Revealing the Origin of Cooperative Adsorption of Chains on Nanoparticle Surfaces through Coarse-Grained Simulations.

This work aims to deepen our understanding of the molecular origin of the recently observed phenomenon of polymer cooperative adsorption onto faceted nanoparticle (NP) surfaces. By exploring a large parameter space for polymer/NP interactions through coarse-grained (CG) molecular dynamics (MD) simulations, it is found that consistent with experiments the presence or absence of cooperativity is related to solvent quality and relative interaction strengths between the polymer and the adsorbent. Specifically, positive cooperativity is associated with stronger polymer-polymer interaction than polymer-surface interactions and vice versa for negative cooperativity. This contrast in interaction energies manifests in positive cooperativity (i.e., increased affinity) and negative cooperativity (i.e., decreased affinity) as concentration increases. It is also found that increasing chain length strengthens cooperativity effects and that the nanoscale confinement of polymer chains to the adsorbing facet (due to weaker affinity to corners and edges) enhances positive cooperativity but weakens negative cooperativity. Moreover, adsorption onto a spherical NP shows stronger positive cooperativity but weaker negative cooperativity compared with adsorption onto a cubic NP of equal surface area. It was further found that as polymer bulk concentration increases, the free energy of adsorption decreases in positive cooperativity, increases in negative cooperativity, and is independent of concentration in noncooperative systems consistent with the phenomenological explanation of cooperativity. We further found that positive cooperativity is associated with growing fluctuations in the adsorption density at critical bulk polymer concentrations. This behavior can be attributed to the competition between enthalpic gains and entropic losses upon adsorption. Overall, our results shed light on the microscopic origin of cooperative adsorption and the role of solvent quality, which can be leveraged in, for example, controlling NP growth into target shapes and designing NP catalysts with improved performance.

Effect of particle anisotropy on the thermodynamics and kinetics of ordering transitions in hard faceted particles.

Monte Carlo simulations were used to study the influence of particle aspect ratio on the kinetics and phase behavior of hard gyrobifastigia (GBF). First, the formation of a highly anisotropic nucleus shape in the isotropic-to-crystal transition in regular GBF is explained by the differences in interfacial free energies of various crystal planes and the nucleus geometry predicted by the Wulff construction. GBF-related shapes with various aspect ratios were then studied, mapping their equations of state, determining phase coexistence conditions via interfacial pinning, and computing nucleation free-energy barriers via umbrella sampling using suitable order parameters. Our simulations reveal a reduction of the kinetic barrier for isotropic-crystal transition upon an increase in aspect ratio, and that for highly oblate and prolate aspect ratios, an intermediate nematic phase is stabilized. Our results and observations also support two conjectures for the formation of the crystalline state from the isotropic phase: that low phase free energies at the ordering phase transition correlate with low transition barriers and that the emergence of a mesophase provides a steppingstone that expedites crystallization.

Computing free energy barriers for the nucleation of complex network mesophases.

A previously introduced framework to identify local order parameters (OPs) distinctive of incipient complex mesophases, such as bicontinuous network phases, is used in this work to evaluate nucleation free-energy barriers. The sampling techniques considered are the mean-first-passage-time (MFPT) method and novel variants of umbrella sampling, including Hybrid Monte Carlo (HMC) and a dual-OP-method that uses a blunter global OP for the umbrella bias while keeping record of configurations for analysis with a local OP. These methods were chosen for their ability to minimize or avoid frequent calculation of the expensive local OP, which makes their continuous on-the-fly tracking computationally very inefficient. These techniques were first validated by studying phase-transition barriers of model systems, i.e., the vapor-liquid nucleation of Lennard-Jones argon and a binary nanoparticle model. The disorder-to-order free energy barrier was then traced for the double gyroid and single diamond formed by mesoscopic bead-spring macromolecular models. The dual OP method was found to be the most robust and computationally efficient, since, unlike HMC, it does not require the expensive local OP to be computed on-the-fly, and unlike the MFPT method, it can negotiate large barriers aided by the biased sampling. The dual OP method requires, however, that a cheap global OP be identified and correlated (in a post-processing step) with the local OP that describes the structure of the critical nucleus, a process that can be aided by machine learning.

Revealing the atomic ordering of binary intermetallics using in situ heating techniques at multilength scales.

