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Stanislav
Chizhik
*^{ab},
Anatoly
Sidelnikov
^{ab},
Boris
Zakharov
^{ab},
Panče
Naumov
^{c} and
Elena
Boldyreva
^{ab}
^{a}Institute of Solid State Chemistry and Mechanochemistry, Siberian Branch of Russian Academy of Sciences, ul. Kutateladze, 18, Novosibirsk 620128, Russian Federation. E-mail: csagbox@gmail.com
^{b}Novosibirsk State University, ul. Pirogova, 2, Novosibirsk 630090, Russian Federation
^{c}New York University Abu Dhabi, P.O. Box 129188, Abu Dhabi, United Arab Emirates

Received
11th November 2017
, Accepted 19th January 2018

First published on 22nd January 2018

Photomechanically reconfigurable elastic single crystals are the key elements for contactless, timely controllable and spatially resolved transduction of light into work from the nanoscale to the macroscale. The deformation in such single-crystal actuators is observed and usually attributed to anisotropy in their structure induced by the external stimulus. Yet, the actual intrinsic and external factors that affect the mechanical response remain poorly understood, and the lack of rigorous models stands as the main impediment towards benchmarking of these materials against each other and with much better developed soft actuators based on polymers, liquid crystals and elastomers. Here, experimental approaches for precise measurement of macroscopic strain in a single crystal bent by means of a solid-state transformation induced by light are developed and used to extract the related temperature-dependent kinetic parameters. The experimental results are compared against an overarching mathematical model based on the combined consideration of light transport, chemical transformation and elastic deformation that does not require fitting of any empirical information. It is demonstrated that for a thermally reversible photoreactive bending crystal, the kinetic constants of the forward (photochemical) reaction and the reverse (thermal) reaction, as well as their temperature dependence, can be extracted with high accuracy. The improved kinematic model of crystal bending takes into account the feedback effect, which is often neglected but becomes increasingly important at the late stages of the photochemical reaction in a single crystal. The results provide the most rigorous and exact mathematical description of photoinduced bending of a single crystal to date.

The phenomenological approach to chemomechanical effects in crystals, however, has not been paralleled with quantification and modelling of these effects. Deeper understanding of the related phenomena is lacking. Deformation of molecular crystals, for instance, is often reported as mere observation of bending or twisting of certain slender crystals when they are exposed to light or heat, and only minimal explanation on the non-uniformity of the product generated in the lattice of the reactant is provided as the root-cause for the deformation. In an attempt to describe the reasons for deformation of single crystals induced by external stimuli such as light-induced bending or twisting, several reports have attempted to assess the effect on deformation of parameters such as light wavelength/intensity or crystal thickness by using mathematical models.^{1,36–44} These models rely upon basic engineering principles that describe the deformation of bending beams, and range from very simple and convenient, yet inexact descriptions based on bending of bilayer beams,^{39–42} to more involved and comprehensive mathematical models and calculations.^{1,36,38,43,44} Yet, these mathematical treatments are often partial; they usually take into consideration only the effect of a particular effector and oftentimes their verification involves empirically fitted parameters to reproduce observed trends in experimental data. A universal model is required that accounts for the effects of all intrinsic (to the crystal) and external (experimental conditions) factors, and would be applicable to a broad range of crystals regardless of the underlying solid-state chemistry. Although being of immediate relevance to the assessment of the actuation performance of single crystals, such an overarching model is currently unavailable and is yet to be established. The assessment of the applicability and exactness of such a holistic mathematical model requires comprehensive and systematic kinematic data collected under controllable and systematically varied conditions on a single reaction system.

It has been already demonstrated that the dependence of the macroscopic strain that develops in the interior of the dynamic crystals on the extent of transformation can be described quantitatively.^{45} Measurements of the temperature-dependent crystal strain can further provide the thermal variation of the kinetic parameters. The kinetics is not only central to understanding of the fundamental mechanisms of the solid-state transformations, but is also the key to achieving an external control over the crystal deformation. This article describes approaches to precise measurement of the macroscopic strain in a single crystal that undergoes a solid-state transformation which can be used to determine the exact kinetic parameters. It is demonstrated—unlike the previously proposed approximate semiempirical approaches that use fitted parameters and do not explicitly account for the thermal variation of the kinetics—that the kinetic constants and their temperature dependence of the forward (photochemical) and the reverse (thermally induced) reactions can be extracted with high accuracy. The possibility to detect and quantitatively characterize the effect of the reaction-induced mechanical effects on the kinetic constants of the reaction (the so-called “feedback effect”), which becomes particularly important at high reaction yields, is also demonstrated. The rigorous mathematical treatment of bending of a single crystal reported here provides the most comprehensive theoretical model to date and scrutinizes the key factors which affect the photomechanical bending of crystals. The determination of the kinematic parameters is essential for comparative benchmarking of the currently available and newly reported materials that are being considered as crystalline microactuators for controlled and efficient conversion of thermal, light and mechanical energy into useful work.

(a) The reaction must be homogeneous—it should proceed by formation of a series of true solid solutions without phase separation between the reactant and the product;

(b) For maximal mechanical response, the habit of the crystal should be needle-like or an elongated plate (the largest cross-sectional size should be less than 1:10 of length) which provides conditions for large and easily measured macroscopic deformation like crystal bending;^{1}

(c) For simplicity of the mathematical treatment, the motion should be sole bending, and should be devoid of torsion (twisting). This simple deformation can be accomplished with needle-like crystals, where pure elastic strain (bending, if the crystal is irradiated from only one side, or elongation, if it is irradiated uniformly from both sides) can be induced, provided that specific symmetry requirements for its mechanical properties and the deformation are fulfilled.^{46,47} Particularly, it is necessary that neither the transformation nor the lengthwise strain cause shear deformation within the plane of irradiation. This is fulfilled when the longest axis of the needle-like crystal is aligned with one of the principal axes of strain caused by the transformation and the plane of the needle's normal cross-section is also the symmetry plane of mechanical properties of both the reactant and the product;

(d) To avoid preferred accumulation of the product on the crystal surface and to effectively translate the strain into bending, the ratio between the crystal thickness and the light penetration depth should be ≤10 (Note 1 in the ESI†). The extinction coefficient should be low in the spectral range where the reaction is initiated. Values > 10^{3} L mol^{−1} cm^{−1} result in localized transformation in a sub-micrometer-thick layer, and thus very thin crystals (<10 μm) of such samples would be required that are difficult to manipulate in the experiments.

