Layering, condensation, and evaporation of short chains in narrow slit pores

Gas adsorption and separation in metal-organic frameworks by PC-SAFT based density functional theory

In this work, we examine the theoretical performance of perturbed-chain statistical associating fluid theory based density functional theory (DFT) in predicting gas adsorption and separation in metal-organic frameworks by using simulation and experimental data as the benchmark. Adsorption isotherms of methane and ethane in pure gas and mixtures and selectivities for ethane/methane mixtures are calculated. The predicted isotherms by DFT are in excellent agreement with simulation and experimental data for pure methane and ethane, whereas for the mixture, DFT is in semi-quantitative accordance with simulation results. For fast and high-throughput screening of material purpose, three algorithms including Picard iteration with line search, Anderson mixing, and Picard-Anderson-hybrid algorithm are proposed to calculate the three dimensional density distribution of confined gases. The advantages and limitations of the three algorithms under various conditions are discussed.

Free-Energy-Averaged Potentials for Adsorption in Heterogeneous Slit Pores Using PC-SAFT Classical Density Functional Theory

This study analyzes the adsorption behavior in two-dimensional heterogeneous slit pores using nonlocal density functional theory based on the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state. Both chemical heterogeneity and surface roughness on small atomistic scales are investigated. The solid structure is considered as individual solid interaction sites whereby chemical heterogeneity is introduced through the presence of different solid-fluid sites and molecular roughness by varying the position of the interaction sites in the first solid layers. The effect of both forms of heterogeneity on the adsorption behavior is assessed individually. Effective one-dimensional solid-fluid potentials provide a way to reduce the dimensionality and computational demand of the density functional theory (DFT) calculations. We determine one-dimensional free-energy-averaged (FEA) solid-fluid potentials of methane and n-butane in the low-density limit for solid systems with molecular roughness and chemical heterogeneity. Using this effective one-dimensional solid-fluid potential at any density, we find excellent agreement of adsorption isotherms for both solid descriptions in systems with homogeneous slit pores. Subcritical adsorption isotherms of n-butane in slit pores with surface roughness show deviations at higher pressures due to the formation of fluid layers in the one-dimensional FEA potential. Chemical heterogeneity introduces a shift of the capillary condensation pressure below the saturation pressure of the bulk liquid, which is well described by the free-energy-averaged system.

Prediction of Adsorption Isotherms and Selectivities: Comparison between Classical Density Functional Theory Based on the Perturbed-Chain Statistical Associating Fluid Theory Equation of State and Ideal Adsorbed Solution Theory

This study gives an assessment of the predictive capability of classical density functional theory (DFT) for adsorption processes of pure substances and mixtures of spherical and nonspherical molecular species. A Helmholtz energy functional based on the perturbed-chain statistical associating fluid theory (PC-SAFT) is applied to calculate isotherms and selectivities of multicomponent adsorption. In order to unambiguously assess the accuracy of the DFT model, we conduct molecular simulations. Monte Carlo (MC) simulations are performed in the grand canonical ensemble using the transition matrix. Two types of systems are studied: a model system, where fluid-fluid and solid-fluid interactions are defined as (single-site) Lennard-Jones interactions, and a more realistic methane-n-butane mixture in a graphite-like pore. Differences between a slit-shaped and a cylindrical pore geometry are examined for the model system. Adsorption isotherms and selectivities obtained from DFT calculations and MC simulations are found in very good agreement, particularly at high pressures. Capillary condensation observed along adsorption isotherms containing n-butane was accurately predicted, both, in equilibrium pressure and in density-increase. Comparisons with results from the ideal adsorbed solution theory are presented, confirming powerful predictions of the DFT approach.

