Programming Interfacial Porosity and Symmetry with Escherized Colloids.

We simultaneously designed the porosity and plane symmetry of self-assembling colloidal films by using isohedral tiles to determine the location and shape of enthalpically interacting surface patches on motifs being functionalized. The symmetries of both the tile and motif determine the plane symmetry group of the final assembly. Previous work has either ignored symmetry considerations altogether or accounted for only the tile's properties, applicable only when the motif is asymmetric; this approach provides a complete account and enables the design of symmetric colloids using this tile-based approach, which are often more practical to manufacture. We present the methodology, based on the type of the tile, and provide computational tools that enable the automatic classification of all tiles for a given motif and the optimization of the tile to fit the motif, sometimes referred to as "Escherization".

Building Interpretable Machine Learning Models to Identify Chemometric Trends in Seabirds of the North Pacific Ocean.

Marine environmental monitoring efforts often rely on the bioaccumulation of persistent anthropogenic contaminants in organisms to create a spatiotemporal record of the ecosystem. Intercorrelation results from the origin, uptake, and transport of these contaminants throughout the ecosystem and may be affected by organism-specific processes such as biotransformation. Here, we explore trends that machine learning tools reveal about a large, recently released environmental chemistry data set of common anthropogenic pollutants measured in the eggs of five seabird species from the North Pacific Ocean. We modeled these data with a variety of machine learning approaches and found models that could accurately determine a range of taxonomic and spatiotemporal trends. We illustrate a general workflow and set of analysis tools that can be used to identify interpretable models which perform nearly as well as state-of-the-art "black boxes." For example, we found shallow decision trees that could resolve genus with greater than 96% accuracy using as few as two analytes and a k-nearest neighbor classifier that could resolve species differences with more than 94% accuracy using only five analytes. The benefits of interpretability outweighed the marginally improved accuracy of more complex models. This demonstrates how machine learning may be used to discover rational, quantitative trends in these systems.

Derivable genetic programming for two-dimensional colloidal materials.

We describe a method for deriving surface functionalization patterns for colloidal systems that can induce self-assembly into any chosen periodic symmetry at a planar interface. The result is a sequence of letters, s ∈ {A,T,C,G}, or a gene, that describes the perimeter of the colloidal object and programs its self-assembly. This represents a genome that is finite and can be exhaustively enumerated. These genes derive from symmetry, which may be topologically represented by two-dimensional parabolic orbifolds; since these orbifolds are surfaces that may be derived from first principles, this represents an ab initio route to colloid functionality. The genes are human readable and can be employed to easily design colloidal units. We employ a biological (genetic) analogy to demonstrate this and illustrate their connection to the designs of Maurits Cornelis (M. C.) Escher.

Symmetry-derived structure directing agents for two-dimensional crystals of arbitrary colloids.

We derive properties of self-assembling rings which can template the organization of an arbitrary colloid into any periodic symmetry in two Euclidean dimensions. By viewing this as a tiling problem, we illustrate how the shape and chemical patterning of these rings are derivable, and are explicitly reflected by the symmetry group's orbifold symbol. We performed molecular dynamics simulations to observe their self-assembly and found 5 different characteristics which could be easily rationalized on the basis of this symbol. These include systems which undergo chiral phase separation, are addressably complex, exhibit self-limiting growth into clusters, form ordered "rods" in only one-dimension akin to a smectic phase, and those from symmetry groups which are pluripotent and allow one to select rings which exhibit different behaviors. We discuss how the curvature of the ring's edges plays an integral role in achieving correct self-assembly, and illustrate how to obtain these shapes. This provides a method for patterning colloidal systems at interfaces without explicitly programming this information onto the colloid itself.

Extrapolation and interpolation strategies for efficiently estimating structural observables as a function of temperature and density.

