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

Jack F. Douglas<EMAIL_ADDRESS>Materials Science and Engineering
Division, National Institute of Standards and Technology, Gaithersburg,
Maryland 20899, United States  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

###### Abstract

Numerous experimental and computational studies have established that most
liquids seem to exhibit a remarkable, yet poorly understood, property termed
‘thermodynamic scaling’ in which the structural relaxation time
$\tau_{\alpha}$, and many other dynamic properties, can be expressed in terms
of a ‘universal’ reduced variable, $TV^{\gamma}$, where $T$ is the
temperature, $V$ is the material volume, and $\gamma$ is a scaling exponent
describing how $T$ and $V$ are linked to each other when either quantity is
varied. Here, we show that this scaling relation can be derived by combining
the Murnaghan equation of state (EOS) with the generalized entropy theory
(GET) of glass formation. In our theory, thermodynamic scaling arises in the
non-Arrhenius relaxation regime as a scaling property of the fluid
configurational entropy density $s_{c}$, normalized by its value $s_{c}^{*}$
at the onset temperature $T_{A}$ of glass formation, $s_{c}/s_{c}^{*}$, so
that a constant value of $TV^{\gamma}$ corresponds to a reduced isoentropic
fluid condition. Molecular dynamics simulations on a coarse-grained polymer
melt are utilized to confirm that the predicted thermodynamic scaling of
$\tau_{\alpha}$ by the GET holds both above and below $T_{A}$ and to test
whether the extent $L$ of stringlike collective motion, normalized its value
$L_{A}$ at $T_{A}$, also obeys thermodynamic scaling, as required for
consistency with thermodynamic scaling. While the predicted thermodynamic
scaling of both $\tau_{\alpha}$ and $L/L_{A}$ is confirmed by simulation, we
find that the isothermal compressibility $\kappa_{T}$ and the long wavelength
limit $S(0)$ of the static structure factor do not exhibit thermodynamic
scaling, an observation that would appear to eliminate some proposed models of
glass formation emphasizing fluid ‘structure’ over configurational entropy. It
is found, however, that by defining a low temperature hyperuniform reference
state, we may define a compressibility relative to this condition,
$\delta\kappa_{T}$, a transformed dimensionless variable that exhibits
thermodynamic scaling and which can be directly related to $s_{c}/s_{c}^{*}$.
Further, the Murnaghan EOS allows us to interpret $\gamma$ as a measure of
intrinsic anharmonicity of intermolecular interactions that may be directly
determined from the pressure derivative of the material bulk modulus. Finally,
we show that many phenomenological relationships related to the glass
formation and melting of materials can be understood based on the Murnaghan
EOS and thermodynamic scaling.

###### keywords:

American Chemical Society, LaTeX

IR,NMR,UV

##  1 Introduction

The thermodynamics and dynamics of liquids and solids at equilibrium generally
depend on the interplay of inertial dynamics deriving from the thermal energy
of the molecules and structural effects associated with emergent density
correlations associated with the local tendency of the molecules to form low
energy particle configurations. The delicate balance between the kinetic and
potential energy contributions to the fluid properties can be altered by the
application of an applied pressure $P$ or other type of applied stress. It is
the coupling of these energetic contributions that somehow leads to the
thermodynamic scaling of dynamic properties, such as the structural relaxation
time $\tau_{\alpha}$, in terms of a reduced variable involving a product of
the temperature $T$ times the volume $V$ to a power $\gamma$ that quantifies
the coupling effect, namely, $TV^{\gamma}$. 1, 2, 3, 4 This scaling phenomenon
can be more precisely referred to as the ‘power-law density-temperature
scaling’, but we adopt the more notationally concise term ‘thermodynamic
scaling’, which is also consistent with conventional terminology in
experimental studies of this reduced variable description of the dynamics of
glass-forming (GF) materials. 1, 2, 3, 4

To quantitatively describe the thermodynamic scaling of dynamic properties, we
must have an equation of state (EOS) that describes how $P$ alters the volume
$V$ or density $\rho$ of the material. Such an EOS is expected to provide
insight into the material parameter, $\gamma$. An EOS formulation also offers
the prospect of developing a universal reduced variable description for
properties of the special classes of fluids following Pitzer’s early reasoning
and approach to the reduced properties of liquids and gases near their
critical points. 5, 6, 7, 8 In the present work, however, we are interested in
the properties of fluids in the low $T$ regime where the fluid compressibility
approaches a vanishing small value upon cooling 9, 10, 11, 12, 13 rather than
diverging upon heating so that a more appropriate reference condition must be
chosen for the EOS that addresses the strong interparticle interactions in
highly condensed materials at low temperatures and high pressures.

Although linear elasticity provides an adequate framework for describing the
dynamics and thermodynamics of materials at low $P$, 14, 15, 16 the relatively
incompressible nature of liquids in comparison to gases means that relatively
large changes in $P$ are required to impart large changes in the density of
condensed materials. Thus, we require a theory that allows us to go beyond the
regime where simple linear elasticity is valid. The urgent need for such an
EOS theory for modeling geophysical phenomena 17, 18, 19, 20 and for
understanding high-impact phenomenon 21 in the 1930s led to the required
theoretical framework.

Murnaghan 22, 23, 24, 25 formulated the ‘integrated linear theory of finite
strain’, a theoretical framework that naturally resembles the formulation of
rubber elasticity from a continuum mechanics perspective. 26 This framework
simplifies when combined with assumptions drawn from the Debye theory of
solids and further assumptions drawn from the works of Grüneisen 27 and
Gilvarry 28, 29 quantifying the anharmonic interactions characteristic of real
materials. We discuss these anharmonic interactions and their quantification
below as these interactions play a central role in determining the
thermodynamic scaling exponent, $\gamma$.

Murnaghan’s general EOS for condensed materials subjected to large pressure
changes takes the remarkably simple form, 23, 30, 31

$P=(1/\gamma_{M})B_{o}[(V_{o}/V)^{\gamma_{M}}-1],$ (1)

where the ‘Murnaghan exponent’ $\gamma_{M}$ was originally interpreted as a
phenomenological material constant quantifying the $P$-$V$ relation,
reminiscent of the adiabatic expansion law of gases, and $B_{o}$ is the bulk
modulus $B$ at some reference condition indicated by the subscript ‘$o$’, a
convention which we also adopt for other quantities in the present paper. For
notational economy, eq 1 is termed the ME. $B_{o}$ is the reciprocal of the
isothermal compressibility $\kappa_{T,o}$ at a chosen reference condition,
which is evidently the key thermodynamic property in the above relationship,
along with $\gamma_{M}$. Later works provided a more fundamental
interpretation of the ME. In particular, Gilvarry 28, 29 showed that the EOS
of the Debye theory of solids corresponded to a specialization of the ME in
which $\gamma_{M}\equiv\gamma_{\mathrm{Debye}}=1/3$ and that the ME is a
natural extension of the Debye EOS to account for anharmonic intermolecular
interactions. Initially, anharmonicity was conceived of in terms of the extent
to which the interparticle potential deviated from a quadratic form, but later
it became appreciated that the normal modes of crystalline materials can
exhibit deviations from the ideal harmonic intermolecular interactions
associated with many-body interactions associated with bonding and packing
interactions in complex molecules such as polymers, as we discuss below. This
alternative type of anharmonicity, which we may term ‘configurational
anharmonicity’, can be highly variable in molecular fluids even if the
molecules share essentially the same local potential describing their van der
Waals interactions, as is often the case in polymeric materials. In
particular, configurational anharmonicity is altered by any factor that
influences molecular packing of molecules such as molecular rigidity, monomer
shape, cohesive interaction strength, and chemical heterogeneity. We also show
below that the generalized entropy theory (GET) 32 enables the analytic
calculation of measures of configurational anharmonicity and provides a
framework for understanding variations in this type of anharmonicity from a
molecular standpoint. We return to a discussion of potential anharmonicity in
Section 3.7, a phenomenon of particular significance for metals and other
solids composed of inorganic atomic species where the interatomic species can
be quite different from materials with van der Waals interactions, the main
concern of the present paper. Although the ME can be derived from specialized
models of crystalline solids with non-quadratic local potentials, its
derivation from continuum mechanics of materials subjected to large
deformations makes this EOS suitable for describing materials for which linear
elasticity does not apply. In practice, this means most real solid materials.

In the original derivation of the ME, 23, 30 $B(T)$ was simply expanded in a
Taylor series about $P=0$ and truncated after the leading first-order term and
then integrated to obtain eq 1 where $\gamma$ equals $\partial B/\partial P$
evaluated at $P=0$, as discussed below. Despite the remarkable simplicity of
this continuum theory derivation of the ME, this relation is supported by a
large body of measurements. 33, 34, 35, 36 Here, we point out the significance
of the $P$ dependence of $B$ in quantifying the strength of ‘anharmonic
interactions’ in solid materials, 37 as evidenced by deviations from linear
elasticity theory in macroscopic elasticity and thermodynamic measurements.

