Classical Density Functional Theory Research Articles

Designing the Next Generation of Polymers with Machine Learning and Physics-Based Models

Abstract The development of next-generation polymers necessitates optimizing several key properties simultaneously, a task that is expensive and infeasible using traditional trial-and-error experimental approaches. A promising alternative is employing a combination of machine learning and physics-based tools to rapidly screen the polymer design space and provide suggestions of new polymers that meet the critical properties required for industrial applications. In this study, we introduce a comprehensive workflow that utilizes machine learning and molecular modeling approaches to design new polymers with the focus on improving five polymer properties: (1) glass transition temperature, (2) dielectric constant, (3) refractive index, (4) stress optic coefficient, and (5) linear coefficient of thermal expansion. Using a small dataset (<200 unique polymers), we developed quantitative structure-property relationships (QSPR) models to accurately predict the experimental polymer properties for both homo- and co-polymer systems. We tested several ML algorithms and identified the best models for predicting these polymer properties, achieving test set R2 greater than 0.77 across all properties. We then explored new polymers by creating a library of over ~10,000 homopolymers using R-group enumeration tools and applied the trained QSPR models to rapidly predict the five polymer properties. The predictions of QSPR models were used to create a multi-parameter optimization score, which helped downselect the large polymer space to ~10 promising candidates. The properties of these selected polymer candidates were subsequently validated with classical molecular dynamics simulations and density functional theory, revealing a strong correlation with the QSPR model predictions. Finally, one of the top candidates was validated by experiments, which showed good agreement against QSPR and physics-based models. Our workflow underscores the power of combining data-driven and theoretical methods in the polymer design process given a small dataset size, offering a valuable resource for experimentalists looking to leverage computer-aided strategies in materials innovation.

Bridging electronic and classical density-functional theory using universal machine-learned functional approximations.

The accuracy of density-functional theory (DFT) calculations is ultimately determined by the quality of the underlying approximate functionals, namely the exchange-correlation functional in electronic DFT and the excess functional in the classical DFT formalism of fluids. For both electrons and fluids, the exact functional is highly nonlocal, yet most calculations employ approximate functionals that are semi-local or nonlocal in a limited weighted-density form. Machine-learned (ML) nonlocal density-functional approximations show promise in advancing applications of both electronic and classical DFTs, but so far these two distinct research areas have implemented disparate approaches with limited generality. Here, we formulate a universal ML framework and training protocol to learn nonlocal functionals that combine features of equivariant convolutional neural networks and the weighted-density approximation. We prototype this new approach for several 1D and quasi-1D problems and demonstrate that functionals with exactly the same hyperparameters achieve excellent accuracy for a diverse set of systems, including the hard-rod fluid, the inhomogeneous Ising model, the exact exchange energy of electrons, the electron kinetic energy for orbital-free DFT, as well as for liquid water with 1D inhomogeneities. These results lay the foundation for a universal ML approach to approximate exact 3D functionals spanning electronic and classical DFTs.

Fingerprints of ordered self-assembled structures in the liquid phase of a hard-core, square-shoulder system.

We investigate the phase ordering (pattern formation) of systems of two-dimensional core-shell particles using Monte Carlo (MC) computer simulations and classical density functional theory (DFT). The particles interact via a pair potential having a hard core and a repulsive square shoulder. Our simulations show that on cooling, the liquid state structure becomes increasingly characterized by long wavelength density modulations and on further cooling forms a variety of other phases, including clustered, striped, and other patterned phases. In DFT, the hard core part of the potential is treated using either fundamental measure theory or a simple local density approximation, whereas the soft shoulder is treated using the random phase approximation. The different DFTs are benchmarked using large-scale grand-canonical-MC and Gibbs-ensemble-MC simulations, demonstrating their predictive capabilities and shortcomings. We find that having the liquid state static structure factor S(k) for wavenumber k is sufficient to identify the Fourier modes governing both the liquid and solid phases. This allows us to identify from easier-to-obtain liquid state data the wavenumbers relevant to the periodic phases and to predict roughly where in the phase diagram these patterned phases arise.