(e) To perform a series of measurements using the same crystal, the reaction should be reversible, either thermally or by inducing a reverse photoreaction with light of different wavelength. By eliminating the data scatter that would be introduced by using different crystals related to defects and other crystal-dependent stochastic factors, this effectively improves the overall data precision over multiple experiments.

It is noteworthy that the above requirements should not be considered as strictly mandatory. The model described here is based on a theoretical framework that is generally applicable to phenomena related to photoinduced mechanical response; it could be additionally adapted or expanded to account for particular features of processes in specific cases. Nevertheless, compliance with the above requirements would ensure conditions to accomplish optimal accuracy.

Our choice of a solid-state photochemical reaction that conveniently meets all of the above requirements is the reversible linkage isomerization in the needle-shaped crystals of nitropentaamminecobalt(III) chloride nitrate, [Co(NH_{3})_{5}NO_{2}]Cl(NO_{3}) (1-N). Upon exposure to blue light, the nitro group in the “nitro form” of this compound switches its coordination atom from nitrogen to oxygen to afford the linkage isomer, the “nitrito form” [Co(NH_{3})_{5}ONO]Cl(NO_{3}) (1-O), 1-N → 1-O (Chart 1). The reverse reaction, 1-N ← 1-O, can occur by heating^{48,49} (ESI Note 2†). Under specific crystallization conditions, crystals of 1-N can be grown as needles with the main axis along the [010] (≡b) direction, and with (201) and (20) as prominent axial faces. At the local absorption maximum of ∼460 nm, the molar extinction coefficient of a solution of 1-N in water is ε ≈ 100 L mol^{−1} cm^{−1},^{48,50,51} which when extrapolated to “concentration” of complex cations in the crystal phase corresponds to characteristic light penetration depth x_{0} ≈ 6.5 μm (ESI Note 3†). According to earlier results described in ref. 1, the most prominent mechanical response can be accomplished with crystals of thickness that exceeds x_{0} at most several times. This consideration brings a practical convenience with the possibility to use crystals with thickness of several tens of microns that can be easily manipulated by hand. An earlier study^{52,53} has established that due to uncompensated internal stress irradiated crystals of 1-N thicker than 40–50 μm are readily disintegrated,^{54} and thus ∼40 μm was set as the upper limit for the thickness of the samples in the experiments where uniform irradiation of the crystal from both sides was used to determine the crystal expansion. The crystals used in the experiments with irradiation from one side to determine the degree of crystal bending were thinner than ∼30 μm.

2.2.1. Reverse thermal isomerization after prior photoisomerization on uniform irradiation from both sides.
A crystal of 1-N affixed to a glass plate was uniformly irradiated from both sides at −20 °C for complete transformation to 1-O by the photochemical reaction 1-N → 1-O. The transformed crystal was transferred into a thermal camera maintained at a selected temperature (Fig. 1a). The crystal elongation induced by the photoreaction and subsequent shrinking induced by heating at different temperatures due to the reverse thermal reaction 1-N ← 1-O was followed by optical microscopy. This method provided the temperature-dependent rate constant (k_{th}) and activation energy (E_{th}) of the reverse thermal isomerization. Six different crystals from the same crystallization batch, with lengths ranging between 446 and 1700 μm and thickness between 20 and 30 μm were studied in one or two irradiation-heating cycles. These experiments were performed to assess the reproducibility of the results under conditions of identical temperature for the same crystal, identical temperature for different crystals, different temperatures for the same crystal and different temperatures for different crystals (for the actual combinations between the recorded crystal samples and temperatures, see ESI Table S1†).

2.2.2. Phototransformation and reverse thermal isomerization on uniform irradiation from one side.
The crystal was irradiated from one side (one of the long crystal faces was exposed uniformly) to induce bending, and the dynamics of bending during continuous irradiation at various temperatures was determined to calculate the temperature-dependent rate constants k_{ph} of the forward reaction, 1-N → 1-O. Alternatively, to obtain the constants of the reverse thermal isomerization k_{th} and the corresponding activation energies, the kinetics of straightening of the crystal due to reverse thermal isomerization was monitored from a photochemically bent crystal. With the goals of the current study requiring the highest precision that could be attained, the same single crystal was used to measure the macroscopic strain (bending) in multiple irradiation-heating cycles in the experiments using the one-side irradiation protocol. We report here only the results obtained for a selected crystal; for a comparison, the data obtained for another crystal are also provided in the ESI (Fig. S5 and S6†).

Single crystals from the same crystallization batch were also used to determine the crystal structure by using single crystal X-ray diffraction before, during and after irradiation—conditions that correspond to pure 1-N, solid solutions with varying ratio of the two isomers, and pure 1-O, respectively. The variation in unit cell parameters of pure 1-N and 1-O within the temperature range 175–300 K was also studied. Blue LED (λ = 465 nm) was used for excitation (further details are available in the Experimental section and in the ESI†).

(1) |

2.4.1. Thermal isomerization after irradiation from both sides.
At complete transformation of 1-N to 1-O, the elongation of the crystal along its longest axis on uniform irradiation at ambient temperature was always close to 3.4%, in good agreement with the change in the lattice parameter b (Section 2.3). On subsequent heating, as the thermal reverse reaction 1-N ← 1-O occurs, the change ΔL = L − L_{N} in the crystal length L, relative to the crystal length L_{N} after complete thermal isomerization of 1-O to 1-N (Fig. 3a) follows the exponential function of time

where ε_{init} is the initial elongation of the crystal caused by the early uniform photoisomerization to 1-O. The crystal strain ΔL/L_{N} is linearly related to C_{ONO} (see Section 2.3). Therefore, the time-dependence of ΔL/L_{N} given by eqn (2) means that the thermal isomerization 1-N ← 1-O is a homogeneous monomolecular reaction described by the first-order kinetic law:

