From sticky-hard-sphere to Lennard-Jones-type clusters

Tuning higher order structure in colloidal fluids

Colloidal particles self assemble into a wide range of structures under external AC electric fields due to induced dipolar interactions [Yethiraj and Van Blaaderen, Nature, 2003, 421, 513]. As a result of these dipolar interactions, at low volume fraction the system is modulated between a hard-sphere like state (in the case of zero applied field) and a "string fluid" upon application of the field. Using both particle-resolved experiments and computer simulations, we investigate the emergence of the string fluid with a variety of structural measures including two-body and higher-order correlations. We probe the higher-order structure using three-body spatial correlation functions and a many-body approach based on minimum energy clusters of a dipolar-Lennard-Jones system. The latter constitutes a series of geometrically distinct minimum energy clusters upon increasing the strength of the dipolar interaction, which are echoed in the higher-order structure of the colloidal fluids we study here. We find good agreement between experiment and simulation at the two-body level. Higher-order correlations exhibit reasonable agreement between experiment and simulation, again with more discrepancy at higher field strength for three-body correlation functions. At higher field strength, the cluster population in our experiments and simulations is dominated by the minimum energy clusters for all sizes 8 ≤ m ≤ 12.

Energetics and spectroscopic studies of CNO (-) (H 2 O) n clusters and the temperature dependencies of the isomers: An approach based on a combined recipe of parallel tempering and quantum chemical methods

A system associated with several number of weak interactions supports numerous number of stable structures within a narrow range of energy. Often, a deterministic search method fails to locate the global minimum geometry as well as important local minimum isomers for such systems. Therefore, in this work, the stochastic search technique, namely parallel tempering, has been executed on the quantum chemical surface of theCNO (-) (H 2 O) n system for n = 1 -8 to generate global minimum as well as several number of local minimum isomers. IR spectrum can act as the fingerprint property for such system to be identified. Thus, IR spectroscopic features have also been included in this work. Vertical detachment energy has also been calculated to obtain clear information about number of water molecules in several spheres around the central anion. In addition, in a real experimental scenario, not only the global but also the local minimum isomers play an important role in determining the average value of a particular physically observable property. Therefore, the relative conformational populations have been determined for all the evaluated structures for the temperature range between 20K and 400K. Further to understand the phase change behavior, the configurational heat capacities have also been calculated for different sizes.

Identification and classification of clusters of dipolar colloids in an external field

Colloids can acquire a dipolar interaction in the presence of an external AC electric field. At high field strength, the particles form strings in the field direction. However, at weaker field strength, competition with isotropic interactions is expected. One means to investigate this interplay between dipolar and isotropic interactions is to consider clusters of such particles. Therefore, we have identified, using the GMIN basin-hopping tool, a rich library of lowest energy clusters of a dipolar colloidal system, where the dipole orientation is fixed to lie along the z axis and the dipole strength is varied for m-membered clusters of 7 ≤ m ≤ 13. In the regime where the isotropic and dipolar interactions are comparable, we find elongated polytetrahedral, octahedral, and spiral clusters as well as a set of non-rigid clusters, which emerge close to the transition to strings. We further implement a search algorithm that identifies these minimum energy clusters in bulk systems using the topological cluster classification [J. Chem. Phys. 139 234506 (2013)]. We demonstrate this methodology with computer simulations, which show instances of these clusters as a function of dipole strength.

Assessment of Anharmonicities in Clusters: Developing and Validating a Minimum-Information Partition Function

Precise thermodynamic calculations are essential for understanding the dynamics of cluster systems and new particle formation. However, the widely employed harmonic statistical mechanical approach often falls short in terms of accuracy. In this study, we present an improved statistical model that incorporates vibrational anharmonicity via a novel partition function that requires only one additional system-specific input parameter. In addition to considering vibrational aspects, we also account for anharmonicity related to the configurational space. The role of anharmonicities is thoroughly examined in the case of general clusters, where the complete sets of conformers, mechanically stable spatial arrangements, are known up to clusters composed of 14 monomers. By performing consistent Monte Carlo simulations on these systems, we benchmark the statistical model's efficacy in reproducing key thermodynamic properties (formation free energy and potential energy) in the classical limit. The model exhibits exceptional alignment with simulations, accurately reproducing free energies within a precision of 2kBT and reliably capturing cluster melting temperatures. Furthermore, we demonstrate the significance and applicability of the model by reproducing thermodynamic barriers in homogeneous gas-phase nucleation of larger clusters. The transferability of our developed approach extends to more complex molecular systems and bears relevance for atmospheric multicomponent clusters, in particular.

