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

Wen-Sheng Xu<EMAIL_ADDRESS>State Key Laboratory of Polymer Physics and
Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of
Sciences, Changchun 130022, P. R. China  Jack F. Douglas
<EMAIL_ADDRESS>Materials Science and Engineering Division, National
Institute of Standards and Technology, Gaithersburg, Maryland 20899, United
States  Zhao-Yan Sun<EMAIL_ADDRESS>State Key Laboratory of Polymer Physics
and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of
Sciences, Changchun 130022, P. R. China

###### Abstract

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.

###### keywords:

American Chemical Society, LaTeX

IR,NMR,UV

##  1 Introduction

Glasses have been central to fabrication technologies since the dawn of
civilization, and these materials are becoming increasingly important in
modern technologies. Glasses are not only encountered as structural materials
in our environment, such as window panes, vessels, and plastic polymeric
materials, but they are also used in high technology applications as materials
for non-volatile electronic memory, 1 organic light emitting diodes and
organic electronics, 2, 3 commercial aircraft 4, molecular separations in
manufacturing and water desalination and purification, 5, 6, 7, 8, 9, 10 etc.
The physics of glass formation is also highly relevant to biological
substances whose properties and functions are dictated by their complex
molecular structure and intermolecular interactions. 11, 12, 13, 14

Polymeric materials have a general propensity to form glasses at low
temperatures ($T$) and exhibit glassy dynamics over a large $T$ range, even if
some of them ultimately form semicrystalline materials composed of a mixture
of crystalline and noncrystalline regions in their solid state. Since polymers
exhibit many of the same features in glass formation as other glass-forming
(GF) materials while at the same time the many complications associated with
crystallization that often arise in GF liquids composed of atomic mixtures 15
can be avoided, these materials have been widely used to investigate the
fundamental nature of glass formation for decades. On the practical side,
polymer glasses have the advantage of being light, ductile, easy to process,
and, of course, relatively inexpensive, giving rise to numerous applications
in daily life and emerging technologies. We are truly living in the ‘Polymer
Age’. 16 From our own standpoint, the rich chemistry of polymers allows for an
exploration of the large molecular parameter space of possible types of glass
formation, the same attribute that makes these materials so attractive for
applications in materials science. As a complement to our work emphasizing the
entropy theory of glass formation, 17, 18, 19, 20 and simulations for purposes
of validation and refinement of the present theoretical perspective on the
dynamics of GF liquids, we mention some excellent summaries of the
experimental observations on polymer glass formation. 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33

Experimental and computational studies over the past decades have established
the rich phenomenology of GF liquids with an emphasis on the approach to the
glass transition temperature $T_{\mathrm{g}}$ from above where the fluid is
still in equilibrium and where strong changes in the fluid dynamics still
occur. 34, 35, 36, 37, 38, 39 The most basic phenomenology is that the dynamic
properties of GF liquids, such as shear viscosity $\eta$ and structural
relaxation time $\tau_{\alpha}$, display a dramatic $T$ dependence upon
cooling toward $T_{\mathrm{g}}$. For example, the shear viscosity of a GF
material can alter by over $15$ orders of magnitude over a relatively narrow
$T$ range. Angell 40, 41 has introduced the concept of ‘fragility’ to quantify
the strength of the $T$ dependence of the dynamics of GF liquids. The dynamics
of ‘strong’ liquids, such as $\text{SiO}_{2}$, has a nearly Arrhenius $T$
dependence, and in contrast, ‘fragile’ liquids, such as o-terphenyl, exhibit
dynamic properties whose $T$ dependence is highly non-Arrhenius. For polymers,
$T_{\mathrm{g}}$ can be tuned roughly from $150$ K to $500$ K, and fragility
varies by nearly an order of magnitude from approximately $25$ to over $200$.
42, 43, 44

While models and theories are being constantly introduced for understanding
the origin and nature of the slowing down of the dynamics of GF liquids, some
ideas remain invariant for qualitatively understanding the basic phenomenology
of glass formation. It has been generally appreciated since the works of Simon
45 and Kauzmann 46 that the rapid increase in $\eta$ and $\tau_{\alpha}$ upon
cooling is accompanied by a drop in the fluid entropy. Much of the early
interesting literature attempting to define the glass transition and
understand the role of the fluid entropy in the glass transition is difficult
to access, but Schmelzer and Tropin 47 have provided an informative review of
these early studies of glass formation, along with a discussion of how these
early studies relate to more recent works. Kauzmann discussed the fate of
equilibrium cooled liquids based on the excess entropy, $S_{\text{exc}}\equiv
S_{\text{liq}}-S_{\text{xtal}}$, where $S_{\text{liq}}$ and $S_{\text{xtal}}$
are the entropies of a material in the liquid and crystalline states,
respectively. The drops in $S_{\text{exc}}$ upon lowering $T$ are so rapid
that, if $S_{\text{exc}}$ in the $T$ regime above $T_{\mathrm{g}}$ is formally
extrapolated to much lower $T$ where the material can no longer remain in
equilibrium, $S_{\text{exc}}$ would vanish at a finite characteristic
temperature called the ‘Kauzmann temperature’ $T_{K}$, below which
$S_{\text{exc}}$ would formally become negative. The inferred vanishing of
$S_{\text{exc}}$ is termed the ‘Kauzmann paradox’ or the ‘entropy crisis’, but
this situation may instead represent the result of an unwarranted
extrapolation, and thus have no physical meaning. Nonetheless, the reality of
the drop in the entropy upon lowering $T$ is an unequivocal feature of GF
liquids, and the general idea that there is some sort of ‘entropy crisis’
associated with intrinsic glass formation remains a prevalent model of the
physical nature of glass formation, an approach that we develop in our own
work. In more recent works, Martinez and Angell 48 have reviewed the
parallelism between the $T$ dependences of the viscosity and fluid entropy in
diverse GF liquids, and Angell 49 has emphasized the role of the fluid entropy
in understanding the fragility of GF liquids.

Another more recently identified feature that is often invoked to explain the
dynamics of GF liquids is the occurrence of dynamic heterogeneity or
collective motion of particles. 34, 35, 36, 37, 38, 39 A significant body of
evidence has indicated the presence of transient clusters of particles with
excessively high or low mobilities relative to simple Brownian particles. In
many cases, the specific nature of dynamic heterogeneity has often been rather
vague, and there are currently many uncertainties about how the various types
of dynamic heterogeneity that one can imagine or identify might be related to
the dynamics of GF liquids. This list seems to be essentially endless and ever
growing, and we believe that the time has come to cull this proliferation by
investigating which of these heterogeneity measures have a direct relation to
the overall dynamics of the fluids. Interestingly, both mobile and immobile
particles are found to exhibit the property of forming ‘fractal’ dynamic
polymeric clusters, regardless of the molecular bond connectivity of
individual molecules, and the temperature and structural characteristics of
these types of heterogeneity conform to well-known phenomenologies of self-
assembling systems. 50, 51, 52, 20 This finding gives us hope for an overall
generalized theoretical treatment of GF materials. It must be acknowledged at
the outset that the theory of glass formation is still in a model building
stage, however. The entropy theory is just one of a number of promising
theoretical frameworks for glass formation. For instance, the nonlinear
Langevin equation (NLE) theory 53, 54, 55 and the more recent elastically
collective nonlinear Langevin (ECNLE) equation theory 56, 57, 58, 59 of
Schweizer and coworkers emphasize the central role of the long wavelength
limit of the static structure factor, $S(0)$, in determining the dynamics of
GF liquids. The free volume model of the dynamics of GF liquids also remains
popular in the polymer science community. 60, 61 Schweizer and coworkers 62
have previously discussed many of these models, so we do not reproduce such a
discussion in this Perspective.

Here, we focus on models of glass formation where the entropy and collective
particle exchange motion are taken to be of primary importance in the
description of the dynamics of GF liquids, as postulated by Adam and Gibbs
(AG). 18 The present perspective on the nature of polymer glass formation thus
emphasizes the consequence of changes in the fluid entropy on the dynamics of
cooled liquids in the low $T$ regime where the $T$ dependences of relaxation
and diffusion are observed to be non-Arrhenius. We strictly avoid the term
‘supercooled’ since it is clear that not all GF liquids, and particularly not
all polymeric liquids, can crystallize, even in principle. 63, 64, 65 It is
then possible that the glass state of some materials at low $T$ is the true
thermodynamic state rather than just a metastable thermodynamic condition.

We first discuss transition state theory (TST) 66, 67, 68 describing the
dynamics of liquids in the high $T$ Arrhenius regime, which, in our view,
should be a natural starting point for any theory of glass formation based on
the idea of thermally activated transport. We then summarize the historical
development of the entropy theory of glass formation which has been ‘under
construction’ for almost a century, a development that started from
observations of correlations between the fluid entropy and heats of
vaporization and dynamics in the 1930s, to the organization and interpretation
of these data by Kauzmann 46 in 1948, and then the formulation of the Gibbs-
DiMarzio (GD) theory 17 in 1958 based on the then newly developed theory of
polymer thermodynamics formulated by Flory 69 in his classical work aimed
originally at understanding the crystallization and equation of state
properties of polymer materials. A precise fundamental link between dynamics
and thermodynamics was first discovered by Bestul and Chang 70 in 1964 by
comparing data for relaxation and excess entropy obtained from specific heat
measurements of molecular and polymer GF liquids, which, in turn, led to the
introduction of the AG model 18 in 1965 to rationalize these observations. The
AG model later gave rise to the current prevailing conceptual framework and
language in which the dynamics of GF liquids is characterized by fragility 40,
41 and other metrics related to the $T$ variation of the configurational
entropy $S_{c}$ associated with a complex energy landscape that gives rise to
the complex dynamics of cooled liquids. These developments led to the
generalized entropy theory (GET) 19 and the string model, 20 which are built
on concepts and methods developed in a vast number of previous experimental
studies, so these newer works are just part of a long historical development
for a description of liquid dynamics to increased precision and validation.
For completeness, we also provide a brief overview of other models of liquid
dynamics emphasizing the fluid entropy, including the random first-order
transition (RFOT) 71, 72, 73, 74 and the Rosenfeld and excess-entropy scaling
approaches 75, 76, 77 to relaxation and diffusion in fluids.

##  2 Transition State Theory

When cooling upon the approach to $T_{\mathrm{g}}$, the $T$ dependence of the
rapidly growing relaxation time of GF liquids is described in most
experimental studies by the Vogel-Fulcher-Tammann (VFT) equation, 78, 79, 80

$\tau_{\alpha}=\tau_{0}\exp\left(\frac{DT_{\mathrm{VFT}}}{T-T_{\mathrm{VFT}}}\right),$
(1)

where $\tau_{0}$ is a prefactor, $D$ is a ‘fragility parameter’ quantifying
the strength of the $T$ dependence of $\tau_{\alpha}$, and $T_{\mathrm{VFT}}$
is the temperature where $\tau_{\alpha}$ extrapolates to infinity. Although
the VFT equation is perhaps the best-known phenomenological relationship in GF
liquids, there have been attempts to challenge the supremacy of this equation.
81, 82, 83, 84, 85, 86, 87, 88, 89, 90 Nonetheless, most researchers persist
in fitting their relaxation time, viscosity, and diffusion data to this simple
equation because of its relative simplicity and apparent effectiveness in
fitting data for a vast range of materials. It should be mentioned, however,
that even some of the earliest quantitative works on relaxation in polymer
materials recognized that the VFT equation, and the mathematically equivalent
and heavily utilized expression of Williams, Landel and Ferry (WLF), 91 are
limited to a $T$ range between $T_{\mathrm{g}}$ and a temperature about $100$
K higher than $T_{\mathrm{g}}$, which corresponds to a temperature comparable
to the crossover temperature $T_{c}$ in our discussion of the GET model in
Section 4. It is notable that the VFT relation may be derived from the GET for
a prescribed $T$ range between $T_{c}$ and $T_{\mathrm{g}}$, along with an
analog of this equation when glass formation is achieved by increasing
pressure ($P$) at fixed $T$. Moreover, the VFT parameters $D$ and
$T_{\mathrm{VFT}}$, or their WLF equivalent, 92 may be calculated from the GET
so these are not just adjustable empirical parameters in the GET. The GET also
specifies an alternative functional form for the $T$ regime above $T_{c}$ but
below the onset temperature $T_{A}$, where relaxation and diffusion become
non-Arrhenius, again with all model parameters fixed in terms of molecular
parameters describing the thermodynamics of polymer fluids.

At sufficiently high $T$, a large body of evidence indicates that the
structural relaxation time $\tau_{\alpha}$ of GF liquids generally exhibits an
Arrhenius $T$ dependence,

$\displaystyle\tau_{\alpha}=A_{0}\exp\left(\frac{\Delta
H_{0}}{k_{B}T}\right),$ (2)

where $A_{0}$ is a prefactor, $k_{B}$ is Boltzmann’s constant, and $\Delta
H_{0}$ is the activation enthalpy. The prefactor $A_{0}$ in eq 2 involves a
vibrational attempt frequency $\tau_{0}$ multiplied by a factor involving the
activation entropy $\Delta S_{0}$, $\exp(-\Delta S_{0}/k_{B})$. We append an
‘$o$’ subscript to the activation energetic parameters to denote that these
quantities are determined in the high $T$ Arrhenius regime where relaxation
times are relatively short and the fluid is relatively ‘dynamically
homogeneous’.

Figure 1: Illustration of the temperature ($T$) dependence of the dynamics of
glass-forming liquids. The figure displays the logarithm of the segmental
structural relaxation time, $\log\tau_{\alpha}$, versus inverse temperature
$10^{3}/T$ for poly(dimethylsiloxane) with the molar mass of $21600$ g/mol at
ambient pressure ($P$). Experimental data were provided to us by the authors
of ref 88. The solid line is a fit to the Vogel-Fulcher-Tammann (VFT) equation
(eq 1). The dashed line is a fit to the Arrhenius equation (eq 2) with the
fitted parameters, $A_{0}=1.3\times 10^{-14}$ s and $\Delta H_{0}/k_{B}=2181$
K. Despite the dramatic increase of $\tau_{\alpha}$ with lowering $T$ upon
approaching the glass transition temperature $T_{\mathrm{g}}$, an Arrhenius
dynamics applies instead at sufficiently high $T$.