Thermodynamic extrapolation has previously been used to predict arbitrary structural observables in molecular simulations at temperatures (or relative chemical potentials in open-system mixtures) different from those at which the simulation was performed. This greatly reduces the computational cost in mapping out phase and structural transitions. In this work, we explore the limitations and accuracy of thermodynamic extrapolation applied to water, where qualitative shifts from anomalous to simple-fluid-like behavior are manifested through shifts in the liquid structure that occur as a function of both temperature and density. We present formulas for extrapolating in volume for canonical ensembles and demonstrate that linear extrapolations of water's structural properties are only accurate over a limited density range. On the other hand, linear extrapolation in temperature can be accurate across the entire liquid state. We contrast these extrapolations with classical perturbation theory techniques, which are more conservative and slowly converging. Indeed, we show that such behavior is expected by demonstrating exact relationships between extrapolation of free energies and well-known techniques to predict free energy differences. An ideal gas in an external field is also studied to more clearly explain these results for a toy system with fully analytical solutions. We also present a recursive interpolation strategy for predicting arbitrary structural properties of molecular fluids over a predefined range of state conditions, demonstrating its success in mapping qualitative shifts in water structure with density.

Grand canonical inverse design of multicomponent colloidal crystals.

Inverse design methods are powerful computational approaches for creating colloidal systems which self-assemble into a target morphology by reverse engineering the Hamiltonian of the system. Despite this, these optimization procedures tend to yield Hamiltonians which are too complex to be experimentally realized. An alternative route to complex structures involves the use of several different components, however, conventional inverse design methods do not explicitly account for the possibility of phase separation into compositionally distinct structures. Here, we present an inverse design scheme for multicomponent colloidal systems by combining active learning with a method to directly compute their ground state phase diagrams. This explicitly accounts for phase separation and can locate stable regions of Hamiltonian parameter space which grid-based surveys are prone to miss. Using this we design low-density, binary structures with Lennard-Jones-like pairwise interactions that are simpler than in the single component case and potentially realizable in an experimental setting. This reinforces the concept that ground states of simple, multicomponent systems might be rich with previously unappreciated diversity, enabling the assembly of non-trivial structures with only few simple components instead of a single complex one.

Symmetry-Based Crystal Structure Enumeration in Two Dimensions.

The accurate prediction of stable crystalline phases is a long-standing problem encountered in the study of conventional atomic and molecular solids as well as soft materials. One possible solution involves enumerating a reasonable set of candidate structures and then screening them to identify the one(s) with the lowest (free) energy. Candidate structures in this set can also serve as starting points for other routines, such as genetic algorithms, which search via optimization. Here, we present a framework for crystal structure enumeration of two-dimensional systems that utilizes a combination of symmetry- and stoichiometry-imposed constraints to compute valid configurations of particles that tile Euclidean space. With mild assumptions, this produces a computationally tractable total number of proposed candidates, enabling multicomponent systems to be screened by direct enumeration of possible crystalline ground states. The python code that enables these calculations is available at https://github.com/usnistgov/PACCS.

Dynamic arrest of adhesive hard rod dispersions.

The phenomenon of dynamic arrest, more commonly referred to as gel and glass formation, originates as particle motion slows significantly. Current understanding of gels and glasses stems primarily from dispersions of spherical particles, but much less is known about how particle shape affects dynamic arrest transitions. To better understand the effects of particle shape anisotropy on gel and glass formation, we systematically measure the rheology, particle dynamics, and static microstructure of thermoreversible colloidal dispersions of adhesive hard rods (AHR). First, the dynamic arrest transitions are mapped as a function of temperature T, aspect ratio L/D≈ 3 to 7, and volume fraction φ≈ 0.1 to 0.5. The critical gel temperature Tgel and glass volume fraction φg are determined from the particle dynamics and rheology. Second, an effective orientation-averaged, short-range attraction between rods is quantified from small-angle scattering measurements and characterized by a reduced temperature τ. Similar τ is found at low rod concentrations, indicating that rod gelation occurs at similar effective attraction strength independent of L/D. Monte Carlo simulations reveal a similar convergence in τ when rods cluster and percolate with an average bond coordination number 〈nc〉≈ 2.4, supporting the link between physical gelation and rigidity percolation. Lastly, AHR results are mapped onto a dimensionless state diagram to compare with previous predictions of attraction-driven gels, repulsion-driven glasses, and liquid crystal phases.

