Ching-Fu Lin

Hsiao-Wen Zan
email: hsiaowen@mail.nctu.edu.tw

Chia-Jung Lu
email: cjlu@ntnu.edu.tw

Hsin-Fei Meng

Olivier Soppera

A Low-Cost Miniaturized Colorimetric Sensor with Vertically-Stacked Semi-Transparent Finger-Type Organic Photo Detector for Formaldehyde Sensing

Keywords: colorimetric-type sensing, formaldehyde sensing, finger type, organic photodetector, vertically integrated

We proposed a sensitive colorimetric sensing system using a paper-like sensing layer on top of a vertically stacked finger-type organic photodetector (OPD) and commercial light-emitting diode (LED). The stacked OPD/LED device was small, with an area of 0.8 cm × 0.5 cm that was even smaller than a coin. Using commercial colorimetric formaldehyde (HCHO) sensing molecules to form the sensing layer, the stacked detecting system successfully detected HCHO of concentrations ranging from 40 to 1000 ppb within 20 min. Critical parameters affecting the sensitivity were investigated. The final system exhibited a stable calibration curve over 60 days. The long lifetime was attributed to using 2,6-Bis(trimethyltin)-4,8-bis(5-(2-ethylhexyl)thiophen-2-yl)benzo[1,2-b :4,5-b' ]dithio phene (PBDTTT-C-T) : phenyl-C71-butyric acid methyl ester (PC 71 BM) to serve as a blended system and using the current variation ratio to represent the sensing response.

Introduction

Owing to the emergence of sensing technology in the Internet-of-things era, studies demonstrating low-cost and miniaturized sensor systems are increasing [START_REF] Baron | Amperometric gas sensors as a low cost emerging technology platform for air quality monitoring applications: A review[END_REF]. Gas sensors that detect toxic or explosive gas species attract significant attention because of their key applications in safety control and environmental monitoring [START_REF] Gillanders | A low-cost, portable optical explosive-vapour sensor[END_REF][START_REF] Urasinska-Wojcik | Ultrasensitive WO3 gas sensors for NO2 detection in air and low oxygen environment[END_REF]. To detect a target gas, several types of colorimetric probing molecules with high selectivity have been developed such as polyethylene-imine-protected copper nanoclusters for detecting trinitrotoluene (TNT), polydiacetylene for detecting dimethylformamide, and potassium disulfitopalladate (II) (K 2 Pd(SO 3 ) 2 ) for detecting carbon monoxide (CO) [START_REF] Aparna | Polyethylene imine capped copper nanoclusters-fluorescent and colorimetric onsite sensor for the trace level detection of TNT[END_REF][START_REF] Wang | A flexible transparent colorimetric wrist strap sensor[END_REF][START_REF] Lin | High Performance Colorimetric Carbon Monoxide Sensor for Continuous Personal Exposure Monitoring[END_REF]. However, a low-cost, miniaturized, and sensitive color reading system has not yet been realized. Devadhasan et al. demonstrated color reading using mobile phones [START_REF] Devadhasan | Smartphone coupled handheld array reader for real-time toxic gas detection[END_REF] that benefitted the development of handheld sensing systems. However, for environmental monitoring such as in a building or factory, we expect low-cost sensors to be used in several key positions such as at the inlets or outlets of gas flowing tubes or in critical contaminant areas. Hence, a concrete, low-cost, miniaturized, and sensitive color reading system is required.

In this study, to demonstrate the sensing ability of the proposed colorimetric sensing system, formaldehyde (HCHO) is chosen as the target gas. HCHO is a colorless, toxic, and carcinogenic volatile organic compound (VOC). It is widely used in indoor decorative materials, building materials, and paint; hence, it is a primary cause of the sick building syndrome (SBS) [START_REF] Kim | Effect of formaldehyde on the expression of adhesion molecules in nasal microvascular endothelial cells: the role of formaldehyde in the pathogenesis of sick building syndrome[END_REF][START_REF] Sahlberg | Airborne molds and bacteria, microbial volatile organic compounds (MVOC), plasticizers and formaldehyde in dwellings in three North European cities in relation to sick building syndrome (SBS)[END_REF]. According to the report from the World Health Organization (WHO), in a 30-min exposure, the HCHO concentration should be below 82 parts-per-billion (ppb) [START_REF] Allouch | Transportable, fast and high sensitive near real-time analyzers: formaldehyde detection[END_REF]. Detecting HCHO in the ppb regime is hence highly required. Particularly, a real-time and low-cost detector is required for the general public to verify the safety of their surrounding environments. Colorimetric sensors using a sensing material that changes color when reacting with HCHO were proposed [START_REF] Wang | A facile and highly sensitive colorimetric sensor for the detection of formaldehyde based on electro-spinning/netting nano-fiber/nets[END_REF][START_REF] Kawamura | Development of a novel hand-held formaldehyde gas sensor for the rapid detection of sick building syndrome[END_REF][START_REF] Wang | Colorimetric sensor strips for formaldehyde assay utilizing fluoral-p decorated polyacrylonitrile nanofibrous membranes[END_REF][START_REF] Maruo | Development of formaldehyde sensing element using porous glass impregnated with β-diketone[END_REF][START_REF] Nakano | An automatic monitor of formaldehyde in air by a monitoring tape method[END_REF][START_REF] Suzuki | Portable sick house syndrome gas monitoring system based on novel colorimetric reagents for the highly selective and sensitive detection of formaldehyde[END_REF][START_REF] Descamps | Real-time detection of formaldehyde by a fluorescence-based sensor[END_REF][START_REF] Toda | Portable system for near-real time measurement of gaseous formaldehyde by means of parallel scrubber stopped-flow absorptiometry[END_REF][START_REF] Meng | Preparation of a natural dye doped Ormosil coating for the detection of formaldehyde in the optical gas sensor[END_REF][START_REF] Qin | A colorimetric chemical sensing platform for real-time monitoring of indoor formaldehyde[END_REF][START_REF] Li | Development of a colorimetric sensor array for the discrimination of aldehydes[END_REF][START_REF] Sekine | Colorimetric monitoring of formaldehyde in indoor environment using built-in camera on mobile phone[END_REF]. Hydroxylamine sulfate with pH indicator, 4-amino hydrazine-5-mercapto-1,2,4-triazole (AHMT), fluoral-P, and -diketones have proven to be successful HCHO-sensing materials exhibiting high selectivity and ppb-regime sensitivity. Even though the colorimetric reaction is typically not reversible, low-cost reproducible test strips can be fabricated easily by casting the reacting material onto fibrous layers, such as papers, glasses, or films [START_REF] Wang | A facile and highly sensitive colorimetric sensor for the detection of formaldehyde based on electro-spinning/netting nano-fiber/nets[END_REF][START_REF] Kawamura | Development of a novel hand-held formaldehyde gas sensor for the rapid detection of sick building syndrome[END_REF][START_REF] Wang | Colorimetric sensor strips for formaldehyde assay utilizing fluoral-p decorated polyacrylonitrile nanofibrous membranes[END_REF][START_REF] Maruo | Development of formaldehyde sensing element using porous glass impregnated with β-diketone[END_REF].

Identifying the color change using naked eye, however, is not reliable. In addition to using the conventional absorbance spectrometer, several groups have attempted to design low-cost and miniaturized systems to provide reliable color reading [START_REF] Nakano | An automatic monitor of formaldehyde in air by a monitoring tape method[END_REF][START_REF] Suzuki | Portable sick house syndrome gas monitoring system based on novel colorimetric reagents for the highly selective and sensitive detection of formaldehyde[END_REF][START_REF] Descamps | Real-time detection of formaldehyde by a fluorescence-based sensor[END_REF][START_REF] Toda | Portable system for near-real time measurement of gaseous formaldehyde by means of parallel scrubber stopped-flow absorptiometry[END_REF][START_REF] Meng | Preparation of a natural dye doped Ormosil coating for the detection of formaldehyde in the optical gas sensor[END_REF][START_REF] Qin | A colorimetric chemical sensing platform for real-time monitoring of indoor formaldehyde[END_REF][START_REF] Li | Development of a colorimetric sensor array for the discrimination of aldehydes[END_REF][START_REF] Sekine | Colorimetric monitoring of formaldehyde in indoor environment using built-in camera on mobile phone[END_REF].

The most popular setup relies on a commercial light-emitting diode (LED) to irradiate the sensing film. The reflected light is collected by a commercial photodetector (PD), as shown in Fig. 1(a) [START_REF] Kawamura | Development of a novel hand-held formaldehyde gas sensor for the rapid detection of sick building syndrome[END_REF][START_REF] Nakano | An automatic monitor of formaldehyde in air by a monitoring tape method[END_REF][START_REF] Suzuki | Portable sick house syndrome gas monitoring system based on novel colorimetric reagents for the highly selective and sensitive detection of formaldehyde[END_REF][START_REF] Descamps | Real-time detection of formaldehyde by a fluorescence-based sensor[END_REF]. An alternative method consists of placing the LED on top, the PD at the bottom, and the sensing layer in between the LED and PD, as shown in Fig. 1(b) [START_REF] Toda | Portable system for near-real time measurement of gaseous formaldehyde by means of parallel scrubber stopped-flow absorptiometry[END_REF][START_REF] Meng | Preparation of a natural dye doped Ormosil coating for the detection of formaldehyde in the optical gas sensor[END_REF][START_REF] Qin | A colorimetric chemical sensing platform for real-time monitoring of indoor formaldehyde[END_REF]. However, the system is still bulky because the LED, sensing layer, and PD are positioned separately to form suitable light paths.

