1. Introduction

Some of the 3-d transition metal complexes show a switching between two or more spins states of the central cation induced by a perturbation of external conditions (T, P, hν) (Kahn & Martinez, 1998; Gütlich et al., 2007; Levchenko et al., 2014). This phenomenon is called spin crossover and was first observed more than 90 years ago (Cambi & Cagnasso, 1931; Cambi & Szegö, 1933). For Fe²⁺ in a (nearly) octahedral ligand field of a certain strength, a low-spin (LS, singlet t_2g⁶) state can be switched to a high-spin (HS, quintet t_2g⁴e_g²) state by heating; here, the entropy associated with a change in electronic multiplicities and vibrational frequencies is a leading driving force (Grandjean et al., 1989). An important parameter is the temperature of spin state equilibrium T_½ = ΔH/ΔS, where ΔH and ΔS are enthalpy and entropy changes, respectively, linked to the change of the spin state. A difference in molecular volumes (LS state shows shorter Fe to ligand distances compared to HS state) makes LS states favorable under external pressure which is, therefore, yet another parameter controlling the spin states (von Ranke, 2017; Gütlich et al., 2007). An external perturbation may induce a gradual conversion of spin states (a crossover) or an abrupt switch with significant hysteresis (a first-order spin state transition). Strength and range of intermolecular interactions are believed to define the shape of the transition curve (i.e. the fraction of complexes in HS state plotted as a function of external perturbation, e.g. temperature or pressure) (Gütlich et al., 2013).

The change in spin state is associated with a change in magnetization, unit-cell volume and color; this is why materials undergoing a spin crossover transition have attracted wide interest for many potential applications, e.g. information storage, optical devices and displays (Létard et al., 2003; Ksenofontov et al., 2004; Tuan, 2012; Brooker, 2015; Kahn et al., 1992; Kahn & Martinez, 1998). Recently, spin-crossover materials have been discussed as solid-state materials for caloric applications (Vallone et al., 2019; Sandeman, 2016; Romanini et al., 2021; von Ranke, 2017; Reis, 2020), as the spin crossover transition is susceptible to pressure (P). For potential barocaloric applications, a requirement of the spin crossover material is a strong dependence of the temperature of spin state equilibrium on pressure; according to Sandeman (2016), the expected caloric effect is proportional to dT_½/dP.

An attractive candidate for barocaloric research is the [Fe(PM-Bia)₂(NCS)₂] complex (Figs. 1, 2 and S1), where PM is N-2′-pyridyl­methyl­ene and Bia is 4-amino­biphenyl (Sandeman, 2016); this is probably the most studied spin crossover material (Létard et al., 1998). The large volume of reported data on structure and properties helps us to identify the genuine spin-crossover response not affected by a particular experimental or sample preparation protocol. [Fe(PM-Bia)₂(NCS)₂] can crystallize in two different polymorphs, an orthorhombic one with space group Pccn (denoted as Bia-P_ortho hereinafter) and a monoclinic one with space group P2₁/c (denoted as Bia-P_mono hereinafter). There is a report on a third polymorph stable at high pressure, an intermediate state (Rotaru et al., 2009), but available structural information is not conclusive. Similar intermediate states induced by variable scan rates have also been reported for other SCO compounds (Ridier et al., 2018; Chakraborty et al., 2012; Fujinami et al., 2015; Li et al., 2022).

Figure 1 View of the crystal packing of [Fe(PM–Bia)2(NCS)2] complex in the LS state (left) and HS state (right). Projection along the c direction for both polymorphs Bia-Pmono (top) and Bia-Portho (bottom). Hydrogen atoms are omitted for clarity. The marker * highlights the difference between the thio­cyanate branch and the phenyl rings in the LS and HS, respectively.

Figure 2 Molecular structure for the two polymorphs in the HS (red) and LS (blue) states: (left) Bia-Pmono and (right) Bia-Portho.

The magnetic behavior of the two polymorphs is different: whereas Bia-P_ortho shows a very abrupt SCO transition in a narrow temperature range of about 1 K at ∼175 K with a thermal hysteresis of 5 K (Rodríguez-Velamazán et al., 2007; Létard et al., 1998, 1999, 2003; Ksenofontov et al., 1998), Bia-P_mono shows a gradual spin crossover with T_½ at approximately 210 K, which stretches over a large temperature range from 150 K to 250 K (Guionneau et al., 1999, 2001; Létard et al., 1999).

In this study, we re-examine single-crystal structures of both polymorphs as a function of temperature, and probe rate-dependent magnetization. We also present synchrotron diffraction data collected in a cyclic mode with a fine temperature sampling uncovering the full temperature evolution of various intra- and intermolecular contacts, as well as lattice deformations and atomic displacement parameters (ADPs). Another goal of our study is to see whether scan rate-dependent measurements lead to an intermediate state or show any sizeable kinetic hysteresis for Bia-P_mono and Bia-P_ortho. In addition, synchrotron powder diffraction data uncover irreversible lattice deformation which can be attributed to radiation damage. The above results of temperature-dependent experiments are parameterized with a phenomenological thermodynamic model.

