Inverse design of crystals and quasicrystals in a non-additive binary mixture of hard disks.

The development of new materials typically involves a process of trial and error, guided by insights from past experimental and theoretical findings. The inverse design approach for soft-matter systems has the potential to optimize specific physical parameters, such as particle interactions, particle shape, or composition and packing fraction. This optimization aims to facilitate the spontaneous formation of specific target structures through self-assembly. In this study, we expand upon a recently introduced inverse design protocol for monodisperse systems to identify the required conditions and interactions for assembling crystal and quasicrystal phases within a binary mixture of two distinct species. This method utilizes an evolution algorithm to identify the optimal state point and interaction parameters, enabling the self-assembly of the desired structure. In addition, we employ a convolutional neural network (CNN) that classifies different phases based on their diffraction patterns, serving as a fitness function for the desired structure. Using our protocol, we successfully inverse design two-dimensional crystalline structures, including a hexagonal lattice and a dodecagonal quasicrystal, within a non-additive binary mixture of hard disks. Finally, we introduce a symmetry-based order parameter that leverages the encoded symmetry within the diffraction pattern. This order parameter circumvents the need for training a CNN and is used as a fitness function to inverse design an octagonal quasicrystal.

Variational umbrella seeding for calculating nucleation barriers.

In this work, we introduce variational umbrella seeding, a novel technique for computing nucleation barriers. This new method, a refinement of the original seeding approach, is far less sensitive to the choice of order parameter for measuring the size of a nucleus. Consequently, it surpasses seeding in accuracy and umbrella sampling in computational speed. We test the method extensively and demonstrate excellent accuracy for crystal nucleation of nearly hard spheres and two distinct models of water: mW and TIP4P/ICE. This method can easily be extended to calculate nucleation barriers for homogeneous melting, condensation, and cavitation.

Nanocrystal Assemblies: Current Advances and Open Problems.

We explore the potential of nanocrystals (a term used equivalently to nanoparticles) as building blocks for nanomaterials, and the current advances and open challenges for fundamental science developments and applications. Nanocrystal assemblies are inherently multiscale, and the generation of revolutionary material properties requires a precise understanding of the relationship between structure and function, the former being determined by classical effects and the latter often by quantum effects. With an emphasis on theory and computation, we discuss challenges that hamper current assembly strategies and to what extent nanocrystal assemblies represent thermodynamic equilibrium or kinetically trapped metastable states. We also examine dynamic effects and optimization of assembly protocols. Finally, we discuss promising material functions and examples of their realization with nanocrystal assemblies.

Finding the differences: Classical nucleation perspective on homogeneous melting and freezing of hard spheres.

By employing brute-force molecular dynamics, umbrella sampling, and seeding simulations, we investigate homogeneous nucleation during melting and freezing of hard spheres. We provide insights into these opposing phase transitions from the standpoint of classical nucleation theory. We observe that melting has both a lower driving force and a lower interfacial tension than freezing. The lower driving force arises from the vicinity of a spinodal instability in the solid and from a strain energy. The lower interfacial tension implies that the Tolman lengths associated with melting and freezing have opposite signs, a phenomenon that we interpret with Turnbull's rule. Despite these asymmetries, the nucleation rates for freezing and melting are found to be comparable.

A mean-field theory for predicting single polymer collapse induced by neutral crowders.

Macromolecular crowding can induce the collapse of a single long polymer into a globular form due to depletion forces of entropic nature. This phenomenon has been shown to play a significant role in compacting the genome within the bacterium Escherichia coli into a well-defined region of the cell known as the nucleoid. Motivated by the biological significance of this process, numerous theoretical and computational studies have searched for the primary determinants of the behavior of polymer-crowder phases. However, our understanding of this process remains incomplete and there is debate on a quantitatively unified description. In particular, different simulation studies with explicit crowders have proposed different order parameters as potential predictors for the collapse transition. In this work, we present a comprehensive analysis of published simulation data obtained from different sources. Based on the common behavior we find in this data, we develop a unified phenomenological model that we show to be predictive. Finally, to further validate the accuracy of the model, we conduct new simulations on polymers of various sizes, and investigate the role of jamming of the crowders.

