Landau Theory of Barocaloric Plastic Crystals

We present a simple Landau phenomenology for plastic-to-crystal phase transitions and use the resulting model to calculate barocaloric effects in plastic crystals that are driven by hydrostatic pressure. The essential ingredients of the model are (i) a multipole-moment order parameter that describes the orientational ordering of the constituent molecules, (ii) coupling between such order parameter and elastic strains, and (iii) the thermal expansion of the solid. The model captures main features of plastic-to-crystal phase transitions, namely large volume and entropy changes at the transition, and strong dependence of the transition temperature with pressure. Using solid C 60 under 0 . 60 GPa as case example, we show that calculated peak isothermal entropy changes of ∼ 58 JK − 1 kg − 1 and peak adiabatic entropy changes of ∼ 23 K agree well with experimental values.

PCs are solids made of molecules with an orientational degree of freedom and whose centres of mass form a lattice with long-range positional order [23].In the parent hightemperature plastic phase, some or all of the constituent molecules perform nearly uncorrelated rotations, forming "molecular globules", and rendering the solid orientationally disordered [21].In the low-temperature crystal phase, the molecules spontaneously align along one or several axes, thus breaking rotational symmetry.Such changes in the orientational order are usually accompanied by very large latent heats and change in volume, which lead to very large BC effects when driving the transition with pressure.
The discovery of colossal BC effects in PCs has sparked theoretical and computational interest primarily using density functional theory [24], molecular dynamics methods [15,[25][26][27], supercell lattice dynamics calculations [11], and mean-field microscopic modelling [28], aimed at understanding plastic-to-crystal transitions and predicting their associated BC response.However, a macroscopic description based on Landau theory is missing.The purpose of this work is to propose such a theory, and illustrate it with a mimimal model that captures the essential features of plastic-to-crystal phase transitions.The main advantage of this approach is that both the molecular and lattice symmetries are incorporated into the model by construction, as well as the couplings between the order parameter (OP) of the transition and the elastic strains.We consider the simplest non-trivial case in which the orientational ordering is described by a quadrupolar OP, as is often done for nematic ordering in liquid crystals.Consequently, our model draws parallels to the Landau-de Gennes theory of phase transitions in liquid crystals [29].
The paper is organized as follows: first, we introduce the orientational OP of plasticto-crystal phase transitions, we develop the relevant Landau free energy density expansion, perform its minimization, and derive expressions for the BC properties (Methods section).
Second, we fit our model to experimental data in solid C 60 and provide a comparison with measured BC quantities (Results and Discussion).Finally, we summarize our findings and discuss the limitations of our model and its possible extensions (Summary and Conclusions).

A. Orientational and strain order parameters
Spontaneous long-range orientational orderings such as those occurring at plastic-tocrystal phase transitions can be described by multipole-moment OPs consistent with both the rotation and inversion symmetries of the constituent molecules [30,31].We consider the simplest possible non-trivial case in which the molecules do not exhibit inversion symmetry, i.e. a quadrupole moment Q.In addition, we consider the elastic strain ϵ that results from the orientational ordering, hydrostatic pressure, and the thermal expansion.

