Comment on "Local accumulation times for source, diffusion, and degradation models in two and three dimensions" [J. Chem. Phys. 138, 104121 (2013)]

In a recent paper, Gordon, Muratov, and Shvartsman studied a partial differential equation (PDE) model describing radially symmetric diffusion and degradation in two and three dimensions. They paid particular attention to the local accumulation time (LAT), also known in the literature as the mean action time, which is a spatially dependent timescale that can be used to provide an estimate of the time required for the transient solution to effectively reach steady state. They presented exact results for three-dimensional applications and gave approximate results for the two-dimensional analogue. Here we make two generalizations of Gordon, Muratov, and Shvartsman’s work: (i) we present an exact expression for the LAT in any dimension and (ii) we present an exact expression for the variance of the distribution. The variance provides useful information regarding the spread about the mean that is not captured by the LAT. We conclude by describing further extensions of the model that were not considered by Gordon,Muratov, and Shvartsman. We have found that exact expressions for the LAT can also be derived for these important extensions..

Molecular Origins of the Thermophysical Properties of Polymers and Modeling of Polymer Permeation by Large Molecules

The molecular origins of the phase transitions of polymers have not been completely understood. The molecular level understanding of polymer behavior is of great technological and scientific value. For example, the melt to glass transition of a polymer Tg is perhaps its most useful quantity describing it. A low Tg polymer will be a useful elastomer and a high Tg polymer will serve for structural purposes. Additionally, sub-glass relaxations are related to polymer aging. Based on a simple poly (ethylene) model, the intramolecular and intermolecular factors governing polymer melting, the glass transition and subglass transitions were investigated through a careful and systematic variation of the torsional potential as well as the cohesive energy of the polymer. The model polymers were studied using constant pressure canonical (Gibbs) dynamics of a system of four polymer chains with one hundred and fifty beads per chain. The advantage of varying systematically the torsional potential is that the morphology of the polymer is controlled, ranging from highly amorphous to highly crystalline, depending on the gauche-trans conformational energy differences. The effect of cohesive energy on the various transitions may also be studied by changing the Van der Waals well depth of each bead in the polymer chain. The first study presented in this chapter is a semi-crystalline case where the gauche energy was +1.14 kcal/mol more stable than the trans energy, and the trans-gauche barrier was 3.01 kcal/mol. A melting point, a glass transition and a tentatively assigned gamma relaxation were characterized. In a second study, the effect of the trans-gauche barrier on the phase transitions of a semi-crystalline polymer (gauche energy=+1.14 kcal/mol) was investigated. In a third study, the effect of the torsional barrier on the glass transition of amorphous polymers (gauche energy=trans energy) was investigated. In a fourth study, the effects of crystallinity on the phase transitions of polymers was investigated by varying the trans-gauche energy differences while maintaining the trans-gauche barrier constant at 4.03 kcal/mol. In the fifth and final study, the effect of the cohesive energy on the polymer phase transitions was investigated by changing the Lennard Jones well depth of each bead while maintaining the torsional potential fixed with a gauche energy of 1.14 kcal/mol relative to the trans energy, and a trans-gauche barrier of 4.03 kcal/mol. Based on these studies, new insights on the general thermo-physical properties of polymers were obtained. A summary of the molecular interpretations of the melting point, glass transition, and sub-glass transitions is provided at the conclusion of this study. Therefore, the strength of this study is its ability to produce numerous phase transitions within a single structural polymer model by a systematic variation of the intermolecular and intramolecular forcefield parameters. This allows an effective comparison of the thermodynamics, the kinetics and morphology of each of the polymer cases.</p