Entropy Theory of Polymer Glass-Formation in Variable Spatial Dimension
Jack F. Douglas, Wensheng Xu, Karl Freed
The importance of packing frustration is broadly appreciated to be an important aspect of glass formation. Recently, great interest has focused on probing the dependence on spatial dimension d as a theoretical tool for exploring essential aspects of glass-formation. We approach this problem based on the generalized entropy theory (GET) , a synthesis of the Adam-Gibbs model with direct computation of the configurational entropy of polymer fluids based on an established statistical mechanical model. This predictive model for the segmental relaxation time of polymeric glass-forming liquids is investigated in the range of dimensions in which a well-defined fluid state exists (i.e., d > 2) to determine how the nature of glass-formation becomes altered with increasing dimension. We find that the relaxation time in the fluid state asymptotically becomes Arrhenius in the limit of infinite d, and we also determine how various measures of fragility and the characteristic temperatures of glass-formation demarking the onset, middle and end of the glass transition vary with d. The ideal glass transition or Kauzmann temperature TK at which the configurational entropy extrapolates to zero in our model is found to not exist above a critical dimensionality, dc. A generalization of the Williams-Landel-Ferry expression for relaxation time is introduced based on the thermodynamically well-defined crossover temperature Tc that avoids the introduction of a glass transition temperature Tg and, steepness index, m, quantities whose fundamental meaning is uncertain even in d = 3. Our basic results compare well with simulations of model glass-forming liquids and many trends predicted by our theory are consistent with simulation observations. Our basic finding is rather obvious physically, namely, as the spatial dimension becomes large, fluctuation effects associated with packing fluctuations become diminished so that Arrhenius relaxation applies over the entire fluid range.
Journal of Chemical Physics
glass-formation, polymers, entropy theory of glass-formation, Arrhenius relaxation, spatial dimension d, fragility