Ordered intermetallic nanoparticles are promising electrocatalysts with enhanced activity and durability for the oxygen-reduction reaction (ORR) in proton-exchange membrane fuel cells (PEMFCs). The ordered phase is generally identified based on the existence of superlattice ordering peaks in powder X-ray diffraction (PXRD). However, after employing a widely used postsynthesis annealing treatment, we have found that claims of "ordered" catalysts were possibly/likely mixed phases of ordered intermetallics and disordered solid solutions. Here, we employed in situ heating, synchrotron-based, X-ray diffraction to quantitatively investigate the impact of a variety of annealing conditions on the degree of ordering of large ensembles of Pt3Co nanoparticles. Monte Carlo simulations suggest that Pt3Co nanoparticles have a lower order-disorder phase transition (ODPT) temperature relative to the bulk counterpart. Furthermore, we employed microscopic-level in situ heating electron microscopy to directly visualize the morphological changes and the formation of both fully and partially ordered nanoparticles at the atomic scale. In general, a higher degree of ordering leads to more active and durable electrocatalysts. The annealed Pt3Co/C with an optimal degree of ordering exhibited significantly enhanced durability, relative to the disordered counterpart, in practical membrane electrode assembly (MEA) measurements. The results highlight the importance of understanding the annealing process to maximize the degree of ordering in intermetallics to optimize electrocatalytic activity.

On the calculation of free energies over Hamiltonian and order parameters via perturbation and thermodynamic integration.

In this work, complementary formulas are presented to compute free-energy differences via perturbation (FEP) methods and thermodynamic integration (TI). These formulas are derived by selecting only the most statistically significant data from the information extractable from the simulated points involved. On the one hand, commonly used FEP techniques based on overlap sampling leverage the full information contained in the overlapping macrostate probability distributions. On the other hand, conventional TI methods only use information on the first moments of those distributions, as embodied by the first derivatives of the free energy. Since the accuracy of simulation data degrades considerably for high-order moments (for FEP) or free-energy derivatives (for TI), it is proposed to consider, consistently for both methods, data up to second-order moments/derivatives. This provides a compromise between the limiting strategies embodied by common FEP and TI and leads to simple, optimized expressions to evaluate free-energy differences. The proposed formulas are validated with an analytically solvable harmonic Hamiltonian (for assessing systematic errors), an atomistic system (for computing the potential of mean force with coordinate-dependent order parameters), and a binary-component coarse-grained model (for tracing a solid-liquid phase diagram in an ensemble sampled through alchemical transformations). It is shown that the proposed FEP and TI formulas are straightforward to implement, perform similarly well, and allow robust estimation of free-energy differences even when the spacing of successive points does not guarantee them to have proper overlapping in phase space.

Molecular Simulations of Laser Spike Annealing of Block Copolymer Lamellar Thin-Films.

We use molecular dynamics simulations to study the phase behavior of a coarse-grained lamella-forming A-b-B diblock copolymer under thin-film soft confinement for different heating cycle lengths, film thicknesses, and substrate-polymer affinities. This model describes the effect on thin-film morphology with a free surface (air-polymer interface) and a solid substrate. Our simulation results were first validated by showing that they capture changes for the order-disorder transition temperature with annealing conditions consistent with those found in laser spike annealing experiments, when the vertical lamella phase formed on neutral substrates. In addition, simulations with a substrate selective for a particular block revealed the formation of other phases, including a mixed vertical-horizontal lamella and a metastable island phase having horizontal but incomplete lamella layers. The nanoscale roughness features of this island phase, and hence its surface wettability, can be tuned with suitable choices of chemistry and annealing conditions.

Correlation between morphology and anisotropic transport properties of diblock copolymers melts.

Molecular simulations of coarse-grained diblock copolymers (DBP) were conducted to study the effect of segregation strength and morphology on transport properties. It was found that in the strong segregation limit (i.e., high χN, where χ is the Flory-Huggins parameter and N is the degree of polymerization), the presence of the DBP interfaces imposes topological constraints similar to those of entanglements as manifested in the rheological signature of the polymer (i.e., a plateau modulus). Furthermore, compared to the behavior of isotropic melts, the crossover from Rouse to reptation scaling of the self-diffusion coefficient (D) parallel to the DBP interface takes place at a smaller N, an effect that depends on temperature and is more pronounced in the Lamellae morphology than in the hexagonal cylinder morphology. Additionally, it is shown that for an entangled melt (i.e., N ≫ Ne where Ne is the entanglement length) block retraction is instrumental for chains to diffuse parallel to the interface of lamellar layers. Lastly, it is found that the anisotropic viscosity of different morphologies is mostly affected by the orientation of the chains relative to the shear flow direction, exhibiting reduced values when chains align in the neutral or flow directions.