As noted before, real condensed materials normally exhibit significant
deviations from the idealized Debye theory arising from the inherently
anharmonic nature of interparticle interactions in the strongly condensed
state. 38 The practical consequences of these interactions are apparent in
observations of the thermal and electrical conductivity, as well as in many
other thermodynamic and dynamic properties of condensed materials. 39, 40
Grüneisen 27 introduced a widely utilized measure of anharmonicity based on a
consideration of the thermal expansion coefficient $\alpha_{P}$, the property
that perhaps best provides the physical expression of these anharmonic
interactions. Note that crystalline materials composed of particles
interacting through ideal harmonic interactions exhibit no thermal expansion
upon heating, and moreover, their stiffness is independent of $T$. 41 In
particular, Grüneisen introduced his famous anharmonicity parameter
$\gamma_{G}$ through the relationship, 27

$\gamma_{G}\equiv\alpha_{P}V/\kappa_{T}C_{V},$ (2)

where $C_{V}$ is the constant volume specific heat. Later, it was realized
that the above definition could be formally obtained from the Debye theory by
taking the lattice normal mode frequencies to scale with the material volume
$V$ to the power $\gamma_{G}$. While no unique measure of anharmonicity exists
even in crystalline materials, Grüneisen’s definition for $\gamma_{G}$ has
been established as a practical measure of anharmonicity. Of course, this
definition of anharmonicity does not consider why the phenomenological scaling
of the vibrational frequencies should scale as $V^{\gamma_{G}}$, as in the
continuum theory definition of anharmonicity through the ME. Barker 42 has
emphasized from general dimensional analysis arguments that the dimensionless
ratio of properties defining $\gamma_{G}$ should make this quantity suitable
for describing anharmonicity in any condensed material, regardless of whether
the material is crystalline or not, and we adopt this perspective in the
present work.

Given the manifold sources of ‘anharmonicity’ arising in real condensed
materials, the abstract nature of the model assumptions underlying eq 2 is
probably a strength rather than a weakness in this definition of material
anharmonicity. It is also stressed that the derivation of the ME by Gilvarry
28, 29 assumes that the Poisson ratio $\nu$ of the material is constant, while
one may expect both $\nu$ and $\gamma_{G}$ to vary with $T$ 43, 44 and
generally with thermodynamic conditions in real materials. As we shall discuss
below, however, $\gamma_{G}$ is often found to weakly depend on $T$ in many
real materials so that the $T$ variation of $\gamma_{G}$ can be reasonably
neglected, at least in particular ranges of $T$.

Gilvarry 28, 29 showed that the ‘Murnaghan exponent’ $\gamma_{M}$ can be
directly interpreted in terms of Grüneisen’s anharmoncity parameter,
$\gamma_{G}$. Specifically, in a generalized EOS to incorporate anharmonicity
of intermolecular interactions, Gilvarry 28, 29 found $\gamma_{M}$ to obey the
following relation,

$\gamma_{M}=2\gamma_{G}+1/3.$ (3)

This equation is sometimes called the ‘Slater equation’ in the recent
literature, because Slater 45, 46 emphasized its importance. Since the ME
reduces to the Debye EOS in the absence of anharmonicity (i.e.,
$\gamma_{G}=0$), 38 the difference, $\Delta\gamma\equiv\gamma_{M}-1/3$,
provides a measure of anharmonicity of interparticle interactions from the
above generalized EOS. Numerous experimental studies have shown that the ME
and its extensions to address attractive and repulsive contributions to the
intermolecular potential provide a markedly good description of the properties
of diverse condensed materials so that we have a framework of sufficient
generality to address the EOS relationships in polymeric and other GF fluids
in their condensed state. For specificity, we note the excellent work of
Anderson 33 for metallic and ionic crystalline materials and precise EOS
measurements by Wang and coworkers 34, 35, 36 for metallic GF materials. These
works show that the ME can give an impressively accurate description of the
$P$-$V$ relationships in real condensed materials without adjustable
parameters. The adaptation of the ME to polymer materials has been discussed
in a previous work by Barker, 47 and we consider some experimental comparisons
to the ME below based on measurements on common polymer materials.

Now, we consider materials in the reference state specified by $V_{o}$ and
$P_{o}$. If we apply a new pressure to the material, we can obtain the
alternative form for condensed polymer materials upon rearrangement of the ME,

$V_{o}/V=[1+\gamma_{M}\kappa_{T,o}(P-P_{o})]^{1/\gamma_{M}}.$ (4)

While it is natural to adopt $P=0$ as a reference condition, as implicit in eq
1, we may just as well expand $B$ in a Taylor series about any reference
pressure $P_{o}$, defining a different reference state. Cook and Rogers 48
later derived the ME by a different route and their work is of interest for
understanding the relation between $\gamma_{M}\kappa_{T,o}$ and the material
cohesive energy density and internal pressure. The work of Cook and Rogers 48
also well illustrates that deviations from the ME arise near phase
transitions, such as crystal melting and insulating-conducting or magnetic
phase transitions, which should come as no surprise since such transitions can
lead to sharp changes in the volume or cohesive energy density. ‘Kinks’ in the
EOS near thermodynamic transitions have independent interests in relation to
the existence of such transitions in GF liquids, which we briefly consider
below.

In practice, there is normally little change in the properties of condensed
materials under ambient pressure in comparison to those at $P=0$, and many
simulations are performed at $P=0$ for this reason (e.g., see refs 49, 50,
51). We then arrive at the relation,

$(V_{o}/V)^{\gamma_{M}}=(\rho/\rho_{o})^{\gamma_{M}}=1+\gamma_{M}\kappa_{T,o}P\equiv
1+P/P^{*},$ (5)

where we have defined a ‘characteristic pressure’ $P^{*}$ for the material.
The regime with $P/P^{*}\gg 1$ defines a high $P$ regime where thermodynamic
properties and relaxation times depend strongly on $P$ and the regime with
$P/P^{*}\ll 1$ corresponds to a low $P$ regime where thermodynamic properties
and relaxation times are close to those at ambient pressure. The $T$
dependence of the EOS parameter $\gamma_{M}$ may be inferred from the original
continuum dynamic mechanics ‘derivation’ of eq 1 by Murnaghan, 23, 30 which
can be immediately deduced from the assumption that the bulk modulus $B$
varies linearly with $P$,

$B=B_{o}+(dB/dP)_{o}P,\ B\equiv 1/\kappa_{T},$ (6)

where consistency requires,

$\gamma_{M}=(dB/dP)_{o}.$ (7)

Experimental estimates of $(dB/dP)_{o}$ can be conveniently obtained from the
well-known Bridgeman equation 27, 43, 33 describing the change of the volume
$V$ from its initial reference volume $V_{o}$ upon the application of a change
in the external pressure $P$,

$V/V_{o}=1-(P/B_{o})+a_{1}(P/B_{o})^{2}+\mathcal{O}[(P/B_{o})^{3}],\
a_{1}=[1+(dB/dP)_{o}]/2,$ (8)

where the higher-order terms $a_{i}$ are specified by Macdonald 43 and
Anderson. 33 However, the higher-order terms in the above Bridgeman equation
are rarely reported except in applications involving extremely large $P$. The
phenomenological Tait equation is obtained by the following approximation, 52

$-(V-V_{o})/V_{o}P=1/(A+DP).$ (9)

$A$ and $D$ are often treated as phenomenological constants in applications of
this formula, and Cho 53 has shown that the ME is recovered for a particular
choice of these constants. The above conventional form of the Tait equation is
so closely related to the ME that the ME is sometimes described as being the
‘modified Tait equation’, 43 despite the fact that the ME has a sounder
theoretical basis through its derivation by Gilvarry 28, 29 as an extension of
the Debye EOS to incorporate anharmonic interactions.

One of the interesting experimental regularities in measurements on anharmonic
interactions in metallic materials, noted first by Slater in the case of
metallic materials, 54 is that $a_{1}$ in eq 8, which links $B_{o}$ to the
nonlinear elasticity term, takes a value close to $2.5\pm 0.5$ for a range of
common metals of importance from a manufacturing standpoint, corresponding to
$(dB/dP)_{o}\approx 4$. In a later careful experimental study, Anderson 33
found the values of $(dB/dP)_{o}$ for a range of metals, salts, and minerals
to be roughly consistent with the earlier work of Slater, 54
$(dB/dP)_{o}\approx 3.5$ to $7.0$. Assuming the ME, a ‘pressure coefficient’
of $(dB/dP)_{o}\approx 4$ then corresponds to $\gamma_{M}\approx 4$. Barker 47
extended this pioneering analysis of the EOS of solid materials to encompass
newly developed polymer materials where the first and second pressure
coefficients $a_{1}$ and $a_{2}$ for polymeric materials were estimated to
take ‘typical’ values in the corresponding ranges, $a_{1}=4.1\pm 0.1$ and
$a_{2}=8.8\pm 0.2$. This corresponds to a rough estimate of $\gamma_{M}=7.0\pm
0.2$ for polymer materials, a value that we will find below to be remarkably
consistent with estimates of the exponent $\gamma$ for thermodynamic scaling
governing the relaxation dynamics of our GF liquid from both the analytic GET
32 and molecular dynamics (MD) simulations. McGowan 55 has also considered the
EOS of polymer materials based on the ME, where emphasis was given to the mass
dependence of the EOS parameters of the ME. Further studies along this line
would be quite interesting.

