A phenomenological model for structural phase transitions in incommensurate alkane/urea inclusion compounds

n-Alkane/urea inclusion compounds are crystalline materials in which n-alkane ‘guest’ molecules are located within parallel one-dimensional ‘host’ tunnels formed by a helical hydrogen-bonded arrangement of urea molecules. The periodic repeat distance of the guest molecules along the host tunnels is incommensurate with the periodic repeat distance of the host substructure. The structural properties of the high-temperature phase of these materials (phase I), which exist at ambient temperature, are described by a (3 + 1)-dimensional superspace. Recent publications have suggested that, in the prototypical incommensurate composite systems, n-nonadecane/urea and n-hexadecane/urea, two low-temperature phases II and ‘III’ exist and that one or both of these phases are described by a (3 + 2)-dimensional superspace. We present a phenomenological model based on symmetry considerations and developed in the frame of a pseudo-spin–phonon coupling mechanism, which accounts for the mechanisms responsible for the I ↔ II ↔ ‘III’ phase sequence. With reference to published experimental data, we demonstrate that, in all phases of these incommensurate materials, the structural properties are described by (3 + 1)-dimensional superspace groups. Around the temperature of the II ↔ ‘III’ transition, the macroscopic properties of the material are not actually associated with a phase transition, but instead represent a ‘crossover’ between two regimes involving different couplings between relevant order parameters.

The (h, k, 0, 0) reflections are main reflections that are common to both the host and guest substructures, the (h, k, l, 0) reflections are main reflections of the host substructure (and also contain information on incommensurate modulations within the guest substructure), the (h, k, 0, m) reflections are main reflections of the guest substructure (and also contain information on incommensurate modulations within the host substructure) and the (h, k, l, m) reflections with l = 0 and m = 0 are satellite reflections [4][5][6]8].
A wide range of techniques have been used to explore structural phase transitions in n-alkane/urea inclusion compounds [1,13,14,[16][17][18][19][20][21][22][23][24][25]. Early studies reported that these materials undergo a single phase transition at a temperature T c1 below ambient temperature [20,26,27], with the phase transition temperature depending on the identity (chain length) of the n-alkane guest molecule. This phase transition is a ferroelastic transition from the hexagonal phase I above T c1 to an orthorhombic phase II below T c1 . In the low-temperature phase, the space group of the basic host structure is P2 1 2 1 2 1 (note that P2 1 2 1 2 1 is a subgroup of P6 1 22). In the low-temperature phase, a 'herringbone' antiferro ordering of the orientations of the guest molecules (projected onto the plane perpendicular to the tunnel  axis) exists between adjacent tunnels [14,16,17]. It was later suggested [22] that this phase transition is governed by an antiferro ordering of the n-alkane guest molecules affecting both the host shearing and the guest orientations. A theoretical analysis [4] described the phase transition within the framework of a (3 + 1)-dimensional superspace group description of all possible group-subgroup-related host and guest substructures. It followed that phase I is described by superspace group P6 1 22(00γ ) and phase II is described by superspace group P2 1 2 1 2 1 (00γ ). In both phases I and II, the host and guest substructures have the same periodicities in the ab-plane (i.e. the plane perpendicular to the tunnel axis), and the misfit parameter γ along the tunnel axis (c-axis) is the same in phases I and II. In the ab-plane, phase II is described by an orthorhombic unit cell which is very similar to the orthohexagonal description of the hexagonal unit cell of phase I but with no C-centring. Thus, for phase II, the orthorhombic unit cell is primitive and the lattice parameters a o and b o are approximately related by a o ≈ b o √ 3 (figure 2).
In this paper, we focus on the low-temperature phase transitions (at ambient pressure) in n-nonadecane/urea and n-hexadecane/urea, as they are the most extensively studied members of the family of incommensurate n-alkane/urea inclusion compounds and are regarded as prototypical incommensurate systems.
For n-nonadecane/urea, the I ↔ II phase transition occurs at T c1 ≈ 157 K (on cooling). However, the 'classical' description of this phase transition, as discussed above, was questioned recently [28], together with the report of another phase transition at a lower temperature T c2 ≈ 140 K (on cooling), corresponding to a weak thermal event in DSC data [29]. The 'new phase' below T c2 was denoted as 'phase III' [28]. Because the (3, 2, 0, 0) reflection (using the orthorhombic setting discussed above), which is a main reflection for both the host substructure and the guest substructure, was not observed in phase II below T c1 , these authors [28] concluded that the orthorhombic unit cell is C-centred (i.e. suggesting that the C-centre is 'preserved' from the orthohexagonal description of the hexagonal phase I), while they concluded that the orthorhombic unit cell of 'phase III' below T c2 is not C-centred. Furthermore, these authors also suggested [30] that an additional modulation with period c i exists along the tunnel axis in both phases II and 'III', leading to a second incommensurate misfit parameter δ = c host /c i in coexistence In the tunnel at the top, an n-alkane guest molecule is included (again in projection) to indicate the preferred orientation of the guest molecules relative to the host tunnel.
