On the Thermodynamics of Supercooled Glass-forming Polymeric Liquids

Nathaniel Jay Craig
Mira Loma High School, Sacramento, CA
2001 Intel STS Second Place Winner

(This is an abridged version of Nathaniel’s report.  The original report included Summary, Abstract, Introduction, Theoretical Model, Results, Discussion, Generalization of Model, Technological Relevance, Conclusion and Acknowledgments sections.)

Summary

A surprising empirical relationship indicates a fundamental relationship between equilibrium and dynamic properties in supercooled polymeric liquids. We explicitly relate these properties by deriving an expression for the fragility index (a dynamic property) in terms of entropy and heat capacity (macroscopic equilibrium properties). Furthermore, we resolve a long-standing weakness of the Adam-Gibbs Model, an entropy theory of supercooled polymeric liquids, by obtaining a new equation for the so-called critical entropy.

Introduction

Supercooled glass-forming polymeric liquids have attracted immense interest in recent years, primarily due to their implications regarding unusual dynamic and equilibrium properties of metastable states.

Under appropriate conditions of temperature and pressure, the various degrees of freedom available for motion become increasingly restricted as polymeric liquid is cooled below its freezing point. The Adam-Gibbs model (1965) describes the temperature dependence of the relaxation time t(T)  in terms of cooperative rearranging regions, defined as independently-rearranging subsystems within the macroscopic ensemble. In an isothermal-isobaric system, the fraction of subsystems permitting rearrangement, n / N, is equal to the ratio of the partition function belonging to the cooperative domain, D and the partition function of the entire macroscopic system, D.

Adam and Gibbs thereby succeeded in obtaining a characteristic equation for relaxation time as a function of temperature, namely t(T) = A exp for the critical size of the cooperative rearranging region sc*, change in chemical potential, Boltzmann constant k, and entropy S(T).

From the Adam-Gibbs expression for relaxation time we develop an explicit equation for fragility index m in terms of heat capacity and entropy:

The above equation constitutes the first significant result of this work. It is necessary, however, to observe that the above result is predicated upon certain questionable assumptions contained within the Adam-Gibbs model. Specifically, the Adam-Gibbs evaluation of Boltzmann statistics yields values for critical entropy on the order of = k ln 2.  Making use of the fundamental relationship between critical entropy and critical size of the cooperative rearranging region, z*=, the Adams-Gibbs assumption produces values of z*~ 1.

However, our intuitive and theoretical understanding of relaxation behavior expects values significantly higher than those provided within the framework of the Adam-Gibbs model; it is this weakness that has created considerable dispute in recent years. In order to address this problem, we identify a constraint upon parameters within the Williams-Landell-Ferry equation, in which:

for WLF coefficients a1(Ts) and a2(T). Manipulation of the obtained constraint produces a novel expression for critical size in which Sc(Tg) represents the configurational entropy around the glass-transition temperature, Tg is the glass-transition temperature, and T2 is a theoretical extrapolation of the Kauzmann temperature. This allows us to write the critical size z* as .  Herein lies the second significant result of this work.

Results and Discussion

We compare results from the above expressions with experimental data for polyvinyl acetate, polymethyl methacrylate, polyvinyl chloride, polystyrene, and polyisobutylene. Our calculations of the fragility index correspond well with both experimental data and ancillary theoretical calculations, offering strong indication of the model’s accuracy. We obtain detailed values for the critical size near the glass transition temperature for a variety of simple and complex liquids in order to affirm the consistency of our equations. Examining z* in terms of m(Tg), we discover that critical size is directly related to the level of intermolecular interactions experienced by the molecular liquid in question.

Conclusions

We have uncovered a deep connection between fragility and macroscopic equilibrium properties in supercooled polymeric liquids. Moreover, we explicitly resolved the long-standing weakness of the Adam-Gibbs configurational entropy model. Our work resolves ongoing debate on the issue and permits the independent determination of critical entropy and critical size based upon characteristic properties. An examination of the relationship between m(Tg) and z* lends us greater insight into the relationship between critical size and the scale of molecular interactions.

Though esoteric, this work may be of widespread significance. Due to their wide range of structural and chemical properties, polymer glasses may be tailored to satisfy the demands of myriad facets of our society and technology. The methodology described in this work is a first step towards understanding the dynamics and physical aging of biological glasses as well as certain polymer glasses important to the design of sensors and solid-state electrolytes. Our work may also be directly applied to elucidating the dynamics of relaxation of borosilicate molecules in a glassy polymer matrix. Such characterization is of central importance to the modeling of glasses utilized in the disposal of high-level nuclear waste.