Effects of chain stiffness and penetrant size on penetrant diffusion in simple polymers: deduced relations from simulation and PRISM theory

Theory of the effect of external stress on the activated dynamics and transport of dilute penetrants in supercooled liquids and glasses

We generalize the self-consistent cooperative hopping theory for a dilute spherical penetrant or tracer activated dynamics in dense metastable hard sphere fluids and glasses to address the effect of external stress, the consequences of which are systematically established as a function of matrix packing fraction and penetrant-to-matrix size ratio. All relaxation processes speed up under stress, but the difference between the penetrant and matrix hopping (alpha relaxation) times decreases significantly with stress corresponding to less time scale decoupling. A dynamic crossover occurs at a critical "slaving onset" stress beyond which the matrix activated hopping relaxation time controls the penetrant hopping time. This characteristic stress increases (decreases) exponentially with packing fraction (size ratio) and can be well below the absolute yield stress of the matrix. Below the slaving onset, the penetrant hopping time is predicted to vary exponentially with stress, differing from the power law dependence of the pure matrix alpha time due to system-specificity of the stress-induced changes in the penetrant local cage and elastic barriers. An exponential growth of the penetrant alpha relaxation time with size ratio under stress is predicted, and at a fixed matrix packing fraction, the exponential relation between penetrant hopping time and stress for different size ratios can be collapsed onto a master curve. Direct connections between the short- and long-time activated penetrant dynamics and between the penetrant (or matrix) alpha relaxation time and matrix thermodynamic dimensionless compressibility are also predicted. The presented results should be testable in future experiments and simulations.

Activated penetrant dynamics in glass forming liquids: size effects, decoupling, slaving, collective elasticity and correlation with matrix compressibility

We employ the microscopic self-consistent cooperative hopping theory of penetrant activated dynamics in glass forming viscous liquids and colloidal suspensions to address new questions over a wide range of high matrix packing fractions and penetrant-to-matrix particle size ratios. The focus is on the mean activated relaxation time of smaller tracers in a hard sphere fluid of larger particle matrices. This quantity also determines the penetrant diffusion constant and connects directly with the structural relaxation time probed in an incoherent dynamic structure factor measurement. The timescale of the non-activated fast dissipative process is also studied and is predicted to follow power laws with the contact value of the penetrant-matrix pair correlation function and the penetrant-matrix size ratio. For long time penetrant relaxation, in the relatively lower packing fraction metastable regime the local cage barriers are dominant and matrix collective elasticity effects unimportant. As packing fraction and/or penetrant size grows, much higher barriers emerge and the collective elasticity associated with the correlated matrix dynamic displacement that facilitates penetrant hopping becomes important. This results in a non-monotonic variation with packing fraction of the degree of decoupling between the matrix and penetrant alpha relaxation times. The conditions required for penetrant hopping to become slaved to the matrix alpha process are determined, which depend mainly on the penetrant to matrix particle size ratio. By analyzing the absolute and relative importance of the cage and elastic barriers we establish a mechanistic understanding of the origin of the predicted exponential growth of the penetrant hopping time with size ratio predicted at very high packing fractions. A dynamics-thermodynamics power law connection between the penetrant activation barrier and the matrix dimensionless compressibility is established as a prediction of theory, with different scaling exponents depending on whether matrix collective elasticity effects are important. Quantitative comparisons with simulations of the penetrant relaxation time, diffusion constant, and transient localization length of tracers in dense colloidal suspensions and cold viscous liquids reveal good agreements. Multiple new predictions are made that are testable via future experiments and simulations. Extension of the theoretical approach to more complex systems of high experimental interest (nonspherical molecules, semiflexible polymers, crosslinked networks) interacting via variable hard or soft repulsions and/or short range attractions is possible, including under external deformation.