(2) |

(3) |

Fig. 3 Variation of the crystal length with time and of the rate constant with temperature for the thermal isomerization reaction 1-N ← 1-O. (a) Change of the relative crystal length ΔL/L_{N}, with time. (b) Temperature dependence of the rate constant (k_{th}) calculated from the data in panel a to eqn (2). |

Application of the Arrhenius equation to the temperature dependence of the rate constant k_{th} (Fig. 3b) gives activation energy E_{th} = 96 ± 2 kJ mol^{−1} and logarithm of the pre-exponential factor ln(k^{0}_{th}) = 31.8 ± 0.8 (k^{0}_{th} = 7 × 10^{13} s^{−1}). These values are consistent with the values ΔH_{a} = 81–110 kJ mol^{−1} and k^{0}_{th} = 10^{10}–10^{16} s^{−1} obtained by using infrared spectroscopy,^{58–61} UV-VIS spectroscopy^{48,62–65} and differential scanning calorimetry^{66} for similar complexes [Co(NH_{3})_{5}ONO]XY, where X and Y are anions (F^{−}, Cl^{−}, Br^{−}, I^{−}, NO_{3}^{−}) (ESI Note 4†).

2.4.2. Thermal isomerization after irradiation from one side.
The kinetics of the thermal isomerization 1-N ← 1-O was also monitored after the 1-O isomer was partially populated by exposing a crystal of 1-N to light from one side, using the setup shown in Fig. 1b. Under such conditions, the forward photoreaction 1-N → 1-O results in bending of the crystal.^{53} The isomerization that corresponds to maximal deflection of the crystal is incomplete, and the concentration of the product (1-O) decreases from the exposed surface toward the interior of the crystal due to finite light absorption. Since the relation between the lattice parameter b and the transformation extent is linear (see Section 2.3), according to the model described in ref. 1 the curvature of such crystal with non-uniform distribution of the product across its thickness is described as

where R is the curvature radius, h is the crystal thickness, and C_{ONO}(x) is the distribution of the concentration of the product (1-O) in the reactant (1-N) normal to the irradiated surface and across the depth through the crystal, x.

(4) |

Provided that by subsequent heating of the irradiated crystal in dark the concentration of 1-O decreases with time as described with eqn (3), the corresponding change in crystal curvature follows the rate law

(5) |

The kinetics of straightening of the crystal is shown in Fig. 4a (for details, see the Experimental section). The time-dependence of the curvature was fitted well by exponential decay functions (ESI Note 5†), and thus the kinetics of the thermal isomerization 1-N ← 1-O can be safely modelled by a first order rate law. Any dependence of k_{th} on the transformation extent could not be detected.

The two different methods to measure the thermal dependence of the kinetics in Fig. 1 provided consistent kinetic constants (ESI Table S4†), activation energy E_{th} = 98 ± 1 kJ mol^{−1} and logarithm of the pre-exponential factor ln(k^{0}_{th}) = 28.3 ± 0.3 (k^{0}_{th} = 3 × 10^{12} s^{−1}) (Fig. 4b). These results confirmed that the kinetics of the homogeneous solid-state transformation can be reliably and accurately monitored by measuring the changes in crystal length and/or curvature.

The equilibrium established between the photoinduced and thermal transformations results in a stationary spatial gradient of the 1-O isomer in the crystal bulk that decreases from the irradiated surface in the direction normal to the surface. For the limiting case of a thin crystal (μh ≪ 1), the model described in ref. 1 predicts the stationary curvature R^{−1} as

(6) |

The photoreaction rate constant at the irradiated surface is k_{ph} = I_{0}αμv_{0}, where I_{0} is the normal incident photon flux density, α is the quantum yield of the reaction, and v_{0} is the reciprocal concentration of the photoreacting species (volume per formula unit). This equation predicts a symmetric bell-like shape of the dependence of the curvature on ln(k_{th}/k_{ph}), with a maximum at k_{th} = k_{ph} (ESI Note 6†).

The temperature dependence of the stationary curvature of the crystal is shown in Fig. 5 along with a fit based on an Arrhenius-type temperature dependence:

(7) |

Fig. 5 Temperature dependence of the stationary curvature 1/R fitted with eqn (6) for Arrhenius-type variation of k_{th}/k_{ph} with temperature according to eqn (7). |

The effective activation energy of the reaction rate ratio, E_{eff} = 97 ± 1.5 kJ mol^{−1}, is very close to E_{th}, the activation enthalpy of k_{th}. This result implies that the photoreaction constant k_{ph} is less affected by the temperature and the transformation extent relative to the thermal rate constant k_{th}, in line with the conclusions in ref. 57. For the light intensity used in this study, the rates of the two reactions (forward and reverse) become identical at T_{0} = 335.4 ± 0.2 K. The effective photoreaction rate constant k_{ph} depends on the light intensity (proportional to I_{0}, as noted above), and due to the dependence of the quantum yield it also depends, to a lower extent, on the temperature. Using the determined temperature dependence of k_{th} and considering that in the stationary state k_{th} = k_{ph}, the value of k_{ph} at this temperature and the given intensity of the radiation source is estimated to ∼1.7 × 10^{−3} s^{−1}. Under the given conditions of temperature and irradiation intensity, this value corresponds to half-life of the nitro isomer 1-N close to the surface of the crystal of about 400 s.

2.7.1. Change of transformation extent with time: average gradient and average value.
The studies of the kinematics of photoinduced bending of crystals are traditionally limited by an analysis of crystal curvature alone.^{39,45} Another independent parameter that characterizes the mechanical response is crystal expansion or contraction along its longest axis but this parameter is usually not considered. The quality of analysis can be enhanced by combining the measurement of the two parameters characterizing the crystal deformation, i.e. the curvature and the length of the crystal (for details of the procedure see Section 4.2).

The relative change in crystal length is defined by the average strain ε_{b} across the crystal thickness:

(8) |

Analysis of the temporal variation of curvature and length using eqn (4) and (8) provides information on two independent characteristics of the distribution of the transformation degree C_{ONO}(x). The crystal elongation in eqn (8) is proportional to the average concentration of 1-O across the crystal thickness. The curvature in eqn (4) is proportional to the bending moment caused by non-uniformity of C_{ONO}(x). A linear approximation such as C_{ONO}(x) ≈ C_{0} + x(dC/dx)_{0} can be used to demonstrate the qualitative meaning of the integral in eqn (4); the curvature then is 1/R ≈ −ε_{b0}(dC/dx)_{0}, that is, it is approximately proportional to the gradient of C_{ONO}(x) over the crystal depth. Thus, the temporal dependence of the crystal elongation and curvature carry information on the average transformation degree and gradient of C_{ONO}(x) as they change over time.

