Data-driven predictions of complex organic mixture permeation in polymer membranes.

Membrane-based organic solvent separations are rapidly emerging as a promising class of technologies for enhancing the energy efficiency of existing separation and purification systems. Polymeric membranes have shown promise in the fractionation or splitting of complex mixtures of organic molecules such as crude oil. Determining the separation performance of a polymer membrane when challenged with a complex mixture has thus far occurred in an ad hoc manner, and methods to predict the performance based on mixture composition and polymer chemistry are unavailable. Here, we combine physics-informed machine learning algorithms (ML) and mass transport simulations to create an integrated predictive model for the separation of complex mixtures containing up to 400 components via any arbitrary linear polymer membrane. We experimentally demonstrate the effectiveness of the model by predicting the separation of two crude oils within 6-7% of the measurements. Integration of ML predictors of diffusion and sorption properties of molecules with transport simulators enables for the rapid screening of polymer membranes prior to physical experimentation for the separation of complex liquid mixtures.

Hydrophobic polyamide nanofilms provide rapid transport for crude oil separation.

Hydrocarbon separation relies on energy-intensive distillation. Membrane technology can offer an energy-efficient alternative but requires selective differentiation of crude oil molecules with rapid liquid transport. We synthesized multiblock oligomer amines, which comprised a central amine segment with two hydrophobic oligomer blocks, and used them to fabricate hydrophobic polyamide nanofilms by interfacial polymerization from self-assembled vesicles. These polyamide nanofilms provide transport of hydrophobic liquids more than 100 times faster than that of conventional hydrophilic counterparts. In the fractionation of light crude oil, manipulation of the film thickness down to ~10 nanometers achieves permeance one order of magnitude higher than that of current state-of-the-art hydrophobic membranes while retaining comparable size- and class-based separation. This high permeance can markedly reduce plant footprint, which expands the potential for using membranes made of ultrathin nanofilms in crude oil fractionation.

Dry Glass Reference Perturbation Theory Predictions of the Temperature and Pressure Dependent Separations of Complex Liquid Mixtures Using SBAD-1 Glassy Polymer Membranes.

In this work we apply dry glass reference perturbation theory (DGRPT) within the context of fully mutualized diffusion theory to predict the temperature and pressure dependent separations of complex liquid mixtures using SBAD-1 glassy polymer membranes. We demonstrate that the approach allows for the prediction of the membrane-based separation of complex liquid mixtures over a wide range of temperature and pressure, using only single-component vapor sorption isotherms measured at 25 °C to parameterize the model. The model was then applied to predict the membrane separation of a light shale crude using a structure oriented lumping (SOL) based compositional model of petroleum. It was shown that when DGRPT is applied based on SOL compositions, the combined model allows for the accurate prediction of separation performance based on the trend of both molecular weight and molecular class.

Use of Boundary-Driven Nonequilibrium Molecular Dynamics for Determining Transport Diffusivities of Multicomponent Mixtures in Nanoporous Materials.

The boundary-driven molecular modeling strategy to evaluate mass transport coefficients of fluids in nanoconfined media is revisited and expanded to multicomponent mixtures. The method requires setting up a simulation with bulk fluid reservoirs upstream and downstream of a porous media. A fluid flow is induced by applying an external force at the periodic boundary between the upstream and downstream reservoirs. The relationship between the resulting flow and the density gradient of the adsorbed fluid at the entrance/exit of the porous media provides for a direct path for the calculation of the transport diffusivities. It is shown how the transport diffusivities found this way relate to the collective, Onsager, and self-diffusion coefficients, typically used in other contexts to describe fluid transport in porous media. Examples are provided by calculating the diffusion coefficients of a Lennard-Jones (LJ) fluid and mixtures of differently sized LJ particles in slit pores, a realistic model of methane in carbon-based slit pores, and binary mixtures of methane with hypothetical counterparts having different attractions to the solid. The method is seen to be robust and particularly suited for the study of study of transport of dense fluids and liquids in nanoconfined media.

A resummed thermodynamic perturbation theory for positive and negative hydrogen bond cooperativity in water.

