1 Introduction

The use of photons to transport energy is appealing thanks to their ability to propagate long distances with low loss, their resistance to electromagnetic interference, and their wide range of controllable parameters like wavelength, polarization, and coherence. Once they reach their destination, however, photons must be converted to a more useful form of energy, like heat or mechanical work. To generate mechanical work, one option is to convert them into an electrical potential that can be harnessed to drive actuator devices. This strategy requires two elements: a photovoltaic module to generate current, and a device to convert this electrical energy into a mechanical output, like an electric motor or piezoelectric crystal. Another option is to utilize a material that directly converts the absorbed photon into mechanical motion without relying on free electrons and external circuitry. Photomechanical materials have the property that their constituent nuclei change position after photon absorption, generating a force and displacement that can be harnessed to perform mechanical work [1,2,3]. This change in atomic coordinates could result from heating (photothermal) [4,5,6], a change in electronic state (photostrictive) [7], or a chemical reaction (photochemical) [8].

The photochemical approach to photomechanical materials relies on harnessing molecular reactions, like cis–trans isomerization, to drive deformations in solid-state systems like polymers [9,10,11] and crystals [12,13,14,15,16,17]. To be useful in practical actuator devices, the photochemical product should be able to return to the reactant state either by thermal fluctuations (T-type reversibility) or by a second photon absorption (P-type reversibility) [18, 19]. To assess the potential of this class of photomechanical materials, specifically their efficiency and work output, it would be useful to have a simple theoretical framework that could be used to analyze such systems at the molecular level. The Bell model for mechanochemistry [20] and the Marcus–Hush model for electron transfer [21] are examples of semiclassical approaches based on displaced harmonic oscillators. An analogous model for the photomechanical response of a molecule could provide the basis for interpretation of experimental results and the design of improved materials. Such a framework could also help address practical questions, like how the photomechanical approach to energy conversion compares to the photovoltaic approach in terms of figures of merit like efficiency.

The goal of this paper is to develop a simple one-dimensional (1D) model that can serve as a starting point for more realistic models of the photomechanical response. First, we introduce the harmonic model with two states, denoted \(|a\rangle\) and \(|b\rangle\), that correspond to two different nuclear configurations. When an external force is applied, this simple model generates predictions that are qualitatively consistent with standard mechanochemistry models. We next analyze the photomechanical process and derive expressions for the work output, blocking force, and absorbed photon-to-work efficiency of the \(|a\rangle\) →\(|b\rangle\) reaction. If the starting state \(|a\rangle\) is the stable isomer (lower energy than \(|b\rangle\)), we find that it is possible to attain photon-to-work efficiencies of > 50%. If \(|a\rangle\) is higher in energy, i.e., a metastable isomer, then one-way efficiencies > 100% are possible by releasing the stored potential energy. We also analyze the effects of nonadiabatic electronic coupling, unequal frequency potentials, and the \(|a\rangle\) →\(|b\rangle\) →\(|a\rangle\) cycling efficiency. We conclude that photomechanical materials have the potential to surpass a photovoltaic-piezoelectric combination and approach that of a photovoltaic–motor combination. Given that most measured photomechanical efficiencies are orders of magnitude below the theoretical limits derived in this paper, our results suggest that there is substantial room for improvement in this class of materials.

2 Results

The model used for our calculations is shown in Fig. 1 [22]. Two harmonic potential energy surfaces represent electronic states \(|a\rangle\) and \(|b\rangle\). We always assume the system starts in state \(|a\rangle\). The force constants are ka and kb, respectively, and the coordinate system is chosen, so that the minima lie at ± x0 along the reaction coordinate in the absence of an applied force. State \(|b\rangle\) is offset in energy by an amount ∆. If ∆ is positive, then the initial state \(|a\rangle\) is the lowest energy, stable state and state \(|b\rangle\) can be associated with either a relaxed excited state or an isomerized molecule. If ∆ < 0, then state \(|a\rangle\) is a higher energy, metastable state that must be populated by some other process, e.g., previous absorption of a photon. Finally, states \(|a\rangle\) and \(|b\rangle\) are coupled by an interaction term V, which we assume to be a real number. The overall Hamiltonian is given by

$$ \hat{H} = |a\rangle E_{a} \langle a| + |b\rangle E_{b} \langle b| + V\left( {|a\rangle \langle b\left| { + |b\rangle \langle a} \right|} \right) $$
Fig. 1
figure 1

a Two uncoupled diabatic harmonic surfaces, centered at ± x0, with energy offset ∆, that serve as the starting point for the calculations. b Adiabatic λ± energy surfaces created by electronic coupling term V, along with the relevant energy gaps


$$ E_{a} = \frac{1}{2}k_{a} \left( {x + x_{0} } \right)^{2} , $$
$$ E_{b} = \frac{1}{2}k_{b} \left( {x - x_{0} } \right)^{2} + \Delta . $$