B. Free-energy density
We consider an isotropic solid subjected to a hydrostatic pressure P , and near a plasticto-crystal transition with a homogeneous quadrupole Q, and a homogeneous strain ϵ.We propose a free-energy density G given as follows, where G 0 = G 0 (T, P ) is the free-energy density of the background, G Q is the rotational energy, G ϵ is the strain energy, which includes thermal expansion, and G ϵQ is the strain-orientation coupling energy, The parameters a, b, c, d, e and g are independent of temperature and pressure, and a and c are assumed to be positive; T Q is the limit of stability of the symmetric phase in the absence of strain-orientation coupling; and α, K and µ are, respectively, the volumetric coefficient of thermal expansion in the parent phase, the bare bulk modulus, and the bare shear modulus.
We define the undeformed state as the volume of the material at atmospheric pressure at temperature T 0 [32].We then set the value of this reference undeformed volume equal to one, and normalize relevant thermodynamic quantities with respect to this volume.We note that the elastic strains ϵ are generated by the acoustic modes of the solid [33], which implies that such vibrations are implicitly incorporated into our model.
At equilibrium, the strain ϵ is given by the minimization ∂G/∂ϵ = 0 [34,35], which gives the following result, We then substitute Eq. ( 2) into Eq.( 1), which yields, where ã, b, c and TQ are renormalized parameters given as follows, We have thus arrived at a free-energy density that, in the absence of pressure and thermal expansion, shares an identical form to the Landau-de Gennes theory of isotropic-to-nematic phase transitions in liquid crystals [29], and therefore, exhibits similar features, the most important of which is the first-order character of the phase transition due to the cubic Equation ( 3) is the starting point for finding the equilibrium configuration Q(T, P ), and from which we compute the relative volume, and the entropy density, where V 0 = (∂G 0 /∂P ) T and S 0 = − (∂G 0 /∂T ) P are the relative volume and entropy density associated with the background, respectively.The background G 0 is an input to the model, which we will discuss in Sec.II D.

C. Plastic-to-crystal phase transition
We consider a plastic-to-crystal phase transition in which the orientational ordering occurs along a single axis n = (0, 0, 1).The quadrupole is thus given as follows [29], where A is a variational parameter that determines the degree of molecular alignment around n.In the plastic phase, the orientation of the molecules is random, thus A = 0.In the crystal phase, the molecular rotations are correlated, and tend to align along n, which leads to A > 0. The case A < 0 involves an ordered phase in which the molecules prefer to align perpendicular to n, but such phase turns out to be metastable at all temperatures and pressures.
Substitution of Eq. ( 6) into Eq.( 3) gives the following free-energy density, where we have rescaled the model parameters of Eq. ( 3) as follows: ã → ã/3, g → g/3, b → −4 b/9, and c → c/9.By minimizing Eq. ( 7) with respect to A, we obtain the following result, where T c is the pressure-induced transition temperature, and T 0 c = TQ + 2c b2 9ã is the transition temperature at ambient pressure.We note that in deriving Eq. ( 8), we have expanded A(T, P ) to linear order in (T c − T ) for T ≤ T c .By setting T 0 = T 0 c and substituting Eqs. ( 6) and ( 8) into Eq.( 4), we obtain the relative volume, where α + ∆α is the coefficient of thermal expansion in the ordered phase with ∆α given as follows, ∆V t is the change in relative volume at the phase transition, with ∆A t being the change in A(T, P ) at T c , and, Substitution of Eqs. ( 6) and ( 8) into Eq.( 5), gives the entropy density, where ∆S t is the change in entropy at the phase transition, We note ∆V t , ∆S t and dT c /dP are not independent as they obey the Clausius-Clapeyron equation, which we recover from Eqs. ( 12), ( 14) and ( 16).
From Eq. ( 15), we obtain the volumetric heat capacity, where C 0 = T (∂S 0 /∂T ) P is the volumetric heat capacity of the background.

D. Thermodynamic properties of the background
Equations ( 10), (15), and (18) depend on the thermodynamic properties of the background V 0 , S 0 and C 0 , which are formally derived from the background free-energy density G 0 .While in Landau theory such contributions are unimportant for describing thermodynamic properties associated solely with the phase transition, they are important in our model for calculating BC effects.
We determine G 0 by first assuming that V 0 is independent of temperature and pressure; second, by parametrizing C 0 with a function linear in temperature and independent of pressure; and third, by integrating V 0 = (∂G 0 /∂P ) T and C 0 = −T (∂ 2 G 0 /∂T 2 ) P .Explicit expressions for C 0 , G 0 , and S 0 are given in Sec.III A 1.
Our parametrization of C 0 is justified by noting that a linear function in temperature approximates well the ambient-pressure heat capacity of the disordered phase of BC PCs [7,16,[36][37][38]; and that the assumption of V 0 independent of temperature implies that C 0 is independent of pressure, as follows from the thermodynamic relation (∂C 0 /∂P ) T = −T (∂ 2 V 0 /∂T 2 ) P .We note that V 0 independent of temperature also implies that S 0 does not depend on pressure, as follows from the Maxwell relation (∂S 0 /∂P ) T = − (∂V 0 /∂T ) P .
We justify the assumption of V 0 independent of temperature and pressure by the overall agreement between our model and experiments, most notably, between the predicted and observed thermal expansion, see Sec.III A 2 and Fig. 2 (c).Such assumption holds in the vicinity of the phase transition, and would fail away from it as the thermal expansion of the individual plastic and crystal phases becomes non-linear in temperature [39].