We herein propose a new design to realize a miniaturized, compact sensing system by vertically stacking the colorimetric sensing layer, semitransparent photodetector, and commercial LED. The semitransparent finger-type organic photodetector was used to allow the LED light to penetrate and detect the reflective light as the sensing signal. It is noteworthy that, in our previous work, we demonstrated such semitransparent OPDs to serve as proximity sensors [23]. In this study, we successfully verified that the detection of the reflective light can be used to perform sensitive colorimetric sensing. The proposed OPD utilized air-stable 2,6-Bis(trimethyltin)-4,8-bis(5-(2-ethylhexyl)thiophen-2-yl)benzo[1,2-b :4,5-b' ]dithio phene (PBDTTT-C-T): phenyl-C71-butyric acid methyl ester (PC 71 BM) as the blended active layer. With glass encapsulation, the OPD exhibited a long lifetime ( > 90 days) in air. By placing the finger-type OPD in between a commercial surface-mount device (SMD) LED and a fibrous-paper sensing film using AHMT as the HCHO colorimetric dye [START_REF] Dickinson | A new sensitive and specific test for the detection of aldehydes: formation of 6-mercapto-3-substituted-s-triazolo [4, 3-b]-s-tetrazines[END_REF], the integrated system is even smaller than a US dime (Fig. 1(c)). Subsequently, the proposed integrated sensor is placed into a flow tube to detect HCHO in the ppb regime (Fig. 1(d)). A reliable and reproducible sensing calibration curve can be obtained for HCHO with concentration ranging from 40 to 1000 ppb. Moreover, even when the OPD has slightly decayed after 60 days, the sensor provides an almost unchanged sensing response when the current variation ratio is used instead of the current difference as the sensing parameter.

Experimental setup

Materials

To fabricate the finger-type semi-transparent OPD, PBDTTT-C-T was purchased from Solarmer Energy Inc. and PC 71 BM was obtained from UNI-ONWARD Corp.

To prepare the colorimetric HCHO-sensing paper, AHMT was purchased from Alfa Aesar Inc., potassium hydroxide (KOH) from Duksan Pure Chemicals Co., hydrochloric acid (HCl) from J.T. Baker Inc., and the porous glass filter used was Whatman GF/C.

Finger-type semi-transparent OPD fabrication

To fabricate the solution-processed finger-type semitransparent OPD that is shown in Fig. 2(a), we first prepared a wet-etched indium tin oxide (ITO) glass substrate (3 cm × 3 cm), which was cleaned by a standard cleaning method using acetone, isopropanol, and deionized water. After a 15 min UV ozone exposure process, we placed the ITO glass on a hot plate at 80 °C, and subsequently formed the blending active layer with PBDTTT-C-T and PC 71 BM (Fig. 1(c)) by blade coating. Next, the specially designed finger-type aluminum electrode of width 60 µm, interval 110 nm, and thickness 100 nm, was evaporated through a shadow mask by a thermal evaporator. Finally, this device was sealed with a UV adhesive glue in a glove box in which the water and oxygen concentrations were maintained below 0.1 ppm. The layer-by-layer device structure is shown in Fig. 2(b).

For the active layer material, PBDTTT-C-T is a well-known polymer that typically acts as an electron donor and absorbs visible light well. PC 71 BM is a material that can facilitate electron-hole separation, which plays an important role in enhancing the efficiency of photoinduced electron transfer. To prepare the PBDTTT-C-T:PC 71 BM blending solution, we used a weight ratio 1:1.5 of PBDTTT-C-T:PC 71 BM; subsequently, we dissolved the material in chlorobenzene (2.25 wt%) and maintained the temperature at 60 °C for 24 h until a thorough dissolution is achieved. Blade coating was used by depositing 30 µL of the active layer solution on the blade, after which the blade was moved at a fixed speed of 300 mm/s to form a 400-nm-thick active layer.

Gaseous HCHO preparation

To prepare gaseous HCHO, we first used a microsyringe to draw formaldehyde solution (3.80-4.35 %W/V, in aqueous Phosphate Buffer, H121-08, Macron Fine Chemicals), which was subsequently injected into a 3 L Tedlar sealed bag filled with high-purity gaseous nitrogen. It is noted that the HCHO solution concentration is set to be 4% in the following calculation. After the HCHO solution had completely evaporated in the bag, gaseous HCHO was formed. The gaseous HCHO concentration was calculated as follows:

(

where is the HCHO concentration in ppm, is the concentration of gaseous HCHO in mg/m 3 , is the pressure in Pa, is the temperature of the environment in °C, and is the molecular weight of HCHO. We used detector tube (10% RSD, purchased from GASTEC Corp.) to confirm the HCHO concentration in Tedlar bag.

This detector tube was calibrated by diffusion tube method according to manufacturer.

As we used the detector tube to check the 16 ppm (calculated concentration was around 15.47 ~ 17.7 ppm) HCHO we prepared in Tedlar bag, the result showed 17 1 ppm (< 10% RSD). This deviation is acceptable in this work for demonstrating the feasibility of HCHO sensing. If a calibration in ppb-regime is required, 2,4-dinitrophenylhydrazine-high performance liquid chromatography (DNPH-HPLC) method may be adopted [START_REF] Gillett | Measurement of indoor formaldehyde concentrations with a passive sampler[END_REF].

Fabrication method of the colorimetric HCHO-sensing paper

To

Formation of the vertically-integrated-type gaseous HCHO-sensing system

In our primary work, we designed a vertically integrated-type gaseous HCHO-sensing system using an AHMT colorimetric sensing paper for the HCHO reaction, a finger-type OPD for the electrical signal readout, and a commercialized 530 nm SMD LED chip for the light source. As shown in Fig. 1(c), the size of the OPD is only 0.8 cm × 0.5 cm, which is similar to the size of the LED. With the special finger-type electrode design of the OPD, the light source can be integrated on the finger-type electrode side of the OPD, thus allowing light to partially pass through the device. The OPD can subsequently detect the light reflected from the colorimetric sensing film. Fig. 1(d) illustrates the vertically integrated-type HCHO-sensing system and the gas-tight syringe with gaseous HCHO, which can be pumped using a stepper motor. The total gas flow rate was set at 1 L/min. As the pumping rate of the stepper motor is controlled by the ambient air dilution, the concentration of HCHO in the sensing chamber can be controlled. The colorimetric reaction between the AHMT paper and gaseous HCHO can change the reflection spectrum and hence vary the current signal in the OPD to provide a sensing signal proportional to the HCHO concentration.

It is noted that, if a high selectivity to HCHO is needed, choosing the wavelength as 540 nm is beneficial to improve the selectivity. This is because that AHMT reacts with different aldehyde materials to derivate different final products with different colors.

The absorption spectrum changes among 520-555 nm. Choosing 540 nm, as reported by Kawamura et al. [START_REF] Kawamura | Development of a novel hand-held formaldehyde gas sensor for the rapid detection of sick building syndrome[END_REF], delivers an improved selectivity to HCHO among various kinds of aldehyde such as acetaldehyde.

Results and discussion

Primary operation mechanisms of the vertically integrated-type gaseous HCHO-sensing system

The performance of the finger-type OPD is first evaluated. Fig. S2(a),(b) show the basic light/dark current performance and lifetime tracing result of the finger-type OPD used in this study, respectively. As show in Fig. S2(a), the OPD performs well with a light/dark current ratio of approximately 10 5 at -1.5 V when a 10000 lux white light source is used. Furthermore, the OPD stably maintains a light/dark current ratio of 10 4 for over 90 days (Fig. S2(b)). This indicates that this OPD is both sensitive and reliable for applications related to light-intensity detection. Another characteristic of the OPD is its ultrasmall light-absorbing area (0.2 cm × 0.2 cm). Therefore, this device can be easily sliced to a size suitable for embedding into systems for light-intensity detection. In this study, we sliced the OPD from its typical size of 3 cm × 3 cm to 0.8 cm × 0.5 cm. Nevertheless, it maintained its light-intensity detection ability, as shown in Fig. S2(c).

To build a gaseous HCHO-sensing system based on colorimetric sensing, we first performed a colorimetric sensing ability test on the AHMT filter paper. We set the glass filter on a glass substrate and used polytetrafluoroethylene tape to adhere both sides of the filter paper on the glass substrate, with the gas exposure area of 1.5 cm × 1 cm on the sensing chip. After depositing the AHMT sensing solution onto the filter paper, we moved the chip into the sensing chamber as in Fig. 1(d), and subsequently commenced the colorimetric sensing evaluation test. Fig. 3(a) shows the color variation performance measured after 12 min of background air flow and 20 min gaseous HCHO flow. Similar to the color transformation of the solution, the sensing paper changed from white to violet when the HCHO concentration is greater than 150 ppb. However, it is difficult for the human eye to perceive this color change if the HCHO concentration is less than 90 ppb. Therefore, a tool that can detect the slight color change must be developed. Furthermore, the tool should be sufficiently small for portable applications. Embedding the finger-type OPD with an SMD LED may be an excellent solution owing to its high sensitivity when detecting reflective light.