Exotic self-assembly of hard spheres in a morphometric solvent

The self-assembly of spheres into geometric structures, under various theoretical conditions, offers valuable insights into complex self-assembly processes in soft systems. Previous studies have utilized pair potentials between spheres to assemble maximum contact clusters in simulations and experiments. The morphometric approach to solvation free energy that we utilize here goes beyond pair potentials; it is a geometry-based theory that incorporates a weighted combination of geometric measures over the solvent accessible surface for solute configurations in a solvent. In this paper, we demonstrate that employing the morphometric model of solvation free energy in simulating the self-assembly of sphere clusters results, under most conditions, in the previously observed maximum contact clusters. Under other conditions, it unveils an assortment of extraordinary sphere configurations, such as double helices and rhombohedra. These exotic structures arise specifically under conditions where the interactions take multibody potentials into account. This investigation establishes a foundation for comprehending the diverse range of geometric forms in self-assembled structures, emphasizing the significance of the morphometric approach in this context.

Brownian dynamics simulations of hard rods in external fields and with contact interactions

We propose a simulation method for Brownian dynamics of hard rods in one dimension for arbitrary continuous external force fields. It is an event-driven procedure based on the fragmentation and mergers of clusters formed by particles in contact. It allows one to treat particle interactions in addition to the hard-sphere exclusion as long as the corresponding interaction forces are continuous functions of the particle coordinates. We furthermore develop a treatment of sticky hard spheres as described by Baxter's contact interaction potential.

Nanoparticle cluster formation mechanisms elucidated via Markov state modeling: Attraction range effects, aggregation pathways, and counterintuitive transition rates

Nanoparticle clusters are promising candidates for developing functional materials. However, it is still a challenging task to fabricate them in a predictable and controllable way, which requires investigation of the possible mechanisms underlying cluster formation at the nanoscale. By constructing Markov state models (MSMs) at the microstate level, we find that for highly dispersed particles to form a highly aggregated cluster, there are multiple coexisting pathways, which correspond to direct aggregation, or pathways that need to pass through partially aggregated, intermediate states. Varying the range of attraction between nanoparticles is found to significantly affect pathways. As the attraction range becomes narrower, compared to direct aggregation, some pathways that need to pass through partially aggregated intermediate states become more competitive. In addition, from MSMs constructed at the macrostate level, the aggregation rate is found to be counterintuitively lower with a lower free-energy barrier, which is also discussed.

Instability of the body-centered cubic lattice within the sticky hard sphere and Lennard-Jones model obtained from exact lattice summations

A smooth path of rearrangement from the body-centered cubic (bcc) to the face-centered cubic (fcc) lattice is obtained by introducing a single parameter to lattice vectors of a cuboidal unit cell. As a result, we obtain analytical expressions in terms of lattice sums for the cohesive energy where the interaction is described by a Lennard-Jones (LJ) interaction potential or a sticky hard-sphere (SHS) model with a r^{-n} long-range attractive term. These lattice sums are evaluated to computer precision by expansions in terms of a fast converging Bessel function series. Applying the whole range of lattice parameters for the SHS and LJ potentials we prove that the bcc phase is unstable (or, at best, metastable) toward distortion into the fcc phase in the low temperature and pressure limit. Even if more accurate potentials are used, such as the extended LJ potential for argon or chromium, the bcc phase remains unstable. This strongly indicates that the appearance of a low temperature bcc phase for several elements in the periodic table is due to higher than two-body forces in atomic interactions.

Development of a Structural Comparison Method to Promote Exploration of the Potential Energy Surface in the Global Optimization of Nanoclusters

A structural comparison method (SCM) was created to quantify the structural diversity of nanoclusters and was implemented into a global optimization algorithm to evaluate structural diversity between generated clusters on the fly and promote exploration of the potential energy surface. The SCM evaluated topological differences between clusters using the common neighbor analysis and provided a numerical measure of similarity between the two clusters. The SCM was implemented into a genetic algorithm by integrating it into a new structure + energy fitness operator such that structural diversity of clusters in the population and their energies were used to assign fitness values to clusters. The efficiency of the genetic algorithm with this new fitness operator was benchmarked against several Lennard-Jones clusters (LJ₃₈, LJ₇₅, and LJ₉₈) known to be difficult cases for global optimization algorithms. For LJ₃₈ and LJ₇₅, this new structure + energy fitness operator performed equally well or better than the energy fitness operator. However, the efficiency of locating the global minimum of LJ₉₈ decreased using this new structure + energy fitness operator. Further analysis of the genetic algorithm with this fitness operator showed that the algorithm did indeed promote exploration of the PES of LJ₉₈ as desired but hindered refinement of clusters, preventing it from locating the global minimum even if the energy funnel of the global minimum had been located.

