Design strategies for the self-assembly of polyhedral shells.

The control over the self-assembly of complex structures is a long-standing challenge of material science, especially at the colloidal scale, as the desired assembly pathway is often kinetically derailed by the formation of amorphous aggregates. Here, we investigate in detail the problem of the self-assembly of the three Archimedean shells with five contact points per vertex, i.e., the icosahedron, the snub cube, and the snub dodecahedron. We use patchy particles with five interaction sites (or patches) as model for the building blocks and recast the assembly problem as a Boolean satisfiability problem (SAT) for the patch-patch interactions. This allows us to find effective designs for all targets and to selectively suppress unwanted structures. By tuning the geometrical arrangement and the specific interactions of the patches, we demonstrate that lowering the symmetry of the building blocks reduces the number of competing structures, which in turn can considerably increase the yield of the target structure. These results cement SAT-assembly as an invaluable tool to solve inverse design problems.

Networks of Limited-Valency Patchy Particles.

Equilibrium gels provide physically attractive counterparts of nonequilibrium gels, allowing statistical understanding and design of the equilibrium gel structure. Here, we assemble two-dimensional equilibrium gels from limited-valency "patchy" colloidal particles and follow their evolution at the particle scale to elucidate cluster-size distributions and free energies. By finely adjusting the patch attraction with critical Casimir forces, we let a mixture of two-valent and pseudo-three-valent patchy particles approach the percolated network state through a set of equilibrium states. Comparing this equilibrium route with a deep quench, we find that both routes approach the percolated state via the same equilibrium states, revealing that the network topology is uniquely set by the particle bond angles, independent of the formation history. The limited-valency system follows percolation theory remarkably well, approaching the percolation point with the expected universal exponents.

Identification of local structures in water from supercooled to ambient conditions.

Studies of water thermodynamics have long been tied to the identification of two distinct families of local structures, whose competition could explain the origin of the many thermodynamic anomalies and the hypothesized liquid-liquid critical point in water. Despite the many successes and insights gained, the structural indicators proposed throughout the years were not able to unequivocally identify these two families over a wide range of conditions. We show that a recently introduced indicator, Ψ, which exploits information on the hydrogen bond network connectivity, can reliably identify these two distinct local environments over a wide range of thermodynamic conditions (188-300 K and 0-13 kbar) and that close to the liquid-liquid critical point, the spatial correlations of density fluctuations are identical to those of the Ψ indicator. Our results strongly support the idea that water thermodynamic properties arise from the competition between two distinct and identifiable local environments.

Potential energy landscape of a coarse grained model for water: ML-BOP.

We quantify the statistical properties of the potential energy landscape for a recently proposed machine learning coarse grained model for water, machine learning-bond-order potential [Chan et al., Nat. Commun. 10, 379 (2019)]. We find that the landscape can be accurately modeled as a Gaussian landscape at all densities. The resulting landscape-based free-energy expression accurately describes the model properties in a very wide range of temperatures and densities. The density dependence of the Gaussian landscape parameters [total number of inherent structures (ISs), characteristic IS energy scale, and variance of the IS energy distribution] predicts the presence of a liquid-liquid transition located close to P = 1750 ± 100 bars and T = 181.5 ± 1 K.

Free-energy landscape and spinodals for the liquid-liquid transition of the TIP4P/2005 and TIP4P/Ice models of water.

Continued increases in computational power now make it possible to evaluate the free-energy landscape associated with the first-order liquid-liquid transition in realistic models of water for which an accurate estimate of the liquid-liquid critical point exists, and to explore its change with pressure near the coexistence line. We report the results of 50 µs-long NPT umbrella sampling simulations for two realistic models for water, TIP4P/2005 and TIP4P/ice, 3-9 K below their critical temperatures. The free energy profile at different pressures clearly shows the presence of two well-defined free energy basins and makes it possible to identify the liquid-liquid spinodal points, the limits of stability that define the (temperature dependent) pressure range within which two distinct free energy basins exist. The results show that for temperatures less than 10 K below the critical temperature, metastable states are possible across a very limited pressure interval, information that is relevant to the interpretation of experiments probing the metastable phase behavior of deeply supercooled water in the so-called no-man's land.

Facile self-assembly of colloidal diamond from tetrahedral patchy particles via ring selection.