2. Experimental

3. Results

3.1. Magnetic properties

The temperature-dependent magnetization measurement for the monoclinic polymorph Bia-P_mono shows a gradual spin transition ranging from 250 K to 150 K covering a wide transition region; the orthorhombic polymorph Bia-P_ortho shows an abrupt transition with T_½↑ = 173 K and T_½↓ = 167 K (thermal hysteresis of 6 K). Both results are in good agreement with the literature (Létard et al., 2003, 1998; Ksenofontov et al., 1998; Guionneau et al., 1999). This confirms that our material meets the standards set by previous groups in terms of purity and properties.

Measurements on single crystals of both polymorphs using different scan rates (Figs. 3, views a1 and a2) show a negligible scan rate dependence of the thermal hysteresis with nearly constant values of 0.8 K for Bia-P_mono and 6.5 K for Bia-P_ortho (Fig. 3, views b1 and b2). On the other hand, measurements on the polycrystalline samples of both polymorphs show that, while the shape of the transition curve remains similar (Fig. 3, views a3 and a4) with increasing the scan rate from 0.2 K min⁻¹ to 10 K min⁻¹, the width of the hysteresis curve undergoes a monotonic increase from 0.6→12.3 K (ΔT = 11.7 K) for Bia-P_mono and from 3.2→17.9 K (ΔT = 14.7 K) for Bia-P_ortho (Fig. 3, views b3 and b4). Furthermore, the measurement on the polycrystalline sample of Bia-P_ortho reveals a two-step transition with an intermediate state observed only during the warming cycles (see below) (Fig. 3, view a4). The two-step character becomes more pronounced with lower scan rates. As previous investigations of the magnetic properties were focused on fast scan rates, this behavior has not been described earlier (Létard et al., 1998, 2003; Ksenofontov et al., 1998; Capes et al., 2000). It is, however, noteworthy that a visually similar two-step behavior was observed in diffuse reflectivity data at elevated hydro­static pressures and was correlated with a new phase of the compound (polymorph III) (Rotaru et al., 2009).

Figure 3 (Left) χMT as a function of temperature at two different scan rates at 0.5 K min−1 (blue) and 5 K min−1 (red) for Bia-Pmono (views a1, a3) and Bia-Portho (views a2, a4). (Right) The observed T½, during cooling (blue) and warming (red), as a function of the scan rate for Bia-Pmono (views b1, b3) and Bia-Portho (views b2, b4) respectively. The solid and the dashed lines correspond to the warming and cooling cycles, respectively. The abbreviations poly and SC stand for polycrystalline and single crystal, respectively.

3.2. Thermal properties

While DSC measurement on Bia-P_ortho is reported in the literature (Létard et al., 1998), providing the value of T_½, ΔH, and ΔS (as listed in Table 1), DSC data for Bia-P_mono have not been reported earlier. Our DSC data for Bia-P_mono exhibits a broad exothermic anomaly at approximately T_½ = 205.6 K upon cooling and an endothermic anomaly on heating at about T_½ = 208.7 K (Fig. 4). The value of T_½ is estimated as the temperature which divides the integrated area under the peak by half. The measurement was performed at a rate of 10 K min⁻¹ and a thermal hysteresis is visible, similar to the observations in the magnetization data measured at fast scan rates (discussed later in Section 8).

Table 1 T1/2 and thermodynamic parameters for Bia-Pmono and Bia-Portho obtained from the crystal structure, magnetization, and DSC calorimetry measurements by fitting the data to the Slichter–Drickamer model The ΔS, ΔH, T1/2 and Γ values estimated from the magnetic and crystal structure data for Bia-Pmono are within the confidence intervals of 45.0–49.4 J K−1 mol−1, 9.5–10.3 kJ mol−1, 206.8–210.9 K and 2.2–3.4 kJ mol−1, respectively.   Thermodynamics values ΔS (J mol−1 K−1) ΔH (kJ mol−1) T1/2 (K) Γ (kJ mol−1) 2RT1/2 (kJ mol−1) Monoclinic (Bia-Pmono) DSC 48 (3) 10.2 (9) 208.5 (5) 2.85 (2) 3.46 (9) Magnetization 47.3 (1) 9.9 (1) 208.3 (9) 2.6 (3)   Crystal structure 46.3 (5) 9.7 (7) 209.8 (2) 3.1 (2)   Orthorhombic (Bia-Portho) Crystal structure 59† 10† 177.0 (5) 5.0 (1) 2.94 (5) †Reference: Létard et al. (1998).

Figure 4 DSC data for polycrystalline Bia-Pmono in cooling and heating mode at a rate of 10 K min−1 showing heat flow as a function of temperature processed using the instrument software. The transition temperature is given.

The enthalpy associated with the spin transition corresponds to the area under the peak, and since ΔG = 0 at the transition temperature, the overall entropy variation upon spin transition is calculated at T_½ using the relation T_½ = ΔH/ΔS, by considering the limits of integration corresponding to ΔT₈₀.³ The calculated entropy change ΔS was found to be 47 ± 2 J mol⁻¹ K⁻¹ and 48 ± 2 J mol⁻¹ K⁻¹ during the cooling and warming processes, respectively.

