Efficient sampling of solvent free energies in polymers

Modelling Sorption and Transport of Gases in Polymeric Membranes across Different Scales: A Review

Professor Giulio C. Sarti has provided outstanding contributions to the modelling of fluid sorption and transport in polymeric materials, with a special eye on industrial applications such as membrane separation, due to his Chemical Engineering background. He was the co-creator of innovative theories such as the Non-Equilibrium Theory for Glassy Polymers (NET-GP), a flexible tool to estimate the solubility of pure and mixed fluids in a wide range of polymers, and of the Standard Transport Model (STM) for estimating membrane permeability and selectivity. In this review, inspired by his rigorous and original approach to representing membrane fundamentals, we provide an overview of the most significant and up-to-date modeling tools available to estimate the main properties governing polymeric membranes in fluid separation, namely solubility and diffusivity. The paper is not meant to be comprehensive, but it focuses on those contributions that are most relevant or that show the potential to be relevant in the future. We do not restrict our view to the field of macroscopic modelling, which was the main playground of professor Sarti, but also devote our attention to Molecular and Multiscale Hierarchical Modeling. This work proposes a critical evaluation of the different approaches considered, along with their limitations and potentiality.

Atomistic Modeling of Hydrogen and Oxygen Solubility in Semicrystalline PA-6 and HDPE Materials

Hydrogen is a clean and sustainable energy carrier which plays a major role in the transition of the global energy market to a less fossil fuel dependent future. Polymer-based materials are crucial in the production, storage, transportation, and energy extraction of hydrogen. More insights in the hydrogen-polymers interactions are required to guide material design and product development, especially for hydrogen solubility in polymers, which is crucial in many applications. The current study aims at rationalizing the determining factors of hydrogen solubility in two relevant polymers: polyamide-6 (PA-6) and high density polyethylene (HDPE). Based on atomistic molecular dynamics simulations and experimental data, we have reached several conclusions related to hydrogen and oxygen solubility in these two polymers. The crystal phases of PA-6 and HDPE are impenetrable to hydrogen and oxygen at elevated pressures, despite the small molecular size of hydrogen and oxygen. The practical implication for gas barrier applications is that polymer crystals act as impermeable obstacles and gas migration takes place primarily in the amorphous phase. Experimental hydrogen and oxygen solubilities in PA-6 and HDPE at elevated pressures can be predicted in a semiquantitative manner by molecular simulations. The discrepancies between experimental and predicted values could be attributed to neglect of the effect of crystal regions on the amorphous polymer domains. Although hydrogen is smaller than oxygen, it has been experimentally observed that hydrogen has a lower solubility in PA-6 and HDPE than oxygen. This observation has been confirmed by molecular simulations and attributed to the more favorable energetic interactions of oxygen with PA-6 and PE than of hydrogen. These interactions dominate the solubility behavior over the distribution of the accessible volume in the polymers.

Molecular Modeling Investigations of Sorption and Diffusion of Small Molecules in Glassy Polymers

With a wide range of applications, from energy and environmental engineering, such as in gas separations and water purification, to biomedical engineering and packaging, glassy polymeric materials remain in the core of novel membrane and state-of the art barrier technologies. This review focuses on molecular simulation methodologies implemented for the study of sorption and diffusion of small molecules in dense glassy polymeric systems. Basic concepts are introduced and systematic methods for the generation of realistic polymer configurations are briefly presented. Challenges related to the long length and time scale phenomena that govern the permeation process in the glassy polymer matrix are described and molecular simulation approaches developed to address the multiscale problem at hand are discussed.

Exploration of bulk and interface behavior of gas molecules and 1-butyl-3-methylimidazolium tetrafluoroborate ionic liquid using equilibrium and nonequilibrium molecular dynamics simulation and quantum chemical calculation

Ionic liquids (ILs) show brilliant performance in separating gas impurities, but few researchers have performed an in-depth exploration of the bulk and interface behavior of penetrants and ILs thoroughly. In this research, we have performed a study on both molecular dynamics (MD) simulation and quantum chemical (QC) calculation to explore the transport of acetylene and ethylene in the bulk and interface regions of 1-butyl-3-methylimidazolium tetrafluoroborate ([BMIM]-[BF4]). The diffusivity, solubility and permeability of gas molecules in the bulk were researched with MD simulation first. The subdiffusion behavior of gas molecules is induced by coupling between the motion of gas molecules and the ions, and the relaxation processes of the ions after the disturbance caused by gas molecules. Then, QC calculation was performed to explore the optical geometry of ions, ion pairs and complexes of ions and penetrants, and interaction potential for pairs and complexes. Finally, nonequilibrium MD simulation was performed to explore the interface structure and properties of the IL-gas system and gas molecule behavior in the interface region. The research results may be used in the design of IL separation media.

