Photothermally induced natural vibration for versatile and high-speed actuation of crystals

The flourishing field of soft robotics requires versatile actuation methodology. Natural vibration is a physical phenomenon that can occur in any material. Here, we report high-speed bending of anisole crystals by natural vibration induced by the photothermal effect. Rod-shaped crystal cantilevers undergo small, fast repetitive bending (~0.2°) due to natural vibration accompanied by large photothermal bending (~1°) under ultraviolet light irradiation. The natural vibration is greatly amplified by resonance upon pulsed light irradiation at the natural frequency to realise high frequency (~700 Hz), large bending (~4°), and high energy conversion efficiency from light to mechanical energy. The natural vibration is induced by the thermal load generated by the temperature gradient in the crystal due to the photothermal effect. The bending behaviour is successfully simulated using finite element analysis. Any light-absorbing crystal can be actuated by photothermally induced natural vibration. This finding of versatile crystal actuation can lead to the development of soft robots with high-speed and high-efficient actuation capabilities.


Introduction
Natural vibration is a physical phenomenon during which a material continues to vibrate at a certain frequency (the natural frequency) upon application of an external force.Additionally, when a periodic force is applied near the natural frequency, the vibrational motion is ampli ed by resonance.For example, during an earthquake, if the natural frequency of a shaking building matches the frequency of the earthquake, the building will resonate and shake violently, which may lead to collapse of the structure. 1uipment and devices must be designed so that resonance does not occur, otherwise they may be damaged or become less durable. 2 Most musical instruments, such as the guitar and saxophone, create sound (natural vibration) via the movement of strings or reeds and amplify the sound via resonance in a box or tube. 3Thus, resonated natural vibrations are widely related to our daily life.
We have focused on crystal actuation because mechanically responsive molecular crystals have potential application in actuators and soft robots. 4Over the past decade, many mechanically responsive crystals have been developed that display bending, [5][6][7] twisting, 8 jumping 9 and agella-like motion. 10st of these motions have been induced via photoisomerisation.Phase transitions can also induce mechanical motions, such as expansion/contraction, 11 bending, 12 jumping 13 and locomotion. 146][17][18][19] However, photoisomerization and phase transitions are not versatile due to the limitation that they can only occur in certain crystals.Furthermore, photoisomerisation-induced motion tends to be slow and thick crystals do not actuate. 191][22][23] The photothermal effect is a phenomenon by which thermal energy (heat) is produced by the nonradiative deactivation of photoexcited materials; it occurs in any light-absorbing crystal.However, photothermally driven bending is small, with a bend angle typically less than 0.5°.One strategy to enhance photothermal bending is to focus on crystals with a large thermal expansion coe cient.
Herein, we report that b-phase crystal (1b) 24 of 2,4-dinitroanisole (1) (Fig. 1a), which has relatively large thermal expansion coe cient, exhibit small but fast natural vibration associated with larger photothermally driven bending during ultraviolet (UV) laser irradiation.More surprisingly, when irradiated with a pulsed UV laser of the same frequency as the natural frequency, the bend angle is considerably ampli ed by resonance, achieving high-speed, large bending, and high energy conversion e cient.
Simulation of this high-speed bending is also successful.Resonated natural vibrations can expand the potential and versatility of crystals as actuation materials.

Physical properties of the crystals
Colourless hexagonal rod-shaped 1b crystals were obtained by fast cooling of saturated methanol solution in a refrigerator at ~5°C (Fig. 1b).Differential scanning calorimetry (DSC) measurements con rmed that the obtained crystals were b-phase, exhibiting a reversible phase transition a ↔ b at -5.1°C and -8.0°C on heating and cooling, respectively (Supplementary Fig. 2), in close agreement with literature values. 24X-ray crystallographic analysis at 20°C further con rmed that the rod-shaped crystal belonged to the monoclinic crystal system and the space group P2 1 /n (Supplementary Table 1), consistent with the reported b-phase structure. 24The 1b molecules were aligned in a herringbone motif along the b-axis due to the two-fold helical axis and π-π stacking of 3.588 Å along the a-axis on the (001) plane (Fig. 1d).This resulted in the formation of the (010) plane as the top face and the longitudinal direction along the a-axis (Fig. 1c).The temperature dependence of the lattice constants indicated that the a-axis length increased linearly from 3.972 Å at 10°C to 4.021 Å at 60°C; the thermal expansion coe cient of the longitudinal direction along the a-axis was calculated as 247 MK -1 (Fig. 1e and Supplementary Fig. 1), which is several times larger than that (average value: 71.4 MK -1 ) of many molecular crystals. 25e ultraviolet-visible (UV-vis) diffuse re ectance spectrum of powdered 1b crystals measured at room temperature showed strong absorption in the UV region, with an absorption peak at 370 nm and a shoulder at 450 nm (Fig. 1f).