Macdonald 43 has reviewed estimates of $\gamma_{M}$ for molecular fluids
including water, hydrocarbon fluids, and metals, along with the $T$ dependence
of $\gamma_{M}$. The precise agreement between the measurements of Anderson 33
and the ME for such a wide range of materials is highly encouraging regarding
the general applicability of the ME, at least as a good first approximation,
to diverse condensed materials. As a general trend, we expect that anharmonic
effects, as quantified by $\gamma_{M}$, tend to become larger with increasing
molecular structural and energetic complexity.

At a purely mathematical level, the derivation of the ME simply requires the
integration 43, 33, 56 of a linear equation for $B(P)$. It is evident that eq
6 neglects the higher-order terms in the $P$ expansion of $B$ whose
incorporation has been the focus in refinements of the ME in applications to
astrophysical and geophysical phenomena where $P$ can be extremely large, 57
but these corrections are often relatively small, at least in small-molecule
liquids, 56 and the $T$ dependence of $(dB/dP)_{o}$ of simple molecules is
often modest. 43, 58 The reader is warned, however, that measurements of the
anharmonicity parameter $\gamma_{G}$, based on the defining equation in eq 2,
can be appreciably dependent on $T$ in polymers 47 so that the $T$ dependence
of the $P$ derivative $(dB/dP)_{o}$ in polymers of different types should be
carefully examined. It is also notable that Poisson ratio $\nu$, 59 which is
assumed to be constant in the Gilvarry’s more fundamental formulation of the
ME, 28, 29 varies in polymeric and other GF materials. We might naturally
expect $\gamma_{G}$, and thus $\gamma_{M}$, to be $T$ dependent. We return to
this discussion in Section 3.

Figure 1: Pressure dependence of the density for polymer materials. The plot
shows the reduced density $\rho/\rho_{o}$ versus pressure $P$ for different
polymers at a temperature of $T=140$ K. $\rho_{o}$ is the density at ambient
pressure. PIB, PIP, PS, PMPS, and PBD are short for polyisobutylene,
polyisoprene, polystyrene, poly(methylphenyl siloxane), and 1,2-polybutadiene,
respectively. Lines correspond to the equation:
$\rho/\rho_{o}=(1+P/P^{*})^{1/\gamma_{M}}$. Experimental data were taken from
ref 60, where the fitted parameters and experimental uncertainties are
discussed.

As many of the references just mentioned are not recent, and oriented to
metallic and inorganic materials of geophysical interest, it is worth pointing
out the applicability of the ME to common polymeric and other GF materials of
relatively recent interest, such as metallic glasses, as a basic ‘reality
check’ given the importance of the ME in our development below. In Figure 1,
we show the variation of the reduced density $\rho/\rho_{o}$ with $P$ for
different polymers measured by the Sokolov group, 60 including polyisobutylene
(PIB), polyisoprene (PIP), polystyrene (PS), poly(methylphenyl siloxane)
(PMPS), and 1,2-polybutadiene (PBD). The ME exponents $\gamma_{M}$ were found
to equal $9.16$, $8.66$, $7.48$, $8.73$, and $9.12$ for PIB, PIP, PS, PMPS,
and PBD, respectively, and the isothermal compressibility parameter $P^{*}$
was tabulated for these polymers by Sokolov and coworkers. 60 These estimates
are to be compared with Baker’s earlier estimates of a ‘typical’ value of
$7.0\pm 0.2$ for polymer materials, 47 as mentioned above. We shall see below
that these values of $\gamma_{M}$ are also comparable to those in our GET
calculations and MD simulations of model polymeric GF liquids. We point out
that the ME also provides an excellent description of relatively recent
measurements on metallic GF materials. 34, 35, 36

As a final point in this section, we mention that the EOS discussed above has
implications for understanding many other important properties of condensed
materials under high pressure conditions, such as the refractive index 61 and
dielectric properties 62 and the pressure dependence of the melting
temperature 63, 64, 52, 65 and the glass transition temperature, 66, 67 an
application that we briefly consider in the present paper.

##  2 Methods

###  2.1 Generalized Entropy Theory

The GET essentials have been described in our recent works, 68, 51, 69 so we
only present a brief summary here. The GET enables analytically investigating
the influence of basic molecular parameters on polymer glass formation. 32 The
basic idea is to merge the lattice cluster theory (LCT) 70, 71 for the
thermodynamics of polymer systems and the Adam-Gibbs (AG) relation 72 between
the segmental structural relaxation time and the configurational entropy. The
LCT yields an analytic expression for the Helmholtz free energy $f$ as a
function of temperature $T$, polymer filling fraction $\phi$, molecular mass
$M$, microscopic cohesive energy parameter $\epsilon_{lm}$, bending energy
parameter $E_{b}$, and a set of geometrical indices that reflect the size,
shape, and bonding patterns of the monomers.

Computations of the characteristic properties of polymer glass formation are
achieved by first analyzing the $T$ dependence of the configurational entropy
per lattice site, $s_{c}=-\left.\partial f/\partial T\right|_{\phi}$. For
instance, the onset temperature $T_{A}$ signals the onset of non-Arrhenius
behavior of $\tau_{\alpha}$ and can be determined by a temperature
corresponding to the maximum $s_{c}^{*}$ of $s_{c}$. The crossover temperature
$T_{c}$ separates two regimes of $T$ with qualitatively different dependences
of $\tau_{\alpha}$ on $T$ and is estimated from $\partial^{2}(Ts_{c})/\partial
T^{2}=0$. 32 The determination of the glass transition temperature
$T_{\text{g}}$ follows its operational definition based on the condition,
$\tau_{\alpha}(T_{\text{g}})=100$ s, where $\tau_{\alpha}$ is computed from
the AG relation, 72

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

The GET assumes that the high $T$ vibrational prefactor equals
$\tau_{o}=10^{-13}$ s, a typical experimental estimate for polymers. 73
$\Delta G_{o}$ is the activation free energy at high $T$, which is anticipated
from transition state theory (TST) 74, 75 to contain both enthalpic $\Delta
H_{o}$ and entropic $\Delta S_{o}$ contributions, i.e., $\Delta G_{o}=\Delta
H_{o}-T\Delta S_{o}$. Motivated by the approximation made by AG 72 that the
entropic contribution to the activation free energy is negligible and the
experimental evidence 32 that $\Delta H_{o}\approx 6k_{B}T_{c}$, the GET
simply assumes that $\Delta G_{o}=6k_{B}T_{c}$. The reasoning for such a
relationship between $\Delta H_{o}$ and $T_{c}$ has been discussed at length
in ref 32, and we refer the reader to this reference for details. We note that
$\Delta H_{o}$ can alternatively be estimated from experiment or simulation,
76 but the approximation adopted by the GET has the advantage that
$\tau_{\alpha}$ for polymer melts can be predicted based solely on molecular
and thermodynamic parameters required to describe their thermodynamics.
Finally, the characteristic temperature $T_{0}$ demarking the ‘end’ of glass
formation, 32 along the fragility index $D$, can be obtained from the Vogel-
Fulcher-Tammann (VFT) equation, 77, 78, 79

$\tau_{\alpha}=\tau_{0}\exp\left(\frac{DT_{0}}{T-T_{0}}\right),$ (11)

where $\tau_{0}$ is a prefactor. The VFT equation has been found to describe
the $T$ variation of $\tau_{\alpha}$ calculated from the GET 32 rather well
over a $T$ range above $T_{\text{g}}$, but below $T_{c}$. Notably, the GET
allows for the direct calculation of $\tau_{0}$, $D$, and $T_{0}$ in terms of
molecular parameters governing the thermodynamics of the fluid. 32 There is a
vibrational contribution to the prefactor $\tau_{0}$ that scales inversely to
an average phonon frequency, which is assumed to take a ‘typical’ value in
molecular liquids, $10^{-13}$ s, although this time scale can be estimated
precisely from the initial decay of the velocity autocorrelation function of
the polymer segments or from an instantaneous normal mode analysis. 80

The present paper considers a melt of chains with the structure of
polypropylene (PP) under a range of fixed $P$ conditions, where the $P$ range
is selected to ensure that thermodynamic scaling holds. 81 In all our
calculations, 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 $\epsilon_{lm}/k_{B}=200$ K, and the bending
energy parameter is $E_{b}/k_{B}=640$ K. The selected chain length is typical
of high molecular masses, for which $\tau_{\alpha}$ remains nearly unchanged
with varying $N_{c}$. The choices of $\epsilon_{lm}$ and $E_{b}$ yield a
thermodynamic scaling exponent that is consistent with the value estimated
from our simulations, as shown in Section 3.