Framework for Inverse Mapping Chemistry-Agnostic Coarse-Grained Simulation Models into Chemistry-Specific Models.

Coarse-grained (CG) models have allowed molecular simulations to access large enough time and length scales to elucidate relationships between macroscale properties and microscale molecular interactions. However, an unaddressed inverse-design problem concerns the identification of an optimal chemistry-specific (CS) molecule that the generic CG model represents. This has been addressed here by introducing new tools for automatically generating and refining the mapping of CS-molecule candidates to the constraints of a CG model, based on representative optimization criteria. With these tools, for each CS-molecule from a candidate group, the best mapping of that molecule onto the CG model is found and their fit is assessed by an objective function designed to emphasize matching key properties of the CG model. We employ this methodology to a range of CG models from small solvent molecules up to block copolymer systems to show its ability to find optimal candidates and to uncover the underlying length scale of some of the CG models. For instances where the identity of the CG model is known a priori, the methodology identifies the correct AA chemistry. For instances where the identity is unknown and a pool of candidates is provided, the method selects a chemistry that aligns well with physical intuition. The best candidate chemistry is also found to be sensitive to changes to the CG model.

Assembly of porous smectic structures formed from interlocking high-symmetry planar nanorings.

Materials comprising porous structures, often in the form of interconnected concave cavities, are typically assembled from convex molecular building blocks. The use of nanoparticles with a characteristic nonconvex shape provides a promising strategy to create new porous materials, an approach that has been recently used with cagelike molecules to form remarkable liquids with "scrabbled" porous cavities. Nonconvex mesogenic building blocks can be engineered to form unique self-assembled open structures with tunable porosity and long-range order that is intermediate between that of isotropic liquids and of crystalline solids. Here we propose the design of highly open liquid-crystalline structures from rigid nanorings with ellipsoidal and polygonal geometry. By exploiting the entropic ordering characteristics of athermal colloidal particles, we demonstrate that high-symmetry nonconvex rings with large internal cavities interlock within a 2D layered structure leading to the formation of distinctive liquid-crystalline smectic phases. We show that these smectic phases possess uniquely high free volumes of up to ∼95%, a value significantly larger than the 50% that is typically achievable with smectic phases formed by more conventional convex rod- or disklike mesogenic particles.

Solid-phase nucleation free-energy barriers in truncated cubes: interplay of localized orientational order and facet alignment.

The nucleation of ordered phases from the bulk isotropic phase of octahedron-like particles has been studied via Monte Carlo simulations and umbrella sampling. In particular, selected shapes that form ordered (plastic) phases with various symmetries (cubic and tetragonal) are chosen to unveil trends in the free-energy barrier heights (ΔG*'s) associated with disorder to order transitions. The shapes studied in this work have truncation parameter (s) values of 0.58, 0.75, 0.8 and 1. The case of octahedra (s = 1.0) is studied to provide a counter-example where the isotropic phase nucleates directly into a (Minkowski) crystal phase rather than a rotator phase. The simulated ΔG*'s for these systems are compared with those previously reported for hard spheres and truncated cubes with s = 0.5 (cuboctahedra, CO) and s = 2/3 (truncated octahedra, TO). The comparison shows that, for comparable degrees of supersaturation, all rotator phases nucleate with smaller ΔG*'s than that of the hard sphere crystal, whereas the octahedral crystal nucleates with a larger ΔG*. Our analysis of near-critical translationally ordered nuclei of octahedra shows a strong bias towards an orientational alignment which is incompatible with the tendency to form facet-to-facet contacts in the disordered phase, thus creating an additional entropic penalty for crystallization. For rotator phases of octahedra-like particles, we observe that the strength of the localized orientational order correlates inversely with ΔG*. We also observe that for s > 0.66 shapes and similar to octahedra, configurations with high facet alignment do not favor high orientational order, and thus ΔG*'s increase with truncation.

Nucleus-size pinning for determination of nucleation free-energy barriers and nucleus geometry.