Fig. 6 compares the changes in crystal curvature and length over time at different temperatures during continuous irradiation. As it can be inferred from there, while the crystal expands until it reaches a plateau (Fig. 6b), the curvature reaches a maximum and decreases afterwards (Fig. 6a). This change reflects the trend in transformation—while the average transformation extent grows monotonically on irradiation, the absolute value of its gradient initially increases and subsequently decreases. The final curvature corresponds to the stationary state described above. As shown in Fig. 6a, while below ambient temperature the crystal straightens almost completely on prolonged irradiation, at higher temperatures it remains bent. The peak in curvature decreases with temperature and vanishes completely above 350 K. The bending kinetics at 333 K corresponds to conditions close to the maximum stationary bending (Fig. 5).

Fig. 6 Time- and temperature-dependence of the crystal curvature, R^{−1} (a) and the relative elongation, ΔL/L_{N} (b) under continuous irradiation. In the experiments performed above 326 K, the light was switched off once the stationary state had been reached to record the changes caused by the reverse (thermal) reaction (hence, sharp drops are seen on the respective curves). All experimental data (solid line) and calculations based on the model described by eqn (4), (8) and (11) (broken lines) shown here refer to only one single crystal, in the main text referred to as crystal 1 (similar data for a second crystal, crystal 2, are available as Fig. S5 from the ESI†). The inset in panel a shows a zoomed image of the plot of the curvature at short irradiation times. |

The behavior of the crystals at and below ambient temperature is in agreement with the results from the diffraction experiments which confirmed that at these temperatures, 1-N is completely transformed to 1-O. At T < 300 K, the reaction can be considered pure photoisomerization, and the reverse (thermal) reaction can be neglected. The maximum elongation of the crystal increases as the temperature decreases (Fig. 6b), in accord with the results in Fig. 2b. The initial rates of the change of crystal length and curvature decrease with decreasing temperature (Fig. 6a), indicating that the rate of photoisomerization decreases. This result is in a qualitative agreement with the dependence of the quantum yield on temperature.^{57}

The preceding analysis is rather qualitative and does not provide information on the actual dependence of the absorption coefficient μ on temperature or the concentration C_{ONO}. The rate of change of crystal length and curvature (characterized by the constant k_{ph} ∼ μα), and the maximum curvature R^{−1}_{max} ∼ με_{b0} are proportional to μ. However, they exhibit opposite dependence on the temperature, in agreement with the temperature variation of α and ε_{b0}. This result warrants quantitative analysis of the dynamics of mechanical response with transformation.

2.7.2. A general kinetic reaction model.
The kinetic model proposed in ref. 1 is based on the equation

that describes the simultaneous occurrence of two opposite monomolecular reactions. The rate of transformation at distance x′ from the surface of the crystal is proportional to the local intensity of the penetrating radiation:

(9) |

(10) |

Generally, the absorption coefficient μ can vary with the depth because the absorption depends on the extent of transformation (the reactant and product generally have different absorption coefficients). The kinetics of transformation is fully accounted for by this model through three parameters (k_{ph}, k_{th} and μ) and by their dependence on the transformation extent. The strain ε_{b0} relates the kinetics of transformation to the observed mechanical response (eqn (4) and (8)). Thus, only four parameters are necessary for quantitative description of the dynamics of the bending induced by one-side irradiation. As shown above, of these four parameters, k_{th} and ε_{b0}, depend only on temperature and can be determined experimentally by using other experimental methods. As the photoreaction constant is defined as k_{ph} = I_{0}αμv_{0} and the values of v_{0} and I_{0} are available independently, the most essential parameters that need to be determined are the quantum yield of the photoreaction α and the absorption coefficient μ, together with their dependencies on the temperature and on the conversion extent.

The dependence of μ on the extent of transformation is mainly determined by the different propensity of the two isomers for light absorption. Other contributions could arise from the different propensity for light absorption of different chromophores (different [Co(NH_{3})_{5}NO_{2}]^{2+} cations in the case of the reaction studied here) in the lattice due to their orientation relative to the direction of the incident light. The quantum yield α depends on the lattice strain, as has been previously demonstrated by the effect of external mechanical load^{52} and thermal strain^{57} on the photoisomerization rate. As the progressing transformation also causes strains inside the crystal, it is anticipated that α is also affected by the transformation degree.

The inclusion of both thermal and compositional variations of μ and α in the model adds significantly to the complexity of the mathematical description. In case when μ and α depend only on temperature and are independent of C_{ONO}, the differential equation given by eqn (9) can be solved by analytical integration,^{1} which greatly simplifies the subsequent analysis. However, when μ and α are also affected by the composition, the system of nonlinear equations (eqn (9) and (10)) can only be solved numerically to account for the effect of C_{ONO} (this situation is equivalent with the inclusion of the feedback effect, as described above). In order to assess the relevance of these additional contributions, in what follows, a simplified model where μ and α depend only on temperature is considered first. This simplified model is then followed by a more elaborate analysis that takes into account the variation of the kinetic parameters with the transformation extent.

2.7.3. Quantitative analysis of the kinetics of transformation (μ and α depend only on temperature).
Assuming that μ and α depend only on temperature, the solution to eqn (9) is

(11) |

The experimental data on crystal curvature and elongation upon photoreaction at different temperatures fitted with eqn (4), (8) and (11) to obtain temperature dependencies of ε_{b}_{0}, μ, and k_{ph} are shown in Fig. 6 (the values of k_{th} were taken from the results obtained in Section 2.4). As it can be inferred from there, although this simplified treatment provides reasonable quantitative description of the experimental data, there is discrepancy of about 10% in the values of both the curvature and elongation (note the difference between the solid and the dashed lines on both panels in Fig. 6). This discrepancy indicates that the neglecting the dependence of μ and α on C_{ONO} is an oversimplification, and calls for a more sophisticated model that would relate these two parameters with the product concentration. Assuming that the volume per one complex cation is v_{0} = 0.26 nm^{3} based on crystallographic data, and approximating the incident photon flux density to I_{0} = 1.5 × 10^{17} s^{−1} cm^{−2} (465 nm, photon energy of 2.67 eV), the quantum yield of photoisomerization estimated from the temperature dependence of the fit is α = k_{ph}/I_{0}μv_{0} and ranges between 6% and 16% at temperatures from 190 K to 357 K.

