Theory of the Effects of Specific Attractions and Chain Connectivity on the Activated Dynamics and Selective Transport of Penetrants in Polymer Melts

We generalize and apply a microscopic force level statistical mechanical theory of activated spherical penetrant dynamics in glass-forming liquids to study the influence of semiflexible polymer connectivity and penetrant–polymer attractive interactions on the penetrant hopping rate. The detailed manner that attractions of highly variable strength and spatial range modify the penetrant size and polymer melt density (from the rubbery state to slightly beyond the kinetic glass transition) dependences of penetrant activation barriers is established. Of special interest are possible nonadditive consequences of physical bonding and steric caging, the degree of coupling of penetrant hopping and the Kuhn segment scale alpha relaxation process, the relative importance of local caging and long-range matrix collective elasticity as a function of penetrant size, and implications for optimizing transport selectivity. With increasing attraction strength, the repulsive caging-restriction effect on penetrant mobility is predicted to grow, in contrast to the effect of the equilibrium penetrant–matrix solvation shell size, which decreases. The former dynamical effect results in a significant enhancement of the importance of the local cage barrier, while the latter effect results in a decrease of the importance of the nonlocal collective elastic barrier. These two competing effects have a very strong influence on selective penetrant transport for different sized penetrants: selectivity varies nonmonotonically with attraction strength in the deeply supercooled state but decreases monotonically in the rubbery state and at fixed attraction strength, exhibits a nonmonotonic variation with the matrix packing fraction. By comparing results based on modeling the matrix as semiflexible polymer chains with analogous calculations using the same dynamical theory but for a disconnected hard sphere matrix, the effect of chain connectivity is revealed and found to have quantitative, but not qualitative, consequences on penetrant-activated dynamics

Theory of Entanglements and Tube Confinement in Rod–Sphere Nanocomposites

We formulate a microscopic theory for the polymer transverse confinement length and associated dynamic potential for a mixture of infinitely thin rods and hard spheres based solely on topological entanglements and excluded volume constraints. For fixed spheres, the needle effective tube diameter decreases with particle loading, and is largely controlled by a single dimensionless parameter involving all three key length-scales in the problem. A crossover from polymer entanglement to nanoparticle-controlled tube localization with increased loading is predicted. A preliminary extension to chain melts exhibits reasonable agreement with a recent simulation, and experimentally testable predictions are made. This work establishes a first-principles theoretical foundation to investigate a variety of dynamical problems in entangled polymer nanocomposites

Theory for the Elementary Time Scale of Stress Relaxation in Polymer Nanocomposites

We construct a microscopic theory for the elementary time scale of stress relaxation in dense polymer nanocomposites. The key dynamical event is proposed to involve the rearrangement of cohesive segment-nanoparticle (NP) tight bridging complexes via an activated small NP dilational motion, which allows the confined segments to relax. The corresponding activation energy is determined by the NP bridge coordination number and potential of mean force barrier. The activation energy varies nonlinearly with interfacial cohesion strength and NP concentration, and a universal master curve is predicted. The theory is in very good agreement with experiments. The underlying ideas are relevant to a variety of other hybrid macromolecular materials involving hard particles and soft macromolecules

Unified Theory of Activated Relaxation in Liquids over 14 Decades in Time

We formulate a predictive theory at the level of forces of activated relaxation in hard-sphere fluids and thermal liquids that covers in a unified manner the apparent Arrhenius, crossover, and deeply supercooled regimes. The alpha relaxation event involves coupled cage-scale hopping and a long-range collective elastic distortion of the surrounding liquid, which results in two inter-related, but distinct, barriers. The strongly temperature and density dependent collective barrier is associated with a growing length scale, the shear modulus, and density fluctuations. Thermal liquids are mapped to an effective hard-sphere fluid based on matching long wavelength density fluctuation amplitudes, resulting in a zeroth-order quasi-universal description. The theory is devoid of fit parameters, has no divergences at finite temperature nor below jamming, and captures the key features of the alpha time of molecular liquids from picoseconds to hundreds of seconds

Statistical Mechanical Theory of Penetrant Diffusion in Polymer Melts and Glasses

