Thermal hysteresis in spin-crossover compounds studied within the mechanoelastic model and its potential application to nanoparticles

The recently developed mechanoelastic model is applied to characterize the thermal transition in spin-crossover complexes, with special attention given to the case of spin-crossover nanoparticles. In a two-dimensional system, hexagonal-shaped samples with open boundary conditions are composed of individual molecules that are linked by springs and can switch between two states, namely, the high-spin (HS) and the low-spin (LS) states. The switching of an individual molecule during the spin transition is decided by way of a Monte Carlo standard procedure, using transition probabilities depending on the temperature, the energy gap between the two states, the enthalpy difference, the degeneracy ratio, and the local pressure determined by the elongation or compression of its closest springs. The influence of external parameters, such as temperature sweeping rate and pressure, or intrinsic features of the system, such as the value of its spring constant, on the width of the thermal hysteresis, its shape, and its position are discussed. The particular case of spin-crossover nanoparticles is treated by considering them embedded into a polymer environment, which essentially affects the molecules situated at the edges and faces by decreasing their transition probabilities from HS to LS. Finally, the pressure hysteresis, obtained by varying the external pressure at constant temperature is discussed.

Figure 1

The hexagonal network of particles connected with springs (snapshot taken during the transition): big blue circles, HS molecules; small red circles, LS molecules; and open circles, HS molecules in the initial state at $n_{\text{HS}}=1$.

Figure 2

Thermal hysteresis: dependence on temperature sweep rate for a system with 10 981 molecules. The other parameters are $D=1100\text{K}$, $g=1096$, $E_a=400\text{K}$, $\kappa =2000\text{K}/\text{N}$, $k=0.7\text{N}/\text{m}$, $k_B=1$, and $\mu =1\text{N}\text{s}/\text{m}$.

Figure 3

Hysteresis width dependence as a function of the temperature sweep rate on a logarithmic scale: Monte Carlo simulation (full circles), fit with a power function of type $y=A_1{x}^{-p_1}+y_0$ with parameters $A_1=97.84$K, $p=0.58$, and $y_0=15.69$ K (full line).

Figure 4

Relaxation curves starting from different HS fractions inside and outside the hysteresis loop: (a) for temperatures below the quasistatic hysteresis the relaxation curves converge to HS fractions close to zero; (b) bistability inside the quasistatic thermal hysteresis loops: depending on the high spin fraction at the starting point, one of the two steady states is obtained; and (c) for temperatures higher than the quasistatic hysteresis loop the relaxation curves converge to a HS fraction close to unity.

Figure 5

Thermal transition for various spring constant interactions. If the spring constant is higher than a threshold value, then the transition is accompanied by a hysteresis. All the parameters are those from Fig. 2, except the number of Monte Carlo steps considered at every temperature unit, which is 100 steps/K.

Figure 6

Size dependence of the thermal hysteresis for systems composed from 37 to 20 419 molecules. The plotted curves corresponding to systems with less than 10 000 molecules have been obtained as the average of several calculated curves (from 10 for the system with 1261 molecules to 1000 for the system with 37 molecules). Number of Monte Carlo steps = 100 steps/K. The other parameters are those from Fig. 2.

Figure 7

(a) Hexagonal spin-crossover nanoparticles (full circles) embedded in a polymer matrix (open circles). (b) Thermal hysteresis for bulk systems (thick full line) and nanoparticles in polymer (thin dashed and dotted lines). (c) Snapshots of spin configurations for polymer-embedded nanoparticles for $n_{\text{HS}}=0.9$ (I), $n_{\text{HS}}=0.7$ (II), and $n_{\text{HS}}=0.5$ (III), and for open boundaries nanoparticles at $n_{\text{HS}}=0.7$ (IV). Black circles, HS molecules; yellow circles, LS molecules.

Figure 8

Thermal hysteresis for different external pressures. Inset: the hysteresis width as a function of external pressure. The other parameters are those given in the previous figures.

Figure 9

Simulated pressure hysteresis loops at constant temperatures. For lower temperature, the hysteresis is wider and shifts to lower pressures.