Entropy Theory of Polymer Glass‐formation in Variable Spatial Dimension

Abstract

We explore the nature of glass-formation in variable spatial dimensionality (d) based on the generalized entropy theory, a synthesis of the Adam-Gibbs model with direct computation of the configurational entropy of polymer fluids using an established statistical mechanical model. We find that structural relaxation in the fluid state asymptotically becomes Arrhenius in the d → ∞ limit and that the fluid transforms upon sufficient cooling above a critical dimension near d = 8 into a dense amorphous state with a finite positive residual configurational entropy. Direct computations of the isothermal compressibility and thermal expansion coefficient, taken to be physical measures of packing frustration, demonstrate that these fluid properties strongly correlate with the fragility of glass-formation.PACS numbers: 64.70.QGrowing evidence indicates that molecular packing plays an essential role in both the physics of crystallization [1][2][3][4][5] and glass-formation [6][7][8][9][10][11][12][13][14][15][16][17], prompting a consideration of variable spatial dimensionality (d) as a conceptual probe of these solidification processes. In particular, the "decorrelation principle", recently proposed by Torquato and Stillinger [18,19], states that unconstrained correlations in hard hyperspheres diminish with increasing d and vanish in the d → ∞ limit. Likewise, recent simulations [7][8][9] have indicated that certain often cited aspects of glass-formation, such as the dynamic heterogeneity and the decoupling between structural relaxation and diffusion, become diminished at elevated d. While prior studies [11][12][13][14][15] have focused mainly on how critical packing fractions associated with glass-formation in hard hyperspheres vary with d, little is known concerning the d dependence of the characteristic temperatures and fragility of glass-formation in molecular fluids. No prior work has considered complex fluids, such as polymeric liquids, where many molecular parameters may be tuned to control the nature of glass-formation.In this Letter, we consider the glass-formation of a model polymer glass-forming (GF) liquid in d dimensions. Our approach is based on the generalized entropy theory (GET) [20], which is a combination of the lattice cluster theory (LCT) [21] and the Adam-Gibbs (AG) theory [22] linking the configurational entropy s c (i.e., the entropy devoid of the vibrational component) to the structural relaxation time τ α . The LCT employs a ddimensional hypercubic lattice model, whose use facilitates the development of an expansion about the meanfield limit of infinite d. The LCT thus naturally provides * Contribution of the National Institute of Standards and Technology -Work not subject to copyright in the United States. † wsxu@uchicago.edu ‡ jack.douglas@nist.gov § freed@uchicago.edu a framework for considering the d dependence of polymer thermodynamic properties, thereby making the GET suitable for exploring glass-formation in d dimensions.As a primary result, we find that structural relaxation in the fluid st...

Generalized entropy theory of glass-formation in fully flexible polymer melts

“…1. In contrast to the results for semiflexible polymer melts with 24 s r is found to be always positive for E b /k B = 0 K, a result having significant implications for the dynamics of glass-forming polymers from the viewpoint of entropy theory, 1,4 as further discussed in Sec. III.…”

Configurational entropy of glass-forming liquids

“…75 The entropy may or may not vanish, depending on the approximations used and the ingredients entering the model. 52,76 An entropy crisis is thus not always present within the Gibbs-DiMarzio line of thought, but one cannot draw general conclusions about the existence of an entropy crisis in supercooled liquids. Moreover, the distinction between individual configurations and freeenergy minima is generally not considered, which may be problematic.…”

Polymer-Solvent Phase Separation and Criticality in the Formation of Porous Polymeric Solids: A Modeling Study

“…The emergence of this heterogeneity is incompatible with the assumptions that underlie traditional self-consistent field theories, 22,23 yet it has important thermodynamic consequences that cannot be described with network percolation models. 18,24,25 Here, we present a model for describing polymer-solvent demixing starting from a finite-element representation of the fluctuating microscopic polymer density field. 26,27 In the vicinity of criticality this model can be iteratively renormalized to determine the mesoscale interactions that govern material structure and viscoelastic properties.…”

“…where n i specifies the state of binary lattice in the site that contains position r, and δφ(r) is the rapidly varying component of the local density field. Following the theoretical developments of Lum, Chandler, and Weeks, 25,[30][31][32][33][34] we assume that the long-range density field is governed by nearest-neighbor (Ising-like) interactions and the fluctuations in the short-range density field obey Gaussian statistics. The energy of a given configuration of the density field is expressed in terms of the Hamiltonian,…”

“…In the vicinity of a sol-gel critical point, physical quantities such as the cluster size, gel fraction, viscosity, and shear modulus have been observed to exhibit power-law scaling. 17,18,18,24,25,44 If polymer-solvent demixing and phase inversion bring about critical gelation, then divergent correlation lengths become manifest in the long-range density, excluded volume, and auxiliary fields. The scale invariant structure that is responsible for this scaling can be computed through the iterative renormalization of our model.…”