In this paper a resummed thermodynamic perturbation theory is developed which accounts for both positive and negative hydrogen bond cooperativity in water. The theory is developed in Wertheim's multi-density statistical mechanics. We demonstrate that the hydrogen bonded structure of water is controlled by positive hydrogen bond cooperativity in homodromic hydrogen bonded clusters. Inclusion of negative anti-cooperativity in antidromic hydrogen bonded structures has little effect on the underlying hydrogen bond structure of liquid water. The resummed perturbation theory is shown to give hydrogen bond statistics consistent with first principles ab initio molecular dynamics simulations. In addition, the theory is shown to be in good agreement with experiment in the prediction of the average number of hydrogen bonds per water molecule in a liquid at ambient conditions. Finally, we develop a full thermodynamic model for water by including contributions to the free energy for isotropic square well attractions. We demonstrate that the model gives good representation of saturated liquid density, hydrogen bonding and vapor pressure. In addition, we show the full model predicts the anomalous minima in isothermal compressibility and isobaric heat capacity.

A molecular equation of state for alcohols which includes steric hindrance in hydrogen bonding.

In this paper, we develop the first equation of state for alcohol containing mixtures which includes the effect of steric hindrance between the two electron lone pair hydrogen bond acceptor sites on the alcohol's hydroxyl oxygen. The theory is derived for multi-component mixtures within Wertheim's multi-density statistical mechanics in a second order perturbation theory. The accuracy of the new approach is demonstrated by application to pure methanol and ethanol and binary ethanol/water mixtures. It is demonstrated that the new approach gives a substantial improvement in the prediction of the hydrogen bonding structure of both pure alcohol and alcohol/water mixtures, as compared to conventional approaches which do not include steric effects between the alcohol association sites. Finally, it is demonstrated that the inclusion of steric effects allows for more accurate binary phase equilibria and heats of mixing prediction with water.

A general mixture equation of state for double bonding carboxylic acids with ≥2 association sites.

In this paper, we obtain the first general multi-component solution to Wertheim's thermodynamic perturbation theory for the case that molecules can participate in cyclic double bonds. In contrast to previous authors, we do not restrict double bonding molecules to a 2-site association scheme. Each molecule in a multi-component mixture can have an arbitrary number of donor and acceptor association sites. The one restriction on the theory is that molecules can have at most one pair of double bonding sites. We also incorporate the effect of hydrogen bond cooperativity in cyclic double bonds. We then apply this new association theory to 2-site and 3-site models for carboxylic acids within the polar perturbed chain statistical associating fluid theory equation of state. We demonstrate the accuracy of the approach by comparison to both pure and multi-component phase equilibria data. It is demonstrated that the 3-site association model gives substantially a different hydrogen bonding structure than a 2-site approach. We also demonstrate that inclusion of hydrogen bond cooperativity has a substantial effect on a liquid phase hydrogen bonding structure.

A theory for the effect of patch/non-patch attractions on the self-assembly of patchy colloids.

In this paper, we develop a thermodynamic perturbation theory to describe the self-assembly of patchy colloids which exhibit both patch-patch attractions as well as patch/non-patch attractions. That is, patches attract other patches as well as the no patch region (we call this region Ψ). In general, the patch-patch and patch-Ψ attractions operate on different energy scales allowing for a competition between different modes of attraction. This competition may result in anomalous thermodynamic properties. As an application, we tune the patch parameters to reproduce the liquid density (suitably scaled) maximum of water. It is then shown that the liquid branch of the colloids phase diagram has liquid densities consistent with both saturated and super-cooled liquid water. Finally, it is shown that the colloids reproduce water's anomalous minimum in isothermal compressibility and negative volume expansivity.

A second order thermodynamic perturbation theory for hydrogen bond cooperativity in water.

It has been extensively demonstrated through first principles quantum mechanics calculations that water exhibits strong hydrogen bond cooperativity. Equations of state developed from statistical mechanics typically assume pairwise additivity, meaning they cannot account for these 3-body and higher cooperative effects. In this paper, we extend a second order thermodynamic perturbation theory to correct for hydrogen bond cooperativity in 4 site water. We demonstrate that the theory predicts hydrogen bonding structure consistent spectroscopy, neutron diffraction, and molecular simulation data. Finally, we implement the approach into a general equation of state for water.