with the misfit parameter γ = c host /c guest along the same direction. Within this interpretation, five integer indices (h, k, l, m, n) are required to index each reflection in the diffraction pattern: The indices m and n relate to the misfit parameters γ and δ, respectively, with c * guest = γ c * host and c * i = a * ± δc * host , where a* is the reciprocal lattice vector corresponding to the orthorhombic a o lattice vector in real space (figure 2). The experimental values reported in [30] for n-nonadecane/urea are γ = 0.418 and δ = 0.090. Space group C222 1 (00γ )(10δ) was assigned [30] to phase II and space group P2 1 2 1 2 1 (00γ )(00δ) was assigned [30] to 'phase III'.
However, a very recent paper [31] (based on a high-resolution synchrotron single-crystal X-ray diffraction study of n-nonadecane/urea at ambient pressure) refuted the recently reported [28,30] interpretations (summarized above) of the structural properties of n-nonadecane/urea and instead demonstrated conclusively that the symmetries of both phases II and 'III' can be described by (3 + 1)dimensional superspace groups, which provide a complete description of the structural properties of these phases. In particular, the misfit parameters γ and δ in the (3 + 2)-dimensional superspace group description are not independent, as the following relation was proved [31] to exist (within experimental accuracy): so that δ = -2 + 5γ . Correspondingly, the indices (l, m, n) in the (3 + 2)-dimensional superspace group description [30,32,33] are also not independent. On this basis, a (3 + 1)-dimensional superspace group can be defined with only four independent indices (h, k, l , m ), where l and m are related to the indices l, m and n in the (3 + 2)-dimensional superspace description (equation (1.2)) by the following equation: l = l − 2n and m = m + 5n. (1.4) It follows that both phases II and 'III' of n-nonadecane/urea are correctly described by the same (3 + 1)-dimensional superspace group P2 1 2 1 2 1 (00γ ). Similarly, the symmetry of 'phase III' of n-hexadecane/urea was shown [31] to be correctly described by a (3 + 1)-dimensional superspace group rather than the (3 + 2)-dimensional superspace group suggested previously [12], as the following relation was proved [31] to exist (within experimental accuracy): δ = 2 -4γ .
A comment [34] has been published in response to paper [31] refuting some of our interpretations, but admitting explicitly the existence of the relationships between the two misfit parameters γ and δ for n-nonadecane/urea and n-hexadecane/urea quoted above and in paper [31]. Given that the (3 + 2)-dimensional superspace groups proposed previously [12,28,30] for n-nonadecane/urea and n-hexadecane/urea have been shown [31] to be incorrect, some general observations can be made based on the true degrees of freedom in these systems. Clearly, the orthorhombic distortion is achieved by orientational ordering of the guest molecules about the tunnel axis, coupled with the elastic properties of the host substructure [22]. By contrast, the translational motions of the guest molecules along the tunnel axis do not interact strongly with the host substructure, as evidenced by the molecular transport properties of these materials [35][36][37][38]. However, interactions between n-alkane guest molecules in adjacent tunnels play a role in the three-dimensional ordering of the guest molecules in the composite host-guest structure, evidenced by the fact that a well-defined value of g (defined above) is observed for n-alkane/urea inclusion compounds (for most n-alkane guest molecules, g = 0, but values of g = 0 have been reported [32,39] for the n-heptane/urea and n-dodecane/urea inclusion compounds). Furthermore, an important feature of incommensurate n-alkane/urea inclusion compounds is that the misfit parameter γ is independent of temperature [12,30].
Within the context of these observations, this paper presents a phenomenological model that accounts for the mechanisms responsible for the I ↔ II ↔ 'III' sequence of phase transitions in the n-nonadecane/urea and n-hexadecane/urea inclusion compounds. Our model is based on symmetry considerations and is developed within the frame of a pseudo-spin-phonon coupling mechanism, as suggested previously [22]. In particular, we show that this model is sufficient to explain the very low intensity (which others [28] may have interpreted as zero intensity) of the (h, k, 0, 0) reflections with h + k odd in phase II of n-nonadecane/urea. Moreover, we demonstrate that the differences between phases II and 'III' are explained by different couplings that exist between the relevant pseudo-spin variables and the translational motions of the n-alkane guest molecules.

