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.

Mesoscale simulations of diffusion and sedimentation in shape-anisotropic nanoparticle suspensions.

We determine the long-time self-diffusion coefficient and sedimentation coefficient for suspensions of nanoparticles with anisotropic shapes (octahedra, cubes, tetrahedra, and spherocylinders) as a function of nanoparticle concentration using mesoscale simulations. We use a discrete particle model for the nanoparticles, and we account for solvent-mediated hydrodynamic interactions between nanoparticles using the multiparticle collision dynamics method. Our simulations are compared to theoretical predictions and experimental data from existing literature, demonstrating good agreement in the majority of cases. Further, we find that the self-diffusion coefficient of the regular polyhedral shapes can be estimated from that of a sphere whose diameter is the average of their inscribed and circumscribed sphere diameters.

Sequence-dependent material properties of biomolecular condensates and their relation to dilute phase conformations.

Material properties of phase-separated biomolecular condensates, enriched with disordered proteins, dictate many cellular functions. Contrary to the progress made in understanding the sequence-dependent phase separation of proteins, little is known about the sequence determinants of condensate material properties. Using the hydropathy scale and Martini models, we computationally decipher these relationships for charge-rich disordered protein condensates. Our computations yield dynamical, rheological, and interfacial properties of condensates that are quantitatively comparable with experimentally characterized condensates. Interestingly, we find that the material properties of model and natural proteins respond similarly to charge segregation, despite different sequence compositions. Molecular interactions within the condensates closely resemble those within the single-chain ensembles. Consequently, the material properties strongly correlate with molecular contact dynamics and single-chain structural properties. We demonstrate the potential to harness the sequence characteristics of disordered proteins for predicting and engineering the material properties of functional condensates, with insights from the dilute phase properties.

Sequence-Dependent Conformational Transitions of Disordered Proteins During Condensation.

Intrinsically disordered proteins (IDPs) can form biomolecular condensates through phase separation. It is recognized that the conformation of IDPs in the dense and dilute phases as well as at the interfaces of condensates can critically impact the resulting properties associated with their functionality. However, a comprehensive understanding of the conformational transitions of IDPs during condensation remains elusive. In this study, we employ a coarse-grained polyampholyte model, comprising an equal number of oppositely charged residues-glutamic acid and lysine-whereby conformations and phase behavior can be readily tuned by altering the protein sequence. By manipulating the sequence patterns from perfectly alternating to block-like, we obtain chains with ideal-like conformations to semi-compact structures in the dilute phase, while in the dense phase, the chain conformation is approximately that of an ideal chain, irrespective of the protein sequence. By performing simulations at different concentrations, we find that the chains assemble from the dilute phase through small oligomeric clusters to the dense phase, accompanied by a gradual swelling of the individual chains. We further demonstrate that these findings are applicable to several naturally occurring proteins involved in the formation of biological condensates. Concurrently, we delve deeper into the chain conformations within the condensate, revealing that chains at the interface show a strong sequence dependence, but remain more collapsed than those in the bulk-like dense phase. This study addresses critical gaps in our knowledge of IDP conformations within condensates as a function of protein sequence.

Sequence-Dependent Material Properties of Biomolecular Codensates and their Relation to Dilute Phase Conformations.

Material properties of phase-separated biomolecular assemblies, enriched with disordered proteins, dictate their ability to participate in many cellular functions. Despite the significant effort dedicated to understanding how the sequence of the disordered protein drives its phase separation to form condensates, little is known about the sequence determinants of condensate material properties. Here, we computationally decipher these relationships for charged disordered proteins using model sequences comprised of glutamic acid and lysine residues as well as naturally occurring sequences of LAF1's RGG domain and DDX4's N-terminal domain. We do so by delineating how the arrangement of oppositely charged residues within these sequences influences the dynamical, rheological, and interfacial properties of the condensed phase through equilibrium and non-equilibrium molecular simulations using the hydropathy scale and Martini models. Our computations yield material properties that are quantitatively comparable with experimentally characterized condensate systems. Interestingly, we find that the material properties of both the model and natural proteins respond similarly to the segregation of charges, despite their very different sequence compositions. Condensates of the highly charge-segregated sequences exhibit slower dynamics than the uniformly charge-patterned sequences, because of their comparatively long-lived molecular contacts between oppositely charged residues. Surprisingly, the molecular interactions within the condensate are highly similar to those within a single-chain for all sequences. Consequently, the condensate material properties of charged disordered proteins are strongly correlated with their dense phase contact dynamics and their single-chain structural properties. Our findings demonstrate the potential to harness the sequence characteristics of disordered proteins for predicting and engineering the material properties of functional condensates, with insights from the dilute phase properties.

