Noncontact evaluation of full elastic constants of perovskite MAPbBr3 via Photoacoustic eigen-spectrum analysis in one test

Elasticity is one basic property of hybrid organic-inorganic perovskites. It highly relates to many fundamental processes in solid physics. The investigation of elasticity is of interest not only to explore the intrinsic properties of a material, but also to improve their potential application performance. In this study, we predict photoacoustic eigen-spectrum (PAES) of single crystal. Then by solving the inverse problem of the generation of PAES, we propose a noncontact method to determine a complete set of elastic constants of single crystal in one test. Experiments confirm the proposed method accurately determines all elastic constants of MAPbBr3. Since this method is totally noncontact and does not require multiple specimens cutting along different crystal axes, it could be more competent for rare, tiny and brittle specimen, or when the specimen is immersed in turbid or opaque medium. Benefitting from these superiorities, the proposed method might be found prominent values in materials science and applications.

elastic constants derived from pAeS. The elastic constants can be estimated from the measured eigenfrequencies by solving the inverse problem. Figure 2 shows the convergence procedure of elastic constants estimation, where Fig. 2(a-c) are the deviation functions in C 11 -C 12 space, C 11 -C 44 space, and C 12 -C 44 space, respectively. Here, the deviation function is defined in Eq. (2). The measured frequencies f i m are extracted from the photoacoustic spectrum as shown in Fig. 1. The predicted eigenfrequencies f i c (x) are calculated by solving Eq. (1) with the elastic constants given by the x-axis and y-axis. The color represents the frequency offset between the prediction values and the measured frequencies. The initial values to predict the eigenfrequencies are set as x = [C 11 , C 12 , C 44 ] = [45 GPa, 30 GPa, 2 GPa]. Initially, the predicted frequencies are far away from the measured frequencies F(x 0 ) = 0.63414, since the preset parameters do not match the accurate elastic constants of the specimen. Then, we optimize the elastic constants by solving the inverse problem. The empty dots in Fig. 2 show the convergence process of approaching the best-fitting parameters. After 10 times of iterations, the parameters are converged at x opt = [C 11 , C 12 , C 44 ] = [30.63 GPa, 11.6 GPa, 3.63 GPa] and the deviation function is reduced to F(x opt ) = 0.001. It is seen that the parameters arrive at the valley of the deviation function, as shown by the stars in Fig. 2. The predicted frequencies match the measurements best. The optimized parameters x opt measure the elastic constants of the specimen.  www.nature.com/scientificreports www.nature.com/scientificreports/ Figure 3 illustrates the results of elasticity evaluation. Figure 3(a) shows the measured frequencies f i m against the predicted eigen-frequencies f i c (x opt ) for seven specimens. All frequencies agree with each other very well after solving the inverse problem. The optimized parameters x opt can accurately predict the eigen-vibration of specimens, in other words, x opt precisely measure elasticity of specimens. Figure 3(b) presents the measured elastic constants C 11 = 30.7 ± 2.46 GPa, C 12 = 10.53 ± 1.62 GPa, and C 44 = 3.51 ± 0.11 GPa of MAPbBr 3 , and error bars indicate standard deviation of multiple measurements. The results agree with measurements by laser ultrasonic(LU), INS and BS method, reported in the previous studies, as shown in Table 1 11,12,30 . Moreover, the measurements of seven specimens have good consistency. The proposed noncontact method measures all elastic constants of MAPbBr 3 .

Discussion
In summary, we have predicted the PAES of a single crystal. Then by solving the inverse problem of the generation of PAES, we proposed a noncontact method to determine a full set of elastic constants of MAPbBr 3 from the detected PAES in one test. In comparison to other methods used for the measurement of elastic constants, PAES has its own unique advantages. Firstly, PAES is totally noncontact, whether the vibration generation or the signal detection. The specimen is only simply immersed in liquid. In contact RUS system, the specimen need to be clamped by transducers with an additional force. Therefore, PAES could be applicable for brittle materials and tiny specimen. Secondly, it can probe all elastic constants of the specimen in one test. INS and BS method require multiple specimens cutting along different crystal axes. These merits make PAES method be more competent for rare specimen. Thirdly, PAES is also different from laser ultrasound that it detects the photoacoustic wave emitted from the vibrating specimen, instead of directly measuring the vibrational surface of the specimen. This method is still available when the specimen is immersed in turbid or opaque medium. Benefitting from these characteristics (or superiorities), PAES might be found valuable in some practical applications, including a precise evaluation of elasticity of tiny rare specimen, online monitoring the elastic constants during the growth process of a crystal in solutions, and testing elasticity of materials implanted in biological tissue, etc. There are also some limitations of PAES. Firstly, soft elastomer could have very low eigen-frequencies, which may be difficult to excite its vibration and detect PA eigen-spectral signals. Secondly, if specimen has big damping factor, eigen vibrations will attenuate quickly and few frequencies could be extracted. Therefore, a high quality factor of the specimen is required. Finally, the proposed method is operated in liquid, a suitable medium is required to avoid dissolution or other chemical reactions between the specimen and medium. To extract elastic constants precisely, we can enhance intensity of PA signals by increasing laser fluence and choosing laser wavelength with high light absorption. Besides, an accurate measurement of size is always important when irregularly shaped specimen is tested. Along with improving technical details, this method would be applied to more materials.  www.nature.com/scientificreports www.nature.com/scientificreports/