Perturbation theory for water with an associating reference fluid.

The theoretical description of the thermodynamics of water is challenged by the structural transition towards tetrahedral symmetry at ambient conditions. As perturbation theories typically assume a spherically symmetric reference fluid, they are incapable of accurately describing the liquid properties of water at ambient conditions. In this paper we address this problem by introducing the concept of an associated reference perturbation theory (APT). In APT we treat the reference fluid as an associating hard sphere fluid which transitions to tetrahedral symmetry in the fully hydrogen bonded limit. We calculate this transition in a theoretically self-consistent manner without appealing to molecular simulations. This associated reference provides the reference fluid for a second order Barker-Henderson perturbative treatment of the long-range attractions. We demonstrate that this approach gives a significantly improved description of water as compared to standard perturbation theories.

On the cooperativity of association and reference energy scales in thermodynamic perturbation theory.

Equations of state for hydrogen bonding fluids are typically described by two energy scales. A short range highly directional hydrogen bonding energy scale as well as a reference energy scale which accounts for dispersion and orientationally averaged multi-pole attractions. These energy scales are always treated independently. In recent years, extensive first principles quantum mechanics calculations on small water clusters have shown that both hydrogen bond and reference energy scales depend on the number of incident hydrogen bonds of the water molecule. In this work, we propose a new methodology to couple the reference energy scale to the degree of hydrogen bonding in the fluid. We demonstrate the utility of the new approach by showing that it gives improved predictions of water-hydrocarbon mutual solubilities.

Equilibrium adsorption and self-assembly of patchy colloids in microchannels.

A theory is developed to describe the equilibrium adsorption and self-assembly of patchy colloids in microchannels. The adsorption theory is developed in classical density functional theory, with the adsorbed phase and fluid phase chemical potentials modeled using thermodynamic perturbation theory. Adsorption of nonpatchy colloids in microchannels is typically achieved through nonequilibrium routes such as spin coating and evaporation. These methods are required due to the entropic penalty of adsorption. In this work we propose that the introduction of patches on the colloids greatly enhances the temperature dependent and reversible adsorption of colloids in microchannels. It is shown how bulk fluid density, temperature, patch size, and channel diameter can be manipulated to achieve the adsorption and self-assembly of patchy colloids in microchannels.

Dual chain perturbation theory: A new equation of state for polyatomic molecules.

In the development of equations of state for polyatomic molecules, thermodynamic perturbation theory (TPT) is widely used to calculate the change in free energy due to chain formation. TPT is a simplification of a more general and exact multi-density cluster expansion for associating fluids. In TPT, all contributions to the cluster expansion which contain chain-chain interactions are neglected. That is, all inter-chain interactions are treated at the reference fluid level. This allows for the summation of the cluster theory in terms of reference system correlation functions only. The resulting theory has been shown to be accurate and has been widely employed as the basis of many engineering equations of state. While highly successful, TPT has many handicaps which result from the neglect of chain-chain contributions. The subject of this document is to move beyond the limitations of TPT and include chain-chain contributions to the equation of state.

Thermodynamic perturbation theory for associating fluids confined in a one-dimensional pore.

In this paper, a new theory is developed for the self-assembly of associating molecules confined to a single spatial dimension, but allowed to explore all orientation angles. The interplay of the anisotropy of the pair potential and the low dimensional space results in orientationally ordered associated clusters. This local order enhances association due to a decrease in orientational entropy. Unlike bulk 3D fluids which are orientationally homogeneous, association in 1D necessitates the self-consistent calculation of the orientational distribution function. To test the new theory, Monte Carlo simulations are performed and the theory is found to be accurate. It is also shown that the traditional treatment in first order perturbation theory fails to accurately describe this system. The theory developed in this paper may be used as a tool to study hydrogen bonding of molecules in 1D zeolites as well as the hydrogen bonding of molecules in carbon nanotubes.

Thermodynamic perturbation theory for self-assembling mixtures of divalent single patch colloids.