The temperature dependence of ε_{b0}, μ^{−1} and α obtained from the fit is shown in Fig. 7. The values of the maximum deformation ε_{b0} are in agreement with the experimentally determined structure, although the deformation predicted by analysis of the macroscopic strain is systematically underestimated at low temperatures (Fig. 6b), and the trend is different at T > 300 K. The temperature trend of the quantum yield corresponds to that obtained in a previous study.^{57} According to ref. 51 and 67 the total quantum yield of photodecomposition and photoisomerization of the complex in water or 50% ethanol in water mixture is 0.15 for excitation around 465 nm. This value is a reasonable estimate for the total quantum yield in the solid state, because both processes ostensibly are initiated from the same excited state, and the decomposition in the solid state is negligible.^{68} Thus, this estimation of the photoisomerization quantum yield from the macroscopic strain is reasonably close to the experimental values obtained by measurements in solution.

Fig. 7 Temperature dependence of the maximum deformation ε_{b0} (a), characteristic penetration depth (reciprocal of the absorption coefficient μ) (b), and quantum yield α (c) obtained using the simple kinetic model given by eqn (11) that accounts only for the dependence on the temperature and disregards the variation with the composition. |

The temperature dependence of the quantum yield α and the absorption coefficient μ (Fig. 7) can now be rationalized. The increase of α with temperature can be the consequence of the lattice thermal expansion.^{57} Additional contribution to the high value of α at high temperatures could come from the decrease in the overall isomerization extent (ESI Note 7†). The stronger absorption μ at higher temperature can also be a consequence of the decreased average reaction extent. As shown in Fig. 7b, the temperature dependence of the penetration depth has two characteristic regimes separated around 330 K. This temperature is close to the point at which the rates of the forward (photochemical) and reverse (thermal) reactions become equal. At lower temperatures, the light is mainly absorbed by the 1-O form while at higher temperatures it is mainly absorbed by the initial 1-N form. Therefore, the temperature dependence of the absorption coefficient can be a consequence of its variation with composition. This conclusion is confirmed by the lower absorption of the 1-O relative to the 1-N form around 465 nm.^{51}

2.7.4. Improved model for the dynamics of photoinduced bending.
With the extensive experimental results at hand, the model proposed several years ago^{1} can now be improved to describe quantitatively the macroscopic deformation of crystals upon photoinduced and thermally induced transformation at variable temperature. The major improvement is the introduction of the dependence of quantum yield α and absorption coefficient μ on temperature and transformation extent. In the improved model we assume that the quantum yield depends on temperature and transformation that has been already initiated in the crystal:

and the kinetic equation is

where k_{ph} = I_{0}α_{0}(T)μ_{0}v_{0}.

where q_{T} = 55 ± 5. This dependence can be traced back to the dependence of the quantum yield on the thermal strain, as described in ref. 57. The dependence of α on the degree of transformation (eqn (12) and (14)) can be represented in a similar form

that corresponds to the effect of strain along axis a which results from the transformation and equals to ε_{a0}C_{ONO}. The value of the coefficient q_{a}/ε_{a0}, 32 ± 3, is comparable to that of q_{T}. The difference in the values of q_{T} and q_{a}/ε_{a0} may be due to the difference in anisotropy of the deformation tensors corresponding to strain caused by thermal expansion and by the photoisomerization.^{57} This value of q_{a} indicates that the quantum yield of photoisomerization decreases approximately twice by the time the reaction is completed as compared to its onset.

α = α_{0}(T)f_{α}(C_{ONO}, ε_{r}) | (12) |

The factors in eqn (12) approximate the environmental effect that the deformation of the crystal structure imposes on the photoreaction.

The first term, the coefficient α_{0}(T), is uniform through the crystal bulk and reflects the dependence of α in the initial (non-reacted) crystal on temperature due to thermal expansion (it was shown earlier^{57} that in the linkage isomerization reaction studied here, α is linearly related to the change of the unit parameter a with temperature). The second term, f_{α}, on the other hand, varies across the crystal from the surface to the bulk. It reflects the dependence of α on the local structure deformation caused by the local transformation extent C_{ONO}(x) and by the residual strain ε_{r}(x) of the crystal along its main axis (b) related to nonlinearity in C_{ONO}(x). The dependence of f_{α} on the local transformation degree is qualitatively analogous to the effect of thermal strains described above, but varies across the crystal. The dependence on ε_{r} reflects the experimental observation of the effect of external load on the photoisomerization rate.^{52} The residual strain is calculated from the elastostatic solution to the problem of beam bending due to non-uniform distribution of the transformation extent through the crystal bulk:

(13) |

The factor f_{α} defines the so-called “feedback effect” of the solid state reaction.^{69–73} To facilitate the computations, it is convenient to represent f_{α} as

f_{α} = exp(q_{a}C_{ONO} + q_{b}ε_{r}) | (14) |

This function approximates the linear dependence of the quantum yield on the strain, provided the arguments of the exponential function are small.^{57} The feedback coefficient q_{a} can be treated as a fitting parameter of the improved model, while the value of the coefficient q_{b} is taken to be 40 from the direct measurement in ref. 45. The residual deformation ε_{r} does not exceed 0.2ε_{b0} for crystals that are not thicker than 5μ^{−1}. Thus, the second term in the exponent for crystals used in the present study is not higher than 0.3.