Depletion-induced crystallization of anisotropic triblock colloids.

The intricate interplay between colloidal particle shape and precisely engineered interaction potentials has paved the way for the discovery of unprecedented crystal structures in both two and three dimensions. Here, we make use of anisotropic triblock colloidal particles composed of two distinct materials. The resulting surface charge heterogeneity can be exploited to generate regioselective depletion interactions and directional bonding. Using extensive molecular dynamics simulations and a dimensionality reduction analysis approach, we map out state diagrams for the self-assembly of such colloids as a function of their aspect ratio and for varying depletant features in a quasi two-dimensional set-up. We observe the formation of a wide variety of crystal structures such as a herringbone, brick-wall, tilted brick-wall, and (tilted) ladder-like structures. More specifically, we determine the optimal parameters to enhance crystallization, and investigate the nucleation process. Additionally, we explore the potential of using crystalline monolayers as templates for deposition, thereby creating complex three-dimensional structures that hold promise for future applications.

The effect of shape, polydispersity, charge, and fraction of crystallite bundles on the cholesteric pitch of cellulose nanocrystal suspensions.

Using Onsager-Straley's second-virial theory, we investigate the cholesteric pitch of cellulose nanocrystal (CNC) suspensions. We model the CNCs as hard chiral bundles of microfibrils and examine the effect of the shape of these chiral bundles, characterized by aspect ratio and chirality, on the cholesteric pitch. Additionally, we explore the impact of length polydispersity and surface charge on the cholesteric phase of CNCs. Furthermore, we consider binary mixtures of twisted bundles and achiral primary crystallites to provide a more realistic representation of CNC suspensions. Our findings reveal that the degree of bundle twisting significantly affects the helical twisting of the cholesteric phase. We also observe that the average particle length and length polydispersity have substantial effects on strongly twisted bundles but minimal effects on weakly twisted ones. Finally, our study indicates that as the range of electrostatic interactions increases, the transfer of chirality from the microscopic to macroscopic length scales becomes masked, resulting in an increase in the cholesteric pitch. In the case of binary mixtures, the bundles act as chiral dopants, and an increasing fraction of bundles progressively enhances the helical twisting of the cholesteric phase.

Coarse-Grained Many-Body Potentials of Ligand-Stabilized Nanoparticles from Machine-Learned Mean Forces.

Colloidal nanoparticles self-assemble into a variety of superstructures with distinctive optical, structural, and electronic properties. These nanoparticles are usually stabilized by a capping layer of organic ligands to prevent aggregation in the solvent. When the ligands are sufficiently long compared to the dimensions of the nanocrystal cores, the effective coarse-grained forces between pairs of nanoparticles are largely affected by the presence of neighboring particles. In order to efficiently investigate the self-assembly behavior of these complex colloidal systems, we propose a machine-learning approach to construct effective coarse-grained many-body interaction potentials. The multiscale methodology presented in this work constitutes a general bottom-up coarse-graining strategy where the coarse-grained forces acting on coarse-grained sites are extracted from measuring the vectorial mean forces on these sites in reference fine-grained simulations. These effective coarse-grained forces, i.e., gradients of the potential of mean force or of the free-energy surface, are represented by a simple linear model in terms of gradients of structural descriptors, which are scalar functions that are rotationally invariant. In this way, we also directly obtain the free-energy surface of the coarse-grained model as a function of all coarse-grained coordinates. We expect that this simple yet accurate coarse-graining framework for the many-body potential of mean force will enable the characterization, understanding, and prediction of the structure and phase behavior of relevant soft-matter systems by direct simulations. The key advantage of this method is its generality, which allows it to be applicable to a broad range of systems. To demonstrate the generality of our method, we also apply it to a colloid-polymer model system, where coarse-grained many-body interactions are pronounced.

A colloidal viewpoint on the sausage catastrophe and the finite sphere packing problem.