Diamond-structured crystals, particularly those with cubic symmetry, have long been attractive targets for the programmed self-assembly of colloidal particles, due to their applications as photonic crystals that can control the flow of visible light. While spherical particles decorated with four patches in a tetrahedral arrangement-tetrahedral patchy particles-should be an ideal building block for this endeavor, their self-assembly into colloidal diamond has proved elusive. The kinetics of self-assembly pose a major challenge, with competition from an amorphous glassy phase, as well as clathrate crystals, leaving a narrow widow of patch widths where tetrahedral patchy particles can self-assemble into diamond crystals. Here we demonstrate that a two-component system of tetrahedral patchy particles, where bonding is allowed only between particles of different types to select even-member rings, undergoes crystallization into diamond crystals over a significantly wider range of patch widths conducive for experimental fabrication. We show that the crystallization in the two-component system is both thermodynamically and kinetically enhanced, as compared to the one-component system. Although our bottom-up route does not lead to the selection of the cubic polytype exclusively, we find that the cubicity of the self-assembled crystals increases with increasing patch width. Our designer system not only promises a scalable bottom-up route for colloidal diamond but also offers fundamental insight into crystallization into open lattices.

A neural network potential with self-trained atomic fingerprints: A test with the mW water potential.

We present a neural network (NN) potential based on a new set of atomic fingerprints built upon two- and three-body contributions that probe distances and local orientational order, respectively. Compared with the existing NN potentials, the atomic fingerprints depend on a small set of tunable parameters that are trained together with the NN weights. In addition to simplifying the selection of the atomic fingerprints, this strategy can also considerably increase the overall accuracy of the network representation. To tackle the simultaneous training of the atomic fingerprint parameters and NN weights, we adopt an annealing protocol that progressively cycles the learning rate, significantly improving the accuracy of the NN potential. We test the performance of the network potential against the mW model of water, which is a classical three-body potential that well captures the anomalies of the liquid phase. Trained on just three state points, the NN potential is able to reproduce the mW model in a very wide range of densities and temperatures, from negative pressures to several GPa, capturing the transition from an open random tetrahedral network to a dense interpenetrated network. The NN potential also reproduces very well properties for which it was not explicitly trained, such as dynamical properties and the structure of the stable crystalline phases of mW.

Two-step nucleation in a binary mixture of patchy particles.

Nucleation in systems with a metastable liquid-gas critical point is the prototypical example of a two-step nucleation process in which the appearance of the critical nucleus is preceded by the formation of a liquid-like density fluctuation. So far, the majority of studies on colloidal and protein crystallization have focused on one-component systems, and we are lacking a clear description of two-step nucleation processes in multicomponent systems, where critical fluctuations involve coupled density and concentration inhomogeneities. Here, we examine the nucleation process of a binary mixture of patchy particles designed to nucleate into a diamond lattice. By combining Gibbs-ensemble simulations and direct nucleation simulations over a wide range of thermodynamic conditions, we are able to pin down the role of the liquid-gas metastable phase diagram on the nucleation process. In particular, we show that the strongest enhancement of crystallization occurs at an azeotropic point with the same stoichiometric composition of the crystal.

The physics of empty liquids: from patchy particles to water.

Empty liquids represent a wide class of materials whose constituents arrange in a random network through reversible bonds. Many key insights on the physical properties of empty liquids have originated almost independently from the study of colloidal patchy particles on one side, and a large body of theoretical and experimental research on water on the other side. Patchy particles represent a family of coarse-grained potentials that allows for a precise control of both the geometric and the energetic aspects of bonding, while water has arguably the most complex phase diagram of any pure substance, and a puzzling amorphous phase behavior. It was only recently that the exchange of ideas from both fields has made it possible to solve long-standing problems and shed new light on the behavior of empty liquids. Here we highlight the connections between patchy particles and water, focusing on the modelling principles that make an empty liquid behave like water, including the factors that control the appearance of thermodynamic and dynamic anomalies, the possibility of liquid-liquid phase transitions, and the crystallization of open crystalline structures.

Designing Enhanced Entropy Binding in Single-Chain Nanoparticles.

Single-chain nanoparticles (SCNPs) are a new class of bio- and soft-matter polymeric objects in which a fraction of the monomers are able to form equivalently intra- or interpolymer bonds. Here we numerically show that a fully entropic gas-liquid phase separation can take place in SCNP systems. Control over the discontinuous (first-order) change-from a phase of independent diluted (fully-bonded) polymers to a phase in which polymers entropically bind to each other to form a (fully-bonded) polymer network-can be achieved by a judicious design of the patterns of reactive monomers along the polymer chain. Such a sensitivity arises from a delicate balance between the distinct entropic contributions controlling the binding.

Liquid-liquid criticality in the WAIL water model.