Diagonalization of this Hamiltonian generates two adiabatic potential energy surfaces (PESs) whose energies λ± are given by

$$ \lambda_{ \pm } \left( x \right) = \frac{1}{2}\left[ {E_{a} \left( x \right) + E_{b} \left( x \right) \pm \sqrt {\left( {E_{b} \left( x \right) - E_{a} \left( x \right)} \right)^{2} + 4V^{2} } } \right]. $$

If a constant force A is applied that pushes the system to the left in Fig. 1a, i.e., toward state \(|a\rangle\), we have an additional potential energy term U(x) = − Ax. This force resists the \(|a\rangle\)\(|b\rangle\) reaction and modifies the Hamiltonian. If we neglect V, the new \(|a\rangle\) and \(|b\rangle\) PES curves are given by

$$ E_{a} \left( x \right) = \frac{1}{2}k_{a} \left( {x - \left( { - x_{0} - \frac{A}{{k_{a} }}} \right)} \right)^{2} + Ax_{a} - \frac{{A^{2} }}{{2k_{a} }} , $$
$$ E_{b} \left( x \right) = \frac{1}{2}k_{b} \left( {x - \left( { + x_{0} - \frac{A}{{k_{b} }}} \right)} \right)^{2} + Ax_{b} - \frac{{A^{2} }}{{2k_{b} }} + \Delta . $$

The modified PESs are shown in Fig. 1b. The applied force shifts the potential minima to new positions xa and xb. The barrier height between the new \(|a\rangle\) and \(|b\rangle\) minima (xa and xb) is lowered by the force and disappears when the potential curve of \(|a\rangle\) crosses through the minimum of the \(|b\rangle\) potential well, as shown in Fig. 2a. The force at which the potential minimum at xb disappears is denoted the blocking force or stop force Astop and represents the maximum force under which the system can maintain two stable points. Note that as A increases, the optical gaps between the \(|a\rangle\) and \(|b\rangle\) surfaces at stable points xa and xb also change, with the gap at xa increasing and that at xb decreasing (Fig. 2b).

Fig. 2
figure 2

a As the force acting against photoisomerization increases, both the ground state (λ) and excited state (λ+) energy minima shift to the left. Eventually, the barrier between the ground state minima vanishes. b The shifts in the optical gaps (ΔE) as a function of applied force A. For these calculations, x0 = 11, ka = kb = 1, Astop = 11

At this point, it is useful to place this model in the context of mechanochemical systems, where the reaction rate depends on the ground state energy barrier between xb and xa [23]. If we assume that \(|b\rangle\) is the reactant that must reach the lower energy product state \(|a\rangle\), we can calculate the barrier as a function of A for various parameter values. Figure 3 plots the barrier height ∆E versus A for various nonadiabatic coupling values V. For low A values, there is a linear dependence of ∆E on A, which saturates at larger forces. This linear dependence of ∆E on A is consistent with most theoretical treatments of the effects of an applied force on reaction rates [24, 25]. Note that the net effect is to change the thermal \(|b\rangle\) →\(|a\rangle\) rate. No thermodynamic work is performed, since the motion along the reaction coordinate is parallel to the applied force.

Fig. 3
figure 3

The dependence of the ground state activation energy ΔEb→a as the applied force A is increased for different values of the electronic coupling V. For these calculations, ka = kb = 1, x0 = − 5, and Δ = 0. Inset: illustration of the ground state PES (red) showing ΔEb→a

We now turn to the photomechanical process in which the system is forced to work against the applied force by photoexcitation from \(|a\rangle\) to \(|b\rangle\). The system starts at xa, absorbs a photon to go to the upper surface, and relaxes to xb, as shown in Fig. 1b. The work output W in this case is given by the usual definition of force × distance

$$ W_{a \to b} = A\left( {x_{b} - x_{a} } \right) = A\Delta x. $$

The input photon energy is just the difference between the \(|a\rangle\) and \(|b\rangle\) potentials at the starting position xa:

$$ \Delta E_{a \to b} = E_{b} \left( {x_{a} } \right) - E_{a} \left( {x_{a} } \right). $$

With V = 0, we can use Eqs. (46) to find

$$ E_{a \to b} = 2k_{b} x_{0}^{2} + 2Ax_{0} \left( {\frac{{k_{b} }}{{k_{a} }}} \right) + \frac{{A^{2} \left( {k_{b} - k_{a} } \right)}}{{2k_{a}^{2} }} + \Delta , $$
$$ W_{a \to b} = 2Ax_{0} + A^{2} \left( {\frac{1}{{k_{b} }} - \frac{1}{{k_{a} }}} \right). $$