Structure Formation in Supraparticles Composed of Spherical and Elongated Particles.

We studied the evaporation-induced formation of supraparticles from dispersions of elongated colloidal particles using experiments and computer simulations. Aqueous droplets containing a dispersion of ellipsoidal and spherical polystyrene particles were dried on superamphiphobic surfaces at different humidity values that led to varying evaporation rates. Supraparticles made from only ellipsoidal particles showed short-range lateral ordering at the supraparticle surface and random orientations in the interior regardless of the evaporation rate. Particle-based simulations corroborated the experimental observations in the evaporation-limited regime and showed an increase in the local nematic ordering as the diffusion-limited regime was reached. A thin shell of ellipsoids was observed at the surface when supraparticles were made from binary mixtures of ellipsoids and spheres. Image analysis revealed that the supraparticle porosity increased with an increasing aspect ratio of the ellipsoids.

Cell-Sized Confinements Alter Molecular Diffusion in Concentrated Polymer Solutions Due to Length-Dependent Wetting of Polymers.

Living cells are characterized by the micrometric confinement of various macromolecules at high concentrations. Using droplets containing binary polymer blends as artificial cells, we previously showed that cell-sized confinement causes phase separation of the binary polymer solutions because of the length-dependent wetting of the polymers. Here, we demonstrate that the confinement-induced heterogeneity of polymers also emerges in single-component polymer solutions. The resulting structural heterogeneity also leads to a slower transport of small molecules at the center of cell-sized droplets than that in bulk solutions. Coarse-grained molecular simulations support this confinement-induced heterogeneous distribution by polymer length and demonstrate that the effective wetting of the shorter chains at the droplet surface originates from the length-dependent conformational entropy. Our results suggest that cell-sized confinement functions as a structural regulator for polydisperse polymer solutions that specifically manipulates the diffusion of molecules, particularly those with sizes close to the correlation length of the polymer chains.

Conformational Properties of Polymers at Droplet Interfaces as Model Systems for Disordered Proteins.

Polymer models serve as useful tools for studying the formation and physical properties of biomolecular condensates. In recent years, the interface dividing the dense and dilute phases of condensates has been discovered to be closely related to their functionality, but the conformational preferences of the constituent proteins remain unclear. To elucidate this, we perform molecular simulations of a droplet formed by phase separation of homopolymers as a surrogate model for the prion-like low-complexity domains. By systematically analyzing the polymer conformations at different locations in the droplet, we find that the chains become compact at the droplet interface compared with the droplet interior. Further, segmental analysis revealed that the end sections of the chains are enriched at the interface to maximize conformational entropy and are more expanded than the middle sections of the chains. We find that the majority of chain segments lie tangential to the droplet surface, and only the chain ends tend to align perpendicular to the interface. These trends also hold for the natural proteins FUS LC and LAF-1 RGG, which exhibit more compact chain conformations at the interface compared to the droplet interior. Our findings provide important insights into the interfacial properties of biomolecular condensates and highlight the value of using simple polymer physics models to understand the underlying mechanisms.

Confined Dynamics in Spherical Polymer Brushes.

We investigate the dynamics of polymers grafted to spherical nanoparticles in solution using hybrid molecular dynamics simulations with a coarse-grained solvent modeled via the multiparticle collision dynamics algorithm. The mean-square displacements of monomers near the surface of the nanoparticle exhibit a plateau on intermediate time scales, indicating confined dynamics reminiscent of those reported in neutron spin-echo experiments. The confined dynamics vanish beyond a specific radial distance from the nanoparticle surface that depends on the polymer grafting density. We show that this dynamical confinement transition follows theoretical predictions for the critical distance associated with the structural transition from confined to semidilute brush regimes. These findings suggest the existence of a hitherto unreported dynamic length scale connected with theoretically predicted static fluctuations in spherical polymer brushes and provide new insights into recent experimental observations.

Aggregation of amphiphilic nanocubes in equilibrium and under shear.

