Elastic Properties and Energy Loss Related to the Disorder–Order Ferroelectric Transitions in Multiferroic Metal–Organic Frameworks [NH4][Mg(HCOO)3] and [(CH3)2NH2][Mg(HCOO)3]

Elastic properties are important mechanical properties which are dependent on the structure, and the coupling of ferroelasticity with ferroelectricity and ferromagnetism is vital for the development of multiferroic metal–organic frameworks (MOFs). The elastic properties and energy loss related to the disorder–order ferroelectric transition in [NH4][Mg(HCOO)3] and [(CH3)2NH2][Mg(HCOO)3] were investigated using differential scanning calorimetry (DSC) and dynamic mechanical analysis (DMA). The DSC curves of [NH4][Mg(HCOO)3] and [(CH3)2NH2][Mg(HCOO)3] exhibited anomalies near 256 K and 264 K, respectively. The DMA results illustrated the minimum in the storage modulus and normalized storage modulus, and the maximum in the loss modulus, normalized loss modulus and loss factor near the ferroelectric transition temperatures of 256 K and 264 K, respectively. Much narrower peaks of loss modulus, normalized loss modulus and loss factor were observed in [(CH3)2NH2][Mg(HCOO)3] with the peak temperature independent of frequency, and the peak height was smaller at a higher frequency, indicating the features of first-order transition. Elastic anomalies and energy loss in [NH4][Mg(HCOO)3] near 256 K are due to the second-order paraelectric to ferroelectric phase transition triggered by the disorder–order transition of the ammonium cations and their displacement within the framework channels, accompanied by the structural phase transition from the non-polar hexagonal P6322 to polar hexagonal P63. Elastic anomalies and energy loss in [(CH3)2NH2][Mg(HCOO)3] near 264 K are due to the first-order paraelectric to ferroelectric phase transitions triggered by the disorder–order transitions of alkylammonium cations located in the framework cavities, accompanied by the structural phase transition from rhombohedral R3¯c to monoclinic Cc. The elastic anomalies in [NH4][Mg(HCOO)3] and [(CH3)2NH2][Mg(HCOO)3] showed strong coupling of ferroelasticity with ferroelectricity.


Powder XRD
The powder XRD patterns of [NH 4 ][Mg(HCOO) 3 ] and [(CH 3 ) 2 NH 2 ][Mg(HCOO) 3 ] were collected through a Bruker D8 Advance diffractometer (Bruker, Billerica, MA, USA) using Cu Kα radiation with a wavelength of 1.5406 Å at 40 kV and 40 mA. The diffraction angle 2 was in the range of 10-60 • and the step size was 0.02 • . The Rietveld fit of the XRD patterns was obtained using GSAS.

DSC
The DSC measurements of [NH 4 3 ] were carried out in the compression mode using PerkinElmer Instruments Diamond DMA (PerkinElmer Instruments, Waltham, MA, USA) in the range of 140-300 K at 2 K/min during heating processes, as described in [39,53].  3] were carried out in the compression mode using PerkinElmer Instruments Diamond DMA (PerkinElmer Instruments, Waltham, MA, USA) in the range of 140-300 K at 2 K/min during heating processes, as described in [39,53].

