Elastic Properties and Energy Dissipation Related to the Disorder-Order Ferroelectric Transition in a Multiferroic Metal-Organic Framework [(CH3)2NH2][Fe(HCOO)3] with a Perovskite-Like Structure

The elastic properties and the coupling of ferroelasticity with ferromagnetism and ferroelectricy are crucial for the development of multiferroic metal-organic frameworks (MOFs) with strong magnetoelectric coupling. Elastic properties and energy dissipation related to the disorder-order ferroelectric transition in [(CH3)2NH2][Fe(HCOO)3] were studied by differential scanning calorimetry (DSC), low temperature X-ray diffraction (XRD) and dynamic mechanical analysis (DMA). DSC result indicated the transition near 164 K. XRD showed the first-order structural transition from rhombohedral R3−c to monoclinic Cc at ~145 K, accompanied by the disorder-order transition of proton ordering in the N–H…O hydrogen bonds in [(CH3)2NH2]+ as well as the distortion of the framework. For single crystals, the storage modulus was ~1.1 GPa and the loss modulus was ~0.02 GPa at 298 K. DMA of single crystals showed quick drop of storage modulus and peaks of loss modulus and loss factor near the ferroelectric transition temperature ~164 K. DMA of pellets showed the minimum of the normalized storage modulus and the peaks of loss factor at ~164 K with weak frequency dependences. The normalized loss modulus reached the maximum near 145 K, with higher peak temperature at higher frequency. The elastic anomalies and energy dissipation near the ferroelectric transition temperature are caused by the coupling of the movements of dimethylammonium cations and twin walls.


Room Temperature Powder XRD
A DSC 200F3 calorimeter (Netzsch, Selb, Germany) was used to perform t measurements of [(CH3)2NH2][Fe(HCOO)3] crystals (around 16 mg) in a N2 filled e ment from 140 K to 300 K at the heating rate of 5 K/min.

Low Temperature Powder XRD
A Smartlab powder XRD system (Rigaku, Tokyo, Japan) was used to collect th der XRD patterns from 300 K to 10 K. 2θ was between 10° and 70°, and the step s 0.02°. The temperature step of 20 K was set between 300 K and 180 K, step of 5 K b 180 K and 120 K, step of 20 K between 120 K and 20 K, and step of 10 K between 2 10 K. The temperature accuracy was 0.1 K using a Lakeshore temperature sens temperature XRD patterns were refined using GSAS.

DMA
The temperature and frequency dependences of elastic properties and energ pation of [(CH3)2NH2][Fe(HCOO)3] single crystals were determined using DMA strument (PerkinElmer Instruments, Waltham, MA, USA) in the single cantileve from 150 K to 320 K at the rate of 2 K/min at frequencies of 1, 5 and 10 Hz.

DSC
As shown in Figure  , the temperature dependences of dielectric and pyroelectric properties showed a sudden jump near 164 K with a hysteresis in the transition temperature during heating and cooling processes [17,28,34], indicating a first order ferroelectric transition from paraelectric to antiferroelectric caused by the disorder-order transition of the hydrogen bonding. The polar hydrogen bonds were antiparallel in the antiferroelectric state [34]. Maczka et al. reported anomalies in the temperature dependences of Infra-Red and Raman spectra near 160 K [32], due to the first-order structural transition from rhombohedral R3 c at high temperature to monoclinic Cc at low temperature, accompanied by the disorder-order transition of proton ordering in the N-H…O hydrogen bonds in [(CH3)2NH2] + as well as the distortion of the metal-formate framework. Below the transition temperature, the ordered phase showed proton ordering in the N-H…O hydrogen bonds in [(CH3)2NH2] + [32].

DSC
As shown in Figure 3, DSC curve of [(CH 3 ) 2 NH 2 ][Fe(HCOO) 3 ] from 140 K to 300 K indicated an endothermic peak at 164 K. The average enthalpy ∆H was 1202 J mol −1 , and the average entropy ∆S was 7.2 J mol −1 K −1 . The ratio of the configuration numbers in the disordered and ordered systems, N, was 2.35. N would be 3 for a simple 3-fold order-disorder model. Therefore, the transition in [(CH 3 ) 2 3 ], the temperature dependences of dielectric and pyroelectric properties showed a sudden jump near 164 K with a hysteresis in the transition temperature during heating and cooling processes [17,28,34], indicating a first order ferroelectric transition from paraelectric to antiferroelectric caused by the disorder-order transition of the hydrogen bonding. The polar hydrogen bonds were antiparallel in the antiferroelectric state [34]. Maczka et al. reported anomalies in the temperature dependences of Infra-Red and Raman spectra near 160 K [32], due to the first-order structural transition from rhombohedral R3c at high temperature to monoclinic Cc at low temperature, accompanied by the disorder-order transition of proton ordering in the N-H· · · O hydrogen bonds in [(CH 3 ) 2 NH 2 ] + as well as the distortion of the metal-formate framework. Below the transition temperature, the ordered phase showed proton ordering in the N-H· · · O hydrogen bonds in [(CH 3 ) 2 NH 2 ] + [32]. Materials 2021, 14, x FOR PEER REVIEW 5 of 12