Group to subgroup relations
In phase I of n-nonadecane/urea and n-hexadecane/urea, the space group of the basic host structure is P6 1 22 and the space group of the basic guest structure is P622. Thus, the superspace group based on the host sublattice is P6 1 22(00γ ) with lattice parameters (a, b, c host ). In both low-temperature phases II and 'III', the orthorhombic unit cell is similar to the orthohexagonal description of the hexagonal unit cell of phase I, but with no C-centring (figure 2). Thus, we deduce that the transitions to phases II and 'III' are related to a lattice instability occurring at a zone boundary point M(½, 0, 0), which is a reciprocal point common to both host and guest sublattices in the hexagonal Brillouin zone [40]. This point M is replaced at the zone centre (Γ point) in the low-temperature phases II and 'III'. The wavevector point group at point M is 222 (D 2 ) and the corresponding irreducible representations, denoted as M 1 /A, M 2 /B 1 , M 3 /B 2 and M 4 /B 3 , are three-dimensional as there are three arms in the star of this wavevector [40]. Thus, the order parameter associated with each representation has three components q i (i = 1, 2, 3) relative to the three arms of the star of the wavevector. However, in the present case, we only need to consider solutions that belong to the subspaces for which q j = Q = 0 for only one of the indices j (with q i = 0 for i = j), as only one arm of the star of point M is involved in the phase transition [40]. It follows that each equivalent point M 1 (½, 0, 0), M 2 (0, ½, 0) and M 3 (−½, ½, 0) generates a ferroelastic domain, and the three domains are related to each other by the threefold axis collinear with the c-axis (tunnel axis) in phase I, which is lost in the orthorhombic low-temperature phase. In fact, an additional splitting of each domain (by about 2°a t 90 K) is observed, due to the energy cost of domain walls, resulting in six orthorhombic domains [19]. Table 1 reports the character table of the critical wavevector group at point M(½, 0, 0) corresponding to the subspace q 1 (only the generating symmetry elements are shown). The subgroups induced by each irreducible representation for both the host and guest sublattices [41] and their representation in (3 + 1)dimensional superspace groups [4] are also shown. It turns out that superspace group P2 1 [40] of the critical wavevector group at point M(½, 0, 0) (only generating symmetry elements are shown) together with the subgroups induced by each irreducible representation for both the host and guest substructures [41] and their description in (3 + 1)-dimensional superspace groups on the basis of the host sublattice [4].
irreducible representation  order parameter of the phase transition (i.e. all symmetry properties of the low-temperature phase are determined by this order parameter, provided the misfit parameter γ remains constant). In figure 3, we report the compatibility relations that exist between the irreducible representations of space group P6 1 22 and those of the zone centre in the P2 1 2 1 2 1 subgroup, including the irreducible representations at point M (table 1), as well as those of the Γ point of phase I which remains as a zone centre point in the P2 1 2 1 2 1 subgroup. It transpires that the identity representation A of the P2 1 2 1 2 1 subgroup (which necessarily represents the symmetry of the order parameters in the distorted low-temperature phase) is compatible not only with M 2 /B 1 but also compatible with the E 2 representation at the zone centre of P6 1 22 (figure 3). Such associated order parameters are generally considered as secondary order parameters [42]. Indeed, we note that, when considered alone, the zone centre E 2 representation induces the C-centred C222 1 space group, of which P2 1 2 1 2 1 is a subgroup [41]. Such a transition would necessarily be a first-order transition, as the symmetrized third power [E 2 ] 3 = A 1 + A 2 + E 2 contains the totally symmetric representation, allowing the presence of a third-order invariant in the Landau free-energy development.

Pseudo-spins (reorientations)
Single-crystal 2 H NMR studies [14] of n-nonadecane/urea have shown that the n-nonadecane guest molecules adopt preferred orientations in both the hexagonal and orthorhombic phases, such that the projection of the plane of the carbon 'skeleton' onto the ab-plane points towards opposite corners of the hexagonal projection of the host tunnel (figure 2). For n-alkanes with odd chain length (e.g. n-nonadecane), the point symmetry is mm2 (C 2v ), and the C 2 axis of the n-alkane guest molecule coincides with one of the C 2 axes of the hexagonal host structure (figure 2). For n-alkanes with even chain length (e.g. n-hexadecane), the point symmetry is 2/m (C 2h ), and the C 2 axis of the n-alkane guest molecule coincides with one of the C 2 axes of the hexagonal host structure (figure 2).  We consider a simple Frankel model of jump reorientations of a rigid n-alkane guest molecule with odd chain length about the tunnel axis (c-axis), between six energetically equivalent orientations (figure 4) which are related to each other by 60°rotation around the sixfold axis of the tunnel. We use p i to denote the occupation probability of orientation i, with the condition that 6 i=1 p i = 1 in each phase. In the hexagonal phase I, the statistical sixfold site symmetry of the n-alkane guest molecule in each tunnel is achieved when p i = 1/6 (i = 1, 2, . . . , 6); (figure 4a), which describes the time-averaged and space-averaged orientational distribution of the n-alkane guest molecules irrespective of the details of the dynamics of the reorientation processes [43][44][45][46][47]. Following the procedure based on the method of projection operators [48], the pseudo-spin variables s i associated with the orientation of an n-alkane guest molecule of odd chain length at a site with 622 (D 6 ) symmetry in phase I are found to have the following form: and (2.1) The coordinate s 1 belonging to the totally symmetric representation A 1 is merely the identity and hence cannot be a symmetry-breaking order parameter. By contrast, all other coordinates s i (i = 1) are symmetry-breaking coordinates and may represent order parameters. We note that, for n-alkanes of even chain length, coordinates s i of the same form are established, except that the coordinate s 2 now belongs to the representation B 2 instead of B 1 . This does not change anything in our model, as neither the B 1 representation nor the B 2 representation is involved in the phase transition mechanism, as shown in the following discussion.