To highlight the presence of Arrhenius dynamics at high $T$, we show the
segmental structural relaxation time $\tau_{\alpha}$ of poly(dimethylsiloxane)
versus $10^{3}/T$ at ambient pressure measured by the Rössler group 88 in
Figure 1. Measurements of dynamics in the Arrhenius regime are generally
challenging experimentally due to the thermal degradation of polymers, but the
field-cycling 1H nuclear magnetic resonance relaxometry has recently changed
this situation, at least for some polymers. 86, 87, 88 It is thus well
established that the fluid dynamics is observed to be homogeneous in the high
$T$ regime where the Arrhenius equation formally holds, and the dynamic
heterogeneity sets in below the onset temperature $T_{A}$ for non-Arrhenius
relaxation where the dynamics deviates from the Arrhenius equation.

While the physical mechanisms for explaining the dynamics of fluids must be
different in the high $T$ Arrhenius and low $T$ non-Arrhenius regimes, it is
essential to build an experimentally validated theoretical foundation for
understanding and ultimately controlling the dynamics of cooled liquids, where
collective many-body effects are prevalent, based on a thorough understanding
of the dynamics in the high $T$ Arrhenius regime. In our view, this is a
natural starting point for any theory of glass formation based on the idea of
thermally activated transport. Molecular dynamics (MD) modeling becomes a
particularly attractive tool at present because of the relative small
magnitude of structural relaxation and relatively short equilibration times in
this regime. It should then be possible to make relatively rapid progress in
understanding the relation between molecular structure and other molecular
parameters and factors controlling structural relaxation and diffusion in the
high $T$ Arrhenius regime. We expect that this information should be useful
for understanding the dynamics of polymers, even though many polymer materials
would physically degrade before reaching such high $T$ at which the Arrhenius
equation describes the dynamics of the materials. These observations again
serve to underscore the fact that we need to be concerned about more than just
the glass transition temperature $T_{\mathrm{g}}$, since $T_{A}$ is often more
than twice $T_{\mathrm{g}}$ 19 so that glass formation encompasses the entire
$T$ regime in which polymer materials are normally utilized and processed and
thermally stable to chemical decomposition. This large $T$ range above
$T_{\mathrm{g}}$ thus has fundamental importance for processing and field use
applications of polymers, and for understanding the nature of glass formation,
regardless of whether the polymer materials crystallize or not at temperatures
lower than $T_{A}$.

Historically, TST of Eyring and others 66, 67, 68 has provided significant
advances for understanding the dynamics of liquids in the Arrhenius regime. If
viscous flow is viewed as a chemical reaction in which the elementary process
is the passing of a single molecule from one equilibrium position to anther
over a potential barrier, viscosity, plasticity, and diffusion of fluids can
all be treated as examples of absolute reaction rates within TST. 93, 94 The
central quantity in TST is the activation free energy $\Delta G_{0}$,

$\Delta G_{0}=\Delta H_{0}-T\Delta S_{0},$ (3)

which contains both enthalpic and entropic contributions. The TST-based
approach has proven to be a reliable general theoretical framework for
describing the dynamic properties of condensed fluids at elevated $T$, 95 even
though the actual development of Eyring’s absolute rate theory 67, 68, 94 for
the dynamics of real liquids is based on the rather idealized view of the
dynamics of real fluids that enables transforming the problem of explaining
gas phase dynamics into a model of the dynamics of condensed liquids in terms
of assumptions more acceptable for gases than for pure fluids. A rigorous TST
has been developed for idealized crystalline materials, 96 in which the
particles interact with purely harmonic interactions. Viewing a liquid as a
highly defective crystal has provided an alternative model of the Arrhenius
dynamics of liquids. 97 This alternative view of structural relaxation and
diffusion in condensed materials emphasizes the collective nature of transport
in the condensed state, but there is still no satisfying analytic theory of
dynamics, even in the Arrhenius regime where TST is expected to apply.
Numerical implementations of TST in ordered condensed materials have shown
that the TST framework still applies even though it currently does not allow
any direct analytic treatment. 98, 99, 100 We thus adopt TST as a working
framework for understanding the dynamics of cooled liquids. Notably, the AG
theory 18 implicitly adopts the TST framework in its formulation, and thus,
this assumption extends to the GET, 19 which combines the AG relation with a
thermodynamic theory that enables calculating the configurational entropy of
polymer fluids. While the TST-based approach to glass formation does not
require the many simplifying assumptions of Eyring, 66, 67, 68 its practical
implementation requires MD simulations 101 to estimate the activation free
energy parameters. Thus, the treatment of the dynamics of condensed fluids
remains largely semi-empirical and requires simulation studies or experimental
measurements to determine parameters characterizing the free energy of
activation for fluids with very different chemical natures. Over the years,
correlations between the energetic parameters of activation and thermodynamic
properties have been observed, such as a strong correlation between $\Delta
H_{0}$ and the heat of vaporization, 66, 102 and changes in $\Delta H_{0}$ and
$\Delta S_{0}$ have been found to occur in a parallel fashion when changing
molecular mass, 103, 104, 105, 106, 67, 101 film thickness, 107, 108 or
additive concentration, 109, 110, 111, 112, 113 a phenomenon termed the
‘entropy-enthalpy compensation’ (EEC) effect. 114, 111, 115, 116 These
correlations provide some sign posts indicating what properties control the
basic energetic parameters, $\Delta H_{0}$ and $\Delta S_{0}$, which are
attractive targets for future simulation and theoretical modeling efforts.

We have observed the EEC effect to be a significant factor in simulations of
polymer systems 107, 108, 117, 118, 119, 120, 121, 122 and experiments of
polymer nanocomposites and thin films. 117, 123, 124 Notably, the magnitude of
the EEC effect on relaxation and diffusion can be rather significant, even
meriting the term ‘astounding’. For example, the prefactor of eq 2 can change
by a factor of about $10^{47}$ as the activation energy grows upon approaching
the melting temperature from below. Extremely large changes in the prefactor
are also encountered through this mechanism in the diffusion coefficient of
metallic glasses; e.g., see Figure 13 in ref 125. Calculations based on TST
have provided insight into these astronomical changes of the prefactor in the
context of surface dislocation nucleation of metallic materials. 126 Numerical
TST calculations have also given some insight into the EEC phenomenon in the
interfacial dynamics of crystalline materials. 98, 99, 100 An implication of
these results is that the neglect of the activation entropy $\Delta S_{0}$,
which is a basic assumption in the classical AG theory 18 and the more recent
GET, 19 can lead to serious errors in the TST description of the dynamics of
GF liquids. This is perhaps the most serious assumption in the GET 19 that we
must address in the future to improve this model. This point is further
discussed in Section 4.

##  3 Classical Entropy Theories of Glass Formation

In this section, we introduce the classical entropy theories of glass
formation, namely, the theories of GD 17 and AG. 18 We present main ideas
behind these theories and discuss their strengths and weaknesses. The current
section augments a previous discussion given in ref 19, so our discussion is
relatively brief.

###  3.1 Gibbs-DiMarzio Theory

In 1958, GD 17 developed a systematic statistical mechanical theory of polymer
glass formation based on the lattice model of semiflexible polymers that had
been developed earlier by Flory 69 for the description of polymer
crystallization. Since the entropy and other thermodynamic properties are
purely configurational in the lattice model, the term ‘configurational
entropy’ has been widely adopted in the literature. The GD theory provided the
first theoretical underpinning for how estimates of the fluid entropy might be
utilized to make predictions of relevance for understanding the dynamics of GF
liquids by identifying an ‘ideal glass transition’ with a thermodynamic event,
the vanishing of the configurational entropy $S_{c}$. This result was a direct
theoretical ‘echo’ of the Kauzmann’s concept of an entropy catastrophe
possibly underlying glass formation. 46 In addition to providing a
conceptually clear picture of the origin of glass formation upon cooling at
some point where the system has ‘no place to go’ dynamically and thus becomes
arrested, the GD theory allowed for quantitative predictions for the glass
transition temperature $T_{\mathrm{g}}$, which was heuristically identified by
GD as being a non-equilibrium analog of the true ideal glass transition
temperature $T_{0}$ that could not be reached because of the extreme slowing
of the dynamics upon approaching this temperature, but which ‘tracks’ $T_{0}$.
DiMarizo and Yang 127 have reviewed the many successes of the GD theory, which
is commonly invoked in experimental studies to the present time.

In a more recent work, 127 DiMarzio revised his interpretation of the model
prediction of a vanishing of the configuraional entropy in light of
information that later became available. In particular, DiMarzio 127 suggested
that $S_{c}$ became ‘critically small’ at the glass transition, rather than
actually vanishing, so that the spirit of the GD theory is preserved. Corsi
and Gudrati 128 have reassessed the initial negative view of the GD theory by
formulating a theory of semiflexible polymers on Husimi lattices to allow
calculations without approximation and their work has provided an expanded
view of the original work by GD 17 that confirms the essential idea of an
entropy crisis in this type of thermodynamic model. We further discuss the
technical revision of the GD picture and its ramifications for glass formation
based on the GET 19 below in Section 4.

The central quantity in the GD model is the configurational entropy $S_{c}$,
defined as the total entropy with vibrational contributions being excluded. In
physical terms, the configurational entropy is related to the number of
distinct configurational states of the fluid. Unfortunately, the conceptual
clarity of the notion of configurational entropy remains a question even in
current studies of glass formation, posing difficulties and challenges for
determining this quantity both experimentally and computationally. Berthier
and coworkers 129 have recently reviewed the methods of estimating $S_{c}$
from a computational perspective. In particular, a new Monte Carlo swap
algorithm has been developed to estimate $S_{c}$ at extremely low $T$, 130,
131 and these simulation results have been interpreted as suggesting the
existence of a Kauzmann temperature in the particular model fluids studied.
132, 133 Unfortunately, this novel simulation method seems to be only
applicable to fluids of spherical particles having a high size polydispersity,
a property that is not encountered in real atomic or molecular materials. It
seems unlikely to us that the swap algorithm can be extended to polymeric and
molecular GF liquids generally. There is still no proof that $S_{c}$ actually
vanishes at a finite $T$. Personally, we are skeptical about the existence of
a finite Kauzmann temperature in real molecular fluids, even though the GET
itself predicts such a temperature rather generally in semiflexible polymer
melts, as we discuss further in Section 4.

In addition, strong criticism has been often raised against fundamental tenets
of the GD theory concerning the identification of a vanishing of $S_{c}$ with
a second-order phase transition. Even more seriously, the vanishing of $S_{c}$
has been suggested to be an artifact of the inaccuracy of the mean-field
calculations of $S_{c}$ for dense polymer fluids based on the lattice model.
Gujrati and Goldstein 134 were the first to demonstrate that the GD model
violated rigorous bounds on the entropy of polymer melts, clearly bringing the
validity of the GD model into question, despite its impressive empirical
success in identifying essential trends between the experimental
$T_{\mathrm{g}}$ and the theoretical ‘ideal glass transition temperature’
$T_{0}$ calculated from the lattice model. Binder and coworkers 135 clarified
the situation somewhat based on simulations of a bond fluctuation lattice
model of flexible polymers, which suggest that $S_{c}$ approaches a constant
positive value at low $T$, but the validity of these results has remained a
question because of the inherently difficulty of estimating thermodynamic
properties reliably from simulations at such low $T$.

The original GD theory of glass formation in polymer liquids involves other
assumptions that significantly limit the predictive capacity of this
pioneering model of polymer glass formation. First, the theory is preoccupied
with the general philosophical problem of locating and explaining the
existence of an ‘ideal glass transition temperature’, $T_{0}$, which, as we
have discussed above, probably does not really exist. Unfortunately, a fluid
cannot remain in equilibrium near $T_{0}$ because the structural relaxation
time becomes astronomical in magnitude near this temperature so this issue is
a bit academic. A more recent work suggests that the vanishing of $S_{c}$ at a
finite temperature $T_{0}$ in the lattice model is just an artifact of the
high $T$ expansion involved in the lattice model calculations for the free
energy. 19 Explicit calculations of $S_{c}$ for flexible polymer melts show
that $S_{c}$ does not vanish at any finite positive $T$ when the high $T$
expansion associated with the treatment of chain stiffness is avoided. 136
When combined with the AG model, 18 this finding also implies that no
divergence generally exists in the AG theory and its extensions such as the
GET. 19 Second, we also emphasize that the GD model is based on a highly
simplified description of polymer chains as semiflexible self-avoiding walks
interacting with uniform short-range interactions and composed of
structureless monomer units. This extremely coarse-grained model of polymer
fluids does not allow one to consider how monomer structure and chain topology
affect polymer glass formation. Moreover, there is more to glass formation
than knowing $T_{\mathrm{g}}$. Large complex changes in the dynamics of GF
liquids initiate at $T$ well above $T_{\mathrm{g}}$, but below $T_{A}$ for
non-Arrhenius dynamics. Thus, we really need a theoretical framework that can
predict the breadth of the transition temperature range by estimating
temperatures characterizing the beginning, middle, and end of this broad
transition phenomenon having both well-defined dynamic and thermodynamic
signatures. Any discussion of this kind requires something like the AG model
18 to interrelate the structural relaxation time to $S_{c}$ or some other
thermodynamic property. The original GD theory describes glass formation from
a purely thermodynamic perspective. It is the ‘marriage’ of the powerful
lattice cluster theory (LCT) 137, 138 extension of the lattice model of type
studied by GD to treat polymer melts having general monomer structure and the
AG model 18 formally linking the melt thermodynamics to the segmental
structural relaxation time. The resulting GET 19 is a highly predictive model
of the segmental dynamics of polymeric liquids over the entire temperature and
pressure regimes of glass formation. Of course, the question remains of
whether this framework is true to the physics of real and simulated materials.
This is why validation by simulation and measurement is so important in
further developments of the entropy theory of glass formation.

###  3.2 Adam-Gibbs Theory

Although a large body of evidence indicates that the Arrhenius equation
describes the $T$ dependence of relaxation and diffusion in liquids at
elevated $T$, 66, 67, 68 this description ceases to apply as liquids are
cooled to low $T$ (e.g., see Figure 1). The term ‘fragility’ has been
introduced to quantify the degree to which the dynamics deviates from the
Arrhenius behavior. 40, 41 The structural relaxation time provides an
important quantity whose change in magnitude can be as large as $10^{15}$
between the high $T$ Arrhenius regime and the low $T$ non-Arrhenius regime
where the material progressively acquires the rheological properties of a
solid as $T_{\mathrm{g}}$ is approached. Of course, the apparently universal
and practical nature of this phenomenon has naturally prompted many
speculations as to its cause.