Low Temperature Powder XRD
The     The differences in the transition temperatures determined by DSC and low temperature XRD may be due to the different heating rates employed. The temperature dependences of lattice parameters obtained by Rietveld refinement of low temperature XRD patterns are shown in Figure 5.  The differences in the transition temperatures determined by DSC and low temperature XRD may be due to the different heating rates employed. The temperature dependences of lattice parameters obtained by Rietveld refinement of low temperature XRD patterns are shown in Figure 5.  The differences in the transition temperatures determined by DSC and low temperature XRD may be due to the different heating rates employed. The temperature dependences of lattice parameters obtained by Rietveld refinement of low temperature XRD patterns are shown in Figure 5.  Above 145 K, Rietveld refinements of XRD patterns are consistent with rhombohedral R3c with fitting parameters R wp ≤ 11.19% and R p ≤ 7.76%. Below 145 K, Rietveld refinements of XRD patterns are in agreement with monoclinic Cc with fitting parameters R wp ≤ 24.22% and R p ≤ 15.97%. Jain et al. reported that due to twinning, XRD patterns of [(CH 3 ) 2 NH 2 ][M(HCOO) 3 ] (M = Mn, Fe, Co, Ni) below the ferroelectric transition temperatures could not be well refined, but the low temperature structure was monoclinic [17].

DMA
The storage modulus E is the real part of the complex modulus of the viscoelastic material, and it is related to the elastic energy storage. The loss modulus E" is the imaginary part of the complex modulus, and it is related to the internal energy dissipation. The loss factor tanδ is the ratio of the loss modulus to the storage modulus. Figure 6 shows the changes of storage modulus E , loss modulus E" and loss factor tan δ of [(CH 3 ) 2 NH 2 ][Fe(HCOO) 3 ] single crystals with temperature from 150 K to 320 K at frequencies of 1, 5, and 10 Hz. Near the ferroelectric transition temperature~164 K, the storage modulus dropped quickly, and the loss modulus and loss factor reached the maximum. For [(CH 3 ) 2 NH 2 ][Fe(HCOO) 3 ] single crystals, the storage modulus was 1.1 GPa, the loss modulus was~0.02 GPa, and the loss factor was~0.015 near 298 K.