We generalize our microscopic, force-level, self-consistent nonlinear Langevin equation theory of activated diffusion of a spherical particle in a dense hard sphere fluid to treat molecular penetrant diffusion in homopolymer melts and nonaging glasses. A coarse-grained mapping is developed where polymer chains are modeled as disconnected, noninterpenetrating Kuhn scale hard spheres (diameter, σ), and the penetrant is modeled as an effective hard sphere (diameter, <i>d</i>) which can be attracted to the polymer segment. The polymer mapping is a priori carried out by enforcing the effective hard sphere fluid reproduces the specific polymer liquid or glass long wavelength dimensionless collective density fluctuation amplitude. The theory predicts that penetrant diffusivity exhibits supra-Arrhenius temperature dependence in supercooled polymer melts and (near) Arrhenius temperature dependence in quenched nonequilibrium polymer glasses. Polymer–penetrant attraction slows down penetrant diffusivity to a degree that is strongly enhanced as penetrants become smaller. By treating <i>d</i>/σ as the only adjustable material-specific parameter, the theory is in good agreement with experimental diffusivity data spanning more than 10 decades for a wide range of penetrants (from small gas to large organic molecules), amorphous polymers, and temperatures. Optimal <i>d</i>/σ values are consistent with a priori physical estimations of effective space-filling molecular and Kuhn segment diameters. Through comparative studies, two different a priori choices of penetrant–matrix attraction strength are established for small gas and large organic penetrants. System parameter transferability is examined. The theory represents a microscopic-based statistical mechanical approach for penetrant diffusion in polymers and provides a foundation for treating time-dependent penetrant diffusivity in aging polymer glasses, collective effects induced by finite penetrant loading, and diffusion in heterogeneous polymeric materials

Theory of Anisotropic Diffusion of Entangled and Unentangled Polymers in Rod–Sphere Mixtures

We present a microscopic self-consistent theory for the long-time diffusion of infinitely thin rods in a hard sphere matrix based on the simultaneous dynamical treatment of topological uncrossability and finite excluded volume constraints. Distinctive regimes of coupled anisotropic longitudinal and transverse diffusion are predicted, and steric blocking of the latter leads to a tube-like localization transition largely controlled by the ratio of the sphere diameter to rod length and tube diameter. For entangled polymers, in a limited regime of strongly retarded dynamics a “doubly renormalized” reptation law is predicted where the confinement tube is compressed and longitudinal motion is partially blocked. At high sphere volume fractions, strong suppression of rod motion results in dynamic localization in the unentangled regime. The present advance provides a theoretical foundation to treat differential mobility effects and flexible chain dynamics in diverse polymer–particle mixtures

Long Wavelength Thermal Density Fluctuations in Molecular and Polymer Glass-Forming Liquids: Experimental and Theoretical Analysis under Isobaric Conditions

We establish via an in-depth analysis of experimental data that the dimensionless compressibility (proportional to the dimensionless amplitude of long wavelength thermal density fluctuations) of one-component normal and supercooled liquids of chemically complex nonpolar and weakly polar molecules and polymers follows extremely well a surprisingly simple and general temperature dependence over an exceptionally wide range of pressures and temperatures. A theoretical basis for this behavior is shown to exist in the venerable van der Waals model and its more modern interpretations. Although associated hydrogen-bonding (and to a lesser degree strongly polar) liquids display modestly more complex behavior, rather simple temperature and pressure dependences are also discovered. A new approach to collapse the temperature- and pressure-dependent dimensionless compressibility data onto a master curve is formulated that differs from the empirical thermodynamic scaling approach. As a practical matter, we also find that the dimensionless compressibility scales well as an inverse power law with temperature with an exponent that is system dependent and decreases with pressure. At very high pressures and low temperatures, the thermal liquid behavior appears to approach (but not reach) a repulsion-dominated random close packing limit. All these findings are relevant to our recent theoretical work on the problem of activated relaxation and vitrification of supercooled molecular and polymeric liquids

Experimental Tests of a Theoretically Predicted Noncausal Correlation between Dynamics and Thermodynamics in Glass-forming Polymer Melts