###  2.2 Molecular Dynamics Simulation

Our simulation study of polymer glass formation is based on a coarse-grained
bead-spring model of polymer melts. 82, 83 Details for this model have been
given elsewhere, 50 along with the simulation methodology, so our description
is brief. This model assumes that each polymer chain is composed of a number
of connected beads, where bond connectivity along neighboring beads is
maintained by the finitely extensible nonlinear elastic (FENE) potential, 82,
83

$\displaystyle
U_{\text{FENE}}(r)=-\frac{1}{2}k_{b}R_{0}^{2}\ln\left[1-\left(\frac{r}{R_{0}}\right)^{2}\right]+4\varepsilon\left[\left(\frac{\displaystyle\sigma}{\displaystyle
r}\right)^{12}-\left(\frac{\displaystyle\sigma}{\displaystyle
r}\right)^{6}\right]+\varepsilon,$ (12)

where $r$ denotes the distance between two beads and $\varepsilon$ and
$\sigma$ are the length and energy scales associated with the Lennard-Jones
(LJ) potential. The firs term of eq 12 extends to $R_{0}$, and the second term
has a cutoff at $2^{1/6}\sigma$. Common choices are adopted for the parameters
$k_{b}$ and $R_{0}$, namely, $k_{b}=30\varepsilon/\sigma^{2}$ and
$R_{0}=1.5\sigma$. Chain stiffness is controlled by an angle potential with
the following form, 84

$\displaystyle U_{\text{bend}}(\theta)=-a_{\theta}\sin^{2}(b_{\theta}\theta),\
0<\theta<\pi/b_{\theta},$ (13)

where the bond angle is given by
$\theta=\cos^{-1}[(\mathbf{b}_{j}\cdot\mathbf{b}_{j+1})/(|\mathbf{b}_{j}||\mathbf{b}_{j+1}|)]$
in terms of the bond vector $\mathbf{b}_{j}=\mathbf{r}_{j}-\mathbf{r}_{j-1}$
between two neighboring beads $j$ and $j-1$. We adopt $b_{\theta}=1.5$ based
on our previous work. 85 Nonbonded pair interactions are described by a
truncated-and-shifted LJ potential,

$\displaystyle
U_{\text{LJ}}(r)=4\varepsilon\left[\left(\frac{\displaystyle\sigma}{\displaystyle
r}\right)^{12}-\left(\frac{\displaystyle\sigma}{\displaystyle
r}\right)^{6}\right]+C(r_{\text{cut}}),\ r<r_{\text{cut}},$ (14)

where $C(r_{\text{cut}})$ is a constant to ensure that $U_{\text{LJ}}$ varies
smoothly to zero at the cutoff distance $r_{\text{cut}}$. In the present work,
we use $r_{\text{cut}}=2.5\sigma$ to include attractive nonbonded
interactions.

Each chain has $120$ beads and the total bead number is $N=12000$ in our
polymer system. In accord with our calculations based on the GET, the chain
length with $120$ beads lies in a mass regime where the segmental dynamics is
insensitive to changes in molecular mass. We describe all quantities and
results from MD simulations in standard reduced LJ units. Specifically,
length, time, temperature, and pressure are, respectively, given in units of
$\sigma$, $\tau$, $\varepsilon/k_{B}$, and $\varepsilon/\sigma^{3}$, where
$\tau=\sqrt{m\sigma^{2}/\varepsilon}$ with $m$ being the bead mass and $k_{B}$
is Boltzmann’s constant. We utilize the HOOMD-blue simulation package 86, 87,
88 to perform all our simulations, and details can be found in refs 50 and 85.

##  3 Results and Discussion

###  3.1 Thermodynamic Scaling in the Arrhenius Relaxation Regime

We first consider a relationship between dynamics and thermodynamics in the
high $T$ Arrhenius regime where TST in its classical form 74, 75 is
applicable. The application of $P$ modifies the enthalpy $\Delta H_{A}(P)$ of
activation as,

$\Delta H_{A}(P)=\Delta H_{A}(0)+PV_{\mathrm{act}},$ (15)

where $V_{\mathrm{act}}$ is the activation volume and the ‘A’ subscript on
$\Delta H_{A}(P)$ and its zero pressure analog $\Delta H_{A}(0)$ serve as a
reminder that our discussion is limited to the Arrhenius regime where $\Delta
H_{A}(P)$ is independent of $T$. As noted by Keyes, 89 the results of Slater,
46 based on the elasticity theory of Eshelby, 90 and the Grüneisen’s extension
of the Debye model of anharmonic solids indicate that $V_{\mathrm{act}}$ can
be directly estimated from $\Delta H_{A}(0)$ in terms of $\gamma_{M}$ and
$\kappa_{T,o}$,

$V_{\mathrm{act}}=\gamma_{M}\kappa_{T,o}\Delta H_{A}(0).$ (16)

$\Delta H_{A}(P)$, in turn, takes the following form,

$\Delta H_{A}(P)=\Delta H_{A}(0)(1+\gamma_{M}\kappa_{T,o}P)=\Delta
H_{A}(0)(1+P/P^{*})=\Delta H_{A}(0)(V/V_{o})^{-\gamma_{M}},$ (17)

where we have used eq 5. See ref 91 for a discussion of eq 16 and its
applicability to polymer materials. The effect of anharmonicity on the
activation energy for relaxation thus leads to a rescaling of $\Delta
H_{A}(P)$ in the $P=0$ reference state by a $V$-dependent factor. We then see
that the dimensionless activation energy $\Delta H_{A}(P)/k_{B}T$ scales as,

$\Delta
H_{A}(P)/k_{B}T\sim(\epsilon/k_{B}T)(V/V_{o})^{-\gamma_{M}}\sim(\epsilon/k_{B}T)(\rho/\rho_{o})^{\gamma_{M}},$
(18)

where $\epsilon$ is a cohesive interaction parameter describing the strength
of attractive interparticle interactions (e.g., the well depth of
intermolecular potential) required to make the $T$ factor dimensionless, 92,
93, 51 which is denoted $\epsilon_{lm}$ in our lattice model. The appropriate
dimensionless structural relaxation time and diffusion coefficient in the
Arrhenius regime are then a function of
$(\epsilon/k_{B}T)(\rho/\rho_{o})^{\gamma_{M}}$. The existence of this reduced
scaling indicates that it is problematic to naively replace $T$ by reciprocal
volume fraction 94 or pressure 95 in models or correlative expressions of the
$T$ dependence of the relaxation time or diffusion coefficient of molecular GF
liquids. This scaling has been confirmed in simulations of model simplified
metallic GF liquids in the Arrhenius regime by Tarjus and coworkers, 96 where
$\gamma_{M}$ was estimated to be about $4$, as in classical estimates for
metals. This is a typical value of $\gamma_{M}$ in metallic materials and
Keyes 89 took this a general estimate of the proportionality factor in eq 16
(See Figure 1 in ref 89). Alba-Simionesco et al. 97 and Dreyfus et al. 98
observed that the scaling relation holds over a wide range of $T$ for a wide
range of GF liquids. In the next section, we attempt to understand the origin
of this unexpected phenomenon.

We remark that the argument for the ‘activation energy’ $\Delta H_{A}$ by
Keyes 89 is an early variant of the ‘shoving model’ 99, 100, 101 in which
$\Delta H_{A}$ is modeled as being proportional to the shear modulus $G$ based
on the continuum theory arguments of Eshelby 90 for the energetic cost of
displacing molecules in a condensed material as a description of the
activation energy. Other authors have suggested that the activation energy
should be more appropriately proportional to the bulk modulus $B$. 102, 103,
104 Given that the Poisson ratio has been assumed to be constant in the
Gilvarry formulation 28, 29 of the ME extending the Debye theory and the $P$
dependence on the differential change of the shear modulus, this alternative
model of $\Delta H_{A}$ leads to exactly the same result as noted by Keyes 89.
Of course, the variation of the Poisson ratio in cooled liquids at low $T$,
which has been observed experimentally in PS and other GF liquids, means that
we must be prepared to consider a $T$-dependent anharmonicity in the
description of cooled liquids in the $T$ regime below the onset temperature
$T_{A}$ for non-Arrhenius relaxation and diffusion. We consider the
implications of a $T$-dependent anharmonicity in Section 3.7.