Microscopic density functional theory for monolayers of diblock copolymers

We propose density functional theory for diblock copolymers in two dimensions. Our theoretical framework is based on Wertheim's first order thermodynamic perturbation theory. Using the proposed approach, we investigate the structure and phase behavior of monolayers of symmetric diblock copolymers. We find that the phase behavior of symmetric diblock copolymer monolayers is similar to that in 3D. This includes the scaling of the equilibrium lamellar width with chain length. We find that the topology of the resulting phase diagrams depends on the chain length and the unlike segment interaction incompatibility and involves either one, two, or three triple points (one of them being the peritectic point). We expect that a similar phase behavior could be obtained for monolayers of colloidal suspensions with carefully tuned interparticle interactions.

Characterization of Methane Excess and Absolute Adsorption in Various Clay Nanopores from Molecular Simulation

In this work, we use grand canonical Monte Carlo (GCMC) simulation to study methane adsorption in various clay nanopores and analyze different approaches to characterize the absolute adsorption. As an important constituent of shale, clay minerals can have significant amount of nanopores, which greatly contribute to the gas-in-place in shale. In previous works, absolute adsorption is often calculated from the excess adsorption and bulk liquid phase density of absorbate. We find that methane adsorbed phase density keeps increasing with pressure up to 80 MPa. Even with updated adsorbed phase density from GCMC, there is a significant error in absolute adsorption calculation. Thus, we propose to use the excess adsorption and adsorbed phase volume to calculate absolute adsorption and reduce the discrepancy to less than 3% at high pressure conditions. We also find that the supercritical Dubinin-Radushkevich (SDR) fitting method which is commonly used in experiments to convert the excess adsorption to absolute adsorption may not have a solid physical foundation for methane adsorption. The methane excess and absolute adsorptions per specific surface area are similar for different clay minerals in line with previous experimental data. In mesopores, the excess and absolute adsorptions per specific surface area become insensitive to pore size. Our work should provide important fundamental understandings and insights into accurate estimation of gas-in-place in shale reservoirs.

Pressure Profile for an Associating Lennard-Jones Fluid Confined in a Spherical Cavity

We present the pressure tensor of an associating Lennard-Jones (LJ) fluid confined in a spherical cavity of hard wall, where a high-order density correlation has been taken into account. To give the two-body association potential for calculating the pressure tensor, an angle-average of site-site attraction over all orientations of two particles is performed. Furthermore, the classical density functional theory is employed to obtain the density profile of the confined fluid, by which the normal and tangential pressure profiles are illustrated under various conditions to show the dependence of the pressure tensor on the association strength, number of associating sites, radius of cavity, and bulk density. As an application, the corresponding surface tension is calculated. It is shown that under a strong association interaction (both association strength and the number of associating sites are large), the pressure profiles are depleted from the wall of the cavity instead of the oscillatory behavior under a weak association interaction. Such a tendency is mainly determined by the competition between association interaction and excluded volume interaction. Therefore, the aggregation state and related properties of an associating LJ fluid within a confinement of nanoscale can be efficiently regulated by the association interaction.

Confinement-Induced Supercriticality and Phase Equilibria of Hydrocarbons in Nanopores

For over a century, the phase behavior of bulk fluids has been described as PVT (pressure-volume-temperature) three-dimensional properties, but it has become increasingly clear that the liquid-vapor phase behavior in confined geometries is significantly altered from the bulk. Efforts have been devoted to accessing confined phase transitions using sorption, molecular simulations, and theoretical methods. However, a comprehensive picture of PVT relationships for confined hydrocarbons remains uncertain. Herein, we introduce d (confining pore diameter) as a fourth dimension, and we present PVT-d behavior of confined fluids in nanopores. For the first time, a T-d phase diagram is presented for n-hexane, n-octane, and n-decane under multiple confinement scales (37.9, 14.8, 9.8, 6.0, 4.1, 3.3, and 2.2 nm cylindrical pore diameter) using experimental differential scanning calorimetry and PVT-d equation of state theory at atmospheric pressure. As pore diameter decreases from 37.9 to 4.1 nm, the bubble point increases by as much as 15 K above bulk, until we observe behavior consistent with a supercritical state, pointing to confinement-induced supercriticality. Remarkably, experimental and theoretical findings overlap very well, showing that this approach effectively captures the phase boundaries between the liquid, vapor, and supercritical fluid regions. The model and completed EOS are additionally extended to calculation of isothermal capillary adsorption, and its validity is discussed.

