Generalized Entropy Theory Research Articles

Advances in the generalized entropy theory of polymer glass formation

Advances in the generalized entropy theory of polymer glass formation

Thermodynamic–Dynamic Interrelations in Glass-Forming Polymer Fluids

We utilize the lattice cluster theory (LCT) of polymer fluids to show that the configurational entropy, enthalpy, and internal energy are all closely interrelated, as suggested by recent measurements by Caruthers and Medvedev, so that the generalized entropy theory (GET) of glass formation, a combination of the LCT and Adam-Gibbs model, can be recast in terms of any of these thermodynamic properties as a matter of convenience. Thermodynamic scaling, a form of density-temperature scaling exhibited by dynamic and some thermodynamic properties, is used to assess which thermodynamic properties are most naturally linked to dynamics, and we explore the origin of this scaling by both direct calculations based on the GET and molecular dynamics simulations of a coarse-grained polymer model. Through a combination of our comprehensive modeling of thermodynamic properties using the LCT and the highly predictive GET model for how the fluid thermodynamics relate to its dynamics, along with simulation results confirming these theoretical frameworks, we obtain insights into thermodynamic aspects of collective motion and the slow $\beta$-relaxation processes of glass-forming liquids.

Deep glassy state dynamic data challenge glass models: Configurational entropy models

There is accumulating evidence that the dynamics of glass-forming systems below the conventional glass transition temperature Tg do not diverge at a finite temperature as conventional extrapolations of the temperature dependence of the dynamics suggest. Here we explore the possibility that theories often interpreted to support this extrapolation may be able to account for the experimentally observed apparent turn-over to a more physically acceptable divergence at zero Kelvin. We examine three different entropy-based glass transition models: the generalized entropy theory (GET), the random first order transition theory (RFOT), and the DiMarzio and Yang model. These are evaluated and compared with the non-diverging deep glassy state dynamic data reported for a 20-million-year-old amber and a vapor deposited ultra-stable amorphous Teflon obtained previously in our laboratories. We make different assumptions about the temperature dependences for configurational entropy for the model evaluations. We find that all configurational entropy models can give non-divergent dynamics with an appropriate configurational entropy and can, then, qualitatively agree with the deep glassy state dynamic data for the ancient amber and the amorphous Teflon. However, all the models still predict somewhat higher relaxation times than the experimentally reported values, which were upper bounds to the equilibrium response.

Equation of State and Entropy Theory Approach to Thermodynamic Scaling in Polymeric Glass-Forming Liquids

We show that thermodynamic scaling can be derived by combining the Murnaghan equation of state (EOS) with the generalized entropy theory (GET) of glass formation. In our theory, thermodynamic scaling arises in the non-Arrhenius relaxation regime as a scaling property of the fluid configurational entropy density $s_c$, normalized by its value $s_c^*$ at the onset temperature $T_A$ of glass formation, $s_c / s_c^*$, so that a constant value of $TV^{\gamma}$ corresponds to a \textit{reduced isoentropic} fluid condition. Molecular dynamics simulations on a coarse-grained polymer melt are utilized to confirm that the predicted thermodynamic scaling of $\tau_{\alpha}$ by the GET holds both above and below $T_A$ and to test whether the extent $L$ of stringlike collective motion, normalized its value $L_A$ at $T_A$, also obeys thermodynamic scaling, as required for consistency with thermodynamic scaling. While the predicted thermodynamic scaling of both $\tau_{\alpha}$ and $L/ L_A$ is confirmed by simulation, we find that the isothermal compressibility $\kappa_T$ and the long wavelength limit $S(0)$ of the static structure factor do not exhibit thermodynamic scaling, an observation that would appear to eliminate some proposed models of glass formation emphasizing fluid `structure' over configurational entropy. It is found, however, that by defining a low temperature hyperuniform reference state, we may define a compressibility relative to this condition, $\delta \kappa_T$, a transformed dimensionless variable that exhibits thermodynamic scaling and which can be directly related to $s_c / s_c^*$. Further, the Murnaghan EOS allows us to interpret $\gamma$ as a measure of intrinsic anharmonicity of intermolecular interactions that may be directly determined from the pressure derivative of the material bulk modulus.