Early experimental studies attempting to rationalize the non-Arrhenius $T$
dependence of the shear viscosity and other fluid transport properties
emphasized that large changes in the dynamics correlate with corresponding
changes in the fluid entropy, 45, 46 and this correlation led some influential
scientists to infer that the thermodynamic changes originate from the
emergence of collective motion within cooled liquids. 46 Following up on an
earlier suggestion of Goldstein 139 that some other thermodynamic property
than density, such as entropy and enthalpy, might be a better ‘control
parameter’ for glass formation, Bestul and Chang 70 took a first step forward
to transform this loose connection of qualitative ideas and observations into
a quantitative relationship between the entropy and the rate of relaxation by
noticing a direct relationship between the excess entropy $S_{\mathrm{exc}}$
and the segmental structural relaxation time $\tau_{\alpha}$. Note that Bestul
and Chang 70 defined an excess entropy by taking the solid reference state to
be the glass state, while some authors later chose the crystalline state as
the reference state, 140, 141 in line with the definition of Kauzmann. 46 This
choice is contingent on what form of the solid exists for a given material,
since some materials have no available, or even perhaps existing, crystalline
form. Regardless of how $S_{\mathrm{exc}}$ is defined, this quantity only
provides a rather rough estimate of $S_{c}$, which is unfortunately not a
directly observable property. It should also be noted that there is another
excess entropy defined as the difference of entropies between the liquid and
gas states, which we designate as $\widehat{S}_{\mathrm{ex}}$ to avoid
confusion with other definitions of the excess entropy discussed above.
$\widehat{S}_{\mathrm{ex}}$ has attracted significant interest in relation to
the dynamics of GF liquids, and Dyre 77 has recently provided a comprehensive
review on this topic. We discuss this alternative theory of the dynamics of
liquids built around $\widehat{S}_{\mathrm{ex}}$ in Section 6.2.

Bestul and Chang 70 also introduced a practical method of estimating $S_{c}$
from $S_{\mathrm{exc}}$ from specific heat measurements, in addition to
performing such measurements at high resolution. They clearly made many
notable contributions to the emergence of a quantitative entropy theory of
glass formation, and, accordingly, we believe that their pioneering
contribution to understanding of the dynamics of GF liquids deserves greater
recognition.

The seminal observations of Bestul and Chang 70 were later rationalized by AG
18 with formal arguments that remain highly influential to the present. In the
AG model, 18 the relaxation of the fluids is assumed to be described by an
extension of TST 66, 67, 68 with a $T$-dependent barrier height $\Delta G$,

$\tau_{\alpha}=\tau_{0}\exp\left(\frac{\Delta G}{k_{B}T}\right),$ (4)

where $\tau_{0}$ is the high $T$ limit of $\tau_{\alpha}$. This is the first
postulate of the AG model. The central idea expressed by this extension of TST
to describe GF liquids, which was widely accepted as a general formulation of
liquid dynamics at the time, is that the dynamic slowing down of cooled
liquids has its origin in the presence of dynamic clusters, which AG termed
the ‘cooperatively rearranging regions’ (CRR) to embody the prevailing idea of
why simple TST ‘breaks down’ at low $T$. AG were rather vague about the form
of these clusters and about how they might be identified in practice, but they
postulated that these hypothetical clusters should have properties consistent
with experimental observations. In particular, AG made a second postulate that
the average number of particles in these CRR should be proportional to the
activation free energy $\Delta G$. The same hypothesis was introduced earlier
by Mott 142; see the paper of Nachtreib and Handler 143 for a discussion of
this model of the $T$ dependence of the activation energy and its application
to understand the strongly non-Arrhenius $T$ dependence of the measured
diffusion coefficient of white phosphorous. It is notable that Nachtreib and
Handler 143 also invoked EEC with the melting point being the compensation
temperature in a heated crystalline material. Finally, AG assumed that the
average number of particles in these hypothetical CRR should be inversely
proportional to the fluid configurational entropy $S_{c}$, thereby tying the
concepts of growing collective motion in cooled liquids to the observed
decreasing fluid entropy, whereupon they adopted an estimate of $S_{c}$
introduced by Bestul and Chang 70 to finish their model of the dynamics of
glass formation.

We may translate the AG arguments into a more formal symbolic representation.
AG first posited that the activation free energy $\Delta G_{0}$ of TST at high
$T$ should be multiplied by a factor $z$, corresponding to the number of
molecules or segments participating in the abstract CRR. AG then further
assumed that the size of the CRR should scale inversely with $S_{c}$,
$z=S_{c}^{*}/S_{c}$, where $S_{c}^{*}$ is the high $T$ limit of $S_{c}$, a
quantity that is also assumed to exist by AG. 18 In other words, AG basically
postulated that the activation free energy $\Delta G$ is a $T$-dependent
quantity,

$\Delta G(T)=z(T)\Delta G_{0}=[S_{c}^{*}/S_{c}(T)]\Delta G_{0},$ (5)

which reduces to standard TST at high $T$ where $\Delta G(T)$ reduces to
$\Delta G_{0}$. AG were silent about the characteristic temperature or the $T$
range at which the change to non-cooperative liquid dynamics should occur,
however. This physically attractive picture of glass formation then indicates
that collective motion simply renormalizes the activation energy by a factor
equal to the number of particles involved in the collective particle exchange
motion, $z$, which basically rationalized the empirical relation of Bestul and
Chang. 70

It is evident that the AG model is clearly more a series of hypotheses than a
real theory, but this model has provided a highly influential general
conceptual framework for understanding the dynamics of GF liquids, despite its
apparent theoretical weaknesses. The AG model has entered a new phase in
recent years as MD simulations enable estimates of collective motion and
calculations of the thermodynamic properties of polymeric and other GF
liquids, so we can now test this model quantitatively and extend the model as
necessary. It is truly remarkable how much of the AG theory has survived in
the more modern and validated description of GF liquids, which is a testament
to the deep physical understanding that AG had of the essential nature of GF
liquids. In particular, the AG theory has held up remarkably well over the
last 50 years in comparison with numerous experiments 144, 49, 140, 145 and
simulations 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158,
159, 160, 161, 162, 163 of diverse GF liquids. Efforts have also been made to
place the AG model on a sounder theoretical foundation by extending TST to
account for barrier crossing processes that involve many particles, 164, 165,
73, 166 basically formalizing the heuristic ideas introduced long before Mott
and AG. The development of TST by Bonzel 167 in the context of understanding
the non-Arrhenius dynamics of crystalline materials is particularly a detailed
reformulation of TST to treat this type of problem in an analytic framework,
and the theoretical framework developed by Freed 166 shares many mathematical
features, but this theory is not formulated in a chemically specific way, so
we view this theory as a promising work in progress rather than a complete
theory. The RFOT theory 71, 72, 73, 74 also offers great potential for
developing a sounder foundation for the AG model, as we discuss in Section
6.1.

While the AG theory 18 represents an important step in our understanding of
the dynamics of GF liquids, in the sense that it provides a rationale for a
quantitative link between $\tau_{\alpha}$ and $S_{c}$, there are several
drawbacks of the AG model that limit its predictive capacity. First, no effort
was made by AG to calculate $S_{c}$ from theoretical considerations.
Originally, AG 18 argued for approximating $S_{c}$ by the excess entropy
$S_{\mathrm{exc}}$ to make contact with the VFT equation 78, 79, 80 and to
rationalize the correlation of Bestul and Chang, 70 while Martinez and Angell
48 emphasized the use of $S_{\mathrm{exc}}$ as a thermodynamic predictor of
the dynamics of GF liquids in the spirit of the AG model. Second, the
combination of the AG relation with a thermodynamic model, such as the GD
model, 17 was never made, so there have been no previous predictions regarding
the influence of molecular structure and interactions on polymer glass
formation, a problem of both fundamental and practical importance. Thus, the
AG theory has been essentially used as a framework for understanding
qualitative features of glass formation, such as the origin of the VFT
equation 78, 79, 80 and the origin of fragility, 40, 41 a measure of the
degree of non-Arrhenius relaxation. The AG model in its original form has
limited value in relation to the design of new materials and prior predictions
of the properties of materials in terms of given molecular parameters.
Moreover, AG offered no quantitative picture of the structure and
polydispersity of the hypothetical CRR so that the original AG theory has some
serious shortcomings. Finally, as we have mentioned in Section 2, AG made the
completely unwarranted assumption of neglecting the activation entropy $\Delta
S_{0}$, presumably for reasons of mathematical expediency. It is known from
even standard TST that $\Delta G_{0}$ contains an entropic contribution that
can sometimes be quite large. 103 This is just one of the things that must be
fixed in any extended model of TST that incorporates the ideas of AG for the
purpose of making quantitative comparisons with relaxation data obtained from
real GF materials.

We again emphasize that despite the above problems in the classical theories
of GD 17 and AG, 18 these models of glass formation have been quite
influential in understanding the nature of glass formation and general trends
of $T_{\mathrm{g}}$ with molecular structure. In particular, these classical
theories 17, 18 have set the stage for the development of new models probing
the origin of the slowing down of the dynamics of GF liquids from a molecular
perspective. In the following, we discuss the development of the entropy
theory along two lines of research. The first line of research merges the AG
relation with explicit computations of the configurational entropy based on a
statistical mechanical model of polymers, yielding the GET of segmental
relaxation in polymeric GF liquids, 19 which we discuss in Section 4. The
second thrust of our work is built on the platform of MD simulations and tools
drawn from liquid state theory to quantify collective motion and calculate
structural relaxation time, along with other dynamical and thermodynamic
properties. The identification of well-defined cooperative motion in cooled
liquids that appears to conform exactly to the main hypotheses of AG 18 and
the finding that the geometrical properties of these clusters accord with
equilibrium polymerization 50, 51, 52, 20 led to the development of the highly
quantitative string model of glass formation, 20 as discussed in Section 5.
The development of these more recent works has continuously addressed the
shortcomings of the classical GD 17 and AG 18 models. Again we underscore that
like all extant theories of glass formation, the entropy theory of glass
formation remains in an exploratory stage so that some ‘loose ends’ are to be
expected. As a complement to our works on the GET and the string model, we
also discuss the RFOT 71, 72, 73, 74 and the Rosenfeld and excess-entropy
scaling approaches 75, 76, 77 to liquid dynamics since these models are
likewise based on the fluid entropy, and may thus be characterized as
alternative entropy theories of glass formation.

##  4 Generalized Entropy Theory (GET) of Polymer Glass Formation

###  4.1 GET Essentials

The GET 19 takes advantage of the LCT 137, 138 for the thermodynamics of
polymer systems, which significantly extends the models of Flory 69 and GD 17
in the sense that it includes explicitly the description of monomer structure.
The LCT represents individual monomers in terms of a set of united-atom groups
with each occupying a single lattice site of volume $V_{\text{cell}}$ and
enables thermodynamic descriptions of the basic molecular characteristics of
polymers, including excluded volume interactions, chain connectivity, cohesive
interaction strength, chain stiffness, and monomer molecular structure. For
simplicity, we discuss the LCT for single-component polymer melts with the
reminder that this theory enables the description of multi-component polymer
systems, 137 such as polymer blends. Extensions of the GET has been made for
the glass formation of polymers with additives 168 and polymer blends 169.

For a compressible polymer melt, the LCT yields an analytic expression for the
Helmholtz free energy $f$ per lattice site in the following general form, 137,
138

$\beta f=\beta f^{mf}-\sum_{i=1}^{6}C_{i}\phi^{i},$ (6)

where $\beta=(k_{B}T)^{-1}$ and $\phi$ is the lattice ‘filling fraction’
defined by the ratio of the total number of united-atom groups to the total
number $N_{l}$ of lattice sites. The term $\beta f^{mf}$ corresponds to a
contribution from the zeroth-order mean-field approximation,

$\displaystyle\beta
f^{mf}=\frac{\phi}{M}\ln\left(\frac{2\phi}{z^{L}M}\right)+\phi\left(1-\frac{1}{M}\right)+(1-\phi)\ln(1-\phi)-\phi\frac{N_{2i}}{M}\ln(z_{b}),$
(7)

where $M$ is the molecular mass corresponding to the total number of united-
atom groups in a single chain, $z$ represents the lattice coordination number,
which is related to the spatial dimension via $z=2d$ for a $d$-dimensional
hypercubic lattice, $L$ is the number of subchains, $N_{2i}$ is the number of
sequential bond pairs lying along the identical subchains,
$z_{b}=(z/2-1)\exp(-\beta E_{b})+1$ with $E_{b}$ being the bending energy
parameter. The second term on the right-hand side of eq 6 represents
corrections to the zeroth-order mean-field contribution, where the
coefficients $C_{i}$ $(i=1,...,6)$ are obtained by collecting terms
corresponding to a given power of $\phi$,

$C_{i}=C_{i,0}+C_{i,\epsilon}(\beta\epsilon)+C_{i,\epsilon^{2}}(\beta\epsilon)^{2}.$
(8)

Here, $\epsilon$ is the microscopic cohesive energy parameter describing the
net attractive van der Waals interactions between nearest-neighbor united-atom
groups. $C_{i,0}$, $C_{i,\epsilon}$, and $C_{i,\epsilon^{2}}$ are tabulated in
the Appendix A of ref 138 and are generally functions of $z$, $T$, $E_{b}$,
$M$, as well as a set of geometrical indices that reflect the size, shape, and
bonding patterns of monomers. The free energy enables computations of all
thermodynamic properties.

Figure 2: Illustration of the $T$ dependence of the two different definitions
of configurational entropy predicted by the generalized entropy theory (GET).
The figure displays the configurational entropy per lattice site versus per
occupied lattice site as a function of $T$ at zero pressure. The symbols
cross, triangle, square, and diamond indicate the positions of the onset
$T_{A}$, crossover $T_{c}$, glass transition $T_{\text{g}}$, and ideal glass
transition temperatures $T_{0}$, respectively. The vanishing of $s_{c}$ for
semiflexible polymer melts is probably spurious due to the high $T$ expansion
associated with the treatment of chain stiffness in the lattice cluster
theory. The calculations are performed for a melt of chains with the structure
of polypropylene (PP), where the cell volume parameter is
$V_{\text{cell}}=2.5^{3}\text{\AA}^{3}$, the chain length is $N_{c}=8000$, the
cohesive energy parameter is $\epsilon/k_{B}=200$ K, and the bending energy
parameter is $E_{b}/k_{B}=600$ K. The above parameter set is utilized for the
calculations considered in Figures 3, 8, 9, and 10.

