recent study on a metallic GF liquid, where the estimation of $S_{c}$ is less
difficult than for polymers because the large vibrational contributions
associated with molecular bonding to the fluid entropy are absent, has
revealed the presence of an extended low $T$ plateau in $S_{c}$.
2020_PRB_101_014113

Combing the functional form for $L$ provided by the living polymerization
model with eq 20, the string model goes beyond the relatively high $T$ regime
of glass formation accessible to present simulations and make predictions for
$\tau_{\alpha}$ over the entire $T$ regime of glass formation. This is
illustrated in Figure 14c, where we show the Angell plot of
$\log\tau_{\alpha}$ determined from simulations, along with the descriptions
based on the string model. It is evident that the full string model indicate a
return to Arrhenius relaxation in the glass state, albeit with a higher
effective activation free energy. We emphasize that the string model 20
implies no divergence in $\tau_{\alpha}$ at any finite $T$. This is in accord
with the GET predictions for flexible polymer melts 136 and the observations
on a number of liquids, such as water, silica, and metallic GF liquids, 248
which show a ‘fragile-to-strong’ transition upon sufficient cooling,
corresponding to a return to Arrhenius relaxation. The interesting thing about
GF liquids exhibiting a fragile-to-strong transition is that the Arrhenius
relaxation seems to occur in a $T$ regime where the liquid is apparently fully
in equilibrium. Note that the position of the transition to a low $T$
Arrhenius behavior as well as the fragility of glass formation and the low $T$
plateau in the activation energy all depend on $\Phi_{A}$ in the string model,
a quantity determined in the Arrhenius regime.

###  5.4 String Model and Phenomenological Models of Glass Formation

While the string model 20 aims to describe the dynamics of GF liquids in terms
of parameters and quantities with well-defined meanings, there are popular
models of glass formation that are highly phenomenological in nature. Here, we
utilize the string model to understand some of these phenomenological models,
which has been discussed in refs 20 and 122. In particular, we show that the
string model enables the rationalization of the empirical fitting function for
$\tau_{\alpha}$ introduced by Rössler and coworkers 86, 87, 88 and the Mauro-
Yue-Ellison-Gupta-Allan (MYEGA) model. 83

As discussed in Section 5.3, the order parameter $\Phi$ for the assembly of
mobile particles into strings can be derived from the equilibrium
polymerization theory. However, the exact mathematical description of $\Phi$
is complicated. Douglas and coworkers 51, 52 found that a simple two-state
model can be used to describe $\Phi$ in the cooperative polymerization model,

$\Phi=\frac{1}{1+\exp[(\Delta H_{p}-T\Delta S_{p})/k_{B}T]}.$ (26)

This two-state expression arises in many physical contexts and has often been
further approximated by expanding around the free energy of association about
the polymerization transition point $T_{p}$ corresponding to the condition of
$\Phi=1/2$, which gives rise to the following approximation, 52

$\Phi\approx\frac{1}{1+\exp[-\Delta
H_{p}(T-T_{p})/k_{B}T_{p}^{2}]}\equiv\frac{1}{1+\exp[(T-T_{p})/\mathcal{D}_{0}]},$
(27)

where $\mathcal{D}_{0}\equiv k_{B}T_{p}^{2}/|\Delta H_{p}|$ reflects the width
of the clustering transition. Reference 51 provides more details regarding the
above argument. The two-state model predicts that $L/L_{A}$ should equal $2$
at $T_{p}$ at which the $T$ dependences of both $\Phi$ and $s_{c}$ exhibit an
inflection point, in accord with the observation of Ngai and coworkers 249,
250 that the apparent activation energy near $T_{c}$ is normally about twice
the high $T$ activation energy. In the Stockmayer fluid of dipolar particles
251 and tabletop measurements of driven magnetic particles thermalized by
vertical shaking, 252 the average string length $L$ has been observed to be
generally near $2$. In the GET, the crossover temperature $T_{c}$ of glass
formation is identified by the temperature at which the $T$ dependence of
$Ts_{c}$ has an inflection point, so we may expect a close connection between
$T_{p}$ and $T_{c}$.