Probably the simplest way to account for the dependence of the absorption upon the transformation extent is to introduce two absorption coefficients for different isomers, μ_{0} for 1-N and μ_{1} for 1-O. Then, the dependence of the local photon flux density on the depth x is

(15) |

(16) |

Fig. 8 compares the trends calculated by using the advanced kinetic model (eqn (15) and (16)) and the experimental data. It is apparent that the advanced model provides improved description of the experimental data, and indeed the relative residuals do not exceed 2%. Assuming that the absorption coefficient μ_{1} and the feedback coefficient q_{a} are common fitting parameters for all curves, the fitting procedure results in values for the characteristic penetration depth and feedback coefficient μ_{1} = 217 ± 23 cm^{−1} (penetration depth μ_{1}^{−1} = 46 ± 5 μm) (ESI Note 8†) and q_{a} = −0.78 ± 0.05, respectively.

Fig. 8 Comparison of the experimentally measured crystal curvature (a) and linear strain (b) with simulations based on the advanced model (eqn (15) and (16)). Note the significant improvement in the fits compared to Fig. 6. |

The feedback coefficient q_{a} is related to the effect of lattice strain on the reaction course caused by isomerization. This effect can be compared with the effect of thermal strain on the photoisomerization rate.^{57} The photoisomerization reaction rate and the thermal deformation ε_{aT}(T) of the lattice along axis a are related as^{57}

k_{ph}(T) = k_{ph}(363 K)[1 + q_{T}(ε_{aT}(T) − ε_{aT}(363))] | (17) |

(18) |

The fitted temperature dependence of the strongest deformation (ε_{b0}), the characteristic penetration depth μ_{0}^{−1}, and the quantum yield for two different crystals are compared in Fig. 9. The values of ε_{b0} for the two different crystals are consistent with each other (Fig. 9a). Moreover, their deviation from the results from X-ray diffraction experiments does not exceed 0.1% across the whole temperature range. The values of the absorption coefficient μ_{0} are also similar (Fig. 9b), and are more consistent relative to μ predicted using a less sophisticated model that neglects the difference in the absorption of the two isomers (Fig. 7b). The variation of μ_{0} with temperature confirms that the nitro form 1-N absorbs light stronger than the nitrito form 1-O (seen in the initial stages of the forward reaction 1-N → 1-O at T > 330 K).

Fig. 9 Temperature dependence of the strongest deformation (a), characteristic penetration depth (b), and quantum yield (c) obtained by fitting the experimental data for two different crystals to the curves simulated with the kinetic model given by eqn (15) and (16). For comparison, data on ε_{b0} obtained from single crystal X-ray diffraction experiment are also plotted in panel (a). |

The temperature dependence of the quantum yield estimated using the advanced model (Fig. 9c) qualitatively resembles that obtained using the simplified model (Fig. 7c), although the quantitative differences are apparent. The thermal variation of the quantum yield can be directly compared to that described by eqn (17). The corresponding correlation coefficients q_{T} are slightly different when estimated for different crystals. The values for the two crystals for which the data are plotted in Fig. 9, 65 ± 4 and 108 ± 5, are comparable to the value 55 ± 5 reported in ref. 57. The discrepancy between different crystals may be a consequence of crystal imperfections, as it is normally anticipated for real solids. This difference could also be related to small variations in the irradiation conditions between different experiments (see ESI Note 9†). Assuming identical quantum yields but different photoisomerization rates (due to different light intensity) for different crystals, the experimental data for different crystals imply nearly 1.7 times difference in light intensity (compare with the estimated factor of 2 in the ESI Note 9†), giving an average value of q_{T} = 70 ± 7.

The preceding analysis confirms that all parameters and their temperature dependence are reasonably modeled by analysis of the temperature-dependent macroscopic strain using the advanced model, and can be assessed against complementary experimental data. The difference in the values of α_{0} obtained from different crystals in particular illustrates the importance of using the same crystal in all measurements. As outlined above, this is possible only if the photoreaction is reversible and the crystal is capable of multiple photothermal reaction cycles.

2.7.5. The effect of chromophore orientation with respect to light.
The fits can be additionally improved by taking into account the difference in light absorption between the different chromophores in the crystal (in case of the reaction studied here, the complex cations). In the structure of 1-N, the complex cations [Co(NH_{3})_{5}NO_{2}]^{2+} are oriented in four different dispositions with respect to the crystallographic axes, and two pairs with different orientations are related by inversion. The inversion does not change the interaction of a molecule with light; thus, only two types of cations interact with light. The orientations of these two different species are related by reflection over a glide plane (glide a in the space group of the crystal structure) normal to the crystallographic direction c (normal to the long crystal axis and to axis a in crystal cross-section, Fig. 10).

where C′ and C′′ are the transformation degrees of the two different types of nitro isomers. The absorption by the two different types of nitro isomers can be represented as

where e is the direction of the polarization vector with the polarization angle θ_{p}, m′ and m′′ are the unit vectors along the directions of the transition dipole moments of these species, μ_{0} is the absorption coefficient of the nitro isomer oriented so that the transition dipole moment is lying along the wave polarization vector.

where C_{b} is the operator of rotation around the b axis at the angle between the a axis and the normal vector to (201) face, φ_{0} = 0.652, M_{c} is the operator of the mirror reflection normal to the c axis (symmetry connection between the two complex cations), m_{0} is the direction of the transition dipole moment of one of the cations in the crystal coordinate system. The kinetic equation given by eqn (16) transforms into the system of two equations for two types of the nitro cations

where the other coefficients are defined above.

If the irradiation normal to the crystal axis was directed along the a axis, there would not be any difference in the interaction of these two types of cations with light. However, the irradiated face of the crystal is (201) (or the symmetrically equivalent (20)) so that the orientations of the transition dipole moments of the two cations with respect to the light direction normal to the axial crystal face are different. The interaction of the two different types of complex cations [Co(NH_{3})_{5}NO_{2}]^{2+} can be described by two different absorption coefficients, μ′_{0} and μ′′_{0}. The initial absorption coefficient of the crystal is defined by the average, (μ′_{0} + μ′′_{0})/2. On irradiation, the cations having larger absorption coefficient are transformed into the nitrito isomer faster. This results in effective decrease in the total absorption by the remaining nitro isomers. This effect can explain the dependence of μ_{0} on the temperature (Fig. 9b; see also ESI Note 10†).

