Density-functional theory study of the body-centered-cubic and cI16 hard-sphere crystals.

The properties of the body-centered-cubic (bcc) solid phase of hard spheres are challenging to compute because of its lack of mechanical and thermodynamic stability, yet this structure remains of interest for theoretical and practical reasons. Density-functional theory (DFT) studies of the bcc hard-sphere solid, using the most accurate functionals from fundamental measure theory, have yielded results with unphysical behaviors in structural and thermodynamic properties. We recently reported [Warshavsky et al., J. Chem. Phys. 148, 024502 (2018)] a Monte Carlo (MC) simulation study of hard spheres initiated in a bcc structure. We observed that such systems, even under constant-volume and single-occupancy-cell constraints, rapidly evolved into either a crystalline state with the cI16 structure or one of a few amorphous states. With these observations in mind, we revisited the DFT calculations of the bcc hard-sphere system by allowing for a bcc-to-cI16 structural transformation. Specifically, the free energy functional was minimized with respect to a density profile having two scalar parameters: the traditional alpha parameter characterizing the width of the Gaussian density distribution around each lattice site and a geometric parameter characterizing the bcc-to-cI16 structural transition. The numerical solutions were physically reasonable across the entire density range. At all densities above ρbσ3 = 1.0, a cI16 structure had lower free energy than the corresponding perfect bcc structure. The degree of lattice distortion from bcc to cI16 increased with density up to the close-packing limit. Finally, the predicted values of the structural and thermodynamic properties were in excellent agreement with those extracted from our previous MC simulations.

Morphology of Fluids Confined in Physically Reconstructed Mesoporous Silica: Experiment and Mean Field Density Functional Theory.

Three-dimensional physical reconstruction of the random mesopore network in a hierarchically structured, macroporous-mesoporous silica monolith via electron tomography has been used to generate a lattice model of amorphous, mesoporous silica. This geometrical model has subsequently been employed in mean field density functional theory (MFDFT) calculations of adsorption and desorption. Comparison of the results with experimental sorption isotherms for nitrogen at 77 K shows a good qualitative agreement, with both experiment and theory producing isotherms characterized by type H2 hysteresis. In addition to the isotherms, MFDFT provides the three-dimensional density distribution for the fluid in the porous material for each state studied. We use this information to map the phase distribution in the mesopore network in the hysteresis region. Phase distributions on the desorption boundary curve are compared to those on the adsorption boundary curve for both fixed pressure and fixed density, revealing insights into the relationship between phase distribution and hysteresis.

On the mechanical stability of the body-centered cubic phase and the emergence of a metastable cI16 phase in classical hard sphere solids.

The stability of the body-centered cubic (bcc) solid phase of classical hard spheres is of intrinsic interest and is also relevant to the development of perturbation theories for bcc solids of other model systems. Using canonical ensemble Monte Carlo, we simulated systems initialized in a perfect bcc lattice at various densities in the solid region. We observed that the systems rapidly evolved into one of four structures that then persisted for the duration of the simulation. Remarkably, one of these structures was identified as cI16, a cubic crystalline structure with 16 particles in the unit cell, which has recently been observed experimentally in lithium and sodium solids at high pressures. The other three structures do not exhibit crystalline order but are characterized by common patterns in the radial distribution function and bond-orientational order parameter distribution; we refer to them as bcc-di, with i ranging from 1 to 3. We found similar outcomes when employing any of the three single occupancy cell (SOC) restrictions commonly used in the literature. We also ran long constant-pressure simulations with box shape fluctuations initiated from bcc and cI16 initial configurations. At lower pressures, all the systems evolved to defective face-centered cubic (fcc) or hexagonal close-packed (hcp) structures. At higher pressures, most of the systems initiated as bcc evolved to cI16 with some evolving to defective fcc/hcp. High pressure systems initiated from cI16 remained in that structure. We computed the chemical potential of cI16 using the Einstein crystal reference method and found that it is higher than that of fcc by ∼0.5kT-2.5kT over the pressure range studied, with the difference increasing with pressure. We find that the undistorted bcc solid, even with constant-volume and SOC restrictions applied, is so mechanically unstable that it is unsuitable for consideration as a metastable phase or as a reference system for studying bcc phases of other systems. On the other hand, cI16 is a mechanically stable structure that can spontaneously emerge from a bcc starting point but it is thermodynamically metastable relative to fcc or hcp.

