Leveraging uncertainty estimates and derivative information in Gaussian process regression for efficient collection and use of molecular simulation data.

We introduce Gaussian Process Regression (GPR) as an enhanced method of thermodynamic extrapolation and interpolation. The heteroscedastic GPR models that we introduce automatically weight provided information by its estimated uncertainty, allowing for the incorporation of highly uncertain, high-order derivative information. By the linearity of the derivative operator, GPR models naturally handle derivative information and, with appropriate likelihood models that incorporate heterogeneous uncertainties, are able to identify estimates of functions for which the provided observations and derivatives are inconsistent due to the sampling bias that is common in molecular simulations. Since we utilize kernels that form complete bases on the function space to be learned, the estimated uncertainty in the model takes into account that of the functional form itself, in contrast to polynomial interpolation, which explicitly assumes the functional form to be fixed. We apply GPR models to a variety of data sources and assess various active learning strategies, identifying when specific options will be most useful. Our active-learning data collection based on GPR models incorporating derivative information is finally applied to tracing vapor-liquid equilibrium for a single-component Lennard-Jones fluid, which we show represents a powerful generalization to previous extrapolation strategies and Gibbs-Duhem integration. A suite of tools implementing these methods is provided at https://github.com/usnistgov/thermo-extrap.

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.

Systematic control of collective variables learned from variational autoencoders.

Variational autoencoders (VAEs) are rapidly gaining popularity within molecular simulation for discovering low-dimensional, or latent, representations, which are critical for both analyzing and accelerating simulations. However, it remains unclear how the information a VAE learns is connected to its probabilistic structure and, in turn, its loss function. Previous studies have focused on feature engineering, ad hoc modifications to loss functions, or adjustment of the prior to enforce desirable latent space properties. By applying effectively arbitrarily flexible priors via normalizing flows, we focus instead on how adjusting the structure of the decoding model impacts the learned latent coordinate. We systematically adjust the power and flexibility of the decoding distribution, observing that this has a significant impact on the structure of the latent space as measured by a suite of metrics developed in this work. By also varying weights on separate terms within each VAE loss function, we show that the level of detail encoded can be further tuned. This provides practical guidance for utilizing VAEs to extract varying resolutions of low-dimensional information from molecular dynamics and Monte Carlo simulations.

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.

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.

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.

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.

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.

Chemometric Outlier Classification of 2D-NMR Spectra to Enable Higher Order Structure Characterization of Protein Therapeutics.

Protein therapeutics are vitally important clinically and commercially, with monoclonal antibody (mAb) therapeutic sales alone accounting for $115 billion in revenue for 2018.[1] In order for these therapeutics to be safe and efficacious, their protein components must maintain their high order structure (HOS), which includes retaining their three-dimensional fold and not forming aggregates. As demonstrated in the recent NISTmAb Interlaboratory nuclear magnetic resonance (NMR) Study[2], NMR spectroscopy is a robust and precise approach to address this HOS measurement need. Using the NISTmAb study data, we benchmark a procedure for automated outlier detection used to identify spectra that are not of sufficient quality for further automated analysis. When applied to a diverse collection of all 252 1H,13C gHSQC spectra from the study, a recursive version of the automated procedure performed comparably to visual analysis, and identified three outlier cases that were missed by the human analyst. In total, this method represents a distinct advance in chemometric detection of outliers due to variation in both measurement and sample.

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.

Improving the efficiency of Monte Carlo simulations of ions using expanded grand canonical ensembles.

While ionic liquids have promising applications as industrial solvents, predicting their fluid phase properties and coexistence remains a challenge. Grand canonical Monte Carlo simulation is an effective method for such predictions, but equilibration is hampered by the apparent requirement to insert and delete neutral sets of ions simultaneously in order to maintain charge neutrality. For relatively high densities and low temperatures, previously developed methods have been shown to be essential in improving equilibration by gradual insertion and deletion of these neutral sets of ions. We introduce an expanded ensemble approach which may be used in conjunction with these existing methods to further improve efficiency. Individual ions are inserted or deleted in one Monte Carlo trial rather than simultaneous insertion/deletion of neutral sets. We show how charge neutrality is maintained and show rigorous quantitative agreement between the conventional and the proposed expanded ensemble approaches, but with up to an order of magnitude increase in efficiency at high densities. The expanded ensemble approach is also more straightforward to implement than simultaneous insertion/deletion of neutral sets, and its implementation is demonstrated within open source software.

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.

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.

A methodology to calculate small-angle scattering profiles of macromolecular solutions from molecular simulations in the grand-canonical ensemble.

The theoretical framework to evaluate small-angle scattering (SAS) profiles for multi-component macromolecular solutions is re-examined from the standpoint of molecular simulations in the grand-canonical ensemble, where the chemical potentials of all species in solution are fixed. This statistical mechanical ensemble resembles more closely scattering experiments, capturing concentration fluctuations that arise from the exchange of molecules between the scattering volume and the bulk solution. The resulting grand-canonical expression relates scattering intensities to the different intra- and intermolecular pair distribution functions, as well as to the distribution of molecular concentrations on the scattering volume. This formulation represents a generalized expression that encompasses most of the existing methods to evaluate SAS profiles from molecular simulations. The grand-canonical SAS methodology is probed for a series of different implicit-solvent, homogeneous systems at conditions ranging from dilute to concentrated. These systems consist of spherical colloids, dumbbell particles, and highly flexible polymer chains. Comparison of the resulting SAS curves against classical methodologies based on either theoretical approaches or canonical simulations (i.e., at a fixed number of molecules) shows equivalence between the different scattering intensities so long as interactions between molecules are net repulsive or weakly attractive. On the other hand, for strongly attractive interactions, grand-canonical SAS profiles deviate in the low- and intermediate-q range from those calculated in a canonical ensemble. Such differences are due to the distribution of molecules becoming asymmetric, which yields a higher contribution from configurations with molecular concentrations larger than the nominal value. Additionally, for flexible systems, explicit discrimination between intra- and inter-molecular SAS contributions permits the implementation of model-free, structural analysis such as Guinier's plots at high molecular concentrations, beyond what the traditional limits are for such analysis.