4. Crystal structure

The crystal structures of both polymorphs have been studied in detail as a function of temperature previously (Buron-Le Cointe et al., 2012; Daubric et al., 2000; Létard et al., 2003; Marchivie et al., 2003, 2005). In the following Sections 4.1 to 4.3, we therefore only describe our observations briefly, relying on the detailed descriptions given in the earlier studies.

Figs. 1 and 2 show the crystal packing and the overlapped HS–LS crystal structures, respectively, for both polymorphs. The structures are formed by layers of molecular units (Fig. 1), arranged in the bc plane for Bia-P_mono and in the ac plane for Bia-P_ortho. The molecular layers are stacked for Bia-P_mono along the a axis and Bia-P_ortho along the b axis. The NCS⁻ groups remain almost linear, with N≡C—S angles of 179.4 (2)° for Bia-P_ortho and 179.3 (3)° for Bia-P_mono. The main differences in the HS spin state of both compounds are:

(a) In Bia-P_mono one of the external phenyl groups is disordered⁴ over two positions, whereas in Bia-P_ortho both external phenyl groups are ordered (Fig. 2).⁴

(b) The Fe—N—C(S) angles for Bia-P_mono diverge significantly more from linearity [154.7 (2)° and 159.0 (1)°] than in Bia-P_ortho 167.7 (3)° (Fig. 2).

(c) Intermolecular S⋯H—C contacts are significantly shorter in Bia-P_ortho [3.430 (1) Å] than in Bia-P_mono [3.5126 (2) Å].

4.1. Temperature dependence of the unit-cell parameters and unit-cell volume

The normalized unit-cell parameters⁵ of both polymorphs as a function of temperature are in excellent agreement with the available literature data (Fig. 5). For Bia-P_ortho, the evolution of the unit-cell parameters upon cooling shows abrupt changes in a very narrow temperature region around the HS–LS transition, whereas changes in unit-cell parameters of Bia-P_mono are smoother and stretch out over a larger temperature range. As expected, for both compounds, the unit-cell volume decreases at the HS–LS transitions. The decrease is significantly larger in Bia-P_mono (−4.92%) than in Bia-P_ortho (−3.97%) (Fig. S3). A strongly anisotropic behavior of the unit-cell parameter is observed, which is strikingly different in both compounds.

Figure 5 Normalized unit-cell parameters and unit-cell volume (normalized to the value at 300 K) as a function of temperature for the Bia-Portho (right) and Bia-Pmono (left). Filled symbols (black, red, and blue) are from the current study. Open star symbols are from ref [a] (Marchivie et al., 2003), and ref [b] (Daubric et al., 2000). Lines are guides to the eye. Error bars are the same size or smaller than the symbols.

For both polymorphs (Fig. 1), within the molecular plane the thio­cyanate group points along the c direction. At the spin transition, c_mono decreases (−1.53%), whereas c_ortho sharply increases (+3.2%) with decreasing temperature. The other perpendicular direction within the molecular layer (b in Bia-P_mono and a in Bia-P_ortho) decreases in both polymorphs, although to a significantly different extent (−1.53% and −4.1% for Bia-P_mono and Bia-P_ortho). The unit-cell parameter corresponding to the stacking direction of the sheets (a in Bia-P_mono and b in Bia-P_ortho) increases at the HS–LS transition in the monoclinic phase (+0.97%), whereas it decreases in the orthorhombic phase (−0.6%).

4.2. Temperature dependence of intramolecular geometry

The main structural changes associated with the spin transition are at the level of the coordination sphere of the central Fe²⁺ ion (Gütlich et al., 2013; Collet & Guionneau, 2018; Lakhloufi et al., 2016) which is surrounded by three pairs of nitro­gen atoms: the pyridyl­methyl­ene ligand (N_PM), the amino­biphenyl ligand (N_Bia) and the thio­cyanate ligand (N_CS) (Fig. 2).

The small temperature step that was chosen in our XRD measurements allows us to map the temperature evolution of the structural transition and clearly highlights the differences in the nature of the transition of the two polymorphs. The HS–LS transition leads to a shortening of the Fe—N distances by approximately 0.2 Å on average in both compounds (Fig. 6). The Fe—N distances are in good agreement with the two/three available data points from the literature (Marchivie et al., 2003; Létard et al., 1998; Guionneau et al., 2001). It is particularly striking that the sharp reduction in the Fe—N distances in Bia-P_ortho happens in a very narrow temperature range of about 1 K difference, between 177 K and 178 K (Fig. S4). In both polymorphs, the deviation of the Fe—N≡C(S) angles from linearity is less in the LS state than in the HS state (Fig. S5). While the degree of linearity of the thio­cyanate group (NCS)⁻ is only slightly decreased at the HS–LS transition for Bia-P_ortho, more noticeable changes are observed for Bia-P_mono where one of the branches is significantly less linear in the LS state and the second branch exhibits an anomalous change around the transition region (Fig. S5), which is reflected in the irregular behavior of the a unit-cell parameter in the transition region (Fig. S6).

