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. These studies have focused primarily on simulating molecular systems exposed to external forces by treating the system as though it moves on a force-modified potential energy surface that incorporates the work performed on a chemical system as it undergoes structural changes in the presence of an external force. These studies have examined the rupture forces of bonds [32], the reactivities of molecules subjected to tensile stresses and strains [38, 39, 40, 41], the effects of strains on pericyclic reactions [14, 29, 30, 31, 33, 34, 42, 43, 44], the differences between thermochemistry and mechanochemistry [45], and the effects of chain length on the transduction of external forces at atomic levels [46, 47].

Interestingly, despite the longer history of mechanochemical activation of bulk systems in experimental research [2], e.g., through processes such as grinding and milling, fewer theoretical studies have focused specifically on the mechanochemistry of bulk systems. However, the stress-induced processes involving bulk systems have been modeled extensively in the context of tribology, which is the study of friction, wear, and lubrication [48, 49]. As such, potential exists to apply existing techniques used to study systems such as sliding contacts in the context of the mechanochemical activation of reactions occurring in bulk systems where shear and compressive stresses play key roles in activating reactions.

Herein, we describe the different types of chemical simulation methods that can be used to study mechanochemical reactions at molecular and bulk levels, with the goal of providing basic information regarding these simulation techniques. Section 2 focuses on molecular mechanochemistry and describes models that can be used to predict the energies and properties of systems exposed to applied forces. Section 3 provides an overview of techniques used to study bulk systems that are exposed to compressive and shear stresses. Concluding remarks are provided in Sect. 4.

2 Methods for Simulating Molecular Mechanochemistry

Molecular mechanochemistry involves subjecting molecules to a tensile force of magnitude F that is applied between two regions in the molecule. This is illustrated schematically in Fig. 1 for the application of F between a pair of hydrogen atoms in cyclobutene, which can be used to model the ring-opening of this molecule under mechanochemical conditions. In this figure, F indicates the applied force and R is the distance between the two groups subjected to the force. From an experimental standpoint, the application of F is achieved through the use of single molecule manipulation techniques, e.g., optical or magnetic tweezers, atomic-force microscopes, by subjecting mechanophores embedded within polymers to ultrasound pulses, or by incorporating mechanophores into molecular force probes, or other means. These techniques have been used to study a wide range of processes such as the unfolding of proteins [25, 26, 27, 50, 51], ring opening reactions [14, 19, 44, 52], and detaching anchored molecules from surfaces [53, 54]. In general, it is found that the application of F can alter the thermodynamic and kinetic properties of reactions. The ability to affect these aspects of reactions is derived from the effects of F on molecular structures and energies. In particular, the application of F alters molecular geometries, whereas the work performed in the presence of F as a system changes structure during a reaction supplies energy that can be used to promote reactions.
Fig. 1

Application of an external force of magnitude, F, to two hydrogen atoms in cyclobutene which are separated by a distance, R. The general concept of subjecting groups in a molecule to external forces while they undergo changes in separation underpins the methods used to simulate mechanochemical reactions at the molecular level. Silver and turquoise spheres indicate carbon and hydrogen atoms, respectively

The effect of subjecting molecules to F can be described through the concept of a force-modified potential energy surface (FMPES). If one considers the typical case in which F is applied between two positions in a molecule, the FMPES takes on the form of
$$ {E}_F\left(\mathbf{q},F\right)={E}_0\left(\mathbf{q}\right)-FR, $$
where q represents the nuclear coordinates that define the geometry of the molecule, E0(q) is the potential energy of the system on the zero-F PES, and R is the distance between the atoms or groups that are subjected to F. In the context of simulating reactions involving changes in bonding, it is necessary obtain E0(q) using quantum chemical (QC) methods, reactive force fields (FFs), or QM/MM methods that can account for the formation and/or dissociation of bonds.

The FMPES is the mechanochemical analogue of the Born–Oppenheimer PES used to study changes in potential energies during reactions with QC calculations, which can then be related to thermodynamic and kinetic quantities, and thus the evaluation of the FMPES is important in the context of understanding the effects of F on reactions. Methods for evaluating and examining the FMPES and making connections to changes in free energy are described in what follows. Section 2.1 describes means of directly evaluating properties of the FMPES. Section 2.2 describes models in which information related to the FMPES is obtained indirectly from features of the zero-F PES.

2.1 Direct Evaluation of the FMPES

Access to the FMPES in (1) is central to the computational study of mechanochemical processes. The FMPES is straightforward to evaluate using potential energies obtained through QC or FF calculations, which can provide E0(q), in conjunction with knowledge of the molecular structure, which provides R. Although a discussion of QC and FF methods is beyond the scope of this review, the suitability of different QC methods for modeling mechanochemical processes involving bond rupture has been examined recently [55]. Several different approaches exist for treating the mechanical work term in (1). The differences between these methods arise from the manner in which F is applied in the calculations. Despite the differences in the specific details of these techniques, all mechanochemical simulation methods that directly calculate the FMPES employ some form of (1), which is differentiable with respect to the nuclear positions. This allows one to examine the FMPES using procedures that are commonly used in conventional computational studies of chemical systems such as geometry optimizations, frequency calculations, intrinsic reaction coordinate (IRC), or minimum energy pathway (MEP) calculations and molecular dynamics (MD) simulations.

In the context of experiments, subjecting a molecule to F typically involves the mechanical manipulation of groups within that molecule. For example, a polymer subjected to mechanochemical conditions in an AFM experiment may be attached to a surface at one end and the AFM tip at the other. In this context, the polymer corresponds to the molecule being exposed to mechanochemical conditions, and the surface and AFM tip may be thought of as external groups used to impose those conditions. To do this, the distance between the AFM tip and the surface is increased, which in turn subjects the polymer to F. Similar scenarios occur in other experimental approaches to mechanochemistry, where large structural changes in macromolecules are used to apply F to molecules attached to force probes, and the movement of polymers attached to reactive species is used to apply F in sonication experiments, for example.

In the context of simulation, computational expense typically prevents the use of model systems that are exact replicas of those used in experiments. As a result, the model systems used in calculations of mechanochemical processes generally employ truncated or coarse-grained representations of the external groups used to apply F, or even exclude these groups entirely. The limited treatment of the external groups, or even their complete elimination from the calculations, requires the use of approximate schemes for subjecting molecules to F in calculations.

In general, two classes of methods exist for subjecting molecules to F in calculations. In the first class of methods, selected atoms in the simulated system are subjected to forces that are directed toward artificial points that are external to the molecule. These points correspond to the locations at which the external groups used to apply F in experiments would be located. Consider the extension of a surface-bound polymer in an AFM experiment as outlined above. In that case, the polymer, or a small portion thereof, would be simulated explicitly, whereas the surface and AFM tip would be replace by artificial points at appropriate locations around that molecule. Mechanochemical conditions could then be simulated by subjecting the atoms at either end of the polymer to forces directed toward the nearest artificial external point. Methods that employ artificial external points to apply F are described in Sect. 2.1.1.

The second class of methods for modeling mechanochemical conditions applies F between atoms in the system without employing artificial external groups. Once again, consider the AFM experiment described above. The application of F leads to an increase in the distance between the ends of the polymer. The extension of the molecule can be simulated without employing any external groups by fixing the distances between the atoms at the end of the molecule to specific values or by applying F along the vector connecting these atoms to induce a change in distance. Techniques that apply F by using groups that are internal to the molecule being simulated are described in Sect. 2.1.2.

In addition to considerations regarding the manner in which F is applied, it is also important to note that experimental mechanochemistry is performed somewhere along the spectrum ranging from controlled separation to isotensional conditions. Controlled separation corresponds to fixing the distance, R, between the groups associated with the degree of freedom to which F is applied and measuring the F required to maintain R. Such conditions apply, for example, to force-extension curves obtained in AFM experiments in which the end-to-end distance of a molecule is fixed and the F required to keep the distance fixed is measured. Measuring F for multiple values of R yields a force-extension curve, which can be useful in determining the values of F needed to induce processes such as the unfolding proteins [35, 56, 57] or the rupture of polymers [58, 59]. At the other end of the spectrum, isotensional conditions involve subjecting a molecule to a constant tensile force, F, and allowing the affected degrees of freedom to change as needed to accommodate F. Such conditions have been achieved in AFM experiments of stretched macromolecules [7, 60] and the ultrasonic cleavage of metal-ligand coordination complexes [61, 62, 63], for example. Of course, in practice, the need to move between different values of F or R, as well as phenomena such as thermal fluctuations, lead to deviations from strictly controlled separation or isotensional conditions in experiments. Instead, molecules are subjected to mechanochemical conditions in ways that cause F and/or R to change in a time-dependent manner. Capturing the specific mechanochemical conditions experienced by a system is important to ensure a meaningful connection between simulation and experiment. As discussed below, achieving this in calculations requires different treatments of the work term (−FR) appearing in (1).

2.1.1 Application of F via Artificial External Groups

Molecular mechanochemistry involves subjecting a molecule to F via groups that are external to it. An explicit treatment of these external groups can be avoided in calculations by devising a model system that includes an explicit representation of the molecule of interest and adding a set of artificial external points, called pulling points (PPs), at locations around the molecule where the external groups would reside in the real system of interest. The PPs are each connected to atoms called attachment points (APs) within the simulated molecule in a manner that allows F to be applied between the PPs and APs.

A model that could be used to study the mechanochemical ring-opening of cyclobutene produced by the application of F between APs and PPs is shown in Fig. 2. In this case, the APs correspond to one of the hydrogen atoms bonded to each of the carbon atoms involved in the scissile bond and the PPs are placed at locations corresponding to the positions at which groups, such as polymers, that would be used to apply F are attached to the ring. In this model, each AP is associated with one of the PPs and is subjected to an external force of magnitude F applied along the vector connecting the AP to the PP with which it is associated.
Fig. 2

Application of F in the steered molecular dynamics model. In this case, two hydrogen atoms in cyclobutene are treated as attachment points each attached to one external PP (indicated by blue circles). Silver and turquoise spheres indicate carbon and hydrogen atoms, respectively

Significant flexibility exists in terms of the manner in which F is applied using models such as that shown in Fig. 2. For example, it is possible to keep the PPs at fixed positions and apply a constant value of F along the vector connecting each PP to its associated AP. Alternatively, the magnitude of F can be obtained through the introduction of an interaction potential for each AP-PP pair. For example, if one employs a harmonic potential for this purpose, the force associated with a given AP-PP pair is \( F=-k\left(r-{r}^{\mathrm{eq}}\right) \), where k is the stiffness of the potential, r is the distance between the AP and its associated PPs, and req is the equilibrium value for this distance. Employing interaction potentials to apply F to the APs is particularly useful if the PPs are moved over time, e.g., to simulate the manner in which the external groups are moved to impose F in experiments, where the changes in the positions of the PPs lead to time-dependent forces of the form \( F(t)=-k\left(r-{r}^{\mathrm{eq}}+vt\right) \), where v represents the velocity at which the PP is moved.

Regardless of the specific manner in which the magnitude of F is determined, the external force applied to the system by the PP-AP interactions is given by
$$ {\mathbf{F}}_{\mathrm{ext}}={\displaystyle \sum_{i=1}^{n_{\mathrm{AP}}}F{\mathbf{n}}_i}, $$
where nAP is the number of APs, which is usually 2 when modeling mechanochemical processes, and ni is the unit vector along the direction connecting the ith AP-PP pair. The FMPES arising from the application of F in this manner is then given by
$$ {E}_F\left(\mathbf{q},F\right)={E}_0\left(\mathbf{q}\right)-{\displaystyle \sum_{i=1}^{n_{\mathrm{PP}}}F\left({r}_i-{r}_i^0\right)}, $$
where q represents the atomic coordinates in the system, E0(q) is the energy of the system on the zero-F PES, ri represents the current distance between the ith AP and its associated PP, and ri0 is a reference value for this distance, which can in principle take on any value, with sensible choices being zero or rieq.

The application of F through the use of artificial external PPs connected to APs in a simulated molecule is called steered molecular dynamics (SMD) [64, 65, 66]. SMD methods are available in some chemical simulation packages, and in cases where such techniques have not been implemented it is straightforward to add the F-dependent terms appearing in the energies and nuclear forces to existing packages to enable simulations on the FMPES. A drawback of the method lies in the fact that the energy in (3) is dependent upon the positions of the PPs, which must be chosen by the person performing the calculation. A sensible choice is to define the positions of the PPs to lie along directions between the APs and atoms that would be present in the devices used to subject the molecule to F.

SMD simulations have a long history, with initial applications focusing primarily on the study of folding and unfolding of proteins [65, 66, 67, 68, 69]. It is worth noting that, despite the emphasis on MD simulations in the SMD label, it is also possible to optimize molecular structures, perform frequency calculations, and locate minimum energy pathways (MEPs) on the FMPES given in (3). In recent years, SMD simulations have become increasingly used in the context of simulating mechanochemical reactions in a broader sense. Selected applications using SMD methods in the context of studying molecular mechanochemistry are described in what follows.

Modern interest in molecular mechanochemistry was stimulated in part by experiments demonstrating how subjecting benzocyclobutene moieties with polymeric substituents to F via ultrasound pulses could induce the ring opening of this molecule in a manner that circumvented the Woodward–Hoffman rules [19]. In these experiments, the polymers were bonded to the carbon atoms of the scissile bond in cyclobutene and the conrotatory ring-opening pathway was favored when these polymers were attached in a trans arrangement with respect to the plane of the ring, whereas the disrotatory pathway was favored if the polymers were attached in a cis orientation.

The F-induced ring-opening of cyclobutene and benzocyclobutene was studied via SMD simulations by Martinez and coworkers [30]. Their simulations employed models such a that shown in Fig. 2, with the hydrogen atoms on the scissile C–C bond acting as APs. APs in cis and trans arrangements were examined to focus on the experimental conditions that favored the disrotatory and conrotatory pathways, respectively. Each AP was attached to an external PP that remained at a fixed location and a constant F was applied between each PP and its associated AP. The zero-F potential energy, E(q), was evaluated at the CASMP2(4,4)/6-31G** level of theory, which is suitable for describing the underlying changes in electronic structure that occurs along the disrotatory pathway. These models were used to perform MD simulations on the FMPES, with the results indicating that disrotatory ring-opening occurred with cis PPs within the simulated period of 1.0 ps for F > 1.5 nN. The ability of the system to follow the disrotatory pathway with cis PPs was rationalized by evaluating the MEPs along the conrotatory and disrotatory pathways at different F. The results of the calculations showed that the barriers along both reaction pathways were lowered by the application of F; however, that along the disrotatory pathway was more affected by F and thus became kinetically favored as F was increased. Overall, these results were consistent with the results of the ultrasound experiments, and provided insights into the results of those experiments in terms of the underlying features of the FMPES.

The Martinez group has applied SMD methods in additional studies aimed at understanding the F-induced chemistry of mechanophores embedded in polymers [42, 43, 44, 58, 70, 71]. For example, SMD simulations have been used in conjunction with constrained distance calculations (see below) to help design and understand the functionality of visible stresses sensors based on polymers bonded to spiropyrans, which undergo a stress-induced conversion to merocyanines along with a change in color [58]. SMD simulations were also used to find how applied stresses induce the opening of strained rings to form cyanoacrylates [44]. Additional studies used SMD simulations and MEP calculations on the FMPES to explain the F-induced change in the cistrans product ratios of the ring opening of gem-difluorocyclopropanes [42, 43]. Experiments show that the trans product of this reaction is favored in the absence of F, which is consistent with the relative stabilities of the cis and trans forms, yet the cis product was formed in increasing amounts under the application of ultrasound pulses. The calculations showed that the change in the product ratios was caused by F-induced changes in the PES that caused a diradical species corresponding to the TS on the zero-F PES to become a minimum on the FMPES, with this species becoming the global minimum at high F. The change in the nature of the diradical species introduced a new step along the reaction pathway, with calculations showing that at high F the ring opening of both cis and trans forms of the reactant yielded the diradical structure followed by progression from this species to the cis product as F is removed. Overall, this work illustrates the ability of SMD calculations to provide insights that can be used to explain experimental observations. In addition, the calculations illustrated that the F-induced stabilization of the zero-F TS may provide a means of trapping and probing TS structures in experiments.

The manner in which oligorotaxanes unfold in AFM experiments has been examined with SMD simulations by Ratner and coworkers [72, 73]. In their studies, one end of a molecule representing an oligorotaxane was attached to a PP via a stiff harmonic potential to mimic the attachment of this molecule to a surface in AFM experiments. The other end of the molecule was then attached to a PP by another harmonic potential. F was applied by moving the latter PP at a constant speed along the direction connecting the terminal atoms of the molecule. E0(q) was evaluated with FF methods, because bond rupture was not expected to occur, and hence the authors could examine relatively large models over a wide range of pulling speeds. The results demonstrated that pulling speeds on the order of 10−3 Å/ps were sufficient to obtain reversible behavior when F was applied and then removed. Force-extension isotherms were obtained in the simulations and showed that regions exist in which the molecule is mechanically unstable, which leads to unfolding and results in transitions between high-F and low-F regimes. The development of these mechanically unstable regions and the transitions between force regimes were found to be related to fluctuations in F, which in turn are related to the stiffness of the harmonic potential used to apply F. Additional simulations by Vilgis and coworkers have modeled F-induced polymer chain scission using similar models [74, 75, 76, 77].

As noted above, SMD simulations were originally used in the context of examining the mechanisms associated with protein unfolding and this remains an active area of research, with many recent studies focused on the reduction of disulfide bonds, which exhibit a variety of responses to F. For example, Gräter and coworkers have used SMD simulations to account for F-induced changes in the mechanism by which thioredoxin catalyzes the reduction of S–S bonds, where experimental studies showed that the reaction rate decreases with low F before increasing again at higher F [78]. The simulations performed to explore the origins of this behavior employed a variant of SMD termed force probe MD. In this method, the PPs are attached to the APs via harmonic potentials and F is applied by moving the PPs at constant velocities. In this case, standard FFs were used to obtain the zero-F energies of systems representing the active site and relevant model protein residues, and the PPs were attached to Cα atoms of the terminal residues. The results of the simulations showed that applying F caused the disulfide to reorient such that it was aligned to within ~20° of the coordinate along which F was applied. This reorientation requires a large rotation to occur in order for the disulfide bond to become aligned with a third sulfur in the SN2 TS for this reaction. This rotational motion results in a decrease in the length coordinate upon moving from the reactant to the TS and thus causes a reduction in the rate within increasing F. This study was unable to account for the subsequent increase in the rate at higher F, but it was hypothesized that the reduction in the rate (or underlying increase in the reaction barrier) of the standard SN2 process with F may render other competing mechanisms for this reaction kinetically favorable at higher F.

