Conventional approaches to the capture of CO2 by metal-organic frameworks focus on equilibrium conditions, and frameworks that contain little CO2 in equilibrium are often rejected as carbon-capture materials. Here we use a statistical mechanical model, parameterized by quantum mechanical data, to suggest that metal-organic frameworks can be used to separate CO2 from a typical flue gas mixture when used under nonequilibrium conditions. The origin of this selectivity is an emergent gas-separation mechanism that results from the acquisition by different gas types of different mobilities within a crowded framework. The resulting distribution of gas types within the framework is in general spatially and dynamically heterogeneous. Our results suggest that relaxing the requirement of equilibrium can substantially increase the parameter space of conditions and materials for which selective gas capture can be effected.

Three-dimensional protein structures usually contain regions of local order, called secondary structure, such as α-helices and β-sheets. Secondary structure is characterized by the local rotational state of the protein backbone, quantified by two dihedral angles called ϕ and ψ. Particular types of secondary structure can generally be described by a single (diffuse) location on a two-dimensional plot drawn in the space of the angles ϕ and ψ, called a Ramachandran plot. By contrast, a recently-discovered nanomaterial made from peptoids, structural isomers of peptides, displays a secondary-structure motif corresponding to two regions on the Ramachandran plot [Mannige et al., Nature 526, 415 (2015)]. In order to describe such 'higher-order' secondary structure in a compact way we introduce here a means of describing regions on the Ramachandran plot in terms of a single Ramachandran number, [Formula: see text], which is a structurally meaningful combination of ϕ and ψ. We show that the potential applications of [Formula: see text] are numerous: it can be used to describe the geometric content of protein structures, and can be used to draw diagrams that reveal, at a glance, the frequency of occurrence of regular secondary structures and disordered regions in large protein datasets. We propose that [Formula: see text] might be used as an order parameter for protein geometry for a wide range of applications.

Cryogenic electron microscopy at atomic length scales was used to study the structure of self-assembled crystalline nanosheets obtained from a series of polypeptoids with the same chain architecture but with different end groups. While long-range order is enhanced by slowing down the self-assembly process, the dominant crystalline motif was found to be a sensitive function of both processing details and end group chemistry. In some cases, adjacent rows of polypeptoid molecules adopt anti-parallel V-shaped side chain conformations. In other cases, adjacent rows of polypeptoid molecules adopt parallel V-shaped side chain conformations. Interestingly, the unit cell is rectangular in both cases with dimensions a = 4.5 Å and c = 50 Å. In all cases, long-range order, quantified by the average number of concatenated unit cells of the same type, is more prevalent along the a direction.

Cooperative adsorption of gases by porous frameworks, which permits more efficient uptake and removal than the more usual noncooperative (Langmuir-type) adsorption, usually results from a phase transition of the framework. Here we show how cooperativity emerges in the class of metal-organic frameworks mmen-M_{2}(dobpdc) in the absence of a phase transition. Our study provides a microscopic understanding of the emergent features of cooperative binding, including the position, slope, and height of the isotherm step, and indicates how to optimize gas storage and separation in these materials.

Diamine-appended metal-organic frameworks (MOFs) of the form Mg₂(dobpdc)(diamine)₂ adsorb CO₂ in a cooperative fashion, exhibiting an abrupt change in CO₂ occupancy with pressure or temperature. This change is accompanied by hysteresis. While hysteresis is suggestive of a first-order phase transition, we show that hysteretic temperature-occupancy curves associated with this material are qualitatively unlike the curves seen in the presence of a phase transition; they are instead consistent with CO₂ chain polymerization, within one-dimensional channels in the MOF, in the absence of a phase transition. Our simulations of a microscopic model reproduce this dynamics, providing a physical understanding of cooperative adsorption in this industrially important class of materials.

Peptoid polymers are often crystalline in the solid-state as examined by X-ray scattering, but thus far, there has been no attempt to identify a common structural motif among them. In order to probe the relationship between molecular structure and crystal structure, we synthesized and analyzed a series of crystalline peptoid copolymers, systematically varying peptoid side-chain length (S) and main-chain length (N). We also examined X-ray scattering data from 18 previously reported peptoid polymers. In all peptoids, we found that the unit cell dimensions, a, b, and c, are simple functions of S and N: a (Å) = 4.55, b (Å) = [2.98]N + 0.35, and c (Å) = [1.86]S + 5.5. These relationships, which apply to both bulk crystals and self-assembled nanosheets in water, indicate that the molecules adopt extended, planar conformations. Furthermore, we performed molecular dynamics simulations (MD) of peptoid polymer lattices, which indicate that all backbone amides adopt the cis conformation. This is a surprising conclusion, because previous studies on isolated molecules indicated an energetic preference for the trans conformer. This study demonstrates that when packed into supramolecular lattices or crystals, peptoid polymers prefer to adopt a regular, extended, all-cis secondary structure.

Properties of soft crystalline materials such as synthetic polymers are governed by locations of constituent atoms. Determining atomic-scale structures in these materials is difficult because they degrade rapidly when studied by electron microscopy, and techniques such as X-ray scattering average over volumes much larger than coherent blocks of the unit cells. We obtained cryo-electron microscopy images of self-assembled nanosheets of a peptoid polymer, made by solid-phase synthesis, in which we see a variety of crystalline motifs. A combination of crystallographic and single-particle methods, developed for cryo-electron microscopy of biological macromolecules, was used to obtain high-resolution images of the crystals. Individual crystals contain grains that are mirror images of each other with concomitant grain boundaries. We have used molecular dynamic simulations to build an atomic model of the crystal structure to facilitate the interpretation of electron micrographs. Direct visualization of crystalline grains and grain boundaries on atomic length scales represents a new level of information for the polymer field.