Parenthetically, we also note that Keyes 89 suggested a tentative expression
for the activation entropy $\Delta S_{A}$ in which the anharmonicity parameter
$\gamma_{M}$ likewise arises, $\Delta S_{A}=\gamma_{M}\alpha_{P}\Delta H_{A}$,
a relation that draws upon the earlier arguments of Lawson 105 and the highly
influential work of Wert and Zener 106. This relation is quite interesting
since any change of molecular structure or interaction that alters the
anharmonicity should alter the entropy and enthalpy of activation in a
proportionate fashion, a phenomenon termed the ‘entropy-enthalpy compensation’
(EEC). 107, 108, 109 Moreover, for materials in the low $T$ solid state,
$\alpha_{P}$ can be taken to be essentially constant, so that we can use the
near constancy $\alpha_{P}T_{m}$ for crystalline materials and
$\alpha_{P}T_{\mathrm{g}}$ for glassy materials to arrive at an approximate
proportionality between $\Delta S_{A}$ and $1/T_{m}$ and $1/T_{\mathrm{g}}$,
respectively. Such a phenomenology, whose origin is discussed below in Section
3.8, is widely observed in diverse crystalline and glass materials. 110, 111,
112, 113, 114, 115, 116, 117, 118, 119, 120, 121 In particular, it seems
suitable to choose $\alpha_{P}T_{\mathrm{g}}=0.16$ based on the empirical rule
of Boyer and Bondi for polymers, which indicates that
$\alpha_{P}T_{\mathrm{g}}$ lies in the range $0.016$ to $0.19$, 122 which
implies $\Delta S_{A}\approx\Delta H_{A}/T_{\mathrm{g}}$ if we take
$\gamma_{M}$ to have a representative value of about $6$, a value roughly
consistent with our results from the GET and MD simulations below for coarse-
grained models of polymer fluids. The product $\alpha_{P}T_{\mathrm{g}}$ is
also nearly constant in metallic GF materials 123 and $\alpha_{P}T_{m}$ is
well-known to be fairly constant in metallic crystalline materials. 124
Curiously, the Keyes relation for $\Delta S_{A}$ implies that the compensation
temperature correlates with $T_{m}$ for crystalline materials and
$T_{\mathrm{g}}$ for materials having a strong propensity to form glasses. We
plan to investigate $\Delta S_{A}$ systematically in the future in view of
evidence that this quantity is closely related to the dynamics of condensed
materials. We note that the AG model 72 and the GET 32 simply neglect the
activation entropy, which is arguably a major shortcoming of these models.

###  3.2 Thermodynamic Scaling in the Non-Arrhenius Relaxation Regime

It is not immediately clear whether or not the scaling in eq 18 applies in the
$T$ regime below the onset temperature $T_{A}$ for non-Arrhenius relaxation, a
regime that is evidently the most practically interesting for polymeric and
other GF liquids. We can anticipate that anharmonic interaction effects become
more prevalent in the $T$ regime below $T_{A}$ and that these interactions
presumably give rise to collective particle motion underlying the non-
Arrhenius relaxation. 125, 126, 127, 128, 129, 130, 131, 120, 93, 132, 133,
134, 135, 136, 137, 138, 139, 140, 141, 49, 50 In Section 3.7, we show that
the growth of collective motion identified in our simulated polymer melts
follows an increase in the anharmonic interaction strength, as quantified by
the derivative of the bulk modulus with respect to $P$.

An impressive body of evidence from both experiment and simulation indicates
that the thermodynamic scaling exponent $\gamma$ obtained from estimates of
viscosity and relaxation time lies in the broad range of $0.18$ to $8.5$. 1
Notably, thermodynamic scaling appears to hold over the entire $T$ range of
glass formation in diverse materials, 1, 3, 4 leading to different theoretical
rationalizations of this fact. Some authors 142, 143 have suggested that this
scaling property derives from a ‘hidden symmetry’ associated with the power-
law nature of the pairwise repulsive intermolecular potential that is typical
of molecular fluids, along with the assumption that attractive intermolecular
interactions can be neglected in fluids in comparison to the harsh repulsive
interactions in liquids. 142, 143 Alternatively, Casalini, Roland, and
coworkers 144, 145, 146 have emphasized that thermodynamic scaling arises from
a scaling property of the fluid configurational entropy, a thermodynamic
explanation with no obvious relation to the functional form of the
intermolecular potential. 142, 143 Recent simulation studies of numerous GF
liquids, including polymeric, 147, 148, 149 ionic and molecular, 150, 151 and
metallic GF liquids, 152 have indicated that the pressure-energy correlations
associated with the power-law potential and the associated virial theorem
linking the potential and kinetic energies of the fluid do not generally exist
so that the power-law potential argument cannot be a general explanation for
thermodynamic scaling. We also note an instructive study of thermodynamic
scaling in polymeric fluids based on the GET 81 where the thermodynamic
scaling of the segmental structural relaxation time was observed and the
scaling exponent $\gamma$ was found to depend on the chain length, chain
rigidity, cohesive interaction strength, and side-group length, factors that
evidently influence the fragility of glass formation. The GET is based on a
lattice model to enable analytically tractable calculations of the
configurational entropy density $s_{c}$, where the intermolecular potential
roughly has the form of a square-well potential. The potential is invariant in
polymer melts having different molecular parameters that obviously greatly
influence the thermodynamic scaling exponent, $\gamma$. This provides
unequivocal evidence for the existence and practical importance of
configurational anharmonicity on the properties of polymers and other
molecular materials. Calculations based on the GET 81 also confirmed the
approximate inverse relation between $\gamma$ and the constant volume
fragility parameter $m_{V}$ and the nearly linear relation between $m_{V}$ and
its constant pressure fragility counterpart $m$ found experimentally by
Casalini and Roland. 153, 154 These results also strongly support the view of
thermodynamic scaling proposed by Casalini and Roland, 145 which emphasizes
that thermodynamic scaling reflects the influence of anharmonic interactions
on the $T$ dependence of the configurational entropy. We next probe the extent
to which thermodynamic scaling applies to the configurational entropy based on
the GET and the extent of cooperative exchange motion based on MD simulations
to examine the interpretation of Casalini and Roland 153, 154 more
quantitatively.

Casalini and Roland 145 have previously investigated the thermodynamic scaling
of excess entropy estimates of model GF liquids, a quantity that is often
taken as an experimental counterpart of the configurational entropy, which
cannot be directly measured experimentally, and they found that the
thermodynamic scaling exponent for the configurational entropy was somewhat
different than that for dynamic properties. It was not clear how much the same
conclusion was affected by their approximate estimate of the configurational
entropy, however. We can completely avoid the uncertainties in estimates of
the configurational entropy in our calculations based on the GET. The MD
simulations should be helpful in establishing the extent to which the scaling
property of the configurational entropy is transferable to the dynamics of GF
liquids.

###  3.3 Thermodynamic Scaling based on the GET

The GET involves a combination of the LCT of polymer thermodynamics, developed
initially by Freed and coworkers, 70, 71 to address the thermodynamics of
multicomponent polymers having nontrivial monomer structure, with the AG
relation, 72 which provides a formal link between the thermodynamics and
dynamics of GF materials. We have calculated $s_{c}$ for many systems in the
past, but we never tested whether or not $s_{c}$ exhibits thermodynamic
scaling. The observation of thermodynamic scaling over a wide $T$ range and
the consistency of the same scaling with the GET evidently require that
thermodynamic scaling should exist. Moreover, the scaling found in the high
$T$ Arrhenius regime must match the thermodynamic scaling in the $T$ regime
below the onset temperature $T_{A}$ for non-Arrhenius relaxation and
diffusion. This is not at all obviously true, and examining thermodynamic
scaling provides a strong test for the GET framework and an opportunity to
learn more about this mysterious phenomenon.

Figure 2: Generalized entropy theory test of thermodynamic scaling of the
configurational entropy density. (a) Configurational entropy density ratio
$s_{c}^{*}/s_{c}$ versus inverse temperature $10^{3}/T$ for a range of $P$.
The cross, triangle, and square symbols indicate the positions of the onset
$T_{A}$, crossover $T_{c}$, and glass transition temperatures $T_{\mathrm{g}}$
of glass formation, respectively. (b) $s_{c}^{*}/s_{c}$ versus
$10^{3}\phi^{\gamma}/T$ with $\gamma=7.0$ for this particular polymer model.
The exponent $\gamma$ can be tuned by changing molecular parameters, such as
chain length, cohesive interaction strength, chain stiffness, monomer
structure, etc. The inset in panel (b) shows that $s_{c}$ by itself does not
exhibit thermodynamic scaling. Figure 3: Generalized entropy theory test of
thermodynamic scaling of the structural relaxation time. (a) Logarithm of the
structural relaxation time $\tau_{\alpha}$ versus inverse temperature
$10^{3}/T$ for a range of $P$. The cross, triangle, and square symbols
indicate the positions of the onset $T_{A}$, crossover $T_{c}$, and glass
transition temperatures $T_{\mathrm{g}}$ of glass formation, respectively. (b)
$\log\tau_{\alpha}$ versus $10^{3}\phi^{\gamma}/T$ with $\gamma=7.0$. Note
that the values of $\tau_{\alpha}$ at $T_{A}$ and $T_{c}$ are relatively
insensitive to $P$ for the range of $P$ investigated.

In Figure 2a, we show the configurational entropy density ratio
$s_{c}^{*}/s_{c}$ versus $10^{3}/T$ calculated from the GET for a model of
polymer fluids over a wide range of $T$ for a range of fixed $P$ values
ranging from $0$ MPa to $50$ MPa. We see that altering $P$ greatly changes the
$T$ dependence of $s_{c}^{*}/s_{c}$.