The effect of F on the thiol-disulfide exchange was examined further using force probe MD simulations based on QM/MM representations of a system corresponding to a dithiothreitol reducing agent interacting with a truncated model of I27 [79]. The force probe MD simulations were performed in conjunction with transition path sampling methods to examine ensembles of reactive trajectories of the force-modified free energy surface. The results of these simulations demonstrated that low F shifts the TS toward the reactant, thus leading to an increase in the reaction rate. The application of higher F was found to populate TS structures that are unfavorable at lower F, causing the system to follow preferentially an alternate reaction path. Other uses of force probe MD simulations in the context of the mechanochemistry of disulfide bonds have examined the electrochemistry of this reaction [80].

2.1.2 Application of F via Internal Groups

The application of F via external groups, e.g., artificial PPs in SMD simulations or external devices used to manipulate molecules in experiments, leads to changes in the structure of the molecule being subjected to F. If the molecule is subjected to a tensile stress, which is the typical case in molecular mechanochemistry, the distance between the atoms that interact with the external groups increases relative to the analogous distance in the absence of F. The extension of the molecule can be interpreted in the context of the application of a force along the vector, R, connecting the atoms attached to these devices. When an equilibrium structure is reached, the force applied along R is equivalent to the restoring force of the molecule at that length; however, the internal and external forces may not be balanced when the system is not at equilibrium. The fact that the external application of F induces changes in the distances and forces between atoms contained entirely within the molecule exposed to F suggests that it is possible to simulate mechanochemical conditions without employing artificial external PPs to apply F.

Simulations of mechanochemical conditions that apply F using groups contained in the molecule of interest are discussed in what follows. The section “Application of F Through Constrained Geometries” discusses techniques in which the molecular geometry is constrained to mimic the application of F. The section “Application of F Between Atoms” describes an approach termed external force is explicitly included (EFEI) in which a constant F is applied along the vector connecting a selected pair of atoms in the molecule. Constrained geometry and EFEI calculations employ different controlled variables, i.e., distances and forces, respectively, to subject the system to F and thus model different experimental conditions. These differences and their potential impacts on the results of calculations are discussed in section “Comparison of Internal Group Models”.

Application of F Through Constrained Geometries
The application of F to a specific pair of atoms in a molecule can be modeled by constraining the distance between these atoms, R, to some specified value, R0, at the same time allowing the system to move along the remaining 3N – 7 nuclear degrees of freedom using methods such as geometry optimizations or MD simulations. The satisfaction of the constraint R=R0 is equivalent to subjecting the system to an external force, F, that is equivalent in magnitude, but opposite in direction, to the internal force, \( {F}_{\mathrm{int}}=\partial {E}_0\left(\mathbf{q}\right)/\partial R \), acting along the vector connecting the constrained pair of atoms. Note that Fint is dependent upon the positions of the nuclei along the 3N – 7 unconstrained coordinates in addition to the value of the constrained distance, R0. This leads to the definition of the FMPES as
$$ {E}_F\left({\mathbf{q}}^{*};{R}_0\right)={E}_0\left({\mathbf{q}}^{*};{R}_0\right)+{F}_{\mathrm{int}}{R}_0, $$
where q* represents the 3N – 7 degrees of freedom that are not constrained, R0 is a parameter corresponding to the value of the constrained distance, and E0 is the energy of the system with the set of 3N – 6 atomic coordinates {q*,R0} on the zero-F PES.

The use of geometric constraints has the advantage that such constraints are used in many types of QC simulations unrelated to mechanochemistry. For example, constrained distances (or other degrees of freedom) are used extensively in applications such as exploring the features of the zero-F PES or evaluating changes in free energies during MD simulations [81]. As a result, many simulation software packages have the ability to constrain the distances between atoms, which permits the use of geometric constraints without any modification of those codes. Moreover, the application of F between specific pairs of atoms removes any subjectivity associated with the choice of the locations and velocities of PPs in SMD simulations.

Constrained distance methods are most commonly used in conjunction with geometry optimizations or MD simulations. In the former case, the unconstrained atoms in the system relax such that the system adopts a structure that is a local energy minimum on an FMPES defined by the 3N – 7 unconstrained degrees of freedom with the fixed value of R=R0. In the latter case, the system samples regions of this 3N – 7 dimensional FMPES around this local minimum. Because R0 is typically associated with an interatomic separation which changes significantly during the reaction of interest, it is common practice to study the FMPESs at different values of R0 via relaxed PES scans or by changing R0 linearly in time to simulate the extension of a molecule during MD simulations.

Constrained geometries have been used extensively in the context of mechanochemistry, with applications including studies of bond rupture [82, 83, 84, 85, 86], reactivity of disulfide bonds [87], unfolding of supramolecular polymers [88], mechanochemical synthesis of phenyl cations [89], extraction of gold nanowires [90], evaluation of restoring forces in force probes [14, 91], and calculation of free energy barriers [92, 93, 94, 95, 96, 97, 98]. Examples of selected applications are described below.

Frank and coworkers have employed a constrained distance method in conjunction with Car–Parrinello MD (CPMD) simulations [99] to examine a variety of mechanochemical processes [33, 39, 40, 41, 100, 101]. During these simulations, F is applied by increasing the distance between a pair of atoms at a constant velocity. They have used this approach to explore the changes in electronic structure that occur when solvated polymers are stretched to the point of rupture [41]. Their studies showed that bond rupture occurred through a heterolytic process involving solvent molecules. Interestingly, their simulations showed that the weakest bond does not necessarily correspond to the site of bond rupture. Rather, the bond that is made most accessible to attack by solvent via F-induced changes in structure most frequently corresponds to the site at which the polymer dissociates.

CPMD simulations in which F was applied by increasing the separation of a pair of atoms at a constant velocity in conjunction with detailed analyses of the electronic structure have also been used to study the dissociation of disulfide bonds in the presence of reducing agents [100]. Those simulations showed that this reaction does not necessarily involve electron transfer, as is common for redox processes, but rather can occur through heterolytic dissociation of the S–S bond followed by proton transfer. The mechanisms available for the F-induced reduction of the S–S bond in the presence of various nucleophiles has also been examined [101], with results illustrating that a wide range of mechanisms exist in addition to the SN2 mechanism favored at zero F. The existence of multiple mechanisms, and the change in the kinetically preferred mechanism with F, was suggested to be the origin of the F-dependence of rates of disulfide reduction observed in experiments.

Boulatov and coworkers have employed constrained geometry calculations of reactive sites coupled to a weak harmonic constraining potential [7] that represents the external device used to evaluate free energy barriers to reactions [92, 93, 94, 95, 96, 97, 98]. In this approach, constrained geometry optimizations are used to obtain reactant and TS structures, energies, and internal forces, and the effect of the external system is subsequently included via the compliance and length of the external harmonic potential. The potential energies of the system comprising the reactive site and constraining potential were augmented with standard thermochemical corrections and conformational averaging to yield free energy barriers, which are of greater relevance to experiments than potential energies. In addition, these constrained calculations were used to examine the F-dependence of the barrier upon the length coordinate used in the calculations. It was found that the F-dependence can be described in terms of a conveniently chosen local coordinate, e.g., the distance between groups within the reactive site, instead of using a length coordinate associated with the vector connecting the ends of groups such as polymers that are used to apply F in experiments, which may not be accurately represented in the truncated model systems.

Although constrained distance methods employ the distance R0 as a controlled variable, they can be used to model conditions in which F is constant. To do this, it is necessary to vary R0 to locate different structures on the FMPES where the internal force, Fint, associated with R is equivalent. If pairs of these structures correspond to species that are relevant for a given reaction, e.g., the reactants, products, or TS, the differences in their energies on the FMPES correspond to reaction energies or barriers under constant F conditions. As discussed in what follows, this approach has been used to examine phenomena such as F-induced changes in proton affinity [38] and reaction barriers [83]. Unfortunately, such calculations can be difficult because it is necessary to locate pairs of structures that have the same values of Fint, which typically involves performing scans over large regions of the FMPES.

Beyer provided an early example of this approach by using constrained distance calculations to study the F-dependence of the proton affinity of dimethyl ether [38]. In these calculations, a hydrogen atom on each of the methyl groups was used as a PP and the distance between the PPs was constrained to a discrete set of values to one-dimensional relaxed scans of the zero-F PES for the protonated and unprotonated forms of the molecule. These energies were then fitted to fourth-order polynomials to yield functions that could be differentiated analytically to locate PP separations at which the internal force associated with the constrained distance was equal in the reactants and products. These data were used to evaluate the relative energies of the protonated and unprotonated forms of the molecule according to (4), with results showing that the proton affinity increased steadily as F increased. The authors attributed the increase in proton affinity to the lower strain energy of the protonated relative to that of the unprotonated form. This difference was attributed to the fact that the protonated is less stiff than the unprotonated form of the molecule, and thus extended to a greater extent through the application of F.

Uggerud and coworkers have used a similar approach to evaluating F-dependent barriers to chemical reactions. For example, one-dimensional projections of the FMPES were obtained by relaxed PES scans in which the distances between PPs in the reactants and TS structures associated with the ring opening and decomposition of substituted triazoles [83] were varied over a wide range of values. The calculations showed that the kinetically preferred mechanism changed with increasing F, with ring opening preferred at low F and C–C bond dissociation becoming preferred as F was increased. The same group has also used constrained distance calculations to assess the factors that determine which bond in a polymer breaks in response to applied F [82]. The results of that work showed that the weakest bond in the system does not necessarily correspond to the rupture site, which is consistent with the work of Frank mentioned above, but rather the bond that dissociates is dependent on bond strength, bond stiffness, and the orientation of the bond with respect to the direction along which F is applied.

The approaches to constraining geometries described above all employ procedures in which a single interatomic distance is fixed to a value, which may be changed to scan the PES or induce reactions in MD simulations. Boulatov and coworkers have employed an alternative approach to constraining geometries to evaluate the restoring forces of molecules in molecular force probes [14, 91]. The force probes used in their experiments corresponded to macrocycles consisting of a cyclobutene moiety attached to a cis stilbene through linkers of different lengths. F was applied by photoisomerization of the stilbene, which led to the ring opening of the cyclobutene moiety. It was found that the rate of ring opening was F-dependent, with shorter linkers leading to greater accelerations. The F applied during the photoisomerization process was calculated by first optimizing the reactant and TS structures for the ring opening process using full models of each macrocycle. The cyclobutene moieties in the reactant and TS for each macrocycle were then excised from these models, the resulting dangling bonds were capped appropriately, and the forces on all atoms in these truncated models were evaluated. The forces acting along different length coordinates were evaluated by vector addition of the atomic forces, and were associated with the forces exerted by the stilbene component on the cyclobutene moiety in the full models of the macrocycle. A comparison of the calculated restoring forces with the rates obtained in the experiments showed that the F-dependence of the rates could be described adequately by a single-coordinate model in which F was applied between groups directly bonded to the scissile bond of cyclobutene.

Application of F Between Atoms
Modeling isotensional experimental conditions involves subjecting atoms to a constant external force. One means of achieving this is by treating specific pairs of atoms in a molecule as PPs and applying an external force, F, along the vector, R, connecting these atoms. This approach is known as the external force is explicitly included (EFEI) model [31]. The application of the F along this direction leads to the FMPES:
$$ {E}_F\left(\mathbf{q};F\right)={E}_0\left(\mathbf{q}\right)-FR\left(\mathbf{q}\right), $$
where q represents the nuclear degrees of freedom, which either include R explicitly as an internal coordinate or can be used to calculate R as the distance between the atoms corresponding to the PPs, E0 is the energy of the structure on the zero-F PES, and R is the distance between the atoms used as PPs. Equation (5) is differentiable with respect to the nuclear positions, which allows procedures such as geometry optimizations, frequency calculations, IRC calculations, and MD simulations to be performed directly on the FMPES. These abilities allow one to examine the F-dependence of molecular structures and energies by changing the value of F used in the calculation. For example, in the case of studying F-dependent reaction kinetics, one can locate reactant and TS structures on the FMPES defined by (5) at different values of F to assess how reactions barriers depend on F. In addition, MD simulations can be performed at different constant values of F or with time-dependent values of F to gain insight into the F-dependent behavior of chemical systems.

Unlike the holonomic constraints used to constrain geometries, the EFEI approach is not implemented in the distributed versions of any software package to our knowledge. However, the modifications to such codes needed to implement the definition of the energy in (5) along with the associated additions to the first and second derivatives of the potential energy with respect to nuclear positions are straightforward to add to existing simulation software. The use of (5) requires the definition of which atoms act as PPs, yet eliminates the need to choose directions along which F is applied. In fact, by applying F along the vector connecting two atoms in the system, one ensures that no net external force is added to the molecule. Despite these benefits, the application of F along the vector connecting two specific atoms in a molecule is clearly an approximation to the manner in which mechanochemistry is achieved under isotensional conditions, because the devices used to subject the molecule to F are not incorporated into the model in any way. EFEI calculations have been used to model processes such as the rupture of covalent bonds in pericyclic reactions [29, 34, 47], cyclizations [102], and the design of optical force probes [103]. Selected studies are described in what follows.

Marx and coworkers have used EFEI methods to examine the ring opening of cyclobutene, and substituted variants thereof [31, 34]. As discussed in greater detail in the section “Comparison of Internal Group Models,” these simulations showed that F can alter the barrier to this reaction in a way that circumvents the Woodward–Hoffman rules. In addition, they used EFEI calculations to examine the transmission of F via oligomers bonded to the carbon atoms in the scissile bond of cyclobutene [46, 47]. The factors that affect the transmission of F through these chains is of fundamental importance in mechanochemistry because oligomers are used for this purpose in many experimental techniques for subjecting molecules to F. They showed that the applied force, Fmax, at which the reaction occurs without additional thermal activation, i.e., the value of F at which the reaction is barrierless on the FMPES, is dependent on the chain length, and that alternating between even and odd numbers of units in the oligomers can drastically alter Fmax for short chains. It was also found that Fmax depends on the angle between the chain and the reactive bonds, the manner in which this angle changes in response to F, as well as the stiffness of the degrees of freedom within the chain. All these factors can be controlled experimentally, and thus this study provides insights that can be used to rationally design groups for applying F in experiments.

Uggerud and coworkers used EFEI-based MD simulations in conjunction with DFT calculations and models consisting of a chain of Morse potentials to study the basic features of mechanochemical processes under dynamic conditions [104]. The results of this study showed that the simple model consisting of Morse potentials is adequate for describing the F-dependent dynamics of polymers, which may reduce the computational costs of simulations of mechanochemical processes. In addition, the study demonstrated significant differences between the results of simulations performed with F held fixed and with F applied suddenly. In particular, sudden-F approaches preferentially led to bond rupture in the central portion of the polymer, whereas fixed-F simulations preferentially led to the rupture of terminal bonds. These results illustrate that care must be taken in terms of simulating F-induce bond rupture processes, with a particular need to ensure that the manner in which F is applied mimics that found in experiments.

Comparison of Internal Group Models

The methods employing constrained geometries implicitly apply F, whereas the SMD and EFEI methods model the application of F explicitly. As such, one may anticipate that simulations using these methods would yield different results. In what follows, we compare how the outcomes of calculations can be affected by the manner in which F is applied in the context of geometry optimizations, and MD simulations.

Geometry optimizations performed on the EFEI FMPES obtained with a given F yield stationary points in which the length associated with the coordinate R adopts a value, Ropt, at which the internal force along this degree of freedom, Fint, equals F. Meanwhile, the forces acting along all other degrees of freedom are zero. The energies of these stationary points are identical to those obtained through constrained geometry optimizations in which R0 is fixed to Ropt. As such, constrained geometry optimizations and EFEI calculations yield identical values of quantities such as reaction energies or activation energies, which are based on the relative energies of stationary points. This equivalence allows either method to be used for the purposes of exploring thermodynamics and kinetics on the basis of comparing the energies of stationary points on the FMPES.

Meanwhile, many computational studies of reactions occurring under mechanochemical conditions have employed MD simulations. These simulations can provide detailed atomic-level information regarding the changes in structure that occur during reactions, and are thus useful in elucidating reaction mechanisms. Such insights can be particularly valuable in the context of mechanochemistry, where the reaction mechanism followed by the system can be dependent on F. The identification of such mechanisms is important in the context of ensuring that the relevant reaction steps are examined in static calculations, i.e., geometry optimizations, which generally require a priori knowledge of the reaction mechanism to locate any intermediate and TS structures along the reaction pathway.

MD simulations of mechanochemical processes have been reported in which either F is treated as a controlled parameter or an interatomic distance is constrained to mimic the application of F. In cases where F is applied explicitly, the system can move along all the 3N – 6 nuclear degrees of freedom to progress from reactants to products on the FMPES. In some cases, these simulations also increase F over time to mimic AFM experiments, for example, or to promote the occurrence of reactions on the timescales accessible in MD simulations. Meanwhile, constraining a distance to a particular value of R0 only allows the system to move along the remaining 3N – 7 nuclear degrees of freedom. Assuming the constrained distance is related to the reaction of interest, fixing it to one value precludes the occurrence of this reaction during the simulation. To overcome this limitation, it is common practice to change the value of R0 over time at a predefined rate, v, from some initial value, R0(0), i.e., R0 = R0(0) + vt [33, 39, 40, 41, 90, 100, 101]. Changing R0 in this manner causes the system to sample the series of FMPESs defined by (4) with the different values of R0 encountered in the simulation. This approach is computationally convenient because it employs thermodynamic integration techniques available in most MD simulation packages. However, changing R0 over time causes the constraint to mimic the application of external forces which change dramatically during the course of the reaction. For R0 below the value of the constrained distance in the TS, the constraint mimics a tensile force, whereas the constraint mimics a compressive force as the system moves from the TS to the product. Such changes in F do not correspond to the conditions imposed in experiments and differ significantly from those modeled in simulations where F is explicitly applied.

The differences in the outcomes obtained in MD simulations in which F is applied explicitly or implicitly through the application of a time-dependent distance constraint are evident from studies of the ring-opening of cyclobutene along conrotatory and disrotatory pathways [29, 30, 31, 33, 34]. As described above, this reaction has been examined extensively in light of experiments illustrating that mechanochemical conditions can be used to alter the major product of the ring-opening reaction by biasing the system along either of the conrotatory or disrotatory directions irrespective of the Woodward–Hoffmann rules.

MD simulations in which F was applied explicitly yield results consistent with these experiments [29, 30, 31, 34]. For instance, an SMD study [30], in which 20 trajectories of cyclobutene were subjected to F using the hydrogen atoms on the carbon atoms of the scissile as APs, showed that the conrotatory pathway was consistently followed if the APs were in a trans configuration, whereas the disrotatory pathway was consistently followed if the APs were in a cis configuration. These results were rationalized in terms of the underlying energetics, which showed that the application of F using cis or trans APs preferentially reduced the barriers along the disrotatory and conrotatory pathways, respectively. Analogous results have been obtained in static calculations and MD simulations using the EFEI approach [29, 31, 34].

