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Register a new userFolding mechanisms at finite temperature
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Folding mechanisms are zero elastic energy motions essential to the deployment of origami, linkages, reconfigurable metamaterials and robotic structures. In this paper, we determine the fate of folding mechanisms when such structures are miniaturized so that thermal fluctuations cannot be neglected. First, we identify geometric and topological design strategies aimed at minimizing undesired thermal energy barriers that generically obstruct kinematic mechanisms at the microscale. Our findings are illustrated in the context of a quasi one-dimensional linkage structure that harbors a topologically protected mechanism. However, thermal fluctuations can also be exploited to deliberately lock a reconfigurable metamaterial into a fully expanded configuration, a process reminiscent of order by disorder transitions in magnetic systems. We demonstrate that this effect leads certain topological mechanical structures to exhibit an abrupt change in the pressure -- a bulk signature of the underlying topological invariant at finite temperature. We conclude with a discussion of anharmonic corrections and potential applications of our work to the the engineering of DNA origami devices and molecular robots.
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Many physical systems including lattices near structural phase transitions, glasses, jammed solids, and bio-polymer gels have coordination numbers that place them at the edge of mechanical instability. Their properties are determined by an interplay between soft mechanical modes and thermal fluctuations. In this paper we investigate a simple square-lattice model with a $phi^4$ potential between next-nearest-neighbor sites whose quadratic coefficient $kappa$ can be tuned from positive negative. We show that its zero-temperature ground state for $kappa <0$ is highly degenerate, and we use analytical techniques and simulation to explore its finite temperature properties. We show that a unique rhombic ground state is entropically favored at nonzero temperature at $kappa <0$ and that the existence of a subextensive number of floppy modes whose frequencies vanish at $kappa = 0$ leads to singular contributions to the free energy that render the square-to-rhombic transition first order and lead to power-law behavior of the shear modulus as a function of temperature. We expect our study to provide a general framework for the study of finite-temperature mechanical and phase behavior of other systems with a large number of floppy modes.
We consider the zero-energy deformations of periodic origami sheets with generic crease patterns. Using a mapping from the linear folding motions of such sheets to force-bearing modes in conjunction with the Maxwell-Calladine index theorem we derive a relation between the number of linear folding motions and the number of rigid body modes that depends only on the average coordination number of the origamis vertices. This supports the recent result by Tachi which shows periodic origami sheets with triangular faces exhibit two-dimensional spaces of rigidly foldable cylindrical configurations. We also find, through analytical calculation and numerical simulation, branching of this configuration space from the flat state due to geometric compatibility constraints that prohibit finite Gaussian curvature. The same counting argument leads to pairing of spatially varying modes at opposite wavenumber in triangulated origami, preventing topological polarization but permitting a family of zero energy deformations in the bulk that may be used to reconfigure the origami sheet.
One account of two-dimensional (2D) structural transformations in 2D ferroelectrics predicts an evolution from a structure with Pnm2$_1$ symmetry into a structure with square P4/nmm symmetry and is consistent with experimental evidence, while another argues for a transformation into a structure with rectangular Pnmm symmetry. An analysis of the assumptions made in these models is provided here, and six fundamental results concerning these transformations are contributed as follows: (i) Softened phonon modes produce rotational modes in these materials. (ii) The transformation to a structure with P4/nmm symmetry occurs at the lowest critical temperature $T_c$. (iii) The hypothesis that one unidirectional optical vibrational mode underpins the 2D transformation is unwarranted. (iv) Being successively more constrained, a succession of critical temperatures ($T_c<T_c<T_c$) occurs in going from molecular dynamics calculations with the NPT and NVT ensembles onto the model with unidirectional oscillations. (v) The choice of exchange-correlation functional impacts the estimate of the critical temperature. (vi) Crucially, the correct physical picture of these transformations is one in which rotational modes confer a topological character to the 2D transformation via the proliferation of vortices.
The maximum pressure a two-dimensional surfactant monolayer is able to withstand is limited by the collapse instability towards formation of three-dimensional material. We propose a new description for reversible collapse based on a mathematical analogy between the formation of folds in surfactant monolayers and the formation of Griffith Cracks in solid plates under stress. The description, which is tested in a combined microscopy and rheology study of the collapse of a single-phase Langmuir monolayer of 2-hydroxy-tetracosanoic acid (2-OH TCA), provides a connection between the in-plane rheology of LMs and reversible folding.
We consider the Haldane model, a 2D topological insulator whose phase is defined by the Chern number. We study its phases as temperature varies by means of the Uhlmann number, a finite temperature generalization of the Chern number. Because of the relation between the Uhlmann number and the dynamical transverse conductivity of the system, we evaluate also the conductivity of the model. This analysis does not show any sign of a phase transition induced by the temperature, nonetheless it gives a better understanding of the fate of the topological phase with the increase of the temperature, and it provides another example of the usefulness of the Uhlmann number as a novel tool to study topological properties at finite temperature.
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