The Lennard-Jones Potential Revisited: Analytical Expressions for Vibrational Effects in Cubic and Hexagonal Close-Packed Lattices

Analytical formulas are derived for the zero-point vibrational energy and anharmonicity corrections of the cohesive energy and the mode Grüneisen parameter within the Einstein model for the cubic lattices (sc, bcc, and fcc) and for the hexagonal close-packed structure. This extends the work done by Lennard-Jones and Ingham in 1924, Corner in 1939, and Wallace in 1965. The formulas are based on the description of two-body energy contributions by an inverse power expansion (extended Lennard-Jones potential). These make use of three-dimensional lattice sums, which can be transformed to fast converging series and accurately determined by various expansion techniques. We apply these new lattice sum expressions to the rare gas solids and discuss associated critical points. The derived formulas give qualitative but nevertheless deep insight into vibrational effects in solids from the lightest (helium) to the heaviest rare gas element (oganesson), both presenting special cases because of strong quantum effects for the former and strong relativistic effects for the latter.

Atlasing of Assembly Landscapes using Distance Geometry and Graph Rigidity

This Article describes a novel geometric methodology for analyzing free energy and kinetics of assembly driven by short-range pair-potentials in an implicit solvent and provides a proof-of-concept illustration of its unique capabilities. An atlas is a labeled partition of the assembly landscape into a roadmap of maximal, contiguous, nearly-equipotential-energy conformational regions or macrostates, together with their neighborhood relationships. The new methodology decouples the roadmap generation from sampling and produces: (1) a queryable atlas of local potential energy minima, their basin structure, energy barriers, and neighboring basins; (2) paths between a specified pair of basins, each path being a sequence of conformational regions or macrostates below a desired energy threshold; and (3) approximations of relative path lengths, basin volumes (configurational entropy), and path probabilities. Results demonstrating the core algorithm's capabilities and high computational efficiency have been generated by a resource-light, curated open source software implementation EASAL (Efficient Atlasing and Search of Assembly Landscapes, ACM Trans. Math. Softw. 2018 44, 1-48. 10.1145/3204472; see software, Efficient Atlasing and Search of Assembly Landscapes, 2016. https://bitbucket.org/geoplexity/easal; video, Video Illustrating the opensource software EASAL, 2016. https://cise.ufl.edu/~sitharam/EASALvideo.mpeg; and user guide, EASAL software user guide, 2016. https://bitbucket.org/geoplexity/easal/src/master/CompleteUserGuide.pdf). Running on a laptop with Intel(R) Core(TM) i7-7700@3.60 GHz CPU with 16GB of RAM, EASAL atlases several hundred thousand conformational regions or macrostates in minutes using a single compute core. Subsequent path and basin computations each take seconds. A parallelized EASAL version running on the same laptop with 4 cores gives a 3× speedup for atlas generation. The core algorithm's correctness, time complexity, and efficiency-accuracy trade-offs are formally guaranteed using modern distance geometry, geometric constraint systems and combinatorial rigidity. The methodology further links the shape of the input assembling units to a type of intuitive and queryable bar-code of the output atlas, which in turn determine stable assembled structures and kinetics. This succinct input-output relationship facilitates reverse analysis and control toward design. A novel feature that is crucial to both the high sampling efficiency and decoupling of roadmap generation from sampling is a recently developed theory of convex Cayley (distance-based) custom parametrizations specific to assembly, as opposed to folding. Representing microstates with macrostate-specific Cayley parameters, to generate microstate samples, avoids gradient-descent search used by all prevailing methods. Further, these parametrizations convexify conformational regions or macrostates. This ratchets up sampling efficiency, significantly reducing number of repeated and discarded samples. These features of the new stand-alone methodology can also be used to complement the strengths of prevailing methodologies including Molecular Dynamics, Monte Carlo, and Fast Fourier Transform based methods.