Before the development of the GET, 19 the LCT has been primarily used to
explain enigmatic observations in polymer blends, 170 including the existence
of an entropic contribution to the effective Flory interaction parameter
$\chi$. The generalization of the LCT to consider polymer glass formation was
made possible first by arriving at an analytical expression 171 for the
configurational entropy in terms of the natural logarithm of the density of
states $\Omega(U)$ in the microcanonical ensemble, 17

$S_{c}(T)=k_{B}\ln\Omega(U)|_{U=U(T)},$ (9)

where $U(T)$ is the internal energy at the temperature $T$. The explicit form
for $S_{c}$ derived from the LCT turns out to be quite complicated
mathematically, 171 but a more recent work 172 indicates that $S_{c}$
approximates very closely the ordinary entropy $S$ calculated from the free
energy $F=N_{l}f$, i.e., $S_{c}\approx S=-\partial F/\partial T|_{\phi}$,
greatly simplifying the evaluation of $S_{c}$ from the LCT. While the ability
to calculate $S_{c}$ within the LCT already enables investigating the ideal
glass transition temperature $T_{0}$, which is the primary focus in the GD
theory, 17 the GET combines the LCT with the AG relation 18 to explore other
key aspects of polymer glass formation. This is achieved by realizing that one
has to utilize the physically consistent configurational entropy when invoking
the AG relation. We discuss below what we mean by ‘physically consistent’. For
simplicity, here we perform illustrative calculations for a melt of chains
with the structure of polypropylene (PP) at zero pressure, where a single
bending energy $E_{b}$ is required for the backbone. As shown in Figure 2, the
configurational entropy per occupied lattice site increases progressively with
increasing $T$, showing no tendency to level off at high $T$, as AG suggested.
18 Note that AG did not consider fluids at constant pressure. 18 On the other
hand, the $T$ dependence of the configurational entropy per lattice site or
the configurational entropy density ($s_{c}=S_{c}/N_{l}$) does plateau at high
$T$, making it a sensible candidate for use in the AG relation. This point has
been elucidated in detail in ref 19, and we briefly discuss some of the
ramifications of the $T$ dependence of the configurational entropy density
$s_{c}$ that serves to define the characteristic temperatures within this
model of glass formation.

Figure 3: Determination of the characteristic temperatures and fragility of
glass formation in the GET. (a) Configurational entropy density $s_{c}/k_{B}$
versus $T$. $T_{A}$ and $T_{0}$ are determined from the temperatures at which
$s_{c}$ displays a maximum and is extrapolated to zero, respectively. (b)
$\partial(Ts_{c})/\partial(k_{B}T)$ versus $T$. $T_{c}$ is determined from the
temperature at which the $T$ dependence of $Ts_{c}$ displays an inflection
point. (c) $\log\tau_{\alpha}$ versus $T$. $T_{\mathrm{g}}$ is determined from
the temperature at which $\tau_{\alpha}=100$ s. (d) Angell plot of
$\log\tau_{\alpha}$. The fragility index $m$ is determined from its standard
definition. 40, 41 The calculations are performed at zero pressure and utilize
the parameter set specified in the caption of Figure 2.

The GET predicts that polymer glass formation is a broad thermodynamic
transition with four characteristic temperatures. 19 Figure 3 summarizes the
determinations of these characteristic temperatures, along with the fragility
index. The first one is called the Arrhenius or onset temperature, $T_{A}$,
which signals the onset of non-Arrhenius behavior of $\tau_{\alpha}$. In
particular, $s_{c}$ exhibits a maximum $s_{c}^{*}$ at a temperature identified
with $T_{A}$ (Figure 3a), and the maximum $s_{c}^{*}$ is chosen to be the high
$T$ limit of $s_{c}$ appearing in the AG relation. 18 Following GD, 17 the
hypothetical ‘ideal glass transition temperature’ $T_{0}$ is identified from
the condition of $s_{c}(T_{0})\rightarrow 0$ (Figure 3a). The crossover
temperature $T_{c}$ separates two regimes of $T$ with qualitatively different
dependences of $\tau_{\alpha}$ on $T$ and is estimated from an inflection
point in the $T$ dependence of $Ts_{c}$ (Figure 3b). To determine
$\tau_{\alpha}$ and thus $T_{\text{g}}$, the GET combines information on
$s_{c}$ with the AG relation, 18

$\tau_{\alpha}=\tau_{0}\exp\left(\frac{\Delta
G_{0}}{k_{B}T}\frac{s_{c}^{*}}{s_{c}}\right),$ (10)

where $s_{c}^{*}$ and $s_{c}$ are inputs directly determined from the LCT, and
$\tau_{0}$ and $\Delta G_{0}$ are the high $T$ limit of $\tau_{\alpha}$ and
the high $T$ activation free energy of TST, respectively.

Because the LCT does not provide a method for evaluating $\tau_{0}$ and
$\Delta G_{0}$, these fundamental parameters have been obtained empirically so
that assumptions are necessarily invoked. 19 The GET takes $\tau_{0}=10^{-13}$
s, following typical experimentally reported values for polymers 173 and
arguments based on TST 68, 66 suggest that this time scale should be on the
order of an inverse molecular vibrational frequency. This time scale can be
refined by calculating the decay time of the velocity autocorrelation
function. In practice, the variations of $\tau_{0}$ with molecular parameters
such as chain stiffness 121 have been found to be rather limited. As
anticipated from TST, 68, 66 there should be both enthalpic and entropic
contributions to the activation free energy (eq 3). However, as $\Delta S_{0}$
is notoriously difficult to estimate from a theoretical viewpoint, the GET
follows the original treatment by AG 18 and assumes that the entropic
contribution to the activation free energy is negligible, 19 i.e., $\Delta
S_{0}=0$. The GET also assumes based on experimental evidence 19 that $\Delta
G_{0}=\Delta H_{0}\approx 6k_{B}T_{c}$. We have discussed this relation at
length in previous works, 19, 122 so the reader is referred to these
necessarily technical discussions for details. We emphasize here that these
simplifying assumptions are not required by the GET, but they allow for the
prediction of polymer properties without any further information than the
determination of molecular parameters (e.g., monomer structure, cohesive
interaction strength, and chain stiffness) and thermodynamic parameters (e.g.,
$T$ and $P$) required to specify the thermodynamics of polymer melts,
properties that the LCT has been established to describe with good
reliability. The enthalpy $\Delta H_{0}$ and entropy $\Delta S_{0}$ of
activation can be independently determined from either simulation or
measurement, and in Section 4.5 we explore an extension of the GET where these
parameters are determined from experiment to gain some insight into general
trends in these energetic parameters. Jeong and Douglas 101 have studied the
dependence of these energetic parameters on molecular mass in unentangled
linear alkane chains by MD simulations, for which there are substantial
experimental data to compare to and validate the simulation results. We expect
this activity of studying the activation free energy parameters of polymers
and GF liquids to expand in the future, given the fundamental importance of
these energetic parameters revealed by recent simulation and experimental
studies. 107, 108, 117, 118, 119, 120, 121, 122, 123, 124

Once $\tau_{\alpha}$ is computed as a function of $T$, then $T_{\text{g}}$ can
be determined using the common empirical definition,
$\tau_{\alpha}(T_{\text{g}})=100$ s (Figure 3c). The fragility index proposed
by Angell, 40, 41

$m=\left.\frac{\partial\log\tau_{\alpha}}{\partial(T_{\text{g}}/T)}\right|_{P,T=T_{\text{g}}},$
(11)

is also readily obtained. Alternatively, we can determine $m$ by fitting
$\tau_{\alpha}$ in the $T$ range between $T_{c}$ and $T_{\text{g}}$ to the VFT
equation (eq 1). 78, 79, 80 Since the VFT equation turns out to be a rather
good description for the $T$ variation of $\tau_{\alpha}$ calculated from the
GET in the $T$ range between $T_{c}$ and $T_{\text{g}}$, $T_{\mathrm{VFT}}$ is
basically equivalent to $T_{0}$ so that we do not differentiate them. $m$ is
related to $D$ simply via
$m=DT_{\text{g}}T_{\mathrm{VFT}}/[(T_{\text{g}}-T_{\mathrm{VFT}})\ln 10]$.

Because the LCT 137, 138 is a powerful theoretical framework for addressing
the thermodynamics of a vast array of polymer systems, a number of important
problems relevant to polymer glass formation can be investigated by the GET.
19 For instance, the GET has been demonstrated to quantitatively describe the
characteristic properties of poly($\alpha$-olefins), 174 to clarify the
meaning of activation volume of polymer liquids, 175 to elucidate the
influence of cohesive energy and chain stiffness on polymer glass formation,
176, 177, 172 and to better understand a variety of phenomena related to
polymer glass formation, such as plasticization and antiplasticization of
polymer melts by molecular additives, 168 density-temperature scaling of the
segmental dynamics of polymer melts, 178 two glass transitions in miscible
polymer blends, 169 and the interpretation of the ‘universal’ WLF parameters
of polymer liquids 92 and the related observation that the structural
relaxation time depends nearly universally on $T-T_{\mathrm{g}}$ in related
families of GF liquids. 179 The LCT has also been extended to model polymers
with specific interactions 138, 177 where cohesive interaction strengths are
different between united-atom groups and telechelic polymers 180, 181, 182
where associative groups at the chain ends have strong interactions, enabling
predictions of glass formation in these polymer systems when combined with the
AG model. 18 It is also worth mentioning that the GET has been shown to be
useful for coarse-grained modeling of polymeric GF materials. 183, 184 Thus,
the GET has the unique capacity to investigate the influence of structural
details and interactions on polymer glass formation and better understand a
variety of important problems in polymer glass formation.

As we have shown above, a particular advantage of the GET is that the
characteristic temperatures ($T_{A}$, $T_{c}$, $T_{\mathrm{g}}$, and $T_{0}$)
of glass formation can be calculated precisely in the theory because of the
analytic definitions of these fundamental characteristic temperatures of glass
formation. This stands in contrast to experimental and computational studies
of glass formation, where large uncertainties often arise in the estimates of
these temperatures. It is also interesting to note that the ECNLE theory of
Schweizer and coworkers 56, 57, 58 allows for predictions of the
characteristic temperatures of glass formation. It would be quite interesting
to compare predictions of this model with the GET to better understand the
physical meaning of the characteristic temperatures of glass formation.

###  4.2 Temperature Dependence of Configurational Entropy

From the AG relation (eq 10), it is evident that the $T$ dependence of
$s_{c}$, along with $\Delta G_{0}$, determines the dynamics of GF liquids
within the GET. It is thus crucial to know how $s_{c}$ varies with $T$ and
then better comprehend the implications of this information on glass
formation. While GD 17 focused primarily on a low $T$ regime where the
configurational entropy extrapolates to zero and AG 18 essentially made some
guesses about the $T$ dependence of the configurational entropy, one advantage
of the GET 19 is that it allows us to analyze $s_{c}$ over the entire $T$
regime of glass formation in great detail. In particular, the GET has made
detailed predictions for the variation of $s_{c}$ with $T$ in both the high
and low $T$ regimes of glass formation. This has been discussed at length in
the original review on the GET. 19 However, our more recent calculations 185,
136 based on the GET have revealed certain features in the $T$ dependence of
$s_{c}$ that are not anticipated from the original GET. Here, we summarize
these features, along with those basic aspects identified in the original GET.
19

Figure 4: $T$ dependence of the configurational entropy in the entire $T$
regime of glass formation. (a) $s_{c}/k_{B}$ as a function of $T$ predicted by
the GET for polymer melts with and without bending constraints at zero
pressure, where $E_{b}/k_{B}=600$ K and $0$ K, respectively. Dashed lines are
the descriptions based on eq 12. The symbols cross, triangle, and square
indicate the positions of $T_{A}$, $T_{c}$, and $T_{\text{g}}$, respectively.
The calculations are performed for a melt of chains with the structure of PP
at zero pressure, where $V_{\text{cell}}=2.5^{3}\text{\AA}^{3}$, $N_{c}=8000$,
and $\epsilon/k_{B}=200$ K. (b) Configurational entropy $S_{c}$ determined
from the heat capacity measurements as a function of $T$ for various molecular
liquids. 3MP, MTHF, ETB, and PC are short for 3-methylpentane, 2-methyl-
tetrahedrofuran, ethylbenzene, and propylene carbonate, respectively. Panel
(b) was adapted with permission from ref 186.

We begin by noting the behavior of $s_{c}$ at $T$ well below $T_{\mathrm{g}}$.
While the vanishing of $s_{c}$ at low $T$ shown in Figures 2 and 3 was found
to be a typical behavior in the original GET for polymer melts with bending
constraints, 19 as in the GD model, 17 our more recent calculations 185, 136
have demonstrated that this potentially spurious behavior can be avoided in
certain models. Specifically, $s_{c}$ was found to level off rather than
vanish at low $T$ in polymer models with variable spatial dimension $d$ when
$d$ is above a critical value in the vicinity of $d=8$ 185 and in fully
flexible polymer models in the physical dimension of $d=3$. 136 Here, we take
the fully flexible polymer model to illustrate our point, as shown in Figure
4a, where we compare the $T$ dependence of $s_{c}$ of a polymer melt with
bending constraints to that without bending constraints. As can be seen,
$s_{c}$ for fully flexible polymer melts no longer vanishes, but instead
evidently approaches a positive constant $s_{r}$ at low $T$, providing a
different perspective for the nature of glass formation than the original GD
theory. 17 This finding is consistent with the suggestion of DiMarzio in his
later work 127 that $S_{c}$ should become ‘critically small’ rather than
vanishing at the glass transition. The explicit expression for $s_{r}$ from
the GET has been derived in ref 136 as a function of molecular parameters,
such as chain length and monomer structure. Interestingly, $S_{c}$ has also
been estimated to be effectively constant at low $T$ in the heat capacity
measurements of a number of molecular liquids by Yamamuro and coworkers. 186,
187 Although this type of observation is often interpreted to be an inherently
nonequilibrium phenomenon, this result is highly attractive as a possibly
equilibrium phenomenon from a viewpoint of the entropy theory, so we reproduce
the excess entropy data of ref 186 for an array of materials in Figure 4b.