DMA
The storage modulus E' is the real part of the complex modulus of the viscoelastic material, and it is related to the elastic energy storage. The loss modulus E'' is the imaginary part of the complex modulus, and it is related to the internal energy dissipation. The loss factor tanδ is the ratio of the loss modulus to the storage modulus. Figure 6 shows the changes of storage modulus E', loss modulus E'' and loss factor tan δ of [(CH3)2NH2][Fe(HCOO)3] single crystals with temperature from 150 K to 320 K at frequencies of 1, 5, and 10 Hz. Near the ferroelectric transition temperature ~164 K, the storage modulus dropped quickly, and the loss modulus and loss factor reached the maximum. For [(CH3)2NH2][Fe(HCOO)3] single crystals, the storage modulus was ~1.1 GPa, the loss modulus was ~0.02 GPa, and the loss factor was ~0.015 near 298 K. For [(CH3)2NH2][Fe(HCOO)3] pellets, the storage modulus was ~55 MPa, the loss modulus was ~3.5 MPa, and the loss factor was ~0.065 near 298 K. Figure 7 shows the  3 ] pellets, the storage modulus was~55 MPa, the loss modulus was~3.5 MPa, and the loss factor was~0.065 near 298 K. Figure 7 shows the changes of the normalized storage modulus, i.e., the ratio of storage modulus at temperature T to that at 298 K, E T /E 298 , the normalized loss modulus, i.e., the ratio of loss modulus at T to that at 298 K, E" T /E" 298 , and loss factor tanδ of [(CH 3 ) 2 NH 2 ][Fe(HCOO) 3 ] pellets with temperature from 130 K to 300 K at frequencies of 0.5, 1, 2, 5, and 10 Hz. The normalized storage modulus gradually dropped with the increase of temperature, from 1.25 at 130 K to~0.45 at~164 K, and then gradually increased with temperature. The minimum in the normalized storage modulus occurred near the ferroelectric transition temperature~164 K, and the softening reached~64%. With the increase of temperature, the normalized loss modulus gradually increased and then decreased. The peak temperature for the normalized loss modulus was near 145 K, and the peak temperature increased with the increase of the frequency, from 144.5 K at 0.5 Hz to 152.4 K at 10 Hz. With the increase of the temperature, loss factor gradually increased and then decreased. The peak temperature for the loss factor was near the ferroelectric transition temperature~164 K with weak frequency dependences, which is the feature of first-order phase transition. The peak height of the normalized loss factor and loss factor increased with the increase of the frequency. changes of the normalized storage modulus, i.e., the ratio of storage modulus at temperature T to that at 298 K, E'T/E'298, the normalized loss modulus, i.e., the ratio of loss modulus at T to that at 298 K, E''T/E''298, and loss factor tanδ of [(CH3)2NH2][Fe(HCOO)3] pellets with temperature from 130 K to 300 K at frequencies of 0.5, 1, 2, 5, and 10 Hz. The normalized storage modulus gradually dropped with the increase of temperature, from ~1.25 at 130 K to ~0.45 at ~164 K, and then gradually increased with temperature. The minimum in the normalized storage modulus occurred near the ferroelectric transition temperature ~164 K, and the softening reached ~64%. With the increase of temperature, the normalized loss modulus gradually increased and then decreased. The peak temperature for the normalized loss modulus was near 145 K, and the peak temperature increased with the increase of the frequency, from 144.5 K at 0.5 Hz to 152.4 K at 10 Hz. With the increase of the temperature, loss factor gradually increased and then decreased. The peak temperature for the loss factor was near the ferroelectric transition temperature ~164 K with weak frequency dependences, which is the feature of first-order phase transition. The peak height of the normalized loss factor and loss factor increased with the increase of the frequency. The anomalies near 280 K and 310 K in Figure 6 and the anomalies around 220-270 K in Figure 7 have no physical meaning, and they may be caused by the instability of the DMA measurements after the samples were under low frequency and high stress and strain conditions for some time. Figure 8 shows the fitting of ln(f) vs. 1/T for the peaks of the temperature dependences of the normalized loss modulus near 145 K as shown in Figure 7b, using Arrhenius equation f = f0exp[-Ea/(RT)], where R is the gas constant, T is the peak temperature for the The anomalies near 280 K and 310 K in Figure 6 and the anomalies around 220-270 K in Figure 7 have no physical meaning, and they may be caused by the instability of the DMA measurements after the samples were under low frequency and high stress and strain conditions for some time. Figure 8 shows the fitting of ln(f) vs. 1/T for the peaks of the temperature dependences of the normalized loss modulus near 145 K as shown in Figure 7b, using Arrhenius equation f = f 0 exp[−E a /(RT)], where R is the gas constant, T is the peak temperature for the normalized loss modulus near 145 K, and f is the frequency. The activation energy E a was~63 kJ/mol.   Figure 9 shows the fitting of the double logarithmic plot ln(tan δ) vs. ln(f) for the peak height of tan δ near 164 K as shown in Figure 7c, using power law tan δ = Af n , where A is the constance, and f is the frequency.     Figure 9 shows the fitting of the double logarithmic plot ln(tan δ) vs. ln(f) for the peak height of tan δ near 164 K as shown in Figure 7c, using power law tan δ = Af n , where A is the constance, and f is the frequency.  The peak height of tan δ near 164 K was obtained using two methods. The first was relative to zero base line, i.e., no base line correction was used. The second was relative to the baseline, which was tangential to data points near 130 K and 180 K. n was determined to be between 0.026 and 0.114. The reported n value for [(CH 3 ) 2 NH 2 ][Mn(HCOO) 3 ] was between −0.382 and −0.078 [25]. The elastic anomalies and energy dissipation detected by DMA near the ferroelectric transition~164 K with relaxation time 0.1-2 s are due to the coupling of the movement of dimethylammonium cations and the mobility of twin walls.  3 ] single crystals and pellets showed elastic anomalies and large energy dissipation near the ferroelectric transition temperature~164 K. Near the transition temperature, the normalized storage modulus reached the minimum, and the normalized loss modulus and the loss factor reached the maximum. The softening in normalized storage modulus reaches~64%. When the frequency increases from 0.5 Hz to 10 Hz, the peak temperature for the normalized loss modulus increases from 144.5 K to 152.4 K, and the activation energy is~63 kJ/mol. The peak temperature for the loss factor showed weak frequency dependences, which is a feature of first-order phase transitions. The elastic anomalies and energy dissipation are caused by the coupling of the movements of dimethylammonium cations and twin walls. Author Contributions: Conceptualization, Z.Z.; methodology, X.S., H.Y., X.W., L.S., S.Y., D.C. and H.T.; formal analysis, X.S., X.W., L.S., S.Y. and D.C.; investigation, Z.Z., X.S. and D.C.; resources, X.S., H.Y., L.S., S.Y., D.C. and H.T.; data curation, X.S., X.W. and D.C.; writing-original draft preparation, Z.Z., X.S. and D.C.; writing-review and editing, Z.Z., X.S., H.Y., X.W., L.S., S.Y., D.C. and H.T.; visualization, X.S. and D.C.; supervision, Z.Z.; funding acquisition, Z.Z.. Material synthesis was carried out by X.S., H.Y., L.S., S.Y., D.C., and H.T. Room temperature XRD was carried out by X.S., H.Y., D.C. and H.T. DSC was carried out by X.S. and D.C. Low temperature XRD was carried out by X.W., X.S., H.Y., L.S. and S.Y. DMA was carried out by X.S., H.Y., D.C. and H.T. All authors have read and agreed to the published version of the manuscript.

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