It is commonly believed that the most efficient way to pack a finite number of equal-sized spheres is by arranging them tightly in a cluster. However, mathematicians have conjectured that a linear arrangement may actually result in the densest packing. Here, our combined experimental and simulation study provides a physical realization of the finite sphere packing problem by studying arrangements of colloids in a flaccid lipid vesicle. We map out a state diagram displaying linear, planar, and cluster conformations of spheres, as well as bistable states which alternate between cluster-plate and plate-linear conformations due to membrane fluctuations. Finally, by systematically analyzing truncated polyhedral packings, we identify clusters of 56 ≤ N ≤ 70 number of spheres, excluding N = 57 and 63, that pack more efficiently than linear arrangements.

Brute-force nucleation rates of hard spheres compared with rare-event methods and classical nucleation theory.

We determine nucleation rates of hard spheres using brute-force molecular dynamics simulations. We overcome nucleation barriers of up to 28 kBT, leading to a rigorous test of nucleation rates obtained from rare-event methods and classical nucleation theory. Our brute-force nucleation rates show excellent agreement with umbrella sampling simulations by Filion et al. [J. Chem. Phys. 133, 244115 (2010)] and seeding simulations by Espinosa et al. [J. Chem. Phys. 144, 034501 (2016)].

Real-Time In Situ Observation of CsPbBr3 Perovskite Nanoplatelets Transforming into Nanosheets.

The manipulation of nano-objects through heating is an effective strategy for inducing structural modifications and therefore changing the optoelectronic properties of semiconducting materials. Despite its potential, the underlying mechanism of the structural transformations remains elusive, largely due to the challenges associated with their in situ observations. To address these issues, we synthesize temperature-sensitive CsPbBr3 perovskite nanoplatelets and investigate their structural evolution at the nanoscale using in situ heating transmission electron microscopy. We observe the morphological changes that start from the self-assembly of the nanoplatelets into ribbons on a substrate. We identify several paths of merging nanoplates within ribbons that ultimately lead to the formation of nanosheets dispersed randomly on the substrate. These observations are supported by molecular dynamics simulations. We correlate the various paths for merging to the random orientation of the initial ribbons along with the ligand mobility (especially from the edges of the nanoplatelets). This leads to the preferential growth of individual nanosheets and the merging of neighboring ones. These processes enable the creation of structures with tunable emission, ranging from blue to green, all from a single material. Our real-time observations of the transformation of perovskite 2D nanocrystals reveal a route to achieve large-area nanosheets by controlling the initial orientation of the self-assembled objects with potential for large-scale applications.

Flexible Colloidal Molecules with Directional Bonds and Controlled Flexibility.

Colloidal molecules are ideal model systems for mimicking real molecules and can serve as versatile building blocks for the bottom-up self-assembly of flexible and smart materials. While most colloidal molecules are rigid objects, the development of colloidal joints has made it possible to endow them with conformational flexibility. However, their unrestricted range of motion does not capture the limited movement and bond directionality that is instead typical of real molecules. In this work, we create flexible colloidal molecules with an in situ controllable motion range and bond directionality by assembling spherical particles onto cubes functionalized with complementary surface-mobile DNA. By varying the sphere-to-cube size ratio, we obtain colloidal molecules with different coordination numbers and find that they feature a constrained range of motion above a critical size ratio. Using theory and simulations, we show that the particle shape together with the multivalent bonds creates an effective free-energy landscape for the motion of the sphere on the surface of the cube. We quantify the confinement of the spheres on the surface of the cube and the probability to change facet. We find that temperature can be used as an extra control parameter to switch in situ between full and constrained flexibility. These flexible colloidal molecules with a temperature switching motion range can be used to investigate the effect of directional yet flexible bonds in determining their self-assembly and phase behavior, and may be employed as constructional units in microrobotics and smart materials.

Shape-induced crystallization of binary DNA-functionalized nanocubes.