Using symmetry to elucidate the importance of stoichiometry in colloidal crystal assembly.

We demonstrate a method based on symmetry to predict the structure of self-assembling, multicomponent colloidal mixtures. This method allows us to feasibly enumerate candidate structures from all symmetry groups and is many orders of magnitude more computationally efficient than combinatorial enumeration of these candidates. In turn, this permits us to compute ground-state phase diagrams for multicomponent systems. While tuning the interparticle potentials to produce potentially complex interactions represents the conventional route to designing exotic lattices, we use this scheme to demonstrate that simple potentials can also give rise to such structures which are thermodynamically stable at moderate to low temperatures. Furthermore, for a model two-dimensional colloidal system, we illustrate that lattices forming a complete set of 2-, 3-, 4-, and 6-fold rotational symmetries can be rationally designed from certain systems by tuning the mixture composition alone, demonstrating that stoichiometric control can be a tool as powerful as directly tuning the interparticle potentials themselves.

On the virial coefficients of confined fluids: Analytic expressions for the thermodynamic properties of hard particles with attractions in slit and cylindrical pores to second order.

We present analytic expressions for the second virial coefficient of hard particles in slit and cylindrical pores with arbitrary square-well fluid-fluid and fluid-solid interactions, number of fluid components, and pore sizes. We derive multiple methods to predict the isotherms in confined materials based on the virial expansion and find that the thermodynamic and structural properties calculated from the virial expansion to second order in the dilute limit display excellent qualitative agreement with previous simulation results.

Flat-Histogram Monte Carlo as an Efficient Tool To Evaluate Adsorption Processes Involving Rigid and Deformable Molecules.

Monte Carlo simulations are the foundational technique for predicting thermodynamic properties of open systems where the process of interest involves the exchange of particles. Thus, they have been used extensively to computationally evaluate the adsorption properties of nanoporous materials and are critical for the in silico identification of promising materials for a variety of gas storage and chemical separation applications. In this work we demonstrate that a well-known biasing technique, known as "flat-histogram" sampling, can be combined with temperature extrapolation of the free energy landscape to efficiently provide significantly more useful thermodynamic information than standard open ensemble MC simulations. Namely, we can accurately compute the isosteric heat of adsorption and number of particles adsorbed for various adsorbates over an extremely wide range of temperatures and pressures from a set of simulations at just one temperature. We extend this derivation of the temperature extrapolation to adsorbates with intramolecular degrees of freedom when Rosenbluth sampling is employed. Consequently, the working capacity and isosteric heat can be computed for any given combined temperature/pressure swing adsorption process for a large range of operating conditions with both rigid and deformable adsorbates. Continuous thermodynamic properties can be computed with this technique at very moderate computational cost, thereby providing a strong case for its application to the in silico identification of promising nanoporous adsorbents.

Predicting structural properties of fluids by thermodynamic extrapolation.

We describe a methodology for extrapolating the structural properties of multicomponent fluids from one thermodynamic state to another. These properties generally include features of a system that may be computed from an individual configuration such as radial distribution functions, cluster size distributions, or a polymer's radius of gyration. This approach is based on the principle of using fluctuations in a system's extensive thermodynamic variables, such as energy, to construct an appropriate Taylor series expansion for these structural properties in terms of intensive conjugate variables, such as temperature. Thus, one may extrapolate these properties from one state to another when the series is truncated to some finite order. We demonstrate this extrapolation for simple and coarse-grained fluids in both the canonical and grand canonical ensembles, in terms of both temperatures and the chemical potentials of different components. The results show that this method is able to reasonably approximate structural properties of such fluids over a broad range of conditions. Consequently, this methodology may be employed to increase the computational efficiency of molecular simulations used to measure the structural properties of certain fluid systems, especially those used in high-throughput or data-driven investigations.