Figure 6 Fe—N bond length as a function of temperature for Bia-Portho (right) and Bia-Pmono (left). The square symbols (green, red, blue) are from this study. Star symbols represent the two data points from ref [a] (Marchivie et al., 2003), ref [b] (Létard et al., 1998), and open circles are from ref [c] (Guionneau et al., 2001). The inset shows a schematic diagram of Fe(PM-Bia)2(NCS)2. The solid lines are guides to the eye. Error bars are the same size or smaller than the symbols.

In both polymorphs, the N_PM—Fe—N_PM and N_Bia—Fe—N_CS angles show an increase in linearity in the low-spin state (Fig. S7). N_Bia—Fe—N_Bia and N_Bia—Fe—N_CS angles get closer to the ideal value of 90° (Fig. S8); however, in Bia-P_mono, one of the N_PM—Fe—N_CS branches, which is directly related to the disordered phenyl ring, shows again anomalies around the transition temperature (Fig. S9).

Calculation of the octahedral distortion parameters, namely ζ (bond length distortion), Σ (angular distortion), and θ [the deviation from a perfectly octahedral geometry, O_h, to a trigonal prismatic structure, D_3h (Ketkaew et al., 2021)] reveals a higher distortion of Bia-P_ortho compared to Bia-P_mono in the high-spin state (see Fig. S10). During the spin state transition, the maximum relative change is observed in Δζ_HL. In the low-spin state, the values of all the distortion parameters decrease and remain nearly constant, leading to more symmetrical octahedra.

Due to the large number of temperature points and smaller standard deviations when compared with the available literature data, significant temperature-dependent changes are visible for the thio­cyanate ligand: in general, the N≡C(—S) triple bond shows an increase at the HS–LS transition, which is particularly abrupt for one of the N≡C—S branches in Bia-P_mono (Fig. 7). In addition, all C—S bond lengths of the thio­cyanate ligands show an apparent increase at the HS–LS transition in both polymorphs with changes being abrupt for Bia-P_ortho. Elongation of the N≡C bond length across the HS–LS transition is due to an increase in back bonding in the LS state, i.e. the metal donates electrons to the ligand, which results in a weakening of adjacent bonds.

Figure 7 (Top) The evolution of the N≡C bond length as a function of temperature. (Bottom) The C—S bond length as a function of temperature. Red filled circles (Bia-Portho), and filled blue and open back square symbols (Bia-Pmono) are from this study. Open blue and black stars are from ref [a] (Marchivie et al., 2003). Red open star symbols are from ref [b] (Létard et al., 1998).

As for the other ligands, most of the intramolecular distances within the pyridine and the phenyl­ene rings are only weakly influenced and stay either constant or show a slight increase with decreasing temperature (Fig. S11). However, it is worth noting that the C—C bond lengths in one of the phenyl­ene rings in Bia-P_mono show significant anomalies around the transition temperature (Fig. S12).

In addition, a striking difference between both polymorphs is related to the geometry of the bi­phenyl rings. Whereas the intramolecular torsion angle of the bi­phenyl rings (defined in Fig. 8) strongly and abruptly decreases at the HS–LS transition in Bia-P_ortho, following the abrupt changes in the Fe—N distances, these angles gradually increase in both bi­phenyl ligands at the HS–LS transition in Bia-P_mono (Fig. 8), which is also corroborated by the anomalous behavior of the a unit-cell parameter (Fig. S6). In addition, within Bia-P_mono the torsion angle of the bi­phenyl ring, which is affected by the disorder, is larger than the angle in the ordered bi­phenyl ligand, leading to a slight asymmetry of these two branches of the molecule.

Figure 8 (a) Temperature evolution of the torsion angles of the two bi­phenyl rings for the two polymorphs (black symbols: Bia-Pmono; red symbols: Bia-Portho). Lines are guide to the eyes. Views (b) and (c) illustrate the torsion angles of Bia-Pmono and Bia-Portho, respectively.

4.3. Intermolecular contacts

The two polymorphs studied here are ideal for investigating the role of intermolecular interaction on the nature of the spin transition, as one of the polymorphs (Bia-P_mono) shows a gradual spin transition, while the other one (Bia-P_ortho) exhibits an abrupt spin transition. To describe the efficiency with which structural changes at individual spin crossover metal sites are transmitted throughout the bulk material, Slichter and Drickamer introduced a phenomenological interaction parameter, Γ, called cooperativity (Slichter & Drickamer, 1972). Earlier studies on SCO compounds suggest that the strength of the cooperativity depends directly on the strength of the π–π interaction (Guionneau et al., 1999; Létard et al., 1997), the van der Waals forces (Weber et al., 2008; Buron-Le Cointe et al., 2012; Martinez & Iverson, 2012) and the hydrogen bonding in the system (Real et al., 2003; Shen et al., 2019).