Density functional theory for polymeric systems in 2D

We propose density functional theory for polymeric fluids in two dimensions. The approach is based on Wertheim's first order thermodynamic perturbation theory (TPT) and closely follows density functional theory for polymers proposed by Yu and Wu (2002 J. Chem. Phys. 117 2368). As a simple application we evaluate the density profiles of tangent hard-disk polymers at hard walls. The theoretical predictions are compared against the results of the Monte Carlo simulations. We find that for short chain lengths the theoretical density profiles are in an excellent agreement with the Monte Carlo data. The agreement is less satisfactory for longer chains. The performance of the theory can be improved by recasting the approach using the self-consistent field theory formalism. When the self-avoiding chain statistics is used, the theory yields a marked improvement in the low density limit. Further improvements for long chains could be reached by going beyond the first order of TPT.

Structure and phase behavior of a confined nanodroplet composed of the flexible chain molecules

A polymer density functional theory has been employed for investigating the structure and phase behaviors of the chain polymer, which is modelled as the tangentially connected sphere chain with an attractive interaction, inside the nanosized pores. The excess free energy of the chain polymer has been approximated as the modified fundamental measure-theory for the hard spheres, the Wertheim's first-order perturbation for the chain connectivity, and the mean-field approximation for the van der Waals contribution. For the value of the chemical potential corresponding to a stable liquid phase in the bulk system and a metastable vapor phase, the flexible chain molecules undergo the liquid-vapor transition as the pore size is reduced; the vapor is the stable phase at small volume, whereas the liquid is the stable phase at large volume. The wide liquid-vapor coexistence curve, which explains the wide range of metastable liquid-vapor states, is observed at low temperature. The increase of temperature and decrease of pore size result in a narrowing of liquid-vapor coexistence curves. The increase of chain length leads to a shift of the liquid-vapor coexistence curve towards lower values of chemical potential. The coexistence curves for the confined phase diagram are contained within the corresponding bulk liquid-vapor coexistence curve. The equilibrium capillary phase transition occurs at a higher chemical potential than in the bulk phase.

Capillary condensation in pores with rough walls: A density functional approach

The effect of surface roughness of slit-like pore walls on the capillary condensation of a spherical particles and short chains is studied. The gas molecules interact with the substrate by a Lennard-Jones (9,3) potential. The rough layer at each pore wall has a variable thickness and density and consists of a disordered quenched matrix of spherical particles. The system is described in the framework of a density functional approach and using computer simulations. The contribution due to attractive van der Waals interactions between adsorbate molecules is described by using first-order mean spherical approximation and mean-field approximation.

A Hybrid Approach for Microscopic Properties and Self-Assembly of Dendrimers between Two Hard Walls

Dendrimers are of interest in a number of applications and theoretical studies due to their interesting and complex architectures. We use a hybrid approach to investigate the microstructure of hard dendrimers and self-assembly of diblock dendrimers confined between two hard walls. In the hybrid approach, a single-chain Monte Carlo simulation is used to evaluate the ideal-gas contribution of the Helmholtz energy and a density functional theory is employed to calculate the excess Helmholtz energy. In our calculations, a coarse-grained model is used to represent the dendrimers of generations 1-4. The effects of generation and bulk packing fraction on the microscopic properties of the hard dendrimers are explored. With the increase of generations, the complexity of the dendritic architecture increases. Accordingly, the depletion effect becomes stronger with the generation at etabulk = 0.1. Furthermore, it is found that the more complex the molecular architecture and the higher the molecular stiffness, the smaller is the partitioning coefficient of confined dendrimers. In addition, we also investigate the effects of the width of the slit and the interaction (epsilon*AA) between hydrophilic segments on the self-assembly of diblock dendrimers in the slit. With the increase of epsilon*AA, we observe that the curves of average packing fraction of the dendrimers in the slit exist an abrupt jump, which corresponds to the first-order phase transition from a disordered state to a lamellar ordered structure. In the slit of H = 11sigma, it is at epsilon*AA = 8 rather than epsilon*AA = 10 or epsilon*AA = 12 that the minimum critical bulk packing fraction appears. This observation is distinctively different from the case of self-assembly of rod-like molecules in the slit, where the critical bulk concentration increases with the decrease of the head-head interaction linearly.