These equations can be combined to obtain an expression for the \(|a\rangle\) →\(|b\rangle\) efficiency \(\eta_{a \to b}\)

$$ \eta_{a \to b} = \frac{{2Ax_{0} + A^{2} \left( {\frac{1}{{k_{b} }} - \frac{1}{{k_{a} }}} \right)}}{{2k_{b} x_{0}^{2} + 2Ax_{0} \left( {\frac{{k_{b} }}{{k_{a} }}} \right) + \frac{{A^{2} \left( {k_{b} - k_{a} } \right)}}{{2k_{a}^{2} }} + \Delta }}. $$

The maximum A that can be applied to the system is limited by the necessity that there exist two stable minima that the molecule can be switched between. In other words, the efficiency will be maximized at Astop and this maximum efficiency is ηstop. Again, for V = 0, we calculate

$$ A_{{{\text{stop}}}} = \left( {\frac{{k_{b} }}{{k_{a} - k_{b} }}} \right)\left[ {2k_{a} x_{0} - k_{b} \sqrt {4k_{a} k_{b} x_{0}^{2} - 2\Delta \left( {k_{b} - k_{a} } \right)} } \right]. $$

Substituting this value back into Eq. (9) allows us to obtain a general expression for the efficiency at the stop force (ηstop)

$$ \eta_{{{\text{stop}}}} = \frac{{2k_{a} k_{b} }}{{k_{a}^{2} - k_{b}^{2} }}\left[ {\frac{{\Delta \left( {k_{b} - k_{a} } \right) - 2k_{a} k_{b} x_{0}^{2} + k_{a} x_{0} \sqrt {4k_{a} k_{b} x_{0}^{2} + 2\Delta \left( {k_{a} - k_{b} } \right)} }}{{2k_{a} k_{b} x_{0}^{2} + \Delta \left( {k_{a} - k_{b} } \right)}}} \right]. $$

Note that this expression is valid for any value of ∆, positive or negative. In Fig. 4, we plot ηa→b as a function of A for ka/kb = 1. For this condition, ηa→b is an increasing function of A and is maximized at Astop for all values of ∆. In fact, for the most common scenario where ∆\(\ge { 0}\), the efficiency is always maximized at Astop. Only when ∆ < 0 and ka > kb do, we find that the maximum ηa→b does not occur at Astop but at an intermediate force. In this case, the maximum ηa→b can even surpass 1.0 (Supporting Information). For ∆ < 0, the large ηa→b values are an artifact of our neglect of the energetic cost of preparing this state but serve to illustrate how the 1D system can be “pre-loaded” to produce more mechanical energy than the input photon.

Fig. 4
figure 4

The photomechanical efficiency ηa→b plotted as a function of applied force A and different ∆ energy offsets. These calculations were done with x0 = 11, ka = kb = 1, and V = 0. For these parameters, the maximum efficiency occurs at Astop for all these \(\Delta\) values

For the remainder of this paper, we will concentrate on the ∆\(\ge { 0}\) case that is most relevant for practical materials. Note that our model assumes that the quantum yield for isomerization is unity in all cases. The molecular parameters that determine efficiency are ka, kb, ∆, and V. To examine the role of the first three parameters in the limit of V = 0, in Fig. 5, we plot ηstop versus the ratio ka/kb for various ∆ values. The first observation is that ηstop is maximized for ∆ = 0. For ∆ > 0, the efficiency decreases as the input energy cost increases, as predicted by Eq. (9). The behavior of ηstop as a function of the force constants is more complex. When ka = kb = k, Eq. (11) reduces to

$$ \eta = \frac{{2Ax_{0} }}{{2kx_{0}^{2} + 2Ax_{0} + \Delta }}. $$
Fig. 5
figure 5

The photomechanical efficiency ηstop plotted as a function of ka/kb for different values of Δ. When Δ > 0, smaller ratios of ka/kb become inaccessible (and thus η = 0) as only one stable minimum exists along under the absence of applied force

In this limit, Astop = kx0 – Δ/(2kx0) and we find

$$ \eta_{{{\text{stop}}}} = \frac{{2kx_{0}^{2} - \frac{\Delta }{k}}}{{4kx_{0}^{2} + \Delta \left( {1 - \frac{1}{k}} \right)}}. $$

This expression leads to a maximum ηstop = 0.5 when ∆ = 0. However, the largest ηstop values are obtained for ka/kb > 1 for all ∆ values, meaning that the \(|b\rangle\) PES has a lower frequency and a shallower well. We found a maximum efficiency ηstop of 0.554 for ka/kb = 2.315 and ∆ = 0. At larger ∆ values, the maximum ηstop decreases and shifts to slightly larger ka/kb ratios; for example, a maximum occurs at ka/kb = 3.307 when ∆ = 100.