Polymer Glass Formation: Role of Activation Free Energy, Configurational Entropy, and Collective Motion

We provide a perspective on polymer glass formation, with an emphasis on models in which the fluid entropy and collective particle motion dominate the theoretical description and data analysis. The entropy theory of glass formation has its origins in experimental observations relating to correlations between the fluid entropy and liquid dynamics going back nearly a century ago, and it has entered a new phase in recent years. We first discuss the dynamics of liquids in the high-temperature Arrhenius regime, where transition state theory is formally applicable. We then summarize the evolution of the entropy theory from a qualitative framework for organizing and interpreting temperature-dependent viscosity data by Kauzmann to the formulation of a hypothetical “ideal thermodynamic glass transition” by Gibbs and DiMarzio, followed by seminal measurements linking entropy and relaxation by Bestul and Chang and the Adam–Gibbs (AG) model of glass formation rationalizing the observations of Bestul and Chang. These developments laid the groundwork for the generalized entropy theory (GET), which merges an improved lattice model of polymer thermodynamics accounting for molecular structural details and enabling the analytic calculation of the configurational entropy with the AG model, giving rise to a highly predictive model of the segmental structural relaxation time of polymeric glass-forming liquids. The development of the GET has occurred in parallel with the string model of glass formation in which concrete realizations of the cooperatively rearranging regions are identified and quantified for a wide range of polymeric and other glass-forming materials. The string model has shown that many of the assumptions of AG are well supported by simulations, while others are certainly not, giving rise to an entropy theory of glass formation that is largely in accord with the GET. As the GET and string models continue to be refined, these models progressively grow into a more unified framework, and this Perspective reviews the present status of development of this promising approach to the dynamics of polymeric glass-forming liquids.

Understanding Activation Volume in Glass-Forming Polymer Melts via Generalized Entropy Theory

The generalized entropy theory (GET) of glass formation provides an analytic framework for the structural relaxation time and thermodynamic properties of glass-forming polymers in terms of molecula...

Energy Renormalization for Coarse‐Graining Polymers with Different Fragilities: Predictions from the Generalized Entropy Theory

Front Cover: Coarse-grained (CG) modeling is necessary to access large spatiotemporal scales for computational design of glass-forming polymers. The recently proposed “energy-renormalization” strategy is theoretically investigated by Wen Sheng Xu and Wenjie Xia in article 1900051 for polymers under varying degrees of coarse-graining, showing promise in achieving temperature-transferable CG modeling.

Energy Renormalization for Coarse‐Graining Polymers with Different Fragilities: Predictions from the Generalized Entropy Theory

AbstractBottom‐up prediction of physical performance of glass‐forming (GF) polymers via coarse‐grained (CG) modeling is challenging because these CG models normally experience significantly altered dynamics that strongly vary with temperature. Building upon the recently developed energy‐renormalization (ER) coarse‐graining method based on molecular dynamics simulations, generalized entropy theory (GET) is employed to theoretically investigate the influence of fundamental molecular parameters on CG modeling of polymers having different glass “fragilities” Taking a linear polymer melt as a model system within the GET framework, it is shown that the chain bending rigidity and cohesive interaction play critical roles in the glass formation of polymers and their CG analogs. To coarse‐grain polymers having a higher fragility index, it requires greater magnitudes of ER factor εCG to rescale the cohesive interaction strength under coarse‐graining and thus recover the atomistic relaxation dynamics over a wide temperature range. The GET further predicts that a higher degree of coarse‐graining generally requires greater magnitudes of εCG due to the influence of loss of configuration entropy sc on the dynamics. GET analyses herein theoretically demonstrate the efficacy of the ER method toward building a multiscale temperature transferable modeling framework for GF polymers, and confirm the importance of preserving sc in CG modeling of dynamics of soft materials.

Energy renormalization for coarse-graining polymers having different segmental structures.

Multiscale coarse-grained (CG) modeling of soft materials, such as polymers, is currently an art form because CG models normally have significantly altered dynamics and thermodynamic properties compared to their atomistic counterparts. We address this problem by exploiting concepts derived from the generalized entropy theory (GET), emphasizing the central role of configurational entropy s c in the dynamics of complex fluids. Our energy renormalization (ER) method involves varying the cohesive interaction strength in the CG models in such a way that dynamic properties related to s c are preserved. We test this ER method by applying it to coarse-graining polymer melts (i.e., polybutadiene, polystyrene, and polycarbonate), representing polymer materials having a relatively low, intermediate, and high degree of glass "fragility". We find that the ER method allows the dynamics of the atomistic polymer models to be faithfully described to a good approximation by CG models over a wide temperature range.