Effect of attractions on the structure of polymer solutions confined between surfaces: A density functional approach

A density functional theory is presented to study the effect of attractions on the structure of polymer solutions confined between surfaces. The polymer molecules have been modeled as a pearl necklace of freely jointed hard spheres and the solvent as hard spheres, both having Yukawa-type attractions and the mixture being confined between attractive Yukawa-type surfaces. The present theory treats the ideal gas free energy functional exactly and uses weighted density approximation for the hard chain and hard sphere contributions to the excess free energy functional. The attractive interactions are calculated using the direct correlation function obtained from the polymer reference interaction site model theory along with the mean spherical approximation closure. The theoretical predictions on the density profiles of the polymer and the solvent molecules are found to agree quite well with the Monte Carlo simulation results for varying densities, chain lengths, wall separations, and different sets of interaction potentials.

Surface phase transitions in athermal mixtures of hard rods and excluded volume polymers investigated using a density functional approach

Using fundamental measures' density functional framework based on Wertheim's first order perturbation theory [J. Chem. Phys. 87, 7323 (1987)] we study the surface phase transitions in athermal polymer-needle mixtures, which demix in bulk into the isotropic polymer-rich (rod-poor) and polymer-poor (rod-rich) phases. We find that the polymer-rich (rod-poor) phase wets the hard wall at coexistence and the wetting transition is of first order. In the partial wetting regime we find a sequence of layerings but these transitions are gradually suppressed as the chain length increases. For long enough chains we detect the prewetting line. Rods exhibit pronounced ordering at the wall in the polymer-rich phases. Our results imply that experiments on the (isotropic) wetting transition for colloidal rod-polymer mixtures should be easier to carry out than those for the colloidal rod-sphere mixtures because the wetting transition occurs at lower rod densities. On the other hand, layerings in sphere-needle mixtures may turn out to be difficult to observe experimentally because some of them will be metastable with respect to the freezing transition, whereas the remaining ones are located very close to the binodal.

Microstructure and Self-Assembly of Inhomogeneous Rigid Rodlike Chains between Two Neutral Surfaces: A Hybrid Density Functional Approach

We use a hybrid density functional approach to investigate the microstructure and self-assembly of inhomogeneous rigid rodlike chains between two neutral surfaces, i.e., two hard walls. In the calculation, the rodlike molecule is modeled as a rigid rod linearly connected by the tangent sphere beads. The hybrid method combines a single-chain Monte Carlo (MC) simulation for the ideal-gas part of Helmholtz energy and a DFT approach for the excess Helmholtz energy. The DFT approach includes a modified fundamental measure theory for the excluded-volume effect, the first order thermodynamics perturbation theory for chain connectivity, and the mean field approximation for the van der Waals attraction. We investigate the effect of the chain length (i.e., aspect ratio) of the rodlike molecule and the separation between two surfaces on the microstructure and self-assembly of inhomogeneous rigid rodlike chains. For the athermal systems, the rodlike chain fluids present a smaller partitioning coefficient compared to the flexible chain fluids. For the thermal systems, lamellar thin films formed by the rigid rodlike molecules perpendicular to the neutral surface are observed. The effects of the head-head interaction and the separation on the self-assembly of the rodlike chain fluids in the slit are investigated.