A microscopic approach to crystallization: Challenging the classical/non-classical dichotomy.

We present a fundamental framework for the study of crystallization based on a combination of classical density functional theory and fluctuating hydrodynamics that is free of any assumptions regarding order parameters and that requires no input other than molecular interaction potentials. We use it to study the nucleation of both droplets and crystalline solids from a low-concentration solution of colloidal particles using two different interaction potentials. We find that the nucleation pathways of both droplets and crystals are remarkably similar at the early stages of nucleation until they diverge due to a rapid ordering along the solid pathways in line with the paradigm of "non-classical" crystallization. We compute the unstable modes at the critical clusters and find that despite the non-classical nature of solid nucleation, the size of the nucleating clusters remains the principle order parameter in all cases, supporting a "classical" description of the dynamics of crystallization. We show that nucleation rates can be extracted from our formalism in a systematic way. Our results suggest that in some cases, despite the non-classical nature of the nucleation pathways, classical nucleation theory can give reasonable results for solids but that there are circumstances where it may fail. This contributes a nuanced perspective to recent experimental and simulation work, suggesting that important aspects of crystal nucleation can be described within a classical framework.

A classical density functional theory for solvation across length scales.

A central aim of multiscale modeling is to use results from the Schrödinger equation to predict phenomenology on length scales that far exceed those of typical molecular correlations. In this work, we present a new approach rooted in classical density functional theory (cDFT) that allows us to accurately describe the solvation of apolar solutes across length scales. Our approach builds on the Lum-Chandler-Weeks (LCW) theory of hydrophobicity [K. Lum et al., J. Phys. Chem. B 103, 4570 (1999)] by constructing a free energy functional that uses a slowly varying component of the density field as a reference. From a practical viewpoint, the theory we present is numerically simpler and generalizes to solutes with soft-core repulsion more easily than LCW theory. Furthermore, by assessing the local compressibility and its critical scaling behavior, we demonstrate that our LCW-style cDFT approach contains the physics of critical drying, which has been emphasized as an essential aspect of hydrophobicity by recent theories. As our approach is parameterized on the two-body direct correlation function of the uniform fluid and the liquid-vapor surface tension, it straightforwardly captures the temperature dependence of solvation. Moreover, we use our theory to describe solvation at a first-principles level on length scales that vastly exceed what is accessible to molecular simulations.

Adsorption and dynamics in cylindrical pore: Molecular dynamics and classical density functional theory study

In this work, we investigate the adsorption of single-component and binary neutral fluids in cylindrical pore using molecular dynamics simulations combined with classical density functional theory (cDFT). For the binary case, we also consider scenarios where one component exhibits a non-spherical structure. We investigated the density distribution curves of fluid components in the pore and found that the cDFT calculations without any adjustable parameter yielded results consistent with molecular dynamics simulations. This consistency becomes more pronounced as the temperature increases. At lower temperatures, the theoretical accuracy declines, but it still remains quantitatively reliable. We have developed a method for calculating diffusion coefficient in porous media involving exchange of particles between the exterior and interior of the pores, and applied the method to compute the diffusion coefficients for molecules from outside to inside the pore, as well as within the pore itself. Based on the calculated diffusion coefficients, we can draw several main conclusions: intrapore diffusion along the axial direction always decreases with increasing pore radius; increasing the surface force field strength enhances diffusion in narrow pores while reducing it in wider pores. Moreover, increasing the attraction strength between particles consistently leads to slower diffusion. These findings provide valuable insights into the factors affecting the diffusion process and can be used to optimize porous materials for various applications.