We now attempt to rescale all our data for $s_{c}^{*}/s_{c}$ by invoking the
existence of thermodynamic scaling in Figure 2b. All our curves for
$s_{c}^{*}/s_{c}$ indeed reduce to a remarkably universal master curve in
terms of $\phi^{\gamma}/T$ with $\gamma=7.0$, where this exponent is dependent
on the particular polymer model. 81 Note that $\phi$ is the dimensionless
filling fraction defined by the ratio of the total number of united-atom
groups to the total number of lattice sites. In the GET, the density,
analogous to $\rho$ in our simulations, may be defined by
$\phi/V_{\text{cell}}$ with $V_{\text{cell}}$ being the volume of a single
lattice site, but we adopt $\phi$ in our analysis for convenience. Evidently,
the GET gives rise to a thermodynamic scaling of $s_{c}^{*}/s_{c}$.
Thermodynamic scaling for $s_{c}^{*}/s_{c}$ also holds if we vary molecular
parameters, such as chain length, cohesive interaction strength, chain
stiffness, and monomer structure, and the exponent $\gamma$ clearly depends on
the polymer molecular structure and other factors that influence molecular
packing in space and the ‘anharmonicity’ of the intermolecular interactions.
81 We emphasize again that these anharmonic interactions are of a many-body
nature rather than being related to the form of the local pair potential.
However, thermodynamic scaling does not hold for $s_{c}$ itself, as shown in
the inset of Figure 2b, since the high $T$ limit of $s_{c}$, $s_{c}^{*}$,
depends strongly on monomer structure, chain stiffness, and cohesive
interaction strength. 32, 155 It would then appear that $s_{c}^{*}$, which can
be estimated from simulation, 126 provides a measure of the configurational
anharmonicity of the intermolecular interactions that might have useful
applications in the determination of optimal intermolecular potentials for
simulations of fluid properties. The GET allows for the calculation of
$s_{c}^{*}$ as a function of molecular and thermodynamic parameters. The
results in the inset of Figure 2b indicate that $s_{c}^{*}$ has significance
for understanding thermodynamic scaling and deserves further systematic
investigation.

We next consider how thermodynamic scaling impacts the segmental structural
relaxation time $\tau_{\alpha}$ in the GET to gain insight about the
consistency of the exponent $\gamma$ in both the Arrhenius and non-Arrhenius
relaxation regimes. The GET involves no free parameters other than molecular
parameters and thermodynamic variables, such as $P$ and $T$, but the model
does involve some assumptions that we have discussed elsewhere. 32, 68, 69
Figure 3a shows our predictions of $\tau_{\alpha}$ over a range of $T$ for a
range of fixed $P$, and Figure 3b shows the corresponding reduced variable
plot indicating that all our data can be reduced to a universal scaling curve
in the entire range of glass formation from $T_{A}$ to a $T$ range below the
glass transition temperature $T_{\mathrm{g}}$, where equilibrium measurements
are normally not even possible. We view the reduction Figure 3b as being truly
remarkable. Importantly, the reduced density-temperature variable is the same
in the $T$ regime below $T_{A}$ as in the Arrhenius regime so that our
arguments above for the calculation of $\gamma_{M}$ apply to the entire liquid
state regime.

###  3.4 Thermodynamic Scaling based on Molecular Dynamics Simulation

We further examine the remarkable thermodynamic scaling through a
consideration of the extent of collective motion in GF liquids, as quantified
by the average length $L$ of stringlike cooperative motion observed generally
in simulations of GF liquids, 125, 126, 127, 128, 129, 130, 131, 120, 93, 132,
133, 134, 135, 136, 137, 137, 138, 139, 140, 141, 49, 50, 51 which has been
found to play a key role in determining the activation energy for relaxation
and which is closely related to the configurational entropy in the GET. 32 In
particular, many past works have shown that $L/L_{A}$, where $L_{A}$ is a
residual collective motion occurring at $T_{A}$ and is something not
envisioned in the original AG model, 72 corresponds to the AG counterpart,
$s_{c}^{*}/s_{c}$. The determinations of $T_{A}$ and $L$ from simulations can
be found in our previous works. 50, 51 Notably, AG also did not consider
constant $V$ versus constant $P$ conditions, which can greatly influence how
the model is ‘interpreted’. 32 Note also that even apart from the difference
between the configurational entropy and the excess entropy, the usual
experimental choice, molar excess entropy, instead of its entropy density
counterpart, certainly does not approach a plateau at high $T$ for fluids
under constant $P$ conditions. 32 We stress again that only the
configurational entropy density ratio exhibits thermodynamic scaling in the
GET.

Figure 4: Simulation test of thermodynamic scaling of the extent of stringlike
cooperative motion. (a) Normalized average string length $L/L_{A}$ versus
inverse temperature $\varepsilon/k_{B}T$ for a range of $P$. The cross symbols
indicate the positions of the onset temperature $T_{A}$ of glass formation.
(b) $L/L_{A}$ versus $(\rho\sigma^{3})^{\gamma}\varepsilon/k_{B}T$ with
$\gamma=7.0$. Figure 5: Simulation test of thermodynamic scaling of the
structural relaxation time. (a) Logarithm of the structural relaxation time
$\tau_{\alpha}$ versus inverse temperature $\varepsilon/k_{B}T$ for a range of
$P$. The cross symbols indicate the positions of the onset temperature $T_{A}$
of glass formation. (b) $\log\tau_{\alpha}$ versus
$(\rho\sigma^{3})^{\gamma}\varepsilon/k_{B}T$ with $\gamma=7.0$.

Accordingly, we plot $L/L_{A}$ versus $\varepsilon/k_{B}T$ in Figure 4, based
on the simulations of a model of flexible polymer melts over a wide range of
$T$ for a range of fixed $P$ that we have described in a previous work. 50 As
in Figure 2, we first show our results for $L/L_{A}$ for a range of fixed $P$
in Figure 4a and then in the thermodynamic scaling form in Figure 4b. It is
evident that $L/L_{A}$ in our simulated polymer melt exhibits thermodynamic
scaling as in our GET analysis of $s_{c}^{*}/s_{c}$ where the scaling exponent
$\gamma$ is again near $7.0$, a value consistent with typical values of
$\gamma_{M}$ from EOS studies. 47, 60 The observation of thermodynamic scaling
is also consistent with the ‘identification’ of the strings with the abstract
‘cooperatively rearranging regions’ hypothesized by AG. 72

Again mirroring our analysis based on the GET, Figure 5 shows our simulation
estimates of $\tau_{\alpha}$ from the intermediate scattering function 50 over
a range of $T$ for a range of fixed $P$, where we first show
$\log\tau_{\alpha}$ versus $\varepsilon/k_{B}T$ in Figure 5a and the
corresponding reduction based on thermodynamic scaling in Figure 5b. The
thermodynamic scaling reduction of $\tau_{\alpha}$ is excellent, as found for
the viscosity, diffusion coefficient, and other dynamical properties for
diverse GF liquids. 1 We can’t help but marvel about this remarkable
phenomenon and its mathematical and physical origin. In particular, it is not
clear why $s_{c}^{*}/s_{c}$ should exhibit the thermodynamic scaling property,
given that the GET results are completely insensitive to the form of the
intermolecular potential. Could there be some effective many-body potential
responsible for this remarkable scaling? It is notable that the original
continuum mechanics type of arguments by Murnaghan 23, 30 to deduce his highly
successful EOS for condensed matter only required the $P$ dependence of the
bulk modulus $B$ to be nearly linear over a wide range of $P$. This result, in
conjunction with the theoretical derivation of the ME by Gilvarry, 28, 29
based on an extension of the Debye theory to include anharmonicity and the
quantification of anharmonicity suggested by Grüneisen, 27 indicates that
anharmonic interactions are apparently the key to understanding thermodynamic
scaling and by extension variability in collective motion and fragility of GF
liquids, as these interactions are critical for understanding the existence of
thermal expansion of solids and $T$-dependent moduli 156 and the thermal
conductivity of non-metallic solid materials. 157, 158

We note that the work of Sastry and coworkers159 indicates that the AG model
exhibits thermodynamic scaling for the structural relaxation time in the Kob-
Anderson model if the density scaling of the high temperature activation
energy parameter of the AG model (designated as $A$) is considered. This is
broadly consistent with our results, along with the experimental observations
of Masiewicz et al. 160 These findings are purely empirical, but they support
the origin of thermodynamic scaling as being related to the scaling of
$s_{c}$.

