Organic molecular compounds with modulated crystal structures

Aperiodic crystals in biology

Biological systems display a broad palette of hierarchically ordered designs spanning over many orders of magnitude in size. Remarkably enough, periodic order, which profusely shows up in non-living ordered compounds, plays a quite subsidiary role in most biological structures, which can be appropriately described in terms of the more general aperiodic crystal notion instead. In this topical review I shall illustrate this issue by considering several representative examples, including botanical phyllotaxis, the geometry of cell patterns in tissues, the morphology of sea urchins, or the symmetry principles underlying virus architectures. In doing so, we will realize that albeit the currently adopted quasicrystal notion is not general enough to properly account for the rich structural features one usually finds in biological arrangements of matter, several mathematical tools and fundamental notions belonging to the aperiodic crystals science toolkit can provide a useful modeling framework to this end.

Entropy-Driven Incommensurability: Chemical Pressure-Guided Polymorphism in PdBi and the Origins of Lock-In Phenomena in Modulated Systems

Incommensurate order, in which two or more mismatched periodic patterns combine to make a long-range ordered yet aperiodic structure, is emerging as a general phenomenon impacting the crystal structures of compounds ranging from alloys and nominally simple salts to organic molecules and proteins. The origins of incommensurability in these systems are often unclear, but it is commonly associated with relatively weak interactions that become apparent only at low temperatures. In this article, we elucidate an incommensurate modulation in the intermetallic compound PdBi that arises from a different mechanism: the controlled increase of entropy at higher temperatures. Following the synthesis of PdBi, we structurally characterize two low-temperature polymorphs of the TlI-type structure with single crystal synchrotron X-ray diffraction. At room temperature, we find a simple commensurate superstructure of the TlI-type structure (comm-PdBi), in which the Pd sublattice distorts to form a 2D pattern of short and long Pd-Pd contacts. Upon heating, the structure converts to an incommensurate variant (incomm-PdBi) corresponding to the insertion of thin slabs of the original TlI type into the superstructure. Theoretical bonding analysis suggests that comm-PdBi is driven by the formation of isolobal Pd-Pd bonds along shortened contacts in the distorted Pd network, which is qualitatively in accord with the 18-n rule but partially frustrated by the population of competing Bi-Bi bonding states. The emergence of incomm-PdBi upon heating is rationalized with the DFT-Cemical Pressure (CP) method: the insertion of TlI-type slabs result in regions of higher vibrational freedom that are entropically favored at higher temperatures. High-temperature incommensurability may be encountered in other materials when bond formation is weakened by competing electronic states, and there is a path for accommodating defects in the CP scheme.

Superspace description of trimethyltin hydroxide at T = 100 K

At low temperatures the metalorganic compound trimethyltin hydroxide, (CH3)3SnOH, possesses a commensurately modulated crystal structure, the modulation wave vector can be described as q   =   1 2 c ∗ . ${\bf{q}}\; = \;{1 \over 2}{{\bf{c}}^ * }.$ The crystal structure is studied by analysing single-crystal X-ray diffraction data within the (3+1)-dimensional superspace approach and superspace group P21212(00γ)00s. The corresponding twofold superstructure has space group symmetry P212121. The structure is characterised by polymeric chains running along c-axis, generated by Sn–O–Sn bridges between neighbouring Sn atoms and packed in a distorted hexagonal pattern and linked via C–H···O interstrand hydrogen bonds along the (orthorhombic) directions [110] and [11̅0], but not along [100].

Cimetidine, C10H16N6S, form C: crystal structure and modelling of polytypes using the superspace approach

The H2 antagonist cimetidine forms many polymorphs, several of which have resisted structural analysis thus far. Using single-crystal X-ray measurements obtained from synchrotron radiation, the crystal structure of cimetidine form C has been solved. This layered structure crystallizes in space groupC2/cwith an unusually large lattice parameter,a= 82.904 Å. The thickness of each layerLis equal toa′ =a/6 = 13.82 Å, anda= 6a′ originates from a sixfoldLLLL′L′L′ sequence withLandL′ differing by 0.5b. This packing is reminiscent of polytypic stacking in metals. A (3 + 1)-dimensional superspace model is derived and used to explain and predict many polytypic modifications. This model is characterized by (i) the (3 + 1)-dimensional symmetry groupX2/c(α0γ)00, whereX= 0\textstyle{1 \over 2}0\textstyle{1 \over 2}; (ii) the lattice parametera′ and modulation vectorq= 1/n(a′*); (iii) the atomic positions of a single molecule of cimetidine form C; (iv) the primary variable, 1/n. The model reproduces the previously solved structure, the 6M polytype, and generates the related polytypesnM with lattice parameteranM =na′ forn= 1, 2, 3, 4 and 5. A comparison of powder X-ray diffraction patterns available for cimetidine form C with those simulated for thenM polytypes suggests that the powder samples published previously probably contain a mixture of various polytypes.

Single-crystal investigation of L-tryptophan with Z' = 16

A complex, disorder-free structure in the space group P1 has been established for L-tryptophan, for which no crystal structure has previously been available. The 16 molecules in the asymmetric unit can be divided into two groups of eight; one where the side chains have gauche orientations and one with trans orientations. Molecules within each group have almost identical molecular geometries. The unit-cell parameters mimic a hexagonal cell, but deviations from 90° for the cell angles α = 84.421 (4) and β = 87.694 (4)° give a small tilt that rules out hexagonal symmetry. The hydrogen-bonding pattern resembles that found in the crystal structure of the racemic structure of DL-tryptophan, but a lower density combined with longer hydrogen bonds and inter-aromatic interactions show that the enantiomeric structure is less efficiently packed.

Confined linear molecules inside an aperiodic supramolecular crystal: the sequence of superspace phases in n-hexadecane/urea

High-resolution studies of the host-guest inclusion compound n-hexadecane/urea are reported at atmospheric pressure, using both cold neutrons and x-ray diffraction. This intergrowth crystal presents a misfit parameter, defined by the ratio c(h)/c(g) (c(host)/c(guest)), which is temperature independent and irrational (γ = 0.486 ± 0.002) from 300 to 30 K. Three different structural phases are reported for this aperiodic crystal over this temperature range. The crystallographic superspaces are of rank 4 in phases I and II, whereas phase III is associated with an increase in rank to 5, with a supplementary misfit parameter (δ = 0.058 ± 0.002) that is constant throughout this phase. The superspace group of phase I is hexagonal P6(1)22(00γ) down to T(c1) = 149.5 ± 0.5 K; phase II, which persists down to T(c2) = 127.8 ± 0.5 K is orthorhombic P2(1)2(1)2(1)(00γ), and phase III is orthorhombic P2(1)2(1)2(1)(00γ)(00δ).