Classical Nucleation Theory (CNT) has recently been used in conjunction with a seeding approach to simulate nucleation phenomena at small-to-moderate supersaturation conditions when large free-energy barriers ensue. In this study, the conventional seeding approach [J. R. Espinosa et al., J. Chem. Phys. 144, 034501 (2016)] is improved by a novel, more robust method to estimate nucleation barriers. Inspired by the interfacial pinning approach [U. R. Pedersen, J. Chem. Phys. 139, 104102 (2013)] used before to determine conditions where two phases coexist, the seed of the incipient phase is pinned to a preselected size to iteratively drive the system toward the conditions where the seed becomes a critical nucleus. The proposed technique is first validated by estimating the critical nucleation conditions for the disorder-to-order transition in hard spheres and then applied to simulate and characterize the highly non-trivial (prolate) morphology of the critical crystal nucleus in hard gyrobifastigia. A generalization of CNT is used to account for nucleus asphericity and predict nucleation free-energy barriers for gyrobifastigia. These predictions of nuclei shape and barriers are validated by independent umbrella sampling calculations.

Molecular dynamics simulation of thermotropic bolaamphiphiles with a swallow-tail lateral chain: formation of cubic network phases.

T-shaped bolaamphiphiles (TBA) with a swallow-tail lateral chain have been found to provide a fertile platform to produce complex liquid crystalline phases that are accessible through changes of temperature and lateral chain length and design. In this work, we use molecular simulations of a simple coarse-grained model to map out the phase behavior of this type of molecules. This model is based on the premise that the crucial details of the fluid structure stem from close range repulsions and the strong directional forces typical of hydrogen bonds. Our simulations confirm that TBAs exhibit a rich phase behavior upon increasing the length of their lateral chain. The simulations detect a double gyroid phase and an axial-bundle columnar phase which bear some structural resemblance to those found in the experiment. In addition, simulations predict two cocontinuous phases with 3D-periodicity: the "single" diamond and the "single" plumber's nightmare phase. Our analysis of energetic and entropic contributions to the free energy of phases formed by TBA with either swallow-tail or linear side-chains suggest that the 3D-periodic network phases formed by the former are stabilized by the large conformation entropy of the side-chains.

Optimizing the formation of colloidal compounds with components of different shapes.

By introducing favorable inter-species interactions, stoichiometric compound phases (C*), akin to intermetallic alloys, can be formed by binary mixtures of nanoparticle components of different shapes. The stability of such C* phases is expected to be affected by asymmetries in both the energetics of like vs. unlike species contacts, and the packing entropy of components, as captured by their shapes and relative sizes. Using Monte Carlo simulations, we explore the effect of changes in size ratio (for fixed contact energy) and in binding energy (for fixed size ratio) in the stability of the CsCl compound phase for equimolar mixtures of octahedra and spheres and of the NaCl compound for equimolar mixtures of cubes and spheres. As a general design rule, it is proposed that enhanced compound stability is associated with inter-species interactions that minimize the free-energy of the C* phase at coexistence with the (disordered) phase that is stable at lower concentrations. For the systems studied, this rule identifies optimal relative particle sizes and inter-species binding energies that are consistent with physically grounded expectations.

Optimizing the formation of solid solutions with components of different shapes.

A key challenge to engineer ordered solids from the co-assembly of two differently shaped building blocks is to predict the key particle characteristics that lead to maximal mutual ordered-phase compatibility (MaxOC). While both entropy disparity, as captured by the relative size of the components, and energetic inter-species selectivity affect MaxOC, it is the former whose effect is less intuitive and the main focus of this work. Such MaxOC predictive rules are formulated and validated by using Monte Carlo simulation results for hard-core mixtures of octahedra and spheres and of other previously studied mixtures. Specifically, it is proposed that component size ratios should maximize their "substitutional symmetry" and hence minimize the combined free-energy cost associated with mutating a host-particle into a guest-particle in each of the solid phases. For the hard-core mixtures examined, packing entropy stabilizes substitutionally disordered solid solutions but not stoichiometric compounds. Additional molecular simulations were hence used to demonstrate, consistent with recent experimental findings, that such compounds can be formed by strengthening the inter-species compatibility via orientation-dependent attractions.

Phase behavior of polyhedral nanoparticles in parallel plate confinement.