Philosophically, the presence of a positive residual configurational entropy
has important ramifications for glass formation within the entropy theory.
First, the Kauzmann’s ‘entropy crisis’ 46 is naturally avoided in the GET for
fully flexible polymer melts. Moreover, when combined with the AG relation in
eq 10, a finite residual configurational entropy evidently indicates that
structural relaxation becomes of an Arrhenius form. In this case,
$\tau_{\alpha}$ does not diverge at any finite $T$, and thus, the material is
a ‘liquid’ at low $T$ from a mathematical standpoint. However, since the
activation energy $\Delta G(T)$ in the low $T$ regime below $T_{\mathrm{g}}$
is higher than that which the fluid has at high $T$ above $T_{A}$ and since
liquids are cooled to such low $T$, the relaxation times at low $T$ are much
larger than those in the high $T$ regime, and by all practical measures, the
polymer melt in the low $T$ regime can be considered to be a ‘solid’ in a
rheological sense. We are thus tempted to term the low $T$ state as being a
type of a ‘glass’. O’Connell and McKenna 188 and Novikov and Sokolov 89 have
tentatively suggested that relaxation in GF materials generally approaches an
Arrhenius behavior at low $T$, consistent with the prediction of the GET for
fully flexible polymers. While our finding of a finite residual
configurational entropy and a corresponding highly viscous ‘glass state’ at
low $T$ in flexible polymer melts provides a clear mechanism of how the
‘entropy crisis’ might be avoided in real polymer melts and a counterexample
to the claim that the AG model predicts a diverging structural relaxation time
at a finite low $T$, we must admit that it does not imply that $s_{c}$ always
exhibits a plateau at low $T$. There is also no doubt that it would be
extremely difficult for materials to reach equilibrium over ‘reasonable’
timescales at low $T$ due to the very long relaxation times, making it quite
challenging to separate equilibrium from non-equilibrium contributions to the
‘effective’ configurational entropy. This situation makes the utility of the
GET of uncertain value below $T_{\mathrm{g}}$.

We can analyze the variation of $s_{c}$ with $T$ more quantitatively. In
particular, we have shown in ref 136 that $T$ dependence of $s_{c}$ can be
reasonably described by the following empirical equation describing the
generally sigmoidal variation of $s_{c}$,

$s_{c}=s_{r}+(s_{c}^{*}-s_{r})[1+\exp(a+b/T)]^{c},$ (12)

where the constants $a$, $b$, and $c$ are obtained by numerically fitting
$(s_{c}-s_{r})/(s_{c}^{*}-s_{r})$ as a function of $T$ since this normalized
quantity varies from zero to unity as $T$ varies, thereby motivating the
fitting functional form, $[1+\exp(a+b/T)]^{c}$. The description based on eq 12
is shown as a dashed line in Figure 4a, where we see that this functional form
provides an excellent description of $s_{c}(T)$ for polymer melts without
bending constraints over the entire $T$ regime of glass formation. The origin
of eq 12, and a similar expression utilized in our recent study of polymer
coarse-graining, 183 can be traced back to the string model of glass
formation, 20 which establishes a relation between emergent elasticity and
cooperative motion in polymeric GF liquids 117 and a quantitative inverse
relation between the extent of cooperative motion and the configurational
entropy, 161 as suggested by AG. 18 A relation between the activation free
energy and material stiffness is further found to qualitatively accord with
the shoving model of Dyre and coworkers. 189 Motivated by these observations
and arguments, eq 12 is obtained by first invoking a simple two-state model of
the shear modulus $G$ of amorphous materials, 190
$G(T)/G(0)=1/[1+\exp(a+b/T)]$, and then by utilizing the observation that the
activation free energy for relaxation scales approximately as a positive power
of $G$ in a model GF material. 191 While these arguments are admittedly
intuitive, eq 12 turns out to provide a good numerical approximation for
$s_{c}(T)$ calculated from the full GET, whose complexity does not lend itself
readily to the simple analytic forms that seem to be highly desired by
experimentalists and engineers. Since the vanishing of $s_{c}$ in polymer
melts with bending constraints is likely an artifact of the high $T$ expansion
involved in the lattice model calculations for the free energy when the
bending constraints are introduced, we may also invoke eq 12 to describe the
variation of $s_{c}$ with $T$ in polymer melts with bending constraints, for
which eq 12 is also a reasonable approximation, as evidenced in Figure 4a.
Note that we choose a somewhat arbitrary value of $s_{r}/k_{B}=0.025$ for the
polymer melt with bending constraints shown in Figure 4a.

Given the good description of eq 12 for $s_{c}(T)$ predicted by the GET, we
can suggest the following functional form for $\tau_{\alpha}$, 136

$\tau_{\alpha}=\tau_{0}\exp\left\\{\frac{\Delta G_{0}/k_{B}T}{\delta
s+(1-\delta s)[1+\exp(a+b/T)]^{c}}\right\\},$ (13)

where $\delta s\equiv s_{r}/s_{c}^{*}$ is the ratio of the activation energy
in the high $T$ regime to that in the low $T$ regime. This approximation is
expected to work in the entire $T$ regime of glass formation below $T_{A}$.

Figure 5: $T$ dependence of the configurational entropy in the high and low
$T$ regimes of glass formation predicted by the GET. (a) $s_{c}^{*}/s_{c}-1$
versus $(T_{A}-T)/T_{A}$ and (b) $Ts_{c}/k_{B}$ versus $T$ for polymer melts
with $E_{b}/k_{B}=0$ K and $600$ K, respectively. Dashed lines in panels (a)
and (b) correspond to eqs 14 and 15, respectively. The symbols cross,
triangle, and square indicate the positions of $T_{A}$, $T_{c}$, and
$T_{\text{g}}$, respectively. The calculations are performed for a melt of
chains with the structure of PP at zero pressure, where
$V_{\text{cell}}=2.5^{3}\text{\AA}^{3}$, $N_{c}=8000$, and
$\epsilon/k_{B}=200$ K.

While the mathematical forms given by eqs 12 and 13 are a bit complicated in
comparison to the VFT relation (eq 1) and may not be utilized readily for
describing experimental and simulation results, the GET predicts that the
functional form for describing the $T$ variation of $s_{c}$ can be simplified
greatly when we focus on certain restricted regimes of glass formation. For
instance, in the high $T$ regime of glass formation between $T_{A}$ and
$T_{c}$, the GET predicts a simple functional form for $s_{c}(T)$ as, 19

$s_{c}^{*}/s_{c}-1=C_{s}[(T_{A}-T)/T_{A}]^{2},\ T_{c}<T_{A}-100\
\mathrm{K}<T<T_{A},$ (14)

where the quantity $C_{s}$ measures the steepness of the $T$ dependence of
$s_{c}$. $C_{s}$ appears to provide a measure of fragility in the high $T$
regime of glass formation where the VFT equation is not valid. Figure 5a
indicates that eq 14 provides a good description for both polymer melts with
and without bending constraints. Equation 14 has been confirmed by simulations
of a coarse-grained polymer melt with and without antiplasticizer additives,
192 and this functional form has been applied in studies of metallic GF
materials as a method for estimating $T_{A}$. 193, 191, 194 The quantity
$C_{s}$ should be of great interest from a simulation viewpoint since
simulations are mostly restricted to a $T$ regime above $T_{c}$ in GF liquids
because of the very long relaxation times at lower $T$, so we think that eq 14
deserves greater consideration in the future studies.

Kivelson and coworkers 81, 82 have suggested a similar form as eq 14 based on
experimental estimates of the reduced activation energy for diverse fluids,
and the particular exponent was reported to be $8/3$ for their analog of eq
14, a value motivated by a ‘frustrated-limited cluster model’ of glass
formation. Chandler and coworkers 84, 85 have also argued for the same $T$
dependence of the activation energy, as the GET predicts for eq 14, which is
termed the ‘parabolic law’ involving a characteristic temperature comparable
to $T_{A}$, along with an adjustable parameter corresponding to $C_{s}$, and
they have made comparisons with experimental data for structural relaxation
times for numerous GF liquids, where a good ‘fitting’ was claimed. In
comparison of the parabolic law proposed by Chandler and coworkers to
measurements, 195, 196 a high $T$ Arrhenius term is added to the parabolic law
to improve the comparison of the model to experiments at high $T$ where
relaxation is Arrhenius, and this modification leads to an expression for
$\Delta G(T)$ exactly in accord with eq 14. We also mention that the parabolic
functional form for the activation energy can be derived from the NLE and
ECNLE theories of Schweizer and coworkers, 53, 54, 55, 56, 57, 58, 59 and the
details have been given in a previous review on the GET. 19 The above
comparisons seem to indicate that the GET has much in common with other models
of glass formation. Note that $C_{s}$ and $T_{A}$ are derived in terms of
molecular structure and interactions in the GET. We again emphasize that eq 14
is only applicable in the high $T$ regime above $T_{c}$ but below $T_{A}$. The
GET predicts that the VFT relation applies in the low $T$ regime of glass
formation below $T_{c}$ where the VFT parameters are likewise calculated from
the theory as a function of molecular parameters, as we discuss below based on
the $T$ dependence of $s_{c}$.

Figure 5b shows the prediction from the GET for the product $Ts_{c}/k_{B}$ as
a function of $T$ for both polymer melts with and without bending constraints.
Over a limited $T$ range in the low $T$ regime of glass formation between
$T_{c}$ and $T_{0}$, a linear relation between $Ts_{c}/k_{B}$ and $T$
approximately holds with the slope $K_{T}$, 197

$Ts_{c}/k_{B}=K_{T}(T/T_{K}-1),$ (15)

where $T_{K}$ is the Kauzmann temperature at which $s_{c}$ is extrapolated to
zero. Since $K_{T}$ bears no direct relation to the strength of the $T$
dependence of the relaxation time, we prefer to call this quantity the low $T$
fragility parameter. We see from Figure 5b that eq 15 holds for $T$ down to
$T_{0}$ for the polymer melt with bending constraints. In the absence of
bending constraints, however, the $T$ range where eq 15 is applicable seems to
be limited, which clearly arises from the sigmoidal variation of $s_{c}$ with
$T$, as discussed earlier.

###  4.3 Packing Frustration and Glass Formation

Despite the fact that the GET 19 is essentially a thermodynamic theory of
glass formation that relies on predictions of fluid dynamics based on the
configurational entropy, calculations based on this model have provided
molecular insight into the physical nature of polymer glass formation and the
most relevant factors governing this phenomenon. Here, we discuss how the GET
is utilized to understand the fragility and $T_{\mathrm{g}}$ of GF polymers,
two of the most important parameters of polymer materials from a practical
viewpoint. A previous work 174 based on the GET investigated the influence of
molecular parameters, such as cohesive interaction strength, chain stiffness,
and molecular mass, on fragility quantified by the index $m$ or $1/D$ in the
VFT relation (eq 1), and this work indicated a remarkable regularity.
Specifically, the changes in fragility arising from varying molecular
parameters can be understood at fixed measures of cohesive interaction
strength from how these parameters influence the packing frustration of the
molecules, as measured quantitatively by the ‘free volume’ at $T_{\text{g}}$,
i.e., $\phi_{v}(T_{\text{g}})$, where $\phi_{v}$ characterized by the extent
to which the lattice is not occupied by chain segments, i.e.,
$\phi_{v}=1-\phi$. The concept of free volume has a long history in modeling
the dynamics of cooled liquids going back to Batschinski, 198 Hildebrand, 199
Doolittle, 200 Ferry, 201 and many others, and this concept remains popular in
the polymer science community. 60, 61 The most widely used definitions of free
volume in the literature and the differences between different definitions
have been recently reviewed by White and Lipson, 61 who have developed a
‘cooperative free volume’ model to explain the dynamics of GF liquids. 202,
203, 204, 205

Figure 6: Schematic illustration of chain configurations for general classes
of polymers. Polymers are categorized on the basis of the relative rigidities
of the backbone and side chains, leading to flexible-flexible (F-F), flexible-
stiff (F-S), stiff-flexible (S-F), and stiff-stiff (S-S) polymers.

While the general concept of packing frustration is rather vague, the GET 19
allows us to quantify this property of fluids in terms of how molecular
parameters, such as the relative rigidities of the polymer backbone and side
chains, chain length, monomer structure, and cohesive interaction strength,
influence macroscopic properties such as the thermal expansion coefficient and
isothermal compressibility that are highly dependent on the efficiency of
molecular packing and the complex intermolecular interactions of molecular
fluids. Early studies with the GET 19 indicated that these molecular
parameters can greatly influence packing efficiency, and, accordingly,
polymers were classified into general classes based on the results of the
model 19 and measurements that also suggested such a classification.
Specifically, this early work 19 led to a classification of polymers into
classes of polymers sharing similar packing characteristics. In particular,
the flexible-flexible (F-F), flexible-stiff (F-S), stiff-flexible (S-F), and
stiff-stiff (S-S) classes of polymers were defined to be chains with a
flexible backbone and flexible side groups, chains with a flexible backbone
and relatively rigid side branches, chains with a relatively stiff backbone
and flexible side groups, and chains with both a stiff backbone and stiff side
groups, respectively. The S-S class of polymers has not been encountered
frequently in the past, but such polymers have recently become of interest in
connection with the applications of molecular filtration in which the large
free volume and $T_{\text{g}}$ of such ‘polymers of intrinsic microporsity’
(PIMs) are highly physical attributes, 6, 5, 7, 8 motivating the theoretical
study of S-S polymers in more detail. For illustrative purposes, Figure 6
displays representative chain configurations for these general classes of
polymers.

Intuitively, F-F polymers can pack efficiently in space, and accordingly,
these polymer systems exhibit a relatively weak $T$ dependence of $s_{c}$ and
are relatively strong glass-formers. In comparison, polymers with either a
stiff backbone or stiff side groups exhibit more packing frustration, and
hence, these polymers have a relatively strong variation of $s_{c}$ with $T$,
provided that the cohesive interaction strength is fixed, as in the case of
molecules having van der Waals or other fixed types of intermolecular
interactions. The GET then predicts that variable fragility is a direct
outgrowth of the extent of packing frustration of the molecules composing the
material, 19 a property that can be engineered through exertion of control of
molecular structure by chemical synthesis. We note that the experimental
measurements of Sokolov and coworkers 43 have confirmed the general trends
predicted by the GET of fragility with varying molecular structure.

Figure 7: Correlations between packing frustration and the characteristic
properties of glass formation predicted by the GET. (a–c) $m$ and (d–f)
$T_{\text{g}}$ versus thermodynamic metrics for packing frustration for
varying individual molecular parameters. The proposed metrics, $\phi_{v}$,
$\tilde{\alpha}_{P}$, and $\tilde{\kappa}_{T}$, are the ‘free volume’ defined
in terms of lattice occupancy, reduced thermal expansion coefficient, and
reduced isothermal compressibility, respectively, which are all given at
$T_{\text{g}}$. In these calculations, the variables $E_{b}$, $\epsilon$, and
$N_{c}$ are systematically varied for polymer melts with the structure of PP,
and the variable side group length $n$ is systematically varied for F-F, F-S,
and S-F polymers. Adapted with permission from ref 175.