Methods
Measurement system. Figure 4(a) gives the experimental setup. The specimen was stored in a beaker and immersed in dodecane. Since the specimen contacts was freely placed in the beaker and no additional force was applied on them, the free boundary condition is approximately satisfied. A Q-switched Nd: YAG laser with a wavelength of 532 nm and a pulse width of about 8~10 ns was used to illuminate the specimen and generate photoacoustic signals. Light absorption of MAPbBr3 starts from about 570 nm and has strong optical absorption at 532 nm 31 . Based on rough prior knowledge of specimen, it can be estimated that the eigen-frequencies of the specimen are about from 0.2 MHz to 5 MHz. An ultrasound transducer (V310, Panametrics) with a central frequency of 4.39 MHz and a relative bandwidth of 100.1% at −6 dB was immersed into dodecane in the beaker to receive the photoacoustic signals. This transducer has enough wide bandwidth to cover the main frequency range of the PA signals of specimens. The detected signals were amplified 46 dB (SA-230F5, NF) and sampled by a data acquisition board (PCI-5105, NI) with a frequency of 30 MHz. For each specimen, the measurement was repeated 200 times and averaged to reduce noise. Figure 4(b) is the photograph of seven specimens of MAPbBr 3 measured in the experiment which are prepared by the inverse temperature crystallization method 31,32 . Equimolar mixture of the CH 3 NH 3 Br and PbBr 2 were dissolved in dimethylformamide (DMF) to obtain a 1 mol/L solution of CH 3 NH 3 PbBr 3 . The above solution was heated and kept at 80 °C for several hours, then large number of small crystals can be readily harvested. For the sake of convenience, they are numbered from 1 to 7. Their sizes were measured and given in Table 2. The volume of the smallest specimen (No. 6) is only 0.5735 mm 3 and the biggest one (No. 5) is 15.9 mm 3 . Density is 3.8 g/cm 3 used in this study. pAeS prediction of elastomer. Let's consider an elastic specimen exposed to a pulsed laser. The specimen would absorb optical energy, arise transient thermoelastic expansion, and emit wideband ultrasound to the surrounding media. This is the so-called photoacoustic effect. As pulsed laser illumination ceases, the elastomer will keep free-vibrating and emitting ultrasound for a period of time, because of its elasticity and inertance. In this stage, the frequencies of free vibrations are the eigenfrequencies of the elastomer. Moreover, the emitted ultrasound has the same frequencies as the vibrations of the elastomer. Therefore, the PAES can be predicted by analyzing eigen-vibrations of the elastomer.
Considering an elastomer with free boundaries, its eigen vibration obeys the following matrix eigenvalue equation 33 ρ π Γ = f Ea a (2 ) (1) 2 where ρ is the density, the components of the vector a = [a iλ ] are eigen vector consists of the expansion coefficients of displacement u i = ∑a iλ Φ λ which is expanded in a suitable set of basis functions Φ λ , and the eigenvalues correspond to the eigen frequencies f. E and Γ are matrices whose elements are E λiλ′i′ = δ ii′ ∫Φ λ Φ λ′ dV, Γ λiλ′i′ = C iji′j′ ∫Φ λ,j Φ λ′,j′ dV respectively. C iji′j′ is the elastic tensor and subscript indices following a comma denotes   www.nature.com/scientificreports www.nature.com/scientificreports/ differentiation with respect to that coordinate. The forward problem of PAES generation can be solved from Eq.
(1), that is, the PAES of elastomer can be predicted with the given elastic constants.
inversion of elastic constants. If the PAES is detected, elastic constants could be estimated by solving the inverse problem, that is, to search a set of optimized elastic constants x opt = [C ij ] to minimize the deviation function 16,17 x n is the parameter vector after n-th iteration, m ]/ f i m is the deviation vector, J = dh/dx is Jacobian matrix, and α is a nonnegative damping factor. With the given initial x 0 and α, we can predict eigenfrequencies and calculate the first x 1 according to Eqs. (2)(3). With the calculated x 1 the next iteration is repeated. Finally, x converges at its optimized value, which makes F(x) reach the minimum value. The inverse problem is solved and x opt is the estimated elastic constants.

Verification of PAES.
Four spherical isotropic particles made of polystyrene divinylbenzene (PSDVB), stannum (Sn), brass (Brs), and steel (Stl) with known elasticity parameters are used to verify the accuracy of PAES. Figure 5 shows the estimated elastic constants values match the actual values very well. Zener anisotropy index A = 2C 44 /(C 11 -C 12 ) 37 of PSDVB, Sn, Brs, Stl are 0.87, 1.00, 1.07, 1.06 respectively that exhibit an elastic isotropic nature. The result confirms the accuracy of PAES method in elasticity measurement of isotropic materials.
Based on the verification of PAES, we proposed a noncontact method to measure the elasticity of the relevant member in HOIPs, named as MAPbBr 3 that has 3 independent elastic constants x = [C 11 C 12 C 44 ] in the cubic phase. A 2d 1 × 2d 2 × 2d 3 rectangular shaped specimen is illuminated by a pulse laser and the generated photoacoustic signals are detected. Since the photoacoustic signals contain the same frequencies as the eigen-vibrations of the specimen. The elastic constants of the specimen can be evaluated by the above inverse problem.

Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.