A universal machine learning framework to automatically identify high‐performance covalent organic framework membranes for CH4/H2 separation

AbstractA universal machine learning framework is proposed to predict and classify membrane performance efficiently and accurately, achieved by combining classical density functional theory and string method. Through application of this framework, we conducted high‐throughput computations under industrial conditions, utilizing an extensive database containing nearly 70,000 covalent organic framework (COF) structures for CH4/H2 separation. The best‐performing COF identified surpasses the materials reported in the previously documented MOF and COF databases, exhibiting an impressive adsorption selectivity for CH4/H2 exceeding 82 and a membrane selectivity reaching as high as 248. More impressively, some of the best candidates identified from this framework have been verified through previous experimental works. Furthermore, the automated machine learning framework and its corresponding scoring system not only enable rapid identification of promising membrane materials from a vast material space but also contribute to a comprehensive understanding of the governing mechanisms that determine separation performance.

Classical density functional theory: The local density approximation

We prove that the lowest free energy of a classical interacting system at temperature [Formula: see text] with a prescribed density profile [Formula: see text] can be approximated by the local free energy [Formula: see text], provided that [Formula: see text] varies slowly over sufficiently large length scales. A quantitative error on the difference is provided in terms of the gradient of the density. Here [Formula: see text] is the free energy per unit volume of an infinite homogeneous gas of the corresponding uniform density. The proof uses quantitative Ruelle bounds (estimates on the local number of particles in a large system), which are derived in an appendix.

The Impact of Metal Centers in the M-MOF-74 Series on Formic Acid Production.

The confinement effect of porous materials on the thermodynamical equilibrium of the CO2 hydrogenation reaction presents a cost-effective alternative to transition metal catalysts. In metal-organic frameworks, the type of metal center has a greater impact on the enhancement of formic acid production than the scale of confinement resulting from the pore size. The M-MOF-74 series enables a comprehensive study of how different metal centers affect HCOOH production, minimizing the effect of pore size. In this work, molecular simulations were used to analyze the adsorption of HCOOH and the CO2 hydrogenation reaction in M-MOF-74, where M = Ni, Cu, Co, Fe, Mn, Zn. We combine classical simulations and density functional theory calculations to gain insights into the mechanisms that govern the low coverage adsorption of HCOOH in the surrounding of the metal centers of M-MOF-74. The impact of metal centers on the HCOOH yield was assessed by Monte Carlo simulations in the grand-canonical ensemble, using gas-phase compositions of CO2, H2, and HCOOH at chemical equilibrium at 298.15-800 K, 1-60 bar. The performance of M-MOF-74 in HCOOH production follows the same order as the uptake and the heat of HCOOH adsorption: Ni > Co > Fe > Mn > Zn > Cu. Ni-MOF-74 increases the mole fraction of HCOOH by ca. 105 times compared to the gas phase at 298.15 K, 60 bar. Ni-MOF-74 has the potential to be more economically attractive for CO2 conversion than transition metal catalysts, achieving HCOOH production at concentrations comparable to the highest formate levels reported for transition metal catalysts and offering a more valuable molecular form of the product.

Multiscale Modeling of Vinyl-Addition Polynorbornenes: The Effect of Stereochemistry.

Vinyl-addition polynorbornenes are candidates for designing high-performance polymers due to unique characteristics, which include a high glass transition temperature associated with a rigid backbone. Recent studies have established that the processability and properties of these polymers can be fine-tuned by using targeted substitutions. However, synthesis with different catalysts results in materials with distinct properties, potentially due to the presence of various stereoisomers that are difficult to quantify experimentally. Herein, we develop all-atom models of polynorbornene oligomers based on classical force fields and density functional theory. To establish the relationship between chemical architecture, chain conformations, and melt structure, we perform detailed molecular dynamics simulations with the fine-tuned atomistic force field and propose simpler coarse-grained descriptions to address the high molecular weight limit. All-atom simulations of oligomers suggest high glass transition temperatures in the range of 550-600 K. In the melt state (800 K), meso chains form highly rigid extended coils (C∞≈11) with amorphous structural characteristics similar to the X-ray diffraction data observed in the literature. In contrast, simulations with racemo chains predict highly helical tubular chain conformations that could promote assembly into crystalline structures.