To form the finger-type OPD HCHO-sensing system, we must consider a good match among three aspects: (1) reflectance spectrum of the sensing layer, (2) OPD absorption spectrum, and (3) luminance peak of the SMD LED. Fig. 3(b) shows the reflectance spectrum measured at the conditions given in Fig. 3(a). As shown, the signal variation ranges from approximately 450 to 650 nm. Consequently, a commercial SMD LED with a luminance peak at 530 nm was chosen as the operating light source of the system. Notably, based on the absorbance spectrum of the active layer material of the OPD shown in Fig. 3(c), this type of light can be well absorbed by the OPD. This suggests that the SMD LED is an appropriate choice for this system.

We integrated the SMD LED with the OPD by embedding it at the bottom of the sensing chamber, as shown in Fig. 1(d), to produce the vertically integrated-type HCHO-sensing system.

Regarding the sensing mechanism of the system, the AHMT paper serves as both the HCHO-sensing layer and the reflective surface for the OPD. Once the system detects HCHO, resulting in a color change in the paper, the reflective light intensity from the paper weakens owing to the variation in the reflectance value. This process can cause the output current of the OPD to decrease, thus allowing for the system to determine the precise gaseous HCHO concentration.

Discussion of the reliability and calibration method

To investigate the sensing behavior more carefully, the real-time sensing curve when detecting 90 ppb HCHO is shown in Fig. 4(a). As previously mentioned, we first measured the background signal for 12 min, and subsequently inject 90 ppb HCHO for 20 min. It is observed that the background current remains nearly constant.

When HCHO was injected, the current of the system decreased gradually. The test flow in Fig. 4(a) is set to be a standard flow for a single test. To evaluate the reproducibility and reliability of the sensing system, we used the vertically integrated sensing system to measure gaseous HCHO with concentrations ranging from 40 to 1000 ppb, all repeated thrice (n = 3). In every round of sensing, similar to the test flow shown in Fig. 4(a), we first measured the background signal for 12 min, and subsequently injected HCHO with a fixed concentration for 20 min.

Evaluating the long-term operation of the system

Although the vertically integrated HCHO-sensing system can detect low concentrations of HCHO, the lifetime performance of the system cannot be neglected.

In this study, the calibration curves using current difference and current variation ratio as sensing signals were compared. Fig. 5(a) shows the current difference when detecting 70 ppb HCHO at days 6 and 60. Obviously, the current difference decreases significantly at day 60. This phenomenon indicates that the sensitivity of the OPD to light intensity variation decays gradually with time. As shown in Fig. S4, even though the fabricated OPD exhibits a good light/dark current ratio over 90 days, the output current at -1.5 V decreases from ~8 × 10 -6 A to ~4 × 10 -6 A from day 6 to day 60.

Consequently, using current difference to serve as the sensing signal results in degradation with time. However, if we use the current variation ratio to serve as the sensing response, a rather stable calibration curve can be obtained even after 60 days, as shown in Fig. 5(b). Therefore, the sensing response analysis method is more appropriate for prolonging the operating lifetime of this vertically integrated HCHO-sensing system. With the same condition, the current variation ratio increases from 0.6% to 1.4%. This verifies that, given a light source with suitable intensity, this vertically integrated gaseous HCHO-sensing system can detect HCHO in the ppb regime.

Conclusion

In this study, we successfully demonstrated a sensitive colorimetric sensing system using a paper-like colorimetric HCHO-sensing layer on top of a vertically stacked semitransparent OPD and commercial SMD LED. The area of the miniaturized stacked OPD/LED device was only 0.8 cm × 0.5 cm, which was suitable for developing portable or wide-spread sensing devices. When HCHO reacted with the sensing layer to cause a color change, the OPD/LED system detected the reflectance light of the sensing layer. For HCHO lower than 90 ppb, the color change cannot be identified easily by the human eyes, but the system successfully delivered a reliable current change to serve as the sensing signal. We confirmed the repeatability and the reliability of the proposed sensing system. Furthermore, we noticed that using current variation ratio instead of the current difference to serve as the sensing response improved the system lifetime significantly to longer than 60 days. Increasing the LED light intensity enhanced the sensitivity in low-concentration HCHO. The stacked system successfully detected HCHO of concentration ranging from 40 to 1000 ppb within 20 min; this satisfied the requirement of WHO. In future, considering real applications, pushing the HCHO detection limit to be less than 10 ppb is beneficial to further improve the sensing accuracy. Possible methods include enhancing the light/dark current ratio of the OPD to enlarge the sensing dynamic range and forming a HCHO sensing film with higher sensitivity. According to literatures, coating the sensing reagent on substrates with nanostructures such as silica porous substrate [START_REF] Qin | A colorimetric chemical sensing platform for real-time monitoring of indoor formaldehyde[END_REF] or electrospinning fibrous film [START_REF] Wang | Colorimetric sensor strips for formaldehyde assay utilizing fluoral-p decorated polyacrylonitrile nanofibrous membranes[END_REF] may increase the gas reacting area and hence the sensitivity. Moreover, in addition to HCHO sensing, we expect the proposed semitransparent OPD/LED system to be easily integrated with other colorimetric chemical sensing films to provide sensitive color-reading.

Fig. 1 :

 1 Fig. 1: (a) Conventional type-1, (b) conventional type-2, and (c) our proposed integrated design using a stacked fibrous sensing film, semitransparent finger-type OPD and SMD LED. (d) Integrated sensor in the flow tube.

Fig. 2 :

 2 Fig. 2:(a) Fabrication process (b) Layer-by-layer structure of the finger-type OPD

  design a gaseous HCHO-sensing paper based on colorimetric detection, the selection of the sensing reagent is vital. Kawamura et al. demonstrated that an AHMT-based sensing design could achieve a higher color transformation rate compared to other traditional sensing designs, such as hydroxylamine-based sensing [12]. Therefore, in our work, we chose the method utilized by Kawamura et al. and prepared two types of reagent solutions. The first was 0.1 g AHMT dissolved in 9% HCl, and the other was 2.3 g KOH dissolved in 10 mL deionized water. The two reagent solutions were mixed in a volume ratio of 1:1, after which 100 of the mixture solution was injected onto the Whatman GF/C glass filter paper to form the colorimetric sensing paper. Dickinson et al. [25] demonstrated the principle behind the colorimetric HCHO sensing of AHMT, as follows: After AHMT (Fig. S1 (I)) reacts with HCHO, the formyl group of HCHO will form an intermediate (Fig. S1(II)), this intermediate should be unstable, and oxygen labile. Then, it will continuously turn quickly (< 1min) to the final product 6-mercapto-s-triazolo[4,3-b] -s-tetrazine (Fig. S1(III)) between the liquid-air interface by aerial oxidation. This enables the color of the sensing solution to transform gradually from transparent to violet.

Fig. 3 (

 3 Fig. 3(d) shows the real-time OPD current tracing under gaseous HCHO flow detection with concentrations from 40 to 1000 ppb. The light source intensity was 35 W/m 2 . It is noteworthy that the current difference, which is the current minus the initial current, is used in the y-axis to compare the current change under different HCHO concentrations. During each round of sensing, before injecting HCHO, air was flown in the background for 12 min until the background signal remained nearly constant. Subsequently, gaseous HCHO with a fixed concentration at a fixed flow rate of 1 L/min was injected for 20 min. As shown in Fig. 3(d), when the HCHO concentration increases, the slope of the current difference increases as well. At a

Fig. 3 :

 3 Fig. 3: (a) Color variation of AHMT paper after 12 min background air flow and 20 min gaseous HCHO flow (b) Corresponding reflectance spectrum comparison of AHMT paper measured after completion of HCHO flow (c) Absorbance spectrum of the active layer material of OPD (d) The current difference as a function of time when detecting HCHO ranging from 40 to 1000 ppb.

Fig. 4 (

 4 b) shows the dynamic repeated test for 90 ppb HCHO. The measurements corresponding to other concentrations are shown in Fig. S3. In every test, a new AHMT sensing paper was used. In both Fig. 4(b) and Fig. S3, we obtained good repeated signals. Subsequently, we plotted the sensing response as a function of HCHO concentration to form the calibration curves.In this study, two types of calibration curves were compared. The first one used the current difference and the second one used the current variation ratio to represent the sensing response. For the first type of calibration curve, as shown in Fig.4(c), the absolute value of the current difference ( ) at different sensing times (5, 10, 15, and 20 min) are plotted as a function of HCHO concentration. For the second type of calibration curve, we calculate the current variation ratio by dividing the absolute value of the current difference by the original current amount (I 0 ), as expressed by Eq. current variation ratio with sensing times of 5, 10, 15, and 20 min are plotted as a function of HCHO concentration in Fig. 4(d). In both Figs. 4(c) and (d), the calibration curves exhibit greater sensing signals when the sensing time is increased. For a 20 min sensing time, the 40 ppb detecting signals can be clearly identified. For high HCHO concentrations such as 500 ppb or 1000 ppb, the system can deliver large sensing signals within 5 min.