The above discussion naturally leads us to expect that $m$ increases with the
extent of packing frustration, as quantified by the thermodynamic measures
discussed above. This general trend is verified in Figure 7a, where
$\phi_{v}(T_{\text{g}})$ is utilized as a candidate for a quantitative measure
of packing frustration. In Figure 7, we have systematically varied $E_{b}$,
$\epsilon$, and $N_{c}$ for polymer melts with the structure of PP and the
side group length $n$ for F-F, F-S, and S-F polymers. Since S-S polymers
exhibit the same trends in $m$ and $T_{\mathrm{g}}$ as F-F polymers, this
class of polymers is not included here for our analysis. Reference 175
provides more details regarding the calculations. We next discuss these
measures of packing frustration in detail and their determination in the GET
as well as their correlation with the fragility of glass formation as the
structural properties of polymers are modified.

From an experimental point of view, $\phi_{v}(T_{\text{g}})$ is difficult to
measure, and it is thus natural to compare $m$ to properties that are related
to $\phi_{v}(T_{\text{g}})$, but more readily measured, such as the thermal
expansion coefficient $\alpha_{P}$ and isothermal compressibility
$\kappa_{T}$,

$\alpha_{P}=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_{P},\
\kappa_{T}=-\frac{1}{V}\left.\frac{\partial V}{\partial P}\right|_{T}.$ (16)

These properties are of both fundamental and practical interest and can be
measured experimentally for polymers and other materials. 206 For equilibrium
fluids, $\kappa_{T}$ is related to the long wavelength limit of the static
structure factor, $S(0)$, via $\kappa_{T}=\rho k_{B}T/S(0)$ with $\rho$ being
the number density. The density is given by $\rho=\phi/V_{\text{cell}}$ in the
GET. Instead of focusing directly on $\alpha_{P}$ and $\kappa_{T}$, the GET
suggests that the reduced properties be considered as measures of packing
frustration, 185, 175, 122

$\tilde{\alpha}_{P}=T\alpha_{P},\ \tilde{\kappa}_{T}=\rho k_{B}T\kappa_{T}.$
(17)

The experimental studies of Simha and Boyer 207 have demonstrated that the
reduced thermal expansion coefficient $\tilde{\alpha}_{P}$ can be used to
estimate the dependence of $T_{\text{g}}$ on both cohesive interaction
strength and chain stiffness for many polymers, thereby emphasizing the
significance of $\tilde{\alpha}_{P}$ rather than $\alpha_{P}$ itself. The NLE
and ECNLE theories of glass formation of Schweizer and coworkers 53, 54, 55,
56, 57, 58, 59 also emphasize the central role of $S(0)$ in predicting the
dynamics of GF liquids. Sanchez 208 has utilized the dimensionless
thermodynamics to discuss the liquid state properties, and this interesting
work is recommended to the reader for further details on reduced thermodynamic
properties. We have investigated the dimensionless thermodynamic properties in
our simulation studies of polymer glass formation. 209, 120, 121, 122

We show a comparison $m$ versus $\tilde{\alpha}_{P}$ and $\tilde{\kappa}_{T}$
in Figures 7b and 7c, respectively, where both quantities are given at
$T_{\text{g}}$. We again see the expected trend that $m$ varies with packing
frustration, as quantified by $\tilde{\alpha}_{P}$ and $\tilde{\kappa}_{T}$.
We emphasize that our discussion here is based on the reduced thermal
expansion coefficient and isothermal compressibility.

We may also utilize the concept of packing frustration to understand
$T_{\text{g}}$, a quantity that is naturally appreciated from the famous WLF
relation 201, 91 and its universal parameters $C_{1}$ and $C_{2}$, which are
often invoked in the description of the dynamics of polymer materials. The WLF
relation can be derived from the VFT expression for $\tau_{\alpha}$ if the
fragility parameter is taken to be simply proportional to $T_{\text{g}}$, an
assumption that accounts for why fragility is not an explicit parameter in the
WLF equation and for why the dynamics depends on the temperature difference
$T-T_{\text{g}}$ for materials for which the WLF equation applies. 92 A
detailed discussion of the WLF relation along with references relevant to
$C_{1}$ and $C_{2}$ has been given in ref 92. Figures 7d–f show that
$T_{\text{g}}$ strongly correlates with the metrics for packing frustration,
provided that materials are compared at fixed cohesive interaction strength,
which informs us that materials with inherently high packing frustration
exhibit an inherently higher $T_{\text{g}}$ and fragility. This finding
accords with the use of high free volume polymer materials or PIMs 6, 5, 7, 8
for applications in gas separation processes, heterogeneous catalysis, and
hydrogen storage as well as the socially important problem of large-scale
water desalination, to name a few emerging applications of these new
materials. $T_{\text{g}}$ in some commercial PIMs is so high that its
estimates have previously been difficult, but recent techniques have overcome
this difficulty, at least in some PIMs. 10, 9 Finally, it is worth noting in
Figures 7d–f that an inversion of the trend of $T_{\text{g}}$ with the
measures of packing frustration occurs when varying $\epsilon$, a result first
identified in ref 174. This more complicated behavior should be borne in mind
when discussing the glass formation of polar and charged polymer fluids, which
generally exhibit high cohesive interaction strength.

###  4.4 Glass Formation under Pressure

It has been well established that glass formation can often be induced by
increasing $P$ at constant $T$. 206, 21, 22 The large alterations in the
dynamics of GF liquids upon the application of pressure are evidently of
significance in numerous manufacturing applications. In particular, there has
been intense interest in separating out aspects of glass formation relating to
the effects of attractive intermolecular interactions, temperature, and
density, 210, 211, 21 These efforts have naturally led to systematic studies
of the dynamics of GF liquids over a wide range of $P$ 21, 212 to complement
the traditional studies where glass formation occurs upon cooling. The GET is
naturally suitable for addressing polymer glass formation under applied
pressures since calculations can be performed under different constant $P$
conditions. In this section, we utilize the GET to discuss several interesting
aspects of polymer glass formation when $P$ is a variable.

Figure 8: Density-temperature scaling of $\tau_{\alpha}$ predicted by the GET.
The main plot shows $\log\tau_{\alpha}$ versus $10^{3}\phi^{\gamma}/T$ with
$\gamma=7.4$ for a range of fixed $P$. The inset displays $\log\tau_{\alpha}$
versus $10^{3}/T$ for the same $P$. The calculations utilize the parameter set
specified in the caption of Figure 2.

We begin by noting the extremely interesting density-temperature scaling
phenomenon, 206, 21, 22 which indicates that dynamic properties such as
structural relaxation time and diffusion coefficient in suitably reduced units
become universal functions of $T$ times density to a power $\gamma$
characteristic of the material. Dyre and coworkers 213 have suggested that the
existence of this type of scaling should play the role of a ‘filter’ for
acceptable models of GF liquids. In a previous study based on the GET, 178 we
indeed found density-temperature scaling to hold for $\tau_{\alpha}$ over the
entire $T$ regime of glass formation in polymer melts. The GET thus passes
this test. For illustrative purposes, we show the density-temperature scaling
of $\tau_{\alpha}$ for a range of fixed $P$ predicted by the GET in Figure 8,
the inset of which displays $\log\tau_{\alpha}$ as a function of $10^{3}/T$.
Experimental and computational studies indicate that the scaling exponent
$\gamma$ obtained from estimates of viscosity and relaxation time lies in the
broad range of $0.18$ to $8.5$. 21 Our previous study 178 indicates that the
scaling exponent $\gamma$ varies in a large range when varying molecular
parameters, such as chain length, chain stiffness, cohesive interaction
strength, and side group length, factors that evidently influence the
fragility of glass formation. Our calculations based on the GET 178 also
confirmed the approximate inverse relation between $\gamma$ and the constant
volume fragility parameter $m_{V}$ found experimentally by Casalini and
Roland. 214, 215

Recently, we have made additional progress on the mysterious density-
temperature scaling phenomenon based on the GET and simulations. We have also
noticed that some other popular models of glass formation do not pass the test
associated with density-temperature scaling. We will report on these findings
in a separate paper, where we discuss the origin of density-temperature
scaling and its many implications. It is notable that the GET is based on a
lattice model where the polymer intersegment interactions are described by the
rough equivalent of a square potential in off-lattice liquids. The scaling
exponent $\gamma$ depends on molecular parameters discussed above, but it has
nothing to do with the shape of the pair potential in our calculations.
Density-temperature scaling in polymer and other molecular fluids then seems
to arise from the variation of the anharmonic intermolecular interactions that
arise from packing frustration, and the GET provides a powerful computational
framework for studying this phenomenon that fundamentally links the
thermodynamics and dynamics of fluids and the relative balance of repulsive
and attractive interparticle interactions governing the dynamics of real
molecular fluids.

Figure 9: $P$ dependence of $\tau_{\alpha}$ at constant $T$ predicted by the
GET. Panels (a) and (b) show $\ln[\tau_{\alpha}(P)/\tau_{\alpha}(0)]$ as a
function of $P$ for a range of $T$ in a wide $P$ range and in the low $P$
limit, respectively. Lines in panels (a) and (b) are the descriptions based on
eqs 18 and 19, respectively. Panels (c) and (d) show the universal reduction
of the $P$ dependence of $\tau_{\alpha}$. Lines indicate
$\ln[\tau_{\alpha}(P)/\tau_{\alpha}(0)]=D_{P}P/(P_{0}-P)$ in panel (c) and
$\ln[\tau_{\alpha}(P)/\tau_{\alpha}(0)]=P\Delta V^{\\#}/k_{B}T$ in panel (d),
respectively. The calculations utilize the parameter set specified in the
caption of Figure 2.

The GET also allows us to quantitatively analyze the variation of
$\tau_{\alpha}$ with $P$ at constant $T$. Experimental 206, 21, 22 and
computational 119, 120 studies have indicated that the $P$ dependence of
$\tau_{\alpha}$ generally displays a pressure analog of the VFT equation
(PVFT) in non-associating GF liquids, in which $P$ and the critical pressure
$P_{0}$, respectively, replace $T$ and the critical temperature
$T_{\mathrm{VFT}}$ in the conventional VFT equation given in eq 1,

$\tau_{\alpha}(P)=\tau_{\alpha}(0)\exp\left(\frac{D_{P}P}{P_{0}-P}\right),$
(18)

where $\tau_{\alpha}(0)$ is the structural relaxation time at zero pressure.
The GET predicts the PVFT relation and further provides a rationale based on
the variation of the configurational entropy of the fluid with $P$. 174 We
show an illustrative result for the $P$ dependence of $\tau_{\alpha}$
calculated from the GET for a range of $T$ in a wide $P$ range in Figure 9a,
along with a universal reduction of the data based on the PVFT relation in
Figure 9c.

While the PVFT relation is evidently required to describe the $P$ variation of
$\tau_{\alpha}$ in a large $P$ range, there is a simple linear variation of
$\ln\tau_{\alpha}$ with $P$ in the low $P$ limit,

$\displaystyle\tau_{\alpha}(P)=\tau_{\alpha}(0)\exp\left(\frac{P\Delta
V^{\\#}}{k_{B}T}\right),$ (19)

where $\Delta V^{\\#}$ defines the activation volume. Figure 9b displays
$\ln[\tau_{\alpha}(P)/\tau_{\alpha}(0)]$ as a function of $P$ in a much
narrower regime of $P$ than that in Figure 9a. We see that the GET predicts
the linear variation of $\ln\tau_{\alpha}$ with $P$ when $P$ is restricted to
small values. Again, we show a universal reduction of the data from the GET
based on eq 19 in Figure 9d. This analysis evidently indicates an onset
pressure $P_{A}$ above which the variation of $\ln\tau_{\alpha}$ with $P$
deviates from the linear relation given by eq 19, the analog of the Arrhenius
regime of GF liquids when $T$ is instead varied at fixed $P$. We have also
found that the $P$ dependence of $\tau_{\alpha}$ calculated from the GET seems
to follow a power law in a restricted $P$ range,
$\tau_{\alpha}(P)\sim(P_{c}-P)^{-\gamma_{c}}$, which $P_{c}$ may be defined by
a crossover pressure and $\gamma_{c}$ is an effective $T$-dependent ‘crossover
exponent’. We do not think that this exponent estimated from the GET has
anything to do with the mode-coupling theory, 216 but it is sometimes
attributed to this theory because it likewise predicts a power-law relation of
this kind, albeit in a $T$ range closer to $T_{A}$ than $T_{c}$. 19 Moreover,
a ‘glass transition pressure’ $P_{\mathrm{g}}$ can be identified for a given
$T$ by the pressure at which $\tau_{\alpha}=100$ s, and as discussed above, we
may also define a pressure analog $P_{0}$ of the VFT temperature by a limiting
pressure at which $\tau_{\alpha}$ extrapolates to infinity. Therefore, the GET
allows us to estimate the different characteristic pressures ($P_{A}$,
$P_{c}$, $P_{\mathrm{g}}$, and $P_{0}$) of glass formation when $P$ is used as
a control variable for glass formation at fixed $T$. There is then an
interesting ‘duality’ between glass formation with variable $T$ at fixed $P$
and with variable $P$ at fixed $T$. This is another topic that we will
investigate in the future.

Parenthetically, the activation volume $\Delta V^{\\#}$ is often encountered
in an industrial and materials science setting when materials are subjected to
large changes in $P$ or applied steady stresses of various kinds, but the
physical meaning of $\Delta V^{\\#}$ is often rather unclear. In our recent
work, 217, 175 we have systematically investigated $\Delta V^{\\#}$ based on
the GET and simulations. Our study indicates that $\Delta V^{\\#}$ is related
to the differential change of the activation free energy as a function of $T$,
and thus bears a close relationship to the fragility of glass formation.