The mechanochemical ring opening of cyclobutene was also investigated using a time-dependent distance constraint to mimic the application of F during MD simulations [33]. In those simulations, the constraint was applied between hydrogen atoms bonded to the carbon atoms in the scissile bond of cyclobutene that were in a cis arrangement with respect to the ring in an attempt to induce disrotatory opening. The target separation of these atoms was then increased at a rate of 2.0 Å/ps during the simulation to mimic the application of an external force. A total of 20 independent simulations were performed using DFT methods. The results of the MD simulations showed that the conrotatory product was formed for all 20 trajectories, which is inconsistent with the results of the sonication experiments [19], SMD simulations [30], and EFEI calculations [29, 31, 34].

The discrepancy between the results of simulations of cyclobutene subjected to constrained distance and isometric conditions motivated us to explore the differences between the methods in greater detail. To accomplish this, we performed MD simulations for the ring opening of cyclobutene using a constrained distance or EFEI methods to apply F implicitly or explicitly, respectively. The outcomes of the simulations were also analyzed through relaxed potential energy surface scans. In these calculations, the QC energy was evaluated at the CASSCF(4,4)/6-31G(d,p) level of theory. The MD simulations were performed with a version of the GAMESS-US software package that we modified to perform MD simulations with time-dependent distance constraints or the explicit application of F through the EFEI method [105], whereas the PES scans were performed with Gaussian09 [106].

The MD simulations of cyclobutene that employed a constrained distance were performed by increasing the distance between a pair of hydrogen atoms in a cis configuration at rates of 0.5, 1.5, 2.5, and 5.0 Å/ps. Five independent trajectories were evaluated at each pulling rate. All AIMD simulations showed that ring opening of cyclobutene occurred exclusively through the conrotatory pathway, which is in agreement with the previous study using similar methods [33], but is inconsistent with EFEI calculations [29, 31, 34], previous SMD simulations [30], and experiments performed under mechanochemical conditions [19]. Structures observed during the simulation performed with a pulling rate of 0.5 Å/ps are shown in Fig. 3 to illustrate the conrotatory ring opening process. At t = 0.0 ps, the system is in the form of cyclobutene. The carbon–carbon scissile bond ruptures at t = 2.8 ps. At this point, the orientation of both methylene groups is consistent with the opening along the disrotatory pathway. At around t = 3.0 ps, however, one of the methylene groups rotates such that system starts to follow a conrotatory pathway. As the system progresses along the conrotatory pathway, rotation occurs about the central carbon–carbon bond, leading to the formation of trans-1,3-butadiene.
Fig. 3

Structures for the ring opening of cyclobutene during an MD simulation at a pulling rate of 0.5 Å/ps. Hydrogen atoms in a cis configuration were selected as the PPs and are indicated with red stars. The motion of the system is indicated using dark blue arrows. Silver and turquoise spheres indicate carbon and hydrogen atoms, respectively

To illustrate further that the system followed the conrotatory pathway, the changes in the length of the carbon–carbon scissile bond and the torsions associated with the conrotatory and disrotatory motions were monitored during the simulations. These quantities are shown in Fig. 4, along with definitions of the conrotatory and disrotatory simulations, for a simulation performed with a pulling rate of 0.5 Å/ps. The data show that the length of the carbon–carbon scissile bond increases slowly at the beginning of the simulation, and then increases rapidly at approximately 3.0 ps, because of bond rupture. Prior to the point of bond rupture, the conrotatory angle fluctuates around zero and the disrotatory angle increases steadily. At the point of bond rupture, however, the conrotatory angle increases sharply and the disrotatory angle decreases to approximately zero. The changes in angles indicate that the ring opening of cyclobutene proceeds toward the disrotatory product at the initial stages of the simulation until the rupture of the carbon–carbon scissile bond. After this point, the system progresses along the conrotatory pathway. Although it may be possible for the system to follow the disrotatory pathway if higher extension rates were used, progression along the conrotatory pathway was observed in all simulations performed with constrained distances, even when the extension rate was ten times faster than that used to generate the data in Fig. 4.
Fig. 4

(a) Schematic representation of the angles defining the ring opening of cyclobutene along conrotatory and disrotatory pathways. Changes in either angle toward 360° indicate that the associated ring-opening pathway (i.e., conrotatory or disrotatory) is followed. (b) Length of carbon–carbon scissile bond distance, dC–C, conrotatory angle, and disrotatory angle during an MD simulation of the ring opening of cyclobutene under mechanochemical conditions imposed by increasing the distance between the cis-PPs at a pulling rate of 0.5 Å/ps

An analogous set of MD simulations were performed on the EFEI FMPES applying F to the same hydrogen atoms that were used to define the constrained distance in the MD simulations discussed above. Five independent trajectories were calculated at six different values of F ranging from 2,500 to 3,000 pN in 100 pN intervals. All the simulations showed that ring opening of cyclobutene proceeds exclusively through the disrotatory pathway, which is inconsistent with the simulations that employed distance constraints, yet is consistent with the sonication experiments [19], in which the major product formed resulted from the disrotatory pathway. Structures observed during the MD simulation at F = 2900 pN are shown in Fig. 5. The structure at 0.1 ps shows that the carbon–carbon scissile bond is quite extended and the methylene groups are rotated in a manner consistent with motion along the disrotatory pathway. This deformation of the structure is because of the application of F. The scissile bond ruptured at approximately 1.6 ps, with the methylene groups moving in a manner that yields the disrotatory product. Rotation about the central carbon–carbon bond followed. Ultimately, this series of processes yields trans-1,3-butadiene along a disrotatory pathway.
Fig. 5

Snapshots taken during an MD simulation for the ring opening of cyclobutene at F = 2900 pN. cis-PPs were used in the simulation and are indicated with red stars. As the reaction progresses, both methylene groups rotate in opposite directions to yield the cis disrotatory product observed at around 1.8 ps. Rotation around the carbon–carbon single bond leads to the formation of the trans disrotatory product at 2.5 ps. The motion of the system is indicated using dark blue arrows. Silver and turquoise spheres indicate carbon and hydrogen atoms, respectively

To illustrate the formation of the disrotatory product in the EFEI-based MD simulations, the changes in the carbon–carbon scissile bond distance and conrotatory and disrotatory angle were monitored over time. These quantities are shown in Fig. 6. At the beginning of the MD simulation, the carbon–carbon scissile bond distance fluctuated around the equilibrium bond length until increasing rapidly around when this bond ruptured at approximately 1.5 ps. The conrotatory and disrotatory angles fluctuate around 0 and 100°, respectively at the beginning of the simulation. At approximately 1.6 ps, the disrotatory angle increased sharply and the conrotatory angle dropped to approximately 0°. The analysis of the angles indicates that the disrotatory angle dominates the behavior of the system throughout the MD simulation. The results obtained for the carbon–carbon scissile bond length as well as the conrotatory and disrotatory angles were observed for the MD simulations at the other values of F.
Fig. 6

Length of the carbon–carbon scissile bond, dC–C, conrotatory angle, and disrotatory angle during an MD simulation of the ring opening of cyclobutene under EFEI mechanochemical conditions using cis-PPs at F = 2,900 pN

Portions of the FMPESs obtained with F applied either explicitly through the EFEI method or implicitly via distance constraints were examined to gain insights into the origins of the differences in the results of the MD simulations performed with these two approaches. To do this, a series of relaxed PES scans were performed at the CASSCF(4,4)/6-31G(d,p) level of theory. The scanned coordinates corresponded to the H-H distance associated with the PPs used in the MD simulations and the torsions associated with the movement of the methylene groups along conrotatory or disrotatory directions. Specifically, conrotatory movement was examined by fixing the dihedrals α and δ in Fig. 4a equal to each other, whereas the disrotatory surface was examined by setting the dihedrals α and β equal to each other. Overall, this corresponds to performing a series of geometry optimizations in which one distance (R0) and two dihedrals were constrained, at the same time allowing the remaining degrees of freedom to relax. The BO energies of these constrained structures were obtained through these optimizations. The internal forces acting along the direction between the hydrogen atoms used as PPs in the MD simulations were then used in conjunction with the distance between these atoms for each scanned structure to construct the slices of the constrained-distance FMPESs associated with varying the dihedral angles associated with conrotatory or disrotatory ring-opening at different values of R0. Similarly, a constant external force, F, was multiplied by the PP separation for each structure to construct the EFEI FMPESs at different values of F.

The set of slices of the constrained-distance FMPESs associated with conrotatory and disrotatory movement of the methylene groups in cyclobutene are shown in Fig. 7a, b, respectively. The label ‘PP separation’ corresponds to the parameter, R0, that was varied in the MD simulations that applied F via constrained distances. The torsions associated with the minimum energy structure for each value of R0 sheds light on the sequence of structures that the system follows preferentially during MD simulations in which R0 is increased in a time-dependent manner. The minimum energy paths (MEPs) associated with this series of structures are indicated as solid black lines on the surfaces in Fig. 7a, b.
Fig. 7

Portions of the FMPESs associated with rotating methylene groups of cyclobutene at different PP separations under constrained distance conditions for (a) the disrotatory pathway and (b) the conrotatory pathway. The energies on the surfaces are plotted relative to that of the structures on the disrotatory surface with a PP separation of 3.0 Å and torsion of 180°. Solid lines indicate the MEP on each surface. The dashed line indicates the path the system follows upon moving from the disrotatory to conrotatory surface after the dissociation of the scissile bond. Locations of transition states and products are indicated as ‘TS’ and ‘P,’ respectively

The MEP on the conrotatory surface shows that the system follows a path connecting the reactants and products in which the PP separation increases to 3.4 Å without any large change in the torsions associated with the methylene groups. Increasing the PP separation from 3.4 Å to 4.2 Å results in the torsions increasing from 120° to 180° to follow a pathway corresponding to the conrotatory ring opening to yield a structure similar to the optimized structure of cis-1,3-butadiene on the BO PES. As the PP distance increases further, the system progresses past minima corresponding to the cis and trans forms of 1,3-butadiene.

The minimum energy pathway on the disrotatory surface does not connect cyclobutene with the 1,3-butadiene. Instead, increasing the PP separation from 3.0 Å to 3.7 Å increases the torsions to 150°. These angles then drop to 120° when the carbon–carbon scissile bond ruptures at a distance of 4.3 Å, which is consistent with the results of the MD simulations. In the structure where the constrained torsions reach 120°, all four values of α, β, γ, and δ are 120°. As such, the system can move from this point toward either the conrotatory or disrotatory products. The data indicate that the lowest energy path from this structure toward larger PP separations on the disrotatory surface does not lead to the 1,3-butadiene product. Instead, once the torsions reach 120°, it is energetically favorable for the system to move to the conrotatory surface (Fig. 7b), with the torsions increasing in a manner that involves both methylene groups rotating in the same direction. This pathway is designated by the dashed line on the conrotatory FMPES, which is a continuation of the solid black line on the disrotatory PES. These features of the conrotatory and disrotatory surfaces obtained through calculations employing constrained distances account for the fact that only conrotatory products were observed during the MD simulations.

The portion of the EFEI FMPES calculated with F = 1,500 pN is shown in Fig. 8. The data show that the structures corresponding to conrotatory and disrotatory products are minima on their respective surfaces. MEPs connecting the reactants with these products could be identified on each surface, and are indicated by the solid lines. The changes in structure that occur along these MEPs are consistent with those observed during the MD simulations performed under EFEI conditions.
Fig. 8

Portion of the FMPES associated with rotating methylene groups of cyclobutene at different PP separations under EFEI mechanochemical conditions at F = 1,500 pN for (a) the disrotatory pathway and (b) the conrotatory pathway. The energies on the surfaces are plotted relative to that of the structures on the disrotatory surface with a PP separation of 3.0 Å and torsion of 180°. Solid lines indicate the MEP on each surface. Locations of transition states and products are indicated as ‘TS’ and ‘P,’ respectively

Overall, the results of these calculations illustrate that MD simulations in which F is applied explicitly can provide different results than those in which F is applied by varying R0 over time. In the case of the former, a constant F is applied and the system is allowed to move on a single FMPES along all 3N – 6 degrees of freedom to follow the MEP connecting reactants to products. In the case of the latter, the system can move along 3N – 7 degrees of freedom to sample a series of FMPESs defined by the values of R0 spanned during the simulation. These differences are evident from an examination of the MEPs in Figs. 7 and 8, as well as the differences in the outcomes of the MD simulations described above. The ring-opening of cyclobutene is likely to be an extreme example of these artifacts because the product formed during the reaction depends on the rupture of a C–C bond and rotation of two methylene groups. The rupture of the scissile bond and rotation along the disrotatory pathway are both favored in simulations where F is applied explicitly in a tensile manner because both of these processes increase the separation of the PPs. Meanwhile, increasing R0 from its equilibrium value in cyclobutene imposes a tensile F that favors the rupture of the C–C bond, but increasing R0 from the point at which bond scission occurs leads to the application of a compressive F that disfavors rotation of the methylene groups along the disrotatory path. In other cases where the reaction is dominated by the change in a given interatomic distance, e.g., the dissociation of strained polymers via the rupture of an individual bond or the dissociation of S–S bonds in proteins, the differences between MD simulations in which F is applied explicitly and those in which F is applied via time-dependent distance constraints may be less significant. Nonetheless, one should be aware of these differences when selecting an approach for imposing F in MD simulations of mechanochemical processes.

2.2 Indirect Evaluation of the FMPES

A qualitative understanding of the manner in which F alters the PES can be useful in developing strategies for the mechanochemical activation of reactions and for understanding the outcomes of experiments performed under mechanochemical conditions. For example, understanding how F affects the relative energies of reactant and TS species, as well as the structures of those species, can aid in the selection of groups that are subjected to F in order to activate reactions under mechanochemical conditions. Such information can be obtained by locating the structures of these species on the FMPESs associated with different values of F through calculations performed directly on the FMPES using the methods describe above. However, such calculations must be performed at each value of F and can thus be quite laborious. Instead, useful insights regarding the relationships between F and the activation of reactions can be obtained by using knowledge of the zero-F PES to predict indirectly features of the FMPES. Kauzmann and Eyring developed one of the first ‘indirect’ models to describe the reactivity of chemical systems on the FMPES [107]. That model involved extending transition state theory to incorporate the effect of F on the activation energy, ΔE, with ΔE being reduced from its zero-F value in a manner that depends linearly on F. This model was found to account for the increased rate of homolytic bond cleavage of polymers with mechanical force. More recently, additional methods for indirectly predicting the features of the FMPES on the basis of knowledge of the zero-F PES have been reported. These techniques are discussed in what follows.

2.2.1 Bell’s Model

Bell developed a theoretical framework to predict the effects of F on the adhesion between cell surfaces [108]. In Bell’s model, it is assumed that applying F does not affect the structures of reactants, TS, and products of a reaction, but rather the sole effect of F is to perform work on the system as it moves between these species during the course of a reaction. In the context of reaction kinetics, the work performed on the system occurs as a result of a change in the separation of groups subjected to F as the system moves from the reactants to the TS. Within the assumption that F does not affect the reactant and TS structures, it is possible to determine this change in separation, ΔR(0), using knowledge of these structures on the zero-F PES, and it follows that the barrier on the FMPES can be approximated as
$$ \Delta {E}^{\ddagger }(F)=\Delta {E}^{\ddagger }(0)-F\Delta R(0), $$
where ΔE(0) is the barrier on the zero-F PES.

The simplicity of Bell’s model is useful in the context of chemical simulation because its use only requires the evaluation of reactant and TS structures on the zero-F PES, yet provides the ability to predict properties on the FMPES. Of course, the assumption that the reactant and TS structures are invariant to F is only reliable (even qualitatively) for low values of F, which limits the range of F over which Bell’s model can provide reliable predictions regarding the relationships between reaction barriers and F. Nonetheless, the simplicity of Bell’s model combined with its ability to provide a qualitatively accurate description of the relationship between reaction barriers and F has led to its application in many practical contexts. For instance, Bell’s model has been used extensively to study the mechanochemical response of several biological systems such as the rupture of disulfide bonds in proteins [109], the distortion of extracellular matrices in cells and tissues [110], and the force generation in actin-binding protein motors [111].

2.2.2 Tilted Potential Energy Profile Model

The assumption that the reactant and TS structures are invariant to F is a key limitation of Bell’s model. The movement of these structures produced by the application of F is addressed in a limited manner by the tilted potential energy profile model [112]. In this model, it is assumed that F is aligned exactly with the zero-F reaction coordinate. The application of F then shifts the energies of all structures along this coordinate by –FR, where R is the distance between the atoms subjected to F at each point along the reaction coordinate. This modification of the energies along the reaction coordinate is illustrated schematically in Fig. 9. As shown in that figure, the modification of the energies has the effect of allowing the reactant and TS structures to move to new locations along the reaction coordinate. In particular, if F is positive and R increases along the reaction coordinate, the reactant and TS structures move closer to one another and the activation energy is reduced, which is consistent with the Hammond postulate. However, by restricting structural changes to occur only along the zero-F reaction coordinate, the tilted potential energy profile model fails to account for anti-Hammond effects, which can influence the activation energy and even cause the system to follow alternate reaction pathways. The tilted potential energy profile model can also be relatively demanding from a computational standpoint, because the energies of multiple structures along the zero-F reaction coordinate must be evaluated to use it. This model has been applied to study the effect of F on the kinetics of several biochemical processes [112, 113, 114].
Fig. 9

A general schematic showing the effect of F on the PES on which a molecule moves as described in the tilted potential energy profile model. The black curve illustrates the change in energy along the reaction coordinate in the absence of an external force. The labels R, TS, and P indicate the positions of the zero-F reactants, transition state, and products along this reaction coordinate, respectively. The red curve is obtained by adding –FR to the zero-F energies. The addition of this work term lowers the barrier relative to its value in the absence of applied F. The application of F also shifts the locations of R, TS, and P from their zero-F positions, but otherwise this model assumes the series structures comprising the reaction coordinate in the absence and presence of F are identical

2.2.3 Extended Bell’s Model

Bell’s model provides a qualitatively correct description of the effect of F on reaction barriers. However, the assumption that the reactant and TS structures are unaffected by the application of F restricts the range of F over which Bell’s model applies. The F-induced movements of the reactant and TS structures along the zero-F reaction coordinate are captured partially by the tilted potential energy profile model; however, in general, the application of F can induce changes in the reactant and TS structures that are orthogonal to the zero-F reaction coordinate. Such changes in structure, and the consequent effects on reaction barriers, can be captured by expanding the FMPES as a Taylor series with respect to the nuclear positions [37, 97, 115].