If the above approximations for $\Phi$ are combined with the string model, an
approximation for the activation free energy emerges as,

$\Delta G_{0}(L/L_{A})\approx\Delta
G_{0}\\{1+\exp[-(T-T_{p})/\mathcal{D}_{0}]\\},$ (28)

which is the same form as the expression for the activation free energy
proposed by Rössler and coworkers. 88 In this two-state model expression, the
activation free energy is somewhat artificially broken into a ‘local’
contribution $\Delta G_{0}$ and a ‘cooperative’ contribution representing the
stringlike collective motion. This type of two-state model has a long history
in the modeling of GF liquids. 253, 254 The ECNLE model of Schweizer and
coworkers 56, 57, 58 assumes a similar decomposition of the activation free
energy in local and collective parts where the collective motions are
conceived to arise from the energetic costs arising from the emergence of
caging in liquids at low $T$ or high densities. Tanaka and coworkers 255 have
recently invoked a mean-field two-state model of liquid dynamics that seems
similar in spirit for the activation energy for structural relaxation in
water. Note that eq 28 is a rough mean-field model that neglects the
distribution of barrier heights associated with the structural organization of
the dynamic heterogeneity, an aspect that is emphasized in the string model.
20

We may also formally recover the recent fashionable MYEGA expression 83 for
the $T$ dependence of $\tau_{\alpha}$ by approximating $L$ by its high $T$
approximation in its well-known equilibrium polymerization model estimate,
256, 251, 257, 258

$L\approx L_{0}\exp[\Delta H_{p}/(2k_{B}T)],$ (29)

where $L_{0}$ is a constant determined by molecular parameters of the
polymerization model. The Arrhenius variation of $L$ holds exactly for purely
uncooperative or ‘isodesmic’ equilibrium polymerization and provides a generic
approximation for $L$ for equilibrium polymerization processes near and above
the thermodynamic transition temperature $T_{p}$. However, this simple
relation for $L$ no longer holds generally at low $T$ in the case in which the
polymerization process becomes cooperative. 20 Correspondingly, we may thus
expect the MYEGA equation, 83 as well as the popular VFT relation, 78, 79, 80
to only hold over a limited $T$ range. Therefore, we can also understand the
success of these phenomenological fitting functions for $\tau_{\alpha}$ based
on the string model.

###  5.5 Opportunities and Challenges

Although stringlike collective motion is a conspicuous feature of the dynamics
of simulated GF liquids broadly in the regime where the dynamics becomes non-
Arrhenius and has also been observed in GF colloidal fluids, 259, 260 the
grain boundaries of colloidal and granular systems, 261, 262 and colloidal
fluids approaching their melting, 263, 264 there is currently no obvious
experimental method for identifying the strings in molecular fluids. There is
clearly a need for getting information about the strings in materials of
practical interest.

In our recent works, 217, 175 we hypothesized that the activation volume
$\Delta V^{\\#}$ might be related to the average string length $L$, based on a
number of studies 265, 225, 266, 267, 268, 229, 269, 270, 271, 272, 273
showing that the growth of $\Delta V^{\\#}$ upon cooling occurs in parallel
with the apparent activation energy of GF liquids below $T_{A}$. If such a
relation could be established between $\Delta V^{\\#}$ and $L$, then it would
provide an accessible method for determining $L$, the central quantity in the
string model of the dynamics of GF liquids. Since measurements of the length
scales characterizing the extent of collective motion are often quite
challenging to make and involve advanced instrumentation that is often not
available in an industrial research setting, while $\Delta V^{\\#}$ can be
straightforwardly measured experimentally, a direct relation between $\Delta
V^{\\#}$ and $L$ is particularly appealing from an experimental viewpoint.
Unfortunately, our study based on simulations and the GET indicates that there
is no direct relation between $\Delta V^{\\#}$ and the extent of collective
motion quantified by $L$. 217, 175