Within the orthorhombic unit cell of the low-temperature phase, there are two n-alkane guest molecules (labelled k = 1 and k = 2) and two urea tunnels per unit cell when projected onto the ab-plane It follows that the relevant pseudo-spin variables S i developed on the basis of the site coordinates s i at the zone centre (i.e. those variables belonging to E 2 symmetry; figure 3) have the following form: where the superscript (k = 1, 2) of the coordinates s k i refers to the labels for the two guest sites specified above. At point M(½, 0, 0) of the Brillouin zone, the sole S i pseudo-spin variable with M 2 /B 1 symmetry ( figure 3) is Following the arguments developed in §2.1, we establish that the S 6 pseudo-spin variable with M 2 /B 1 symmetry (equation (2.3)) acts as a primary order parameter for the hexagonal-to-orthorhombic phase transition, while S 5 (equation (2.2)) is a secondary order parameter ( figure 3). From equations (2.1-2.3), we also establish that the ground state corresponding to the expected 'herringbone' arrangement of the guest molecules in the low-temperature phase (see §1) [14,16,17] is achieved under the concomitant actions of S 5 and S 6 (with S 5 = 0 and S 6 = 0), corresponding to As shown in figure 4b, this situation does indeed correspond to a 'herringbone' arrangement of the nalkane guest molecules in the low-temperature orthorhombic phase. This description also implies that the ground state is disordered, with the n-alkane guest molecules occupying two equivalent orientations in each site, related by the {C 2z |000} symmetry operation of the orthorhombic P2 1 2 1 2 space group of the basic guest structure (table 1), in agreement [4] with orientational disorder of the n-alkane guest molecules in superspace group P2 1 2 1 2 1 (00γ ). As a consequence, a 'frozen' orientational disorder resulting in a 'glassy' crystalline state is expected at very low temperature; indeed, an excess heat capacity has been detected [49] between T = 0.1 K and T = 0.7 K for n-nonadecane/urea, which is well described by a two-level system model and could then be the signature of an orientational 'glassy' state due to the guest substructure.

Translations (acoustic modes)
For the urea host substructure, the pure translational motions close to the Brillouin zone centre are the acoustic modes. The propagation of these modes depends on the elastic constants of the crystal, and hence are related to the spontaneous strain that takes place at a ferroelastic phase transition (see §3, equation 3.4). In the present case, the elastic tensor components in the hexagonal phase transform as A 1 (e 1 +e 2 , e 3 ), E 1 (e 4 , e 5 ) and E 2 (e 1 -e 2 , e 6 ) symmetries [50,51]. It follows that the spontaneous strain e s associated with the (e 1 -e 2 ) strain tensor component responsible for an orthorhombic distortion is a secondary order parameter together with the S 5 pseudo-spin coordinate (figure 3). The associated elastic constants are (c 11 -c 12 ) = 2 c 66 . For the n-alkane guest substructure, the unique pure translational degree of freedom at the zone centre is the 'sliding mode', which represents translation of the n-alkane guest substructure as a whole relative to the urea host substructure along the tunnel axis. The sliding mode is analogous to the phason mode in modulated incommensurate systems, and hence, it is expected to exhibit very low (but non-zero) activation energy. Indeed, a diffusive or over-damped mode in the range of 0.8 meV that could be related to a sliding mode with a gap has been reported at the zone centre [25,52]. In any case, the sliding mode, which is polarized along the c-axis and hence belongs to A 2 symmetry at the zone centre, cannot be related to any order parameter, provided that the misfit parameter γ remains constant ( figure 3).
In both the host and guest substructures, the longitudinal acoustic (LA) and transverse acoustic (TA) modes at point M(½, 0, 0) transform as M 4 /B 3 (TA x polarized along the a-axis), M 3 /B 2 (LA y polarized along the b-axis) and M 2 /B 1 (TA z polarized along the c-axis). The TA z mode of the guest substructure (denoted as T z in figure 3) is of particular interest as it belongs to the M 2 /B 1 representation of the order parameter responsible for the loss of C-centring in the orthorhombic low-temperature phase. Furthermore, it corresponds to pure translational motions of the n-alkane guest molecules along the tunnel axis, which are alternately out of phase between successive tunnels along the [110] direction of the orthorhombic unit cell ( figure 4). This mode can therefore be viewed as a zone boundary sliding mode, probably associated with a low activation energy. 'Freezing' of such a motion will result in a static configuration in which the n-alkane guest molecules in successive tunnels along the [110] direction are alternately displaced along the c-axis by +δ g /2 or −δ g /2 from their average positions.