In general, π–π interactions can be classified into three categories based on the interplanar angle and on the distances between the centroids of aromatic rings: strong, moderate and weak interactions (Martinez & Iverson, 2012). An investigation on the strength of these interactions with the Mercury (Macrae et al., 2020) program shows that for Bia-P_ortho, no strong π–π interaction between the phenyl rings is observed, neither in the HS nor in the LS state.

For Bia-P_mono, on the other hand, a number of strong π–π interactions between two phenyl rings are observed, and they are similar in the LS and in the HS state. In addition, in the transition region at ∼225 K, some of the moderate π–π interactions become stronger, while at lower temperatures, these interactions are weakened again (Fig. S13, Table S2). As these interactions point along the a direction of the crystal structure, they contribute to the observed anomalies in the temperature dependence of the a unit-cell parameter.

Both polymorphs show short C⋯C contacts [< 3.4 Å, i.e. smaller than the sum of van der Waals radii (Batsanov, 2001)], suggesting the presence of van der Waals interactions in the system. Of these, the shortest C⋯C contacts [3.326 (11) Å] are observed in Bia-P_mono in the HS state and they also persist in the LS states. The short C⋯C contacts in the case of Bia-P_mono lead to the formation of a three-dimensional network of strong van der Waals interactions in this polymorph. For Bia-P_ortho, fewer C⋯C contacts are observed, which are weaker in the HS state than the ones observed in Bia-P_mono [3.428 (9) Å], yet some of them become stronger in the LS state [3.324 (9) Å].

The third important interaction which might influence the cooperativity is the intermolecular hydrogen-bonding interaction involving the sulfur atoms of the NCS⁻ branches with the closest (H)—C atom in one of the internal bi­phenyl rings. For Bia-P_ortho, a short S⋯ (H)—C contact [less than 3.5 Å (Batsanov, 2001)] exists in the HS state, whereas for Bia-P_mono, the shortest contact in the HS state is larger than 3.5 Å, and a value less than 3.5 Å is only reached at lower temperatures (Fig. 9). It is striking that the length of the intermolecular S⋯C contacts for both polymorphs becomes almost identical in the LS state due to an increase of this distance when Bia-P_ortho passes the HS–LS transition.⁶ As the S⋯(H)—C contacts show no significant change in bond lengths, we assume that the hydrogen-bonding network is not substantially changed during cyclic measurements.

Figure 9 (a) Evolution of the shortest intermolecular S⋯C distance for Bia-Portho (small filled and open red circles while cooling and heating, respectively) and Bia-Pmono (black squares). Views (b) and (c) show the S⋯C shortest distance between two molecules in Bia-Pmono and Bia-Portho, respectively. Open large symbols are from ref [a] (Marchivie et al., 2003), ref [b] (Buron-Le Cointe et al., 2012) and ref [c] (Guionneau et al., 2001). The solid lines in (a) are guides to the eye. Error bars are the same size or smaller than the symbols.

Intermolecular interactions can be classified into two categories: those within the molecular plane, the so-called intrasheet contact, and those that involve molecular units belonging to different sheets called intersheet contact. For Bia-P_mono in the HS state, several intra- and intersheet contacts correspond to van der Waals interactions. For Bia-P_ortho, in the HS state, there is no intrasheet contact corresponding to van der Waals interactions, yet there is one intersheet contact which corresponds to hydrogen bonding. In the LS state of both polymorphs, the intersheet contacts are formed by hydrogen bonding, whereas the intrasheet contacts correspond to van der Waals interactions. The variation of the various intra- and intersheet contacts upon cooling is different with some of them decreasing through the HS–LS transition and others increasing (Fig. S15). It is worth noting that due to the lower symmetry of Bia-P_mono, the nature of the intermolecular contacts is different on the two sides of the molecular layers.

5. Coexistence of HS and LS domains, HS and LS states, and atomic displacement parameters

Reconstructions of reciprocal space based on single-crystal diffraction data (Fig. 10) for Bia-P_ortho show a splitting of the Bragg peaks in the region of the spin crossover transitions, which can be attributed to the formation and coexistence of domains of the HS and LS states, hallmarks of a first-order transition. The superposition of the crystal structures corresponding to the two spin states results in the observed splitting of the Bragg reflections. The intensities corresponding to the two states were integrated together. As a consequence, in the refinement an increase in the ADP values of the atoms is visible, which is clearly seen, in particular for the Fe atom (see Fig. S4).

Figure 10 Temperature dependence across the HS to LS transition of a line of diffraction spots: (a) Bia-Pmono upon cooling along (0, k, ) with k varying from 7 to 12; Bia-Portho upon cooling (b) and heating (c) along (h, 0, ) with h varying from −3 to 3. Note the different temperature scales.

On the other hand, for Bia-P_mono, the gradual crossover from HS to LS leads to a continuous shift of positions of the Bragg peaks. This indicates that in this polymorph the HS and LS states are not distributed into larger domains, but instead there is probably a random distribution of HS/LS molecules throughout the crystal.