Although the strings can be directly observed in measurements of colloidal
suspensions, we continue the search for an accessible experimental metric for
$L$ based on correlation studies. For instance, previous works 274, 275 on
energy and mobility fluctuations in the glassy interfacial dynamics of
nanoparticles indicate that the color of these fluctuations correlates
strongly with $L$. The basic idea here is that mobility fluctuations
associated with mobile particles, such as strings, should lead to observable
effects on properties that measure fluctuations in the mobility, such as
electrical conductivity. 276, 220, 277 In the future work, we need to check
these relationships between noise color associated with mobility fluctuations
and $L$ in cooled liquids below $T_{A}$ and see if we can develop an
accessible method for estimating $L$ in molecular and polymeric liquids. There
is also the prospect of observing stringlike collective motion on the surface
of crystalline and GF materials in real space based on ultrahigh resolution
imaging. 278, 279 The imaging studies, even in the case of the strings in the
colloidal systems, 280, 281 needs improvement in order to compare
quantitatively with the string model. Another promising possibility seems to
be offered by single molecule fluorescence microscopy. 282, 283 This technique
allows for investigations of the rotational motion of single fluorescence
probes with different sizes doped in a GF polymer, leading to the
identification of a characteristic length scale that might be related to $L$,
but much work remains to be done to establish such a relationship
quantitatively. We look forward to participating in these developments in the
future.

##  6 Other Entropy-Based Models of Liquid Dynamics

The strong correlation between the fluid entropy and the dynamics of GF
liquids has led to alternative models to the AG model 18 and its antecedents,
the GET and the string model discussed above. For completeness, we provide a
brief overview of these alternative models, with an emphasis on the hints
provided by these model into how the GET might be developed in the future.

###  6.1 Random First-Order Transition Theory

The RFOT theory 71, 72, 73, 74 is close in spirit to the AG model 18 in the
sense that it is predicated on the idea of dynamic domains in cooled liquids
which grow upon cooling, accompanied by a corresponding drop in the
configurational entropy, $S_{c}$. This mean-field model, initially formulated
in the context of spin glass materials, 71 has also elements similar to the GD
model 17 of polymer glass formation in the sense that it predicts $S_{c}$ to
vanish at a Kauzmann temperature, $T_{K}$.

The RFOT theory 71, 72, 73, 74 envisions that the liquid is divided into
metastable regions with a characteristic correlation length $\xi$, describing
the average ‘entropic droplet’ or ‘mosaic’ size. A scaling relation between
$\xi$ and $S_{c}$ is suggested by a consideration of the balance between the
surface and bulk free energies of these regions. This theory also assumes that
the activation energy for relaxation scales with $\xi$ with a power. It should
thus be appreciated that the general qualitative findings of this interesting
mean-field model have many points of contact with the AG 18 and GET models. 19
A particularly interesting aspect of the RFOT theory is that it focuses on the
geometrical size of what may be identified with the CRR of the AG model,
rather than the number of particles within these hypothetical clusters.