A second-order expansion of the FMPES at a stationary point on the zero-F PES yields
$$ {E}_F\left({\mathbf{q}}_0+\delta \mathbf{q};F\right)={E}_0\left({\mathbf{q}}_0\right)+\frac{1}{2}\delta {\mathbf{q}}^T{\mathbf{H}}_{\mathrm{int}}\delta \mathbf{q}-F\left({R}_0+\delta R\right), $$
where q0 represents the nuclear coordinates corresponding to a stationary point on the zero-F PES, δq represents the changes in these coordinates upon the application of F, R0 is the distance between the atoms subjected to F when F is not applied, and δR is the change in R produced by the application of F. All first derivatives of E(q) with respect to nuclear coordinates are zero and are therefore not included in (7). The second derivatives of E0 with respect to nuclear coordinates are grouped into the term Hint, which is the Hessian in terms of internal coordinates. The distance between atoms that are subjected to F, termed R, represents one of the internal coordinates used to construct Hint.
The change in the structure of a stationary point upon moving from the zero-F PES to the FMPES can be quantified as
$$ \delta \mathbf{q}=\mathbf{C}\mathbf{F}, $$
where \( \mathbf{C}={\mathbf{H}}_{\mathrm{int}}^{-1} \) is the compliance matrix and F is a vector whose components are all zero except for that associated with internal coordinate R, which has a magnitude of F. Substituting (8) into (7) and noting that F is the only non-zero component of F yields
$$ {E}_F\left({\mathbf{q}}_0+\delta \mathbf{q};F\right)={E}_0\left({\mathbf{q}}_0\right)-F{R}_0-\frac{F^2}{2}{C}_{RR}, $$
where CRR is a compliance term that accounts for the relationship between δR and F while taking into account the relaxation along all other internal coordinates. Equation (9) can be applied to reactant and TS geometries to estimate the reaction barrier on the FMPES as
$$ \Delta {E}^{\ddagger }(F)=\Delta {E}^{\ddagger }(0)-F\Delta {R}_0-\frac{F^2}{2}\Delta {C}_{RR}, $$
where \( \Delta {R}_0={R}_0^{TS}-{R}_0^r \) is the change in R as the system progresses from the reactant to the TS and \( \Delta {C}_{RR}={C}_{RR}^{TS}-{C}_{RR}^r \) is the change in compliance matrix element along the direction R between the TS and the reactant. The first two terms in (10) are equivalent to Bell’s model – see (6) – whereas the last term incorporates changes in energy that arise from F-induced changes in the structures of the reactants and TS.

The extended Bell’s model outlined in (10) provides an estimate of the reaction barrier on the FMPES without performing a large number of quantum chemical calculations. The terms in this model can provide information in the selection of the atoms that are subjected to F to achieve a desired mechanochemical response. For instance, this model indicates that lowering the barriers on the FMPES, and hence activating reactions, can be achieved by selecting pairs of atoms whose separations increase upon moving from the reactant to TS and/or for which the TS is more compliant than the reactant \( \left({C}_{RR}^{TS}>{C}_{RR}^r\right) \). This model has been used to describe the effect of applied force on the rupture of bonds in pericyclic reactions [37] and the influence of anti-Hammond effects on barriers [116], and to explore how the regiochemistry of polymers attached to mechanophores can affect F-dependent changes in reaction rates. [117].

One of the main limitations of this model is that the barrier on the FMPES is predicted using structures, energies, and Hessians obtained with zero-F reactant and TS structures. As such, the predictions made with (10) become increasingly unreliable as F is increased. It may be possible to improve the abilities of extended Bell’s models to predict accurately the structures of reactant and TS structures, as well as barriers, at higher F by increasing the order to which the zero-F energy is expanded with respect to changes in structure; however, no proof showing that this Taylor series expansion converges exists to our knowledge. Regardless, calculating the higher order derivatives of the zero-F energy with respect to the nuclear positions is generally computationally intractable.

An extended version of Bell’s model that is applicable to a wider range of F than that in (10) has been reported by Makarov and coworkers [36]. Their model straddles the line between indirect and direct evaluation of the FMPES by using Hessians obtained at non-zero F to propagate numerically the reactant and TS structures over a range of F. This approach involves representing the structure of a stationary point on the FMPES as
$$ \Delta {E}^{\ddagger }(F)=\Delta {E}^{\ddagger }(0)-{\displaystyle {\int}_0^F\left[{\mathbf{q}}_{TS}\left({F}^{\prime}\right)-{\mathbf{q}}_R\left({F}^{\prime}\right)\right]}\mathbf{1}d{F}^{\prime }, $$
where l is a vector indicating the direction along which F is applied. The structures obtained at different values of F can then be used to find reaction barriers at higher F:
$$ \Delta {E}^{\ddagger }(F)=\Delta {E}^{\ddagger }(0)-{\displaystyle {\int}_0^F\left[{\mathbf{q}}_{TS}\left({F}^{\prime}\right)-{\mathbf{q}}_R\left({F}^{\prime}\right)\right]}\mathbf{1}d{F}^{\prime }, $$
where qTS and qR are the structures of the TS and reactant, respectively, at F. Although this approach overcomes the limited abilities of (10) at higher F, its use imposes additional computational demands because a Hessian must be generated for the reactant and TS at each F considered. The additional information obtained by generating Hessians at higher F makes it possible to identify and characterize F-induced instabilities in reactant and TS structures in terms of F-induced changes to the eigenvalues of the Hessian. However, the F-induced changes in the reactant and TS structures arising from these instabilities can also be determined from geometry optimizations of these structures on the FMPES at a comparable or even lower computational cost than that associated with performing multiple calculations of the Hessian.
The models described above aim to obtain approximate details regarding the FMPES using information from the zero-F PES. However, chemical processes are governed by free energies. Boulatov and coworkers have explored the ability to access free energy barriers using truncated Taylor series expansions [97, 118]. The contribution of the potential energy is obtained by treating the system as reactive site coupled to an external harmonic constraining potential as discussed in the section “Application of F through Constrained Geometries,” which incorporates the effects of polymers used to subject the reactive site to F without employing an explicit treatment of those polymers. The constraining potential has a compliance, Cc, and equilibrium length, Rc, which are incorporated into the potential energy to yield. Incorporation of the vibrational partition functions, Z, which are dependent upon Cc, yields the free energy barrier for the dominant reactant conformer as
$$ \Delta {G}^{\ddagger}\left({C}^c,F\right)=\Delta {G}^{\ddagger}\left(\infty, 0\right)+\left(F\Delta {R}_0+\frac{F^2\left(\Delta {C}_{RR}\right)}{2}\right)\;\left(\frac{C^c+{C}_{RR}^r}{C^c+{C}_{RR}^{TS}}\right)+\frac{\Delta {R}_0^2}{C^2+{C}_{RR}^T}-{k}_BT \ln {\left(\frac{Z^0}{Z^c}\right)}^{\ddagger }{\left(\frac{Z^c}{Z^0}\right)}^r $$
where terms with a superscript ‘c’ refer to quantities that incorporate the compliance of the confining potential. Equation (13) reduces to (10) in the limit that \( {C}^c\to \infty \) and \( {R}^c\to \infty \).
Equation (13) is intended to describe the free energy barrier of a reactive site embedded in a polymer, where ΔR should correspond to the distance between the PPs in the experimental system where F is applied two polymers that are grafted onto a reactive site. Computational limitations prevent the use of model systems that accurately include polymers of sufficient length to ensure that (13) reliably reproduces experimental conditions. To overcome this limitation, Boulatov and coworkers employed the previously demonstrated relationship between the F applied along a conveniently defined local coordinate that applied between the PPs to cast (13) in a form amenable to use to quantities obtained through QC calculations of small model systems in the limit that \( {C}^c\to \infty \) and \( {R}^c\to \infty \):
$$ \Delta {G}^{\ddagger}\left({C}^c,{F}_l\right)=\Delta {G}^{\ddagger}\left(\infty, 0\right)+\left({F}_l\Delta {R}_0^l+\frac{F_l^2\left(\Delta {C}_{ll}\right)}{2}\right), $$
where the forces, separations, and compliances correspond to those associated with the local coordinate, l, as opposed to the coordinate connecting the ends of the polymers used in experiments. Equation (14) is compatible with QC calculations of small model systems, where the necessary distances and compliances can be obtained using a local mechanochemical reaction coordinate that is relevant to the reaction being studied, e.g., a bond length that changes during a reactions. The comparison of this model with Bell’s model and a full statistical mechanical treatment of reaction kinetics has shown that it is suitable for describing mechanochemical processes over the range of F encountered in experiments [118].

2.2.4 Comparison of Indirect Models for Predicting the FMPES

The conceptual models outlined above all describe the effects of F on reaction barriers to different levels of approximation. The abilities of these models can be assessed by comparing the predicted barriers to those obtained from QC calculations of the barriers on the FMPES. Such a comparison is provided in Fig. 10, which compares the barriers obtained with different conceptual methods for the ring-opening of 1,3-cyclohexadiene with F applied to the atoms indicated with asterisks in Fig. 10a. QC structures, energies, and frequencies of the reactant and TS for this reaction were calculated at different F using CASSCF(6,6)/6-31G(d,p) methods. These data were then used to predict the barriers at different F using (6), (10), and (12). The comparison of the reaction barriers obtained with these different methods is given in Fig. 10b.
Fig. 10

(a) Reaction scheme for the ring opening of cyclohexadiene along the disrotatory (allowed) pathway. The atoms indicated with asterisks were used to apply F. (b) Reaction barriers as a function of F applied at the CASSCF(6,6)/6-31G(d,p) level and using (6), (10), and (12)

The results demonstrated that the agreement between the models and the QC values increases with the amount of information incorporated into the models. For instance, Bell’s model – see (6) – captures the increase in ΔE at low F, but fails to describe the reduction in ΔE at higher F. This is clearly a result of using a linear model to describe the F-induced changes in ΔE. The extended Bell’s model based on zero-F data – see (10) – qualitatively captures the changes in ΔE, with this quantity increasing at low F and decreasing at high F. However, the quantitative agreement between the barriers predicted with those model and the QC barriers becomes increasing poor as F is increased. Meanwhile, the barriers predicted using an extended Bell’s model with F-dependent parameters – see (12) – agrees well with the QC barriers over the entire range of F considered. In general, the agreement between the conceptual models and the QC barriers varies with the nature of the reaction and the atoms used to apply F. However, the relative agreement between the results obtained with the conceptual models and the QC data illustrated in this example are qualitatively consistent with what one would expect.

Although the agreement between the barriers predicted with these models and the QC data improves as (6) < (10) < (12), the computational requirements associated with these methods trend in the opposite direction. Specifically, Bell’s model can be directly used if the zero-F structures and energies of the reactants and TS are known. The extended Bell’s model given by (10) requires information regarding the zero-F compliance matrices of the reactant and TS. The Hessians associated with these structures are generally available after QC optimizations of these structures (assuming one has confirmed the natures of these stationary points via frequency calculations); however, a small amount of additional effort is needed to transform the Hessians to the coordinate systems associated with a set of internal coordinates that explicitly contain R. The use of (12) has significantly greater computational requirements because the Hessian must be evaluated at each F. This can become particularly costly if one is examining the dependence of the reaction barriers upon the different pairs of atoms used to apply F because a series of Hessian calculations would be required for each pair of atoms considered.

3 Methods for Simulating Bulk Mechanochemistry

The methods discussed in Sect. 3 are suited to modeling mechanochemical experiments in which individual molecules are subjected to tensile stresses applied between regions in those molecules. However, a large branch of mechanochemical experiments employ techniques such as milling and grinding in which shear and/or compressive stresses applied at the macroscopic levels induce reactions. Methods that focus on the action of tensile stresses at the molecular level, such as those described in Sect. 3, are not well-suited to modeling the shear conditions relevant for this type of mechanochemical activation. For instance, separating two regions of an individual molecule via a one-dimensional PES scan, SMD, or EFEI methods in an effort to mimic shear at the level of an individual molecule induces rotation as opposed to shear. In addition, shear and compressive stresses are typically transmitted via intermolecular interactions, and thus single molecule representations are not appropriate for modeling mechanochemical processes induced by shear and/or compression.

Invoking mechanochemical reactions via the application of shear and/or compressive stresses is similar to the field of tribochemistry, which involves studying reactions that are induced by the conditions experienced when surfaces slide past one another [119]. These conditions include stresses that reach the theoretical yield strengths of the materials in contact (which can reach several GPa) and local temperatures that reach the melting points of these materials (which can reach hundreds or even thousands of Kelvin depending on the materials). The high temperatures achieved can induce thermochemical processes, whereas the extreme stresses experienced in sliding contacts can promote mechanochemical reactions. As such, mechanochemical processes induced via shear and compression can be thought of as a subset of tribochemical reactions [120], and simulation techniques that are suitable for studying tribochemical processes can also be used to model these types of mechanochemical reactions. Despite this potential, far fewer simulations of these ‘bulk’ mechanochemical processes have been reported than their molecular counterparts, and thus the application of methods used in tribochemistry to study mechanochemical processes represents a relatively open avenue for further research.

To account for conditions of shear and compression, tribochemical simulations employ models that are sufficiently large to account for the interactions between molecules and, potentially, surfaces that impose and transmit these stresses. Such models often include surfaces that can be moved relative to one another to induce stresses and/or place the system in a simulation cell that can be deformed to impose stresses and strains. Schematics of such models are given in Fig. 11. Figure 11a shows a system containing two slabs that are separated by material that is to be sheared. This is achieved by moving the upper slab related to the bottom slab. Figure 11b shows a system in which an interface has been placed in a simulation cell. Compression can be invoked by subjecting the cell to stresses or strains perpendicular to the interface, whereas shear can be achieved by applying stresses or strains parallel to the interface.
Fig. 11

(a) Two slabs of alumina separated by a collection of aldehydes. Shear stress is imposed by moving the upper slab a distance, Δx, relative to the lower slab. (b) A hydroxylated alumina interface contained within a simulation cell. Compression can be achieved by subjecting the system to a stress (L) or strain (εzz) perpendicular to the interface. Shear can be achieved by applying a shear stress (τ) or a shear strain (εxz) parallel to the interface

In what follows, we describe how models such as those in Fig. 11 can be used to study stress-activated processes. As indicated above, these techniques have been used primarily to investigate tribochemical reactions; however, these techniques should be transferrable to studies of mechanochemical reactions. Section 3.1 focuses on describing techniques that allow tribochemical reactions to be modeled with static calculations such as geometry optimizations, potential energy surface scans, and minimum energy path calculations. Section 3.2 focuses on using MD simulations to examine the chemical behavior of systems that are exposed to compressive and/or shear stresses.

3.1 Static Simulation Methods

Static calculation methods include techniques such as single point energy calculations and geometry optimizations, which can be used to examine features of the PES without actually simulating the dynamics of the system. The use of static simulation methods to study the changes in energy and structure that occur during tribochemical processes has a long history [48, 49]. These methods have been used to optimize the reactants and products of tribochemical reactions, to examine the PES associated with the movement of surfaces relative to one another, and to identify the MEPs connecting the reactants and products of tribochemical reactions. The calculations provide insight into the changes in energies and stresses associated with these reactions, which are then used to predict properties such as friction forces and friction coefficients. Methods for mapping PESs are described in Sect. 3.1.1 and techniques for locating the MEPs of sheared systems are described in Sect. 3.1.2. In the context of mechanochemistry, these types of calculations may be useful for determining the structural transformations that occur during reactions and in evaluating the stresses to which molecules in contact are exposed.

3.1.1 Mapping Potential Energy Surfaces

As noted above, static computational models do not attempt to simulate the dynamics associated with chemical reactions. However, by performing a series of static calculations on systems whose structural features vary in a systematic manner, one can develop an understanding of the changes in structure and energetics that occur during a reaction. In the context of tribochemistry, such calculations are often used to map out the two-dimensional PES associated with the movement of one surface relative to another using models of the form shown in Fig. 11, which may or may not contain material between the slabs depending on the particular application. The construction of the PES is achieved by moving the upper surface relative to the bottom one over a series of fixed increments and evaluating the energies of each structure. The changes in energy on the resultant PES can be used to identify low energy pathways along this surface, which should correspond to the preferred directions for slip and shear, and can also be used to estimate the energetic barriers associated with slip along these directions. An example of a PES constructed via this approach and the consequent estimation of slip barriers along competing directions is shown in Fig. 12. The usefulness of the information gained from these PES scans is dependent upon how closely the changes in structure permitted during the simulations, e.g. relaxation of atomic coordinates and/or components of the lattice vectors orthogonal to the cell deformation, represent those which would occur if the system was simply allowed to evolve naturally.
Fig. 12

Upper panel: A two-dimensional PES constructed by moving one slabs of MoS2 relative to another along the x and y directions. Possible slip paths on the surface are shown with the solid and dashed lines. Numbers indicate energies in eV/atom relative to the energy of the optimized structure of MoS2 at various positions on the surface. Lower panel: Changes in energy along the paths designated I and II in the upper panel as a function of sliding distance. Figure reprinted with permission from Liang et al. [121]. Copyright (2008) by the American Physical Society

Mapping the PES using a set of static calculations, as opposed to exploring the reaction via dynamic methods, is advantageous from the standpoint of computational requirements. The overall computational cost can easily be moderated by selecting the density of the points in the scan. A low density of grid points can be used to obtain a coarse description of the shape of the PES and then a set of calculations using a higher density of grid points can be used to gain a refined representation of the PES along important directions such as slip pathways. This approach also has built-in parallelism. Because the points of the surface are decoupled, calculations of individual points be can be performed independently of one other and the limit of parallelism is effectively nonexistent.

An early example of fully mapping a PES between two materials was reported by Smith et al. in 1998 [122]. In that study, they use the ACRES method [123], a parallelizable grid-based method for calculating DFT energies, to calculate the PES associated with movement along the interface between molybdenum trioxide (MoO3) and molybdenum disulfide (MoS2). To do this, they built slabs of each material, brought the slabs into contact to form an interface that spanned the xy plane, and then minimized the energy of the system with respect to the interfacial separation at 16 lateral positions in the xy plane. Because of computational costs, only the interfacial distance was varied at each point on the PES while the atomic positions were held fixed. The PES obtained through these calculations led to the identification of a clearly preferred slip pathway along the interface, which corresponded to a groove on the MoS2 surface into which the oxygen atoms at the surface of the MoO3 slab fit. Because the interfacial distance was relaxed, it provided some information regarding the manner in which the surfaces interact at various lateral positions as well. It was found that orienting the surfaces in relative positions that caused the oxygen atoms in the MoO3 surface to move out of the groove in the MoS2 surface led to an interfacial separation that was 0.5 Å greater than the minimum energy interfacial separation for this system. Meanwhile, moving along the grooves showed that the interfacial separation varied by only 0.01 Å.

Phillpot et al. [121] expanded on the work of Smith and coworkers by evaluating the PESs of MoO3/MoO3, MoS2/MoS2, and MoO3/MoS2 interfaces using model systems contained in periodically-repeated simulation cells in conjunction with plane-wave DFT calculations. In the approach used in their study, the entire simulation cell was first optimized to ensure the system was at a minimum configuration before scanning the PES. This optimization was performed by allowing the atoms in the system to relax while keeping the lattice vectors defining the simulation cell fixed. An external load was then applied by compressing the system in the direction perpendicular to the interface, fixing the top and bottom atomic layers, and allowing the remaining atoms to relax. This procedure was continued until the magnitude of the system’s internal stress along the compressed direction was equal to the desired normal load. The application of a load in this context can be useful to determine the load-dependence of properties such as shear strengths and slip barriers. After the target load was reached, the PES was mapped by rigidly moving the top half of the system in the xy plane, relaxing the atoms at each point, and comparing the energy to that of the untranslated system. When moving the system in this manner, it is important to tilt the c lattice vector according the Lees–Edwards boundary conditions [124] to avoid introducing a second slip plane at the periodic boundary of the cell.