This is reflected in the temperature dependence of the ADPs of the N atoms of Bia-P_mono (Fig. 11). Outside the SCO range, the ADPs exhibit a monotonic decrease with decreasing temperature. Around the transition, a clear λ-type anomaly is observed for U₂₂ of two of the nitro­gen atoms, in particular for the one attached to the thio­cyanate group at T_½, which indicates a large displacement in a direction perpendicular to the Fe—N bond (see Fig. 11 and inset therein). It should be noted that some SCO compounds exhibit anomalous ADPs of the N atoms along the Fe—N bonds (Chernyshov et al., 2009). This anomaly is a consequence of the fact that disorder is present at T_½ with half of the Fe²⁺ cations in the high-spin state and the other half in the low-spin state. As the instrumental resolution of the diffraction experiment is only 0.8 Å, the disorder cannot be resolved but is instead modeled in terms of the average between HS and LS positions, and the disorder contribution to the atomic displacement parameter (Chernyshov et al., 2003). For Bia-P_mono the ADPs normal to the Fe—N bond are not sensitive to the disorder in Fe—N distances. The observed increase of the U₂₂ parameter of nitro­gen near T_½ might be due to a disordered component related to the difference in angles and tilting of the entire complex for two co-existing spin states.

Figure 11 Temperature dependence of U22 (Å2) for the N atoms of (a) Bia-Pmono and (b) Bia-Portho. The gray dashed line represents the transition temperature. The schematic diagram insets in (a1), (a2) and (b2) show the molecular structure of the Fe(PM-Bia)2(NCS)2 and the corresponding ADPs of the N atoms bonded to Fe atom, respectively.

6. Thermal cycling

A systematic analysis of the reproducibility of the spin transition as a function of temperature is of fundamental importance for the future caloric application of a material. To elucidate this aspect, we carried out consecutive cooling and heating cycles, both for the magnetic susceptibility and X-ray diffraction measurements.

The magnetic susceptibility measurement, which was carried out for Bia-P_mono in two consecutive cycles (warming, cooling, warming, cooling and warming again), indicates a nearly perfect reproducibility. On the other hand, the unit-cell volume and unit-cell parameters (Fig. 12) extracted from the diffraction data are not perfectly reproducible on cycling. The volume in the second warming cycle increases by about 0.3% compared to the first warming cycle, whereas the unit-cell parameters show different deviation in the LS and HS states with respect to the first warming cycle.

Figure 12 Evolution of relative change in the high-spin fraction calculated on the basis of unit-cell parameters and unit-cell volume upon consecutive thermal cycles for Bia-Portho and Bia-Pmono (right and left, respectively). Calculated using = [x(LS)cycle1 − x(T)][x(LS)cycle1 − x(HS)cycle1]−1. Filled blue and red triangle symbols represent the value of the first cooling, followed by the first warming, respectively. Open blue and red triangles represent the second cooling followed by the second warming. Lines are guides to the eyes. Error bars are the same size or smaller than the lines.

Despite a significant change in volume, the unit-cell volume of Bia-P_ortho is perfectly reproduced in all measured cycles using X-ray diffraction (2× cooling and 2× warming; Fig. 12), although the same is not true for the behavior of individual unit-cell parameters. In the first cooling and warming cycle, all the unit-cell parameters are fully reproducible in the LS state, yet they are not in the HS state. During the second warming and cooling cycle, all the unit-cell parameters are badly reproduced both in the HS and LS states. As the parameters describing the mosaicity of the crystal exhibit a nearly constant and perfectly reproducible behavior on cycling (Fig. S16), indicating that the crystal maintains its quality, the observed changes must be attributed to underlying changes in the crystal structure.

On careful inspection of the structural and the intra–intermolecular features (obtained from the single crystal data), during the cyclic measurements for Bia-P_ortho, they do not exhibit a significant change when taking into account the resulting standard deviations (see Fig. S5). Thus, the overall change of the unit-cell parameters must be a consequence of several very small changes in the crystal structure across the lattice, which add up to the observed differences on cycling.⁷

7. Entropy changes and cooperativity

The spin crossover behavior is characterized by the high-spin fraction (γ_HS), which is also the order parameter for the HS↔ LS transition (Gütlich et al., 2013; Chernyshov et al., 2004). It can be obtained from the crystal structure data, via the Fe—N bond lengths, and from the magnetization data, via the magnetic susceptibility χ_M, using the following expressions,

An estimate of the thermodynamic parameters, the enthalpy (ΔH) and entropy (ΔS) associated with the spin-crossover, could be obtained by fitting the values of γ_HS, obtained from equations (1) and (2), to the Slichter–Drickamer (1972) model:

Here the parameter Γ corresponds to the cooperativity, which is assumed to be temperature independent and traces its origin to the interaction between individual spins and the average magnetization of the crystal (Halcrow, 2013; Kreutzburg et al., 2017). Further R, ΔH and ΔS denote the universal gas constant, the enthalpy and entropy changes associated with the HS↔LS transition, respectively. At the equilibrium temperature T_½, corresponding to γ_HS = 0.5 (Nicolazzi & Bousseksou, 2018), the enthalpy in equation (3) can be substituted with the relation ΔH = ΔST_½, allowing the modification of equation (3) as:

Using equation (4), a fit was carried out for γ_HS, keeping ΔS and Γ as the variable parameters. For Bia-P_mono and Bia-P_ortho, Figs. 13(a), 13(c) illustrate the fit obtained from crystal structure data and Figs. 13(b), 13(d) show the corresponding fit from magnetization data, respectively.