Comparing gas transport in three polymers via molecular dynamics simulation

Molecular dynamics (MD) simulation was employed to study the transport of methane and n-butane molecules in the bulk and interface region of polyethylene (PE), poly(4-methyl-2-pentyne) (PMP) and polydimethylsiloxane (PDMS).

Charge-Neutral Constant pH Molecular Dynamics Simulations Using a Parsimonious Proton Buffer

In constant pH molecular dynamics simulations, the protonation states of titratable sites can respond to changes of the pH and of their electrostatic environment. Consequently, the number of protons bound to the biomolecule, and therefore the overall charge of the system, fluctuates during the simulation. To avoid artifacts associated with a non-neutral simulation system, we introduce an approach to maintain neutrality of the simulation box in constant pH molecular dynamics simulations, while maintaining an accurate description of all protonation fluctuations. Specifically, we introduce a proton buffer that, like a buffer in experiment, can exchange protons with the biomolecule enabling its charge to fluctuate. To keep the total charge of the system constant, the uptake and release of protons by the buffer are coupled to the titration of the biomolecule with a constraint. We find that, because the fluctuation of the total charge (number of protons) of a typical biomolecule is much smaller than the number of titratable sites of the biomolecule, the number of buffer sites required to maintain overall charge neutrality without compromising the charge fluctuations of the biomolecule, is typically much smaller than the number of titratable sites, implying markedly enhanced simulation and sampling efficiency.

Constant pH Molecular Dynamics in Explicit Solvent with λ-Dynamics

pH is an important parameter in condensed-phase systems, because it determines the protonation state of titratable groups and thus influences the structure, dynamics, and function of molecules in solution. In most force field simulation protocols, however, the protonation state of a system (rather than its pH) is kept fixed and cannot adapt to changes of the local environment. Here, we present a method, implemented within the MD package GROMACS, for constant pH molecular dynamics simulations in explicit solvent that is based on the λ-dynamics approach. In the latter, the dynamics of the titration coordinate λ, which interpolates between the protonated and deprotonated states, is driven by generalized forces between the protonated and deprotonated states. The hydration free energy, as a function of pH, is included to facilitate constant pH simulations. The protonation states of titratable groups are allowed to change dynamically during a simulation, thus reproducing average protonation probabilities at a certain pH. The accuracy of the method is tested against titration curves of single amino acids and a dipeptide in explicit solvent.

Predicting water sorption and volume swelling in dense polymer systems via computer simulation

Atomistic model structures of amorphous polyamide 6 (PA-6) and of an adhesive system consisting of the diglycidyl ether of bisphenol A (DGEBA) as epoxy resin and isophorone diamine (IPD) as a curing agent are generated. For the adhesive, we use a new approach for the generation of the cross-linked polymer networks. It takes into account the chemical reaction kinetics of the curing reaction and, therefore, results in more realistic network structures. On the basis of the corresponding model structures, the equilibrium water content and the swelling ratio of amorphous PA-6 and of the DGEBA+IPD networks are calculated via computer simulation for different thermodynamic conditions. A hybrid method is used combining the molecular dynamics technique with an accelerated test particle insertion method. The results are in reasonable agreement with experiments and, in the case of the PA-6 system, with results obtained via other computer simulation methods.

A reversible minimum-to-minimum mapping method for the calculation of free-energy differences

