Dynamics of small molecules in dense polymers subject to thermal motion

Geometric Analysis of Free and Accessible Volume in Atmospheric Nanoparticles

Computer-generated atomistic microstructures of atmospheric nanoparticles are geometrically analyzed using Delaunay tessellation followed by Monte Carlo integration to compute their free and accessible volume. The nanoparticles studied consist of cis-pinonic acid (a biogenic organic aerosol component), inorganic ions (sulfate and ammonium), and water. Results are presented for the free or unoccupied volume in different domains of the nanoparticles and its dependence on relative humidity and organic content. We also compute the accessible volume to small penetrants such as water molecules. Most of the free volume or volume accessible to a penetrant as large as a water molecule is located in the domains occupied by organics. In contrast, regions dominated by inorganics do not have any cavities with sizes larger than 1 Å. Solid inorganic domains inside the particle are practically impermeable to any small molecule, thereby offering practically infinite resistance to diffusion. A guest molecule can find diffusive channels to wander around within the nanoparticle only through the aqueous and organic-rich domains. The largest pores are observed in nanoparticles with high levels of organic mass and low relative humidity. At high relative humidity, the presence of more water molecules reduces the empty space in the inner domains of the nanoparticle, since areas rich in organic molecules (which are the only ones where appreciable pores are found) are pushed to the outer area of the particle. This, however, should not be expected to affect the diffusive process as transport through the aqueous phase inside the particle will be, by default, fast due to its fluid-like nature.

How Segmental Dynamics and Mesh Confinement Determine the Selective Diffusivity of Molecules in Cross-Linked Dense Polymer Networks

The diffusion of molecules ("penetrants") of variable size, shape, and chemistry through dense cross-linked polymer networks is a fundamental scientific problem broadly relevant in materials, polymer, physical, and biological chemistry. Relevant applications include separation membranes, barrier materials, drug delivery, and nanofiltration. A major open question is the relationship between transport, thermodynamic state, and penetrant and polymer chemical structure. Here we combine experiment, simulation, and theory to unravel these competing effects on penetrant transport in rubbery and supercooled polymer permanent networks over a wide range of cross-link densities, size ratios, and temperatures. The crucial importance of the coupling of local penetrant hopping to polymer structural relaxation and the secondary importance of mesh confinement effects are established. Network cross-links strongly slow down nm-scale polymer relaxation, which greatly retards the activated penetrant diffusion. The demonstrated good agreement between experiment, simulation, and theory provides strong support for the size ratio (penetrant diameter to the polymer Kuhn length) as a key variable and the usefulness of coarse-grained simulation and theoretical models that average over Angstrom scale structure. The developed theory provides an understanding of the physical processes underlying the behaviors observed in experiment and simulation and suggests new strategies for enhancing selective polymer membrane design.

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.

Introducing Memory in Coarse-Grained Molecular Simulations

Preserving the correct dynamics at the coarse-grained (CG) level is a pressing problem in the development of systematic CG models in soft matter simulation. Starting from the seminal idea of simple time-scale mapping, there have been many efforts over the years toward establishing a meticulous connection between the CG and fine-grained (FG) dynamics based on fundamental statistical mechanics approaches. One of the most successful attempts in this context has been the development of CG models based on the Mori-Zwanzig (MZ) theory, where the resulting equation of motion has the form of a generalized Langevin equation (GLE) and closely preserves the underlying FG dynamics. In this Review, we describe some of the recent studies in this regard. We focus on the construction and simulation of dynamically consistent systematic CG models based on the GLE, both in the simple Markovian limit and the non-Markovian case. Some recent studies of physical effects of memory are also discussed. The Review is aimed at summarizing recent developments in the field while highlighting the major challenges and possible future directions.

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.

Modeling gas transport in polymer-grafted nanoparticle membranes

We use coarse-grained molecular dynamics simulations to study gas diffusion within nanocomposites consisting of matrix-free polymer-grafted nanoparticles. We compare the transport of gas penetrants in systems using polymer models with and without an angle potential and show that gas diffusion enhancement occurs in nanocomposite systems only with the angle potential. This enhancement is related to the free volume in the system, but the cage size experienced by the gas penetrant seems to be a more relevant indicator of gas diffusion enhancement. The enhancement seen in our simulations is smaller than that observed in experiments.

Bottom-up approach to represent dynamic properties in coarse-grained molecular simulations

Several molecular coarse-graining methods have been proposed in recent years to derive chemical- and state-point transferable force fields. While these force fields describe structural and thermodynamic properties in good agreement with fine-grained models and experiments, dynamic properties are usually overestimated. Herein, we examine if the long-time dynamic properties of molecular coarse-grained (CG) systems can be correctly represented by employing a dissipative particle dynamics (DPD) thermostat, which is "bottom-up informed" by means of a variant of the Markovian Mori-Zwanzig (MZ) DPD coarse-graining method. We report single-site and multiple-site CG models for a monomer, dimer, and 24mer based on 2,2-dimethyl propane as a chemical repeat unit and report data obtained from MZ-DPD simulations of liquids, polymer solutions, and polymer melts. We find that despite incomplete time scale separation of the molecular CG model, MZ-DPD achieves quantitative accuracy in predicting diffusive dynamics in single-component liquids and polymer solutions (24mers in a dimer solvent). We also find that MZ-DPD simulations of molecular penetrant diffusion in polymer networks do not reach quantitative agreement with the fine-grained model. Modeling diffusion governed by the activated barrier crossing of small molecular penetrants in these dense systems requires an accurate description of energy barriers, presumably combined with the treatment of memory effects. The use of a MZ-DPD thermostat extends the scope and applicability of molecular CG models for multicomponent systems where a correct description of the relative diffusion rates of the different components is important.