The light source used in the experiments was unpolarized, and can be considered a combination of uniformly distributed linearly polarized waves. The angular density of these linearly polarized waves is i_{0}(θ_{p}) = I_{0}/2π. As a result of light absorption and 1-N → 1-O phototransformation, the initial uniform angular density will change to the time-dependent non-uniform distribution i(θ_{p},x,t) inside the crystal. The absorption eqn (15) then becomes

(19) |

(20) |

Two terms in the eqn (20) are related to the fact that the complex cation is located at the mirror plane normal to the long crystal axis (axis b). M_{b} is the corresponding symmetry operator. The form of the eqn (20) assumes that the direction of incident light is strictly normal to the long crystal axis (axis b). In general case, if a crystal is inclined with respect to light, corresponding rotation operators should be applied to both transition dipole moments (ESI Note 11†). The components of m′ and m′′ are defined in the laboratory coordinate system in which the light is parallel to the x direction, the crystal axis is lying along y with the face (201) normal to the light beam. Thus, these vectors can be defined with the following equations

m′ = C_{b}(φ_{0})m_{0}m′′ = C_{b}(φ_{0})M_{c}m_{0} | (21) |

(22) |

Fitting experimental data using this model (eqn (19)–(22)) is computationally demanding. The results depend on the exact orientation of the crystal with respect to light, which may vary between different experiments and different crystals. As an illustration we present in Fig. 11 the result of single fitting for experiment carried at 190 K on one single crystal. The residuals of the fit are solely defined by the noise in the experimental data. The fitting parameters are ε_{b0} = 0.04127 ± 1 × 10^{−5}, q_{a} = −0.628 ± 0.005, α_{0} = 0.0394 ± 1 × 10^{−4}, μ_{0}^{−1} = 3.6 ± 0.2 μm, μ_{1}^{−1} = 34.6 ± 0.2 μm. The transition dipole moment direction in the crystal system is defined by a vector m = (−0.505, −0.77, 0.389) ± (0.13, 0.11, 0.13).

Fig. 11 Fitting the experimental curvature (a) and relative elongation (b) with functions derived from eqn (19)–(22). |

The values of the elongation ε_{b0}, the feedback coefficient q_{a}, the estimated quantum yield α_{0} and the averaged absorption coefficient for the 1-O isomer obtained using this model are close to the values obtained using eqn (15) and (16). The values of the characteristic depth of light absorption (averaged over all polarization planes in the incident light) for the two orientations of the complex cations are 〈μ′_{0}〉_{θp}^{−1} = 7.5 ± 0.5 μm and 〈μ′′_{0}〉_{θp}^{−1} = 12.3 ± 0.8 μm. This result means that on average the cations in a more favorable orientation must isomerize 65% faster. The average absorption of the unpolarized light by the reactant isomer 1-N in the crystal can be estimated from the obtained transition moment direction and the value of μ_{0} to give the characteristic penetration depth x_{0} = 2/〈μ′_{0} + μ′′_{0}〉_{θp} = 9.3 ± 0.6 μm. This value is close to all previous estimations of this parameter.

The above values can be used to explain the results in Fig. 9b. At T > 330 K, that is, at temperatures that favor low conversion 1-N → 1-O, the main contribution to the reaction comes from the cations with higher absorption coefficient (characteristic light penetration depth ∼ 7.5 μm). At T < 330 K, the reaction proceeds almost completely. Both types of cations with different orientations are involved so that the characteristic light absorption is close to the average of the values corresponding to each of the orientations (characteristic light penetration depth ∼ 9.3 μm). Another important parameter which can be calculated from the determined absorption parameters is the molar extinction coefficient corresponding to the absorption, averaged over all possible space orientations of the complex cation, as it appears in a solution. Its calculated value, μ_{0}N_{a}v_{0}/(3ln(10)) = 61 ± 3 L mol^{−1} cm^{−1} is in agreement with the reported value of absorption by the complex cation at the absorption maximum near 460 nm.^{48,50,51}

There are also restrictions on the kind of mechanical response which should not include deformation modes like twisting of crystals and should be restricted by pure uniform bending. The case of twisting can still be modelled.^{1} However, it introduces serious difficulties in experimental techniques (measurement of deformation characteristics of curvature, torsion and elongation or contraction) and in theoretical treatment (complicated change of the irradiation conditions in course of transformation). Whether the twist appears or not, is defined by certain symmetry features of the strain and strength tensors specified earlier.^{1} The optimal thickness of the crystals is comparable with that of the characteristic absorption depth corresponding to the wavelength of light inducing the phototransformation. For much thicker crystals (h/x_{0} > 10) relaxation of the mechanical stresses will occur in the surface layer of the irradiated crystal. In such case the reaction can be studied only up to a small extent of transformation (the experimental setup can be modified using reflection of the laser beam from a mirror fixed at the end of the crystal^{52}). The relatively low coefficient of light absorption (typically, the extinction coefficient less than ∼1000 L mol^{−1} cm^{−1}) is directly related to the restriction on the crystal thickness. Strong light absorption requires manipulation of very thin (submicron crystals), which is technically challenging. Finally, the model requires that the phototransformation is reversible and the crystal does not change its properties after multiple cycles of bending and straightening. This provides the possibility to use the same crystal in multiple measurements under a variety of conditions. This strategy increases the reliability and the reproducibility of measurements, and excludes the data scatter related to factors that pertain to the imperfections of individual real crystals.

In cases of not strict fulfilment of the limitations the quantification of crystal deformation with the model still can be used as an instrument for systematic estimation of main quantities defining the phenomenon of photoinduced bending like apparent transformation rate constants and light penetration depth which can be treated as a characteristic size of the transformation non-uniformity in the thickness of crystals. Moreover, the very fact of the experiment deviating from the model can indicate certain features of the process that may be not obvious from the first sight (such as the contribution of plastic deformation to elastic bending, the existence of shear components of stress, incomplete reversibility, etc.). It is also to note, that the conditions imposed on the phototransformation, in order to enable the best applicability of the model to describe the experimental observations, are the same that are required for the crystals undergoing this transformation to be used in a photomechanical device. Thus, testing the applicability of the model to the experimental behaviour of photobendable crystals can be used to evaluate the potential of these crystals as supramolecular actuators or sensors.