Assembly of three-dimensional binary superlattices from multi-flavored particles.

Binary superlattices constructed from nano- or micron-sized colloidal particles have a wide variety of applications, including the design of advanced materials. Self-assembly of such crystals from their constituent colloids can be achieved in practice by, among other means, the functionalization of colloid surfaces with single-stranded DNA sequences. However, when driven by DNA, this assembly is traditionally premised on the pairwise interaction between a single DNA sequence and its complement, and often relies on particle size asymmetry to entropically control the crystalline arrangement of its constituents. The recently proposed "multi-flavoring" motif for DNA functionalization, wherein multiple distinct strands of DNA are grafted in different ratios to different colloids, can be used to experimentally realize a binary mixture in which all pairwise interactions are independently controllable. In this work, we use various computational methods, including molecular dynamics and Wang-Landau Monte Carlo simulations, to study a multi-flavored binary system of micron-sized DNA-functionalized particles modeled implicitly by Fermi-Jagla pairwise interactions. We show how self-assembly of such systems can be controlled in a purely enthalpic manner, and by tuning only the interactions between like particles, demonstrate assembly into various morphologies. Although polymorphism is present over a wide range of pairwise interaction strengths, we show that careful selection of interactions can lead to the generation of pure compositionally ordered crystals. Additionally, we show how the crystal composition changes with the like-pair interaction strengths, and how the solution stoichiometry affects the assembled structures.

Molecular simulation of capillary phase transitions in flexible porous materials.

We used flat-histogram sampling Monte Carlo to study capillary phase transitions in deformable adsorbent materials. Specifically, we considered a pure adsorbate fluid below its bulk critical temperature within a slit pore of variable pore width. The instantaneous pore width is dictated by a number of factors, such as adsorbate loading, reservoir pressure, fluid-wall interaction, and bare adsorbent properties. In the slit pores studied here, the bare adsorbent free energy was assumed to be biparabolic, consisting of two preferential pore configurations, namely, the narrow pore and the large pore configurations. Four distinct phases could be found in the adsorption isotherms. We found a low-pressure phase transition, driven primarily by capillary condensation/evaporation and accompanied by adsorbent deformation in response. The deformation can be a relatively small contraction/expansion as seen in elastic materials, or a large-scale structural transformation of the adsorbent. We also found a high-pressure transition driven by excluded volume effects, which tends to expand the material and thus results in a large-scale structural transformation of the adsorbent. The adsorption isotherms and osmotic free energies can be rationalized by considering the relative free energy differences between the basins of the bare adsorbent free energy.

Monte Carlo simulation of cylinders with short-range attractions.

Cylindrical or rod-like particles are promising materials for the applications of fillers in nanocomposite materials and additives to control rheological properties of colloidal suspensions. Recent advances in particle synthesis allows for cylinders to be manufactured with short-ranged attractions to study the gelation as a function of packing fraction, aspect ratio and attraction strength. In order to aid in the analysis of small-angle scattering experiments of rod-like particles, computer simulation methods were used to model these particles with specialized Monte Carlo algorithms and tabular superquadric potentials. The attractive interaction between neighboring rods increases with the amount of locally-accessible surface area, thus leading to patchy-like interactions. We characterize the clustering and percolation of cylinders as the attractive interaction increases from the homogenous fluid at relatively low attraction strength, for a variety of aspect ratios and packing fractions. Comparisons with the experimental scattering results are also presented, which are in agreement.

Communication: Predicting virial coefficients and alchemical transformations by extrapolating Mayer-sampling Monte Carlo simulations.

Virial coefficients are predicted over a large range of both temperatures and model parameter values (i.e., alchemical transformation) from an individual Mayer-sampling Monte Carlo simulation by statistical mechanical extrapolation with minimal increase in computational cost. With this extrapolation method, a Mayer-sampling Monte Carlo simulation of the SPC/E (extended simple point charge) water model quantitatively predicted the second virial coefficient as a continuous function spanning over four orders of magnitude in value and over three orders of magnitude in temperature with less than a 2% deviation. In addition, the same simulation predicted the second virial coefficient if the site charges were scaled by a constant factor, from an increase of 40% down to zero charge. This method is also shown to perform well for the third virial coefficient and the exponential parameter for a Lennard-Jones fluid.