Fig. 4 :

 4 Fig. 4: (a) Real-time current variation when detecting 90 ppb HCHO for 20 mins. (b) Repeated real-time measurement when detecting 90 ppb HCHO thrice using three different AHMT sensing papers. (n = 3, R: current variation ratio, M: mean value of sensing response, SD: standard deviation of sensing response) (c) Calibration curves using current difference as the sensing signal (n = 3). Sensing time varies from 0 to 20 min. (d) Calibration curves using current variation ratio as the sensing response (n = 3). Sensing time varies from 0 to 20 min.

Fig. 5 :

 5 Fig. 5: 70 ppb HCHO real-time sensing curves using (a) current difference and (b) current variation ratio to represent the sensing signal, at days 6 and 60.

Fig. 6 :

 6 Fig. 6: (a) Current difference and (b) sensing response plotted as a function of HCHO concentration with light intensities of 35 W/m 2 and 65 W/m 2 (n = 3).

Acknowledgements

This work was supported by the Ministry of Science and Technology (MOST),

Taiwan (Project number 107-2221-E-009 -086), and by MOST Add-on Grant for International Cooperation (Project number 108-2923-E-009 -002 -MY3).

Supporting information


A Amor

A Fernandes
email: amancio.fernandes@sorbonne-universite.fr

J Pouget

Snap-through of elastic bistable beam under contactless magnetic actuation

Keywords: Bistable mechanism, buckling, snap-through, Laplace force, bifurcation

We propose the modeling of contactless switching of a bistable slender beam using Laplace force actuation. The model beam is based on the elastica approach which allows large amplitudes of the cross-section rotation and large elastic transformations. Furthermore, the extensibility of the elastic beam is accounted for and it plays a crucial role in the switching process. The study is devoted particularly to the actuation mechanism of the bistable beam uniformly loaded by a density of lineic force which is permanently perpendicular to the beam deformation. Such actuation can be produced by the Laplace force due to an electric current travelling along the beam placed in a magnetic induction. The mechanism of the bistable switching is analyzed in detail using a variational formulation and stable and unstable equilibria are described. We investigate numerically the post-buckling behavior to capture the exact buckling modes that follow the path in the unstable region. The numerical simulations are also used to obtain the bistable response in terms of actuating force (or electric current amplitude), beam end-shortening and mid-point displacement of the beam. We also perform a finite element computation and provide a validation of the results obtained by solving the set of equations of the boundary value problem.

The second part of the work is focussed on the experimental validation of the switching mechanism of the bistable beam presented in the analytical part. We design an experimental set-up for the fine measurement of the mid-point displacement of the bistable beam as a function of the electric current travelling the beam. The region of bistable instability is revealed by experimental adjustment of the bifurcation point associated with the actuating force. All the results extracted from experimental tests are compared to those coming from the modeling investigations, which ascertains with good accuracy the approach of the proposed model for bistable beam.

Introduction

Consider an elastic slender beam. One end of the beam is fixed, the other one undergoes a small end-shortening caused by a compressive force applied in the direction of the beam axis. As long as the force is weak, less than a certain critical value, the beam is in compression as predicted by linear elasticity. As soon as the force exceeds a critical value, the beam is no longer straight, and undergoes a transverse deformation or deflection [START_REF] Bazant | Stability of structures[END_REF][START_REF] Bigoni | Nonlinear solid mechanics : bifurcation theory and material instability[END_REF][START_REF] Dym | Stability theory and its applications to structural mechanics[END_REF][START_REF] Thomson | A general theory of elastic stability[END_REF]. The amplitude of deflection increases with end-shortening as well as with compressive force. We are faced with a bifurcation phenomenon, resulting in the buckling of the beam (Timoshenko and Gere , 1963). The later can deflect either upwards or downwards.

The elastic structure that we are describing is a bistable system and both positions of the beam (upwards or downwards) are stable states. However, the most interesting aspect of this bistable system is to be able to switch from one stable state to another regardless of how the transition is completed. Although a great emphasis has been paid to beam buckling and several studies are available in the literature, the mechanism of switching between stable positions remains an active topic of research, especially because a linear model is inadequate and recourse to a nonlinear model becomes inescapable. Furthermore, some questions are raised such as how does the bistable system switch between stable positions and what are the buckling modes which are involved in the process? Actuating a bistable beam to force it to switch can be accomplished using various means. A first idea is to load laterally the beam with a punctual force at any point of the beam, or with a punctual moment applied to the beam in the direction perpendicular to plane deformation [START_REF] Cleary | Modeling and experimental validation of actuating a bistable buckled beam via moment input[END_REF]. However, the bistable beam response under an applied punctual force depends on the way of applying the force, more precisely, on whether the force is vertically fixed onto the midpoint of the elastic beam or if the force stays on a central line between beam ends while the bistable snaps. Loading a bistable beam with a vertical force which is always fixed onto the beam has been reported in some studies, among them Pippard [START_REF] Pippard | The elastic arch and its modes of instability[END_REF] and Patricio [START_REF] Patricio | An elastica problem : instability of an elastic arch[END_REF]. Nevertheless, this situation is not convenient for experimental considerations. An extensive study of the actuation of a bistable beam by punctual force along a fixed central line in space (not necessarily on the midline) has been presented in [START_REF] Camescasse | Bistable buckled beam: Elastica modeling and analysis of static actuation[END_REF] and experimental validation of the model has been discussed in [START_REF] Camescasse | Bistable buckled beam and force actuation : Experimental validations[END_REF]. A comparative study has been proposed by [START_REF] Hung | Snapping of an elastica under various loading mechanism[END_REF] according to the way of applying the force onto the beam force while the latter is switched. Then Cheng and Hung examine three cases : (i) the force is attached onto the beam mid-point, (ii) the force is applied along a fixed central line in space and (iii) the force is applied by means of a device such that the force is always perpendicular to the deformed beam and its point of application is allowed to slide on the beam, while staying along the central line in space.

Switching or snapping-through of bistable systems can be obtained by means of active materials such as piezoelectric ceramics, shape memory alloys, electroactive polymers or magnetostrictive materials to cite the most popular. In this situation, the active elements bonded onto the elastic beam are able to induce local strains or stresses onto the buckled beam to control the switching process [START_REF] Maurini | Distributed piezoelectric actuation of a bistable buckled beam[END_REF].

In each the above-mentioned situations, the actuation is in contact with the bistable system and the influence of the actuating mechanism must be accounted for. The present study proposes an innovative means of actuating a bistable system by a uniform density of lineic force, which is able to trigger the snap-through of the buckled beam as soon as the density of force is great enough to make the bistable pass through the zone of instability to reach the other stable position. Here, the loading mechanism is such that the lineic density of force is permanently fixed onto the beam.

Moreover, when the beam is deformed while switching, the lineic force stays perpendicular to the beam deformation. Such a loading mechanism can be properly produced by the Laplace force [START_REF] Griffiths | Introduction to electrodynamics[END_REF][START_REF] Jackson | Classical electrodynamics[END_REF]. More precisely, this is the force applied to an electrically conductive wire traversed by a volume density of electric current placed in a magnetic induction field.

The elementary force is then perpendicular to the plane formed by the magnetic induction vector and vector of the electric current density or, in other words, the tangent vector to the deformed beam.

One of the main purposes of the present study is to investigate the bistable buckled beam on the basis of the elastica model, especially because a rather small disturbance of the actuating work can produce displacement and cross-section rotation of the beam of relatively large amplitudes.

Bistable buckled beam modeling is subject of numerous research studies which can be divided in different approaches according to the starting hypothesis. Elastica model has been extended to elastic arches loaded at its center by Pippard [START_REF] Pippard | The elastic arch and its modes of instability[END_REF]. Vangbo [START_REF] Vangbo | An analytical analysis of a compressed bistable buckled beam[END_REF] investigated the switching response of a bistable beam including the compressive effect by using an expansion of the solution to the beam equation on the buckling modes. In the framework of modal formulation, the detailed behavior of the stable and unstable branches of the bistability response under loading has been proposed by (Cazottes et al. , 2010). These authors have examined on the basis of a similar modeling, that is, buckling modal formulation, the actuation of a bistable buckled beam subject to a moment [START_REF] Cazottes | Design of actuation for bistable structures using smart materials[END_REF]. Beside the elastica beam theory, some weakly nonlinear approches are often considered. In particular, a theory accounting for extensibility of the beam leads to geometric nonlinearity combining bending and stretching deformations. This kind of approach has been developed by [START_REF] Nayfeh | Exact solution and stability of post-buckling configurations of beams[END_REF] for a buckled beam and by Pinto and Gonçalves [START_REF] Pinto | Active non-linear control of buckling and vibrations of a flexible buckled beam[END_REF] for a simply supported shallow elastic arch. The weakly nonlinear approach has been considered also for both static and dynamic snap-through of bistable structures by [START_REF] Chandra | A numerical investigation of snap-through in a shallow arch-like model[END_REF]. The present work is devoted to the bistable beam but we also mention some extensions of bistability effect and even tristability to elastic thin plates or shells. Such studies have been conducted by Schultz et al. [START_REF] Schultz | Snap-through of ansymmetric laminates using piezocomposite actuators[END_REF] for bistability control of composite plates using piezoelectric actuators, or by [START_REF] Vidoli | Tristability of thin orthotropic shells with uniform initial curvature[END_REF] for tristability of thin orthotropic shells, or by [START_REF] Fernandes | Multiparameter actuations for shape control of biatable composite plates[END_REF] for the study of shape control of bistable composite plates.