The PES scans of the MoO3/MoO3, MoS2/MoS2, and MoO3/MoS2 interfaces led to the conclusion that sliding MoO3 on MoO3 has the highest slip energy barrier and consequently requires the largest force to induce slip. Meanwhile, the mixed MoO3/MoS2 interface required a force that was approximately an order of magnitude lower to pass over its slip barrier. This methodology, or variations of it, has become a standard approach to constructing the PESs associated with tribochemical reactions and has been used to study hydrogenated [125, 126] and fluorinated-diamond surfaces [126], graphene oxide [127], graphane and fluorographane [128], and graphene/boron nitride [129], among other systems.

In 2013, Hod proposed a method for predicting slip paths in materials called the registry index (RI) [130], which has lower computational demands than PES scans. This method is based on the concept that the highest energy inter-surface stacking mode is generally the one with the most overlapping atoms between the two surfaces (e.g., in graphite, the highest energy configuration has the carbon atoms in adjacent sheets directly on top of each other, although the lowest energy configuration minimizes this overlap). To calculate the RI, each atomic center is assigned a circle of given radius and the projected overlap between the circles in two adjacent layers, designated the SCC, is calculated. This value has a maximum at the highest energy stacking mode and a minimum at the lowest energy mode. The RI is then evaluated as
$$ \mathrm{R}\mathrm{I}=\frac{S_{\mathrm{CC}}-{S}_{\mathrm{CC}}^{\min }}{S_{\mathrm{CC}}^{\max }-{S}_{\mathrm{CC}}^{\min }}, $$
which is bound to values between zero and one. In essence, RI is a measure of the extent of electron cloud overlap between two layers, which is one of the primary contributors to the sliding energy landscape in materials. Hod has shown the usefulness of RI in studying graphite, hexagonal boron nitride, MoS2, graphene/boron nitride, and multi-walled boron nitride nanotubes [129, 130].

3.1.2 PES Scans Along the Slip Path

Mapping the two-dimensional PES associated with sliding one surface past another can be computationally demanding, with the number of energy calculations increasing quadratically with the number of points considered along each direction. The computational requirements can be reduced considerably in cases where the slip path is known, where it is possible to perform a one-dimensional scan over a series of points along this path. This can be done in a manner similar to that described above, but only stepping along the slip path. Alternatively, optimization schemes, such as the nudged elastic band method, can be used to calculate the minimum energy path that aligns to a particular slip path. Such methods are described below.

One of the earliest reported examples of using one-dimensional scans to study atomic-scale friction with quantum chemical methods was published in 1990 by Zhong and Tománek [131], where they modeled a layer of palladium on graphite. To determine the friction behavior of this system, a one-dimensional PES scan was obtained using DFT by moving the palladium atoms along a specific direction on the graphite surface. The friction force was calculated by differentiating the potential energy with respect to the sliding distance. It was found that friction coefficient was low (on the order of 10−2) for small loads and increased for larger loads, in agreement with previous AFM experiments.

The Pakkanen group has reported a series of studies in which they examined the friction behavior of a variety of materials with known slip paths. For these studies, they developed small models of the material of interest, typically only including a few atomic layers. For example, they modeled hydrogen-terminated diamond surfaces using a C13H22 fully saturated tricyclic model [132]. They placed two of these systems in contact and calculated the energy at various points along the sliding path with a load applied normal to the sliding direction. These methods have been employed by the Pakkanen group to study interactions with hydrogen- [132], methyl- [133], and fluoro-terminated [134] diamond surfaces, other hydrocarbons [135], graphene sheets [136], boron nitride [137], and boron nitride on ice [138]. Others have studied the slip behavior of materials using a similar methodology to that used by the Pakkanen group, but using larger model systems. Using periodically repeating models of 22 metals and ceramics, Ogata et al. calculated the shear strain-stress relationships of these materials to determine the maximum strain a crystal can withstand [139]. Mosey, Liao, and Carter used one-dimensional PES scans to predict the shear strengths of materials such as iron oxides and chromia [140, 141, 142].

3.1.3 Optimization to a Minimum Energy Path

PES scans over a discrete set of points provide insights into the changes in energy and structure that occur during a slip process. However, it is unlikely that a given point on a scanned PES corresponds exactly to the transition state for a reaction or slip process. This uncertainty in the position of the transition state introduces errors in calculated reaction barriers. In addition, the slip path itself may not correspond to the series of structures along the grid used to construct the PES. To gain a more accurate description of the reaction process, it is useful to evaluate the series of structures, including the transition state, corresponding to the MEP the system follows as it moves from reactants to products. In calculations of molecular systems, this is often achieved through the evaluation of the IRC, which starts from a known transition state structure and constructs the MEP by following the steepest descent trajectory from the TS to the reactant and product structures it connects. It is not straightforward to optimize TS structures for condensed phase systems such as those used in tribochemical studies without first optimizing the MEP. As such, one cannot readily employ the IRC procedure used to study reactions involving molecular systems. Instead, MEPs for reactions occurring in the condensed phase are obtained by starting from the reactant and product structures and locating the MEP that connects these structures. An approach for locating MEPs for shear-induced transformations is described in what follows. This method is based on the nudged elastic band (NEB) method, which has been used to study condensed phase reactions in which the simulation cell remains fixed.

The NEB method [143, 144, 145] is part of a family of path minimization techniques known as chain-of-states methods. In these methods, two positions on a PES are connected by a series of images that trace a path on the surface. After the initial path is generated, various optimization techniques can be used to minimize the path to an MEP on the PES. The features of this process are outlined schematically in Fig. 13. In one chain-of-states approach, called the elastic band method, the images are connected by springs and the path is optimized by minimizing the force experienced by each image as the sum of the true force and the spring force. The true force on an image i, which has coordinates ri, is given by \( {F}_i^{\nabla }=-\nabla {E}_0\left({\mathbf{r}}_{\mathbf{i}}\right) \) and pulls the intermediate images toward the ends of the path. Meanwhile, the spring force is given by \( {\mathbf{F}}_i^s=-\nabla {E}_s\left({\mathbf{r}}_i\right) \) where the spring energy, Es, is given by \( {E}_s={k}_s\left[{\left({\mathbf{r}}_{i+1}-{\mathbf{r}}_i\right)}^2+{\left({\mathbf{r}}_i-{\mathbf{r}}_{i-1}\right)}^2\right] \) and ks is a spring constant.
Fig. 13

Optimization to the MEP via the chain-of-states methods. The contour plot designates a PES. The circles indicate images along a path on this surface that are moved to obtain the MEP. The initial path is consistent with series of structures changed linearly to connect the reactants and products. The forces acting on each image are used in conjunction with optimization techniques to convert the initial path into the MEP connecting the reactant and product structures. Figure reprinted with permission from Caspersen and Carter [146]. Copyright (2005) National Academy of Sciences, USA

The elastic band method does not normally find the correct MEP because of two main issues: image sagging and corner cutting. Image sagging arises when the chosen spring constant is too low and results in the images “sliding” down the path toward the reactant and product. Corner cutting arises from the opposite problem, when the spring constant is too high and the path cannot relax enough to minimize to a curved MEP and cuts the corner, frequently giving an energy barrier that is too high. In practice, it is not possible to choose a spring constant that prevents both issues.

The NEB method recognized that the issues with the elastic band technique arose from specific components of the force. The issues with corner cutting arise from components of the force perpendicular to the path, which tend to pull images away from the path. Image sagging can be attributed to components of the true force parallel to the path; the spacing between images becomes uneven to balance out the net force. The simple solution to these issues is to minimize the elastic band with these force components projected out. The NEB force is
$$ {\mathbf{F}}_i^{\mathrm{NEB}}={\mathbf{F}}_i^{\nabla \perp }+{\mathbf{F}}_i^{s\left|\right|}, $$
where \( {\mathbf{F}}_i^{\nabla \perp }={\mathbf{F}}_i^{\nabla }-\left({\mathbf{F}}_i^{\nabla}\cdot {\widehat{\boldsymbol{\uptau}}}_i\right){\widehat{\boldsymbol{\uptau}}}_i \) is the true force perpendicular to the path and \( {\mathbf{F}}_i^{s\left|\right|}=k\left(\left|{\mathbf{r}}_{i+1}-{\mathbf{r}}_i\right|-\left|{\mathbf{r}}_i-{\mathbf{r}}_{i-1}\right|\right){\widehat{\boldsymbol{\uptau}}}_i \) is the spring force parallel to the path, where \( {\widehat{\boldsymbol{\uptau}}}_i \) is the unit-tangent to the path and ri are the Cartesian positions of the atoms in image i. A later modification to the standard NEB, called climbing image NEB [147], allows the highest energy point along the MEP to move along the path to locate the transition state. The force on the climbing image, \( {\mathbf{F}}_i^{CI}={\mathbf{F}}_i^{\nabla }-2\left({\mathbf{F}}_i^{\nabla}\cdot {\widehat{\boldsymbol{\uptau}}}_i\right){\widehat{\boldsymbol{\uptau}}}_i \), does not include spring forces and points up \( {\widehat{\boldsymbol{\uptau}}}_i \) toward the direction in which the energy is increasing. The forces acting on the images in this approach are illustrated in Fig. 14.
Fig. 14

NEB force projections for a typical image i and a climbing image l. Reprinted with permission from Sheppard et al. [148]. Copyright 2012, AIP Publishing LLC

The NEB method is particularly attractive for optimizing the MEP of a variety of chemical processes for a number of reasons, including (1) it is computationally inexpensive, only requiring evaluation of the potential energy and first derivative of the energy with respect to coordinates, (2) it provides a robust, but flexible convergence to an MEP, and (3) it is inherently parallelizable, making it particularly appealing for modern, highly parallel computer resources.

Unfortunately, as originally presented, the NEB method cannot be used to study mechanochemistry in the bulk, as there is no way to allow the simulation cell to change in response to shear or compression. Trinkle et al. [149, 150] proposed a method that employs the NEB method to optimize the MEP for the atomic coordinates and relaxes the cell vectors using Parrinello–Rahman molecular dynamics techniques [151]. Caspersen and Carter proposed a method that extends the Born–Oppenheimer approximation to assume that the motion of the nuclei is decoupled from the motion of the lattice vectors. They used this concept to relax the nuclear coordinates to a zero force state, and then used the NEB formalism to optimize the cell vectors. These methods ultimately suffer from the same issues that the climbing image NEB was introduced to overcome. Additionally, the motions of the atomic and cell degrees of freedom are not fully decoupled and it has been shown that if these are not coupled properly it can lead to non-physical MEPs [152].

The generalized solid-state NEB (GSSNEB) [148] was developed recently to overcome these problems. This method uses the NEB formalism to optimize simultaneously the atomic coordinates and the simulation cell to the MEP. In the GSSNEB method the strains and stresses associated with the simulations cell are used as analogues of atomic positions and forces. These quantities are incorporated into the calculation by concatenating the strain associated with the cell of a particular image to the changes in atomic positions:
$$ \Delta \mathbb{R}=\left\{J\boldsymbol{\upvarepsilon}, \Delta \mathbf{r}\right\}, $$
and the stress, σ, on an image is similarly concatenated to forces on the atoms:
$$ \mathbb{F}=\left\{-\frac{\Omega \boldsymbol{\upsigma}}{\boldsymbol{J}},\mathbf{F}\right\}, $$
where \( J={\Omega}^{1/3}{N}^{1/6} \) is a Jacobian introduced to ensure that the stresses and strains have the same units and scale as the forces and positions, respectively, Ω is the volume of the simulation cell, and N is the number of atoms. This method has been used primarily to study phase transitions [148, 153, 154, 155, 156] and adsorption/desorption processes [157, 158, 159].

3.2 Dynamic Methods

The static methods described above can be used to provide information regarding the changes in structures, energetics, and forces associated with reactions induced through shear and/or compression. However, computational demands typically limit PES scans to examining two dimensions at most, and hence such scans may miss important details of reaction mechanisms that occur along directions that are not scanned. The evaluation of the MEP with techniques such as the GSSNEB approach accounts for the motion of the system along all degrees of freedom as a reaction occurs; however, this method requires a priori knowledge of the reactants and products, and is thus only useful in cases where the reaction to be examined is already known.

MD simulations allow systems to move naturally on the PES and can thus be used to explore known reactions as well as identify new reactions. In order to simulate chemical reactions, i.e., processes involving changes in bonding, it is generally necessary to employ a potential derived from quantum chemical (QC) methods or reactive force fields. These techniques have been used extensively in the context of simulating tribochemical reactions [48, 49]. For instance, Harrison et al. performed a series of studies modeling hydrogen-passivated diamond interfaces using constant strain simulations. In an initial report, they studied the friction behavior of atomically smooth interfaces under various loads and shear rates [160]. Later, they modeled rough interfaces by introducing short alkyl chains to the surface in place of hydrogen atoms [161, 162]. More recently, Harrison used another atomically smooth hydrogen-passivated diamond surface as a model for atomic force microscope tips with different geometries [163]. Luo and coworkers have also used reactive FFs to examine tribochemical processes related to lubrication [164]. Our group has used constant strain conditions to model the friction behavior of hydroxylated alumina surfaces [165, 166] and aldehydes bound to these surfaces [167], as well as aldehydes compressed between these surfaces [168].

As noted above, shear- and compression-induced mechanochemical reactions form a subset of tribochemical reactions, and hence the MD simulation techniques used to study tribochemical processes can be applied to mechanochemical conditions. These techniques involve the introduction of slabs representing surfaces that can be moved relative to one another or placing the system in a simulation cell that is subjected to stresses and/or strains. Techniques for studying stress-induced reactions with MD simulations are described in what follows. Inducing shear stresses by moving surface slabs relative to one another are described in Sect. 3.2.1. The application of shear by subjecting the simulation cell to time-dependent strains is described in Sect. 3.2.2 and the direct application of external stresses to the simulation cell is discussed in Sect. 3.2.3. A comparison of results obtained by subjecting simulations cells to strains and stresses is given in Sect. 4.

In the discussion that follows, we are only looking at ways to impose shear strains and stresses on a simulation cell during molecular dynamics simulations, because this is specifically relevant to mechanochemistry. However, in any MD simulation, it is important to consider the temperature. The energy introduced via the stress or strains used to induce a mechanochemical reaction are released by the system after the reaction occurs. The release of this energy can lead to large increases in the temperature of the system, which accumulate if not dissipated. Dissipation can be achieved through the use of thermostats; however, the thermostat needs to be applied carefully to ensure the energy is removed from the system in a manner that is consistent with the non-equilibrium nature of these simulations [49].

3.2.1 Relative Movement of Slabs

The PES scans described in Sect. 3.1 involved moving slabs representing surfaces relative to one another to mimic shear and/or compression. Although that section focused on static calculations, this approach can also be used in conjunction with MD simulation methods. In this approach, one constructs a system composed of two slabs that are to be moved relative to one another, which are potentially separated by molecular species that are to be subjected to stresses. A typical example would place lubricant molecules between two surfaces to study lubrication, but this approach can be used in the context of studying other stress-induced processes as well.

Once the system is constructed, stresses are imposed by fixing and/or moving some of the atoms in the system at well-defined positions or velocities. For example, to impose a constant load, L, normal to an interface spanning the x–y plane, the outermost atoms in the slabs representing the interface can be fixed at positions that cause the forces acting along the z direction to equal L. Likewise, shear strains and stresses can be imposed by fixing some of the atoms in the lower slab while moving the atoms in the upper slab along a particular direction. This can be achieved by moving the atoms according to a predetermined velocity profile, e.g., the atoms in the upper slab are moved at a constant velocity, or by attaching some of the atoms in the upper slab to a spring which is moved at a constant velocity and applies forces to the atoms in the upper slab that cause them to move along the slip direction [48, 49]. In both cases, only the uppermost layer or few layers of atoms in the upper slab is/are moved in a predefined manner or is/are affected by the external spring. In this manner, the uppermost layer(s) of atoms act as an external driving force that subjects the interior atoms to stresses without causing the interior atoms to undergo artificial dynamics. The forces acting on the atoms in the system can be monitored to assess the stresses that are experienced by the system and the changes in structure and bonding that occur as a result of these stresses can be observed.

An example of a system consisting of two slabs separated by a fluid is shown in Fig. 15. In this case, a subset of the atoms in the bottom slab is fixed at their equilibrium positions. The atoms in the uppermost portion of the upper slab are fixed in a structure consistent with the equilibrium structure of the material forming the slab to yield a rigid unit. However, unlike the rigid portion of the bottom layer, this portion of the upper layer is subjected to a compressive load, L, and moved laterally by connecting it to a spring with a force constant k that is moved at a constant velocity, v. As the length of the spring changes, the rigid portion of the upper slab is subjected to force that causes it to move as a single unit. The remaining atoms in the system, i.e., those designated the mobile portions of the upper and lower slabs, as well as the fluid, interact with the atoms in the rigid portions of the upper and lower slabs. As such, the relative movement of these rigid regions imposes a shear stress on the interior portion of the system, which can drive reactions.
Fig. 15

Schematic of a layered system used to study shear-induced reactions. The system consists of two slabs separated by a fluid. The atoms in the lower portion of the bottom slab, designated ‘fixed rigid layer,’ are fixed at their equilibrium positions within the material forming the slab. The atoms in the uppermost portion of upper slab, designated 'mobile rigid layer,' are also fixed at relative positions consistent with the equilibrium structure of the material forming the slab. This yields a rigid structure that is moved to subject the rest of the system to shear forces. In this example, this mobile rigid unit is subjected to a compressive load, L, and is attached to a harmonic spring with a force constant, k, with an end that is moved at a constant velocity, v. The change in the length of the spring exerts a force on the mobile rigid unit, which causes it to move. The mobile atoms in the upper and lower slabs, as well as those in the fluid, experience a shear force produced by the relative movements of the rigid layers. Shear can also be imposed with similar systems by moving the mobile rigid layer at a constant velocity instead of pulling it with a spring

Techniques in which shear is simulated in MD simulations by moving slabs relative to one another are commonly used in conjunction with force fields. Meanwhile, imposing shear via the relative movement of slabs is not used as commonly in conjunction with QC calculations, particularly planewave DFT calculations, because of the use of periodic simulation cells. If the system consisting of two slabs, and possibly additional molecules between these slabs, is placed in such a cell, the upper portion of the upper slab interacts with the periodic image of the bottom portion of the bottom slab (and vice versa). If the upper slab is moved along some direction while the bottom slab and the simulation cell itself is held fixed, the interaction between the upper slab and the periodic image of the lower slab introduces an additional interface along which slip occurs. One way to minimize the effects of this additional interface is to use simulation cells that contain large amounts of vacuum space between periodic images along the direction normal to the slabs. Unfortunately, this approach leads to a large increase in the volume of the simulation cell, which in turn increases the number of planewave basis functions needed in the calculation to the point at which computational expense renders these calculations intractable. An alternative approach is to deform the simulation cell in a manner consistent with the motion of the atoms in the slabs to ensure that slip does not occur along the interface between the periodic images. This is most easily achieved by deforming the simulation cell along the desired slip direction and allowing the atoms inside the cell to respond to this deformation, as opposed to moving the atoms directly to impose shear stresses and strains. Techniques for deforming the simulation cell are described in the next section.