Dynamic colloidal assembly pathways via low dimensional models.

Here we construct a low-dimensional Smoluchowski model for electric field mediated colloidal crystallization using Brownian dynamic simulations, which were previously matched to experiments. Diffusion mapping is used to infer dimensionality and confirm the use of two order parameters, one for degree of condensation and one for global crystallinity. Free energy and diffusivity landscapes are obtained as the coefficients of a low-dimensional Smoluchowski equation to capture the thermodynamics and kinetics of microstructure evolution. The resulting low-dimensional model quantitatively captures the dynamics of different assembly pathways between fluid, polycrystal, and single crystals states, in agreement with the full N-dimensional data as characterized by first passage time distributions. Numerical solution of the low-dimensional Smoluchowski equation reveals statistical properties of the dynamic evolution of states vs. applied field amplitude and system size. The low-dimensional Smoluchowski equation and associated landscapes calculated here can serve as models for predictive control of electric field mediated assembly of colloidal ensembles into two-dimensional crystalline objects.

Phase behavior of the 38-atom Lennard-Jones cluster.

We have developed a coarse-grained description of the phase behavior of the isolated 38-atom Lennard-Jones cluster (LJ38). The model captures both the solid-solid polymorphic transitions at low temperatures and the complex cluster breakup and melting transitions at higher temperatures. For this coarse model development, we employ the manifold learning technique of diffusion mapping. The outcome of the diffusion mapping analysis over a broad temperature range indicates that two order parameters are sufficient to describe the cluster's phase behavior; we have chosen two such appropriate order parameters that are metrics of condensation and overall crystallinity. In this well-justified coarse-variable space, we calculate the cluster's free energy landscape (FEL) as a function of temperature, employing Monte Carlo umbrella sampling. These FELs are used to quantify the phase behavior and onsets of phase transitions of the LJ38 cluster.

Colloidal cluster crystallization dynamics.

The crystallization dynamics of a colloidal cluster is modeled using a low-dimensional Smoluchowski equation. Diffusion mapping shows that two order parameters are required to describe the dynamics. Using order parameters as metrics for condensation and crystallinity, free energy, and diffusivity landscapes are extracted from brownian dynamics simulations using bayesian inference. Free energy landscapes are validated against Monte Carlo simulations, and mean first-passage times are validated against dynamic simulations. The resulting model enables a low-dimensional description of colloidal crystallization dynamics.

A Smoluchowski model of crystallization dynamics of small colloidal clusters.

We investigate the dynamics of colloidal crystallization in a 32-particle system at a fixed value of interparticle depletion attraction that produces coexisting fluid and solid phases. Free energy landscapes (FELs) and diffusivity landscapes (DLs) are obtained as coefficients of 1D Smoluchowski equations using as order parameters either the radius of gyration or the average crystallinity. FELs and DLs are estimated by fitting the Smoluchowski equations to Brownian dynamics (BD) simulations using either linear fits to locally initiated trajectories or global fits to unbiased trajectories using Bayesian inference. The resulting FELs are compared to Monte Carlo Umbrella Sampling results. The accuracy of the FELs and DLs for modeling colloidal crystallization dynamics is evaluated by comparing mean first-passage times from BD simulations with analytical predictions using the FEL and DL models. While the 1D models accurately capture dynamics near the free energy minimum fluid and crystal configurations, predictions near the transition region are not quantitatively accurate. A preliminary investigation of ensemble averaged 2D order parameter trajectories suggests that 2D models are required to capture crystallization dynamics in the transition region.

Universal features of the free-energy functional at the freezing transition for repulsive potentials.