Figure 13 HS fraction γHS as a function of temperature for Bia-Pmono (a,b), and Bia-Portho (c,d). The red circles are obtained from crystal structure data. Blue circles indicate the magnetization data. The dashed line corresponds to fitting of the Slichter–Drickamer model [see equation (4)]. The inset in (a) shows the fit of the data obtained from crystal structure data fitted with the Slichter–Drickamer model, yet fixing the thermodynamic values to the ones determined by DSC measurement.

The value of cooperativity (Γ) obtained from the single-crystal structural data could be further confirmed using the DSC measurements. For this, using the value of ΔS obtained from the DSC data, a fit of γ_HS obtained from the Fe—N bond lengths was carried out. The value of Γ obtained from the fit was found to be in good agreement with the one obtained on considering both ΔS and Γ as free parameters.

Due to the continuous nature of equation (4), a fit to the abrupt and discontinuous magnetization and bond length curves for Bia-P_ortho, cannot be obtained. We thus fixed the value of ΔS deduced from DSC measurements (Létard et al., 1998) and allowed only Γ as the variable parameter [Fig. 13(c)].

The values obtained from the above analysis are summarized in Table 1. The value of ΔS is significantly larger than the entropy variation resulting from the change of the spin state ΔS_ele = Rln[(2S_HS +1)/(2S_LS +1)] = 13.4 J mol⁻¹ K⁻¹ (for S_HS = 2 and S_LS = 0). The excess entropy can be attributed to the contribution from vibrational, configurational and rotational degrees of freedom present in both polymorphs (Molnár et al., 2019). The obtained values of Γ and T_½ for both polymorphs are in good agreement with the Slichter–Drickamer model, which attributes the gradual transitions to weak interactions with Γ < 2RT_½ and abrupt transitions to strong interactions with Γ > 2RT_½ (see Table 1) (Nicolazzi & Bousseksou, 2018).

8. Discussion

The detailed crystallographic study of the temperature dependence of the two polymorphs shows that while changes in structural parameters of Bia-P_ortho happen in a `one-step' mechanism in a small temperature interval, for Bia-P_mono the observed behavior is not only stretched out over a large temperature interval of the gradual spin transition, but it is also more complex. Several structural parameters (e.g. the a unit-cell parameter) show changes of trend in the region of the transition, which correspond to changes, both in the intramolecular geometry and in the intermolecular interactions, in Bia-P_mono. In addition, the comparison of reciprocal space sections and atomic displacement parameters of both polymorphs suggest that while in Bia-P_ortho the microstructure in the transition region is characterized by the existence of HS and LS state domains, in Bia-P_mono there is most probably rather a random distribution of molecular complexes, which are in the HS or LS states.

With the help of the Slichter–Drickamer (1972) model, the enthalpy (ΔH) and entropy (ΔS) associated with the spin crossover were extracted from the magnetization, DSC and crystal structure data. The cooperativity Γ was also extracted using this model, with higher Γ for Bia-P_ortho in agreement with the abruptness of the SCO transition of this polymorph.

Unfortunately the Slichter and Drickamer model, and in general any thermodynamic model, does not clarify the atomistic origin of cooperativity. To elucidate the relationship between the thermodynamic parameters, the distinct nature of the transitions and the crystal structures, it is useful to discuss the intermolecular interactions. In the literature, it is frequently assumed that the extent of cooperativity increases with the number and strength of intermolecular interactions (Marchivie et al., 2003; Hayami et al., 2003; Guionneau, 2014). However, there are also examples where a large number of short intermolecular contacts inhibit the propagation of the structural changes associated with the spin crossover (Halcrow, 2011; Reger et al., 2005).

A detailed comparison of the intermolecular interactions in the two polymorphs investigated here shows that a differentiated view on the interactions is necessary, as, on the one hand, Bia-P_mono has stronger van der Waals and π–π interactions than Bia-P_ortho, and, on the other hand, hydrogen bonding is only observed in the HS state of Bia-P_ortho and is absent in the HS state of Bia-P_mono. The cooperativity is significantly higher for the HS–LS transition in Bia-P_ortho and thus seems to be mainly due to the hydrogen bonding between the molecular units which serve as a means to transmit the structural deformations associated with SCO throughout the lattice in a highly cooperative manner. The absence of π–π and van der Waals interactions in Bia-P_ortho might then even provide more freedom at the intramolecular level to propagate these changes throughout the lattice. In Bia-P_mono,the stronger van der Waals interaction and π–π interaction are possibly competing and lead to smearing out of the transition over a large temperature range. An examination of number and strength of intermolecular contacts may therefore be insufficient to predict the cooperativity of spin conversion, as different contacts contribute to the cooperativity with different signs.