As in the AG model, 18 a basic problem about the implementation of the RFOT
model 71, 72, 73, 74 for polymeric fluids is that $S_{c}$ can only be roughly
estimated based on similar approximations introduced by Bestul and Chang 70
and AG, 18 where the uncertainties are especially large in polymer fluids
because of the relatively large vibrational entropy associated with molecular
bonding in such fluids. 161 We note that an interesting methodology for
estimating $S_{c}$ for non-molecular GF liquids based on a RFOT framework has
been introduced, 2014_PNAS_111_11668 so this situation could conceivably
change in the future if this methodology can be extended to treat molecular GF
liquids. Steven and Wolynes 284 have performed a test of the RFOT model based
on estimates of $S_{c}$ obtained from specific heat data for a range of small-
molecule liquids where this type of $S_{c}$ estimate is expected to be a
reasonable rough approximation. We view the estimation of $S_{c}$ from
specific heat measurements to be inherently unreliable for quantitative
analysis, however, and we strictly avoid any discussion of this method for
estimating $S_{c}$. Although $S_{c}$ can be estimated for molecular fluids
from simulations, 161 including polymers, the method is limited in the $T$
range in which equilibrium simulations can be performed. The RFOT theory 71,
72, 73, 74 offers no method for calculating $S_{c}$ and other parameters of
the model. The GET 19 is founded on a statistical mechanical model of polymer
fluids, which enables calculations of all the thermodynamic properties in
terms of molecular parameters, including $S_{c}$. The LCT for polymer
thermodynamics, 137, 138 which is the thermodynamic component of the GET, 19
has been validated by many studies. 170 By combining the LCT with the AG
model, 18 the GET 19 is a highly predictive theory with no free parameters
beyond the molecular and thermodynamic parameters governing the thermodynamic
state of polymer fluids. Of course, the extension of the GET to include the
entropy of activation, a quantity neglected by AG, 18 makes the GET no longer
fully predictive. We consider below how this limitation of the GET might be
overcome. An interesting difference between the AG and RFOT models is that the
dimensionless activation free energy is deduced to scale in the RFOT theory
165, 285 as $\Delta G/k_{B}T\sim 1/S_{c}$, which is consistent with the
phenomenological expression first noted by Bestul and Chang rather than the AG
expression, $\Delta G/k_{B}T\sim 1/(TS_{c})$. 165 This RFOT expression is then
consistent with the GET when the enthalpy $\Delta H_{0}$ of activation is set
to zero, in which case the free energy of activation is taken to be dominated
by the entropy of activation, $\Delta S_{0}>0$, an assumption suitable for the
description of systems where particle interactions are dominated by repulsive
interactions, as in the case of hard-sphere fluids at constant density. Note
that the RFOT model emphasizes the configurational entropy per unit mass
rather than the entropy density, which has ramifications for considering the
glass formation of materials at constant pressure.

It might be possible to combine the LCT with the activation free energy
scaling arguments underlying the RFOT approach 71, 72, 73, 74 to glass
formation, but the model seems to involve unspecified parameters that we
currently do not know how to specify and various assumptions that require
careful assessment, as we have tried to perform in the GET. We plan to explore
this alternative approach to the dynamics of GF liquids in the future.

At this stage, we mention an interesting exploration of the compatibility of
the string model of glass formation and the RFOT scaling arguments for the
dynamics of GF liquids by Starr et al. 161 This work adopts the hypothesis
that the ‘string’ clusters exhibiting cooperative exchange motion, identified
successfully as having the main attributes of the hypothetical CRR of AG, 18
can also be identified with the hypothetical ‘entropic droplets’ of the RFOT
model as the logical compatibility of the AG and RFOT models would formally
suggest. Verification of such a relation would be important because the RFOT
theory offers no explicit algorithm for what might constitute an ‘entropic
droplet’ and is equally vague as the AG model regarding the physical nature of
the CRR. It has often been assumed in theoretical discussions of the AG and
RFOT models that the CRR are more or less spherical, 18, 286, 287, 288 in line
with the term droplet, but the basis of this assumption is rather unclear.
Klein and coworkers 289, 290, 291, 292 have considered the role of
fluctuations on the formation of dynamic clusters in model GF liquids,
indicating that while compact clump-like structures are found in mean-field
theory descriptions of GF materials, the clusters become progressively more
ramified, i.e., fractal, when fluctuation effects associated with the short-
range attractive interactions of real liquids are incorporated into the
theory, a finding having profound implications for both glass formation and
crystal nucleation at high undercooling. 293, 294 Resolving the physical
nature of the ‘entropic droplets’ is then a central question in developing the
RFOT theory into a quantitative model of the dynamics of GF liquids and we
discuss this matter below.