One of the most attractive development of bistable systems (or multistable systems by extension)

is their ability to be miniaturized from millimeter scale to dimensions of the order of the micron.

Such bistable systems become particularly interesting to design microelectromechanical devices (MEMS). At the microscopic scale, the actuation making use of Lorentz force is quite attractive.

According to the Lorentz theory of electrons, the force on an electric punctual charge consists of two parts. The first part is a force produced by an electric field on an electric charge (electrostatic force). The second part is the force applied to a moving electric charge placed in a magnetic induction. When the material deals with a continuum of density (per unit of volume or mass) of electric charge, the Lorentz force becomes a kind of effective Lorentz force. Subsequent use of Lorentz force leads to the notion of a so-called ponderomotive force or electromagnetic force. More precisely, the latter force is deduced by applying the classical Lorentzian (space) averaging procedure to an assembly of electric punctual charge contained within a microelement of volume ∆V [START_REF] Maugin | Continuum mechanics of electromagnetic solids[END_REF]. The macroscopic quantities thus obtained by this spatial averaging procedure are considered to be sufficiently smooth functions of space and time. Now, going back to our problem, if the charges are travelling in a conductive wire placed in a magnetic induction, the force acting on the wire is called the Laplace force [START_REF] Griffiths | Introduction to electrodynamics[END_REF][START_REF] Jackson | Classical electrodynamics[END_REF]. Obviously, we then have a contactless actuation of the bistable system. Moreover, such electromagnetic actuations are easily controlled using electronic circuits.

Electromechanical system technology presents important advantages including, small size, light weight, limited energy consumption and integrated electronic components. Electrostatic force has been exploited to excite sensors at their resonant frequencies, based on bistable concept (Herrera-of electric potential between a semi-infinite rigid electrode and the initially curved beam has been analyzed for applications to MEMS [START_REF] Das | Pull-in and snap-through instabilities in transient deformations of microelectromechanical systems[END_REF][START_REF] Pane | Investigation on bistability and fabrication of bistable prestressed curved beam[END_REF][START_REF] Williams | Modeling of initially curved beam structures for design of multistable MEMS[END_REF]. [START_REF] Park | Pre-shaped buckled-beam actuators : theory and experiments[END_REF] present a study of the actuation of pre-shaped beam controlled by electromagnetic force and experimental validation of the snap-through mechanism of the bistable micro-beam is also conducted. Some other interesting works concerning micro-systems based on bistable beam have been conducted mainly by [START_REF] Buchaillot | Post-buckling dynamic behavior of self-assembled 3D microstructures[END_REF], [START_REF] Krylov | Dynamic stability of electrostatically actuated initially curved shallow microbeams[END_REF] or [START_REF] Krylov | Bistability of curved microbeams actuated by fringing electrostatic fields[END_REF].

Actuating a bistable beam by means of active materials is an interesting development in order to control the stability of the bistable beam during the snap-through process by using electric field, gradient of temperature or magnetic induction. Such materials induce local strains leading to trigger switching the bistable beam as soon as the strain is large enough to enter the unstable zone.

Electrostatically actuated bistable systems are well adapted for controlling the bistability and they are perfectly designed for MEMS [START_REF] Krylov | Bistability of curved microbeams actuated by fringing electrostatic fields[END_REF][START_REF] Medina | Bistability criterion for electrostatically actuated initially curved micro plate[END_REF]. Deformation of a bistable beam bonded with a piezoelectric element is induced by elongation or compression in the piezoelectric layer by means of the application of an appropriate electric field. Aimmanee and Tichakorn [START_REF] Aimmanee | Piezoelectrically induced snap-through buckling in a buckled beam bonded with a segmented actuator[END_REF] have proposed such an actuation of a bistable beam using smart materials. A study of distributed multiparameter actuation for the buckling and snap-through control of a pinned elastic beam was reported in [START_REF] Maurini | Distributed piezoelectric actuation of a bistable buckled beam[END_REF]. There, the authors propose an elastic beam sandwiched by two pairs of piezoelectric actuators driven by two differences of electric potential. An extension to the bistable shell made of magnetic rubber composite using the magnetic actuation of the shell cap has been conducted by [START_REF] Seffen | Eversion of bistable shells under magnetic actuation: a model of nonlinear shapes[END_REF].

The present work is based on an extensible elastica beam model. We describe the beam deformation by two strain measures which are the extension strain along the beam axis and the beam curvature.

The beam kinematic is presented in the next section (Section 2). Both descriptions in the fixed frame and local frame attached to the beam deformation are given in that section. We then derive the equations of the buckled beam from the virtual work principle. Section 3 is devoted to this task. Moreover, the boundary conditions -clamped-clamped beam -are specified in the same section. Section 4 deals with the numerical study of the set of nonlinear boundary value problems. The numerical problem thus stated relies on the model equations based on the shooting method associated with a numerical procedure to capture the unknown shooting parameters. In the same section we compare the solutions derived from the model equations to those obtained with the help of the finite element method. One of the main results coming from the numerical investigations is the bistable response, that is, the actuating driving force (or the electric current) as a function of the beam mid-point displacement. The results are discussed according to the chosen parameters (geometry of the beam, the end-shortening). A part of the work is devoted to the experimental validation of the proposed model (Section 5). We report the investigation of the experimental measurements in Section 5 and discuss the results thus obtained and compare them to those deduced from the model and numerical simulations.

2 Description of the bistable system

Kinematics

We consider an elastic beam clamped at each of its ends and having length L 0 at rest. The left end is fixed while the right end is subject to an end-shortening of small amount ∆L. The cross-section of the beam is homogeneous and rectangular with a width b and thickness h. A fixed Cartesian reference frame {A; e 1 , e 2 , e 3 } is attached to the structure in its initial configuration R 0 . The reference frame is such that the unit vector e 1 lies in the direction of the beam axis. Due to the end-shortening, the distance between clamps becomes L = L 0 -∆L. The elastic beam is initially straight and free of stress (see Fig. 1.a for a sketch). The beam undergoes a buckling caused by the imposed end-shortening at the right end which is the current configuration. A compressive force P is associated to the end-shortening and it can be viewed as the conjugate variable of the end-shortening ∆L. This force will be determined once the problem is completely solved. The beam in the reference configuration is described by the set

R 0 = {OG 0 = q 0 (s) = s e 1 , with s ∈ [0, L 0 ]} (1)
where s is the abscissa along the beam axis and q 0 (s) is the position vector of the material point G 0 . The deformed beam in the current configuration is a smooth curve (see Fig. given by We denote by τ the unit vector tangent to the curve describing the deformed beam at the point G(s). The tangent vector is defined by

OG = q(s) = x(s) e 1 + y(s) e 2 , s ∈ [0, L 0 ] . (2
τ = dq ds , ( 3
)
where s is the curvilinear coordinate measured along the deformed beam. In addition, we introduce Λ = ds ds , the ratio of the length of differential line element of the beam in the deformed configuration to that of the undeformed configuration. The parameter Λ measures the extensibility of the beam.

The normal unit vector n is perpendicular to the tangent vector τ . Accordingly, the set of orthogonal unit vectors {τ , n, e 3 } forms the local frame attached to the deformed beam at the curvilinear abscissa s. These vectors can be written in the reference frame {A; e 1 , e 2 , e 3 } using the following transformation τ (s) = cos θ(s) e 1 + sin θ(s) e 2 , (4a)

n(s) =-sin θ(s) e 1 + cos θ(s) e 2 , (4b)
where the angle of rotation θ is θ = (e 1 , τ ). The reader is referred to Fig. 2 for the sketch of different vectorial variables used in the forthcoming sections. The gradient of the beam transformation characterized by the vector q(s) can be easily obtained from Eqn.

T (s) n N (s) τ R (s) F (s) n M (s) e 1 e 2
(3) accounting for the extensibility Λ of the beam and it can be written down as

q = Λτ , (5)
where the prime is the derivative with respect to the curvilinear abscissa s.

The extensional strain ε and the beam curvature κ are chosen as the two strain measures. They are given by

ε = Λ -1 , (6)
κ = θ, s = dτ ds • n = 1 Λ θ, s . (7)
The vector position in the current configuration of the beam is written in the local frame {G; τ , n, e 3 } given by

q(s) = q T (s) τ + q N (s) n , (8)
which can be compared to that of given by Eqn. (2). Consequently, we have the choice to compute the beam transformation either in the reference frame {A; e 1 , e 2 , e 3 } or in the local frame {G; τ , n, e 3 } by sake of convenience. The formulation from one frame to the other can be obtained by using Eqn. ( 4). The geometric relations Eqn. ( 5) become

q T -θ q N = Λ , (9a)
q N + θ q T = 0 . (9b)
In the forthcoming sections, especially for the variational formulation, the use of variables defined in the local frame will be most useful.

Constitutive equations of the beam

The beam cross-section is rectangular and homogeneous and it undergoes internal resultants in force and moment which is defined as distributor of the internal actions given by the set {R(s), M(s)}.