E. BC effects
We calculate isothermal changes in entropy ∆S(T, P ) and adiabatic changes in temperature ∆T (T s , P ) on compression (0 → P ) and decompression (P → 0) following the standard procedure [40], where T s is the starting temperature.In general, both ∆S(T, P ) and ∆T (T s , P ) implicitly involve the background entropy density S 0 , see Eq. ( 15).However, given that S 0 is independent of pressure within our approximations (Sec.II D), only ∆T (T s , P ) depends on S 0 .
We now turn to the BC effects predicted by our model, which are schematically shown in Fig. 1 on compression for a material with dT c /dP > 0. We first note that the BC response depends on a threshold pressure P * at which S 0 PC = S C , where S 0 PC ≡ S 0 (T 0 c ) is the value of the entropy density at ambient pressure as T approaches T 0 c from above, and S C ≡ S 0 (T c )−(1 − KαdT c /dP ) αP +∆S t is the value of the entropy density at finite pressure as T approaches T c from below, see Figs. 1 (a), 1 (b), and Eq.(15).For P ≤ P * , the temperatureindependent isothermal changes in ∆S(T, P ) result solely from the thermal expansion of the individual plastic (T > T c ) and crystal phases (T < T 0 c ), while the temperature-dependent part in ∆S(T, P ) originates from the phase transition (T 0 c < T < T c ), as shown in Fig. 1 (c).For P > P * , ∆S(T, P ) increases in magnitude while retaining its overall shape, see Fig. 1 (d).
For the adiabatic changes ∆T (T s , P ) at P ≤ P * , the temperature-independent responses associated with the individual plastic and crystal phases occur, respectively, for T s < T 0 c and T s > T * s , while the temperature-dependent changes involving the phase transition occur for T 0 c < T s < T * s (Fig. 1 (e)).Here, T * s is the starting temperature at which S(T * s , 0) = S PC , where S PC ≡ S 0 (T c ) − (1 − KαdT c /dP ) αP is the value of the entropy density at finite pressure as T approaches T c from above, see Fig. 1 (e) and Eq.(15).For P > P * , ∆T (T s , P ) exhibits two temperature-dependent regions at T 0 c < T s < T * * s and T * c < T s < T * s , where T * * s is the starting temperature at which S(T * * s , 0) = S C , see Fig. 1 (f).We note that a similar analysis for a material with dT c /dP < 0 would yield inverse BC effects [40].

Fits to experiments
At ambient pressure and above ∼ 260 K, the C 60 molecules in fullerite are orientationally disordered and their centre of masses lie in a face-centered cubic lattice [41].Upon cooling, fullerite undergoes a first-order phase transition to a simple-cubic lattice structure in which the molecular orientations are partially ordered [42,43],with a relative volume reduction of about 1 % [44].For applied hydrostatic pressures in the range in which BC effects have been experimentally observed (0 − 0.60 GPa) [16], the transition temperature increases up to ∼ 360 K.The orientational OP of the phase transition is a tetrahexacontapole (64pole) [45], and while moments of lower order such as the quadrupole Q vanish due to molecular symmetry, we still use our model, as Eq. ( 7) is isomorphic to the free-energy density of fullerite [45].