The RFOT scaling expression for the activation free energy reads $\Delta
G\sim\xi^{\psi}$, where $\psi$ is a surface tension exponent whose value
depends on the assumed geometry of the entropic droplets. See ref 161 for an
extended discussion of $\psi$. As noted above, identifying $\xi$ with the
radius of gyration of the strings $R_{\mathrm{g},\mathrm{string}}$ indicates
that $\Delta G=AR_{\mathrm{g},\mathrm{string}}^{\psi}$ with $A$ and $\psi$
being adjustable constants. 161 This preliminary test of the RFOT model
indicates compatibility between the AG and RFOT models in the qualitative
sense that the free energy barrier should grow with the size of regions
exhibiting cooperative motion. However, the exponent $\psi$ is not compatible
with compact liquid droplet estimates of this scaling exponent, suggesting
that these field excitations should be modeled as being fractal structures, as
in the case of the string model. We may also view this scaling of the
activation free energy with as being compatible with the string model based on
the observed scaling relation, 161, 245 $R_{\mathrm{g},\mathrm{string}}\sim
L^{\nu}$, where $\psi=1/\nu$ in this interpretation of the $\Delta G$ scaling
with $1/\nu$ being the fractal dimension of the self-avoiding polymeric
strings. This interpretation of the free energy scaling with the average
string size would lead to $\psi=5/3$ if we assume the Flory estimate of $\nu$.
295 This estimate of $\psi$ roughly accords with the value of $\psi\approx
1.3$ estimated from simulations of polymer melts by Starr et al. 161

Reference 161 has discussed the theoretical RFOT estimates of $\psi$ in
comparison to computational estimates, along with the implications of the
observed scaling of $\Delta G$ with $R_{\mathrm{g},\mathrm{string}}$, so we do
not elaborate on this matter further here. Our main point here is that the
RFOT approach appears to be promising, provided that a more realistic physical
description of the ‘entropic droplets’ is considered. Despite the apparent
success of the RFOT scaling, there are still issues that must be faced. In
particular, this scaling approach still involves a number of practical issues
regarding predictions of relaxation times for particular materials. We
currently do not understand the meaning of the parameters $A$ and $\psi$ in
the RFOT model so that quantitative predictions of relaxation times remain
elusive. In our view, there seems to be no demonstrated advantage of the RFOT
model 71, 72, 73, 74 over the AG model. 18 This exercise is valuable, however,
in the sense that it provides a new perspective on the AG model and indicates
that further attention should be given to the geometrical structure of the
strings in developing the entropy theory of glass formation into a more
general and validated theoretical framework. This effort remains a work in
progress.

###  6.2 Rosenfeld and Excess-Entropy Scaling

There is another interesting approach to the dynamics of liquids based on an
attempt to interrelate the rate of diffusion and relaxation to the fluid
entropy. Based on a conception going back to Boltzmann that both the
thermodynamics and dynamics of liquids is dominated by hard-core
intermolecular interactions, 296 Rosenfeld 75, 76, 77 developed a scaling
model of the dynamics of liquids emphasizing a formal mapping of liquids onto
hard spheres based on a consideration of the entropy in the liquid state,
$S_{\text{liq}}$, relative to the entropy in the ideal gas state,
$S_{\mathrm{id}}$, rather than the crystal or glass state as in the case of
AG. 18 See ref 77 for a detailed discussion of Rosendeld’s works. This ‘excess
entropy’, relative to the ideal gas, denoted as
$\widehat{S}_{\mathrm{ex}}=S_{\text{liq}}-S_{\mathrm{id}}$, should not be
confused with other definitions of the ‘excess entropy’ $S_{\mathrm{exc}}$ in
the context of the AG model. In particular, Rosenfeld argued that transport
properties of liquids such as the diffusion coefficient $D$ should scale with
$\widehat{S}_{\mathrm{ex}}$ as,

$D\sim\exp(B\widehat{S}_{\mathrm{ex}}),$ (30)

where $B$ is an adjustable positive constant. 77 This particular exponential
variation of dynamic properties is termed ‘Rosenfeld scaling’ and the
description of $D$ and $\tau_{\alpha}$ by more general functional forms is
termed ‘excess-entropy scaling’. 77 These two distinct terms indicate that
Rosenfeld scaling has its limitations, as we shall discuss below. There is
also a popular extension of the Rosenfeld scaling model introduced by Dzugutov
297 based on an approximation of $\widehat{S}_{\mathrm{ex}}$ by its two-body
estimate from the radial distribution function $g(r)$,