We investigate the self-assembly of amphiphilic nanocubes into finite-sized aggregates in equilibrium and under shear, using molecular dynamics (MD) simulations and kinetic Monte Carlo (KMC) calculations. These patchy nanoparticles combine both interaction and shape anisotropy, making them valuable models for studying folded proteins and DNA-functionalized nanoparticles. The nanocubes can self-assemble into various finite-sized aggregates ranging from rods to self-avoiding random walks, depending on the number and placement of the hydrophobic faces. Our study focuses on suspensions containing multi- and one-patch cubes, with their ratio systematically varied. When the binding energy is comparable to the thermal energy, the aggregates consist of only few cubes that spontaneously associate/dissociate. However, highly stable aggregates emerge when the binding energy exceeds the thermal energy. Generally, the mean aggregation number of the self-assembled clusters increases with the number of hydrophobic faces and decreases with increasing fraction of one-patch cubes. In sheared suspensions, the more frequent collisions between nanocube clusters lead to faster aggregation dynamics but also to smaller terminal steady-state mean cluster sizes. The results from the MD and KMC simulations are in excellent agreement for all investigated two-patch cases, whereas the three-patch cubes form systematically smaller clusters in the MD simulations compared to the KMC calculations due to finite-size effects and slow aggregation kinetics. By analyzing the rate kernels, we are able to identify the primary mechanisms responsible for (shear-induced) cluster growth and breakup. This understanding allows us to tune nanoparticle and process parameters to achieve desired cluster sizes and shapes.

Conformational Properties of Polymers at Droplet Interfaces as Model Systems for Disordered Proteins.

Polymer models serve as useful tools for studying the formation and physical properties of biomolecular condensates. In recent years, the interface dividing the dense and dilute phases of condensates has been discovered to be closely related to their functionality, but the conformational preferences of the constituent proteins remain unclear. To elucidate this, we perform molecular simulations of a droplet formed by liquid‒liquid phase separation of homopolymers, as a surrogate model for the prion-like low-complexity domains. By systematically analyzing the polymer conformations at different locations in the droplet, we find that the chains become compact at the droplet interface compared to the droplet interior. Further, segmental analysis revealed that the end sections of the chains are enriched at the interface to maximize conformational entropy, and are more expanded than the middle sections of the chains. We find that the majority of chain segments lie tangential to the droplet surface and only the chain ends tend to align perpendicular to the interface. These trends also hold for the natural proteins FUC LC and LAF-1 RGG, which exhibit more compact chain conformations at the interface compared with the droplet interior. Our findings provide important insights into the interfacial properties of biomolecular condensates and highlight the value of using simple polymer physics models to understand the underlying mechanisms.

Effect of Surfactant Selectivity on Shape and Inner Morphology of Triblock Terpolymer Microparticles.

The confined assembly of block copolymers (BCPs) has become a useful tool to prepare microparticles with controlled anisotropy and inner structure. While a solid understanding of the behavior of AB diblock copolymers exists, knowledge on the parameters affecting ABC triblock terpolymer assembly is much more limited. In this work, the effect of block-selective surfactants, sodium-4-vinylbenzenesulfonate (VBS) and sodium dodecylsulfate (SDS), is analyzed in the evaporation-induced confined assembly (EICA) of a polystyrene-block-polybutadiene-block-poly(methyl methacrylate) triblock terpolymer (SBM). Despite using the same terpolymer and emulsification process, SDS results in ellipsoidal microparticles with axially stacked lamellae, while VBS results in spherical microparticles with concentric lamellae or 3D spiral morphology. This change in morphology upon switching the surfactant is further substantiated by molecular simulations and enhances the understanding of terpolymer microphase separation in confinement.

Role of Strong Localized vs Weak Distributed Interactions in Disordered Protein Phase Separation.

Interaction strength and localization are critical parameters controlling the single-chain and condensed-state properties of intrinsically disordered proteins (IDPs). Here, we decipher these relationships using coarse-grained heteropolymers comprised of hydrophobic (H) and polar (P) monomers as model IDPs. We systematically vary the fraction of P monomers XP and employ two distinct particle-based models that include either strong localized attractions between only H-H pairs (HP model) or weak distributed attractions between both H-H and H-P pairs (HP+ model). To compare different sequences and models, we first carefully tune the attraction strength for all sequences to match the single-chain radius of gyration. Interestingly, we find that this procedure produces similar conformational ensembles, nonbonded potential energies, and chain-level dynamics for single chains of almost all sequences in both models, with some deviations for the HP model at large XP. However, we observe a surprisingly rich phase behavior for the sequences in both models that deviates from the expectation that similarity at the single-chain level will translate to a similar phase-separation propensity. Coexistence between dilute and dense phases is only observed up to a model-dependent XP, despite the presence of favorable interchain interactions, which we quantify using the second virial coefficient. Instead, the limited number of attractive sites (H monomers) leads to the self-assembly of finite-sized clusters of different sizes depending on XP. Our findings strongly suggest that models with distributed interactions favor the formation of liquid-like condensates over a much larger range of sequence compositions compared to models with localized interactions.