Comparison to experiments
We now turn to the results predicted by our model using the above parametrization.
BC effects in solid C 60 were determined on compression and decompression from the entropies on cooling and heating ramps, respectively [16].Therefore, for the purposes of comparison, we calculate the isothermal entropy changes accordingly [40], i.e. ∆S(T, P ) = S cooling (T, P ) − S cooling (T, 0) for 0 → P , and ∆S(T, P ) = S heating (T, 0) − S heating (T, P ) for P → 0; as well as the adiabatic changes in temperature, i.e. ∆T (T s , P ) = T (S cooling , P ) − T s (S cooling , 0) for 0 → P , and ∆T (T s , P ) = T (S heating , 0) − T s (S heating , P ) for P → 0. Here, S heating (T, P ) is the predicted entropy on heating (Fig. 3 (a)) calculated from Eq. (15) with the above parametrization, and S cooling (T, P ) is the predicted entropy on cooling (Fig. 3 (b)) calculated from Eq. ( 15) with the above parametrization, except that T 0 c = 257 K and dT c /dP = 172 K GPa −1 , which we obtained by fitting Eq. ( 9) to the experimental temperature-pressure phase boundary on cooling [16].
We now assess the reversibility of the BC response [9].In order to account for the transition hysteresis, we calculate reversible isothermal changes in entropy ∆S rev and reversible adiabatic changes in temperature ∆T rev using the standard procedure [40], namely, ∆S rev (T, P ) = S heating (T, 0) − S cooling (T, P ) for 0 → P , ∆S rev (T, P ) = S heating (T, 0) − S cooling (T, P ) for P → 0, ∆T rev (T s , P ) = T (S cooling , P ) − T s (S heating , 0) for 0 → P , and ∆T rev (T s , P ) = T (S heating , 0) − T s (S cooling , P ) for P → 0. The results are shown in Fig. 4.
Our model captures the characteristic shape of the BC effects in the individual plastic and crystal phases, as well as the shape associated with the phase transition.On compression and decompression, we obtain identical threshold pressures (P * = 0.12 GPa), and nearly identical temperature spans T * s − T 0 c , T * s − T * * s and T * * s − T 0 c (Fig. 5), which is a consequence of the small hysteresis (≲ 6 K).
We have proposed a Landau theory for PCs in which orientational ordering is the sole driver of the plastic-to-crystal phase transition, thus enabling predictions of associated conventional and inverse BC effects.The key components of our theory are a multiple-moment OP that characterizes the collective alignment of the constituent molecules, its coupling to elastic strains, and the thermal expansion.We have illustrated these ideas with the simplest non-trivial model in which the orientational ordering is described by a quadrupolar OP.
Our theory predicts the general behavior found in several classes of BC PCs [7,13,24] and related orientationally disordered crystals [17][18][19], while also providing a physical framework for understanding their thermodynamic properties.
The most significant limitation of our model is the absence of pressure dependency in the transition entropy change ∆S t and the transition volume change ∆V t , which we attributed to neglecting non-linear invariants beyond those we considered in the orientation-strain coupling energy.Such interactions will add to a more precise description of the plastic-tocrystal transition and associated BC effects, but will not change the main predictions of our theory.
We have used solid C 60 as a case example and obtained excellent agreement with the observed thermal expansion and the temperature-pressure phase boundary, and good agreement with the observed BC effects.We have identified the temperature regions in the BC response associated with the individual plastic and crystal phases, as well as those involving the phase transition.For C 60 single crystals, we anticipate peak isothermal entropy changes and peak adiabatic temperature changes as large as ∼ 90 JK −1 kg −1 and ∼ 35 K, respectively, using 0.60 GPa at 265 K.
Finally, we note that our work provides a starting point for developing theories for electrolyte [15] and ferroelectric [49] PCs where the corresponding ionic diffusion and polarization likely contribute to the BC effects.T (K)