Adsorption Modeling Based on Classical Density Functional Theory and PC-SAFT: Temperature Extrapolation and Fluid Transfer.

Adsorption is at the heart of many processes from gas separation to cooling. The design of adsorption-based processes requires equilibrium adsorption properties. However, data for adsorption equilibria are limited, and therefore, a model is desirable that uses as little data as possible for its parametrization, while allowing for data interpolation or even extrapolation. This work presents a physics-based model for adsorption isotherms and other equilibrium adsorption properties. The model is based on one-dimensional classical density functional theory (1D-DFT) and the perturbed-chain statistical associating fluid theory (PC-SAFT). The physical processes inside the pores are considered in a thermodynamically consistent approach that is computationally efficient. Once parametrized with a single isotherm, the model is able to extrapolate to other temperatures and outperforms the extrapolation capabilities of state-of-the-art models, such as the empirical isotherm models from Langmuir or Toth. Furthermore, standard combining rules can be used to transfer parameters adjusted to an adsorbent/fluid pair to other fluids. These features are demonstrated for the adsorption of N2, CH4, and CO2 in metal-organic frameworks. Thereby, the presented model can calculate temperature-dependent isotherms for various fluids by using data limited to a single isotherm as input.

Solvation Forces Near Hydrophobic Surfaces: A Classical Density Functional Theory Study.

We use classical density functional theory (DFT) to model solvation interactions between hydrophobic surfaces, which we show to be characterized by depletion attraction at small surface to surface separations and a slowly decaying bipower law interaction at large separations. The solvation interaction originates from van der Waals (vdW) and Coulombic interactions between molecules in the polar solvent, e.g., water, and from the molecules thermal motion and finite volume. We investigate model hydrophobic surfaces represented by bubbles and nonpolar solids, e.g., aliphatic particles, and calculate in a DFT fashion the distribution of molecules in the interlaying solvent between two such surfaces and the hydrophobic excess force resulting from it. The interactions are largely attractive, which is well-known in measurement, albeit vdW attraction between molecules in solids and in the solvent may cause repulsion at certain interface to interface separations. We commence our analysis by suggesting an asymptotic analytical bipower law expression for the solvation interaction at large separations. Thereafter we present a full numerical solution, which is in good agreement with the analytical prediction and further explores the interaction at small surface to surface separations. Our theoretical results yield adhesion energies which agree with previous experiments.

Structure and Screening in Confined Electrolytes: The Role of Ion Association.

The effect of ionic association on the structure and property of confined electrolytes is investigated using the classical density functional theory. We find that ionic association strongly affects the ion distribution, surface force, and screening behavior of confined electrolytes. The decay length ξ, which can describe the screening effect of high-concentration electrolytes, satisfies a scaling relationship ξ/λD ∼ (σ/λD)n, with λD being the Debye length and σ representing the ion diameter. We find that n = 1.5 in the nonassociation model, which is contributed by the charge correlation, but n = 3 in the association model, which is contributed by the density correlation. The ion association changes the concentration-dependent characteristics of the screening length by promoting the shift of the decay behavior from the charge-dominated regime to the density-dominated regime. Our result reveals the importance of ion association for electrolyte structure and screening behaviors.

