At the electronic level, the ligand field strength δ acting on the iron centre of an iron(II) complex raises the degeneration of the 3d orbitals, thus affording in an octahedral field, two orbitals (e_g) separated from the other three orbitals (t_2g) (Scheme 1).

According to relative values of δ and the electrons pairing energy ∏, the ground state can be defined as follows:

When δ ≫ ∏, all six electrons occupy the three orbitals of the lowest energy. The total spin is S = 0 and the iron(II) ion is in the low-spin (LS) state.

Figure 1(a) shows the molecular configuration diagram, i.e., a plot of adiabatic energy vs. the distortion coordinate, here the metal-ligand distance, which is believed to be the most perturbed coordinate upon the spin transition.

Thermally induced SCO conditions are met in the intermediate case when the energy difference between the first vibronic levels of the LS and HS states, ΔE_HL⁰, ranges in the order of thermal energy, k_BT [11]. The Fe-ligand bond length increases to about 10%, on going from the LS to the HS state [11]. Other physical properties that are modified by the SCO phenomenon are:

The magnetic properties from a diamagnetic state (S = 0) to a paramagnetic state (S = 2).

The principle of the thermal SCO behaviour can be understood on thermodynamic grounds. The molecular free energy ΔG variation during the LS-HS transformation for non-interacting molecules is given by: (1) Δ G = Δ H − T Δ S (2) where Δ G = G HS − G LS (3) and Δ H = H HS − H LS (4) and Δ S = S HS − S LS

The temperature for which the number of SCO active molecules is in a 50% ratio, is called the transition temperature T_1/2 and is defined by ΔG = 0, which implies: (5) T 1 / 2 = Δ H / Δ S

ΔS is related to the degeneracies ratio g = g_HS/g_LS by the relation: (6) Δ S = R ln ( g HS / g LS ) = R lng

A simple description of a SCO between LS and HS states can be given: below T_1/2, the enthalpy factor dominates, ΔG > 0 and the LS state is the most stable. As far as temperature increases, the entropic factor TΔS increases and above T_1/2 the entropy factor dominates, ΔG < 0 and the HS state becomes more stable. The entropy term is so strong that the population of the LS state is almost zero.

The system is in general described by the HS fraction, usually denoted n_HS or γ_HS which represents the proportion of molecules in the HS state.

We can also estimate the pressure effect on T_1/2: under pressure the free energy ΔG is written as: (7) Δ G = Δ H − T Δ H + P Δ V

It results that the pressure dependence of T_1/2 is: (8) T 1 / 2 ( P ) = T 1 / 2 ( 0 ) + P Δ V / Δ S

For interacting molecules an interaction parameter Γ is introduced: (9) Δ G = Δ H − T Δ S + n H S ( 1 − n H S ) Γ

According to the strength of the Γ parameter, different thermal behaviours for n_HS(T) are obtained. At low temperature, the LS state is the more stable one, which means n_HS = 0. When the temperature is increased, the HS state is populated. Figure 1(b) is obtained for non-interacting molecules resulting in a gradual spin conversion/spin equilibrium as observed in the liquid state or for diluted materials, e.g., [Fe_1-xZn_x(2-picolylamine)₃]Cl₂·EtOH [13]. In the solid state and in the presence of interactions of elastic and electronic origin between the spin changing molecules, a steep phase transition can result, which is termed spin transition (ST). For strongly interacting systems, the ST can be accompanied by a thermal hysteresis loop which is the sign of a first order transition that can be observed provided interactions exceed a threshold value Γ_c, as observed for instance for the 1D chain [Fe(4-amino-1,2,4-triazole)₃] (NO₃)₂ [25]. In this case, critical temperatures T_c^↑ and T_c^↓ can be used to define the transition temperatures. A quite unusual case is when the transition occurs stepwise which may results for instance from the presence of two crystallographic lattice sites, as observed for [Fe(4,4′-bis-1,2,4-triazole)₃] (ClO₄)₂ [26] or intermediate magnetic phases [11]. A spin conversion can also occur with a residual effect at low temperatures which is typical when crystal defects are present, for instance as a result of grinding [11] or when chains of different lengths are identified, for instance for coordination polymers [27]. This phenomenon can also be observed at higher temperatures. Basically, a combination of these curves can also be theoretically predicted. Most interestingly, similar shape curves can be observed when the trigger is pressure [28].

On the basis of the pioneering theoretical work of Wajnsflasz and Pick [29], a Ising-like model for considering the n_HS evolution has been proposed [30–32]. In this phenomenological two-level model, a fictitious spin operator σ is introduced with two eigen values: σ = ± 1 for HS and LS states, respectively with respective degeneracies g_HS and g_LS.