The force and moment read

R(s) = R x (s) e 1 + R y (s) e 2 = N (s) τ + T (s) n , (10a)
M(s) = M (s) e 3 . ( 10b
)
The quantity N is the axial (perpendicular to the cross-section) force along the beam axis, T is the shear force and M is the bending moment with respect to the e 3 direction. The resultants in force and moment acting on the beam cross-section are functions of the strain measures defined in the above section. The elastic behavior of the beam is assumed to be linear with respect to the strain measures. Accordingly, the constitutive equations are stated as

N (s) = EA ε(s) , (11a)
M (s) = EIκ(s) , ( 11b
)
where E is the Young modulus, A the area of the cross-section (in the present case A = b h) and I is the moment of inertia of the cross-section with respect to the e 3 direction. In addition, EA and EI are the axial beam and bending stiffnesses, respectively.

3 Variational formulation for the buckling bistable beam

Principle of virtual work

To obtain the beam equations, we adopt the principle of virtual work. The principle for the present model of beam and configuration is stated as follows (in the static case)

δS = δW i + δW e = 0 . (12)
The different contributions to the principle are W i for the work of internal forces, W e for the work of the applied actions.

The virtual work of internal forces of an elastic beam takes on the general form (the details of the algebraic manipulation can be found in [START_REF] Camescasse | Bistable buckled beam: Elastica modeling and analysis of static actuation[END_REF])

δW i = - L0 0 {R(s) • δq (s) + M(s) • δp (s) -(q (s) × R(s)) δp(s)} ds (13)
where "×" denotes the vector product. We recall that the vector q(s) is the position vector of the material point G (s) (see Eqn. (2)) and p = θ e 3 is the vector rotation of the beam cross-section.

The bistable beam is subject to a lineic density of force applied into the beam staying in (e 1 , e 2 )plane. The virtual work of this density of force reads

δW e = L0 0 F(s) • δq(s)ds (14)
where δq(s) is the virtual displacement of the material point G(s) belonging to the deformed beam.

Dimensionless equations

For the sake of consistency and in order to highlight some key parameters, we introduce dimensionless variables and parameters, defined by

• lengths (S, Q, ∆ ) = (s, q, ∆L) /L 0 , (15a)
• Forces and moments (N, T, P, M) = (N/F 0 , T /F 0 , P/F 0 , M/M 0 ) ,

• Density of forces

F = FL 0 /F 0 , (15c)
• Energy

E = E tot /E 0 , (15d)
where the reference parameters have been defined as

F 0 = EAk, M 0 = EI/L 0 and E 0 = F 0 L 0 .
Moreover, the following key parameter has been introduced

k = I AL 2 0 (16)
which characterizes the ratio of the compression energy over the bending energy. The parameter k ∝ (h/L 0 ) 2 where L 0 /h is the slenderness ratio of the beam. Now, with the new dimensionless variables the position vector takes the form We now propose a new form for the variational equation Eqn. ( 12). However, the variational equation is written with the variables defined in the local frame. It reads as

Q(S) = Q X (S) e 1 + Q Y (S) e 2 = Q T (S) τ + Q N (S) n (
δS[Q, θ] = 1 0 {(N -θ T) (δQ T -Q N δθ) + (θ N + T -F) (δQ N + Q T δθ)}dS + 1 0 {M + (1 + ε) T}δθdS = 0 . (18)
It has been assumed here that F = F n, that is, every element of force density is perpendicular to the deformed bistable beam. Obviously, in the algebraic manipulations leading to Eqn. (18), we accounted for the boundary conditions at the end of the beam. In particular, for clamped boundary conditions we have δθ = 0 and δQ = 0, because the displacement and the rotation of the beam at its ends are given.

Equations for the bistable buckled beam

The variational equation Eqn. ( 18) must be satisfied for any arbitrary variation δQ and δθ which meets the boundary conditions at S = 0 and S = 1 (clamped-clamped beam). We deduce the equations of the bistable beam as

N -θ T = 0 , (19a) θ N + T -F = 0 , (19b)
M + (1 + ε) T = 0 . ( 19c
)
Along with the above equations the associated boundary conditions read as

Q (0) = 0, Q (1) = X B e 1 , θ (0) = 0, θ (1) = 0 . (20)
In the second boundary equation Eqn. ( 20), X B denotes the abscissa of the right end of the beam due to the end-shortening of the clamp B. With the view of completing the above differential equations, it is convenient to write the geometrical equations Eqns. ( 9) in non-dimensional notation

Q T -θ Q N = 1 + ε , (21a)
Q N + θ Q T = 0 . (21b)
A similar transformation holds for equations Eqns. ( 5). Their form using the non-dimensional notation introduced in Eqn. ( 15) can be obtained easily. Now, we have almost all the ingredients to solve the problem of a bistable buckled beam subject to a density of lineic force. It is worth noting that each elementary lineic force is always perpendicular to the deformed beam while the beam is snapping.

At first sight, we have a set of nonlinear differential equations with given boundary conditions.

This kind of problem is not trivial because first of all, there exist unstable equilibrium regions spanned while the beam switches from one stable state to the other one. In addition, the classical method cannot capture the configuration of the beam in the unstable region. The second difficulty is that at each step of the process method, the configuration of the lineic density force is modified because the density force is always normal to the deformed beam which is an unknown of the problem. Therefore, a more sophisticated algorithm to solve the problem becomes necessary.

4 Numerical analysis and results

Problem statement

This section is devoted to the numerical investigation of the nonlinear boundary value problem governing the snap-through of the bistable buckled beam. The equations to be solved were obtained in the previous Section. More precisely, they are the equations of the beam equilibrium Eqns. ( 19) and the geometric relationships Eqns. ( 21).

We recall that the axial strain and bending moment satisfy the constitutive equations Eqns (11) written in dimensionless form as

ε = k N, M = θ . ( 22
θ (0) = 0 , (23a)
Q T (0) = 0 , (23b)
Q N (0) = 0 . (23c)
• At the B end (S = 1) Figure 3 shows the sketch of the clamped-clamped buckled beam for a given density of lineic force while switching. The detail of the lineic density produced by Laplace force is also shown, especially the induction field and the vector of electric current density par unit of area. In classical electrodynamics, the Laplace force acting on a small part of elementary volume dv of the beam is given by [START_REF] Jackson | Classical electrodynamics[END_REF])

θ (1) = 0 , (24a)
Q T (1) = X B , (24b)
Q N (1) = 0 . ( 24c
dF = j × B dv ( 25
)
where j is the density of electric current per volume unit such that dv = Ad (A = b h being the beam cross-section area). On introducing the electric current per unit of length i τ = A j where τ is the unit vector tangent to the deformed beam, the Laplace force acting on a piece of the beam of length d is then

dF = i d τ × B . ( 26
)
The beam is placed between two magnets producing the magnetic induction B assumed to be uniform in the volume between the two magnets. For a given magnetic induction B, the only free parameter is the electric current. It will be the control parameter.

Algorithm for solving the switching bistable beam problem

To solve the set of nonlinear equations Eqns. ( 19) using an algorithm based on a Runge-Kutta method, we need the initial values for N, θ, M and T at S = 0. At first sight, the values N (0) = N A , M (0) = M A and T (0) = T A which are the clamping efforts are not known. Accordingly, a numerical shooting algorithm must be used (see [START_REF] Ascher | Numerical solution of boundary value problem for ordinary differential equations[END_REF] for the implementation method).

The input parameters of the method are given by boundary condition Eqns. ( 23) and we start with an initial guess for the set of parameters {N A , T A , M A = θ A }. The objective parameters are given by Eqns. ( 24). Moreover, we do not know the buckling force P = N A since only the end-shortening of the right end of the beam is given. The buckling force P is therefore obtained by solving the buckled beam problem with loading set to zero (F = 0).

The boundary value problem to be solved is not trivial since we are faced with unstable branches of solution while the beam is switching from one stable state to the other one. To avoid the divergence of the algorithm and discrepancy of the final results, we control the vertical displacement of the material point of the beam at the curvilinear abscissa S C such that

Q X (S C ) = (1 -∆ ) /2.
The idea is to increment the mid-point by a very small amount at each value Q Y (S C ) = Y C that corresponds to an unique deformed beam, while a same density of lineic actuating force can produce, a priori, different beam deformations.

A second numerical method is proposed to ascertain the first one, based on the finite element method. More precisely, we take the advantage of the FEniCS numerical libraries [START_REF] Logg | Automated solution of differential equations by the finite element method : The FEniCS book[END_REF].

The numerical code deals with the minimization of the total potential energy associated to the bistable beam and its actuating energy. The problem at hand is to find the equilibrium configurations, as with the first method. The difficulty is that for a given loading, different beam deformations can be obtained. We must have recourse to a more sophisticated algorithm using a so-called deflated continuation method by varying iteratively the loading parameter [START_REF] Farrel | The computation of disconnected bifurcation diagrams[END_REF] in order to capture the successive bifurcation points and the associated beam equilibria that can be either stable or unstable.