The pseudo-spin phonon coupling mechanism
On the basis of symmetry properties and a pseudo-spin translation coupling mechanism, we have identified two primary order parameters belonging to the zone boundary M 2 /B 1 representation, namely the T z zone boundary sliding mode (denoted as Q 1 ) and the S 6 pseudo-spin coordinate (denoted as Q 2 ). Furthermore, two secondary order parameters belonging to the zone centre E 2 representation have been identified, namely the S 5 pseudo-spin coordinate and the spontaneous strain e s corresponding to the component (e 1 -e 2 ) of the strain tensor (figure 3). Different couplings between these variables are allowed by symmetry. In particular, bilinear coupling between Q 1 and Q 2 (which both belong to M 2 /B 1 symmetry) of the form Q 1 Q 2 is allowed, and bilinear coupling between S 5 and e s (which both belong to E 2 symmetry) of the form S 5 e s is allowed. Furthermore, linear-quadratic coupling terms are allowed between primary and secondary order parameters of the form (Q 1 ) 2 S 5 , (Q 1 ) 2 e s , (Q 2 ) 2 S 5 and (Q 2 ) 2 e s .
A complete treatment would consider four competing 'full' order parameters S 5 , e s , Q 1 and Q 2 , each exhibiting its own temperature dependence and all dependent on each other. Such a complicated situation is unworkable in practice using classical analytical methods, unless some simplifications are adopted. First, we limit our discussion to the subspace of the wavevector group at point M corresponding to q 1 = 0 with q 2 = q 3 = 0 (see §2.1). In this context, starting from the hexagonal high-temperature phase I (S 5 = e s = Q 1 = Q 2 = 0) and provided the misfit parameter γ remains unchanged between the hightemperature and low-temperature phases (see §2.1), only two solutions for the low-temperature phases II and 'III' are induced by the M 2 /B 1 and E 2 representations, specifically the C222 1 (00γ ) superspace group (corresponding to S 5 = 0, e s = 0, Q 1 = 0, Q 2 = 0) and the P2 1 2 1 2 1 (00γ ) superspace group (corresponding to S 5 = 0, e s = 0, Q 1 = 0, Q 2 = 0). Furthermore, it has been shown that both phases II and 'III' belong to the same P2 1 2 1 2 1 (00γ ) superspace group [31]. As the phase with superspace symmetry C222 1 (00γ ) remains a virtual phase at ambient pressure, we adopt a shortcut by considering that, as a first approximation, Q 1 and Q 2 behave as 'full' order parameters (i.e. with their own temperature dependences) and that S 5 and e s with E 2 symmetry behave as 'classical' secondary order parameters. We note that the same approach was adopted by Breczewski et al. [53] to describe the ferroelastic phase transition in n-heptadecane/urea, but they considered only one zone boundary order parameter involved in the transition instead of the two coupled order parameters Q 1 and Q 2 used here.
In the present case, the temperature dependences of S 5 and e s are entirely related to those of Q 1 and Q 2 through the coupling terms (Q 1 ) 2 S 5 , (Q 1 ) 2 e s , (Q 2 ) 2 S 5 and (Q 2 ) 2 e s , which prevent the formation of the phase with C222 1 (00γ ) symmetry. Furthermore, we suppose that the order parameter Q 1 first condenses at T c1 , and that Q 2 is related to the thermal anomaly at T c2 (T c1 > T c2 ). At this stage of our analysis, this choice is arbitrary and could be reversed, as discussed in §4. Accordingly, we express the Landau free-energy potential developed up to sixth order as follows: where c 0 11 and c 0 12 are the 'bare' elastic constants [50] such as (c 0 11 − c 0 12 ) = 2c 0 66 . For simplicity, we have retained only the bilinear coupling term Q 1 Q 2 and the linear-quadratic coupling terms (Q 1 ) 2 S 5 and (Q 1 ) 2 e s . All other allowed couplings between Q 2 , S 5 and e s (i.e. (Q 2 ) 2 S 5 and (Q 2 ) 2 e s ) are neglected because the transition at T c2 does not produce any detectable anomaly in the temperature dependence of e s (see §4). Also, no Lifshitz invariant or gradient terms are considered because the misfit parameter γ (i.e. the parameter that describes all the incommensurate properties of the n-alkane/urea inclusion compounds) remains temperature-independent through the observed phase transitions. As we must account for two thermal anomalies at low temperature, occurring at T c1 and T c2 , we need to introduce two critical temperatures, which we denote as T 1 (the temperature at which A 1 changes sign) and T 2 (the temperature at which A 2 changes sign): with a 1 > 0 and a 2 > 0. The spontaneous strain is defined as follows: so that e s = ((2GE)/(F(c 0 11 − c 0 12 ))) S 5 . Thus, e s is directly proportional to S 5 , as expected. Then, equation (3.1) can be rewritten as follows: . (3.6) A similar form of the free-energy expansion has been considered by Salje and coworkers [51,[54][55][56] to describe the phase transition in sodium feldspar. The condition that the system is in thermodynamic equilibrium for any combination of the order parameters is which leads to two basic equations that must be satisfied simultaneously: and Only two solutions of equations (3.8) and (3.9) exist, which define two thermodynamic phases: and phase II : P2 1 2 1 2 1 (00γ ) : Q 1 = 0, Q 2 = 0. (3.11) No further phase described by only one order parameter is stable for D = 0. However, the uncoupled expression in Q 1 and Q 2 (i.e. with D = 0 in equation (3.5)) would produce two successive phase transitions around T 1 and T 2 (equation (3.2)). Hence, for B 1 > 0 and B 2 > 0, a second-order phase transition will occur at a temperature T c between T 1 and T 2 . For B 1 < 0 and B 2 > 0, a first-order phase transition (as observed experimentally) will occur at a temperature T c = T c1 with T c > T 1 and T c > T 2 (note that the signs of B 1 and B 2 are not imposed by any symmetry constraints). Importantly, the thermal anomaly at T c2 is no longer associated with a phase transition, but should be viewed instead as a 'crossover' point between two regimes [51] involving two-order parameters Q 1 and Q 2 with the same symmetry but different values of the ratio Q 1 /Q 2 . At temperatures T in the range T 2 < T < T c , the order parameter Q 1 dominates over Q 2 , because the non-zero value of Q 2 is induced via the coupling coefficient D, whereas for T < T 2 , the order parameter Q 2 increases because of the change of sign of the coefficient A 2 . Hence, around T 2 , many macroscopic properties of the crystal (such as heat capacity and elastic constants) will show similar types of behaviour to those that would be expected to occur at a structural phase transition. Therefore, in the experimental context (and provided no further structural evidence is available), the 'crossover' described above could readily be confused with a phase transition, with or without symmetry breaking [51]. Now, the minimization of equation (3.5) gives Putting this expression for Q 2 into equation (3.5) leads to a complicated even power expression of degree 15 (not given here) for F in (Q 1 ) 2 , the analytical treatment of which is unworkable in practice. Nevertheless, the true minimum of F together with the equilibrium value of Q 1 can be determined numerically. Then, the equilibrium value of Q 2 is deduced from equation (3.12) and the excess heat capacity ( C p ) can also be simulated numerically from the relation: To proceed further with numerical simulations, we have normalized the order parameters to unity (Q 1 = Q 2 = 1) at absolute zero temperature. In the absence of coupling (i.e. with D = 0), we obtain (3.14) Furthermore, the jump (denoted as Q 1c ) in Q 1 at T c (corresponding to the first-order phase transition) is given by with B 1 < 0 as specified above. Hence, we are left with seven independent free parameters, specifically: (Q 1c ) 2 , T 1 , T 2 , C 1 , B 2 , C 2 and D.

Comparison with experimental data
We recall that the n-nonadecane/urea and n-hexadecane/urea inclusion compounds have been reported to undergo two successive phase transitions at T c1 (about 157 K on cooling for n-nonadecane/urea and about 150 K on cooling for n-hexadecane/urea) and at T c2 (about 140 K on cooling for n-nonadecane/urea and about 125 K on cooling for n-hexadecane/urea), corresponding to the phase sequence I → II → 'III' on decreasing temperature as discussed above [12,28,30]. In §3, we have proposed instead that the 'transition' at T c2 actually corresponds to a 'crossover' between two competing order parameters belonging to the same symmetry. Now, the observed phase transition at T c1 is clearly first order, given the abrupt changes in lattice parameters (and hence in the spontaneous strain e 1 -e 2 ) and the thermal hysteresis observed at this transition [12,29,57], which implies that B 1 < 0 (and C 1 > 0) in equation (3.5). We also recall that two zone boundary order parameters Q 1 and Q 2 both contribute to the new reflections (h, k, l , m ) that appear in the low-temperature phase in positions with h + k odd, including main reflections for the host substructure (with m = 0), main reflections for the guest substructure (with l = 0) and satellite reflections (with l = 0 and m = 0). We emphasize that the respective contributions of Q 1 and Q 2 may be different for different n-alkane/urea inclusion compounds, and hence may explain differences in behaviour such as those observed [12,28] between n-nonadecane/urea and n-hexadecane/urea.