3.2.2 Deforming the Simulation Cell

One common approach to compressing and/or shearing systems contained in periodic simulation cells involves deforming the simulation cell itself as indicated in Fig. 11b. Consider a simulation cell defined by the lattice vectors, a, b, and c, where a and b span the x–y plane. The cell contains a system with two slabs spanning the ab plane, which are separated by some material that is to be subjected to compressive and/or shear stresses. In order to impose compressive stresses, one could reduce the z component of the c lattice vector at a fixed rate to apply a compressive strain. Similarly, shear can be imposed by moving the x and/or y components of the c vector at specific rates along specific directions.

In order for this approach to be effective, it is necessary to ensure that the compressive and/or shear strains to which the simulation cell is subjected are transmitted to the atoms within the cell. Indeed, it is possible in principle to deform the cell in arbitrary ways without altering the positions of the atoms at all and still obtain a perfectly suitable periodically repeated system. To ensure the atoms in the cell move in conjunction with the lattice vectors, it is common practice to represent the atomic positions in fractional coordinates. This approach also ensures that the Lees–Edwards boundary conditions [124] are satisfied to ensure that artificial slip planes are not introduced at the interface between each periodically repeated cell.

The deformation of the cell and its contents leads to changes in the internal stress tensor associated with the system. The internal stresses are in balance with the theoretical external stresses required to induce that deformation of the cell. As such, the internal stresses of the system can be used to ascertain the external stresses experienced by the system, and by correlating these stresses with processes that occur within the system, one can determine the stresses required to induce those processes. This is commonly used, for instance, to determine the stresses needed to induce slip events, which are directly related to friction forces or the strengths of materials. However, this approach could be used to evaluate the stresses required to promote mechanochemical reactions.

Inducing stresses via deformation of the simulation cell is a conceptually simple approach. However, a few details require careful consideration when using this approach. First, one must consider the rate at which the lattice vectors are deformed. This is ultimately determined by the computational requirements of the simulation method being used. In the case of QC-based MD simulations it is only possible to simulate sub-nanosecond timescales and thus it is often necessary to deform the cell vectors at rates of approximately 1 Å/ps (100 m/s) to induce sufficiently large strains to observe reactions in the accessible timescales. The deformation rates may be decreased by two or three orders of magnitude if the simulations are performed using force fields instead of QC methods, but most force fields are unable to describe changes in bonding, and thus this approach may not be well-suited to studying mechanochemical reactions.

In addition to the rate of cell deformation, the manner in which the cell is deformed also requires consideration. It is common practice to deform the cell linearly, although that may not represent the manner in which a system would move during a stress-induced process. For instance, a system undergoing slip moves very slowly when near its equilibrium configuration and subjected to small shear stresses, whereas it moves quite rapidly during the slip process itself. A linear cell deformation causes the system to move at the same velocities during these two events; effectively, the system moves too fast near the equilibrium state and too slow during slip. This issue is difficult to overcome when using cell deformation to impose stresses unless some a priori information exists regarding the manner in which the cell should be deformed.

An additional aspect that must be considered when deforming the cell in conjunction with planewave DFT calculations stems from the fact that the basis set is dependent upon the size and shape of the cell. In principle, one should regenerate a new set of basis functions every time the cell is changed, although that is neither efficient nor convenient from a computational standpoint. An alternative approach would be to employ a constant set of basis functions throughout the simulation. Unfortunately, the quality of the basis set varies with the deformation of the cell, and thus the results of simulations performed with this approach would only be reliable if very large basis sets were used. A solution to this issue is found in the constant kinetic energy cutoff approach of Bernasconi et al. [169] in which a single set of planewave basis functions is used throughout the simulation while the kinetic energies of those planewaves are adjusted with each change in the size and shape of the simulation cell to ensure the quality of the basis set remains constant throughout the entire simulation.

3.2.3 Subjecting the Simulation Cell to External Stresses

MD simulations in which the cell is deformed treat the changes in the cell vectors as a controlled variable and allow the system to respond to the deformation by exerting a stress on the cell, which can then be correlated with the occurrence of reactive events observed during the simulation. An alternative approach involves subjecting the system to an external stress as indicated in Fig. 11b and allowing shape and size of the simulation cell to respond to this external stress. Once again, events occurring within the cell can be correlated with the external stress, which allows one to determine the stresses required to activate the events.

To allow the simulation cell to respond to an external stress, it is necessary to treat the lattice vectors defining the simulation cell as dynamic variables. This is achieved by assigning a fictitious mass, W, to the lattice vectors and allowing them to move according to Newton’s equations of motion using an extended Lagrangian. There are several formulations of this extended Lagrangian approach [151, 170, 171, 172], with the Parrinello–Rahman formulation [151] being used most commonly. The extended Lagrangian used in this formulation is
$$ L=\frac{1}{2}{\displaystyle \sum_{i=1}^N{m}_i{\dot{\mathbf{s}}}_i\mathbf{G}{\dot{\mathbf{s}}}_i}-V\left(\mathbf{r}\right)+\frac{1}{2}W\mathrm{T}\mathrm{r}\left({\dot{\mathbf{h}}}^T\dot{\mathbf{h}}\right)-p\Omega, $$
where i runs over all atoms, h is a matrix containing each lattice vectors as columns, the metric tensor defined as \( \mathbf{G}={\mathbf{h}}^T\mathbf{h},\mathbf{r} \) represents the Cartesian coordinates of the atoms, and s represents the atomic positions represented as fractional coordinates of the lattice vectors \( \left({\mathbf{r}}_i=\mathbf{h}{\mathbf{s}}_i\right),\;V\left(\mathbf{r}\right) \) is the potential energy, p is the external pressure, and Ω is the volume of the simulation cell. Although devised to study systems at a target external pressure, p, the Parrinello–Rahman method can be used to examine systems that are subjected to pressures that vary over time. This is achieved simply by changing the value of p in a predetermined manner. This approach has been used, for example, to study pressure-induced reactions related to polymerization and lubrication [173, 174, 175, 176, 177, 178].
Adjusting p in (19) imposes an isotropic hydrostatic pressure on the system. However, in the context of mechanochemistry it would be more useful to examine shear and compressive stresses. This can be achieved by generalizing the Lagrangian in (19) to allow the system to maintain a target stress tensor, S, as opposed to a pressure. The form of the generalized Lagrangian is
$$ L=\frac{1}{2}{\displaystyle \sum_{i=1}^N{m}_i{\dot{\mathbf{s}}}_i\mathbf{G}{\dot{\mathbf{s}}}_i}-V\left(\mathbf{r}\right)+\frac{1}{2}W\mathrm{T}\mathrm{r}\left({\dot{\mathbf{h}}}^T\dot{\mathbf{h}}\right)-p\Omega -\frac{1}{2}Tr\left(\boldsymbol{\Sigma} \mathbf{G}\right), $$
where \( \varSigma ={\mathbf{h}}_0^{-1}\left(\mathbf{S}-p\mathbf{I}\right){\mathbf{h}}_0^{T-1}{\Omega}_0 \), h0 is a reference matrix that is required to define strains, and Ω0 is the volume of that reference cell. For a given cell, h, the components of the strain tensor are defined as
$$ \boldsymbol{\upvarepsilon} =\frac{1}{2}\left({\mathbf{h}}_0^{T-1}{\mathbf{h}}^T\mathbf{h}{\mathbf{h}}_0^{-1}-1\right), $$
and it follows that the elastic energy is given by
$$ {V}_{el}=p\left(\Omega -{\Omega}_0\right)+{\Omega}_0\mathrm{T}\mathrm{r}\left(\mathbf{S}-p\mathbf{I}\right)\boldsymbol{\upvarepsilon} . $$
By analogy to the constant pressure formulation of the Parrinello–Rahman method, this extended Lagrangian can be used in conjunction with varying external stresses by changing the components of S in a proscribed manner during the simulation. In addition, this approach can be combined with methods that deform the simulation cell to impose shear strains. Indeed, our group has used this approach extensively to study various processes related to friction and tribochemistry, with the Parrinello–Rahman method being used to maintain a constant normal load, and cell deformations being used to impose shear strains [165, 166, 167, 168].

Subjecting systems directly to external stresses, as opposed to strains, is advantageous in terms of determining the stress at which a reaction occurs. For instance, if the time at which a reaction occurs during the simulation is known along with the manner in which the external stress is varied over time, it is quite straightforward to determine the external stresses associated with reactive events. By contrast, subjecting the system to proscribed shear strains requires one to determine the stress required to activate a reaction from the internal stress tensor, which tends to oscillate during simulations, leading to errors in the estimated shear stresses.

The application of external stresses allows the size and shape of the simulation cell to change as needed to accommodate the applied stress, which overcomes the linear deformation of the simulation cell that was a drawback associated with applying linearly increasing strains. For instance, a linearly increasing shear strain causes the system to move at a constant speed along a particular direction. Meanwhile, allowing the system to move in response to a linearly increasing shear stress allows the system to navigate better the lowest energy path on the PES and move at different rates during various stages of the shear/slip process.

The method has the disadvantage that the applied stress continues linearly even after slip has been achieved (assuming the stress was being varied in a linear manner). As such, there is a significant imbalance between the external and internal stresses once the system moves to a new local minimum on the PES after slip occurs. One must also be careful in determining the manner in which shear is imposed. Generally, this is achieved by increasing the external stress linearly along the slip direction, although other means of varying the external stress may be needed to represent different types of experimental conditions. In addition, consideration must be made with respect to the rate at which stresses are varied. In general, this is limited by the timescales accessible in the MD simulations. If QC-based methods are used, it may be necessary to change the stresses at rates of 1–10 GPa/ps, whereas rates two to three orders of magnitude slower may be possible if force fields are used. Finally, one must still employ an approach such as that of Bernasconi et al. [169] to maintain a consistent quality of basis set when changing the shape and/or size of the simulation cell via external stresses when the MD simulations are performed with planewave DFT.

3.2.4 Comparison of Dynamic Methods Involving Deformation of the Simulation Cell

As discussed above, shear-induced processes can be examined with MD simulations by deforming simulation cells in either of two ways. In the first case, one applies a strain along the slip directions, which increases the system’s internal stress until a reaction occurs. In the second case, the system is exposed to an external shear stress and the simulation cells change in order to accommodate this stress. These two methods are similar to the constrained distance and EFEI methods described above in the sense that one method involves controlling changes in structure whereas the other involves controlling changes in forces. As such, one may anticipate differences in the outcomes of simulations performed in which the cells are deformed via strains or stresses. To provide some insight into the results obtained with these two different approaches, AIMD simulations were performed in which bulk alumina (Al2O3) was sheared along the \( \left[21\overline{3}0\right]/(0001) \) direction and plane according to these protocols until slip occurred using a version of the Quantum-Espresso simulation package that was modified to apply stresses and strains that varied over time [179]. The calculations were performed using planewave DFT with the PBE exchange-correlation functional [180], valence electrons expanded to a kinetic energy cutoff of 80 Ry, and core electrons represented by norm-conserving pseudopotentials [181]. The results of these simulations are summarized in Fig. 16, which shows how the stresses and cell change during these simulations.
Fig. 16

Shear stress and shear distance vs time for AIMD simulations of bulk alumina shear to induce slip along the \( \left[21\overline{3}0\right]/(0001) \) direction and plane (a) moving the x and y components of the c vector along the slip direction at a rate of 0.5 Å/ps or (b) by increasing the external stress applied along the slip direction at a rate of 2.5 GPa/ps

The data in Fig. 16a were obtained using a procedure in which the x and y components of the c vector of bulk alumina were varied to move the uppermost portion of the simulation cell along the slip direction at a rate of 0.5 Å/ps. The data in this figure illustrated that the internal stress of the system increases in a relatively linear manner until dropping rapidly at a time of ~4.2 ps. The drop in the internal stress is consistent with the occurrence of a slip event and shear stress reached when slip occurred corresponds to the shear strength of the system, which is in the range of 15–17 GPa according to the data in this figure. The distance the cell is moved along the slip direction is also shown as a function of time. As illustrated in the figure, this distance increases linearly at the proscribed rate of 0.5 Å/ps during all portions of the simulation. As discussed above, this is problematic as the system should likely move at different rates during the shear/slip process, with the system moving slowly during the shearing phase and rapidly during slip as it moves past the slip barrier to the next energy minimum along the slip direction.

The data in Fig. 16b were obtained by subjecting the simulation cell to an external stress that increased at a rate of 2.5 GPa/ps. The data in the figure illustrate how far the end of the c lattice vector has moved from its equilibrium position over time in order to accommodate the applied stress. It should be noted that this movement is not restricted to occur exactly along the slip direction, as is required in the fixed strain calculations, but rather the motion of the c vector is unrestricted. The data show that the shear distance increases in a relatively linear manner at a rate of ~0.4 Å/ps until a time of ~4.8 ps is reached at which point the shear distance increases rapidly. This rapid increase is correlated with the slip event, and the change in the rate at which the shear distance increases illustrates that the system should move at different rates in the shear and slip phases. The data in the figure also show that the external shear stress increases linearly during the simulation and the external stress at the point where slip occurs corresponds to the shear strength, which in this case is ~12.5 GPa. The applied stress continues to increase after slip occurs, which results in an increase in the rate at which the shear distance increases which is not consistent with the system moving into an energy minimum along the slip direction.

Overall, the data in Fig. 16 illustrate the ways these two different approaches to changing the simulation cell can be used to determine the shear stresses associated with reactive events, which in this case correspond to slip events. In both cases it is straightforward to determine the points at which slip events occur. The use of applied strains has the advantage in terms of selecting the rate at which the strain changes, because this quantity can be related to the velocity of sliding surfaces. Meanwhile, it is more difficult to define the rate at which the shear stress is increased in a meaningful manner. The values of the shear strengths obtained in the simulations performed with these methods differ by several GPa. This can be related partially to the restricted movement of the cell associated with applied strains, which limits the ability of the system to accommodate applied loads. Both methods have deficiencies in the manner in which the cell is deformed after slip occurs; however, it may be possible to address these issues by reducing the rates at which shear stresses or strains are applied after a slip event occurs.

4 Concluding Remarks

In this review, we have described techniques that can be used to simulate chemical reactions in molecular and bulk systems that occur under mechanochemical conditions. Techniques for modeling molecular mechanochemistry focus on incorporating into the PES the work performed on a molecule that is subjected to an external force applied between two regions in the molecule. These techniques have been used extensively and are well developed, with models existing that are appropriate for qualitative predictions and interpretations, as well as for modeling isometric and isotensional conditions. The discussion of methods for simulating bulk mechanochemical processes focused on techniques that have been used to study tribochemical processes. These techniques focus on studying systems that are subjected to compressive and shear stresses, and are transferable to mechanochemistry. Interestingly, bulk mechanochemical processes have not been studied with simulations nearly as extensively as their molecular counterparts. As such, the application of techniques used to study tribochemical processes to the specific area of bulk mechanochemistry may be a promising avenue for future research.



Financial support from the Natural Sciences and Engineering Research Council of Canada’s Discovery Grant Program is acknowledged. GSK is grateful for support from the Ontario Graduate Scholarship program.