The hypothesis that the anomalous behavior of liquid water is related to the existence of a second critical point in deeply supercooled states has long been the subject of intense debate. Recent, sophisticated experiments designed to observe the transformation between the two subcritical liquids on nano- and microsecond time scales, along with demanding numerical simulations based on classical (rigid) models parameterized to reproduce thermodynamic properties of water, have provided support to this hypothesis. A stronger numerical proof requires demonstrating that the critical point, which occurs at temperatures and pressures far from those at which the models were optimized, is robust with respect to model parameterization, specifically with respect to incorporating additional physical effects. Here, we show that a liquid-liquid critical point can be rigorously located also in the WAIL model of water [Pinnick et al., J. Chem. Phys. 137, 014510 (2012)], a model parameterized using ab initio calculations only. The model incorporates two features not present in many previously studied water models: It is both flexible and polarizable, properties which can significantly influence the phase behavior of water. The observation of the critical point in a model in which the water-water interaction is estimated using only quantum ab initio calculations provides strong support to the viewpoint according to which the existence of two distinct liquids is a robust feature in the free energy landscape of supercooled water.

Structure of High-Pressure Supercooled and Glassy Water.

We numerically investigate the structure of deep supercooled and glassy water under pressure, covering the range of densities corresponding to the experimentally produced high- and very-high-density amorphous phases. At T=188 K, a continuous increase in density is observed on varying pressure from 2.5 to 13 kbar, with no signs of first-order transitions. Exploiting a recently proposed approach to the analysis of the radial distribution function-based on topological properties of the hydrogen-bond network-we are able to identify well-defined local geometries that involve pairs of molecules separated by multiple hydrogen bonds, specific to the high- and very-high-density structures.

Hydrodynamic instability and flow reduction in polymer brush coated channels.

A polymer brush is a passive medium. At equilibrium the knowledge of its chemical composition and thickness is enough for a full system characterization. However, when the brush is exposed to fluid flow it reveals a much more intriguing nature, in which filamentous protrusions and the way they interact among themselves and with the surrounding fluid are of outmost importance. Here we investigate such a rich behavior via numerical simulations. We focus on the brush hydrodynamic response at low Reynolds numbers, observing a significant fluid flow reduction inside a polymer-brush coated channel. We find that the reduction of the flow inside the channel is significantly larger than what would happen if the brush effect consisted only in reducing the effective channel width. This amplified reduction is understood as being due to the morphological instability of the brush-liquid interface which is shown to have an elastic origin: the mechanical stress acting on the brush due to the imposed flow is partially released by the interface modulation. In turn, this modulation dissipates more energy than a flat interface in the surrounding fluid, causing a reduction of flow velocity. Our results and interpretations provide an explanation for recent experimental measurements.

Advances in the study of supercooled water.

In this review, we report recent progress in the field of supercooled water. Due to its uniqueness, water presents numerous anomalies with respect to most simple liquids, showing polyamorphism both in the liquid and in the glassy state. We first describe the thermodynamic scenarios hypothesized for the supercooled region and in particular among them the liquid-liquid critical point scenario that has so far received more experimental evidence. We then review the most recent structural indicators, the two-state model picture of water, and the importance of cooperative effects related to the fact that water is a hydrogen-bonded network liquid. We show throughout the review that water's peculiar properties come into play also when water is in solution, confined, and close to biological molecules. Concerning dynamics, upon mild supercooling water behaves as a fragile glass former following the mode coupling theory, and it turns into a strong glass former upon further cooling. Connections between the slow dynamics and the thermodynamics are discussed. The translational relaxation times of density fluctuations show in fact the fragile-to-strong crossover connected to the thermodynamics arising from the existence of two liquids. When considering also rotations, additional crossovers come to play. Mobility-viscosity decoupling is also discussed in supercooled water and aqueous solutions. Finally, the polyamorphism of glassy water is considered through experimental and simulation results both in bulk and in salty aqueous solutions. Grains and grain boundaries are also discussed.

Structural and topological changes across the liquid-liquid transition in water.

It has recently been shown that the TIP4P/Ice model of water can be studied numerically in metastable equilibrium at and below its liquid-liquid critical temperature. We report here simulations along a subcritical isotherm, for which two liquid states with the same pressure and temperature but different density can be equilibrated. This allows for a clear visualization of the structural changes taking place across the transition. We specifically focus on how the topological properties of the H-bond network change across the liquid-liquid transition. Our results demonstrate that the structure of the high-density liquid, characterized by the existence of interstitial molecules and commonly explained in terms of the collapse of the second neighbor shell, actually originates from the folding back of long rings, bringing pairs of molecules separated by several hydrogen-bonds close by in space.

Monodisperse patchy particle glass former.

Glass formers are characterized by their ability to avoid crystallization. As monodisperse systems tend to rapidly crystallize, the most common glass formers in simulations are systems composed of mixtures of particles with different sizes. Here, we make use of the ability of patchy particles to change their local structure to propose them as monodisperse glass formers. We explore monodisperse systems with two patch geometries: a 12-patch geometry that enhances the formation of icosahedral clusters and an 8-patch geometry that does not appear to strongly favor any particular local structure. We show that both geometries avoid crystallization and present glassy features at low temperatures. However, the 8-patch geometry better preserves the structure of a simple liquid at a wide range of temperatures and packing fractions, making it a good candidate for a monodisperse glass former.