Phase equilibria and plate-fluid interfacial tensions for associating hard sphere fluids confined in slit pores

The excess Helmholtz free energy functional for associating hard sphere fluid is formulated by using a modified fundamental measure theory [Y. X. Yu and J. Z. Wu, J. Chem. Phys. 117, 10156 (2002)]. Within the framework of density functional theory, the thermodynamic properties including phase equilibria for both molecules and monomers, equilibrium plate-fluid interfacial tensions and isotherms of excess adsorption, average molecule density, average monomer density, and plate-fluid interfacial tension for four-site associating hard sphere fluids confined in slit pores are investigated. The phase equilibria inside the hard slit pores and attractive slit pores are determined according to the requirement that temperature, chemical potential, and grand potential in coexistence phases should be equal and the plate-fluid interfacial tensions at equilibrium states are predicted consequently. The influences of association energy, fluid-solid interaction, and pore width on phase equilibria and equilibrium plate-fluid interfacial tensions are discussed.

Capillary phase transitions of linear and branched alkanes in carbon nanotubes from molecular simulation

Capillary phase transitions of linear (from C(1) to C(12)) and branched (C(5) isomers) alkanes in single-walled carbon nanotubes have been investigated using the gauge-cell Monte Carlo simulation. The isotherm at a supercritical temperature increases monotonically with chemical potential and coincides with that from the traditional grand canonical Monte Carlo simulation, whereas the isotherm at a subcritical temperature exhibits a sigmoid van der Waals loop including stable, metastable, and unstable regions. Along this loop, the coexisting phases are determined using an Maxwell equal-area construction. A generic confinement effect is found that reduces the saturation chemical potential, lowers the critical temperature, increases the critical density, and shrinks the phase envelope. The effect is greater in a smaller diameter nanotube and is greater in a nanotube than in a nanoslit.

Density functional theory for inhomogeneous associating chain fluids

We propose a nonlocal density functional theory for associating chain molecules. The chains are modeled as tangent spheres, which interact via Lennard-Jones (12,6) attractive interactions. A selected segment contains additional, short-ranged, highly directional interaction sites. The theory incorporates an accurate treatment of the chain molecules via the intramolecular potential formalism and should accurately describe systems with strongly varying external fields, e.g., attractive walls. Within our approach we investigate the structure of the liquid-vapor interface and capillary condensation of a simple model of associating chains with only one associating site placed on the first segment. In general, the properties of inhomogeneous associating chains depend on the association energy. Similar to the bulk systems we find the behavior of associating chains of a given length to be in between that for the nonassociating chains of the same length and that for the nonassociating chains twice as large.

Investigation of excess adsorption, solvation force, and plate-fluid interfacial tension for Lennard-Jones fluid confined in slit pores

The excess Helmholtz free energy functional is formulated in terms of a modified fundamental measure theory [Y. X. Yu and J. Z. Wu, J. Chem. Phys. 117, 10156 (2002)] for a short ranged repulsion and a first-order mean-spherical approximation theory [Y. P. Tang, J. Chem. Phys. 118, 4140 (2003)] for a long ranged attraction. Within the framework of the density functional theory, the density profile, excess adsorption, solvation force, and plate-fluid interfacial tension of a Lennard-Jones fluid confined in slit pores are predicted, and the results agree well with the simulation data. The phase equilibria inside the slit pores are determined according to the requirement that temperature, chemical potential, and grand potential in coexistence phases should be equal, and the plate-fluid interfacial tensions at equilibrium states are predicted consequently.

Density functional theory for polyelectrolytes near oppositely charged surfaces

We report a nonlocal density functional theory of polyelectrolyte solutions that faithfully accounts for both short- and long-range correlations neglected in a typical mean-field method. It is shown that for systems with strong electrostatic interactions, the long-range correlations are subdued by direct Coulomb attractions, thereby manifesting strong local excluded-volume effects. The theory has also been used to describe the influence of the polyion chain length and small ion valence on charge inversion due to the adsorption of polyelectrolytes at an oppositely charged surface.