Natural vibration induced by the photothermal effect
When a hexagonal rod-shaped 1β crystal (length 6075 µm, width 151 µm, width of the top face 40 µm, thickness 105 µm; crystal III in Supplementary Table 2) xed at one end was irradiated from the top with the UV laser (375 nm, 1456 mW cm −2 ) (Fig. 2a), the crystal was signi cantly bent due to the photothermal effect.Surprisingly, smaller and faster repetitive bending accompanied this large photothermal bending both under UV irradiation and after irradiation ceased (Fig. 2d and Supplementary Movie 1).
Upon UV irradiation, the photothermally driven bending away from the light source quickly reached a bend angle of 0.85° in 9.4 ms, then gradually increased to 1.22° in 100 ms (Fig. 2c, d).This bend angle is much greater than the 0.2°-0.5°previously reported for other crystals 20 − 23 due to the larger thermal expansion coe cient (247 MK −1 ) of the 1β crystal compared with those (80-130 MK −1 ) reported for other crystals.After stopping the UV exposure at 100 ms, the crystal bent up quickly to 0.32° in 17 ms, then slowly returned to 0.23° in 100 ms.The photothermally driven bending data were tted with an exponential curve (Fig. 2e), giving time constants (τ on , τ off ) of 4.7 and 6.4 ms for UV light on and off conditions, respectively.The surface temperature, which was simultaneously monitored by an infrared (IR) thermography camera, increased from 25.6°C to 35.3°C upon 100 ms of UV irradiation (Fig. 2b) and decreased to 33.1°C in 100 ms after the UV irradiation was stopped (red curve in Fig. 2d).The maximum bend angle and the maximum surface temperature increased linearly with increasing UV intensity (Fig. 2h and Supplementary Fig. 7).
The smaller, faster repetitive bending was extracted by tting photothermal bending curve (Fig. 2e) to provide the vibration-like bending pro le (Fig. 2f).The small bend angle was ~ 0.20° initially, then gradually decreased to a steady angle of 0.02° at 80 ms.When the UV irradiation was stopped, the bend angle immediately recovered to 0.18°, then attenuated to a steady 0.04° in 20 ms.Fourier transform analyses of the time pro le of bending with and without UV irradiation revealed that the frequencies of the small vibration-like bending were both 390 Hz (Fig. 2g).The bend angle of the small vibration increased linearly with increasing UV intensity.In contrast, the frequency did not change and remained at 390 Hz for any UV laser intensity.This con rmed that the small vibration-like bending was the natural vibration (Fig. 2i and Supplementary Fig. 7).
For a hexagonal rod-shaped 1β crystal, the natural frequency f cal can be calculated according to Eq. ( 1): 1 where h is the thickness, l is the length, b is the width, c is the width of the top surface, E is the Young's modulus (1.65 GPa, Supplementary Fig. 6) along the length direction and ρ is the density (1.567 g cm −3 , Supplementary Table 1).The calculated natural frequency f cal of 397 Hz was in close agreement with the measured value (390 Hz).