Modelling Mixed-Gas Sorption in Glassy Polymers for CO₂ Removal: A Sensitivity Analysis of the Dual Mode Sorption Model

In an effort to reduce the experimental tests required to characterize the mixed-gas solubility and solubility-selectivity of materials for membrane separation processes, there is a need for reliable models which involve a minimum number of adjustable parameters. In this work, the ability of the Dual Mode Sorption (DMS) model to represent the sorption of CO₂/CH₄ mixtures in three high free volume glassy polymers, poly(trimethylsilyl propyne) (PTMSP), the first reported polymer of intrinsic microporosity (PIM-1) and tetrazole-modified PIM-1 (TZ-PIM), was tested. The sorption of gas mixtures in these materials suitable for CO₂ separation has been characterized experimentally in previous works, which showed that these systems exhibit rather marked deviations from the ideal pure-gas behavior, especially due to competitive effects. The accuracy of the DMS model in representing the non-idealities that arise during mixed-gas sorption was assessed in a wide range of temperatures, pressures and compositions, by comparing with the experimental results available. Using the parameters obtained from the best fit of pure-gas sorption isotherms, the agreement between the mixed-gas calculations and the experimental data varied greatly in the different cases inspected, especially in the case of CH₄ absorbed in mixed-gas conditions. A sensitivity analysis revealed that pure-gas data can be represented with the same accuracy by several different parameter sets, which, however, yield markedly different mixed-gas predictions, that, in some cases, agree with the experimental data only qualitatively. However, the multicomponent calculations with the DMS model yield more reliable results than the use of pure-gas data in the estimation of the solubility-selectivity of the material.

Size-dependent penetrant diffusion in polymer glasses

Molecular Dynamics simulations are used to understand the underpinning basis of the transport of gas-like solutes in deeply quenched polymeric glasses. As found in previous work, small solutes, with sizes smaller than 0.15 times the chain monomer size, move as might be expected in a medium with large pores. In contrast, the motion of larger solutes is activated and is strongly facilitated by matrix motion. In particular, solute motion is coupled to the local elastic fluctuations of the matrix as characterized by the Debye-Waller factor. While similar ideas have been previously proposed for the viscosity of supercooled liquids above their glass transition, to our knowledge, this is the first illustration of this concept in the context of solute mass transport in deeply quenched polymer glasses.

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).

Determination of the effective diffusivity of water in a poly (methyl methacrylate) membrane containing carbon nanotubes using kinetic Monte Carlo simulations

A kinetic Monte Carlo (kMC) simulation algorithm is developed for computing the effective diffusivity of water molecules in a poly(methyl methacrylate) (PMMA) matrix containing carbon nanotubes (CNTs) at several loadings. The simulations are conducted on a cubic lattice to the bonds of which rate constants are assigned governing the elementary jump events of water molecules from one lattice site to another. Lattice sites belonging to PMMA domains of the membrane are assigned different rates than lattice sites belonging to CNT domains. Values of these two rate constants are extracted from available numerical data for water diffusivity within a PMMA matrix and a CNT pre-computed on the basis of independent atomistic molecular dynamics simulations, which show that water diffusivity in CNTs is 3 orders of magnitude faster than in PMMA. Our discrete-space, continuum-time kMC simulation results for several PMMA-CNT nanocomposite membranes (characterized by different values of CNT length L and diameter D and by different loadings of the matrix in CNTs) demonstrate that the overall or effective diffusivity, D(eff), of water in the entire polymeric membrane is of the same order of magnitude as its diffusivity in PMMA domains and increases only linearly with the concentration C (vol. %) in nanotubes. For a constant value of the concentration C, D(eff) is found to vary practically linearly also with the CNT aspect ratio L/D. The kMC data allow us to propose a simple bilinear expression for D(eff) as a function of C and L/D that can describe the numerical data for water mobility in the membrane extremely accurately. Additional simulations with two different CNT configurations (completely random versus aligned) show that CNT orientation in the polymeric matrix has only a minor effect on D(eff) (as long as CNTs do not fully penetrate the membrane). We have also extensively analyzed and quantified sublinear (anomalous) diffusive phenomena over small to moderate times and correlated them with the time needed for penetrant water molecules to explore the available large, fast-diffusing CNT pores before Fickian diffusion is reached.