Morphology
The            [15]. The enthalpy ∆ determined to be 53.0 -62.4 J⋅mol −1 and the entropy ΔS was determined to be 0.246 J⋅mol −1 ⋅K −1 . The ratio of the number of configurations in the disordered and ordere tems, N, was determined to be 1.025-1.030. It suggests that the transition is more co than a simple 3-fold order-disorder model, with N around 3. Maczka et al. report the λ-type anomaly in the DSC curve indicated that the ferroelectric transit [NH4][Mg(HCOO)3] was second order [15]. Variable-temperature Raman spectr cated no obvious jumps in the characteristic peak positions of the vibration group the transition temperature, indicating the nature of the second-order phase tra [19,22]. Figure 5b illustrates the DSC curves of [(CH3)2NH2][Mg(HCOO)3] with anom 261-264 K at the cooling and heating rate of 5 K/min. This is consistent with the tra temperatures of 258-263 K reported by Pato-Doldan et al. [35] and 259-267 K repor Asaji et al. [36]. The enthalpy ∆H was determined to be 2.596-2.723 kJ⋅mol −1 , and tropy ΔS was determined to be 9.8-10.4 J⋅mol −1 ⋅K −1 . N was determined to be 3.2 indicating a 3-fold order-disorder model for the dimethylammonium cation. Asa reported similar values of ∆H, 2.7 ± 0.2 kJ⋅mol −1 and ΔS, 10 ± 1 J⋅mol −1 ⋅K −1 [36].   3 ] was second order [15]. Variable-temperature Raman spectra indicated no obvious jumps in the characteristic peak positions of the vibration groups near the transition temperature, indicating the nature of the second-order phase transition [19,22].  [15]. Variable-temperature Raman spectra indicated no obvious jumps in the characteristic peak positions of the vibration groups near the transition temperature, indicating the nature of the second-order phase transition [19,22]. Figure 5b illustrates the DSC curves of [(CH3)2NH2][Mg(HCOO)3] with anomalies at 261-264 K at the cooling and heating rate of 5 K/min. This is consistent with the transition temperatures of 258-263 K reported by Pato-Doldan et al. [35] and 259-267 K reported by Asaji et al. [36]. The enthalpy ∆H was determined to be 2.596-2.723 kJ⋅mol −1 , and the entropy ΔS was determined to be 9.    3 ] with anomalies at 261-264 K at the cooling and heating rate of 5 K/min. This is consistent with the transition temperatures of 258-263 K reported by Pato-Doldan et al. [35] and 259-267 K reported by Asaji et al. [36]. The enthalpy ∆H was determined to be 2.596-2.723 kJ·mol −1 , and the entropy ∆S was determined to be 9.8-10.4 J·mol −1 ·K −1 . N was determined to be 3.25-3.49, indicating a 3-fold order-disorder model for the dimethylammonium cation. Asaji et al.