###  4.5 Opportunities and Challenges

The GET is just one of a number of promising models of glass formation which
has the particular advantage of predicting the segmental structural relaxation
time of polymeric GF liquids in terms of essential molecular parameters over
the entire $T$ regime of glass formation. The model is also advantageous
because it describes the equation of state and other basic thermodynamic
properties of polymer melts within a mature and validated theoretical model in
a consistent framework. Moreover, the GET naturally allows for the validation
of thermodynamic and dynamic properties based on the string model 20 of glass
formation in conjunction with MD simulations of coarse-grained polymer melts
that cover a wide range of molecular parameters of interest in understanding
the thermodynamic and dynamic properties of real polymer materials. As
discussed earlier, there are assumptions in the GET relating to the precise
relation between thermodynamic and dynamic properties whose validity needs to
be further assessed and modified if necessary, but this type of limitation
exists for all current models of GF liquids. Here, we discuss how the GET
might be improved.

We first note that, while $\Delta H_{0}$ can be estimated from an Arrhenius
fit to the dynamics of fluids at high $T$ if such information is available, it
remains a difficult matter to determine and understand $\Delta S_{0}$. The
theoretical estimation of $\Delta S_{0}$ is generally appreciated to be the
primary source of uncertainty in TST, 218 and inevitably, $\Delta S_{0}$ must
be estimated from either experiment 103 or simulation. 101 Theoretical
attempts to calculate $\Delta S_{0}$ has a long history in condensed
materials, which we briefly mention here. Vineyard 96 proposed a rigorous TST
for idealized crystalline materials with ideal harmonic interactions, which
has been validated for simple ordered molecular clusters. 219 Unfortunately,
anharmonic effects, as found characteristically in cooled liquids, heated
crystals, and the interfacial dynamics of crystals for $T>2T_{m}/3$ with
$T_{m}$ being the melting temperature, 220 cause the Vineyard theory to break
down. A general discussion of TST for crystalline materials has been provided
by Rice, 221 who emphasized the relation between the Arrhenius prefactor and
the stable and saddle point frequencies. This work is particularly interesting
from a philosophical viewpoint in the sense that it explains why one must
consider a free energy of activation, $\Delta G_{0}$, in condensed materials
rather than just an activation energy, $\Delta H_{0}$. Collective motion
associated with many-body effects is thus essential for barrier crossing
events associated with relaxation and diffusion in condensed materials.

While the first principles analytic estimation of $\Delta S_{0}$ for fluids
from TST has remained challenging, numerical implementations of TST have
recently enabled precise estimations of $\Delta S_{0}$ for model condensed
materials. 99, 98, 126 A large body of semi-empirical works have combined
careful measurements with physical arguments to explain $\Delta S_{0}$ in
terms of a presumed connection with local elastic distortions required for a
particle or defect to move in the condensed state. This line of argument can
be traced back to the pioneering work of Zener 218 in the context of metallic
crystalline materials and was taken further by Lawson 222, 223 by directly
relating the activation volume $\Delta V^{\\#}$ to $\Delta S_{0}$. Keyes 224
argued for a corresponding relation between $\Delta V^{\\#}$ and $\Delta
H_{0}$. These relations have provided a rationalization of the EEC effect
between $\Delta H_{0}$ and $\Delta S_{0}$ that has been validated empirically
in many polymeric and other condensed materials. 225, 226, 227, 228, 229
Exactly solvable toy models of dynamics on hierarchical potential energy
landscapes, 230, 231, 232 as naturally found in disordered condensed
materials, also give rise to the EEC. The large entropy of activation in
condensed materials is found in these models to arise from the rapid
proliferation of transition state paths over high barriers in such
hierarchical spaces, an intuitive physical picture supporting the formal
theoretical results of Yelon and coworkers. 115, 233 A more physically
tangible real-space approach to understanding relaxation in condensed fluids
starts from a consideration of the nature of collective motion in the fluids
required to enable atomic displacement. Based on this point of view, Barrer
105, 234 has offered an interesting heuristic ‘zone theory’ of the EEC
phenomenon in the high $T$ Arrhenius regime of liquids, thus having features
in common with the heuristic arguments of AG in cooled liquids, 18 as
discussed in Section 3.2. Therefore, a physical approach of this kind focusing
on the precise nature of collective motion in dense fluids 161 might offer a
deep understanding of the link between $\Delta H_{0}$ and $\Delta S_{0}$, as
found in the relaxation, diffusion, and chemical reaction processes of so many
condensed materials. 114, 111

Figure 10: Influence of the high $T$ activation free energy parameters on polymer glass formation predicted by the GET. (a) $T_{\mathrm{g}}$ and (b) $m$ as a function of the activation entropy $\Delta S_{0}/k_{B}$ for a range of $\Delta H_{0}$. The calculations utilize the parameter set specified in the caption of Figure 2. Figure 11: Description of experimental results for the $T$ dependence of $\tau_{\alpha}$ for several polymers based on the GET. (a) Poly(isoprene). (b) Poly(propylene glycol). (c) Poly(butadiene). (d) Poly(propylene sulfide). Experimental data were provided to us by the authors of ref 88. Solid and dashed lines represent our best fits based on the GET with and without $\Delta S_{0}$, respectively. The calculations based on the GET are performed for a melt of chains with the structure of PP at zero pressure, where $V_{\text{cell}}=2.5^{3}\text{\AA}^{3}$ and $N_{c}=8000$. The molecular parameters $\epsilon$ and $E_{b}$ and energetic parameters $\Delta H_{0}$ and $\Delta S_{0}$ for the polymers are given in Table 1, along with the basic properties of polymers. Table 1: Basic properties of the polymers considered in Figure 11. The listed properties include the molar mass $M$, glass transition temperature $T_{\mathrm{g}}$, and fragility index $m$, along with the model parameters, $\epsilon$, $E_{b}$, $\Delta H_{0}$, and $\Delta S_{0}$, used for the fits to the experimental results based on the GET. polymer | $M\ [\mathrm{g}/\mathrm{mol}]$ | $T_{\mathrm{g}}\ [\mathrm{K}]$ | $m$ | $\epsilon/k_{B}\ [\mathrm{K}]$ | $E_{b}/k_{B}\ [\mathrm{K}]$ | $\Delta H_{0}/k_{B}T_{c}$ | $\Delta S_{0}/k_{B}$
---|---|---|---|---|---|---|---
PI | 15700 | 207 | 84 | $189.8$ | $357.8$ | $5.0$ | $-1.65$
PPG | 18000 | 202 | 105 | $163.9$ | $395.5$ | $4.0$ | $-2.8$
PB | 87000 | 174 | 90 | $165.2$ | $282.4$ | $4.5$ | $-2.2$
PPS | 44000 | 229 | 116 | $185.5$ | $442.0$ | $4.6$ | $-1.8$

Although the GET does not provide a method for estimating $\Delta H_{0}$ and
$\Delta S_{0}$, we may examine the influence of these basic parameters on
polymer glass formation within this TST-based model. As noted above, we have
utilized simplifying approximations in the GET to avoid estimating these
energetic parameters, i.e., $\Delta H_{0}=6k_{B}T_{c}$ and $\Delta
S_{0}/k_{B}=0$, with the second simplifying assumption being ‘inherited’ from
AG. 18 Here, we tentatively treat both $\Delta H_{0}$ and $\Delta S_{0}$ as
adjustable parameters to better understand their potential relevance to
polymer glass formation. Figure 10 shows both $T_{\mathrm{g}}$ and $m$ as a
function of $\Delta S_{0}$ for a range of $\Delta H_{0}$. Our calculations
also include the predictions based on the assumptions, $\Delta
H_{0}/k_{B}T_{c}=6$ and $\Delta S_{0}/k_{B}=0$, of the original GET. 19 We see
that $T_{\mathrm{g}}$ increases linearly with $|\Delta S_{0}|$, while $m$
drops considerably as $|\Delta S_{0}|$ elevates for the range of $\Delta
S_{0}$ considered here. This result clearly demonstrates that precise
estimations of both $\Delta H_{0}$ and $\Delta S_{0}$ are crucial for
transforming the GET into a quantitative predictive molecular theory. Next, we
present our preliminary results on comparisons between the GET and experiment,
in particular when the activation entropy is present.

Figure 11 shows our GET description of experimental results for the $T$
dependence of $\tau_{\alpha}$ for several polymers, including poly(isoprene)
(PI), poly(propylene glycol) (PPG), poly(butadiene) (PB), and poly(propylene
sulfide) (PPS). These representative polymers are selected to demonstrate the
significance of including $\Delta S_{0}$ in the GET. All the experimental data
were provided to us by the authors of ref 88, and the molar mass for each
polymer corresponds to the highest value reported there. For simplicity, our
calculations based on the GET are performed for a melt of chains with the
structure of PP at zero pressure, where
$V_{\text{cell}}=2.5^{3}\text{\AA}^{3}$ and $N_{c}=8000$, for all the polymers
considered. In an effort to describe the experimental data, we first adopt the
original GET, where $\Delta H_{0}=6k_{B}T_{c}$ and $\Delta S_{0}=0$, to fit
the experimental values of $T_{\mathrm{g}}$ and $m$ for each polymer by
adjusting $\epsilon$ and $E_{b}$ simultaneously. The fitted results are shown
as dashed lines in Figure 11, and the best values for $\epsilon$ and $E_{b}$
are summarized in Table 1. We see that the GET, even without $\Delta S_{0}$,
leads to a reasonable quantitative description of the experimental results in
the $T$ regime near $T_{\mathrm{g}}$, but deviations are noticeable at high
$T$. Based on the obtained molecular parameters $\epsilon$ and $E_{b}$, we
then allowed both $\Delta H_{0}$ and $\Delta S_{0}$ to be adjusted. Utilizing
the values of $\Delta H_{0}$ and $\Delta S_{0}$ in Table 1, Figure 11 exhibits
our calculations from the GET with $\Delta S_{0}$ as solid lines, indicating
that the inclusion of $\Delta S_{0}$ enables a quantitative and accurate
description of the experimental data for the $T$ dependence of
$\log\tau_{\alpha}$ over the entire $T$ regime of glass formation accessible
to experiment.

In the future work, we need to compare the GET predictions to measurements for
more polymers to better understand both $\Delta H_{0}$ and $\Delta S_{0}$, as
well as the interrelation of these two energetic parameters. Jeong and Douglas
101 have explored the determination of $\Delta H_{0}$ and $\Delta S_{0}$ for
alkane melts based on MD simulations, and we have also investigated these
parameters in coarse-grained polymer melts with variable pressure, 119, 120,
217 cohesive interaction strength, 235, 122 and chain stiffness. 121 The
computational machinery certainly exists to make progress in understanding the
activation parameters based on simulations, and we hope that such analyses
will lead to corresponding advances in analytic theories of these fundamental
energetic parameters. We think that a comprehensive understanding of these
energetic parameters will provide the foundation on which a real theory of GF
liquids will ultimately be based. For the present, we must admit that TST of
liquid dynamics has some serious shortcomings that are an impediment to the
GET and other models of glass formation based on the foundation of thermally
activated transport.

##  5 String Model of Glass-Forming Liquids

###  5.1 Historical Background

As in the case of the relation between the fluid entropy and the dynamics in
GF liquids, the idea that the growth of the structural relaxation time should
be associated with increased collective motion that accompanies the reduction
of configurational entropy long predates the AG model. 18 Most of the
arguments along this line in the older literature of GF liquids are rather
qualitative, but we mention a historically interesting suggestion by Mott at a
Solvay conference on condensed materials in 1951, 142 in which the activation
energy of materials in a relatively disordered state was suggested to grow
extensively with the mass of some dynamic cluster in the material. Later, this
suggestion was subjected to experimental tests by Nachtreib and Handler 143 to
interpret their self-diffusion measurements of white phosphorous over a large
$T$ range, and the measurements seemed to confirm the heuristic picture of the
$T$-dependent activation energy of Mott for diffusion in this material.
Although these historical contributions are not often discussed now, we also
acknowledge the historical contributions by Orowan 236 to the understanding of
the nature of molecular motions underlying flow processes in viscous fluids
and solid crystalline materials. See the paper of Goldstein 237 for a
recounting of this now largely inaccessible work. Moreover, the influential
‘zone theory’ of Barrer 105, 234, 238 likewise emphasized the necessity of
collective atomic motion involving many particles in connection with molecular
diffusion. These are the shoulders upon which later models of the dynamics of
GF liquids rest.

With this kind of historical background in view, it seems natural that AG 18
would propose that the activation energy in cooled liquids should grow in
proportion to the number of particles in some sort of hypothetical CRR,
thereby creating an enduring conceptual construct in modeling the dynamics of
liquids. As discussed in Section 3.2, AG 18 further proposed that the number
of particles in the vaguely defined CRR should scale inversely with the
configurational entropy $S_{c}$. The greatest strength of this assumption is
apparently that the VFT relation 78, 79, 80 can be rationalized when this
assumption is combined with rough estimates of $S_{c}$ based on an
approximation suggested earlier by Bestul and Chang. 70 The AG model is not so
much a theory, but rather a set of hypotheses introduced to conform to
prevailing theoretical ideas and correlations between properties seen
experimentally. This is probably a quite good description of more recent
theories of glass formation, so we do not think that the AG model should be
judged harshly given its rather empirical origin. This ‘semi-empirical’ model
has been immensely successful for creating a qualitative picture of the origin
of the strong variation of the relaxation and diffusion properties with $T$
and polymer concentration, and it has largely determined the language used to
describe GF liquids, regardless of whether or not one believes in the
assumptions of the AG model. The RFOT theory 71, 72, 73, 74 is based upon
similar ideas to rationalize the slowing down of the dynamics of GF liquids in
terms of an ‘entropic droplet model’.

It is worth noting that DiMarzio never accepted the AG model, explaining why
he never combined the AG model 18 with the lattice model of GD developed for
calculating $S_{c}$. 17 This situation is apparent in his later work, 127
where DiMarzio introduced an alternative to the AG model by asserting a rather
different relation between the thermodynamics and the structural relaxation
time in cooled liquids. We also mention a formal critique of the assumptions
underlying the AG model by Dyre and coworkers. 239 As a counterweight to these
criticisms, we mention some theoretical discussions that formalize the idea of
a growing activation energy as $T$ is varied in the interfacial dynamics of
crystalline materials 167 and in GF liquids, 166 showing that there are
significant merits to the assumptions of the AG model. 18

Dudowicz et al. 50 have shown that the fluid configurational entropy scales
inversely with the average length of dynamic polymers calculated from a model
of living polymerization, a finding that has taken great significance that
dynamic polymeric structures involving stringlike structures have been
observed in many simulations of GF liquids 240, 241, 179, 113, 161, 107, 20,
117, 108 to date, in which the collective motion has been examined in great
detail and the thermodynamic, geometric, and dynamic properties of the dynamic
stringlike structures closely conform to equilibrium polymerization.
Historically, the findings of Dudowicz et al. 50 served as a great impetus for
the development of the string model of glass formation, in which the dynamic
clusters in cooled liquids were identified as being equilibrium polymers and
realizations of the hypothetical CRR of AG. 18 Douglas et al. 51 have
comparatively discussed the variation of the configurational entropy of fluids
exhibiting equilibrium polymerization and the configurational entropy
calculated from the GET, and strong similarities in the $T$ variation of the
configurational entropies from the two models are indicated, such as analogs
of the onset temperature $T_{A}$, the crossover temperature $T_{c}$, and the
glass transition temperature $T_{\mathrm{g}}$, below which the configurational
entropy saturates to a finite value in the polymerization model. Douglas et
al. 52 have also shown that the ‘cooperativity’ of the equilibrium
polymerization transition, a description of the extent to which the transition
resembles a phase transition, plays the role of fragility in the
correspondence between the polymerization model and the GET.