Gelling without Structuring: A SAXS Study of the Interactions among DNA Nanostars.

We evaluate, by means of synchrotron small-angle X-ray scattering, the shape and mutual interactions of DNA tetravalent nanostars as a function of temperature in both the gas-like state and across the gel transition. To this end, we calculate the form factor from coarse-grained molecular dynamics simulations with a novel method that includes hydration effects; we approximate the radial interaction of DNA nanostars as a hard-sphere potential complemented by a repulsive and an attractive Yukawa term; and we predict the structure factors by exploiting the perturbative random phase approximation of the Percus-Yevick equation. Our approach enables us to fit all the data by selecting the particle radius and the width and amplitude of the attractive potential as free parameters. We determine the evolution of the structure factor across gelation and detect subtle changes of the effective interparticle interactions, that we associate to the temperature and concentration dependence of the particle size. Despite the approximations, the approach here adopted offers new detailed insights into the structure and interparticle interactions of this fascinating system.

Advances in Computational Studies of the Liquid-Liquid Transition in Water and Water-Like Models.

There has been uninterrupted interest in supercooled water ever since the pioneering experiments of Speedy and Angell revealed sharp increases in this substance's response functions upon supercooling. One intriguing hypothesis that was formulated to explain this behavior involves the existence of a metastable liquid-liquid transition (LLT) at deeply supercooled conditions. The preponderance of experimental evidence is consistent with this hypothesis, although no definitive proof exists to date. Computational studies have played an important role in this area, because ice nucleation can in principle be controlled in silico. It has been claimed, controversially, that the LLT is a misinterpreted liquid-solid transition in all models of water. Recent studies disprove this viewpoint by providing unambiguous counter-examples of distinct liquid-liquid and liquid-crystal transitions in tetrahedral models. In one, state-of-the-art sampling methods were used to compute the free energy surface of a molecular model of water and revealed the existence of two liquid phases in metastable equilibrium with each other and a stable crystal phase, at the same, deeply supercooled thermodynamic conditions. Further studies showed that, by tuning the potential parameters of a model tetrahedral system, it is possible to make the LLT evolve continuously from metastability to being thermodynamically stable with respect to crystallization. Most recently, it has been shown that the simulation code used to challenge the hypothesis of an LLT contains conceptual errors that invalidate the results on which the challenge was based, definitively resolving the controversy. The debate has vastly expanded the range of fundamental questions being pursued about phase transitions in metastable systems and ushered the use of increasingly sophisticated computational methods to explore the possible existence of LLTs in model systems.

Connection between liquid and non-crystalline solid phases in water.

The origin of water anomalies hides in an experimentally inaccessible region of the phase diagram known as no-man's land, bounded at low temperature by the domain of stability of amorphous glasses, and at high temperature by the homogeneous nucleation line, below which liquid water loses its metastability. The existence of at least two different forms of glass on one side, i.e., the low-density amorphous (LDA) and the high-density amorphous (HDA) ices, and of one anomalous liquid on the other side, points to a hidden connection between these states, whose understanding has the potential to uncover what happens in no-man's land and shed light on the complex nature of water's behavior. Here, we develop a Neural Network scheme capable of discerning local structures beyond tetrahedrality. Applied over a wide region of the water's phase diagram, we show that the local structures that characterize both LDA and HDA amorphous phases are indeed embedded in the supercooled liquid phase. Remarkably, the rapid increase in the LDA-like population with supercooling occurs in the same temperature and pressure region where thermodynamic fluctuations are maximized, linking these structures with water's anomalies. At the same time, the population of HDA-like environments rapidly increases with pressure, becoming the majority component at high density. Our results show that both LDA and HDA are genuine glasses, and provide a microscopic connection between the non-equilibrium and equilibrium phase diagrams of water.

A structural indicator for water built upon potential energy considerations.

We introduce a parameter-free structural indicator to classify local environments of water molecules in stable and supercooled liquid states, which reveals a clear two-peak distribution of local properties. The majority of molecules are tetrahedrally coordinated (T molecules), via low-energy hydrogen bonds. The minority component, whose relative concentration decreases with a decrease in the temperature at constant pressure, is characterized by prevalently three-coordinated molecules, giving rise to a distorted local network around them (D molecules). The inter-conversion between T and D molecules explains the increasing specific heat at constant pressure on cooling. The local structure around a T molecule resembles the one found experimentally in low-density amorphous ice (a network structure mostly composed by T molecules), while the local structure around a D molecule is reminiscent of the structural properties of high-density amorphous ice (a network structure composed by a mixture of T and D molecules).