On Measuring the Characteristic Length of the Cooperative Molecular Dynamics in the Glass-Forming Liquids

An analysis of the methods for measuring the characteristic length of the cooperative molecular dynamics in glass-forming liquids is submitted. The equations and methods for measuring the size of the cooperatively rearranging regions in post Adam-Gibbs theories are compered, and it is proved that they give one and the same number, but with different interpretation for the relaxed molecular units. The interpretation that the rearranging units defined by Wunderlich are smallest structural units, named beads, is supported by the predicted “universal” value for the cooperatively rearranging range at the glass-transition temperature, as well as from a precise modeling for the relaxations by the generalized entropy theory. It has been found that the characteristic length estimated by random first order transition theory, as well as by the “four-point” dynamic susceptibility method, measured as the number of the bead’s diameters coincide with the length of the cooperatively rearranging region in Adam-Gibbs theory extended at the molecular level.

Autonomous robotic exploration using a utility function based on Rényi’s general theory of entropy

In this paper we present a novel information-theoretic utility function for selecting actions in a robot-based autonomous exploration task. The robot’s goal in an autonomous exploration task is to create a complete, high-quality map of an unknown environment as quickly as possible. This implicitly requires the robot to maintain an accurate estimate of its pose as it explores both unknown and previously observed terrain in order to correctly incorporate new information into the map. Our utility function simultaneously considers uncertainty in both the robot pose and the map in a novel way and is computed as the difference between the Shannon and the Renyi entropy of the current distribution over maps. Renyi’s entropy is a family of functions parameterized by a scalar, with Shannon’s entropy being the limit as this scalar approaches unity. We link the value of this scalar parameter to the predicted future uncertainty in the robot’s pose after taking an exploratory action. This effectively decreases the expected information gain of the action, with higher uncertainty in the robot’s pose leading to a smaller expected information gain. Our objective function allows the robot to automatically trade off between exploration and exploitation in a way that does not require manually tuning parameter values, a significant advantage over many competing methods that only use Shannon’s definition of entropy. We use simulated experiments to compare the performance of our proposed utility function to these state-of-the-art utility functions. We show that robots that use our proposed utility function generate maps with less uncertainty and fewer visible artifacts and that the robots have less uncertainty in their pose during exploration. Finally, we demonstrate that a real-world robot using our proposed utility function is able to successfully create a high-quality map of an indoor office environment.

Influence of Pressure on Glass Formation in a Simulated Polymer Melt

We investigate the thermodynamic and dynamic properties of a model unentangled glass-forming polymer melt over a wide range of pressures (P) and temperatures (T) using molecular dynamics simulation. We show that the T dependence of the reduced thermal expansion coefficient and isothermal compressibility becomes weaker for higher P. Based on the generalized entropy theory, these trends suggest that the isobaric fragility of glass formation decreases with increasing P, consistent with our analysis from the structural relaxation time (τα). Moreover, we confirm previous findings that the onset and end of the glass formation process in our model can be estimated solely based on the static structure factor. We then provide a detailed analysis for the T and P dependence of τα as well as other quantities characterizing the dynamic heterogeneity of glass-forming liquids. In particular, we discuss how T and P influence the average extent L of the stringlike cooperative motion of monomers in our model polymer melt, ...

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

The generalized entropy theory (GET) offers many insights into how molecular parameters influence polymer glass-formation. Given the fact that chain rigidity often plays a critical role in understanding the glass-formation of polymer materials, the GET was originally developed based on models of semiflexible chains. Consequently, all previous calculations within the GET considered polymers with some degree of chain rigidity. Motivated by unexpected results from computer simulations of fully flexible polymer melts concerning the dependence of thermodynamic and dynamic properties on the cohesive interaction strength (ϵ), the present paper employs the GET to explore the influence of ϵ on glass-formation in models of polymer melts with a vanishing bending rigidity, i.e., fully flexible polymer melts. In accord with simulations, the GET for fully flexible polymer melts predicts that basic dimensionless thermodynamic properties (such as the reduced thermal expansion coefficient and isothermal compressibility) are universal functions of the temperature scaled by ϵ in the regime of low pressures. Similar scaling behavior is also found for the configurational entropy density in the GET for fully flexible polymer melts. Moreover, we find that the characteristic temperatures of glass-formation increase linearly with ϵ and that the fragility is independent of ϵ in fully flexible polymer melts, predictions that are again consistent with simulations of glass-forming polymer melts composed of fully flexible chains. Beyond an explanation of these general trends observed in simulations, the GET for fully flexible polymer melts predicts the presence of a positive residual configurational entropy at low temperatures, indicating a return to Arrhenius relaxation in the low temperature glassy state.