###  5.2 String Model Essentials

As just discussed above and elaborated on in ref 19, some of the assumptions
made by AG 18 now have some support from simulations and the analytic theory
of fluids exhibiting self-assembly, providing the starting point for
developing the string model of glass formation. A subsequent work 20 has found
that the stringlike collective motion conforms quantitatively to the
equilibrium polymerization model over the entire $T$ range below $T_{A}$ for
which simulations can currently be performed for GF liquids. As we shall see
below, the only unspecified parameters in these comparisons between the
simulation data and the string model are the activation enthalpy $\Delta
H_{0}$ and activation entropy $\Delta S_{0}$ of TST, which are determined in
the high $T$ Arrhenius regime, i.e., $\Delta H_{0}$ from the slope of the
variation of $\ln\tau_{\alpha}$ with $1/T$ and $\Delta S_{0}$ from the
prefactor in the Arrhenius relation (eq 2). Our lack of a theoretical
understanding of the dynamics of fluids in the Arrhenius regime is the
greatest hurdle in predicting the dynamics of GF liquids from the string
model.

Figure 12: Test of the average string length $L$ as a quantitative realization
of the hypothetical cooperatively rearranging regions (CRR) of AG 18 based on
simulation results of a coarse-grained polymer melt. (a) Snapshot of
stringlike clusters of polymer segments undergoing cooperative exchange motion
over the lifetime over which these clusters persist. Each string is shown by
large spheres in a different color. Only strings of length larger than $4$ are
shown to aid the visualization, and the other segments are displayed as
translucent think cylinders. (b) Configurational entropy $S_{c}$ versus $1/L$.
(c) $\ln\tau_{\alpha}$ versus $z/T$, where the size of the CRR is $z=L/L_{A}$
with $L_{A}$ being the value of $L$ at $T_{A}$. (d) $\ln\tau_{\alpha}$ versus
$1/TS_{c}$. Lines in panels (b–d) are a guide to the eye. Here and in Figures
13 and 14, all quantities from the coarse-grained simulations are expressed in
standard reduced Lennard-Jones units. See ref 217 for details about reduced
units and their mapping to laboratory units. Panels (b–d) were adapted with
permission from ref 161.

As noted briefly above, the development of the string model 20 was motivated
by extensive simulations of both polymeric and other GF liquids 240, 241, 179,
113, 161, 107, 20, 117, 108 indicating that the most mobile particles in GF
liquids form stringlike structures defined in terms of their cooperative
exchange motion over the lifetime in which the strings exist. The technical
details for quantifying the stringlike cooperative motion have been repeatedly
described in previous works and we refer the reader to ref 241 for a
discussion of this procedure. Such cooperative motion is broadly consistent
with the philosophical assumptions of the AG model. 18 Figure 12a shows a
snapshot of stringlike clusters of polymer segments undergoing cooperative
exchange motion over the lifetime over which these clusters persist in a
simulated polymer melt. 113 We note that there is little correlation between
these stringlike motions and chain connectivity, so these stringlike motions
are not ‘reptative’, i.e., the segments in these motions are not correlated
with the chain backbone. Such collective motion occurs in a rather similar
fashion even in metallic GF liquids where there are no chemical bonds. Starr
et al. 161 have investigated what measure of dynamic heterogeneity may
appropriately quantify the size scales of the CRR envisioned by AG based on
simulations of a coarse-grained polymer melt, where $S_{c}$ has been
calculated. It has been shown that $S_{c}$ is inversely proportional to the
average string length $L$ to a good approximation, as illustrated in Figure
12b. Moreover, taking the size of the CRR as $z=L/L_{A}$ with $L_{A}$ being
the value of $L$ at $T_{A}$, the linear relation between $\ln\tau_{\alpha}$
and $z/T$ is confirmed by the analysis in Figure 12b. Figure 12d also verifies
the linear relation between $\ln\tau_{\alpha}$ and $1/TS_{c}$. Further,
$L/L_{A}$ has been shown to track the normalized activation free energy,
$\Delta G/\Delta G_{0}$. 20 Therefore, these results convincingly demonstrate
that the dynamic strings provide a quantitative realization of the
hypothetical CRR of AG. 18

In the string model of glass formation, applicable in the $T$ range below
$T_{A}$ where relaxation is non-Arrhenius, the activation free energy $\Delta
G$ for structural relaxation is proportional to the average string length $L$
normalized by its value at the onset temperature $T_{A}$. As a formal
extension of TST, 68, 66 the string model of glass formation defines the
$T$-dependent activation free energy as $\Delta G(T)=\Delta G_{0}(L/L_{A})$,
leading to the following expression for $\tau_{\alpha}$,

$\tau_{\alpha}=\tau_{0}\exp\left(\frac{\Delta
G_{0}}{k_{B}T}\frac{L}{L_{A}}\right).$ (20)

The parameter $\tau_{0}$ can be eliminated from a knowledge of $\tau_{\alpha}$
at $T_{A}$, 117 and $\Delta H_{0}$ may be determined from the Arrhenius
equation (eq 2) in the high $T$ regime where standard TST is assumed to be
applicable as a descriptive framework for liquid dynamics, resulting in the
following equation with $\Delta S_{0}$ being the only fitting parameter,

$\tau_{\alpha}=\tau_{\alpha}(T_{A})\exp\left(\frac{\Delta H_{0}-T\Delta
S_{0}}{k_{B}T}\frac{L}{L_{A}}-\frac{\Delta H_{0}-T_{A}\Delta
S_{0}}{k_{B}T_{A}}\right).$ (21)

Notably, simulation studies indicate that collective motion does not
completely vanish in the Arrhenius regime, 161 i.e., $L_{A}$ is not equal to
unity. Moreover, the string model asserts that $\Delta S_{0}$ cannot be
neglected, but the spirit of the AG model is certainly preserved in this
model, despite this and other technical differences. 20 Therefore, the essence
of the string model of glass formation is that the activation free energy in
the high $T$ Arrhenius regime is ‘renormalized’ by the $T$-dependent factor
$L/L_{A}$ quantifying the change in the extent of collective motion in the
non-Arrhenius regime below $T_{A}$.

Figure 13: String model description of the relationship between
$\tau_{\alpha}$ and $L$. (a–c) Results for a simulated coarse-grained polymer
melt with variable pressure, chain stiffness, and cohesive interaction
strength, respectively. Lines indicate $\ln(\tau_{\alpha}/\tau_{0})=\Delta
G_{0}(L/L_{A})/k_{B}T$. Adapted with permission from refs 217, 121, and 122.

Equation 20 of the string model has been quantitatively confirmed in
simulations of a number of polymeric and other GF liquids, including knotted
ring and star polymer melts, 242, 243, 244 thin films on solid substrates,
107, 108 polymer nanocomposites having a range of concentrations and polymer-
surface interaction strengths, 117 polymer melts with variable pressure, 119,
120, 217 chain stiffness, 121 and cohesive interaction strength, 235, 122
polymer nanofibers, 118 metallic liquids, 245 and superionic
$\mathrm{UO}_{2}$. 246 For the purposes of illustration, we show in Figure 13
that the string model describes well the simulation results in a coarse-
grained polymer melt, where the pressure, chain stiffness, or cohesive
interaction strength is systematically varied. The string model is further
justified by a theoretical analysis by Freed, 166 where TST is extended to
account for stringlike cooperative barrier crossing events in GF liquids.
However, much work needs to be done to extend the string model to a wide range
of other materials. We again emphasize that even standard TST in the Arrhenius
regime remains in a relatively rudimentary theoretical state.

In passing, we mention that relaxation in GF liquids occurs as a multi-stage
hierarchical process. 247 While we evidently focus on the long-time structural
relaxation process involving both large-scale diffusive molecular motion and
momentum diffusion, there is a ‘fast’ relaxation process dominated by the
inertial motion of the molecules whose amplitude grows upon heating. It is
worth pointing out that the fast dynamics can be considered in a unified way
with the long-time structural relaxation within the string model, as indicated
in a recent work. 247

###  5.3 Temperature Dependence of String Length

The string model 20 not only identifies the CRR with the stringlike
cooperative motion, but it also provides an explanation for the origin and
geometrical nature of these strings based on the framework of treating them as
‘initiated equilibrium polymers’. 50 This has been discussed in detail in ref
20. Here, we briefly review the most interesting predictions from the living
polymerization model of strings.

In the model of initiated equilibrium polymerization, 50 the strings are
dynamic or ‘equilibrium’ polymers that form and disintegrate in equilibrium.
These dynamic polymers have a $T$-dependent average length, or polymerization
index, $L\equiv\langle L\rangle$. The fraction of linked mobile particles
$\Phi$ serves as the order parameter for the self-assembly process of strings,
which is related to $L$ via the relation,

$L=\frac{1}{1-\Phi+\delta r/2},$ (22)

where $\delta r$ is the ratio of the initiator to the monomer volume fraction
$\phi_{0}$. $\Phi$ is limited to the range between $\delta r$ and $1$. Taking
the onset temperature $T_{A}$ as the reference, along with the approximation
of $\delta r\approx\Phi_{A}$ with $\Phi_{A}$ being the value of $\Phi$ at
$T_{A}$, eq 22 can be written as 20

$L=\frac{L_{A}(1-\Phi_{A}/2)}{1-\Phi+\Phi_{A}/2}.$ (23)

In the living polymerization model, $\Phi$ is the extent of polymerization
defined by the fraction of monomers forming polymeric structures, and, in
simulations, this quantity can be interpreted to be the fraction of the highly
mobile particles participating in the strings. The variation of $\Phi$ with
$T$ is well described by the prediction from the polymerization model, 50

$\Phi=1-\phi(T)/\phi_{0},$ (24)

where the explicit expression for $\phi(T)$ is given in refs 20 and 50. $\Phi$
is directly related to $S_{c}$ in the theory of equilibrium polymerization, a
relation that has been validated in simulations of a coarse-grained GF polymer
melt, 20 where $\Phi$ and $S_{c}$ are independently calculated. $\Phi$ is
notably much easier to calculate than $S_{c}$ in simulations. As we have
discussed earlier in Figure 12, a simulation study based on the same polymer
melt has also confirmed an inverse scaling between the average string length
$L$ and $S_{c}$ to a very good approximation, 161 a basic tenet of the AG
theory 18 and the thermodynamic relationship in the equilibrium polymerization
model. 51

While the full expression for $L$ predicted by the string model 20 is
complicated mathematically, this expression can be simplified by using the
high $T$ expansion of $\Phi$, 50

$L=L_{A}\left(1-\frac{\Phi_{A}}{2}\right)\left\\{1+\frac{\Phi_{A}}{2}\left[1+\phi_{0}\exp\left(-\frac{\Delta
G_{p}}{k_{B}T}\right)\right]\right\\},$ (25)

where the free energy $\Delta G_{p}=\Delta H_{p}-T\Delta S_{p}$ describes the
thermodynamics of string polymerization rather than the activation free energy
of the fluid. Equation 25 is expected to be valid in the $T$ range above
$T_{c}$ but below $T_{A}$, which we have confirmed in simulations of coarse-
grained polymer melts with variable cohesive energy and pressure. 235, 120 It
is an important feature of equilibrium polymerization that the excess degree
of polymerization in the high $T$ regime, $L_{A}-1$, is directly related to
the plateau of $L$ in the low $T$ regime, where both the degree of
polymerization and the fluid configurational entropy saturate to finite values
associated with the fully polymerized state. This feature of the
polymerization model has highly non-trivial implications for the string model
of glass formation, since it links collective motion in the high $T$ Arrhenius
regime to the scale of collective motion in the equilibrium glass state, where
relaxation again becomes Arrhenius in the string model of the dynamics of GF
liquids under equilibrium conditions. Of course, equilibrium conditions at low
$T$ are normally difficult to achieve in practice, so this prediction is
somewhat of academic interest, just as the question of whether or not $S_{c}$
actually vanishes at a finite $T$.

Figure 14: $T$ dependence of the thermodynamic and dynamic properties of glass
formation predicted by the string model, along with comparisons with
simulation results of a coarse-grained polymer melt. (a) $L$ and (b) $S_{c}$
as a function of $T$. (c) Angell plot of $\log\tau_{\alpha}$. Solid and dashed
lines correspond to the full form of the string model and its approximant
based on the high $T$ expansion, respectively. The filled regions indicate the
$T$ regime below $T_{\mathrm{g}}$ estimated from the full string model, which
is determined by the condition, $\tau_{\alpha}=100$ s, corresponding to
$10^{14}$ in standard reduced Lennard-Jones units. Adapted with permission
from ref 20.

Figure 14a examines the validity of the polymerization model in describing the
variation of $L$ with $T$ determined from simulations of a coarse-grained
polymer melt. 20 As can be seen, the string model, either in its full form
(given in ref 20) or its simplified form based on the high $T$ expansion (eq
25), provides an excellent description of the available simulation data, which
are restricted to a $T$ range between $T_{A}$ and $T_{c}$. However, large
differences arise at low $T$ between the full and simplified string models,
which has important implications for glass formation. While $L$ from the high
$T$ expansion increases rapidly upon cooling at low $T$, the full form for $L$
instead exhibits a plateau at low $T$. Since $L$ inversely scales with $S_{c}$
in the string model, we see from Figure 14b that the string model makes
similar predictions for $S_{c}$. Note that the plateau of $S_{c}$ at low $T$
in the full string model is reminiscent of the prediction from the GET for
fully flexible polymer melts, as discussed in Section 4.2. We note that a