The free-energy difference between coexisting solid and liquid phases is studied in the context of classical density functional theory (DFT). A bridge function is used to represent the higher-order (n>2) terms in the perturbative expansion of the excess Helmholtz free energy, and the values of this bridge function within the solid lattice are determined by inversion using literature Monte Carlo simulation results. Four potential models, specifically hard-sphere and inverse twelfth-, sixth-, and fourth-power repulsive, are studied. The face-centered cubic (fcc) solid is considered for the hard-sphere and inverse twelfth- and sixth-power potentials, while the body-centered cubic (bcc) solid is considered for the inverse sixth- and fourth-power potentials. For a given solid structure there is a remarkable similarity among the bridge functions for different potentials that is analogous to the universality in the sum of elementary diagrams, or bridge functions, of liquid-state theory as originally observed by Rosenfeld and Ashcroft [Phys. Rev. A 20, 1208 (1979)]. In further analogy with liquid-state theory, the bridge functions in the present problem are plotted as functionals of the second-order convolution term in the perturbative expansion. In each case, the plot indicates a unique functionality in the dense regions of the solid near the lattice sites but a scattered and nonunique behavior in the void regions. Interestingly, knowledge of the functional relationship in the unique region near the lattice sites seems to be sufficient to quantitatively model the solid-fluid phase transition. These qualitative observations are true for both fcc and bcc solid phases, although there are some quantitative differences between them. The findings suggest that pursuit of a closure-based DFT of solid-fluid transitions may be profitable.

Fokker-Planck analysis of separation dependent potentials and diffusion coefficients in simulated microscopy experiments.

Total internal reflection microscopy (TIRM) and video microscopy (VM) are methods for nonintrusively measuring weak colloidal interactions important to many existing and emerging applications. Existing analyses of TIRM measured single particle trajectories can be used to extract particle-surface potentials and average particle diffusion coefficients. Here we develop a Fokker-Planck (FP) formalism to simultaneously extract both particle-surface interaction potentials and position dependent diffusion coefficients. The FP analysis offers several advantages including capabilities to measure separation dependent hydrodynamic interactions and nonequilibrium states that are not possible with existing analyses. The FP analysis is implemented to analyze Brownian dynamic simulations of single particle TIRM and VM experiments in several configurations. Relative effects of spatial and temporal sampling on the correct interpretation of both conservative and dissipative forces are explored and show a broad range of applicability for accessible experimental systems. Our results demonstrate the ability to extract both static and dynamic information from microscopy measurements of isolated particles near surfaces, which provides a foundation for further investigation of particle ensembles and nonequilibrium systems.

Closure-based perturbative density-functional theory of hard-sphere freezing: properties of the bridge functional.

The study of freezing using perturbative classical density-functional theory is revisited, using a bridge functional approach to resum all terms beyond second order in the free energy expansion. More precisely, the first-order direct correlation function of the solid phase is written as a functional expansion about the homogeneous liquid phase, and the sum of all higher-order terms is represented as a functional of the second-order term. Information about the shape and uniqueness of this bridge functional for the case of hard spheres is obtained via an inversion procedure that employs Monte Carlo fluid-solid coexistence data from the literature. The parametric plots obtained from the inversion procedure show very little scatter in certain regions, suggesting a unique functional dependence, but large scatter in other regions. The scatter is related to the anisotropy of the solid lattice at the particle scale. Interestingly, the thermodynamic properties of the phase transition are quite insensitive to the regions where the scatter is large, and several simple closures (i.e., analytical forms of the bridge function) reproduce exactly the liquid-solid coexistence densities and Lindemann parameter from simulation. The form of these closures is significantly different from the usual closures employed in liquid-state integral equation theory.

Thermodynamic and mechanical properties of epoxy resin DGEBF crosslinked with DETDA by molecular dynamics.

Fully atomistic molecular dynamics (MD) simulations were used to predict the properties of diglycidyl ether of bisphenol F (DGEBF) crosslinked with curing agent diethyltoluenediamine (DETDA). This polymer is a commercially important epoxy resin and a candidate for applications in nanocomposites. The calculated properties were density and bulk modulus (at near-ambient pressure and temperature) and glass transition temperature (at near-ambient pressure). The molecular topology, degree of curing, and MD force-field were investigated as variables. The models were created by densely packing pre-constructed oligomers of different composition and connectivity into a periodic simulation box. For high degrees of curing (greater than 90%), the density was found to be insensitive to the molecular topology and precise value of degree of curing. Of the two force-fields that were investigated, cff91 and COMPASS, the latter clearly gave more accurate values for the density as compared to experiment. In fact, the density predicted by COMPASS was within 6% of reported experimental values for the highly crosslinked polymer. The predictions of both force-fields for glass transition temperature were within the range of reported experimental values, with the predictions of cff91 being more consistent with a highly cured resin.

Closure-based density functional theory applied to interfacial colloidal fluids.