Stringlike Cooperative Motion Explains the Influence of Pressure on Relaxation in a Model Glass-Forming Polymer Melt.

Numerous experiments reveal that the dynamics of glass-forming polymer melts are profoundly influenced by the application of pressure, but a fundamental microscopic understanding of these observations remains incomplete. We explore the structural relaxation of a model glass-forming polymer melt over a wide range of pressures (P) by molecular dynamics simulation. In accord with experiments for nonassociating polymer melts and the generalized entropy theory, we find that the P dependence of the structural relaxation time (τα) can be described by a pressure analog of the Vogel-Fulcher-Tammann equation and that the characteristic temperatures of glass formation increase with P, while the fragility decreases with P. Further, we demonstrate that τα for various P can quantitatively be described by the string model of glass formation, where the enthalpy and entropy of activation are found to be proportional, an effect that is expected to apply to polymeric materials under various applied fields.

Communication: The simplified generalized entropy theory of glass-formation in polymer melts.

While a wide range of non-trivial predictions of the generalized entropy theory (GET) of glass-formation in polymer melts agree with a large number of observed universal and non-universal properties of these glass-formers and even for the dependence of these properties on monomer molecular structure, the huge mathematical complexity of the theory precludes its extension to describe, for instance, the perplexing, complex behavior observed for technologically important polymer films with thickness below ∼100 nm and for which a fundamental molecular theory is lacking for the structural relaxation. The present communication describes a hugely simplified version of the theory, called the simplified generalized entropy theory (SGET) that provides one component necessary for devising a theory for the structural relaxation of thin polymer films and thereby supplements the first required ingredient, the recently developed Flory-Huggins level theory for the thermodynamic properties of thin polymer films, before the concluding third step of combining all the components into the SGET for thin polymer films. Comparisons between the predictions of the SGET and the full GET for the four characteristic temperatures of glass-formation provide good agreement for a highly non-trivial model system of polymer melts with chains of the structure of poly(n-α olefins) systems where the GET has produced good agreement with experiment. The comparisons consider values of the relative backbone and side group stiffnesses such that the glass transition temperature decreases as the amount of excess free volume diminishes, contrary to general expectations but in accord with observations for poly(n-alkyl methacrylates). Moreover, the SGET is sufficiently concise to enable its discussion in a standard course on statistical mechanics or polymer physics.

Generalized Entropy Theory of Glass Formation in Polymer Melts with Specific Interactions

Chemical structure has been long recognized to greatly influence polymer glass formation, but a general molecular theory that predicts how chemical structure determines the properties of glass-forming polymers has been slow to develop. While the generalized entropy theory (GET) explains the influence of various molecular details on polymer glass formation, the application of the GET has heretofore been limited to the use of the simplest polymer model in which all united atom groups within the monomers of a species interact with a common monomer averaged van der Waals energy. However, energetic heterogeneities are ubiquitous within the monomers of real polymers, and their implications for polymer glass formation remain to be investigated theoretically. This paper uses an extension of the GET to explore the influence of energetic heterogeneities within monomers upon the nature of polymer glass formation. This extension of the GET is achieved by combining the Adam–Gibbs theory relating the structural relaxation time to the configurational entropy with a recent significant extension of the lattice cluster theory for polymer melts with specific interactions, in particular, for melts where three distinct van der Waals interaction energies are required to describe the energetic heterogeneities within monomers. The present paper focuses on establishing general trends for the variation of characteristic properties of glass formation, such as the isobaric fragility parameter mP and the glass transition temperature Tg, with molecular details, such as the specific interactions and chain stiffness. Our computations confirm that the previously used model with monomer averaged interactions correctly captures general trends in the variation of mP and Tg with various molecular parameters. More importantly, adjustment of the energetic heterogeneities within monomers alone are shown to provide an efficient mechanism for tailoring the properties of glass-forming polymers. The variations of polymer properties along iso-fragility and iso-Tg lines are illustrated as important design tools for exhibiting the combined influence of specific interactions and chain stiffness.