The Ninth Industrial Fluid Properties Simulation Challenge.

The Ninth Industrial Fluid Properties Simulation Challenge aimed to test the ability of molecular modeling approaches to predict water/oil interfacial tension (IFT) at conditions of high temperature and pressure. In particular, the challenge featured water/oil IFT where the oil was n-dodecane, toluene, or a 50:50 n-dodecane/toluene blend at 1.825 MPa and temperatures in the range of 383 K to 443 K. Seven entries were received including approaches such as molecular dynamics (MD) and Monte Carlo (MC) simulations, COSMO-RS, and iSAFT, and they were judged by comparison to pendant drop tensiometer benchmark data. The quality of predictions varied among the entries between approximately 20 % and 70 % of the total points possible with the entries based on MD and MC having the highest scores in most cases. As is often the case in molecular modeling, predictions of the relative trends tended to be reliable even if the absolute values were not.

Quasi-Two-Dimensional Phase Transition of Methane Adsorbed in Cylindrical Silica Mesopores.

Using Monte Carlo and molecular dynamics simulations, we examine the adsorption of methane in cylindrical silica mesopores in an effort to understand a possible phase transition of adsorbed methane in MCM-41 and SBA-15 silica that was previously identified by an unexpected increase in the adsorbed fluid density following capillary condensation, as measured by small-angle neutron scattering (SANS) [Chiang, W-S., et al., Langmuir 2016, 32, 8849]. Our initial simulation results identify a roughly 10 % increase in the density of the liquidlike adsorbed phase for either an isotherm with increasing pressure or an isobar with decreasing temperature and that this densification is associated with a local maximum in the isosteric enthalpy of adsorption. Subsequent analysis of the simulated fluid, via computation of bond-orientational order parameters of specific annular layers of the adsorbed fluid, showed that the layers undergo an ordering transition from a disordered, amorphous state to one with two-dimensional hexagonal structure. Furthermore, this two-dimensional restructuring of the fluid occurs at the same thermodynamic state points as the aforementioned densification and local maximum in the isosteric enthalpy of adsorption. We thus conclude that the densification of the fluid is the result of structural reorganization, which is signaled by the maximum in the isosteric enthalpy. Owing to the qualitative similarity of the structural transitions in the simulated and experimental methane fluids, we propose this hexagonal reorganization as a plausible explanation of the densification observed in SANS measurements. Lastly, we speculate how this structural transition may impact the transport properties of the adsorbed fluid.

Position-Dependent Dynamics Explain Pore-Averaged Diffusion in Strongly Attractive Adsorptive Systems.

Using molecular simulations, we investigate the relationship between the pore-averaged and position-dependent self-diffusivity of a fluid adsorbed in a strongly attractive pore as a function of loading. Previous work (Krekelberg, W. P.; Siderius, D. W.; Shen, V. K.; Truskett, T. M.; Errington, J. R. Connection between thermodynamics and dynamics of simple fluids in highly attractive pores. Langmuir 2013, 29, 14527-14535, doi: 10.1021/la4037327) established that pore-averaged self-diffusivity in the multilayer adsorption regime, where the fluid exhibits a dense film at the pore surface and a lower density interior pore region, is nearly constant as a function of loading. Here we show that this puzzling behavior can be understood in terms of how loading affects the fraction of particles that reside in the film and interior pore regions as well as their distinct dynamics. Specifically, the insensitivity of pore-averaged diffusivity to loading arises from the approximate cancellation of two factors: an increase in the fraction of particles in the higher diffusivity interior pore region with loading and a corresponding decrease in the particle diffusivity in that region. We also find that the position-dependent self-diffusivities scale with the position-dependent density. We present a model for predicting the pore-average self-diffusivity based on the position-dependent self-diffusivity, which captures the unusual characteristics of pore-averaged self-diffusivity in strongly attractive pores over several orders of magnitude.

Assembly of multi-flavored two-dimensional colloidal crystals.

We systematically investigate the assembly of binary multi-flavored colloidal mixtures in two dimensions. In these mixtures all pairwise interactions between species may be tuned independently. This introduces an additional degree of freedom over more traditional binary mixtures with fixed mixing rules, which is anticipated to open new avenues for directed self-assembly. At present, colloidal self-assembly into non-trivial lattices tends to require either high pressures for isotropically interacting particles, or the introduction of directionally anisotropic interactions. Here we demonstrate tunable assembly into a plethora of structures which requires neither of these conditions. We develop a minimal model that defines a three-dimensional phase space containing one dimension for each pairwise interaction, then employ various computational techniques to map out regions of this phase space in which the system self-assembles into these different morphologies. We then present a mean-field model that is capable of reproducing these results for size-symmetric mixtures, which reveals how to target different structures by tuning pairwise interactions, solution stoichiometry, or both. Concerning particle size asymmetry, we find that domains in this model's phase space, corresponding to different morphologies, tend to undergo a continuous "rotation" whose magnitude is proportional to the size asymmetry. Such continuity enables one to estimate the relative stability of different lattices for arbitrary size asymmetries. Owing to its simplicity and accuracy, we expect this model to serve as a valuable design tool for engineering binary colloidal crystals from multi-flavored components.