$S_{2}=-2\pi\rho\int_{0}^{\infty}\\{g(r)\ln g(r)-[g(r)-1]\\}r^{2}dr,$ (31)

where $r$ is the spatial distance. For readers familiar with the AG model, 18
Rosenfeld scaling appears to be rather odd, but it must be remembered that the
sign of $\widehat{S}_{\mathrm{ex}}$ is opposite to $S_{c}$. Note also that
$\widehat{S}_{\mathrm{ex}}$ is in the numerator of eq 30, while $S_{c}$ is in
the denominator in the corresponding AG relation for $D$ and $\tau_{\alpha}$.
It is then easy to appreciate why confusion is sometimes encountered regarding
how the fluid entropy should be related to liquid dynamics. Interestingly,
simulation results have indicated that $\widehat{S}_{\mathrm{ex}}$ actually
scales inversely with the product of the numerically estimated $S_{c}$ and $T$
in model GF liquids, 298 as required by self-consistency between Rosenfeld
scaling and the AG model, a relation that seems rather mysterious to us.
Particularly clear expositions of the excess-entropy scaling approach to
liquid dynamics have been made by Chakraborty and coworkers 299 and Sastry and
coworkers. 300 An extensive and excellent review on Rosenfeld scaling has
recently been given by Dyre. 77 Speedy et al. 301 have discussed the
limitations of the hard-core approximation in describing the dynamics of
liquids.

We can calculate $\widehat{S}_{\mathrm{ex}}$ from the LCT 137, 138 to
implement this type of entropy theory. However, we did not do so because
numerous simulations have shown that while this model seems to hold rather
well in the high $T$ regime, large deviations from Rosenfeld scaling are
normally observed in GF liquids at low $T$. 77 In many fluids, the simple
exponential form of $D$ and $\tau_{\alpha}$ in terms of
$\widehat{S}_{\mathrm{ex}}$ is superseded by other functional forms that can
be applied with impressive accuracy over a wide range of temperatures and
densities, accounting for the intense interest in this approach to
understanding and correlating the dynamic properties of liquids in terms of
thermodynamic properties of liquids. Unfortunately, the particular functional
form relating transport properties to $\widehat{S}_{\mathrm{ex}}$ has been
found to be highly dependent on molecular structure in molecular fluids 77,
302 so that this approach to liquid dynamics offers no direct general
relationship between transport properties and $\widehat{S}_{\mathrm{ex}}$.
Thus, the applicability of this approach is largely limited to atomic and
small-molecule liquids. Chopra et al. 303 have discussed the origin of the
failure of density-temperature scaling in molecular fluids and related this
change of scaling to changes in the density-temperature scaling of transport
properties, a phenomenon that is deeply related to the cooperative dynamics of
GF liquids, as we have recently found elsewhere. We thus do not see how one
could develop a quantitative theory of the dynamics of real polymeric GF
liquids based on this theoretical framework since many polymer materials are
only thermally stable against degradation below $T_{A}$, i.e., non-Arrhenius
collective dynamics is inherent to this class of materials. Nonetheless,
recent works suggest that the excess-entropy scaling approach has some
important conceptual ideas to offer for the future development of the GET, 19
which we briefly discuss below.

Given the apparently general observation that the excess entropy scaling is
restricted to elevated temperatures and the emphasis of this model on
repulsive interactions, it is natural to compare this model to the standard
theory of activated transport where $D$ formally scales as, $D\sim\exp(\Delta
S_{0}/k_{B})$, for molecular diffusion in systems in which attractive
interactions are entirely neglected. This might then provide an approach to
calculating $\Delta S_{0}$, one of the most elusive quantities in TST. 103,
217 We hope to pursue this approach to understanding and numerically
estimating $\Delta S_{0}$ in the future.