###  3.5 Thermodynamic Scaling as Filter for Assessing Theories of Glass
Formation

Dyre and coworkers 143 have argued that thermodynamic scaling, apart from its
theoretical and practical significance, has the additional value as offering a
‘filter’ for assessing theories of glass formation. Even though one may argue
about why thermodynamic scaling exists, theories that do not give rise to this
property should be rejected. We discuss the power of this way of thinking by
considering some alternative models than the AG model 72 in relation to
thermodynamic scaling. In particular, there are alternative models of the
dynamics of GF liquids that emphasize, for example, the dependence of the
structural relaxation time and diffusion coefficient on the static structure
factor in the long wavelength limit, $S(0)$, 161, 162, 163, 164, 165, 166, 167
and the mean square displacement on a caging timescale on the order of a
picosecond, i.e., the so-called ‘Debye-Waller parameter’, $\langle
u^{2}\rangle$. 168, 169 It is not obvious on the surface whether these models
are related to the AG model, and whether the central quantities in these
models should exhibit thermodynamic scaling.

Figure 6: Simulation test of thermodynamic scaling of the Debye-Waller
parameter. (a) Debye-Waller parameter $\langle u^{2}\rangle$ versus inverse
temperature $\varepsilon/k_{B}T$ for a range of $P$. The cross symbols
indicate the positions of the onset temperature $T_{A}$ of glass formation.
(b) $\langle u^{2}\rangle$ versus
$(\rho\sigma^{3})^{\gamma}\varepsilon/k_{B}T$ with $\gamma=7.0$. The result in
panel (b) shows a reasonably successful, but not perfect, transformation of
$\langle u^{2}\rangle$ to a reduced variable description consistent with the
thermodynamic scaling exhibited by $\tau_{\alpha}$. Figure 7: Simulation test
of thermodynamic scaling of the long wavelength limit of the static structure
factor. (a) Long wavelength limit $S(0)$ of the static structure factor versus
inverse temperature $\varepsilon/k_{B}T$ for a range of $P$. The cross symbols
indicate the positions of the onset temperature $T_{A}$ of glass formation.
(b) $S(0)$ versus $(\rho\sigma^{3})^{\gamma}\varepsilon/k_{B}T$ with
$\gamma=7.0$. The result in (b) shows that a transformation to a reduced
variable description based on the thermodynamic scaling exhibited by
$\tau_{\alpha}$ is not successful for $S(0)$.

Figures 6 and 7 show our attempt at finding a reduced scaling for $\langle
u^{2}\rangle$ and $S(0)$ estimated from our simulations. In Figure 6, we find
that thermodynamic scaling provides a reasonable description of our simulation
observations of $\langle u^{2}\rangle$, although the thermodynamic scaling of
$\langle u^{2}\rangle$ is apparently not as perfect as in the case of
$s_{c}/s_{c}^{*}$. Further investigations are required to determine whether
the small deviations from thermodynamic scaling arise from numerical
uncertainties or other factors not yet identified.

The deviation from thermodynamic scaling is well beyond numerical
uncertainties in the case of $S(0)$ in Figure 7. $S(0)$ is proportional to the
isothermal compressibility of the fluid, so one might have thought that such a
quantity would exactly exhibit the pressure-potential energy correlations
emphasized by Dyre and coworkers 143 as a possible origin of thermodynamic
scaling in many real liquids. While the estimates of $S(0)$ based on $S(q)$
may depend on the system size, we have confirmed that varying the system size
over a reasonable range does not change our results for $S(0)$ in any
substantial way. The inordinately large-scale simulations of Chremos 170
indicate that our estimates of $S(0)$ in the present work are reliable.
Apparently, the system cannot be too small to make reliable estimates of
$S(0)$. Our simulations and the work of Chremos show that the lowest
wavenumber $q$ should be $\sim 0.5\sigma^{-1}$ in the present polymer model,
which corresponds to a total bead number of $\sim 2000$ for a typical number
density of $\rho=1$. Moreover, it should be noted that the estimates of $S(0)$
by first calculating $\kappa_{T}$ and $\rho$ by the usual sum rule for $S(0)$
may yield somewhat different results, where the difference should be larger
for lower $T$. There are long-range correlations developing in the system that
cause deviations from the sum rule relating $\kappa_{T}$, $\rho$, and $k_{B}T$
to $S(0)$. Since long-range correlations can be expected in GF liquids, this
matter should be investigated further in the future.

The simple fact that $S(0)$ clearly does not exhibit thermodynamic scaling
means that we may reject the mode-coupling theory 171 and the nonlinear
Langevin equation theory 161, 162, 163 and its recent extension, the
elastically cooperative nonlinear Langevin theory 164, 165, 166, 167 as
acceptable models of GF liquids, based on the ‘filter criterion’ of Dyre and
coworkers 143. Berthier and Tarjus 172 have come to the essentially same
conclusion regarding the physically inadequacy of the mode-coupling theory and
its variants because this class of ‘generalized free volume’ theories presume
that the pair correlation function $g(r)$, or its integral $S(0)$, determines
the dynamics of fluids. In physical terms, our analysis means that
‘structure’, based on $S(0)$ or properties derived from $g(r)$, does not by
itself determine the dynamics of liquids.

Dyre and coworkers 173 have clarified the origin of the ‘breakdown’ of
thermodynamic scaling of $S(0)$ in condensed fluids through the analysis of
many of the virial-related thermodynamic properties of materials composed of
atomic species interacting through the ubiquitous LJ potential. As well-known,
174, 175 thermodynamic scaling is exactly recovered if the attractive
contribution to the LJ potential is formally assumed to vanish, but this
power-law density-temperature scaling no longer holds when the attractive
component to the LJ potential is included. 173 An attractive interaction is
essential for the stability of condensed matter at constant $P$ so we may
rather generally expect virial-related properties, such as $S(0)$, to not
exhibit thermodynamic scaling, a result that accords with our analytic results
based on the GET and our simulations of polymeric GF liquids discussed below.

Of course, the loss of ‘thermodynamic scaling’ does not imply that a more
general density-temperature scaling does not exist. Dyre and coworkers 176,
177 have shown how to derive generalized density-temperature scaling relations
from the same type of potential-virial correlations as found in the case of a
purely repulsive power-law interatomic potential, and they have discussed the
significance of these generalized density-temperature scaling relations, when
they exist, for the dynamics of model condensed materials. In particular, they
have devised a classification scheme of liquids (Roskilde simplicity
classification of liquids based on ‘isomorphs’) based on these extended
density-temperature scaling relationships. 176, 177

Recent computational studies of a range of GF liquids, including polymeric,
147, 148, 149 ionic 150, 151 and certain metallic GF liquids, 152 have shown
that the virial-potential scaling that is central to the existence of
generalized density-temperature scaling does not hold, which minimally seems
to imply that this classification scheme is of limited value for polymeric GF
liquids, which are our main interest. Nonetheless, the work of Dyre and
coworkers 173 provides significant insight into how attractive interactions
‘break the scaling symmetry’ responsible for the simple power-law scaling of
the compressibility factor, and other thermodynamic properties related to the
pair correlation function. This finding has far reaching implications for the
theories of the dynamics of GF liquids, such as the conventional mode-coupling
theory of glass formation, 171 that predict that a relation between the
structural relaxation time and equilibrium ‘structural parameters’ derived
from $g(r)$, such as $S(0)$, and the isothermal compressibility, $\kappa_{T}$.
We note that a lack of consistent density-temperature scaling also arises in
the peak height of the $4$-point density function, a higher-order density
correlation function than $S(0)$, which is often considered to be a measure of
‘dynamic heterogeneity’ in GF liquids. 178

Recently, there have been efforts to address the problems in existing mode-
coupling type formulations of the dynamics of GF liquids just noted. Dell and
Schweizer 179 recognized the general failure of mode-coupling approaches in
which collective interparticle interactions are modeled by $S(0)$, and they
addressed this fundamental problem in an extended mode-coupling approach,
along with a series of approximations, to account for the effect of attractive
interparticle interactions. Interestingly, their extended mode-coupling theory
framework can apparently account for many of the shortcomings identified in
earlier simulations of model atomic GF liquids based on the assumption that
interparticle interactions are dominated by pairwise repulsive interactions
alone. Perhaps most importantly from the standpoint of the present paper, Dell
and Schweizer 179 found that their extended theory was consistent with
thermodynamic scaling when applied to model atomic GF liquids studied
previously. This promising extension of mode-coupling theory, along with novel
additional elements included by Dell and Schweizer, 179 requires further
validation and has not been applied to the much more complicated case of
polymeric GF liquids. At this stage of development, we note that the
structural relaxation time of even these toy atomic fluids is not simply a
function of $S(0)$ in the proposed modified mode-coupling theory so the
absence of thermodynamic scaling for $S(0)$ cannot be used to exclude the new
model of Dell and Schweizer. 179 It will be interesting to see how this
improved mode-coupling theory performs in the future for more complex GF
liquids.

We shall see below that we may recover a consistent thermodynamic scaling by
subjecting the isothermal compressibility to a density dependent
transformation that relates this property to the configurational entropy,
allowing us to obtain a reduced isothermal compressibility that satisfies
thermodynamic scaling. These relations give us hope of a thermodynamic
relation between the dynamics and thermodynamics of GF liquids built around
the configurational entropy density and properties related to the
configurational entropy density.

