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

Abstract

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.

Tutorial on Chemical Pressure Analysis: How Atomic Packing Drives Laves/Zintl Intergrowth in K3Au5Tl

“…This method is based on the recognition that non-optimal interatomic distances are detectable in the local pressures that surround the atoms of a solid-state lattice, as constructed from the output of DFT calculations. In the study of a variety of intermetallic phases, CP analysis has provided explanations for diverse structural phenomena, such as the insertion of interfaces into simple structures [19,20,24,25], the adoption of local icosahedral symmetry in quasicrystal approximants [26][27][28], the emergence in incommensurability [29][30][31][32][33][34], the stabilizing effects of various kinds of single point substitutions [23][24][25]35,36], and the formation of intergrowth structures [27,35,[37][38][39]. The CP maps generated in the process can also be used to analyze the forces involved in chemical bonding and molecular structure [40][41][42][43].…”

Frustrated Packing in Simple Structures: Chemical Pressure Hindrance to Isolobal Bonds in the TiAl3 type and ZrAl2.6Sn0.4

“…Both of these storiesthe preference of TiAl 3 for its own type vs the ZrAl 3 type (at least at high temperatures), and the reverse transition upon incorporating Sn into ZrAl 3 provide illustrations of the Frustrated and Allowed Structural Transitions (FAST) principle − applied recently to the explanation of several incommensurately modulated structures. The FAST principle serves a guide to navigating situations where several factors, such as the electronic and atomic packing factors commonly evoked in intermetallics, simultaneously influence the stabilities of two related structures.…”

DFT-Guided Crystal Structure Redetermination and Lattice Dynamics of the Intermetallic Actinoid Compound UIr

“…Recently, the crystal structure of PdBi was reinvestigated and corrected from space group P2 1 to P2 1 /c. 18 The authors of that study also reported twinning for the room-temperature crystals of PdBi. It originates from a phase transition from the high-temperature CrB structure type into the PdBi structure type at room temperature.…”

“…Similar observations were made in the case of PdBi that crystallizes isotypically with UIr. Recently, the crystal structure of PdBi was reinvestigated and corrected from space group P 2 1 to P 2 1 / c . The authors of that study also reported twinning for the room-temperature crystals of PdBi.…”

“…Surprisingly, structural optimizations of UIr converged in the centrosymmetric space group P 2 1 / c , which is a supergroup of the reported space group of UIr. This, as well as a recent reinvestigation of the crystal structure of the parent compound PdBi that led to the centrosymmetric space group P 2 1 / c , motivated us to reinvestigate the crystal structure of UIr experimentally and quantum chemically.…”