Finally, recent studies appear to indicate that $\widehat{S}_{\mathrm{ex}}$
might enable the estimation of the characteristic temperatures of glass
formation. For instance, refs 304 and 305 suggest that the onset temperature
$T_{A}$ may be calculated as the temperature at which
$\Delta\widehat{S}_{\mathrm{ex}}$, defined as $\widehat{S}_{\mathrm{ex}}$
minus it two-body estimate $S_{2}$, changes sign. It has also been suggested
that the crossover temperature $T_{c}$ might be inferred from the $T$
variation of $\widehat{S}_{\mathrm{ex}}$. 304, 306 It would be an invaluable
advance to the GET to have an alternative thermodynamic-based approach for
calculating $T_{A}$, and we plan to test this proposal against independent
estimates of $T_{A}$ for our polymer model in the future. Over time, we hope
to see the various ‘pieces’ of the entropy theory of glass formation to grow
together to form a highly predictive and validated framework from theoretical,
computational, and experimental perspectives.

##  7 Summary and Outlook

The entropy perspective on the dynamics of glass-forming liquids and the
general conception that collective motion accompanies the drop of
configurational entropy has been developing steadily over the last century,
and over this period, empirical correlations between thermodynamic and dynamic
properties have gradually been superseded by quantitative relations organized
around increasingly sophisticated theoretical frameworks for quantifying the
dynamics and thermodynamics of condensed materials and for quantifying
collective motion and its physical consequences on relaxation and diffusion.
The generalized entropy theory and the string model are just the most recent
manifestations of this conceptual development, along with the necessary
experimental and computational works that give credence to these models
through validation studies.

Although much progress has been made, there are still many ‘gaps’ in the
entropy approach to the dynamics of glass-forming liquids that will require
significant effort in the future to obtain a more predictive model of the
dynamics of glass-forming liquids. For example, our understanding of the
activation free energy parameters has not advanced very much beyond the
descriptions introduced by Eyring and coworkers. 66, 67, 68 A significant
effort also needs to be made to develop a fundamental understanding of the
activation parameters in the Arrhenius regime where the complicated effects of
dynamic heterogeneity are not an important issue. It is very helpful that
relaxation times are generally relatively short in the high temperature
Arrhenius regime, which is a favorable aspect for simulation studies that
could provide much information on which a sound theoretical treatment of the
activation energy parameters could be made. There are also many experimental
data now available for this purpose, so we are hopeful that much progress in
this direction will be made in the near future. Simulations have shown that
molecular topology can greatly influence the dynamics of polymeric glass-
forming materials, 243, 244 so computational and experimental investigations
of how molecular topology influence the high temperature activation energetic
parameters and the overall dynamics of glass-forming liquids promise to be a
future of development. We also anticipate that studies of nanocomposites, thin
supported films, polymers with solvent additives, and polymer blends all offer
promising areas for examining the activation free energy parameters and their
large influence on the dynamics of glass-forming liquids. In short, we
envision this type of study to form the foundation for a fundamental and
predictive theory of glass-forming liquids.

There are also many issues to resolve in the entropy theory relating to the
question of why dynamic heterogeneity in glass-forming liquids, such as the
strings, generally adopts structural forms and obeys the thermodynamics
consistent with existing models of self-assembly. 50, 51, 52, 20 Answering
these questions will require a theoretical understanding of a different order.
Equilibrium polymerization has long been understood to arise in connection
with the description of fluctuation effects in materials undergoing second-
order phase transitions, 307, 308, 309, 310, 311, 109, 312 and we expect that
a deeper understanding of glass formation within appropriate field theoretical
frameworks will naturally evolve in time. We also expect that the theoretical
concepts and methods derived from the alternative entropy theories of glass
formation, namely, the random first-order transition theory 71, 72, 73, 74 and
the excess-entropy scaling approach, 75, 76, 77 will play a large role in
developing the generalized entropy theory further. We await these theoretical
developments and quantitative computational investigations to validate these
new concepts. For the present, the generalized entropy theory and the string
model emphasized in the present work are promising working models. It seems
inevitable that many of the currently disparate models of glass formation will
come together to form a unified theory of glass formation. Hopefully, we will
not have to wait another century for these developments to occur.