We now compare our model potential with experimental data for n-nonadecane/urea, using the following strategy to select suitable values of the parameters [(Q 1c ) 2 , T 1 , T 2 , C 1 , B 2 , C 2 and D] in our model. First, the jump of (Q 1 ) 2 at T c1 [denoted as (Q 1c ) 2 ] was fixed at 0.5 in order to comply qualitatively with the experimental results obtained for e s = (e 1 -e 2 ) at T c1 for n-nonadecane/urea [57] and n-hexadecane/urea [12] (see equations (3.4) and (3.15)). Then, the value of T 1 was set to 126 K, corresponding to T c1 ≈ 157 K (as observed for n-nonadecane/urea), and the value of T 2 was set to 140 K, corresponding to T c2 ≈ 140 K (as observed for n-nonadecane/urea). The coefficients B 2 , C 1 and C 2 were adjusted manually so that the thermal anomaly around T c2 is sufficiently narrow and intense. Finally, we have found that very weak values of the coupling coefficient D are required in order to reproduce the experimental data qualitatively (this observation is completely consistent with previous studies [47,58], which concluded that the reorientational and translational motions of the n-alkane guest molecules in n-alkane/urea inclusion compounds are uncorrelated). The results of our calculations using C 1 = 0.5, B 2 = 0.1 and C 2 = 0.9, together with either D = 10 −4 or D = 5 × 10 −3 , are shown in figure 5.
Clearly, our calculated data for the excess heat capacity C p as a function of temperature (figure 5b) are in good qualitative agreement with experimental DSC data reported for n-nonadecane/urea [29] and n-hexadecane/urea [12]. Thus, a strong thermal anomaly is associated with the first-order transition at T c1 , and only a weak thermal anomaly is associated with the 'crossover' at T c2 .
The temperature dependences of (Q 1 ) 2 and (Q 2 ) 2 , calculated using the phenomenological parameters specified above, are shown in figure 5b. According to the relations in equation (3.4), (Q 1 ) 2 is proportional to both S 5 and e s . No direct measurement of S 5 has been reported so far, but e s can be determined from the lattice parameters a o and b o (equation (3.3)) for n-nonadecane/urea [57] and n-hexadecane/urea [12].  followed by a damped regular increase at lower temperatures, as expected for a 'weak' first-order phase transition. As mentioned above, Q 1 and Q 2 are related to the intensities of the main reflections with h + k odd and the intensities of the satellite reflections with h + k odd in the low-temperature phase. Furthermore, the intensities of these peaks are proportional to the square of the order parameter responsible for the phase transition [59]. Thus, plots of (Q 1 ) 2 versus T and (Q 2 ) 2 versus T calculated using our model (figure 5b) can be compared directly to the intensities of the relevant reflections as a function of temperature.
In the case of n-nonadecane/urea containing fully deuterated guest molecules, an abrupt increase in the intensities of the satellite reflections (h, k, l , m ) with l = 0, m = 0 and h + k odd has been reported [28, fig. 3b] at the transition near T c1 , followed by a regular increase up to a maximum and then a slow decrease down to about T c2 . This reported behaviour is consistent with the results obtained from our ad hoc model, except for the existence of the maximum between T c1 and T c2 . At the same time, the main reflections of the host substructure (h, k, l , 0) with h + k odd and the main reflections of the guest substructure (h, k, 0, m ) with h + k odd are very weak, but it is evident [28, fig. 3b] that their intensities increase as temperature decreases, which is the predicted behaviour for (Q 2 ) 2 according to our model (figure 5b) and is in good agreement with experimental observations that we reported recently [31]. Below 130 K, the intensities of these reflections exhibit a marked and progressive increase [28, fig. 3], as expected from our model (figure 5b).
For n-hexadecane/urea, it is reported [12] that the main reflections with h + k odd first appear abruptly at T c1 ≈ 150 K, whereas the intensities of the satellite reflections with h + k odd grow progressively at temperatures below about 130 K [12, fig. 11]. Significantly, these observations are also in qualitative agreement with our model (figure 5b), but with the roles of Q 1 and Q 2 reversed compared to the situation discussed above for n-nonadecane/urea. Thus, for n-hexadecane/urea, Q 2 must be considered as the driving order parameter instead of Q 1 when the coupling occurs as (Q 2 ) 2 S 5 and (Q 2 ) 2 e s rather than as (Q 1 ) 2 S 5 and (Q 1 ) 2 e s .    Figure 6. Schematic representation, viewed along the tunnel axis (c-axis), of the orientations of the n-alkane guest molecules in phases I and II of the n-nonadecane/urea and n-hexadecane/urea inclusion compounds. The orientations of the guest molecules, projected onto the ab-plane, are shown as arrows. In phase I, the guest molecules are equally distributed among six energetically equivalent orientations. In the initial state of phase II for n-nonadecane/urea (state A), herringbone ordering of the guest molecules is partially achieved, with the lengths of the arrows symbolizing the relative probability of different orientations. Guest molecules displaced by +δ g /2 along the tunnel axis are shown as blue arrows and guest molecules displaced by −δ g /2 are shown as red arrows. In the initial state of phase II for n-hexadecane/urea (state B), complete 'herringbone' ordering of the guest molecules is achieved. In the final state of phase II for both n-nonadecane/urea and n-hexadecane/urea, herringbone ordering is achieved and the guest molecules adopt the same displacements along the tunnel axis as in state A.