Confinement-Induced Fractionation and Liquid-Liquid Phase Separation of Polymer Mixtures.

The formation of (bio)molecular condensates via liquid-liquid phase separation in cells has received increasing attention, as these aggregates play important functional and regulatory roles within biological systems. However, the majority of studies focused on the behavior of pure systems in bulk solutions, thus neglecting confinement effects and the interplay between the numerous molecules present in cells. To better understand the physical mechanisms driving condensation in cellular environments, we perform molecular simulations of binary polymer mixtures in spherical droplets, considering both monodisperse and polydisperse molecular weight distributions for the longer polymer species. We find that confinement induces a spatial separation of the polymers by length, with the longer ones moving to the droplet center. This partitioning causes a distinct increase in the local polymer concentration near the droplet center, which is more pronounced in polydisperse systems. Consequently, the confined systems exhibit liquid-liquid phase separation at average polymer concentrations where bulk systems are still in the one-phase regime.

Rapid Precipitation of Ionomers for Stabilization of Polymeric Colloids.

Polymeric colloids have shown potential as "building blocks" in applications ranging from formulations of Pickering emulsions and drug delivery systems to advanced materials, including colloidal crystals and composites. However, for applications requiring tunable properties of charged colloids, obstacles in fabrication can arise through limitations in process scalability and chemical versatility. In this work, the capabilities of flash nanoprecipitation (FNP), a scalable nanoparticle (NP) fabrication technology, are expanded to produce charged polystyrene colloids using sulfonated polystyrene ionomers as a new class of NP stabilizers. Through experimental exploration of formulation parameters, increases in the ionomer content are shown to reduce the particle size, mitigating a significant trade-off between the final particle size and inlet concentration; thus, expanding the processable material throughput of FNP. Further, the degree of sulfonation is found to impact stabilization with optimal performance achieved by selecting ionomers with intermediate (2.45-5.2 mol %) sulfonation. Simulations of single ionomer chains and their arrangement in multicomponent NPs provide molecular insights into the assembly and structure of NPs wherein the partitioning of ionomers to the particle surface depends on the polymer molecular weight and degree of sulfonation. By combining the insights from simulations with diffusion-limited growth kinetics and parametric fits to experimental data, a simple design formulation relation is proposed and validated. This work highlights the potential of ionomer-based stabilizers for controllably producing charged NP dispersions in a scalable manner.

Single-chain and condensed-state behavior of hnRNPA1 from molecular simulations.

Intrinsically disordered proteins (IDPs) are essential components for the formation of membraneless organelles, which play key functional and regulatory roles within biological systems. These complex assemblies form and dissolve spontaneously over time via liquid-liquid phase separation of IDPs. Mutations in their amino acid sequence can alter their phase behavior, which has been linked to the emergence of severe diseases. We study the conformation and phase behavior of a low-complexity domain of heterogeneous nuclear ribonucleoprotein A1 (hnRNPA1) using coarse-grained implicit solvent molecular dynamics simulations. We systematically analyze how these properties are affected by the number of aromatic residues within the examined sequences. We find a significant compaction of the chains and an increase in the critical temperature with an increasing number of aromatic residues. The local persistence length is determined in single-chain simulations, revealing strong sequence-specific variations along the chain contour. Comparing single-chain and condensed-state simulations, we find many more collapsed polymer conformations in the dilute systems, even at temperatures near the estimated θ-temperature of the solution. These observations strongly support the hypothesis that aromatic residues play a dominant role in condensation, which is further corroborated by a detailed analysis of the intermolecular contacts, and conversely that important properties of condensates are captured in coarse-grained simulations. Interestingly, we observe density inhomogeneities within the condensates near criticality, which are driven by electrostatic interactions. Finally, we find that the relatively small fraction of hydrophobic residues in the IDPs results in interfacial tensions, which are significantly lower compared to typical combinations of immiscible simple liquids.

Crystal Prediction via Genetic Algorithms in a Model Chiral System.

