Phase Diagram of a Strained Ferroelectric Nanowire

Revealing the three-dimensional arrangement of polar topology in nanoparticles

In the early 2000s, low dimensional ferroelectric systems were predicted to have topologically nontrivial polar structures, such as vortices or skyrmions, depending on mechanical or electrical boundary conditions. A few variants of these structures have been experimentally observed in thin film model systems, where they are engineered by balancing electrostatic charge and elastic distortion energies. However, the measurement and classification of topological textures for general ferroelectric nanostructures have remained elusive, as it requires mapping the local polarization at the atomic scale in three dimensions. Here we unveil topological polar structures in ferroelectric BaTiO3 nanoparticles via atomic electron tomography, which enables us to reconstruct the full three-dimensional arrangement of cation atoms at an individual atom level. Our three-dimensional polarization maps reveal clear topological orderings, along with evidence of size-dependent topological transitions from a single vortex structure to multiple vortices, consistent with theoretical predictions. The discovery of the predicted topological polar ordering in nanoscale ferroelectrics, independent of epitaxial strain, widens the research perspective and offers potential for practical applications utilizing contact-free switchable toroidal moments.

Theory of density waves and organization of proteins in icosahedral virus capsids

Understanding the physical principles underlying the structural organization of the proteinaceous viral shells is of major importance to advance antiviral strategies. Here, we develop a phenomenological thermodynamic theory, which considers structures of small and middle-size icosahedral viral shells as a result of condensation of a minimum number of protein density waves on a spherical surface. Each of these irreducible critical waves has icosahedral symmetry and can be expressed as a specific series of the spherical harmonics Ylm with the same wave number l. As we demonstrate, in small viral shells self-assembled from individual proteins, the maxima of one critical density wave determine the positions of proteins, while the spatial derivatives of the second one control the protein orientations on the shell surface. In contrast to the small shells, the middle-size ones are always formed from pentamers and hexamers (referred to as capsomers). Considering all such structures deposited in the Protein Data Bank, we unexpectedly found that the positions of capsomeres in these shells correspond to the maxima of interference patterns produced by no more than two critical waves with close wave numbers. This fact allows us to explain the observed limit size of the icosahedral shells assembled from pentamers and hexamers. We also construct nonequilibrium thermodynamic potentials describing the protein crystallization and discuss the reasons behind the specific handedness of the viral shells.