DMA
DMA was used to determine the complex modulus of the viscoelastic materials. The real part is the storage modulus E', the imaginary part is the loss modulus E", and E"/E' is the loss factor tanδ. The temperature dependence of the storage modulus E', loss modulus E" and loss factor tanδ of [NH 4 ][Mg(HCOO) 3 ] single crystals are exhibited in Figure  6a-c, and the temperature dependences of the normalized storage modulus E' T /E' 280 (i.e., the ratio of the storage modulus at temperature T to that at 280 K), normalized loss modulus E" T /E" 298 (i.e., the ratio of the loss modulus at temperature T to that at (e) (f)    [15]. The resonant ultrasound spectroscopy (RUS) study of the elastic properties and acoustic dissipation associated with the disorder-order ferroelectric transition in [NH 4 ][Zn(HCOO) 3 ] exhibited that, with the increase in temperature, the elastic moduli, which were proportional to the square of the resonant frequencies, gradually decreased. There was also a marked change in the rate of decrease near the ferroelectric transition temperature of 192 K, and acoustic dissipation gradually increased with a peak near 192 K [16]. The Brillouin scattering (BS) study of the elastic properties and acoustic dissipation associated with the ferroelectric transition in [NH 4 ][M(HCOO) 3 ] (M = Mn, Zn) displayed that, with the increase in temperature, the frequency shift for the longitudinal and transverse acoustic phonons propagating along the x axis gradually decreased, and anomalies occurred near the ferroelectric transition temperature, with an estimated relaxation time around 4.6 × 10 −13 s [22]. Figure 7 illustrates the temperature dependence of storage modulus E', loss modulus E" and loss factor tanδ of the [(CH 3 ) 2 NH 2 ][Mg(HCOO) 3 ] single crystals, and the normalized storage modulus E' T /E' 298 (i.e., the ratio of the storage modulus at temperature T to that at 298 K), normalized loss modulus E" T /E" 298 (i.e., the ratio of the loss modulus at temperature T to that at 298 K) and loss factor tanδ of the [(CH 3 ) 2 3 ] was first order, and it was associated with the freezing of the 120 • reorientation of the dimethylammonium ion around the c-c axis through the two methyl groups of the cation, which were non-equivalent below the ferroelectric transition temperature of 267 K [36]. The strong frequency dependence of the dielectric constant and the dielectric loss of [(CH 3 ) 2 NH 2 ][Mg(HCOO) 3 ] were reported, and the peak temperature increased at higher frequencies, indicating a dielectric relaxation mechanism [35]. The RUS study of the elastic properties and acoustic dissipation associated with the disorder-order ferroelectric transition in [(CH 3 ) 2 NH 2 ][M(HCOO) 3 ] (M = Mn, Co, Ni) exhibited a broadening of the resonant peaks above the ferroelectric transition temperature and kinks in the temperature dependence of the resonant frequencies, which are proportional to elastic moduli, near the ferroelectric transition temperature of 185 K (M = Mn), 165 K (M = Co) and 180 K (M = Ni), respectively [40,41]. Figure 8 demonstrates a double logarithmic plot of the frequency dependence of loss factor peak height, ln(tanδ) vs. ln(f), for the peak in the temperature dependence of tanδ near 256 K for [NH 4 3 ] with n between −0.382 and −0.078 [39].
Materials 2021, 14, x FOR PEER REVIEW 9 of 12 peaks above the ferroelectric transition temperature and kinks in the temperature dependence of the resonant frequencies, which are proportional to elastic moduli, near the ferroelectric transition temperature of 185 K (M = Mn), 165 K (M = Co) and 180 K (M = Ni), respectively [40,41]. The elastic anomalies and energy loss associated with the ferroelectric transitions were detected using BS at 0-30 GHz, RUS at 0.1-2.0 MHz and DMA at 0.1-10 Hz. This suggests that multiple relaxation processes may be involved with different relaxation times, e.g., around 10 −12 s detected by BS, around 10 −6 s detected by RUS and around 1 s detected by DMA. showed much narrower peaks of loss modulus and loss factor. The peak temperature was independent of frequency, and the peak height decreased at higher frequencies, indicating the features of the first-order phase transition. The frequency dependence of the loss factor peak height was consistent with the power law tanδ = Af n , where n was between −0.029 and −0.  The elastic anomalies and energy loss associated with the ferroelectric transitions were detected using BS at 0-30 GHz, RUS at 0.1-2.0 MHz and DMA at 0.1-10 Hz. This suggests that multiple relaxation processes may be involved with different relaxation times, e.g., around 10 −12 s detected by BS, around 10 −6 s detected by RUS and around 1 s detected by DMA. showed much narrower peaks of loss modulus and loss factor. The peak temperature was independent of frequency, and the peak height decreased at higher frequencies, indicating the features of the first-order phase transition. The frequency dependence of the loss factor peak height was consistent with the power law tanδ = Af n , where n was between −0.029 and −0. showed strong coupling of ferroelasticity with ferroelectricity. Multiple relaxation processes may be involved with different relaxation times, e.g., around 10 −12 s detected by BS, around 10 −6 s detected by RUS and around 1 s detected by DMA. Structure plays an important role in the elastic properties of MOFs. The study of the elastic anomalies and energy loss related to ferroelectric transitions is important for the development of multiferroic MOFs with high strength, ferroelectric transition temperature near room temperature, strong coupling of ferroelasticity, ferroelectricity and ferromagnetism for use in actuators, magnetic and electric devices.

Data Availability Statement:
The data presented in this study are available on request from the corresponding authors. The data are not publicly available due to privacy reasons.