Resonance ampli cation of a natural vibration
As mentioned above, the bend angle of the natural vibration was one order of magnitude smaller than that of the photothermally driven bending.Surprisingly, however, when crystal III was irradiated with pulsed UV light at the natural frequency of 390 Hz, the natural vibration was dramatically ampli ed by resonance (Fig. 3a, b and Supplementary Movie 2).The bend angle began at 0.20° in the rst cycle and increased with irradiation time to the maximum of 3.4° after 150 ms; the difference in angle between the maximum of 2.4° and the minimum of −1.0° represented a 17-fold resonance ampli cation.Then, the resonated bend angle gradually decreased as the surface temperature increased and reached the steady bend angle of 2.3° when the surface temperature reached the constant 40.8°C after 850 ms under pulsed UV light.Once the UV irradiation was stopped at 1190 ms, the bend angle rapidly decreased and the natural vibration almost disappeared within 200 ms.This resonated bending was observed for at least 460 cycles without any fatigue of the crystal under pulsed UV irradiation.
To determine the exact natural frequency of crystal III, we observed the bending behaviour while irradiating it with UV pulses of a wide range of frequencies; the natural frequency (the most ampli ed frequency) was 390 Hz (black open circle, Fig. 3c).In addition, the tted natural frequency f t determined by the forced vibration model (see the footnote in Supplementary Table 2) was also 390 Hz (red open circle, Fig. 3c).
Furthermore, bending ampli cation due to resonance was examined in detail using another crystal V with a higher natural frequency of 702 Hz (Supplementary Fig. 8 and Table 2).The resonated bend angle increased linearly with intensity of the 702 Hz pulsed UV light (Fig. 3d and Supplementary Fig. 9).Interestingly, when UV light pulsed over a wide range of 5-1000 Hz was used, bending ampli cation by resonance was observed not only at the 702 Hz natural frequency f but also at its odd fractions of 233 (f /3), 139 (f /5), 99 (f /7), 77 (f /9), 63 (f /11) and 53 (f /13) Hz (Fig. 3e), and various bending patterns were created (Fig. 3g-i and Supplementary Fig. 10).
This ampli cation was caused by the start of down-bending and up-bending due to the natural vibration at the same time when the UV light was turned on and off, respectively.To evaluate separately the photothermally driven bending and the natural vibration induced by pulsed UV irradiation of odd fractions of the natural frequency, the bend angle of the natural vibration was extracted by tting the photothermally driven bend angle by the pulse frequency (Supplementary Fig. 11).The tted line described a linear increase in bend angle of the natural vibration with increasing pulse frequency (blue, Fig. 3f).On the other hand, the photothermally driven bend angle decreased nonlinearly with increasing pulse frequency (red, Fig. 3f).

Actuation performance
To clarify the relationship between crystal shape and natural vibration as well as photothermally driven bending, the bending behaviour of ve crystals (I-V) of different shapes were examined, with different geometries ranging from 5 to 8 mm in length and 50 to 220 µm in thickness (summarised in Supplementary Fig. 8 and Supplementary Table 2).When the crystals were continuously irradiated with UV light for 100 ms, they exhibited photothermally driven bending of 1.1°-1.7°,which increased slightly in proportion to the thickness (Fig. 4a).The natural vibrations of 0.06°-0.3°were extracted and were greatly ampli ed to 1.9°-4.0°due to resonance upon irradiation with pulsed UV light at each natural frequency; these were nearly proportional to the aspect ratio (length/thickness) of the crystals (Fig. 4b, c).The natural frequencies were in the range of 200-700 Hz and were nearly proportional to thickness/length 2 (h/l 2 ), in accordance with Eq. (1) (Fig. 4d).Moreover, the resonance ampli cation ratio ranging from 9 to 32 also increased in proportion to the thickness/length 2 (Fig. 4e).Summarising, the frequency and the bend angle could be changed and tuned by changing the crystal shape; larger bending could be realised by resonating thinner and longer crystals, and conversely, higher natural frequency could be realised by using thicker and shorter crystals (Fig. 4f).
Next, we evaluated the energy conversion e ciency and the tip de ection speed of 1β crystals I-V and compared the results with other photomechanical crystals (Fig. 4B and Supplementary Table 3).The tip de ection speed v l of the resonated natural vibration was in the range of 0.2-0.6 m s −1 (black solid circle), which was more than ten times faster than the 0.01-0.03m s −1 (black open circle) of the nonresonated natural vibration.In addition, this speed is ten times faster than the 0.001-0.05m s −1 of the photothermally driven bending of the 1β crystal (red circle) and other crystals (red triangle, 21 square, 22 and rhombus 23 ), and two-to-three orders of magnitude faster than the 10 −4 -10 −3 m s −1 for photoisomerisation (blue triangle, 21 square, 22 and rhombus 23 ).To the best of our knowledge, the bending speed of the resonated natural vibration reported herein is the fastest of reported mechanically responsive crystals.
The energy conversion e ciency η from incident light energy U L to mechanical energy U M was evaluated according to Eq. ( 2): 26 where m e is the effective weight of the crystal ( where m is the crystal weight), I L is the UV irradiation intensity per area, and t I is the irradiation time.The energy conversion e ciency of the resonated natural vibration was 10 −5 -10 −3 (black solid circle), which is two-to-three orders higher than that (10 −8 -10 −6 ) of the non-resonated natural vibration (black open circle) and the 10 −7 -10 −6 of the photothermally driven bending (red circle).Compared with other crystals, this e ciency is two and three orders of magnitude higher than the 10 −7 -10 −6 of bending by the photothermal effect (red triangle, 21 square, 22 and rhombus 23 ) and ten orders of magnitude higher than that the 10 −15 -10 −10 of photoisomerisation (blue triangle, 21 square, 22 and rhombus 23 ).Thus the resonated natural vibration induced actuations with the fastest speed and the highest energy conversion e ciency.

