In recent posts, I discussed how "spin crossover" is a misnomer for the plethora of organometallic compounds that undergo spin-state phase transitions (abrupt, first-order, hysteretic, multi-step,...)
In theory development, it is best to start with the simplest possible model and then gradually add new features to the model until (hopefully) arriving at a minimal model that can describe (almost) everything. Hence, I described how the two-state model can describe spin crossover. An Ising "spin" has values of +1 or -1, corresponding to high spin (HS) and low spin (LS) states. The "magnetic" field is half of the difference in Gibbs free energy between the two states.
The model predicts equal numbers of HS and LS at a temperature
The two-state model is modified by adding Ising-type interactions between the “spins” (molecules). The Hamiltonian is then of the form
The temperature dependence in the field arises because this is an effective Hamiltonian.
The Ising-type interactions are due to elastic effects. The spin-state transition in the iron atom leads to changes in the Fe-N bond lengths (an increase of about 10 per cent in going from LS to HS), changing the size of the metal-ligand (ML6 ) complex. This affects the interactions (ionic, pi-pi, H-bond, van der Waals) between the complexes. The volume of the ML6 complex changes by about 30 per cent, but typically the volume of the crystal unit cell changes by only a few per cent. The associated relaxation energies are related to the J’s. Calculating them is non-trivial and will be discussed elsewhere. There are many competing and contradictory models for the elastic origin of the J’s.
In this post, I only consider nearest-neighbour ferromagnetic interactions. Later, I will consider antiferromagnetic interactions and further-neighbour interactions that lead to frustration.
Slichter-Drickamer model
This model was introduced in 1972 is beloved by experimentalists, especially chemists, because it provides a simple analytic formula that can be fit to experimental data.
The system is assumed to be a thermodynamic mixture of HS and LS. x=n_HS(T) is the fraction of HS. The Gibbs free energy is given by
This is minimised as a function of x to give the temperature dependence of the HS population.
The model is a natural extension of the two-state model, by adding a single parameter, Gamma, which is sometimes referred to as the cooperativity parameter.
The model is equivalent to the mean-field treatment of a ferromagnetic Ising model, with Gamma=2zJ, where z is the number of nearest neighbours. Some chemists do not seem to be aware of this connection to Ising. The model is also identical to the theory of binary mixtures, such as discussed in Thermal Physics by Schroeder, Section 5.4.
Successes of the model.
good quantitative agreement with experiments on many materials.
a first-order transition with hysteresis for T_1/2 < Tc =z J.
a steep and continuous (abrupt) transition for T_1/2 slightly larger than Tc.
Values of Gamma are in the range 1-10 kJ/mol. Corresponding vaules of J are in the range 10-200 K, depending on what value of z is assumed.
Weaknesses of the model.
It cannot explain multi-step transitions.
Mean-field theory is quantitatively, and sometimes qualitatively, wrong, especially in one and two dimensions.
The description of hysteresis is an artefact of the mean-field theory, as discussed below.
Curves show the free energy as a function of the order parameter (magnetisation) in mean-field theory. The dashed lines are the lines of metastability deduced from these free-energy curves. Inside these lines, the free energy has two minima: the equilibrium one and a metastable one. The lines are sometimes referred to as spinodal curves.
The consequences of the metastability for a field sweep at constant temperature are shown in the Figure below, taken from Banerjee and Bar.
How does this relate to thermally induced spin-state transitions?
Consider the phase diagram shown above of a ferromagnetic Ising model in a magnetic field. The red and blue lines correspond to temperature scans for two SCO materials that have different values of the parameters Delta H and DeltaS.
The occurrence of qualitatively different behaviour is determined by where the lines intercept the temperature and field axes, i.e. the values of T_1/2 /J and Delta H/J. If the former is larger than Tc/J, as it is for the blue line, then no phase transition is observed.
The parameter Delta H/J determines whether at low temperatures, the complete HS state is formed.
The figure below is a sketch of the temperature dependence of the population of HS for the red and blue cases.
Note that because of the non-zero slope of the red line, the temperature T_1/2 is not the average of the temperatures at which the transition occurs on the up and down temperature sweeps.
Deconstructing hysteresis.
The physical picture above of metastability is an artefact (oversimplification) of mean-field theory. It predicts that an infinite system would take an infinite time to reach the equilibrium state from the metastable state.
(Aside: In the context of the corresponding discrete-choice models in economics, this has important and amusing consequences, as discussed by Bouchaud.)
In reality, the transition to the equilibrium state can occur via nucleation of finite domains or in some regimes via a perturbation with a non-zero wavevector. This is discussed in detail by Chaikin and Lubensky, chapter 4.
The consequence of this “metastability” for a first-order transition in an SCO system is that the width of the hysteresis region (in temperature) may depend on the rate at which the temperature is swept and whether the system is allowed to relax before the magnetisation (fraction of HS) is measured at any temperature. Emprically, this is observed and has been highlighted by Brooker, albeit without reference to the theoretical subtleties I am highlighting here. She points out that up to 2014, chemists seemed to have been oblivious to these issues and reported results without testing whether their observations depended on the sweep rate or whether they waited for relaxation.
(Aside. The dynamics are different for conserved and non-conserved order parameters. In a binary liquid mixture, the order parameter is conserved, i.e., the number of A and B atoms is fixed. In an SCO material, the number of HS and LS is not conserved.)
In the next post, I will discuss how an antiferromagnetic Ising model can give a two-step transition and models with frustrated interactions can give multi-step transitions.