Chiral crystals and their constituent molecules play a prominent role in theories about the origin of biological homochirality and in drug discovery, design, and stability. Although the prediction and identification of stable chiral crystal structures is crucial for numerous technologies, including separation processes and polymorph selection and control, predictive ability is often complicated by a combination of many-body interactions and molecular complexity and handedness. In this work, we address these challenges by applying genetic algorithms to predict the ground-state crystal lattices formed by a chiral tetramer molecular model, which we have previously shown to exhibit complex fluid-phase behavior. Using this approach, we explore the relative stability and structures of the model's conglomerate and racemic crystals, and present a structural phase diagram for the stable Bravais crystal types in the zero-temperature limit.

Understanding Gas Transport in Polymer-Grafted Nanoparticle Assemblies.

We rationalize the unusual gas transport behavior of polymer-grafted nanoparticle (GNP) membranes. While gas permeabilities depend specifically on the chemistry of the polymers considered, we focus here on permeabilities relative to the corresponding pure polymer which show interesting, "universal" behavior. For a given NP radius, Rc, and for large enough areal grafting densities, σ, to be in the dense brush regime we find that gas permeability enhancements display a maximum as a function of the graft chain molecular weight, Mn. Based on a recently proposed theory for the structure of a spherical brush in a melt of GNPs, we conjecture that this peak permeability occurs when the densely grafted polymer brush has the highest, packing-induced extension free energy per chain. The corresponding brush thickness is predicted to be h max = 3 R c , independent of chain chemistry and σ, i.e., at an apparently universal value of the NP volume fraction (or loading), ϕNP, ϕNP,max = [Rc/(Rc + hmax)]3 ≈ 0.049. Motivated by this conclusion, we measured CO-2 and CH4 permeability enhancements across a variety of Rc, Mn and σ, and find that they behave in a similar manner when considered as a function of ϕNP, with a peak in the near vicinity of the predicted ϕNP,max. Thus, the chain length dependent extension free energy appears to be the critical variable in determining the gas permeability for these hybrid materials. The emerging picture is that these curved polymer brushes, at high enough σ behave akin to a two-layer transport medium - the region in the near vicinity of the NP surface is comprised of extended polymer chains which speed-up gas transport relative to the unperturbed melt. The chain extension free energy increases with increasing chain length, up to a maximum, and apparently leads to an increasing gas permeability. For long enough grafts, there is an outer region of chain segments that is akin to an unperturbed melt with slow gas transport. The permeability maximum and decreasing permeability with increasing chain length then follow naturally.

Self-Assembly of Amphiphilic Cubes in Suspension.

We study the self-assembly of amphiphilic cubic colloids using molecular dynamics as well as rejection-free kinetic Monte Carlo simulations. We vary both the number and location of the solvophobic faces (patches) on the cubes at several colloid volume fractions and determine the resulting size and shape distributions of the self-assembled aggregates. When the binding energy is comparable to the thermal energy of the system, aggregates typically consist of only few spontaneously associating/dissociating colloids. Increasing the binding energy (or lowering the temperature) leads to the emergence of highly stable aggregates, e.g., small dimers in pure suspensions of one-patch cubes or large (system-spanning) aggregates in suspensions of multipatch colloids. In mixtures of one- and multipatch cubes, the average aggregation number increases with increasing number of solvophobic faces on the multipatch cubes as well with increasing fraction of multipatch cubes. The resulting aggregate shapes range from elongated rods over fractal objects to compact spheres, depending on the number and arrangement of solvophobic patches on the cubic colloids. Our findings establish the complex self-assembly pathways for a class of building blocks that combine both interaction and shape anisotropy, with the potential of forming hierarchically ordered superstructures.

Employing Artificial Neural Networks to Identify Reaction Coordinates and Pathways for Self-Assembly.

Capturing the autonomous self-assembly of molecular building blocks in computer simulations is a persistent challenge, requiring to model complex interactions and to access long time scales. Advanced sampling methods allow to bridge these time scales but typically need accurate low-dimensional representations of the transition pathways. In this work, we demonstrate for the self-assembly of two single-stranded DNA fragments into a ring-like structure how autoencoder neural networks can be employed to reliably provide a suitable low-dimensional representation and to expose transition pathways: The assembly proceeds through a two-step process with two distinct half-bound states, which are correctly identified by the neural net. We exploit this latent space representation to construct a Markov state model for predicting the four molecular conformations and their transition rates. We present a detailed comparison with two other low-dimensional representations based on empirically determined order parameters and a time-lagged independent component analysis (TICA). Our work opens up new avenues for the computational modeling of multistep and hierarchical self-assembly, which has proven challenging so far.