Monte Carlo simulations are used to investigate the phase behavior of hard cubes, truncated cubes, cuboctahedra and truncated octahedra when confined between two parallel hard walls. The walls are separated by a distance H* which is varied to accommodate a different number of layers, from a monolayer up to approximately 5 layers, hence allowing us to probe the transitional phase behavior as the system goes from a quasi-2D geometry to a quasi-3D bulk behavior. While our results do reveal some phases whose structures resemble those that have been observed before for such systems in 2D and 3D spaces, other phases are also detected, including buckled phases, rotator plastic phases, and solids with significant translational disorder. Ordered phases formed for H* values that are a little too narrow to accommodate an additional particle layer are particularly interesting as they tend to have complex structures. The maximum density for such frustrated phases is low compared to that of non-frustrated ones for the same system at different H*. As the asphericity in the shapes is reduced, the simulated phases show structural features that approach those of the phases that have been reported for hard spheres under similar confinement.

Effect of inter-species selective interactions on the thermodynamics and nucleation free-energy barriers of a tessellating polyhedral compound.

The phase behavior and the homogeneous nucleation of an equimolar mixture of octahedra and cuboctahedra are studied using thermodynamic integration, Gibbs-Duhem integration, and umbrella sampling simulations. The components of this mixture are modeled as polybead objects of equal edge lengths so that they can assemble into a space-filling compound with the CsCl crystal structure. Taking as reference the hard-core system where the compound crystal does not spontaneously nucleate, we quantified the effect of inter-species selective interactions on facilitating the disorder-to-order transition. Facet selective and facet non-selective inter-species attractions were considered, and while the former was expectedly more favorable toward the target tessellating structure, the latter was found to be similarly effective in nucleating the crystal compound. Ranges for the strength of attractions and degree of supersaturation were identified where the nucleation free-energy barrier was small enough to foretell a fast process but large enough to prevent spinodal fluctuations that can trap the system in dense metastable states lacking long-range order. At those favorable conditions, the tendency toward the local orientational order favored by packing entropy is amplified and found to play a key role seeding nuclei with the CsCl structure.

Simultaneous estimation of free energies and rates using forward flux sampling and mean first passage times.

In this work, a method is proposed to simultaneously compute the transition rate constant and the free energy profile of a rare event along an order parameter connecting two well-defined regions of phase space. The method employs a forward flux sampling technique in combination with a mean first passage time approach to estimate the steady state probability and mean first passage times. These quantities are fitted to a Markovian model that allows the estimation of the free energy along the chosen order parameter. The proposed technique is first validated with two test systems (an Ising model and a model potential energy surface) and then used to study the solid-phase homogeneous nucleation of selected polyhedral particles.

Heuristic rule for binary superlattice coassembly: mixed plastic mesophases of hard polyhedral nanoparticles.

Sought-after ordered structures of mixtures of hard anisotropic nanoparticles can often be thermodynamically unfavorable due to the components' geometric incompatibility to densely pack into regular lattices. A simple compatibilization rule is identified wherein the particle sizes are chosen such that the order-disorder transition pressures of the pure components match (and the entropies of the ordered phases are similar). Using this rule with representative polyhedra from the truncated-cube family that form pure-component plastic crystals, Monte Carlo simulations show the formation of plastic-solid solutions for all compositions and for a wide range of volume fractions.

Localized orientational order chaperones the nucleation of rotator phases in hard polyhedral particles.

The nucleation kinetics of the rotator phase in hard cuboctahedra, truncated octahedra, and rhombic dodecahedra is simulated via a combination of forward flux sampling and umbrella sampling. For comparable degrees of supersaturation, the polyhedra are found to have significantly lower free-energy barriers and faster nucleation rates than hard spheres. This difference primarily stems from localized orientational ordering, which steers polyhedral particles to pack more efficiently. Orientational order hence fosters here the growth of orientationally disordered nuclei.

Engineering entropy in soft matter: the bad, the ugly and the good.

The role of entropic interactions, often subtle and sometimes crucial, on the structure and properties of soft matter has a well-recognized place in the classic and modern scientific literature. However, the lessons learned from many of those studies do not always form part of the standard arsenal of strategies that are taught or used for de novo studies relevant to the engineering of new materials. Fortunately, a growing number of examples exist where entropic effects have been designed a priori to achieve a desired or new outcome. This tutorial review describes some recent such examples, selected to illustrate the potential benefits of a more pro-active approach to harnessing the often overlooked power of entropy.