Theory of activated penetrant diffusion in viscous fluids and colloidal suspensions

We heuristically formulate a microscopic, force level, self-consistent nonlinear Langevin equation theory for activated barrier hopping and non-hydrodynamic diffusion of a hard sphere penetrant in very dense hard sphere fluid matrices. Penetrant dynamics is controlled by a rich competition between force relaxation due to penetrant self-motion and collective matrix structural (alpha) relaxation. In the absence of penetrant-matrix attraction, three activated dynamical regimes are predicted as a function of penetrant-matrix size ratio which are physically distinguished by penetrant jump distance and the nature of matrix motion required to facilitate its hopping. The penetrant diffusion constant decreases the fastest with size ratio for relatively small penetrants where the matrix effectively acts as a vibrating amorphous solid. Increasing penetrant-matrix attraction strength reduces penetrant diffusivity due to physical bonding. For size ratios approaching unity, a distinct dynamical regime emerges associated with strong slaving of penetrant hopping to matrix structural relaxation. A crossover regime at intermediate penetrant-matrix size ratio connects the two limiting behaviors for hard penetrants, but essentially disappears if there are strong attractions with the matrix. Activated penetrant diffusivity decreases strongly with matrix volume fraction in a manner that intensifies as the size ratio increases. We propose and implement a quasi-universal approach for activated diffusion of a rigid atomic/molecular penetrant in a supercooled liquid based on a mapping between the hard sphere system and thermal liquids. Calculations for specific systems agree reasonably well with experiments over a wide range of temperature, covering more than 10 orders of magnitude of variation of the penetrant diffusion constant.

Prediction and validation of diffusion coefficients in a model drug delivery system using microsecond atomistic molecular dynamics simulation and vapour sorption analysis

Diffusion of small to medium sized molecules in polymeric medical device materials underlies a broad range of public health concerns related to unintended leaching from or uptake into implantable medical devices. However, obtaining accurate diffusion coefficients for such systems at physiological temperature represents a formidable challenge, both experimentally and computationally. While molecular dynamics simulation has been used to accurately predict the diffusion coefficients, D, of a handful of gases in various polymers, this success has not been extended to molecules larger than gases, e.g., condensable vapours, liquids, and drugs. We present atomistic molecular dynamics simulation predictions of diffusion in a model drug eluting system that represent a dramatic improvement in accuracy compared to previous simulation predictions for comparable systems. We find that, for simulations of insufficient duration, sub-diffusive dynamics can lead to dramatic over-prediction of D. We present useful metrics for monitoring the extent of sub-diffusive dynamics and explore how these metrics correlate to error in D. We also identify a relationship between diffusion and fast dynamics in our system, which may serve as a means to more rapidly predict diffusion in slowly diffusing systems. Our work provides important precedent and essential insights for utilizing atomistic molecular dynamics simulations to predict diffusion coefficients of small to medium sized molecules in condensed soft matter systems.

Tracer diffusion in a crowded cylindrical channel

Based on a coarse-grained model, we carry out molecular dynamics simulations to analyze the diffusion of a small tracer particle inside a cylindrical channel whose inner wall is covered with randomly grafted short polymeric chains. We observe an interesting transient subdiffusive behavior along the cylindrical axis at high attraction between the tracer and the chains, however, the long-time diffusion is always normal. This process is found to be enhanced for the case that we immobilize the grafted chains, i.e., the subdiffusive behavior sets in at an earlier time and spans over a longer time period before becoming diffusive. Even if the grafted chains are replaced with a frozen sea of repulsive, nonconnected particles in the background, a transient subdiffusion is observed. The intermediate subdiffusive behavior only disappears when the grafted chains are replaced with a mobile background sea of mutually repulsive particles. Overall, the long-time diffusion coefficient of the tracer along the cylinder axis decreases with an increase in system volume fraction, the strength of the attraction between the tracer and the background, and also on freezing the background.

Moving boundary problems governed by anomalous diffusion

Anomalous diffusion can be characterized by a mean-squared displacement 〈x(2)(t)〉 that is proportional to t(α) where α≠1. A class of one-dimensional moving boundary problems is investigated that involves one or more regions governed by anomalous diffusion, specifically subdiffusion (α<1). A novel numerical method is developed to handle the moving interface as well as the singular history kernel of subdiffusion. Two moving boundary problems are solved: the first involves a subdiffusion region to the one side of an interface and a classical diffusion region to the other. The interface will display non-monotone behaviour. The subdiffusion region will always initially advance until a given time, after which it will always recede. The second problem involves subdiffusion regions to both sides of an interface. The interface here also reverses direction after a given time, with the more subdiffusive region initially advancing and then receding.

Pure and Modified Co-Poly(amide-12-b-ethylene oxide) Membranes for Gas Separation Studied by Molecular Investigations

This paper deals with a theoretical investigation of gas transport properties in a pure and modified PEBAX block copolymer membrane with N-ethyl-o/p-toluene sulfonamide (KET) as additive molecules. Molecular dynamics simulations using COMPASS force field, Gusev-Suter Transition State Theory (TST) and Monte Carlo methods were used. Bulk models of PEBAX and PEBAX/KET in different copolymer/additive compositions were assembled and analyzed to evaluate gas permeability and morphology to characterize structure-performance relationships.