  1. 1.
    Ribas-Arino J, Marx D (2012) Covalent mechanochemistry: theoretical concepts and computational tools with applications to molecular nanomechanics. Chem Rev 112:5412–5487CrossRefGoogle Scholar
  2. 2.
    Beyer MK, Clausen-Schaumann H (2005) Mechanochemistry: the mechanical activation of covalent bonds. Chem Rev 105:2921–2948CrossRefGoogle Scholar
  3. 3.
    James SL, Adams CJ, Bolm C et al (2012) Mechanochemistry: opportunities for new and cleaner synthesis. Chem Soc Rev 41(1):413–447CrossRefGoogle Scholar
  4. 4.
    Rosen BM, Percec V (2007) Mechanochemistry: a reaction to stress. Nature 446:381–382CrossRefGoogle Scholar
  5. 5.
    Seidel CAM, Kühnemuth R (2014) Mechanochemistry: molecules under pressure. Nat Nanotechnol 9:164–165CrossRefGoogle Scholar
  6. 6.
    Black AL, Lenhardt JM, Craig SL (2011) From molecular mechanochemistry to stress-responsive materials. J Mater Chem 21:1655–1663CrossRefGoogle Scholar
  7. 7.
    Kreuzer HJ, Payne SH, Livadaru L (2001) Stretching a macromolecule in an atomic force microscope: statistical mechanical analysis. Biophys J 80:2505–2514CrossRefGoogle Scholar
  8. 8.
    Kersey FR, Yount WC, Craig SL, Carolina N (2006) Single-molecule force spectroscopy of bimolecular reactions: system homology in the mechanical activation of ligand substitution reactions. J Am Chem Soc 128:3886–3887CrossRefGoogle Scholar
  9. 9.
    Duwez A-S, Cuenot S, Jérôme C, Gabriel S, Jérôme R, Rapino S, Zerbetto F (2006) Mechanochemistry: targeted delivery of single molecules. Nat Nanotechnol 1:122–125CrossRefGoogle Scholar
  10. 10.
    Wang MD, Yin H, Landick R, Gelles J, Block SM (1997) Stretching DNA with optical tweezers. Biophys J 72:1335–1346CrossRefGoogle Scholar
  11. 11.
    Ashkin A, Dziedzic JM, Bjorkholm JE, Chu S (1986) Observation of a single-beam gradient force optical trap for dielectric particles. Opt Lett 11:288–290CrossRefGoogle Scholar
  12. 12.
    Moffitt JR, Chemla YR, Smith SB, Bustamante C (2008) Recent advances in optical tweezers. Annu Rev Biochem 77:205–228CrossRefGoogle Scholar
  13. 13.
    Grier DG (2003) A revolution in optical manipulation. Nature 424:810–816CrossRefGoogle Scholar
  14. 14.
    Yang Q-Z, Huang Z, Kucharski TJ, Khvostichenko D, Chen J, Boulatov R (2009) A molecular force probe. Nat Nanotechnol 4:302–306CrossRefGoogle Scholar
  15. 15.
    Lundbæk JA, Collingwood SA (2010) Lipid bilayer regulation of membrane protein function: gramicidin channels as molecular force probes. J R Soc Interface 7:373–395CrossRefGoogle Scholar
  16. 16.
    Suslick KS (2004) Sonochemistry. Compr Coord Chem II 1:731–739Google Scholar
  17. 17.
    Basedow AM, Ebert KH (1977) Ultrasonic degradation of polymers in solution. Adv Polym Sci 22:83–148CrossRefGoogle Scholar
  18. 18.
    Thompson LH, Doraiswamy LK (1999) Sonochemistry: science and engineering. Ind Eng Chem Res 38:1215–1249CrossRefGoogle Scholar
  19. 19.
    Hickenboth CR, Moore JS, White SR, Sottos NR, Baudry J, Wilson SR (2007) Biasing reaction pathways with mechanical force. Nature 446:423–427CrossRefGoogle Scholar
  20. 20.
    James SL, Friščić T (2013) Mechanochemistry. Chem Soc Rev 42:7494–7496CrossRefGoogle Scholar
  21. 21.
    Friščić T, Halasz I, Beldon PJ, Belenguer AM, Adams F, Kimber SAJ, Honkimäki V, Dinnebier RE (2013) Real-time and in situ monitoring of mechanochemical milling reactions. Nat Chem 5:66–73Google Scholar
  22. 22.
    Christinat N, To J, Schu C, Scopelliti R, Severin K (2009) Synthesis of molecular nanostructures by multicomponent condensation reactions in a ball mill. J Am Chem Soc 131:3154–3155CrossRefGoogle Scholar
  23. 23.
    Cravotto G, Cintas P (2010) Reconfiguration of stereoisomers under sonomechanical activation. Angew Chem Int Ed Engl 49:6028–6030CrossRefGoogle Scholar
  24. 24.
    Lu H, Isralewitz B, Krammer A, Vogel V, Schulten K (1998) Unfolding of titin immunoglobulin domains by steered molecular dynamics simulation. Biophys J 75:662–671CrossRefGoogle Scholar
  25. 25.
    Oberhauser AF, Hansma PK, Carrion-Vazquez M, Fernandez JM (2001) Stepwise unfolding of titin under force-clamp atomic force microscopy. Proc Natl Acad Sci 98:468–472CrossRefGoogle Scholar
  26. 26.
    Fowler SB, Best RB, Toca Herrera JL, Rutherford TJ, Steward A, Paci E, Karplus M, Clarke J (2002) Mechanical unfolding of a Titin Ig domain: structure of unfolding intermediate revealed by combining AFM, molecular dynamics simulations, NMR and protein engineering. J Mol Biol 322:841–849CrossRefGoogle Scholar
  27. 27.
    Rief M, Gautel M, Oesterhelt F, Fernandez JM, Gaub E, Gaub HE (1997) Reversible unfolding of individual Titin immunoglobulin domains by AFM. Science 276:1109–1112CrossRefGoogle Scholar
  28. 28.
    Weder C (2009) Polymers react to stress. Nature 459:45–46CrossRefGoogle Scholar
  29. 29.
    Kochhar GS, Bailey A, Mosey NJ (2010) Competition between orbitals and stress in mechanochemistry. Angew Chem Int Ed Engl 49:7452–7455CrossRefGoogle Scholar
  30. 30.
    Ong MT, Leiding J, Tao H, Virshup AM, Martinez TJ (2009) First principles dynamics and minimum energy pathways for mechanochemical ring opening of cyclobutene. J Am Chem Soc 131:6377–6379CrossRefGoogle Scholar
  31. 31.
    Ribas-Arino J, Shiga M, Marx D (2009) Understanding covalent mechanochemistry. Angew Chem Int Ed Engl 48:4190–4193CrossRefGoogle Scholar
  32. 32.
    Beyer MK (2000) The mechanical strength of a covalent bond calculated by density functional theory. J Chem Phys 112:7307–7312CrossRefGoogle Scholar
  33. 33.
    Friedrichs J, Lüssmann M, Frank I (2010) Conservation of orbital symmetry can be circumvented in mechanically induced reactions. Chemphyschem 11:3339–3342CrossRefGoogle Scholar
  34. 34.
    Ribas-Arino J, Shiga M, Marx D (2009) Unravelling the mechanism of force-induced ring-opening of benzocyclobutenes. Chem Eur J 15:13331–13335CrossRefGoogle Scholar
  35. 35.
    Makarov DE, Wang Z, Thompson JB, Hansma HG (2002) On the interpretation of force extension curves of single protein molecules. J Chem Phys 116:7760–7765CrossRefGoogle Scholar
  36. 36.
    Konda SSM, Avdoshenko SM, Makarov DE (2014) Exploring the topography of the stress-modified energy landscapes of mechanosensitive molecules. J Chem Phys 140:104114CrossRefGoogle Scholar
  37. 37.
    Bailey A, Mosey NJ (2012) Prediction of reaction barriers and force-induced instabilities under mechanochemical conditions with an approximate model: a case study of the ring opening of 1,3-cyclohexadiene. J Chem Phys 136:044102CrossRefGoogle Scholar
  38. 38.
    Beyer MK (2003) Coupling of mechanical and chemical energy: proton affinity as a function of external force. Angew Chem Int Ed Engl 42:4913–4915CrossRefGoogle Scholar
  39. 39.
    Lupton EM, Achenbach F, Weis J, Bräuchle C, Frank I (2006) Modified chemistry of siloxanes under tensile stress: interaction with environment. J Phys Chem B 110:14557–14563CrossRefGoogle Scholar
  40. 40.
    Lupton EM, Nonnenberg C, Frank I, Achenbach F, Weis J, Bräuchle C (2005) Stretching siloxanes: an ab initio molecular dynamics study. Chem Phys Lett 414:132–137CrossRefGoogle Scholar
  41. 41.
    Aktah D, Frank I (2002) Breaking bonds by mechanical stress: when do electrons decide for the other side? J Am Chem Soc 124:3402–3406CrossRefGoogle Scholar
  42. 42.
    Lenhardt JM, Ong MT, Choe R, Evenhuis CR, Martinez TJ, Craig SL (2010) Trapping a diradical transition state by mechanochemical polymer extension. Science 329:1057–1060CrossRefGoogle Scholar
  43. 43.
    Lenhardt JM, Ogle JW, Ong MT, Choe R, Martinez TJ, Craig SL (2011) Reactive cross-talk between adjacent tension-trapped transition states. J Am Chem Soc 133:3222–3225CrossRefGoogle Scholar
  44. 44.
    Kryger MJ, Ong MT, Odom SA, Sottos NR, White SR, Martinez TJ, Moore JS (2010) Masked cyanoacrylates unveiled by mechanical force. J Am Chem Soc 132:4558–4559CrossRefGoogle Scholar
  45. 45.
    Konôpka M, Turanský R, Reichert J, Fuchs H, Marx D, Štich I (2008) Mechanochemistry and thermochemistry are different: stress-induced strengthening of chemical bonds. Phys Rev Lett 100:115503CrossRefGoogle Scholar
  46. 46.
    Dopieralski P, Anjukandi P, Rückert M, Shiga M, Ribas-Arino J, Marx D (2011) On the role of polymer chains in transducing external mechanical forces to benzocyclobutene mechanophores. J Mater Chem 21:8309–8316Google Scholar
  47. 47.
    Ribas-arino J, Shiga M, Marx D (2010) Mechanochemical transduction of externally applied forces to mechanophores. J Am Chem Soc 132:10609–10614CrossRefGoogle Scholar
  48. 48.
    Harrison JA, Gao G, Schall JD, Knippenberg MT, Mikulski PT (2008) Friction between solids. Philos Trans R Socient A 366:1469–1495CrossRefGoogle Scholar
  49. 49.
    Mosey NJ, Muser M (2007) Atomistic modeling of friction. In: Lipkowitz KB, Larter R, Cundari TR (eds) Reviews in computational chemistry, 25th ed. Wiley-VCH, New York, pp 67–124Google Scholar
  50. 50.
    Schlierf M, Li H, Fernandez JM (2004) The unfolding kinetics of ubiquitin captured with single-molecule force-clamp techniques. Proc Natl Acad Sci USA 101:7299–7304CrossRefGoogle Scholar
  51. 51.
    Fernandez JM, Li H (2004) Force-clamp spectroscopy monitors the folding trajectory of a single protein. Science 303:1674–1678CrossRefGoogle Scholar
  52. 52.
    Potisek SL, Davis DA, Sottos NR, White SR, Moore JS (2007) Mechanophore-linked addition polymers. J Am Chem Soc 129:13808–13809CrossRefGoogle Scholar
  53. 53.
    Grandbois M, Beyer M, Rief M, Clausen-Schaumann H, Gaub HE (1999) How strong is a covalent bond? Science 283:1727–1730CrossRefGoogle Scholar
  54. 54.
    Freitas A, Sharma M (2001) Detachment of particles from surfaces: an AFM study. J Colloid Interface Sci 233:73–82CrossRefGoogle Scholar
  55. 55.
    Iozzi MF, Helgaker T, Uggerud E (2009) Assessment of theoretical methods for the determination of the mechanochemical strength of covalent bonds. Mol Phys 107:2537–2546CrossRefGoogle Scholar
  56. 56.
    Su T, Purohit PK (2009) Mechanics of forced unfolding of proteins. Acta Biomater 5:1855–1863CrossRefGoogle Scholar
  57. 57.
    Ikai A, Alimujiang Y (2001) Force–extension curves of dimerized polyglutamic acid. Appl Phys A 120:117–120CrossRefGoogle Scholar
  58. 58.
    Davis DA, Hamilton A, Yang J et al (2009) Force-induced activation of covalent bonds in mechanoresponsive polymeric materials. Nature 459:68–72CrossRefGoogle Scholar
  59. 59.
    Zemanová M, Bleha T (2005) Isometric and isotensional force-length profiles in polymethylene chains. Macromol Theory Simulations 14:596–604CrossRefGoogle Scholar
  60. 60.
    Keller D, Swigon D, Bustamante C (2003) Relating single-molecule measurements to thermodynamics. Biophys J 84:733–738CrossRefGoogle Scholar
  61. 61.
    Paulusse JMJ, Sijbesma RP (2004) Reversible mechanochemistry of a Pd(II) coordination polymer. Angew Chem Int Ed Engl 43:4460–4462CrossRefGoogle Scholar
  62. 62.
    Karthikeyan S, Potisek SL, Piermattei A, Sijbesma RP (2008) Highly efficient mechanochemical scission of silver-carbene coordination polymers. J Am Chem Soc 130:14968–14969CrossRefGoogle Scholar
  63. 63.
    Piermattei A, Karthikeyan S, Sijbesma RP (2009) Activating catalysts with mechanical force. Nat Chem 1:133–137CrossRefGoogle Scholar
  64. 64.
    Izailev S, Stepaniants S et al (1999) Steered molecular dynamics. In: Deuflhard P, Hermans J, Leimkuhler B, Mark AE, Reich S, Skeel RD (eds) Computational molecular dynamics: challenges, methods, ideas, vol 4. Springer, New York, pp 39–65CrossRefGoogle Scholar
  65. 65.
    Izrailev S, Stepaniants S, Balsera M, Oono Y, Schulten K (1997) Molecular dynamics study of unbinding of the avidin-biotin complex. Biophys J 72:1568–1581CrossRefGoogle Scholar
  66. 66.
    Grubmuiller H, Heymann B, Tavan P (1996) Ligand binding: molecular mechanics calculation of the streptavidin-biotin rupture force. Science 271:997–999CrossRefGoogle Scholar
  67. 67.
    Isralewitz B, Baudry J, Gullingsrud J, Kosztin D, Schulten K (2001) Steered molecular dynamics investigations of protein function. J Mol Graph Model 19:13–25CrossRefGoogle Scholar
  68. 68.
    Isralewitz B, Gao M, Schulten K (2001) Steered molecular dynamics and mechanical functions of proteins. Curr Opin Struct Biol 11:224–230CrossRefGoogle Scholar
  69. 69.
    Stepaniants S, Izrailev S, Schulten K (1997) Extraction of lipids from phospholipid membranes by steered molecular dynamics. J Mol Model 3:473–475CrossRefGoogle Scholar
  70. 70.
    Silberstein MN, Cremar LD, Beiermann BA, Kramer SB, Martinez TJ, White SR, Sottos NR (2014) Modeling mechanophore activation within a viscous rubbery network. J Mech Phys Solids 63:141–153CrossRefGoogle Scholar
  71. 71.
    Diesendruck CE, Peterson GI, Kulik HJ, Kaitz JA, Mar BD, May PA, White SR, Martínez TJ, Boydston AJ, Moore JS (2014) Mechanically triggered heterolytic unzipping of a low-ceiling-temperature polymer. Nat Chem 6:623–628CrossRefGoogle Scholar
  72. 72.
    Franco I, George CB, Solomon GC, Schatz GC, Ratner MA (2011) Mechanically activated molecular switch through single-molecule pulling. J Am Chem Soc 133:2242–2249CrossRefGoogle Scholar
  73. 73.
    Franco I, Schatz GC, Ratner MA (2009) Single-molecule pulling and the folding of donor-acceptor oligorotaxanes: phenomenology and interpretation. J Chem Phys 131:124902CrossRefGoogle Scholar
  74. 74.
    Paturej J, Kuban L, Milchev A, Vilgis TA (2012) Tension enhancement in branched macromolecules upon adhesion on a solid substrate. Europhys Lett 97:58003CrossRefGoogle Scholar
  75. 75.
    Ghosh A, Dimitrov DI, Rostiashvili VG, Milchev A, Vilgis TA (2010) Thermal breakage and self-healing of a polymer chain under tensile stress. J Chem Phys 132:204902CrossRefGoogle Scholar
  76. 76.
    Paturej J, Milchev A, Rostiashvili VG, Vilgis TA (2011) Polymer chain scission at constant tension – an example of force-induced collective behaviour. Europhys Lett 94:48003CrossRefGoogle Scholar
  77. 77.
    Paturej J, Dubbeldam JLA, Rostiashvili VG, Milchev A, Vilgis TA (2014) Force spectroscopy of polymer desorption: theory and molecular dynamics simulation. Soft Matter 10:2785–2799CrossRefGoogle Scholar
  78. 78.
    Wiita AP, Perez-Jimenez R, Walther KA, Gräter F, Berne BJ, Holmgren A, Sanchez-Ruiz JM, Fernandez JM (2007) Probing the chemistry of thioredoxin catalysis with force. Nature 450:124–127CrossRefGoogle Scholar
  79. 79.
    Li W, Gräter F (2010) Atomistic evidence of how force dynamically regulates thiol/disulfide exchange. J Am Chem Soc 132:16790–16795CrossRefGoogle Scholar
  80. 80.
    Baldus IB, Gräter F (2012) Mechanical force can fine-tune redox potentials of disulfide bonds. Biophys J 102:622–629CrossRefGoogle Scholar
  81. 81.
    Ryckaert J-P, Ciccotti G, Berendsen HJ (1977) Numerical integration of the Cartesian equations of motion of a system with constraints: molecular dynamics of n-alkanes. J Comput Phys 23:327–341CrossRefGoogle Scholar
  82. 82.
    Smalø HS, Uggerud E (2013) Breaking covalent bonds using mechanical force, which bond breaks? Mol Phys 111:1563–1573CrossRefGoogle Scholar
  83. 83.
    Smalø HS, Uggerud E (2012) Ring opening vs. direct bond scission of the chain in polymeric triazoles under the influence of an external force. Chem Comm 48:10443–10445CrossRefGoogle Scholar
  84. 84.
    Kryger MJ, Munaretto AM, Moore S (2011) Structure-mechanochemical activity relationships for cyclobutane mechanophores. J Am Chem Soc 133:18992–18998CrossRefGoogle Scholar
  85. 85.
    Lourderaj U, McAfee JL, Hase WL (2008) Potential energy surface and unimolecular dynamics of stretched n-butane. J Chem Phys 129:094701CrossRefGoogle Scholar
  86. 86.
    Schmidt SW, Beyer MK, Clausen-Schaumann H (2008) Dynamic strength of the silicon-carbon bond observed over three decades of force-loading rates. J Am Chem Soc 130:3664–3668CrossRefGoogle Scholar
  87. 87.
    Iozzi MF, Helgaker T, Uggerud E (2011) Influence of external force on properties and reactivity of disulfide bonds. J Phys Chem A 115:2308–3215CrossRefGoogle Scholar
  88. 88.
    Groote R, Szyja M, Pidko EA, Hensen EJM, Sijbesma RP (2011) Unfolding and mechanochemical scission of supramolecular polymers containing a metal-ligand coordination bond. Macromolecules 44:9187–9195CrossRefGoogle Scholar
  89. 89.
    Shiraki T, Diesendruck CE, Moore JS (2014) The mechanochemical production of phenyl cations through heterolytic bond scission. Faraday Discuss. doi:10.1039/C4FD00027G Google Scholar
  90. 90.
    Krüger D, Rousseau R, Fuchs H, Marx D (2003) Towards “mechanochemistry”: mechanically induced isomerizations of thiolate-gold clusters. Angew Chem Int Ed Engl 42:2251–2253CrossRefGoogle Scholar
  91. 91.
    Huang Z, Yang Q, Khvostichenko D, Kucharski TJ, Chen J, Boulatov R (2009) Method to derive restoring forces of strained molecules from kinetic measurements. J Am Chem Soc 131:1407–1409CrossRefGoogle Scholar
  92. 92.
    Kucharski TJ, Yang Q-Z, Tian Y, Boulatov R (2010) Strain-dependent acceleration of a paradigmatic SN2 reaction accurately predicted by the force formalism. J Phys Chem Lett 1:2820–2825CrossRefGoogle Scholar
  93. 93.
    Kucharski TJ, Huang Z, Yang Q-Z, Tian Y, Rubin NC, Concepcion CD, Boulatov R (2009) Kinetics of thiol/disulfide exchange correlate weakly with the restoring force in the disulfide moiety. Angew Chemie 121:7174–7177CrossRefGoogle Scholar
  94. 94.