Results and comments

The forthcoming results have been obtained for a thin elastic beam made of amagnetic stainless steel A316L with a Young modulus of E = 207 GPa, length of 60 mm, width of 5 mm and thickness of 0.05 mm. The average magnetic induction between both magnets is about 0.3 Tesla. One of the first numerical results is the beam response under the density of lineic force produced by the Laplace force. Using the numerical method we compute the actuating force or, more conventently, the electric current, as function of the mid-point displacement of the beam. large (large end-shortening of the order of 3% or great buckling deflection). At first sight, the curves are quite similar in shape. The present results can be compared to those obtained with a punctual force actuation [START_REF] Camescasse | Bistable buckled beam: Elastica modeling and analysis of static actuation[END_REF]. The part of the force-displacement diagram which is almost a line corresponds to the unstable region. Note that all the curves pass through zero (y m = 0, i = 0), and are symmetric with respect to the origin. Moreover, all the curves display a negative slope between extrema, which means that the bistable system possesses a negative stiffness on this part of the branch, which corresponds to instability. An important comment from the present results is that both numerical methods give quite similar results, which confirms the robustness of the modeling of the bistable buckled beam based on elastica theory. The results are applied not only for weak buckling deflection but also for rather large deflection in comparison to the beam slenderness. The concept of Laplace force is not so easy to handle. A question which could be raised is what would be the bistable response if the Laplace force is replaced by vertically fixed force? This point is discussed in Appendix A.

The second result extracted from the numerical simulations is the compressive force P applied at the right end of the beam in order to maintain a given end-shortening as function of the displacement computed at the beam mid-point. The results for three different values of the end-shortening are reported on Fig. 5.

We observe that, for each curve, the compressive force for null loading corresponds to the buckling critical force depending on the chosen end-shortening. As soon as the electric current is switched on and it begins to increase, the compressive force increases rapidly as well until it reaches a plateau.

As in the previous results the comparison of both numerical approaches gives very close results, which confirms, once more, the reliability of the present modeling based on elastica beam theory.

Instructive results can be obtained by computing the total potential energy. We recall that the numerical using FE method is based on the minimization of the total potential energy of the system. The latter reads as

E = 1 0 1 2 Mκ + 1 2k Nε dS . ( 27
)
The first term on the right hand side in Eqn. ( 27) represents the density of bending energy and the second term is the energy density of compression. The potential energy is computed using the results from the numerical simulations of the model equations (see Eqns. ( 19)) for each configuration of the deformed beam while the beam is snapping. Figure 6 shows the variations of the total potential energy as function of the beam deflection at the beam mid-point from the upper stable position to lower one and vice versa. The FE results have been superposed to the results of the first method in Fig. 6. It worth noticing that the potential energy possesses two minima corresponding to the stable positions of the bistable beam. The relative error between both numerical results is less than 0.4%. One of the advantages of the elastica model and the path-following technique used to solve the nonlinear boundary value problem is that we are able to capture with enough accuracy the beam deformation when passing through the zone of instability of the bistable system. However, for rather small end-shortening, the beam cross-section rotation and displacements have small amplitudes. Therefore, the equations for the bistable beam can be linearized. More precisely, the set of equations Eqns. (19 and 21) are linearized with zero actuating loading (F = 0) around the critical buckling force P c for P > P c . Trivial algebraic manipulations lead to the following equation for the beam deflection

Q (4) Y + P (1 -kP) Q Y = 0 . ( 28
)
Here we recover the classical Euler equation for a buckling beam; however, in Eqn. ( 28) the buckling force P is altered by the factor (1 -kP) accounting for the compressibility of the beam in the prebuckling regime. We point out that the deflection of the beam Q Y has been chosen instead of the rotation θ because the deflection must satisfy four boundary conditions at the ends of the beam, which not the case for the rotation. Eqn. ( 28) possesses two kinds of solution (symmetric and asymmetric modes) satisfying the boundary conditions for clamped-clamped beam, namely

Q Y (0) = 0, Q Y (0) = 0, Q Y (1) = 0, Q Y (1) = 0.
Appendix B provides the buckling mode solutions to Eqn. ( 28) along with the associated boundary conditions. Going back to the solution deduced from the numerical simulations of the boundary value problem, we can write the solution as an expansion of the buckling modes

Q Y (S) = J j=1 A j Y j (S) (29)
where Y j (S) are the buckling modes (see Appendix B for their exact forms). The factors A j correspond to the contribution of the modes. The first derivatives (and also the second derivatives) form an orthogonal vectorial base. Therefore, by multiplying the derivative of the expansion Eqn. ( 29) with respect to S by the j's buckling mode, we arrive at

A j = Q Y (S) , Y j (S) (30)
where •, • is the scalar product in the functional space L 2 [0, 1].

The result of the computation is shown in Fig. 7 for the first five modes. We observe that the first mode is predominant in the vicinity of the stable positions, and that its influence decreases rapidly when the bistable is switching as the second buckling mode takes over from the first one.

When the bistable passes through a zero deflection at the mid-point of the beam, the second mode prevails, and the bistable is in the region of instability. Fig. 7 gives interesting information about the buckling mode contribution, especially the third mode, which is an even mode. It plays a significant role in the switching process since 30 per cent are involved in the snapping scenario.

The other modes, the fourth and fifth are not really significant, as their contributions are less then 4%. This result demonstrates that if we want to use a reduced model, the first three buckling modes should be enough to describe the beam switching in the framework of small end-shortenings.

Fig. 8 summarizes the beam deflection at the different step of the snapping process.

On using the path-following procedure with the control parameter defined by vertical coordinate of the crossing point of the beam with the vertical line fixed at the mid-point between both clamps, we are able to capture, step by step, the beam deformation. More precisely, when the bistable is switching, the crossing point moves along the vertically fixed line. To each crossing point corresponds a unique beam deformation. Accordingly, the switching scenario can be seen as follows : (a) when the actuating force is zero the bistable beam deformation corresponds to the first buckling mode; (b) when the electric current increases, the beam mid-point increases as well; however, for a strong enough lineic force, the beam starts snapping, accordingly the beam

-4 -3 -2 -1 0 1 2 3 4 y m [mm] 0 20 40 60 80 100 A i [%] A1 A2 A3 A4 A5
Figure 7: Contributions of the first five buckling modes in the snap-through process (in percent) for an end-shortening of 1% of the total length. profile is mainly a combination of the first three buckling modes with the mode contribution given in Fig. 7; (c) the bistable beam is passing through the domain of instability, the electric current must be decreased, the beam mid-point passes through zero, the second buckling mode is therefore predominant; (d) the beam mid-point continues to increase and we recover the profile of the beam at point (b), but symmetrically inverted and finally; the beam ends its running at the upper stable position (e) and we recover the first buckling mode, but upside down.

4 3 2 1 0 1 2 3 4 y m [mm] 3 2 1 0 1 2 3 i [A]

Experimental validations

This section reports an effort to bridge the modeling approach presented in the above sections with experimental observations. The emphasis is placed on the validation of the bistable response in electric current (lineic density of force) as function of the mid-point vertical displacement of the beam subject to electromagnetic Laplace force. Comparisons of the proposed experimental results to those derived from the bistable beam based on elastica theory will be discussed. A thin elastic beam having the same dimensions and material properties as used in the analytical and numerical studies was fabricated for the experiment (see Section 4.3).

Description of the experimental set up and process of measures

The thin elastic beam is manufactured from a sheet of amagnetic stainless steel A316L using electro-eroding process. Several test beams have been cut with an accuracy of less than 1/100 mm.

Particular attention is paid to obtaining a high quality planar beam with a minimum of residual stress. The beam is thus subject to metallurgical process consisting in heating at 500 0 C in an oven for few hours and maintained absolutely planar and then in cooling gradually to the ambiant temperature.

The thin beam is placed between two clamps which maintain the beam ends very tightly. Fig. 9 shows the photos of the complete experimental set-up. The test set-up is equipped with a micrometric translator at one of the beam ends. The translator allows us to control the displacement of the beam ends (the right end) while the other is fixed. Using travel translator stage we are able to monitor the end-shortening of the beam with 10 µm adjustment (see Fig. 9). Moreover, a digital micrometer with an accuracy of 1 µm is used in order to control the displacement of the platform of the moving clamp.

Experimental measurement and discussion of the results

Bifurcation diagram. One of the first experimental results concerns the bifurcation diagram, namely, the beam deflection at its center as function of the end-shortening when actuation is set to zero. On using the travel translator we impose the desired end-shortening at the right end of the beam. The end-shortening is read directly on the digital comparator. The beam deflection at its center is measured thanks to the laser profilometer and the value is transmitted to the computer. Accordingly, by varying the axial translation we obtain first, the value of the end-shortening and then, the corresponding deflection of the beam. Figure 11 shows the bifurcation diagram. The experimental data are depicted by small red points and they are superimposed to the result deduced from the numerical simulations (the curve in solid line). We notice a strong fit between the model and experimental data. Obviously, because of the high slenderness ratio of the beam L h = 12000, the pre-buckling zone is so tiny that it is not observable.

Discussion and concluding remarks

The main objective is to develop and analyze a model for bistable buckled beam based on an elastica approach of beam, including extensibility. On using the complete model that is, by considering rotations and displacements of large amplitudes, we examine in detail the switching process of the bistable beam. The beam actuation is produced by a lineic density of uniform force. This means that the force configuration follows the beam deformation variation while the bistable transits from one stable position to the other one. As a matter of fact such an actuating force can be produced by an electromagnetic Laplace force. The latter is due to the electric current travelling in the beam placed in a magnetic induction.

The most pertinent results describing the switching process of the bistable buckled beam are obtained first with the use of semi-analytical solution by solving the nonlinear boundary value problem. At last, experimental validation of the bistable beam actuated by an electromagnetic Laplace force is proposed and confirms with a high quality of accuracy the present model, which demonstrates that this apparently simple bistable system actuated by a contactless process is rich enough to provide particularly interesting comments :

• The bistable modeling in the elastica theory framework allows us to write down the governing beam equation for transformation of large amplitudes. The equations of the nonlinear boundary value problem are deduced from a variational formulation. Moreover, the equations of the bistable beam are written in the local referential frame attached to the actual configuration of the beam (deformed beam).