The behavior of dense colloidal fluids near surfaces can now be probed in great detail with experimental techniques like confocal microscopy. In fact, we are approaching a point where quantitative comparisons of experiment with particle-level theory, such as classical density functional theory (DFT), are appropriate. In a forward sense, we may use a known surface potential to predict a particle density distribution function from DFT; in an inverse sense, we may use an experimentally measured particle density distribution function to predict the underlying surface potential from DFT. In this paper, we tested the ability of the closure-based DFT of Zhou and Ruckenstein (J. Chem. Phys. 2000, 112, 8079-8082) to perform forward and inverse calculations on potential models commonly employed for colloidal particles and surfaces. To reduce sources of uncertainty in this initial study, Monte Carlo simulation results played the role of experimental data. The combination of Rogers-Young and modified-Verlet closures consistently performed well across the different potential models. For a reasonable range of choices of the density, temperature, and potential parameters, the inversion procedure yielded particle-surface potentials to an accuracy on the order of 0.1kT.

Interfacial colloidal sedimentation equilibrium. II. Closure-based density functional theory.

In Part I [R. E. Beckham and M. A. Bevan, J. Chem. Phys. 127, 164708 (2007)], results were presented for the sedimentation equilibrium of concentrated colloidal dispersions using confocal scanning laser microscopy experiments, Monte Carlo (MC) simulations, and a local density approximation perturbation theory. In this paper, we extended the modeling effort on those systems to include nonlocal density functional theory (DFT), which is capable of predicting the microstructure of the sediment at length scales comparable to the colloidal particle dimension. Specifically, we use a closure-based DFT formulation to predict interfacial colloidal sedimentation equilibrium density profiles. The colloid-colloid and colloid-surface interactions were modeled with DLVO screened electrostatic potentials using parameters taken directly from the experimental work. The DFT profiles were compared to the experimental and MC results from Part I. Good agreement was found for relatively dilute interfacial colloidal fluids, but agreement was less satisfactory as interfacial layering became more pronounced for conditions approaching the onset of interfacial crystallization. We also applied DFT in an inverse sense using the measured colloid density profile to extract the underlying colloid-surface potential; this can be thought of as a microscopic analog to the well-known procedure of using the macroscopic (coarse-grained) density profile to extract the osmotic equation of state. For the dilute interfacial fluid, the inverse DFT calculations reproduced the true colloid-surface potential to within 0.5kT at all elevations.

Synthesis and characterization of uniform alumina-mesoporous silica hybrid membranes.

The synthesis and characterization of alumina-mesoporous silica (alumina-MS) hybrid membranes are reported. The hybrids are formed using a variation of the evaporative-induced self-assembly (EISA) process reported by Hayward et al. (Langmuir 2004, 20, 5998) based on dip coating of an Anopore 200 nm membrane with a Brij-56/TEOS/HCl/H2O solution. Numerous analytical methods are used to probe both the hybrid material and the silica phase after dissolution of the Anopore substrate. Most importantly, He/N2 permeation measurements show that the effective pore size of the membrane can be tuned from 20 to 5 nm based on the number of dip-coating cycles used. The observed He/N2 permselectivity of 2.7 +/- 0.11 is nearly identical to the theoretical value obtained (2.65) assuming Knudsen diffusion dominates. The selectivity of these membranes is higher than that of most commercial "5 nm" membranes (2.29), which is ascribed to the lack of pinhole defects in the materials reported here. The hybrid membranes as well as the silica obtained after dissolution of the Anopore substrate have been characterized using scanning and transmission electron microscopy and X-ray diffraction. Those results indicate that the silica deposited in the Anopore membrane possesses uniform pores approximately 5 nm in size, consistent with the permeation studies. The current work presents an alternative approach to materials that possess many of the properties of mesoporous silica thin films (i.e., pores of controlled size and topology) without the difficulty of growing mesoporous silica thin films on porous supports.

Enthalpy-entropy compensation is not a general feature of weak association.