Leveraging the anisotropic shape of DNA-functionalized nanoparticles holds potential for shape-directed crystallization of a wide collection of superlattice structures. Using coarse-grained molecular dynamics simulations, we study the self-assembly of a binary mixture of cubic gold nanoparticles, which are functionalized by complementary DNA strands. We observe the spontaneous self-assembly of simple cubic (SC), plastic body-centered tetragonal (pBCT), and compositionally disordered plastic body-centered tetragonal (d-pBCT) phases due to hybridization of the DNA strands. We systematically investigate the effect of length, grafting density, as well as rigidity of the DNA strands on the self-assembly behavior of cubic nanoparticles. We measure the potential of mean force between DNA-functionalized nanocubes for varying rigidity of the DNA strands and DNA lengths. Using free-energy calculations, we find that longer and flexible DNA strands can lead to a phase transformation from SC to the pBCT phase due to a gain in entropy arising from the orientational degrees of freedom of the nanocubes in the pBCT phase. Our results may serve as a guide for self-assembly experiments on DNA-functionalized cubic nanoparticles.

Crystal Polymorph Selection Mechanism of Hard Spheres Hidden in the Fluid.

Nucleation plays a critical role in the birth of crystals and is associated with a vast array of phenomena, such as protein crystallization and ice formation in clouds. Despite numerous experimental and theoretical studies, many aspects of the nucleation process, such as the polymorph selection mechanism in the early stages, are far from being understood. Here, we show that the hitherto unexplained excess of particles in a face-centered-cubic (fcc)-like environment, as compared to those in a hexagonal-close-packed (hcp)-like environment, in a crystal nucleus of hard spheres can be explained by the higher order structure in the fluid phase. We show using both simulations and experiments that in the metastable fluid phase, pentagonal bipyramids, clusters with fivefold symmetry known to be inhibitors of crystal nucleation, transform into a different cluster, Siamese dodecahedra. These clusters are closely similar to an fcc subunit, which explains the higher propensity to grow fcc than hcp in hard spheres. We show that our crystallization and polymorph selection mechanism is generic for crystal nucleation from a dense, strongly correlated fluid phase.

Steering self-organisation through confinement.

Self-organisation is the spontaneous emergence of spatio-temporal structures and patterns from the interaction of smaller individual units. Examples are found across many scales in very different systems and scientific disciplines, from physics, materials science and robotics to biology, geophysics and astronomy. Recent research has highlighted how self-organisation can be both mediated and controlled by confinement. Confinement is an action over a system that limits its units' translational and rotational degrees of freedom, thus also influencing the system's phase space probability density; it can function as either a catalyst or inhibitor of self-organisation. Confinement can then become a means to actively steer the emergence or suppression of collective phenomena in space and time. Here, to provide a common framework and perspective for future research, we examine the role of confinement in the self-organisation of soft-matter systems and identify overarching scientific challenges that need to be addressed to harness its full scientific and technological potential in soft matter and related fields. By drawing analogies with other disciplines, this framework will accelerate a common deeper understanding of self-organisation and trigger the development of innovative strategies to steer it using confinement, with impact on, e.g., the design of smarter materials, tissue engineering for biomedicine and in guiding active matter.

Exploiting anisotropic particle shape to electrostatically assemble colloidal molecules with high yield and purity.

HYPOTHESIS: Colloidal molecules with anisotropic shapes and interactions are powerful model systems for deciphering the behavior of real molecules and building units for creating materials with designed properties. While many strategies for their assembly have been developed, they typically yield a broad distribution or are limited to a specific type. We hypothesize that the shape and relative sizes of colloidal particles can be exploited to efficiently direct their assembly into colloidal molecules of desired valence. EXPERIMENTS: We exploit electrostatic self-assembly of negatively charged spheres made from either polystyrene or silica onto positively charged hematite cubes. We thoroughly analyze the role of the shape and size ratio of particles on the cluster size and yield of colloidal molecules. FINDINGS: Using a combination of experiments and simulations, we demonstrate that cubic particle shape is crucial to generate high yields of distinct colloidal molecules over a wide variety of size ratios. We find that electrostatic repulsion between the satellite spheres is important to leverage the templating effect of the cubes, leading the spheres to preferentially assemble on the facets rather than the edges and corners of the cube. The sixfold symmetry of cubes favors the assembly of molecules with six, four, and two satellite spheres at appropriate size ratios and interaction strength. Furthermore, we reveal that our protocol is not affected by the specific choice of the material of the colloidal particles. Finally, we show that the permanent magnetic dipole moment of the hematite cubes can be utilized to separate colloidal molecules from non-assembled satellite particles. Our simple and effective strategy might be extended to other templating particle shapes, thereby greatly expanding the library of colloidal molecules that can be achieved with high yield and purity.