The connection between slow activated relaxation in glass-forming liquids and various equilibrium thermodynamic properties remains intensely debated. The microscopic elastically collective nonlinear Langevin equation theory, a force-level approach that causally relates the structure and dynamics, describes the activated relaxation as a mixed local–nonlocal process involving local caging constraints coupled with longer-range collective elasticity. Rather surprisingly, we recently showed that this theory predicts a noncausal connection between dynamics and thermodynamics (via the dimensionless compressibility, S0, an equation-of-state property) for the hard-sphere fluid as a consequence of fundamental relations between local and long-wavelength density fluctuations in equilibrium statistical mechanics. The effective activation barrier is predicted to grow in a power law manner with the inverse S0 with an exponent of one in the high-temperature regime and three in the deeply supercooled regime. These predictions have been experimentally verified to hold well in both molecular and inorganic glass-forming liquids. Here, we show that this same basic S0-space behavior also describes segmental relaxation in the more chemically complex case of polymer melts under isobaric atmospheric- and high-pressure conditions. Linear master curves in S0-space are constructed based on a fragility-dependent crossover from local caging to collective elasticity as the primary origin of slow dynamics. Predicted implications in temperature space include a fragile-to-strong crossover as a function of polymer chemistry signaling the unimportance of collective elasticity effects, a power law scaling of the activation barrier with inverse temperature in the deeply supercooled regime (with a polymer-specific exponent determined entirely from thermodynamics), an alternative approach to collapse the temperature- and pressure-dependent dynamic relaxation data onto a master curve, and a new practical method to more accurately determine fragility

Theory of Transient Localization, Activated Dynamics, and a Macromolecular Glass Transition in Ring Polymer Liquids

We construct a segmental scale force level theory for the center-of-mass diffusion constant and corresponding relaxation time for globally compact unconcatenated ring polymer solutions and melts (degree of polymerization N). The approach is based on slowly decaying macromolecular scale intermolecular force dynamic correlations as the origin of their unusual dynamics. Unentangled Rouse, weakly caged, and activated regimes are predicted. The barrier of the activated regime scales linearly with N and as a power law of concentration, which drives a kinetic glass transition on the radius-of-gyration scale. The values of N at the two dynamic crossovers (Rouse to weakly caged, weakly caged to activated) are proportional, with nonuniversality entering mainly via macromolecular volume fraction and dimensionless compressibility. Quantitative comparisons with simulation data reveal good agreement. Aspects of intermediate time dynamics are analyzed, and predictions are made for the conditions required to observe a macromolecular glass transition in the laboratory and on the computer

Fragile Glass Formation and Non-Arrhenius Upturns in Ethylene Vitrimers Revealed by Dielectric Spectroscopy

Vitrimers, dynamic networks with bonds that exchange without breaking, are an emerging class of reprocessable and recyclable polymers. The dynamics in such materials are complex and span from a single bond exchange or alpha relaxation event up to bulk flow. Most prior work has focused on investigations of stress relaxation times or creep experiments, but little has been pursued to investigate more local dynamics over a wide range of temperatures. A series of precise ethylene vitrimers are synthesized with four to seven carbons between dynamic bonds, and broadband dielectric spectroscopy is used to probe the segmental dynamics. Three distinct modes are identified in the dielectric spectraan alpha process, beta process, and a normal mode assigned to strand motion in the network between dynamic bonds. The last mode corresponds within error to the rheological crossover time, indicating that this process is responsible for bulk flow at higher temperatures. At lower temperatures, approaching the glass transition causes a positive deviation of the crossover time from Arrhenius behavior in the networks at roughly the same distance above Tg. Finally, we analyze our networks in the context of a previously developed theory for bond dissociation in associating polymers and find evidence that the non-Arrhenius behavior reflects strong decoupling of the bond exchange barrier crossing event with the segmental or alpha relaxation. This implies the bond exchange event that conserves dynamic cross-link density experiences a local frictional resistance due to the surrounding polymer matrix that is smaller and much less temperature dependent than the primary structural relaxation process, and to a larger degree than observed in most associating copolymer melts where physical bond breaking is a dissociative process