The disordered phenyl group in Bia-P_mono has nearly equal probabilities of occupation of the two positions A and B in the high-spin state. Below the spin transition temperature, one of the two positions shows an increased probability of occupation over the other (Fig. 14). This indicates that the configurational disorder might play a role in the entropy changes during the spin-crossover transition.

Figure 14 Temperature dependence of the occupation fraction of the two disorder positions of the phenyl group in Bia-Pmono.

The observed influence of thermal cycling on both polymorphs can also be directly related to the cooperativity. While magnetization measurements are found to be reproducible for both polymorphs, the same is not true when the structural parameters, measured with synchrotron light, are considered. For Bia-P_ortho the unit-cell volume is reproducible upon cycling and the crystal does not exhibit any apparent deterioration (cracks, fractures), demonstrating its exceptional robustness upon cycling, which can be related to the strong intermolecular contacts. On the other hand, for Bia-P_mono the unit-cell volume increases on cycling (Fig. 12). This can be related to radiation damage in which the dose is accumulated progressively with time and temperature (Chernyshov et al., 2022). The radiation dose seems to affect the spin-crossover process and favors the HS state, shifting the transition region to lower temperatures (from 210.6 K to 208.8 K).

The apparent rate-dependent hysteresis in the magnetization data, observed for the polycrystalline samples of both polymorphs, could be related to the temperature lag between the sample and the temperature controller which increases with higher scan rates, leading to broadening of the thermal hysteresis. The fact that a similar effect is not observed for the single crystals rather points to a grain size dependent phenomenon. A possible explanation might be linked to the grain size dependent intrinsic kinetic behavior of the domain formation (Ridier et al., 2018).

For the slowest scan rates, in the polycrystalline sample the thermal hysteresis still exists for Bia-P_ortho (hysteresis ≃ 4 K), while for Bia-P_mono the thermal hysteresis almost vanishes. This can be explained on the basis of the intrinsic kinetics of the spin conversion, which is slower than the temperature change for Bia-P_ortho. However, due to the gradual nature of transition, a slow scan rate allows more time for Bia-P_mono to overcome the barrier between the LS and HS states (see appendix A1).

An important difference in the kinetic behavior of the two polymorphs is the formation of a scan rate dependent intermediate state on warming for Bia-P_ortho, which is absent for Bia-P_mono. The similarity between our observations for low scan rates and the state observed at hydro­static pressures above 1 kbar (which was attributed by the authors to the co-existence of Bia-P_ortho with unknown polymorph III) (Rotaru et al., 2009) is striking and deserves further investigation. For this, further diffraction experiments are required; however, these are out of the scope of this article.

9. Conclusion

A comprehensive study of the mechanism of the transition in the orthorhombic and monoclinic polymorphs of the spin crossover compound [Fe(PM-Bia)₂(NCS)₂], using magnetization, DSC and synchrotron single-crystal X-ray diffraction, is presented and the explicit role of the hydrogen bonding, π–π and van der Waals interactions on the cooperativity of the transition is highlighted. Based on the atomistic insights obtained from single-crystal diffraction, the role of inter- and intramolecular interactions and their interplay with the anisotropy of the unit-cell parameters, various interactions and symmetry of the crystal across the spin crossover has been explored. The analysis of the data highlights the complexity of the structural processes involved in the spin crossover transition. Our data are analyzed within the framework of the Slichter and Drickamer model to obtain the values of enthalpy, entropy and cooperativity associated with the spin crossover for both polymorphs, which may be utilized as experimental constraints for theoretical calculations.

In the cyclic measurements, Bia-P_ortho exhibits robust reproducibility, while for Bia-P_mono the effect of radiation damage is observed for longer exposure times, which induces a shift in the spin crossover temperature. While knowledge of reproducibility is vital to understand the intrinsic effects of SCO, the scan rate dependence plays an important role in applications that involve a time-dependent use of SCO compounds. A close examination of the scan rate dependence of the thermal hysteresis highlights the difference in the intrinsic nature of spin state dynamics and reveals a grain size dependence of the magnetic properties, thermal exchange and kinetic behavior for both polymorphs. Based on this observation, a theoretical model is presented to describe the key role of non-equilibrium spin state dynamics and the dependence of thermal hysteresis on the scan rate. A new scan rate dependent intermediate state appears in the heating cycle for the orthorhombic polymorph, which has not been reported previously. Further diffraction experiments are mandatory to understand the microscopic picture of this state.

The results of this paper serve as a step in understanding the nature of transition in spin crossover complexes highlighting the role of various interactions and their correlation with the microscopic features of the crystal structures. Since the nature of the transition is strongly interlinked with the atomistic features, one can potentially engineer SCO not only by tuning the number and strengths of various interactions, but also taking into account opposite signs of their contributions in the collective phenomena.