Multivariable extrapolation of grand canonical free energy landscapes.

We derive an approach for extrapolating the free energy landscape of multicomponent systems in the grand canonical ensemble, obtained from flat-histogram Monte Carlo simulations, from one set of temperature and chemical potentials to another. This is accomplished by expanding the landscape in a Taylor series at each value of the order parameter which defines its macrostate phase space. The coefficients in each Taylor polynomial are known exactly from fluctuation formulas, which may be computed by measuring the appropriate moments of extensive variables that fluctuate in this ensemble. Here we derive the expressions necessary to define these coefficients up to arbitrary order. In principle, this enables a single flat-histogram simulation to provide complete thermodynamic information over a broad range of temperatures and chemical potentials. Using this, we also show how to combine a small number of simulations, each performed at different conditions, in a thermodynamically consistent fashion to accurately compute properties at arbitrary temperatures and chemical potentials. This method may significantly increase the computational efficiency of biased grand canonical Monte Carlo simulations, especially for multicomponent mixtures. Although approximate, this approach is amenable to high-throughput and data-intensive investigations where it is preferable to have a large quantity of reasonably accurate simulation data, rather than a smaller amount with a higher accuracy.

Controlling relative polymorph stability in soft porous crystals with a barostat.

We use Monte Carlo simulations to investigate the thermodynamic behavior of soft porous crystal (SPC) adsorbents under the influence of an external barostat. We consider SPCs that naturally exhibit polymorphism between crystal forms of two distinct pore sizes. In the absence of barostatting, these crystals may be naturally divided into two categories depending on their response to stress applied by the adsorbate fluid: those which macroscopically deform and change the volume of their unit cell ("breathing") and those which instead undergo internal rearrangements that change the adsorbate-accessible volume without modifying the unit cell volume ("gate-opening"). When breathing SPCs have a constant external pressure applied, in addition to the thermodynamic pressure of the adsorbate fluid, we find that the free energy difference between the crystal polymorphs is shifted by a constant amount over the entire course of adsorption. Thus, their relative stability may be easily controlled by the barostat. However, when the crystal is held at a fixed overall pressure, changes to the relative stability of the polymorphs tend to be more complex. We demonstrate a thermodynamic analogy between breathing SPCs held at a fixed pressure and macroscopically rigid gate-opening ones which explains this behavior. Furthermore, we illustrate how this implies that external mechanical forces may be employed to tune the effective free energy profile of an empty SPC, which may open new avenues to engineer the thermodynamic properties of these polymorphic adsorbents, such as selectivity.

Relationship between Pore-size Distribution and Flexibility of Adsorbent Materials: Statistical Mechanics and Future Material Characterization Techniques.

Measurement of the pore-size distribution (PSD) via gas adsorption and the so-called "kernel method" is a widely used characterization technique for rigid adsorbents. Yet, standard techniques and analytical equipment are not appropriate to characterize the emerging class of flexible adsorbents that deform in response to the stress imparted by an adsorbate gas, as the PSD is a characteristic of the material that varies with the gas pressure and any other external stresses. Here, we derive the PSD for a flexible adsorbent using statistical mechanics in the osmotic ensemble to draw analogy to the kernel method for rigid materials. The resultant PSD is a function of the ensemble constraints including all imposed stresses and, most importantly, the deformation free energy of the adsorbent material. Consequently, a pressure-dependent PSD is a descriptor of the deformation characteristics of an adsorbent and may be the basis of future material characterization techniques. We discuss how, given a technique for resolving pressure-dependent PSDs, the present statistical mechanical theory could enable a new generation of analytical tools that measure and characterize certain intrinsic material properties of flexible adsorbents via otherwise simple adsorption experiments.