    Akbulatov S, Tian Y, Kapustin E, Boulatov R (2013) Model studies of the kinetics of ester hydrolysis under stretching force. Angew Chem Int Ed Engl 52:6992–6995CrossRefGoogle Scholar
  95. 95.
    Akbulatov S, Tian Y, Boulatov R (2012) Force−reactivity property of a single monomer is sufficient to predict the micromechanical behavior of its polymer. J Am Chem Soc 134:7620–7623CrossRefGoogle Scholar
  96. 96.
    Tian Y, Kucharski TJ, Yang Q-Z, Boulatov R (2013) Model studies of force-dependent kinetics of multi-barrier reactions. Nat Commun 4:2538CrossRefGoogle Scholar
  97. 97.
    Kucharski TJ, Boulatov R (2011) The physical chemistry of mechanoresponsive polymers. J Mater Chem 21:8237–8255CrossRefGoogle Scholar
  98. 98.
    Hermes M, Boulatov R (2011) The entropic and enthalpic contributions to force-dependent dissociation kinetics of the pyrophosphate bond. J Am Chem Soc 133(50):20044–20047CrossRefGoogle Scholar
  99. 99.
    Car R, Parrinello M (1985) Unified approach for molecular dynamics and density-functional theory. Phys Rev Lett 55:2471–2474CrossRefGoogle Scholar
  100. 100.
    Hofbauer F, Frank I (2010) Disulfide bond cleavage: a redox reaction without electron transfer. Chem Eur J 16:5097–5101CrossRefGoogle Scholar
  101. 101.
    Hofbauer F, Frank I (2012) CPMD simulation of a bimolecular chemical reaction: nucleophilic attack of a disulfide bond under mechanical stress. Chem Eur J 18:16332–16338CrossRefGoogle Scholar
  102. 102.
    Krupička M, Sander W, Marx D (2014) Mechanical manipulation of chemical reactions: reactivity switching of Bergman cyclizations. J Phys Chem Lett 5:905–909CrossRefGoogle Scholar
  103. 103.
    Stauch T, Dreuw A (2014) Force-spectrum relations for molecular optical force probes. Angew Chem Int Ed Engl 53:2759–2761CrossRefGoogle Scholar
  104. 104.
    Smalø HS, Rybkin VV, Klopper W, Helgaker T, Uggerud E (2014) Mechanochemistry: the effect of dynamics. J Phys Chem A 118:7683–7694CrossRefGoogle Scholar
  105. 105.
    Schmidt M, Baldridge K (1993) General atomic and molecular electronic structure system. J Comput Chem 14:1347–1363CrossRefGoogle Scholar
  106. 106.
    Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Scalmani G, Barone V, Mennucci B, Petersson GA, Nakatsuji H, Caricato M, Li X, Hratchian HP, Izmaylov AF, Bloino J, Zheng G, Sonnenberg JL, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, Vreven T, Montgomery JA Jr, Peralta JE, Ogliaro F, Bearpark M, Heyd JJ, Brothers E, Kudin KN, Staroverov VN, Keith T, Kobayashi R, Normand J, Raghavachari K, Rendell A, Burant JC, Iyengar SS, Tomasi J, Cossi M, Rega N, Millam JM, Klene M, Knox JE, Cross JB, Bakken V, Adamo C, Jaramillo J, Gomperts R, Stratmann RE, Yazyev O, Austin AJ, Cammi R, Pomelli C, Ochterski JW, Martin RL, Morokuma K, Zakrzewski VG, Voth GA, Salvador P, Dannenberg JJ, Dapprich S, Daniels AD, Farkas O, Foresman JB, Ortiz JV, Cioslowski J, Fox DJ (2009) Gaussian 09, revision D.01. Gaussian, WallingfordGoogle Scholar
  107. 107.
    Kauzmann W, Eyring H (1940) The viscous flow of large molecules. J Am Chem Soc 62:3113–3125CrossRefGoogle Scholar
  108. 108.
    Bell G (1978) Models for the specific adhesion of cells to cells. Science 200:618–627CrossRefGoogle Scholar
  109. 109.
    Dopieralski P, Ribas-Arino J, Anjukandi P, Krupicka M, Kiss J, Marx D (2013) The Janus-faced role of external forces in mechanochemical disulfide bond cleavage. Nat Chem 5:685–691CrossRefGoogle Scholar
  110. 110.
    Lele TP, Thodeti CK, Ingber DE (2006) Force meets chemistry: analysis of mechanochemical conversion in focal adhesions using fluorescence recovery after photobleaching. J Cell Biochem 97:1175–1183CrossRefGoogle Scholar
  111. 111.
    Greenberg MJ, Moore JR (2010) The molecular basis of frictional loads in the in vitro motility assay with applications to the study of the loaded mechanochemistry of molecular motors. Cytoskeleton 67:273–285CrossRefGoogle Scholar
  112. 112.
    Bustamante C, Chemla YR, Forde NR, Izhaky D (2004) Mechanical processes in biochemistry. Annu Rev Biochem 73:705–748CrossRefGoogle Scholar
  113. 113.
    Evans E, Ritchie K (1997) Dynamic strength of molecular adhesion bonds. Biophys J 72(4):1541–1555CrossRefGoogle Scholar
  114. 114.
    Evans E (2001) Probing the relation between force-lifetime and chemistry. Annu Rev Biophys Biomol Struct 30:105–128CrossRefGoogle Scholar
  115. 115.
    Konda SSM, Brantley JN, Bielawski CW, Makarov DE (2011) Chemical reactions modulated by mechanical stress: extended Bell theory. J Chem Phys 135:164103CrossRefGoogle Scholar
  116. 116.
    Konda SSM, Brantley JN, Varghese BT, Wiggins KM, Bielawski CW, Makarov DE (2013) Molecular catch bonds and the anti-Hammond effect in polymer mechanochemistry. J Am Chem Soc 135:12722–12729CrossRefGoogle Scholar
  117. 117.
    Brantley JN, Konda SSM, Makarov DE, Bielawski CW (2012) Regiochemical effects on molecular stability: a mechanochemical evaluation of 1,4- and 1,5-disubstituted triazoles. J Am Chem Soc 134:9882–9885CrossRefGoogle Scholar
  118. 118.
    Tian Y, Boulatov R (2013) Comparison of the predictive performance of the Bell-Evans, Taylor-expansion and statistical-mechanics models of mechanochemistry. Chem Comm 49:4187–4189CrossRefGoogle Scholar
  119. 119.
    Hsu SM, Zhang J, Yin Z (2002) The nature and origin of tribochemistry. Tribol Lett 13:131–139CrossRefGoogle Scholar
  120. 120.
    Kajdas C (2013) General approach to mechanochemistry and its relations to tribochemistry. In: Pihtili H (ed) Tribology in engineering. InTech, pp 209–240Google Scholar
  121. 121.
    Liang T, Sawyer WG, Perry SS, Sinnott SB, Phillpot SR (2008) First-principles determination of static potential energy surfaces for atomic friction in MoS2 and MoO3. Phys Rev B 77:104105CrossRefGoogle Scholar
  122. 122.
    Smith GS, Modine NA, Waghmare UV, Kaxiras E (1998) First-principles study of static nanoscale friction between MoO3. J Comput Mater Des 5:61–71CrossRefGoogle Scholar
  123. 123.
    Modine NA, Zumbach G, Kaxiras E (1997) Adaptive-coordinate real-space electronic structure calculations for atoms, molecules, and solids. Phys Rev B 55:10289CrossRefGoogle Scholar
  124. 124.
    Lees AW, Edwards SF (1972) The computer study of transport processes under extreme conditions. J Phys C Solid State Phys 5:1921–1929CrossRefGoogle Scholar
  125. 125.
    Zilibotti G, Righi MC (2011) Ab initio calculation of the adhesion and ideal shear strength of planar diamond interfaces with different atomic structure and hydrogen coverage. Langmuir 27:6862–6867CrossRefGoogle Scholar
  126. 126.
    Wang J, Want F, Li J, Sun Q, Yuan P, Jia Y (2013) Comparative study of friction properties for hydrogen- and fluorine-modified diamond surfaces: a first-principles investigation. Surf Sci 608:74–79CrossRefGoogle Scholar
  127. 127.
    Wang L-F, Ma T-B, Hu Y-Z, Wang H (2012) Atomic-scale friction in graphene oxide: an interfacial interaction perspective from first-principles calculations. Phys Rev B 86:125436CrossRefGoogle Scholar
  128. 128.
    Wang L-F, Ma T-B, Hu Y-Z, Wang H, Shao T-M (2013) Ab initio study of the friction mechanism of fluorographene and graphane. J Phys Chem C 117:12520–12525CrossRefGoogle Scholar
  129. 129.
    Leven I, Krepel D, Shemesh O, Hod O (2013) Robust superlubricity in graphene/h-BN heterojunctions. J Phys Chem Lett 4:115–120CrossRefGoogle Scholar
  130. 130.
    Hod O (2013) The registry index: a quantitative measure of materials’ interfacial commensurability. ChemPhysChem 14:2376–2391CrossRefGoogle Scholar
  131. 131.
    Zhong W, Tománek D (1990) First-principles theory of atomic-scale friction. Phys Rev Lett 64:3054–3057CrossRefGoogle Scholar
  132. 132.
    Neitola R, Pakkanen TA (2001) Ab initio studies on the atomic-scale origin of friction between diamond (111) surfaces. J Phys Chem B 105:1338–1343CrossRefGoogle Scholar
  133. 133.
    Koskilinna JO, Linnolahti M, Pakkanen TA (2005) Tribochemical reactions between methylated diamond (111) surfaces: a theoretical study. Tribol Lett 20:157–161CrossRefGoogle Scholar
  134. 134.
    Neitola R, Pakkanen TA (2006) Ab initio studies on nanoscale friction between fluorinated diamond surfaces: effect of model size and level of theory. J Phys Chem B 110:16660–16665CrossRefGoogle Scholar
  135. 135.
    Neitola R, Pakkanen TA (2004) Ab initio studies on the atomic-scale origin of friction between hydrocarbon layers. Chem Phys 299:47–56CrossRefGoogle Scholar
  136. 136.
    Neitola R, Ruuska H, Pakkanen TA (2005) Ab initio studies on nanoscale friction between graphite layers: effect of model size and level of theory. J Phys Chem B 109:10348–10354CrossRefGoogle Scholar
  137. 137.
    Koskilinna JO, Linnolahti M, Pakkanen TA (2007) Friction paths for cubic boron nitride: an ab initio study. Tribol Lett 27:145–154CrossRefGoogle Scholar
  138. 138.
    Koskilinna JO, Linnolahti M, Pakkanen TA (2008) Friction and a tribochemical reaction between ice and hexagonal boron nitride: a theoretical study. Tribol Lett 29:163–167CrossRefGoogle Scholar
  139. 139.
    Ogata S, Li J, Hirosaki N, Shibutani Y, Yip S (2004) Ideal shear strain of metals and ceramics. Phys Rev B 70:104104CrossRefGoogle Scholar
  140. 140.
    Liao P, Carter EA (2010) Ab initio density functional theory+U predictions of the shear response of iron oxides. Acta Mater 58:5912–5925CrossRefGoogle Scholar
  141. 141.
    Mosey NJ, Liao P, Carter EA (2008) Rotationally invariant ab initio evaluation of Coulomb and exchange parameters for DFT+U calculations. J Chem Phys 129:014103CrossRefGoogle Scholar
  142. 142.
    Mosey NJ, Carter EA (2009) Shear strength of chromia across multiple length scales: An LDA+U study. Acta Mater 57:2933–2943CrossRefGoogle Scholar
  143. 143.
    Mills G, Jónsson H (1994) Quantum and thermal effects in H2 dissociative adsorption: evaluation of free energy barriers in multidimensional quantum systems. Phys Rev Lett 72:1124–1127CrossRefGoogle Scholar
  144. 144.
    Mills G, Jónsson H, Schenter G (1995) Reversible work transition state theory: application to dissociative adsorption of hydrogen. Surf Sci 324:305–337CrossRefGoogle Scholar
  145. 145.
    Jónsson H, Mills G, Jacobsen KW (1998) Nudged elastic band method for finding minimum energy paths of transitions. In: Berne BJ, Ciccotti G, Coker DF (eds) Class Quantum Dyn Condens Phase Simulations. World Scientific, Singapore, pp 385–404Google Scholar
  146. 146.
    Caspersen KJ, Carter EA (2005) Finding transition states for crystalline solid-solid phase transformations. Proc Natl Acad Sci 102:6738–6743CrossRefGoogle Scholar
  147. 147.
    Henkelman G, Uberuaga BP, Jónsson H (2000) A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J Chem Phys 113:9901–9904CrossRefGoogle Scholar
  148. 148.
    Sheppard D, Xiao P, Chemelewski W, Johnson DD, Henkelman G (2012) A generalized solid-state nudged elastic band method. J Chem Phys 136:074103CrossRefGoogle Scholar
  149. 149.
    Trinkle D, Hennig R, Srinivasan S, Hatch D, Jones M, Stokes H, Albers R, Wilkins J (2003) New mechanism for the α to ω martensitic transformation in pure titanium. Phys Rev Lett 91:025701CrossRefGoogle Scholar
  150. 150.
    Hennig RG, Trinkle DR, Bouchet J, Srinivasan SG, Albers RC, Wilkins JW (2005) Impurities block the α to ω martensitic transformation in titanium. Nat Mater 4:129–133CrossRefGoogle Scholar
  151. 151.
    Parrinello M, Rahman A (1981) Polymorphic transitions in single crystals: a new molecular dynamics method. J Appl Phys 52:7182–7190CrossRefGoogle Scholar
  152. 152.
    Liu J, Johnson D (2009) bcc-to-hcp transformation pathways for iron versus hydrostatic pressure: coupled shuffle and shear modes. Phys Rev B 79:134113Google Scholar
  153. 153.
    Xiao P, Henkelman G (2012) Communication: from graphite to diamond: reaction pathways of the phase transition. J Chem Phys 137:101101CrossRefGoogle Scholar
  154. 154.
    Vu NH, Le HV, Cao TM, Pham VV, Le HM, Nguyen-Manh D (2012) Anatase-rutile phase transformation of titanium dioxide bulk material: a DFT + U approach. J Phys Condens Matter 24:405501CrossRefGoogle Scholar
  155. 155.
    Dong X, Zhou X-F, Qian G-R, Zhao Z, Tian Y, Wang H-T (2013) An ab initio study on the transition paths from graphite to diamond under pressure. J Phys Condens Matter 25:145402CrossRefGoogle Scholar
  156. 156.
    Xiao P, Cheng J-G, Zhou J-S, Goodenough JB, Henkelman G (2013) Mechanism of the CaIrO3 post-perovskite phase transition under pressure. Phys Rev B 88:144102CrossRefGoogle Scholar
  157. 157.
    Dai Y, Ni S, Li Z, Yang J (2013) Diffusion and desorption of oxygen atoms on graphene. J Phys Condens Matter 25:405301CrossRefGoogle Scholar
  158. 158.
    Briquet LGV, Wirtz T, Philipp P (2013) First principles investigation of Ti adsorption and migration on Si(100) surfaces. J Appl Phys 114:243505CrossRefGoogle Scholar
  159. 159.
    Jennings PC, Aleksandrov HA, Neyman KM, Johnston RL (2014) A DFT study of oxygen dissociation on platinum based nanoparticles. Nanoscale 6:1153–1165CrossRefGoogle Scholar
  160. 160.
    Harrison JA, White CT, Colton RJ, Brenner DW (1992) Molecular-dynamics simulation of atomic-scale friction of diamond surfaces. Phys Rev B 46:9700–9708CrossRefGoogle Scholar
  161. 161.
    Harrison JA, Colton RJ, White CT, Brenner DW (1993) Effect of atomic-scale surface roughness on friction: a molecular dynamics study of diamond surfaces. Wear 168:127–133CrossRefGoogle Scholar
  162. 162.
    Harrison JA, White CT, Colton RJ, Brenner DW (1995) Investigation of the atomic-scale friction and energy dissipation in diamond using molecular dynamics. Thin Solid Films 260:205–211CrossRefGoogle Scholar
  163. 163.
    Gao G, Cannara RJ, Carpick RW, Harrison JA (2007) Atomic-scale friction on diamond: a comparison of different sliding directions on (001) and (111) surfaces using MD and AFM. Langmuir 23:5394–5405CrossRefGoogle Scholar
  164. 164.
    Yue D-C, Ma T-B, Hu Y-Z, Yeon J, van Duin ACT, Wang H, Luo J (2013) Tribochemistry of phosphoric acid sheared between quartz surfaces: a reactive molecular dynamics study. J Phys Chem C 117:25604–25614CrossRefGoogle Scholar
  165. 165.
    Carkner CJ, Mosey NJ (2010) Slip mechanisms of hydroxylated α-Al2O3 (0001)/(0001) interfaces: a first-principles molecular dynamics study. J Phys Chem C 114:17709–17719CrossRefGoogle Scholar
  166. 166.
    Carkner CJ, Haw SM, Mosey NJ (2010) Effect of adhesive interactions on static friction at the atomic scale. Phys Rev Lett 105:056102CrossRefGoogle Scholar
  167. 167.
    Haw SM, Mosey NJ (2012) Tribochemistry of aldehydes sheared between (0001) surfaces of α-alumina from first-principles molecular dynamics. J Phys Chem C 116:2132–2145CrossRefGoogle Scholar
  168. 168.
    Haw SM, Mosey NJ (2011) Chemical response of aldehydes to compression between (0001) surfaces of α-alumina. J Chem Phys 134:014702CrossRefGoogle Scholar
  169. 169.
    Bernasconi M, Chiarotti GL, Focher P, Scandolo S, Tosatti E, Parrinello M (1995) First-principle-constant pressure molecular dynamics. J Phys Chem Solids 56:501–505CrossRefGoogle Scholar
  170. 170.
    Wentzcovitch R (1991) Invariant molecular-dynamics approach to structural phase transitions. Phys Rev B 44:2358–2361CrossRefGoogle Scholar
  171. 171.
    Berendsen HJC, Postma JPM, van Gunsteren WF, DiNola A, Haak JR (1984) Molecular dynamics with coupling to an external bath. J Chem Phys 81:3684CrossRefGoogle Scholar
  172. 172.
    Andersen HC (1980) Molecular dynamics simulations at constant pressure and/or temperature. J Chem Phys 72:2384–2393CrossRefGoogle Scholar
  173. 173.
    Serra S (1999) Pressure-induced solid carbonates from molecular CO2 by computer simulation. Science 284:788–790CrossRefGoogle Scholar
  174. 174.
    Mugnai M, Pagliai M, Cardini G, Schettino V (2008) Mechanism of the ethylene polymerization at very high pressure. J Chem Theory Comput 4:646–651CrossRefGoogle Scholar
  175. 175.
    Bernasconi M, Chiarotti G, Focher P, Parrinello M, Tosatti E (1997) Solid-state polymerization of acetylene under pressure: ab initio simulation. Phys Rev Lett 78:2008–2011CrossRefGoogle Scholar
  176. 176.
    Schettino V, Bini R (2007) Constraining molecules at the closest approach: chemistry at high pressure. Chem Soc Rev 36:869–880CrossRefGoogle Scholar
  177. 177.
    Mosey N, Woo T, Müser M (2005) Energy dissipation via quantum chemical hysteresis during high-pressure compression: a first-principles molecular dynamics study of phosphates. Phys Rev B 72:054124CrossRefGoogle Scholar
  178. 178.
    Mosey NJ, Müser MH, Woo TK (2005) Molecular mechanisms for the functionality of lubricant additives. Science 307:1612–1615CrossRefGoogle Scholar
  179. 179.
    Giannozzi P, Baroni S, Bonini N et al (2009) QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. J Phys Condens Matter 21:395502CrossRefGoogle Scholar
  180. 180.
    Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made simple. Phys Rev Lett 77:3865–3868CrossRefGoogle Scholar
  181. 181.
    Troullier N, Martins JL (1991) Efficient pseudopotentials for plane-wave calculations. Phys Rev B 43:1993–2006CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Gurpaul S. Kochhar
    • 1
  • Gavin S. Heverly-Coulson
    • 1
  • Nicholas J. Mosey
    • 1
  1. 1.Department of ChemistryQueen’s UniversityKingstonCanada

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