The true structural periodicities and superspace group descriptions of the prototypical incommensurate composite materials: Alkane/urea inclusion compounds

Abstract

The prototypical family of incommensurate composite materials are the n-alkane/urea inclusion compounds, in which n-alkane guest molecules are arranged in a periodic manner along one-dimensional tunnels in a urea host structure, with an incommensurate relationship between the periodicities of the host and guest substructures along the tunnel. We develop interpretations of the structural periodicities, superspace group descriptions and symmetry properties of the lowtemperature phases of n-alkane/urea inclusion compounds, based in part on a high-resolution synchrotron single-crystal X-ray diffraction study of n-nonadecane/urea. Specifically, we prove that, on passing from phase I to phase II, the C-centering of the orthohexagonal unit cell is lost for both the host and guest substructures, and that the symmetries of all phases I, II and III are described completely by (3+1)-dimensional superspace groups.

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

“…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 5 b ) 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.…”

“…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 ].…”

“…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 high-temperature 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 C 222 1 (00 γ ) superspace group (corresponding to S 5 ≠ 0, e s ≠ 0, Q 1 = 0, Q 2 = 0) and the P 2 1 2 1 2 1 (00 γ ) 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 P 2 1 2 1 2 1 (00 γ ) superspace group [ 31 ]. As the phase with superspace symmetry C 222 1 (00 γ ) 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.…”

“…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 + 5 γ . Correspondingly, the indices ( l , m , n ) in the (3 + 2)-dimensional superspace group description [ 30 , 32 , 33 ] are also not independent.…”

Comment on Couzi et al . (2018): a phenomenological model for structural transitions in incommensurate alkane/urea inclusion compounds

“…Instead of using these careful measurements, M. Couzi et al . used the results from their recently published article, which describes an X-ray diffraction study performed on n -nonadecane-h 40 /urea-h 4 but with measurements reported at only two temperatures (147 K and 100 K) [11]. In a Comment concerning that article [2], which they did not contest, we demonstrated that their data have no relevance.…”