Supercell approximation for an incommensurately modulated crystal of profilin:actin

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.