Atomic and electronic structures of Ni64Zr36 metallic glass under high pressure

Amorphous metallic alloys, also known as metallic glasses (MGs), are materials with unique physical properties resulting from their disordered yet densely packed atomic structure. The packing density of MGs can be further enhanced by external pressure, forcing the decrease of interatomic distances and modifying both the atomic and electronic structure of an alloy. This work reports on classical molecular dynamics (MD) and density functional theory (DFT) studies of Ni64Zr36 MG in a hydrostatic pressure range of 0–120 GPa. The MD simulations revealed that compression leads to enhanced short-range ordering by increasing the contribution of efficiently packed icosahedral-like clusters. According to the DFT calculations, for pressure above 50 GPa, Zr atoms show a significant change in electronic configuration, with a dominant charge transfer from their s and p to d-states and charge redistribution between Ni and Zr atoms. This variation is correlated with the appearance of pairs with significantly shortened interatomic distances, as detected by the MD. We conclude that the enhanced icosahedral ordering in Ni64Zr36 MG is induced not only by the pressure-driven densification of an alloy but also by a variation of its electronic structure.

Density functional theory of freezing transition in a binary mixture of spheres and soft-ellipsoids

ABSTRACT This study uses the classical density functional theory (DFT) of freezing to investigate phase transitions in ellipsoidal-spherical binary mixtures interacting via a generalised Gay-Berne (GB) intermolecular potential, with a focus on I-N, I-SmA and N-SmA transitions. The mixtures, consisting mainly of ellipsoids, contain spherical additives at concentrations of 5 % , 10 % and 15 % . The pair correlation functions, which are crucial for the DFT, are derived from Percus-Yevick integral equation theory. Our results show a significant influence of spherical additives on liquid crystalline phases formed by ellipsoids. At an ellipsoid length-to-width ratio of 3.0, only the I-N transition is observed, with the SmA phase remaining unstable under all conditions. Increasing the ellipsoidal length-to-width ratio to κ = 4.0 introduces the SmA phase, whose stability depends on the concentration and size of the spherical additives. The stability of the SmA phase decreases with increasing concentration and size of the spheres. Phase diagrams, which are shown in temperature-density and pressure-temperature planes, emphasise the influence of the size of the spherical dopants on the phase stability.

Atomistic Insight on Effect of Silica Fume on Intermolecular Interactions between Poly(carboxylate) Superplasticizer and Calcium Ions in Concrete.

Understanding how poly(carboxylate)s of chemical admixtures interact with calcium ions in cement pore solutions in the presence of silica fume is fundamental to developing better chemical admixtures for concrete production. In this work, the intermolecular interactions of calcium ions with a poly(carboxylate) superplasticizer type of chemical admixture was investigated via classical all-atom molecular dynamics (MD) simulations and Density Functional Theory (DFT) calculation methods in the presence of silica fume. The classical all-atom MD simulation and DFT calculation results indicate that calcium ions are interacting with oxygen atoms of the carboxylate group of PCE. The better interaction energy could mean an improved adsorption of the PCE segment with calcium ions. In this regard, it can be noted that the ester-based PCE segment could have a better adsorption onto calcium ions in comparison with the ether-based PCE segment. Moreover, the presence of silicon dioxide could improve the adsorption of the PCE segment onto calcium ions.

Predicting the Morphology of Multiphase Biomolecular Condensates from Protein Interaction Networks

Phase-separated biomolecular condensates containing proteins and RNAs can assemble into higher-order structures by forming thermodynamically stable interfaces between immiscible phases. Using a minimal model of a protein/RNA interaction network, we demonstrate how a “shared” protein species that partitions into both phases of a multiphase condensate can function as a tunable surfactant that modulates the interfacial properties. We use Monte Carlo simulations and free-energy calculations to identify conditions under which a low concentration of this shared species is sufficient to trigger a wetting transition. We also describe a numerical approach based on classical density functional theory to predict concentration profiles and surface tensions directly from the model protein/RNA interaction network. Finally, we show that the wetting phase diagrams that emerge from our calculations can be understood in terms of a simple model of selective adsorption to a fluctuating interface. Our work shows how a low-concentration protein species might function as a biological switch for regulating multiphase condensate morphologies. Published by the American Physical Society 2024

Classical density functional theory in three dimensions with GPU-accelerated automatic differentiation: Computational performance analysis using the example of adsorption in covalent-organic frameworks