The origin of the higher ηstop values for ka/kb > 1 lies in the modified optical properties, rather than the increased force generation. The lower kb value reduces the photon energy required to make the transition between \(|a\rangle\) and \(|b\rangle\) surfaces, as illustrated in Fig. 6a. The different dependences of Eqs. (7) and (8) on the force constants ka and kb mean that it is possible to dramatically reduce the cost of the ∆Ea→b photon by lowering kb while only slightly decreasing the work output. Once kb is much smaller than ka, however, both the work output and photon input energies decline at the same rate. This can be seen from a plot of the Wab and ∆Eab values as a function of the ka/kb ratio, shown in Fig. 6b. The 10% gain in photon-to-work efficiency for ka > kb provides a hint that the work output can be enhanced by tuning the molecular vibrational structure. Previous workers have established that vibrational structure and coherence near conical intersections can have a large impact the quantum yield of a photochemical reaction [26,27,28]. The results presented here assume that the photochemical quantum yield is unity, so the photon-to-work efficiency reflects how vibrational structure affects the light absorption process which occurs far from this type of intersection of the harmonic surfaces. In a real system, the effects of molecular vibrational structure on both the conical intersection and the absorption energy will both have to be considered to accurately calculate the overall photon-to-work efficiency.

Fig. 6
figure 6

a Illustration of how changes in ka/kb lead to different ∆Ea→b energy gaps for A = 5. Both the energy cost and work output decrease as the ka/kb ratio increases, but at different rates, leading to the maximum in the efficiency seen in Fig. 5. b Plots of the available work (black) and photon cost ∆Ea→b (red) as ka/kb increases for x0 = 11, V = 0, and Δ = 0. The photon cost asymptotically approaches 250 a.u. as ka/kb increases, while the work continues to decrease and reduces the one-way efficiency at higher ratios of ka/kb. The maximum efficiency occurs in the dashed region (ka/kb = 2.315) where the photon cost has decreased more rapidly than the work output

If V ≠ 0, the Astop and ηstop values can be evaluated numerically by iteratively solving Eqs. (3), (10), and (11). Details are given in the Supporting Information. The effect of V on ηstop is shown in Fig. 7, which plots ηstop as a function of V for different values of ∆. In all cases, increasing V leads to a roughly linear decrease in ηstop. This can be understood as a consequence of the nonadiabatic coupling leading to a lowered activation barrier on the adiabatic ground state surface, which in turn lowers Astop. At large V values, the second minimum at xb disappears, preventing the calculation of ηstop values.

Fig. 7
figure 7

The photomechanical efficiency ηstop is plotted as a function of the electronic coupling V for various ∆ values. For these calculations, x0 = 5 and ka = kb = 1. Note that larger V couplings and ∆ values result in smaller Astop and, thus, ηstop values as the ground state barrier decreases

Finally, we consider the cycling efficiency of the 1D coupled harmonic system. We have already evaluated the efficiency in the \(|a\rangle\) →\(|b\rangle\) direction. The return \(|b\rangle\) →\(|a\rangle\) stroke does not contribute to the work against the applied force, but does require extra photon energy, so the efficiency will always be decreased. In this case, the overall efficiency of the \(|a\rangle\) →\(|b\rangle\) →\(|a\rangle\) cycle is just

$$ \eta_{{{\text{cycle}}}} = \frac{{W_{a \to b} }}{{\Delta E_{a \to b} + \Delta E_{b \to a} }}. $$

We consider the case of ∆ ≥ 0, where the starting state \(|a\rangle\) is the lowest energy isomer. At the maximum ηstop, the \(|a\rangle\) PES curve intersects the minimum of the \(|b\rangle\) PES curve, and the photon energy required to make the \(|b\rangle\) →\(|a\rangle\) transition becomes negligible. In this case, the maximum ηcycle = ηstop = 0.554 for ka/kb = 2.315. This analysis indicates that the highest cycle efficiency will be attained for T-type materials, since in practice, this return transition would be accomplished by thermal excitation rather than an optical photon. Lower A values not only lead to lower forward \(|a\rangle\) →\(|b\rangle\) efficiencies but also require additional input photon energy to return the system from \(|b\rangle\) to \(|a\rangle\).