Discussion
Photothermal bending is caused by the non-steady thermal elongation difference between the irradiated and back surfaces, which is due to the non-steady temperature gradient formed by the heat conduction of photothermal energy. 21On the other hand, the natural vibration is caused by the thermal load in the crystal generated by the photothermal effect.Despite these different bending mechanisms, we attempted to simulate actual bending behaviour by combining both photothermally driven bending and natural vibration.Figure 5a presents a schematic illustration of the geometrical parameters of a hexagonal crystal cantilever for photothermally driven bending and natural vibration.Upon photoirradiation, heat is generated at the irradiated surface by the photothermal effect, conducting into the thickness direction according to the non-steady heat conduction Eq. ( 3): 21 where α is the thermal diffusivity (1.34 × 10 −7 m 2 s −1 , Supplementary Fig. 3) along the thickness direction, y is the displacement in the thickness direction, t is time and T(y, t) is the temperature at a certain y and t.Consequently, the temperature gradient and the thermal load are generated and trigger the photothermal bending and the natural vibration, respectively.
In the elastic cantilever system, the photothermal bending and the natural vibration are explicitly included in Eq. ( 4): 27 where A is the cross-sectional area, x is the displacement along the length direction, d(x, t) is the de ection at a certain x and t and F(x, t) is the applied thermal load at a certain x and t derived from the temperature gradient.Using the tip de ection d(l, t) calculated according to Eq. ( 4), the combined bend angle q of the photothermal bending and the natural vibration can be calculated using Eq. ( 5): We simulated the bend angle by nite element analysis using the ANSYS software package. 28The temperature gradient was calculated by non-steady heat conduction analysis, and then the actual bending was simulated by coupled thermo-structural analysis (Supplementary 6 and Supplementary Fig. 12).
We conducted two types of simulations as described below.
Simulation I.The bending simulation of crystal III was conducted based on the surface temperature changes measured by infrared (IR) thermography.The measured temperature was applied to the top surface, while the temperatures of two upper slanting surfaces were set according to the tted temperature distribution (Supplementary 6.1 and Supplementary Fig. 13).According to Eq. ( 2), the temperature gradient was formed in the thickness direction and the temperatures of the irradiated and the back surfaces after UV irradiation for 100 ms were calculated to be 35.8°Cand 32.6°C, respectively (Fig. 5b, c), giving a temperature difference of 3.2°C (Fig. 5d).
Coupled thermo-structural analysis was then performed based on the calculated temperature gradient.
The bend angle and time constant of the photothermal bending were well simulated to be 1.26° and 4.2 δ(l,t) l ms, respectively, which are comparable to the 1.22° and 4.7 ms of the measured bending (Fig. 5e and Supplementary Movie 3).The 390 Hz non-resonated natural vibration was also successfully simulated.
Next, the resonated natural vibration under 390 Hz pulsed UV irradiation was simulated.First, the time dependence of the temperature difference between the irradiated and back surfaces was calculated (Fig. 5f).Coupled thermo-structural analysis was then performed; the calculated time pro le of ampli ed bend angle was nearly coincident with the measured one (Fig. 5g and Supplementary Movie 4).
Simulation II.The bending simulation of crystal III, based on the irradiated light energy, was also performed, without the surface temperature measurement.When the upper surface of the crystal was irradiated with UV light (375 nm, 1456 mW cm −2 ), almost all of the irradiated light energy should have been absorbed by the crystal because this 1β crystal emitted very weak uorescence (quantum yield 0.005) (Supplementary Fig. 5).The speci c heat capacity (1.53 J g −1 K −1 , Supplementary Fig. 4) was used to translate heat energy to temperature.Simulation II is detailed in Supplementary 6.2 and Supplementary Fig. 14.The simulated and measured time pro les for bending under UV irradiation for 100 ms were comparable (Fig. 5h and Supplementary Movie 5).Furthermore, the simulated pro le of ampli ed, repeated bending under 390 Hz pulsed UV irradiation was completely coincident with the measured pro le (Fig. 5i and Supplementary Movie 6).Thus, the bending simulation II based on the irradiated light energy was successful, even without using the measured temperature as in Simulation I.
In conclusion, we discovered that molecular crystals cause high-speed natural vibration by the photothermal effect.The natural vibration was dramatically ampli ed by resonance to achieve high frequency, large bending, and high energy conversion e ciency.Any light-absorbing crystal can be actuated by photothermally induced natural vibration.This versatile and fast crystal actuation is expected to contribute to the development of soft robotics.