A blob model to parameterize polymer hole free volumes and solute diffusion

Solute diffusion in solid polymers has tremendous applications in packaging, reservoir, and biomedical technologies but remains poorly understood. Diffusion of non-entangled linear solutes with chemically identical patterns (blobs) deviates dramatically in polymers in the solid-state (α_lin > 1, Macromolecules 2013, 46, 874) from their behaviors in the molten state (α_lin = 1, Macromolecules, 2007, 40, 3970). This work uses the scale invariance of the diffusivities, D, of linear probes D(N·M_blob + M_anchor,T,T_g) = N^{-α_lin(T,T_g)}D(M_blob + M_anchor,T,T_g) comprising N identical blobs of mass M_blob and possibly one different terminal pattern (anchor of mass M_anchor) to evaluate the amounts of hole-free volume in seven polymers (aliphatic, semi-aromatic and aromatic) over a broad range of temperatures (-70 K ≤T-T_g≤ 160 K). The new parameterization of the concept of hole-free volumes opens the application of the free-volume theory (FVT) developed by Vrentas and Duda to practically any polymer, regardless of the availability of free-volume parameters. The quality of the estimations was tested with various probes including n-alkanes, 1-alcohols, n-alkyl acetates, and n-alkylbenzene. The effects of enthalpic and entropic effects of the blobs and the anchor were analyzed and quantified. Blind validation of the reformulated FVT was tested successfully by predicting from first principles the diffusivities of water and toluene in amorphous polyethylene terephthalate from 4 °C to 180 °C and in various other polymers. The new blob model would open the rational design of additives with controlled diffusivities in thermoplastics.

Coarse-grained molecular dynamics simulation of activated penetrant transport in glassy polymers

Membrane separations of gas mixtures strive to maximize the permeability of a desired species while keeping out undesired ones. Permeability vs. selectivity data from many polymer membranes for a given gas pair with diameters d_A and d_B are typically collected in a "Robeson plot"', and are bound from above by a line with a slope λ = (d_B/d_A)² - 1. A microscopic understanding of this relationship, especially λ, is still missing. We perform molecular dynamics simulations of penetrant diffusion using three different coarse-grained polymer models over a wide range of penetrant sizes, temperatures, and monomer densities. The empirically relevant λ = (d_B/d_A)² - 1 is only found for polymers that are either supercooled liquids with caged segmental dynamics or glasses and when the penetrant size is approximately half the Kuhn length of the chains, for which the penetrant diffusion is an activated process.

Correlated matrix-fluctuation-mediated activated transport of dilute penetrants in glass-forming liquids and suspensions

We formulate a microscopic, force-level statistical mechanical theory for the activated diffusion of dilute penetrants in dense liquids, colloidal suspensions, and glasses. The approach explicitly and self-consistently accounts for coupling between penetrant hopping and matrix dynamic displacements that actively facilitate the hopping event. The key new ideas involve two mechanistically (at a stochastic trajectory level) coupled dynamic free energy functions for the matrix and spherical penetrant particles. A single dynamic coupling parameter quantifies how much the matrix displaces relative to the penetrant when the latter reaches its transition state which is determined via the enforcement of a temporal causality or coincidence condition. The theory is implemented for dilute penetrants smaller than the matrix particles, with or without penetrant-matrix attractive forces. Model calculations reveal a rich dependence of the penetrant diffusion constant and degree of dynamic coupling on size ratio, volume fraction, and attraction strength. In the absence of attractions, a near exponential decrease of penetrant diffusivity with size ratio over an intermediate range is predicted, in contrast to the much steeper, non-exponential variation if one assumes local matrix dynamical fluctuations are not correlated with penetrant motion. For sticky penetrants, the relative and absolute influence of caging versus physical bond formation is studied. The conditions for a dynamic crossover from the case where a time scale separation between penetrant and matrix activated hopping exists to a "slaved" or "constraint release" fully coupled regime are determined. The particle mixture model is mapped to treat experimental thermal systems and applied to make predictions for the diffusivity of water, toluene, methanol, and oxygen in polyvinylacetate liquids and glasses. The theory agrees well with experiment with values of the penetrant-matrix size ratio close to their chemically intuitive values.

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.

Molecular flexibility effects upon liquid dynamics

Simulation results for the diffusive behavior of polymer chain/penetrant systems are analyzed. The attractive range and flexibility of simple chain molecules were varied in order to gauge the effect on dynamics. In all cases, the dimensionless diffusion coefficient, D*, is found to be a smooth, single-valued function of the packing fraction, eta. The functions D*(eta) are found to be power laws with exponents that are sensitive to both chain stiffness and particle type. For a specific system type, the D*'s for both penetrant and chain-center-of-mass extrapolate to zero at the same packing fraction, eta0. This limiting packing fraction is interpreted to be the location of the glass transition, and (eta0-eta), the distance to the glass transition.

General relationships between the mobility of a chain fluid and various computed scalar metrics

We performed molecular dynamics simulations of chain systems to investigate general relationships between the system mobility and computed scalar quantities. Three quantities were found that had a simple one-to-one relationship with mobility: packing fraction, potential energy density, and the value of the static structure factor at the first peak. The chain center-of-mass mobility as a function of these three quantities could be described equally well by either a Vogel-Fulcher type or a power law equation.