• Important results concern the response of the bistable beam to the Laplace force. The responses have been computed for several end-shortening values and they describe the evolution of the electric current (through the computed applied lineic force) as function of the vertical displacement of the beam mid-point. The response exhibits an N-shaped curve. The numerical algorithm is efficient enough, which allows us to capture the beam configurations at each step of the vertical increment, even for unstable branches. A second numerical investigation based on the FE method and path following procedure is also reported. The results of the second numerical method confirms perfectly the first numerical results, especially the robustness of the beam modeling based on elastica beam theory.

Among the results which can be extracted from both numerical investigations, we have examined (i) the bistable response, the electric current as function of the displacement of the beam mid-point, (ii) the buckling loading applied to one of the beam ends to maintain the given end-shortening, (iii) the total potential energy exhibiting minima at the each stable position of the buckled beam (upward and downward) and (iv) the analysis of buckling modes and their precise contribution to the snapping process.

• Finally, a section is devoted to experimental tests which have ascertained the validity of the model. The comparisons between the experimental tests and numerical results are quite good, with very small discrepancies. In short, a good correspondance is observed between the bistable response modeling prediction and experimental results, which establishes a reliable validation of the present model.

All the results presented in this study are good reasons to undertake complementary and forthcoming works, on the nonlinear dynamics of the bistable beam in the post-buckling regime when applying time-dependent electric current through the beam. Experimental investigations will have to be considered to confirm the model predictions. The model along with the obtained results opens new views to design innovative devices in the area of micro-robotics, medical engineering, etc.

Appendix A Comparison with bistable switching by means of a lineic density of vertically fixed force.

A question that we might ask ourselves is what would the bistable beam response be if the Laplace force was replaced by a lineic density of force with fixed vertical direction. Would there be a noticeable difference? To understand the problem better, we performed a numerical simulation of the bistable response subject to a lineic fixed force. The result is illustrated in Fig. 13. The curves are superposed for the same end-shortening of 3 % and for the same order of lineic density of force. A difference is clearly observed, as the maximum of lineic density of follower force actuation is little bit greater than that of the vertically fixed force. In addition, the relative difference between maxima have been computed and presented in Fig. 14. The relative difference is defined by where F V is for the vertical force and F L referred to the Laplace force. We notice that δF increases with the end-shortening. For instance, an end-shortening of the order of 5% gives rise to a relative difference between both kinds of actuation of about 5.4%.

Accordingly, we cannot reasonably use the lineic density of vertically fixed force as an approximation of the Laplace force.

Appendix B Buckling modes for a clamped-clamped beam

The buckling modes are obtained by solving Eqn. ( 28 In Eqns (B.1) and (B.3) A j is an amplitude that we have at hand to be adjusted in order that the mode is normalized.

  1.b). The point G 0 in the initial configuration is then transformed into the material point G of the deformed beam

)Figure 1 :

 1 Figure 1: Clamped-clamped elastic beam: (a) Beam in the reference configuration at rest (no load) (b) Beam in its buckled configuration.

Figure 2 :

 2 Figure 2: A part of the beam in the deformed configuration: (i) Internal actions (force and moment) at the material point G (s), (ii) Fixed and local frames.

  17) which is the position vector of any material point of the beam at the curvilinear abscissa S written either in the fixed frame or in the local frame. The different coordinates in Eqn. (17) satisfy the dimensionless definitions introduced in Eqn. (15).

)

  Note that at the ends A and B of the beam we have τ = e 1 and n = e 2 . In the set of equations Eqns. (19), Eqns. (21) have been included to compute the local displacement Q N and Q T as function of θ and N and boundary conditions concern those displacements. The set of equations that meets the boundary conditions (clamped-clamped beam) reads as • At the A end (S = 0)

Figure 3 :

 3 Figure3: Clamped-clamped buckled elastic beam: the Laplace force acting on the deformed beam and detail of an element of beam undergoing Laplace force produced by a volume density of moving electric charges placed in a magnetic induction.

Figure 4 :

 4 Fig.4. A classical N-shaped curve is obtained for three different end-shortenings: (i) weak (small end-shortening 0.3% of the beam length), (ii) intermediate (end-shortening of 1%) and (iii) rather

Figure 5 :

 5 Figure 5: Bistable response: force or buckling loading as function of the beam midpoint displacement for three different values of the beam end-shortening (in solid line: the elastica model, small triangles and diamonds: FE results).

Figure 6 :

 6 Figure 6: Bistable response: Total potential energy as function of the beam mid-point displacement for three different values of the beam end-shortening (in solid line: the elastica model, small triangle and diamonds: FE results).

Figure 8 :

 8 Figure 8: Switching scenario of the bistable beam, (a) beam at the lower stable position, (b) region of instability, combination of buckling modes, (c) second buckling mode, (d) beam profile symmetrically inverted with respect to that at (b), (e) beam at the upper stable position.

Figure 9 :Figure 10 :

 910 Figure 9: Photos of the experimental set-up: (a) the global view of the different apparatus, (b) the detail showing the beam placed between both magnets and (c) the measurement of the mid-point beam displacement owing the laser profilometer.

Figure 11 :

 11 Figure 11: Diagram of bifurcation: beam center deflection as function of the end-shortening. Experimental results: red spot, model: blue solid line

Figure 12 :

 12 Figure12: Bistable response: experimental results superposed onto the numerical simulations of the bistable beam model.

Figure 13 :Figure 14 :

 1314 Figure 13: Lineic density of force as function of the beam mid-point displacement. Comparison between bistable actuations by means of lineic density of Laplace force and vertically fixed force. Laplace force (in solid line) and vertically fixed force (dashed line).

  ), accounting for the boundary conditions for a clamped-clamped beam. The solution splits into two kinds of modes.(i) The symmetric buckling modes read asY j (S) = A j (1cos (α j S)) (B.1)withα j = 2jπ (B.2)Moreover, we have set α 2 = P (1 -kP) in Eqn. (B.1).(ii) The antisymmetric modes are given byY j (S) = A j 1cos (α j S) -2 α j (α j Ssin (α j S)) (B.3)where now the values of α j are the solutions to the implicit equation

Acknowledgements

The research work reported in the paper has been supported by the research project BISCELTECH funded by Fonds Unique Interministériel (FUI-APP21).


Peng Li

Virginie Chamard

Imaging of radiation-induced structural evolution via ptychography

Imaging of radiation-induced structural evolution via ptychography

Peng Li 1 , Virginie Chamard 11 Aix -Marseille Université, CNRS, Centrale Marseille, Institut Fresnel, 13013 Marseille, France peng.li@fresnel.fr Ptychographic microscopy, in forward or Bragg diffraction geometry has produced impressive results based on the ability to image in 3D and at high spatial resolution the structural details of complex samples [1]. It takes advantage of inversion procedures, which retrieve the sample complex-valued scattering contrast by phasing back the coherent diffraction intensity patterns acquired at different sample-beam positions with enough beam overlapping between neighboring positions. This introduces redundancy which is the key for the inversion algorithm to converge. In the Bragg geometry, a ptychography modality our group has pioneered [2], new structural details obtained on a shell calcite prism have allowed us to progress in the understanding of calcareous biomineralization [2]. As they are based on the use of coherent xrays, those inversion methods will further spread out with the forthcoming sources of increased coherent flux (a factor of about 100 is expected with the ESRF EBS program). However, "less successful" results, where experiments have been corrupted by the onset of radiation damages, are reported. Those damages result in a strong modification and ultimately destruction of the diffracted signal, as a function of the x-ray dose [3]. Some attempts to circumvent this problem have been proposed, such as performing fast acquisitions of a given damaged sample state [3]. However, the blurring of the signal, due to the time evolving sample, strongly compromises the quality of the reconstruction [3].

Here we propose a novel strategy, which has been proved to be successful with simulation. We used a simple model for the radiation-induced structural evolution, but a more realistic model might be introduced, based on the ongoing advances regarding the understanding of radiation damage propagation. The proposed strategy is based on a ptychography approach, which has already shown the capability to retrieve additional parameters (like the probe profile and positions, the mode structure, etc.). Our simulation shows that a ptychography scan of a sample experiencing radiation-induced structural evolution (leading to its destruction) contains enough information to ensure the retrieval of the different evolving sample states (5 states in our simulation). The radiation-induced structural evolution is linearly dependent on the photon dose deposited on the sample. Hence, when a Gaussian beam illuminates the sample, the region in the vicinity of the beam maximum evolves rapidly, while the regions with less photons evolve slowly. Thereby, at each beam position, many evolving states coexist, and their diffraction patterns add up incoherently to give the final intensity pattern. With the redundancy introduced by the overlapping scan, the reconstruction is successful, and the retrieved states present good agreement with the original ones. Note that given the fact that a finer sampling of the state evolution is possible, we are confident that the more realistic case of a continuous sample evolution is also solvable with this transient-ptychography approach.

This work was funded by the European Research Council (ERC) under the European Union's Horizon H2020 research and innovation program grant agreement No. 724881.