For non-interacting molecules the Hamiltonian is: (10) H ^ = ∑ i = 1 , N Δ 2 σ ^ iwhere Δ is the electronic gap between LS and HS states, which was labelled as δ in the previous section. The thermal average value of σ is: (11) 〈 σ 〉 = tanh ( − Δ − k B T ln ( g ) 2 k B T )

The HS fraction n_HS is related to <σ> as follow: (12) n H S = 1 + 〈 σ 〉 2

The model is termed “like” because the ratio g may be quite large, up to a few thousands, as it involves both the spin degeneracies and the density of vibrational states [33]. This model can also be viewed as a simple Ising model under a temperature-dependent “effective” field which accounts for the different degeneracies of the levels.

Among the various SCO molecules available, 1D coordination polymers play a growing role as this family of molecules not only can display highly cooperative spin transitions but also can be combined to several other properties in hybrid materials (e.g., SCO and liquid crystal properties [34]). In these molecules short and long range interactions are recognized as being critical to leads to highly cooperative materials [27]. The 1D Hamiltonian for interacting molecules including long- and short-range interactions is given by ([31]): (13) H ^ = ∑ Δ eff − G 〈 σ 〉 2 k B T − J ∑ σ i σ i + 1where J and G are the short and long range interactions, respectively, and Δ_eff = Δ − k_B T ln g.

With Δ = 3127 K, J = 800 K, G = 105.5 K, ln g = 8.448 and using the Hamiltonian given by equation (13), we obtain the temperature dependence of n_HS shown in Figure 2. This result fits the experimental data obtained for the 1D chain compound [Fe(Htrz)₂trz]BF₄ (Htrz = 4-H-1,2,4-triazole, trz = 1,2,4-triazolato) [35], a material which is currently the topic of intense miniaturization efforts and considered for implementation in logistic devices [36–41].

Taking into account the pressure P that can be applied on the material, Δ_eff now can be expressed by: (13) Δ eff = Δ + α P − k B T ln g

This way, the HS fraction now vs. pressure at T = 400 K for the 1D chain compound [Fe(Htrz)trz]BF₄ can be simulated, revealing a square shape hysteresis loop (Figure 3).

Note that this curve has not yet been experimentally observed on [Fe(Htrz)₂trz]BF₄, but is here predicted in this work to demonstrate the usefulness of the Ising-like model in the design of future pressure sensors, and encourage its use to future users.

Each SCO material is characterized by well defined T_1/2 and P_1/2. Applying pressure induces a shift from HS to LS state, which is, the direct consequence of the lower volume for the LS molecule [43], although the reverse situation could also be observed [44]. In both cases, the change from LS to HS is easily detectable by optical means because SCO molecular materials often display thermochromism that is a different colour in the HS and LS states. This is the case for instance for the family of 1D ST chain compounds with 1,2,4-triazole or 1-tetrazole ligands that are deep purple powders in the LS state and white in the HS state [22], and that present the best contrast available today. Here we propose the use of SCO molecular materials with optical reflectivity detection as an alternative for a new range of temperature and pressure sensors [45-47]. The sensor material is deposited either as a thin film or nanostructured by soft-lithography on a sensitive element. A light source and a detector are used within an optical reflectivity set-up (Scheme 2).

One of the attractive aspects of these materials is their pronounced colour change that makes possible the visual detection of the variation of a measured parameter that can alert a given user without the use of sophisticated electronics such as optical detectors or radio frequency identification tags to record experimental data, which can be useful in remote control applications. On the other hand, due to a well defined (T, P) phase diagram [48], the same element can be used to measure temperature at constant pressure and pressure at a constant temperature, which is particularly appealing. But one should take care of the measurement error due the fluctuation of one parameter when detecting the other. An estimation of such error can be made according to the c.a 20 K/kbar rate variation of T_1/2 vs. P_1/2 [48]. Thus, when using SCO materials as a temperature sensor, 1 bar error on pressure will only induce 20 mK on the measured temperature. On the other hand, when using the material as a pressure sensor, 1 K error on temperature will induce 50 bars error on the measured pressure; this error leads to a low relative error in high pressure range. The sensitivity and resolution of SCO sensors also depend on the contrast of the colour change of the materials which can be high when all molecules of the materials are concerned by the switching process [48]. Spatial resolution, repeatability and time response are the biggest advantages of SCO sensors due to the molecular origin of the phenomenon and its associated non fatigability [23]. Most interestingly, the pressure range necessary to switch a SCO material can be tailored through chemical design.

Many iron(II) compounds exhibit sharp transitions with dramatic modification of the magnetic moment with temperature, sometimes with hysteresis necessary for memory use [22]. Those with smooth spin conversion behaviour have a quasi linear response and seem however to be the most simple materials to implement. We describe below a few possible methods depending on the SCO curve of the selected materials.