Methods
Material preparation and characterisation.Commercially available 1 (TCI) was as-received, without further puri cation.A sample (300 mg) was dissolved in MeOH (4 mL) with heating, and the resultant solution was cooled in a refrigerator at 5°C for a few tens of minutes to obtain colourless rod-shaped crystals.
The DSC thermograms of 1β crystals were recorded using a DSC8500 instrument (Perkin Elmer) with an empty aluminium pan as a reference.Enthalpy changes during thermal phase transitions were determined during heating and subsequent cooling between −50°C and 120°C.The same instrument was used to measure the speci c heat capacity during heating.
Single-crystal X-ray diffraction data of crystals of 1β at 20°C were collected using an R-AXIS RAPID diffractometer (Rigaku) equipped with a monochromatic Mo-Kα radiation source (λ = 0.71069 Å) operating at 50 kV and 40 mA.The temperature of the sample was regulated using a nitrogen gas ow cryostat.The crystal structure was solved using the integrated space group and crystal structure determination algorithm of SHELXT 29 software and re ned on F 2 using the full-matrix least-squares SHELXL method. 30The calculation was performed using the Olex2 molecular graphical interface. 31Face indices and unit cell constants over the temperature range of 10−60°C were measured using the same equipment.
UV-vis diffuse re ectance spectrum of powdered 1β crystals was measured using a spectrophotometer (Lambda 650; PerkinElmer) equipped with an integrating sphere.The was mounted on a glass plate and then covered with a quartz plate.The experiment was performed over the wavelength range of 200-800 nm at 1 nm resolution, using a scan speed of 266.75 nm min −1 .
Thermal diffusivity of the 1β single crystal was measured by the TWA technique 32 along the thickness direction (b-axis) at room temperature (25°C).A single crystal was positioned between an indium−tin oxide glass heater and a nickel-gold microthermocouple sensor.The thermal diffusivity was calculated based on the relationship between the phase delay of the temperature waves between the heater and the sensor, and the frequency of the temperature wave.
Young's modulus of the 1β crystal was measured using a universal testing machine (RTG-1210; A&D).
The displacement dependence of applied load was measured by cantilever bending test.The process is described in Supplementary 4.
Observation of crystal bending.Photothermally driven of a rod-shaped 1β crystal was observed using a digital high-speed microscope (VW-6000; Keyence).The entire rod-shaped crystal was irradiated using a UV laser (λ = 375 nm; FOLS-03; SAWAKI Kobo) equipped with a ber line generator (FLG10FC-633; Thorlabs).The surface temperature distribution was recorded using an IR thermography system (FSV-2000; Apiste).After collecting thermography images, the emissivity of a 1β crystal was modi ed by applying a black-body coating (emissivity: 0.96; TA410KS; TASKO) to the rod-shaped crystal and the surface temperature was calibrated using the modi ed crystal emissivity of 0.95.The bending motion and surface temperature distribution were recorded at 8,000 and 120 fps, respectively.Various UV pulse frequencies were generated as square waves (duty ratio: 50%) via transistor-transistor logic control of the UV laser by an Arduino Leonard microcomputer device.The bend angle of the crystal was automatically analysed using Tracker, a video analysis and modelling tool. 33nding simulation.element analysis simulations of the photothermally driven bending were performed using ANSYS software. 28A simpli ed three-dimensional (3D) analysis geometry of the single crystal was created using SpaceClaim direct 3D solid modelling software. 28Material properties of 1β crystals were assigned to the structure.The 3D geometry of the single crystal was divided into elements via the meshing process.The detailed computational process is described in Supplementary 6.

Figures
Figures

Figure 1 Physical
Figure 1