3 Discussion

There are several reasons that the analysis presented above should be considered an upper limit for photomechanical efficiencies in real systems. First, we only considered the T = 0 K limit. When ∆E < kT, we can expect thermal fluctuations to effectively remove the stationary point at the \(|b\rangle\) curve minimum. In practice, this would lower Astop and thus ηstop. Second, we have assumed that the photochemical quantum yield for the \(|a\rangle\) →\(|b\rangle\) reaction is unity. The efficiencies calculated above should be multiplied by a quantum yield factor that in practice is always less than 1.0 due to both radiative and nonradiative decay channels. Third, we have only considered a 1D system with coupled product-reactant modes. In a polyatomic molecule with many modes oriented orthogonal to the reaction coordinate, e.g., azobenzene [29,30,31], much of the input photon energy will be dissipated into vibrations that are not aligned with the applied force, further decreasing the efficiency.

How efficient could a molecular photoisomerization be in practice? Gaub and coworkers investigated the photomechanical response of single oligomer chains composed of azobenzene repeat units attached to an atomic force microscope tip. In these experiments, the applied force was not aligned with the cistrans reaction coordinate, which is usually taken to be the C–N–N dihedral angle. Even given this misalignment, however, measurements of the chain contraction under load yielded an absorbed photon-to-work efficiency of 0.1 for a trans → cis photoisomerization [32]. This value is actually not far from the single-molecule theoretical limit derived in this paper. However, once quantum yields, absorption cross sections, and light propagation were taken into account, the calculated incident photon-to-work efficiency was estimated to be 7.5 × 10–6. [33] Photomechanical crystals composed purely of photoactive molecules typically exhibit absorbed photon-to-work efficiencies of less than 1%. [34, 35], as do most polymer systems [36]. The large gap between the theoretical molecular limit and the measured efficiencies suggests that there remains a substantial room to improve these materials.

The results presented above suggest that directional application of force could result in changes in molecular optical properties. Although redshifts are commonly observed in high-pressure experiments, they usually result from changes in the medium polarizability due to increased density [37, 38]. When the molecular PESs are distorted by the application of a directional force, our model predicts a blueshift of the \(|a\rangle\) →\(|b\rangle\) absorption and a redshift of the \(|b\rangle\) →\(|a\rangle\) fluorescence. Although challenging, the measurement of optical properties during the directional pulling of single molecules could reveal the PES deformations illustrated in Fig. 2. If applying a force leads to substantial absorption shifts, then a fruitful area for improving photomechanical materials might involve tuning of reactant–product vibrational structures to enhance efficiency. The fact that even in this simple model, the maximum efficiency does not occur for ka/kb = 1.0 suggests that careful consideration of how molecular vibrations affect both photon absorption and mechanical response properties may be necessary to optimize these materials.

Finally, we can compare molecular photomechanical elements to photovoltaic approaches for transforming photon energy into work. At 0 K and given an input photon at the semiconductor band edge, the photovoltaic energy conversion efficiency can approach 1.0 [39]. Piezoelectric actuators have a maximum electrical-to-mechanical conversion efficiency of 0.5 [40], so assuming no electrical losses, the PV-piezoelectric approach would yield an overall efficiency of 1.0 × 0.5 = 0.5. If the PV cell is attached to a DC electric motor, whose electrical-to-mechanical efficiency can approach 1.0 [41], then the photon-to-work efficiency will also approach 1.0. Our results show that the photomechanical approach can be competitive with either photovoltaic approach in terms of theoretical efficiency. The photomechanical approach possesses several potential advantages, however, including (1) simplicity, since only a single element with no connections is required; (2) insensitivity to electromagnetic fields, since no free carriers are generated; (3) fast response, since the molecular shape change follows the photoisomerization time, which can be on the order of picoseconds. For a given application, the best approach will likely be determined by factors like device size, environment, and the available light source. It should be emphasized that the field of photomechanical materials is still relatively young compared to the fields of electromagnetic actuators and photovoltaics, so considerable improvement may be expected.

4 Conclusion

The simple 1D model in this paper represents a preliminary step in the development of a molecular model for the photomechanical process. It provides a way to estimate mechanical outputs like the stop force, work, and efficiency from molecular parameters like vibrational frequencies, reaction coordinates, and electronic couplings. A central result is that the theoretical photon-to-work efficiency of a molecule is comparable to that of photovoltaic devices. A second result is that the maximum efficiency is obtained when ka > kb, showing that the interplay between force-induced changes in the optical as well as the mechanical properties must be considered in the design of such molecules. It is hoped that this work will motivate more sophisticated theoretical studies and materials design that enable bulk photomechanical systems to approach the molecular performance limits.