## Biographies

Wen-Sheng Xu is a professor in the State Key Laboratory of Polymer Physics and
Chemistry at Changchun Institute of Applied Chemistry, Chinese Academy of
Sciences in China, where he has been a faculty member since 2019. He received
a B.E. degree in Materials Science and Engineering at Tianjin University in
2007 and a Ph.D. degree in Chemistry and Physics of Polymers at Changchun
Institute of Applied Chemistry, Chinese Academy of Sciences in 2012 under the
guidance of Professors Li-Jia An and Zhao-Yan Sun. From 2013 to 2018, he was a
postdoc first working with Professor Karl F. Freed at the University of
Chicago and then with Dr. Yangyang Wang at Oak Ridge National Laboratory. His
research is focused on the glass formation and rheology of polymeric
materials.

Jack F. Douglas is a NIST Fellow in the Materials Science and Engineering
Division of the National Institute of Standards and Technology (NIST) at the
facility in Gaithersburg, Maryland. He obtained a B.S. degree in Chemistry and
an M.S. degree in Mathematics from Virginia Commonwealth University and then a
Ph.D. degree in Chemistry at the University of Chicago. After receiving his
Ph.D. degree, he was a NATO Fellow at the Cavendish Laboratory, Cambridge and
a NRC postdoctoral fellow at NIST before becoming a research scientist at
NIST. His field of research includes the equilibrium and dynamic properties of
polymer solutions and melts, fractional calculus and path integration, phase
separation and critical phenomena, relaxation processes in glass-forming
liquids, polycrystalline materials and nanoparticles, the elastic properties
of gels, and the thermodynamics and dynamics of self-assembly processes.

Zhao-Yan Sun is a professor at Changchun Institute of Applied Chemistry,
Chinese Academy of Sciences in China and a deputy director of the State Key
Laboratory of Polymer Physics and Chemistry. She received a Ph.D. degree in
Polymer Chemistry and Physics at Jilin University in 2001. She was a
postdoctoral fellow in the Department of Chemistry at University of Dortmund
from 2001 to 2002. Since 2003, she has been a faculty member at the present
institution, initially as an assistant professor and then an associate
professor, before becoming a full professor in 2010. She was awarded the Young
Chemistry Award of the Chinese Chemical Society in 2005. Her research group
focuses on the structure and dynamics of polymers and nanocomposites and the
development of computer simulation methods.

W.-S.X. acknowledges the support from the National Natural Science Foundation
of China (No. 21973089). Z.-Y.S acknowledges the support from the National
Natural Science Foundation of China (Nos. 21833008 and 21790344), the National
Key R&D Program of China (No. 2018YFB0703701), the Jilin Provincial science
and technology development program (No. 20190101021JH), and the Key Research
Program of Frontier Sciences, CAS (QYZDY-SSW-SLH027). J.F.D. thanks his long-
term collaborators, Karl F. Freed, Jacek Dudowicz, Francis W. Starr, Hao
Zhang, and Wen-Sheng Xu, and many postdocs associated with the research groups
at the National Institute of Standards and Technology, the University of
Chicago, the University of Alberta, Wesleyan University, and Changchun
Institute of Applied Chemistry, Chinese Academy of Sciences for the many
collaborative efforts over many years that made his contributions to the
entropy theory of glass formation possible. J.F.D also acknowledges the
National Institute of Standards and Technology for its long-term support of
this research effort. Z.-Y.S thanks Professor Li-Jia An for helpful
discussions on glass formation over the years. W.-S.X. thanks Professor Karl
F. Freed and Dr. Jack F. Douglas for numerous discussions concerning polymer
glass formation over the years. W.-S.X. also thanks Professors Li-Jia An and
Zhao-Yan Sun for their support and encouragement at every stage of his
academic career. We are grateful to the authors of ref 88 for sharing their
experimental data with us and to Dr. Beatriz A. Pazmiño Betancourt for help
with the visualization of strings in simulations.

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