In this work we extend Wertheim's thermodynamic perturbation theory (TPT) to binary mixtures (species A and species B) of patchy colloids were each species has a single patch which can bond a maximum of twice (divalent). Colloids are treated as hard spheres with a directional conical association site. We restrict the system such that only patches between unlike species share attractions; meaning there are AB attractions but no AA or BB attractions. The theory is derived in Wertheim's two density formalism for one site associating fluids. Since the patches are doubly bondable, associated chains, of all chain lengths, as well as 4-mer rings consisting of two species A and two species B colloids are accounted for. With the restriction of only AB attractions, triatomic rings of doubly bonded colloids, which dominate in the corresponding pure component case, cannot form. The theory is shown to be in good agreement with Monte Carlo simulation data for the structure and thermodynamics of these patchy colloid mixtures as a function of temperature, density, patch size and composition. It is shown that 4-mer rings dominate at low temperature, inhibiting the polymerization of the mixture into long chains. Mixtures of this type have been recently synthesized by researchers. This work provides the first theory capable of accurately modeling these mixtures.

Resummed thermodynamic perturbation theory for bond cooperativity in associating fluids with small bond angles: effects of steric hindrance and ring formation.

In this paper we develop a thermodynamic perturbation theory for two site associating fluids which exhibit bond cooperativity (system energy is non-pairwise additive). We include both steric hindrance and ring formation such that the equation of state is bond angle dependent. Here, the bond angle is the angle separating the centers of the two association sites. As a test, new Monte Carlo simulations are performed, and the theory is found to accurately predict the internal energy as well as the distribution of associated clusters as a function of bond angle.

Cluster perturbation theory for the self-assembly of associating fluids into complex structures.

Wertheim's two-density thermodynamic perturbation theory (TPT) has proven to be an indispensable statistical mechanical tool in the description of associating fluids with a single association site. TPT was developed to enforce the monovalence of the hydrogen bond and only recently has been extended to account for divalent association sites. It has been shown through experiment and molecular simulation that certain one-site associating fluids can self-assemble into complex extended supramolecular structures as a result of multiple bonding of association sites. In this paper we reorganize TPT into a form that is more easily applied to complex associated structures. The derived theory is general to all possible self-assemble structures. We obtain the free energy and bonding fractions in a general way in terms of single-cluster partition functions and averages. The new formalism removes any reference to graph theory allowing for the conceptually straightforward application of the two-density formalism to complex self-assembled structures.

Resummed thermodynamic perturbation theory for bond cooperativity in associating fluids.

We develop a resummed thermodynamic perturbation theory for bond cooperativity in associating fluids by extension of Wertheim's multi-density formalism. We specifically consider the case of an associating hard sphere with two association sites and both pairwise and triplet contributions to the energy, such that the first bond in an associated cluster receives an energy -ε((1)) and each subsequent bond in the cluster receives an energy -ε((2)). To test the theory we perform new Monte Carlo simulations for potentials of this type. Theory and simulation are found to be in excellent agreement. We show that decreasing the energetic benefit of hydrogen bonding can actually result in a decrease in internal energy in the fluid. We also predict that when ε((1)) = 0 and ε((2)) is nonzero there is a transition temperature where the system transitions from a fluid of monomers to a mixture of monomers and very long chains.

Molecular theory for self assembling mixtures of patchy colloids and colloids with spherically symmetric attractions: the single patch case.

In this work we develop a new theory to model self assembling mixtures of single patch colloids and colloids with spherically symmetric attractions. In the development of the theory we restrict the interactions such that there are short ranged attractions between patchy and spherically symmetric colloids, but patchy colloids do not attract patchy colloids and spherically symmetric colloids do not attract spherically symmetric colloids. This results in the temperature, density, and composition dependent reversible self assembly of the mixture into colloidal star molecules. This type of mixture has been recently synthesized by grafting of complimentary single stranded DNA [L. Feng, R. Dreyfus, R. Sha, N. C. Seeman, and P. M. Chaikin, Adv. Mater. 25(20), 2779-2783 (2013)]. As a quantitative test of the theory, we perform new monte carlo simulations to study the self assembly of these mixtures; theory and simulation are found to be in excellent agreement.