The meaning of the "universal" WLF parameters of glass-forming polymer liquids.

Although the Williams-Landell-Ferry (WLF) equation for the segmental relaxation time τ(T) of glass-forming materials is one of the most commonly encountered relations in polymer physics, its molecular basis is not well understood. The WLF equation is often claimed to be equivalent to the Vogel-Fulcher-Tammann (VFT) equation, even though the WLF expression for τ(T) contains no explicit dependence on the fragility parameter D of the VFT equation, while the VFT equation lacks any explicit reference to the glass transition temperature Tg, the traditionally chosen reference temperature in the WLF equation. The observed approximate universality of the WLF parameters C1((g)) and C2((g)) implies that τ(T) depends only on T-Tg, a conclusion that seems difficult to reconcile with the VFT equation where the fragility parameter D largely governs the magnitude of τ(T). The current paper addresses these apparent inconsistencies by first evaluating the macroscopic WLF parameters C1((g)) and C2((g)) from the generalized entropy theory of glass-formation and then by determining the dependence of C1((g)) and C2((g)) on the microscopic molecular parameters (including the strength of the cohesive molecular interactions and the degree of chain stiffness) and on the molar mass of the polymer. Attention in these calculations is restricted to the temperature range (Tg < T < Tg + 100 K), where both the WLF and VFT equations apply.

Advances in the generalized entropy theory of glass-formation in polymer melts.

The generalized entropy theory (GET) of polymeric glass-forming liquids is reformulated into a computationally simpler and more natural formalism than the original version of this theory. The new theoretical framework greatly facilitates establishing essential trends in the dependence of the segmental relaxation time τ, fragility, characteristic temperatures of glass-formation, etc., on the combined influences of monomer molecular structure, chain rigidity, and cohesive interaction strength. Special attention is placed on the estimating the parameters of the phenomenological Vogel-Fulcher-Tammann relations for describing segmental relaxation in diverse liquids in the low temperature range of glass-formation, Tg > T > Tc (or Tg < T < Tg + 100 K), where Tg and Tc are, respectively, the glass transition temperature and the crossover temperature separating the high and low temperature regimes of glass-formation. Finally, we discuss how the molecular energetic interaction parameters of the GET can be estimated from experimental data. Illustrative calculations are performed for the stiffness factor σ and the cohesive energy density u as a first step in this direction.

Influence of Cohesive Energy and Chain Stiffness on Polymer Glass Formation

The generalized entropy theory is applied to assess the joint influence of the microscopic cohesive energy and chain stiffness on glass formation in polymer melts using a minimal model containing a single bending energy and a single (monomer averaged) nearest neighbor van der Waals energy. The analysis focuses on the combined impact of the microscopic cohesive energy and chain stiffness on the magnitudes of the isobaric fragility parameter mP and the glass transition temperature Tg. The computations imply that polymers with rigid structures and weak nearest neighbor interactions are the most fragile, while Tg becomes larger when the chains are stiffer and/or nearest neighbor interactions are stronger. Two simple fitting formulas summarize the computations describing the dependence of mP and Tg on the microscopic cohesive and bending energies. The consideration of the combined influence of the microscopic cohesive and bending energies leads to the identification of some important design concepts, such as iso-fragility and iso-Tg lines, where, for instance, iso-fragility lines are contours with constant mP but variable Tg. Several thermodynamic properties are found to remain invariant along the iso-fragility lines, while no special characteristics are detected along the iso-Tg lines. Our analysis supports the widely held view that fragility provides more fundamental insight for the description of glass formation than Tg.

Two glass transitions in miscible polymer blends?

In contrast to mixtures of two small molecule fluids, miscible binary polymer blends often exhibit two structural relaxation times and two glass transition temperatures. Qualitative explanations postulate phenomenological models of local concentration enhancements due to chain connectivity in ideal, fully miscible systems. We develop a quantitative theory that explains qualitative trends in the dynamics of real miscible polymer blends which are never ideal mixtures. The theory is a synthesis of the lattice cluster theory of blend thermodynamics, the generalized entropy theory for glass-formation in polymer materials, and the Kirkwood-Buff theory for concentration fluctuations in binary mixtures.