Relationships between the enthalpy and entropy changes resulting from perturbations of a system have been discussed in the literature for some time. Both positive correlations (compensation) and negative correlations (anti-compensation) between deltaH and DeltaS have been observed in various experimental contexts, including chemical reaction, physical association, solvation, and protein folding. Many examples have been demonstrated to be statistical artifacts, but some are genuine signatures of the perturbations in molecular characteristics. In particular, recent literature claims that compensation is a general feature of bimolecular associations arising from weak intermolecular interactions. We employ a statistical mechanical framework to predict the magnitude and direction of enthalpy-entropy correlation in bimolecular association. The theory links the macroscale thermodynamic correlation to the relationship between the intermolecular potential parameters. Using a harmonic approximation to the Lennard-Jones model and potential parameters taken from the literature, we show examples of both compensation and anti-compensation for gas-phase self-association among five homologous series. Furthermore, an aggregate presentation of data for 48 different chemical species shows no correlation in either direction, for the case of self-association in a dilute gas phase.

Inverse density-functional theory as an interpretive tool for measuring colloid-surface interactions in dense systems.

Recent advances in optical microscopy, such as total internal reflection and confocal scanning laser techniques, now permit the direct three-dimensional tracking of large numbers of colloidal particles both near and far from interfaces. A novel application of this technology, currently being developed by one of the authors under the name of diffusing colloidal probe microscopy (DCPM), is to use colloidal particles as probes of the energetic characteristics of a surface. A major theoretical challenge in implementing DCPM is to obtain the potential energy of a single particle in the external field created by the surface, from the measured particle trajectories in a dense colloidal system. In this paper we develop an approach based on an inversion of density-functional theory (DFT), where we calculate the single-particle-surface potential from the experimentally measured equilibrium density profile in a nondilute colloidal fluid. The underlying DFT formulation is based on the recent work of Zhou and Ruckenstein [Zhou and Ruckenstein, J. Chem. Phys. 112, 8079 (2000)]. For model hard-sphere and Lennard-Jones systems, using Monte Carlo simulation to provide the "experimental" density profiles, we found that the inversion procedure reproduces the true particle-surface-potential energy to an accuracy within typical DCPM experimental limitations (approximately 0.1 kT) at low to moderate colloidal densities. The choice of DFT closures also significantly affects the accuracy.

Engineering nanospaces: ordered mesoporous silicas as model substrates for building complex hybrid materials.

This contribution summarizes investigations of organic-inorganic hybrid materials wherein the inorganic phase is ordered mesoporous silica such as MCM-41 and SBA-15. The review, which covers work performed in the last three years, emphasizes studies of: (1) covalently attached functional groups, (2) new approaches to functionalization, (3) approaches for achieving high densities of uniform functional groups, (4) periodic mesoporous organosilicas (PMOs) with hierarchical ordering, (5) new functional chemistries, and (6) the application of new materials to enantioselective catalysis and emerging areas. The review concludes with the authors outlining some outstanding problems in the field.

Making the business case for process safety using value-at-risk concepts.

An increasing emphasis on chemical process safety over the last two decades has led to the development and application of powerful risk assessment tools. Hazard analysis and risk evaluation techniques have developed to the point where quantitatively meaningful risks can be calculated for processes and plants. However, the results are typically presented in semi-quantitative "ranked list" or "categorical matrix" formats, which are certainly useful but not optimal for making business decisions. A relatively new technique for performing valuation under uncertainty, value at risk (VaR), has been developed in the financial world. VaR is a method of evaluating the probability of a gain or loss by a complex venture, by examining the stochastic behavior of its components. We believe that combining quantitative risk assessment techniques with VaR concepts will bridge the gap between engineers and scientists who determine process risk and business leaders and policy makers who evaluate, manage, or regulate risk. We present a few basic examples of the application of VaR to hazard analysis in the chemical process industry.

Coarse-grained nonequilibrium approach to the molecular modeling of permeation through microporous membranes.

We present a modeling technique that combines a statistical-mechanical coarse-graining scheme with a nonequilibrium molecular simulation algorithm to provide an efficient simulation of steady-state permeation across a microporous material. The coarse-graining scheme is based on the mapping of an atomistic model to a lattice using multidimensional free-energy and transition-state calculations. The nonequilibrium simulation algorithm is a stochastic, lattice version of the recently developed atomistic dual-control-volume grand canonical molecular dynamics. We demonstrate the approach on a model of methane permeating through a bulk portion of siliceous zeolite ZK4 at 300 K under imposed fugacity differences. We predict the coarse-grained (cage-level) density profiles and observe the development of nonlinearities as the magnitude of the fugacity difference is increased. From the net flux of methane we also predict a mean permeability coefficient under the various conditions. The simulation results are obtained over time scales on the order of microseconds and length scales on the order of dozens of nanometers.