A general method is introduced for the calculation of the free-energy difference between two systems, 0 and 1, with configuration spaces omega(0), omega(1) of the same dimensionality. The method relies upon establishing a objective mapping between disjoint subsets gamma(i)(0) of omega(0) and corresponding disjoint subsets gamma(i)(1) of omega(1), and averaging a function of the ratio of configurational integrals over gamma(i)(0) and gamma(i)(1) with respect to the probability densities of the two systems. The mapped subsets gamma(i)(0) and gamma(i)(1) need not span the entire configuration spaces omega(0) and omega(1). The method is applied for the calculation of the excess chemical potential mu(ex) in a Lennard-Jones (LJ) fluid. In this case, omega(0) is the configuration space of a (N-1) real molecule plus one ideal-gas molecule system, while omega(1) is the configuration space of a N real molecule system occupying the same volume. Gamma(i)(0) and gamma(i)(1) are constructed from hyperspheres of the same radius centered at minimum-energy configurations of a set of "active" molecules lying within distance a from the ideal-gas molecule and the last real molecule, respectively. An algorithm is described for sampling gamma(i)(0) and gamma(i)(1) given a point in omega(0) or in omega(1). The algorithm encompasses three steps: "quenching" (minimization with respect to the active-molecule degrees of freedom), "mutation" (gradual conversion of the ideal-gas molecule into a real molecule, with simultaneous minimization of the energy with respect to the active-molecule degrees of freedom), and "excitation" (generation of points on a hypersphere centered at the active-molecule energy minimum). These steps are also carried out in reverse, as required by the bijective nature of the mapping. The mutation step, which establishes a reversible mapping between energy minima with respect to the active degrees of freedom of systems 0 and 1, ensures that excluded volume interactions emerging in the process of converting the ideal-gas molecule into a real molecule are relieved through appropriate rearrangement of the surrounding active molecules. Thus, the insertion problem plaguing traditional methods for the calculation of chemical potential at high densities is alleviated. Results are presented at two state points of the LJ system for a variety of radii a of the active domain. It is shown that the estimated values of mu(ex) are correct in all cases and subject to an order of magnitude lower statistical uncertainty than values based on the same number of Widom [J. Chem. Phys. 39, 2808 (1963)] insertions at high fluid densities. Optimal settings for the new algorithm are identified and distributions of the quantities involved in it [number of active molecules, energy at the sampled minima of systems 0 and 1, and free-energy differences between subsets gamma(i)(0) and gamma(i)(1) that are mapped onto each other] are explored.

Replica exchange with dynamical scaling

A replica exchange method is presented which requires fewer replicas and is designed to be used for large systems. In this method, dynamically scaled replicas are placed between conventional replicas at broadly spaced temperatures. The potential of the scaled replicas is linearly scaled by a dynamical variable which varies between 0 and 1. When the variable is near either end point the replica can undergo exchanges with one of its neighboring replicas. Two different versions of the method are presented for a model system of a small peptide in water. The scaled replica can replace many replicas and the method can be up to ten times more efficient than conventional replica exchange.

A generalized direct-particle-deletion scheme for the calculation of chemical potential and solubilities of small- and medium-sized molecules in amorphous polymers

The development, validation, and first applications of a generalized version of an inverse Widom method are described. It permits the calculation of solubility coefficients for molecules as large as, e.g., benzene in all polymers for which reasonable forcefield parameters exist. Predicting the solubility is a key to the knowledge-based design of materials utilized to solve permeability related problems. For long time, particle insertion methods, such as the Widom method, were the only way to predict solubilities from molecular models, but they, in most cases, only worked well for rather small penetrants (e.g., H2, O2, N2). Therefore, a few years ago, a new particle deletion algorithm "DPD" was introduced by Boulougouris, Economou, and Theodorou to overcome this problem in principle. The related computer code was, however, only applicable to special, relatively simple model systems. As application examples for the generalized version described here, solubility calculations for nitrogen, oxygen, and benzene in poly(dimethyl siloxane) are presented.

Molecular-dynamics evaluation of fluid-phase equilibrium properties by a novel free-energy perturbation approach: Application to gas solubility and vapor pressure of liquid hexane

A novel free-energy perturbation method is developed for the computation of the free energy of transferring a molecule between fluid phases. The methodology consists in drawing a free-energy profile of the target molecule moving across a binary-phase structure built in the computer. The novelty of the method lies in the difference of the definition of the free-energy profile from the common definition. As an important element of the method, the process of making a correction to the transfer free energy with respect to the cutoff of intermolecular forces is elucidated. In order to examine the performance of the method in the application to fluid-phase equilibrium properties, molecular-dynamics computations are carried out for the evaluation of gas solubility and vapor pressure of liquid n-hexane at 298.15 K. The gas species treated are methane, ethane, propane, and n-butane, with the gas solubility expressed as Henry's constant. It is shown that the method works fine and calculated results are generally in good agreement with experiments. It is found that the cutoff correction is strikingly large, constituting a dominant part of the calculated transfer free energy at the cutoff of 8 A.