Our analysis demonstrates that two different contributions govern the intensities of the main reflections with h + k odd and the intensities of the satellite reflections with h + k odd, in relation to the two-order parameters Q 1 and Q 2 . Hence, we propose that the intensity of the satellite reflections is mainly associated with the T z zone boundary sliding mode polarized along the c-axis (Q 1 ), as it is related to the incommensurate properties of the composite crystal along the tunnel axis, and we propose that the intensity of the main reflections is essentially governed by the pseudo-spin coordinate S 6 (Q 2 ), as it is responsible for the commensurate 'herringbone' ordering of the n-alkane guest molecules in the a o b o -plane. Thus, for n-nonadecane/urea, it turns out that the zone boundary sliding mode T z (Q 1 ) first condenses at T c1 together with the pseudo-spin coordinate S 5 , creating the spontaneous strain e s , while the pseudo-spin coordinate S 6 (Q 2 ) is almost inactive. As a result, orientational ordering of the n-nonadecane guest molecules is partially achieved due to S 5 and a small contribution of S 6 (i.e. Q 1 > > Q 2 ), whereas the anti-translations of the n-nonadecane guest molecules along the tunnel direction have already condensed, resulting in the intermediate 'state A' of phase II schematized in figure 6. Below the 'crossover' temperature T 2 , S 6 (Q 2 ) then condenses to achieve the final state of phase II with a complete 'herringbone' orientational ordering of the n-nonadecane guest molecules.
In the case of n-hexadecane/urea, for which the roles of Q 1 and Q 2 are reversed, the pseudospin coordinate S 6 (Q 2 ) first condenses at T c1 together with S 5 , thus achieving complete 'herringbone' ordering of the n-hexadecane guest molecules, while T z (Q 1 ) is almost inactive (i.e. Q 2 > > Q 1 ), resulting in the intermediate 'state B' of phase II schematized in figure 6. Then, below the 'crossover' temperature (T 1 in this case), the zone boundary sliding mode T z (Q 1 ) condenses to give the final state of phase II ( figure 6).
Clearly, the difference between the n-nonadecane/urea and n-hexadecane/urea inclusion compounds may be interpreted in terms of differences in the coupling forces between Q 1 , Q 2 , S 5 and e s .

Concluding remarks
Based on a wide range of published data relating to structural phase transitions in the incommensurate n-nonadecane/urea and n-hexadecane/urea inclusion compounds, and based on symmetry considerations, we have identified the order parameters necessary to explain the I ↔ II ↔ 'III' phase sequence reported recently in the literature for these materials. Specifically, the order parameters are zone centre and zone boundary pseudo-spin coordinates, the coupling of which accounts for 'herringbone' orientational ordering of the n-alkane guest molecules and explains the transition from the orientationally disordered hexagonal P6 1 22(00γ ) phase I to the orientationally ordered orthorhombic P2 1 2 1 2 1 (00γ ) phase II. Also involved are the macroscopic strain (e 1 -e 2 ), which is responsible for the orthorhombic distortion, and the so-called zone boundary sliding mode of the guest substructure relative to the host substructure.
We have elaborated a simplified thermodynamic Landau potential taking into account the fact that phases II and 'III' belong to the same (3 + 1)-dimensional superspace group P2 1 2 1 2 1 (00γ ). This model can account qualitatively for the temperature dependence of the spontaneous strain and the temperature dependence of the intensities of main reflections and satellite reflections. Hence, we have shown that the reported II ↔ 'III' phase transition should actually be regarded as a 'crossover' between two competing order parameters with the same symmetry: specifically, the zone boundary pseudo-spin coordinate and the zone boundary sliding mode. So, we conclude that the herringbone orientational ordering and the translational freezing of the n-alkane guest molecules take place in two steps within a single phase II, with these two steps occurring in the reverse order on decreasing temperature for n-nonadecane/urea and n-hexadecane/urea. Furthermore, our results show conclusively that the physical basis of the phase transition mechanisms in these incommensurate composite materials does not require the introduction of a higher-dimensional [(3 + 2)-dimensional] description [30] in superspace.
Finally, in the context of the (3 + 1)-dimensional description of the superspace group in phase II of n-nonadecane/urea, the fact that satellite reflections are indexed with large values of l and m [31] may appear unusual. However, we note that examples of incommensurate systems with high values of indices for satellite reflections arise when the modulation function becomes discontinuous and a non-analytical 'soliton regime' can exist [60].
Data accessibility. This article has no additional data.