Structure elucidation and construction of isomerisation pathways in small to moderate-sized (6-27) MgO nanoclusters: an adaptive mutation simulated annealing based analysis with quantum chemical calculations

Determination of global minimum structures and elucidation of reaction paths or minimum energy paths between low-lying minima are of great chemical importance. To that end, we have used our own Adaptive Mutation Simulated Annealing method to determine the global minimum and the minimum energy paths for various isomerisation reactions for small to moderate-sized (MgO)n (n = 6-27) clusters, using the Born-Mayer potential with suitable parameter values. The minimum energy structures obtained by us match well with previously reported data and are used as guess structures for further optimisation at the DFT level (using the B3LYP functional and DGDZVP basis set). Our optimised structures are found to match very well with the further DFT optimised structures, where the comparison is done by determining the root mean square deviation values as well as the radial distribution function profiles. A scheme is proposed to determine the minimum energy paths for isomerisation reactions for some cluster sizes where the transition state/s obtained by us, at very low computational cost, match well with those obtained from further optimisation using DFT calculations. We have shown the efficacy of our method in determining the reaction pathways, even for cases that involve multi-step reactions.

From canyons to valleys: Numerically continuing sticky-hard-sphere clusters to the landscapes of smoother potentials

We study the energy landscapes of particles with short-range attractive interactions as the range of the interactions increases. Starting with the set of local minima for 6≤N≤12 hard spheres that are "sticky," i.e., they interact only when their surfaces are exactly in contact, we use numerical continuation to evolve the local minima (clusters) as the range of the potential increases, using both the Lennard-Jones and Morse families of interaction potentials. As the range increases, clusters merge, until at long ranges only one or two clusters are left. We compare clusters obtained by continuation with different potentials and find that for short and medium ranges, up to about 30% of particle diameter, the continued clusters are nearly identical, both within and across families of potentials. For longer ranges, the clusters vary significantly, with more variation between families of potentials than within a family. We analyze the mechanisms behind the merge events and find that most rearrangements occur when a pair of nonbonded particles comes within the range of the potential. An exception occurs for nonharmonic clusters, i.e., those that have a zero eigenvalue in their Hessian, which undergo a more global rearrangement.

Self-organized emergence of folded protein-like network structures from geometric constraints

The intricate three-dimensional geometries of protein tertiary structures underlie protein function and emerge through a folding process from one-dimensional chains of amino acids. The exact spatial sequence and configuration of amino acids, the biochemical environment and the temporal sequence of distinct interactions yield a complex folding process that cannot yet be easily tracked for all proteins. To gain qualitative insights into the fundamental mechanisms behind the folding dynamics and generic features of the folded structure, we propose a simple model of structure formation that takes into account only fundamental geometric constraints and otherwise assumes randomly paired connections. We find that despite its simplicity, the model results in a network ensemble consistent with key overall features of the ensemble of Protein Residue Networks we obtained from more than 1000 biological protein geometries as available through the Protein Data Base. Specifically, the distribution of the number of interaction neighbors a unit (amino acid) has, the scaling of the structure’s spatial extent with chain length, the eigenvalue spectrum and the scaling of the smallest relaxation time with chain length are all consistent between model and real proteins. These results indicate that geometric constraints alone may already account for a number of generic features of protein tertiary structures.

Distinguishing failure modes in oligomeric polymer nanopillars

Brittle failure is ubiquitous in amorphous materials that are sufficiently cooled below their glass transition temperature, T_g. This catastrophic failure mode is limiting for amorphous materials in many applications, and many fundamental questions surrounding it remain poorly understood. Two challenges that prevent a more fundamental understanding of the transition between a ductile response at temperatures near T_g to brittle failure at lower temperatures are (i) a lack of computationally inexpensive molecular models that capture the failure modes observed in experiments and (ii) the lack of quantitative metrics that can distinguish various failure mechanisms. In this work, we use molecular dynamics simulations to capture ductile-to-brittle transition in glass-forming oligomeric polymer systems where we systematically vary both the temperature relative to T_g and the form of the interaction potential to induce a variety of failure modes. We characterized the effects of this new potential on macroscopic mechanical properties as well as microscopic structural and dynamical response during deformation. Finally, we develop several quantitative metrics to distinguish between different failure modes, and we find that the transition between catastrophic brittle failure, necking, and homogeneous plastic flow is gradual as the temperature is increased.

Morphometric Approach to Many-Body Correlations in Hard Spheres

We model the thermodynamics of local structures within the hard sphere liquid at arbitrary volume fractions through the morphometric calculation of n-body correlations. We calculate absolute free energies of local geometric motifs in excellent quantitative agreement with molecular dynamics simulations across the liquid and supercooled liquid regimes. We find a bimodality in the density library of states where fivefold symmetric structures appear lower in free energy than fourfold symmetric structures and from a single reaction path predict a dynamical barrier which scales linearly in the compressibility factor. The method provides a new route to assess changes in the free energy landscape at volume fractions dynamically inaccessible to conventional techniques.