We show how classical density functional theory can greatly benefit from algorithmic advances in machine learning, especially neural networks. By exploiting GPU-accelerated backward automatic differentiation, we overcome the often cumbersome and error-prone implementation of functional derivatives for classical density functional theory computations. This provides an efficient and straightforward solution for computing functional derivatives, opening up a wide range of applications. We show the gain in computational performance by using backward automatic differentiation to compute the functional derivatives on GPUs, and exemplify the use of this easy-to-implement and highly extensible classical density functional theory framework to predict the adsorption isotherms of a methane/ethane mixture described by a Helmholtz energy functional based on the PC-SAFT equation of state in the covalent-organic framework 2,3-DhaTph. Together with this manuscript, we provide the full classical density functional theory code as supplementary material.

Electrode wettability and capacitance of electrical double layer capacitor: a classical density functional theory study

We suggest a coarse-grained water model for use in classical density functional theory (cDFT) to describe aqueous inorganic salt solutions that act as working electrolytes in electrical double-layer capacitors (EDLCs) with electrodes comprising two face-to-face doped carbon monolayers. Focus of the cDFT calculations lies on the influence of solvent electrode wettability (SEW) on capacitance and energy storage behaviors, while also considering its interaction with factors like electrolyte bulk concentration, pore size, electrode voltage, and temperature. New phenomena are disclosed theoretically. Remarkably, this study challenges the traditional notion that energy storage is consistently boosted by enhancing the electrode’s ionophobicity. Contrarily, the SEW effect reduces energy storage below the standard aqueous electrochemical window voltage (around 1.2 V) and only enhances the energy storage as the voltage surpasses a certain threshold up to the optimal window voltage (2 V–2.5 V). Furthermore, a non-monotonic SEW effect on energy storage is demonstrated under appropriate conditions, shedding new light on the complex relationship between ionophobicity and energy storage. Moreover, the present coarse-grained water model enables the prediction of the experimentally observed inverse relationship between temperature and capacitance. In contrast, the widely used electrolyte primitive model predicts the existence of a maximum value. The decisive factor for the impact of SEW on capacitance and energy storage is identified as congestion within the electrode pore, while other factors contribute by affecting this congestion. The present research offers valuable insights, highlighting the significance of SEW in the innovative and strategic design of aqueous inorganic EDLC devices.

Variability of entropy force and its coupling with electrostatic and steric hindrance interactions

We investigated the effective interaction potential (EIP) between charged surfaces in solvent comprised of dipole dimer molecules added with a certain amount of ionic liquid. Using classical density functional theory, the EIP is calculated and decoupled into entropic and energy terms. Unlike the traditional Asakura–Oosawa (AO) depletion model, the present entropic term can be positive or negative, depending on the entropy change associated with solvent molecule migration from bulk into slit pore. This is determined by pore congestion and disruption of the bulk dipole network. The energy term is determined by the free energy associated with hard-core repulsion and electrostatic interactions between surface charges, ion charges, and polarized charges carried by the dipole dimer molecules. The calculations in this article clearly demonstrate the variability of the entropy term, which contrasts sharply with the traditional AO depletion model, and the corrective effects of electrostatic and spatial hindrance interactions on the total EIP; we revealed several non-monotonic behaviors of the EIP and its entropic and energy terms concerning solvent bulk concentration and solvent molecule dipole moment; additionally, we demonstrated the promoting effect of dipolar solvent on the emergence of like-charge attraction, even in 1:1 electrolyte solutions. The microscopic origin of the aforementioned phenomena was analyzed to be due to the non-monotonic change of dipolar solvent adsorption with dipole moment under conditions of low solution dielectric constant. The present findings offer novel approaches and molecular-level guidance for regulating the EIP. This insight has implications for understanding fundamental processes in various fields, including biomolecule-ligand binding, activation energy barriers, ion tunneling transport, as well as the formation of hierarchical structures, such as mesophases, micro-, and nanostructures, and beyond.