Ion Current Rectification and Long-Range Interference in Conical Silicon Micropores.

Fluidic devices exhibiting ion current rectification (ICR), or ionic diodes, are of broad interest for applications including desalination, energy harvesting, and sensing, among others. For such applications a large conductance is desirable, which can be achieved by simultaneously using thin membranes and wide pores. In this paper we demonstrate ICR in micrometer sized conical channels in a thin silicon membrane with pore diameters comparable to the membrane thickness but both much larger than the electrolyte screening length. We show that for these pores the entrance resistance is key not only to Ohmic conductance around 0 V but also for understanding ICR, both of which we measure experimentally and capture within a single analytic theoretical framework. The only fit parameter in this theory is the membrane surface potential, for which we find that it is voltage dependent and its value is excessively large compared to the literature. From this we infer that surface charge outside the pore strongly contributes to the observed Ohmic conductance and rectification by a different extent. We experimentally verify this hypothesis in a small array of pores and find that ICR vanishes due to pore-pore interactions mediated through the membrane surface, while Ohmic conductance around 0 V remains unaffected. We find that the pore-pore interaction for ICR is set by a long-ranged decay of the concentration which explains the surprising finding that the ICR vanishes for even a sparsely populated array with a pore-pore spacing as large as 7 µm.

Splay-bend nematic phases of bent colloidal silica rods induced by polydispersity.

Liquid crystal (LC) phases are in between solids and liquids with properties of both. Nematic LCs composed of rod-like molecules or particles exhibit long-range orientational order, yielding characteristic birefringence, but they lack positional order, allowing them to flow like a liquid. This combination of properties as well as their sensitivity to external fields make nematic LCs fundamental for optical applications e.g. liquid crystal displays (LCDs). When rod-like particles become bent, spontaneous bend deformations arise in the LC, leading to geometric frustration which can be resolved by complementary twist or splay deformations forming intriguing twist-bend (NTB) and splay-bend (NSB) nematic phases. Here, we show experimentally that the elusive NSB phases can be stabilized in systems of polydisperse micron-sized bent silica rods. Our results open avenues for the realization of NTB and NSB phases of colloidal and molecular LCs.

Machine Learning of Implicit Combinatorial Rules in Mechanical Metamaterials.

Combinatorial problems arising in puzzles, origami, and (meta)material design have rare sets of solutions, which define complex and sharply delineated boundaries in configuration space. These boundaries are difficult to capture with conventional statistical and numerical methods. Here we show that convolutional neural networks can learn to recognize these boundaries for combinatorial mechanical metamaterials, down to finest detail, despite using heavily undersampled training sets, and can successfully generalize. This suggests that the network infers the underlying combinatorial rules from the sparse training set, opening up new possibilities for complex design of (meta)materials.

Guiding the self-assembly of colloidal diamond.

The assembly of colloidal cubic diamond is a challenging process since the shape and interaction parameters and the thermodynamic conditions where this structure is stable are elusive. The simultaneous use of shape-anisotropic particles and strong directional interactions has proven to be a successful path to exclusively nucleate this structure. Here, using molecular dynamics simulations, we explore in detail the conditions where the nucleation of cubic diamond from tetrahedral building blocks is favored. In particular, we focus on the effect of depletion and DNA-mediated interactions to form and stabilize this cubic diamond crystal. We find that a particular balance between the strength and the range of the depletion interactions enhances the self-assembly of stable cubic diamond, leading to a narrow region where this structure is nucleated. Moreover, we determine that stronger short-range depletion attractions may arrest the system, leading to the formation of percolating diamond networks or fully disordered gel structures. Accordingly, the internal arrangements of these structures exhibit a distinct variation in terms of fractal dimension and the presence of six-membered rings that increasingly acquire internal strain as the arrest gets more pronounced. With these results, we provide a clear route for the self-assembly of cubic colloidal diamond, toward the realization of crystals with superior photonic properties.