The stationary bending experiment (Section 2.5) was conducted as follows. With the irradiation switched on and the temperature set to a desired value we were waiting for the crystal to stop its deformation. Then we took a photograph of the crystal to measure the deformation reached. We changed the temperature, so that the crystal shape was changing for some time and waited until the deformation stopped, and a photograph was taken again. Thus, after repeating this procedure several times we obtained the dependence of stationary bending on temperature.

Every isothermal bending/unbending experiment (Section 2.7.1) was preceded with thermal annealing of crystals at 360 K for one hour in the darkness. Then the desired temperature was reached and stabilized during another hour. After this procedure the irradiation was switched on and the kinetics was measured. The bending/unbending process was registered as time-lapse photographs. The image series was analyzed (Fig. S1 in the ESI†) with a software developed as a specialized plugin for ImageJ^{74} (the plugin “Bending Crystal Track” is available at ImageJ website http://imagej.net/PhotoBend). The curvature and length change were calculated from coordinates of three points corresponding to two ends and the center of the crystal defined by the image template matching technique. Knowing the three values equal to the length of chords (H – the chord connecting the two ends of the crystal, H_{1} and H_{2} – the chords connecting the end of the crystal with its middle, Fig. S1 in the ESI†), the angle corresponding to the crystal arc is calculated:

(23) |

The value of this angle was used to calculate the crystal curvature

(24) |

L = Rα_{c} | (25) |

The complete set of measurements was carried out with two crystals with slightly different geometrical parameters. The data discussed in this paper correspond to crystals labeled “crystal 1” (706 μm long and 23 μm thick) and “crystal 2” (644 μm long and 20 μm thick). We show the experimental data for the crystal 1 in the main text and for the crystal 2 in ESI.†Fig. 9 shows the comparison of resulting parameters obtained for both crystals: ε and μ are the same, while quantum yields are different (presumably, because of different irradiation intensity because of different installation inside the camera, or different crystal imperfections, e.g. different scattering by the surface).

To obtain 1-O, a crystal of 1-N was irradiated by focused green LED light for 12 hours, followed by irradiation by focused LED blue light for 1 hour (experiment 2). Very low electron density peak corresponding to traces of 1-N were detected, however a reliable refinement of corresponding site occupancy factors was not possible and this component was not included in the refinement. Mixed crystals with intermediate concentrations of both isomers (experiments 3–5) were obtained by irradiation and data collection at ∼2 h intervals at 295 K. In the high-temperature experiments (experiments 6–9) the crystal was additionally annealed at 333 K for 2 minutes. Before experiment 10, the crystal was annealed at 343 K for 50 minutes to obtain 1-N. Before the experiment 11, 1-N was irradiated by focused blue LED light for 2 hours to obtain individual 1-O, and subsequently annealed for 7 minutes at 333 K before experiment 12. The details of the data collection and refinement details are summarized in Table S2.† Complete structural data were deposited in the CSD.^{78}

The temperature dependence of cell parameters of 1-N and 1-O was measured using STOE IPDS II diffractometer with an image-plate detector, MoKα radiation and an Oxford Cryostream cooling device. Diffraction data for cell refinement were collected at 300, 275, 250, 225, 200 and 175 K. Reflection measurements for 0.40 × 0.03 × 0.03 crystal of 1-O were performed during continuous irradiation by focused light of 3 W-blue LED. The same crystal was annealed for 1 h at 353 K to obtain 1-N isomer and used for further cell parameter measurements. The software X-AREA^{79} was used for data collection and cell refinement. The details on data collection and cell refinement are summarized in Table S3.†

In this work, we illustrate the applicability of this approach on a selected example of a solid-state reaction system that undergoes photoinduced reversible linkage isomerization [Co(NH_{3})_{5}NO_{2}]Cl(NO_{3}) ↔ [Co(NH_{3})_{5}ONO]Cl(NO_{3}). The model elaborated here, however, is general and within the assumptions made in respect to the compound, the crystal and the transformation (Section 2.11), it can be applied to a wide variety of other phototransformations in crystals. The method for analysis of the kinetics of phototransformation by measuring the macroscopic strain that develops during elastic bending of a needle-shaped crystal is practically self-sufficient and does require very little additional information. It provides that following parameters: (a) the rate constant of the thermal reaction k_{th} and its dependence on the temperature, (b) the rate constant of the photochemical reaction k_{ph} and the quantum yield α (for known light intensity), (c) the dependence of α on the temperature and transformation extent, (d) the light absorption coefficients for the reactant and the reaction product, and (e) the orientation of the vector of the dipole moment of the electronic transition in the crystal structure.

The only information required from complementary X-ray diffraction experiments is the dependence of lattice strain on the extent of transformation. In fact, the maximal strain achieved at complete transformation can be estimated from measuring the crystal deformation (without XRD experiments). In some selected cases, when strain is linearly related to the transformation degree, and there is a reverse thermal reaction described by the first-order kinetic law, the measurement of the kinetics of mechanical response may be sufficient even without the XRD. However, the XRD analysis gives the unambiguous information on the relation between macroscopic crystal deformation and the degree of photochemical transformation. In the present study, the discrepancy between the estimated lattice strain based solely on the measurements of the macroscopic crystal deformation and on precise single-crystal X-ray diffraction data does not exceed 0.1%. The uncertainty in the measurement of the strain is determined by two factors: the resolution of the analyzed images (the uncertainty with the method used here is about 1 pixel) and possible errors in the analysis of the crystal geometry because the ratio of the crystal thickness to its length is not infinitely small. Taking into account the two factors in the case described in this work gives an estimate in the error of the measured strain of about 0.1%. The method described here is the most comprehensive to date available method for kinetic and kinematic analysis of photobendable crystals, and is the only available means to obtain parameters that are specific for the interaction of light with the crystal and the photochemical reaction in the crystalline state. It provides information on both the average transformation degree and the spatial distribution of the reaction product through the crystal depth, in contrast with the other techniques that provide only average values.

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## Footnote |

† Electronic supplementary information (ESI) available. CCDC 1536477–1536488. For ESI and crystallographic data in CIF or other electronic format see DOI: 10.1039/c7sc04863g |

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