We finally point out that the occurrence of thermodynamic scaling is not
itself a sufficient criterion for the validity of a model of glass formation.
Fluids that exhibit strong correlations between the virial and potential
energy fluctuations from the existence of a power-law intermolecular potential
exhibit thermodynamic scaling, but molecular fluids do not normally exhibit
strongly correlating dynamics because the presence of chemical bonds tends to
lead to the loss of these correlations. This situation would seem to preclude
the use of atomic GF liquids as model systems for describing the dynamics of
molecular GF fluids. Of course, purely density-based ‘free volume’ models of
glass formation can be ruled out because such models are inherently
inconsistent with thermodynamic scaling. Dyre and coworkers 143 are indeed
correct in indicating that thermodynamic scaling offers an important criterion
for checking the validity of models of glass formation. We view the violation
of thermodynamic scaling for $S(0)$ and other properties related to the fluid
pair correlation function $g(r)$ as providing further evidence in favor of the
interpretation of thermodynamic scaling originally proposed by Casalini and
Roland. 145

###  3.6 Relationship between Configurational Entropy and Isothermal
Compressibility

Although the GET is helpful in understanding the origin of thermodynamic
scaling as being fully consistent with this model of the dynamics of GF
liquids, the inherent difficulty, if not plain impossibility, of estimating
the configurational entropy density $s_{c}$ in measurements is certainly a
downside of the entropy model of glass formation. Of course, there is no real
problem if we express the thermodynamic scaling from the GET in terms of
temperature and molecular parameters, but this situation makes us wonder if
there are any other thermodynamic properties that exhibit thermodynamic
scaling, which are more readily accessible experimentally and might serve as a
surrogate for $s_{c}$ in the GET. It is certainly helpful that the GET allows
us to calculate any thermodynamic property that we can think of to examine the
above possibility. The fragility parameter $D$ of the VFT equation or the
steepness parameter $m$, quantifying the strength of the $T$ dependence of the
structural relaxation and diffusion at constant pressure or volume, may be
calculated in terms of only the molecular and thermodynamic parameters
required to specify the thermodynamic properties of the material. It is
clearly of some advantage that we have a fully predictive theory of all these
quantities in our quest for an alternative to $s_{c}$.

As others had done before, 151 we have found that the Debye-Waller parameter
$\langle u^{2}\rangle$ exhibits thermodynamic scaling to a good approximation,
which gives us hope that some other thermodynamic properties might obey
thermodynamic scaling. This observation can further be taken as a major clue
of where we should look for such a property that encodes similar information
as $s_{c}$.

We have long viewed $\langle u^{2}\rangle$ as corresponding to a kind of
‘dynamical free volume’ that is related to the local compressibility rather
than a static geometrical size of cavities on a scale set by the particles by
density fluctuations as in traditional ‘free volume’ models emphasizing
density and the ‘structure’ of the fluid. $\langle u^{2}\rangle$ largely
reflects the thermal energy of the molecules and the same can be said to be
true of the isothermal compressibility $\kappa_{T}$ of the material, which is
the inverse of the bulk modulus, $B$. $\kappa_{T}$ is clearly accessible by
mechanical or sound propagation measurements or through measurements of the
structure factor at long scattering wavelengths, $S(0)$, in conjunction with
estimates of the density through a well-known thermodynamic relation,
$\kappa_{T}=\rho k_{B}T/S(0)$. $\kappa_{T}$ is also an attractive quantity to
consider for developing a description of the dynamics of GF liquids because,
as we have discussed in Section 1, this property is central to defining the
EOS of condensed materials and its derivatives clearly encode information
about the anharmonicity of the interparticle interactions in the condensed
state. While it is clear from a direct analysis based on the GET, and
validating coarse-grained MD simulations, that $\kappa_{T}$ does not normally
exhibit thermodynamic scaling in molecular GF liquids, we might then consider
if any obvious transformation of $\kappa_{T}$ might give rise to the
definition of a reduced compressibility which has the thermodynamic scaling
property and which has some relationship to $s_{c}/s_{c}^{*}$. We next show
that this is indeed the case.

We start by noting that the $T$ variation of $\langle u^{2}\rangle$ varies in
a way that changes as one approaches various temperatures that are
characteristic of glass formation, and thus, our transformation of
$\kappa_{T}$ should show some recognition of these characteristic
temperatures. For example, we have repeatedly observed over the admittedly
limited $T$ range accessible to equilibrium simulations that $\langle
u^{2}\rangle$ at low $T$ scales linearly with $T$ and apparently extrapolates
to zero at a temperature consistent with the VFT temperature $T_{0}$, deduced
from the structural relaxation time. 180, 181, 182, 183, 184 The word
‘extrapolation’ is emphasized here as we do not literally believe that
$\langle u^{2}\rangle$ vanishes at $T_{0}$ any more than we think that $s_{c}$
actually vanishes at its corresponding Kauzmann temperature $T_{K}$, which
likewise relies on an extrapolation from the $T$ regime above
$T_{\mathrm{g}}$. In the GET, $T_{K}$ is basically equivalent to $T_{0}$
because the VFT functional form is predicted to hold rather well over the $T$
range between $T_{\mathrm{g}}$ and the crossover temperature $T_{c}$, which is
typically about $(1.2-1.3)T_{\mathrm{g}}$, depending on chain length, chain
stiffness, cohesive interaction strength, monomer structure, pressure, etc.

The thermodynamic scaling of $\langle u^{2}\rangle$ at low $T$ suggests that
we consider the relative isothermal compressibility,
$\Delta\kappa_{T}\equiv\kappa_{T}-\kappa_{T,0}$ with $\kappa_{T,0}$ being the
value of $\kappa_{T}$ at $T_{0}$, which mathematically acknowledges that there
are ‘special’ temperatures involved in glass formation and which by
construction vanishes at $T_{0}$. It is important to note that $T_{0}$ can be
measured experimentally and $\kappa_{T,0}$ can be computed precisely in the
GET, so we remain in complete theoretical control and are discussing
quantities that can be clearly measured. Note that $\kappa_{T,0}$ depends on
molecular parameters and pressure so that $\Delta\kappa_{T}$ may have very
different scaling properties than $\kappa_{T}$ when both $T$ and $V$ are
varied.

Since $\Delta\kappa_{T}$ has the units of a reciprocal energy density, we are
still comparing thermodynamic ‘apples and oranges’, so to speak, if we attempt
to compare to the configurational entropy density ratio $s_{c}^{*}/s_{c}$. We
then consider some sort of ‘obvious’ reduced variable that might be
interrelated to $s_{c}^{*}/s_{c}$, which is the central quantity in the GET
controlling the $T$ dependence of the activation energy for $\tau_{\alpha}$.
We note here that $s_{c}^{*}$ is highly dependent on molecular parameters, so
reducing $s_{c}$ by $s_{c}^{*}$ is clearly the key to the existence of the
thermodynamic scaling of $s_{c}^{*}/s_{c}$. Given our definition of
$s_{c}^{*}/s_{c}$, it seems natural to define a reduced $s_{c}$ that vanishes
at the onset temperature $T_{A}$ of glass formation, another characteristic
temperature that may be calculated precisely in the GET and estimated
experimentally. In particular, we introduce a reduced relative configurational
entropy density $\delta s_{c}$,

$\delta s_{c}\equiv s_{c}^{*}/s_{c}-1=(s_{c}^{*}-s_{c})/s_{c},$ (19)

which vanishes by construction at $T_{A}$. This definition may be recognized
as being analogous to the type of reduced variable introduced in critical
phenomena in the development of a universal reduced description of
thermodynamic and dynamic properties where $s_{c}$ formally replaces $T$. Note
that $\delta s_{c}$ naturally arises in the formal separation of the
activation free energy of the GET/AG model into uncooperative and cooperative
components, 32 i.e., $\Delta G(T)(s_{c}^{*}/s_{c})=\Delta G_{o}+\Delta
G_{o}\delta s_{c}$.

Figure 8: Relationship between the isothermal compressibility and
configurational entropy density in the generalized entropy theory. (a) Reduced
isothermal compressibility $\delta\kappa_{T}$ versus reduced configurational
entropy density $\delta s_{c}$ for a range of $P$. The cross, triangle, and
square symbols indicate the positions of the onset $T_{A}$, crossover $T_{c}$,
and glass transition temperatures $T_{\mathrm{g}}$ of glass formation,
respectively. Dashed lines are fits to the equation,
$\delta\kappa_{T}=\mathcal{C}\delta s_{c}$, where $\mathcal{C}$ depends on
$P$. (b) $\delta\kappa_{T}/\mathcal{C}$ versus $\delta s_{c}$ for a range of
$P$. The inset in panel (a) displays the thermodynamic scaling of the ratio
$\delta\kappa_{T}/\mathcal{C}$. Symbols in panel (b) have the same meaning as
in panel (a). While there is no strong ‘kink’ in $\delta\kappa_{T}$, as one
would expect for the presence of a phase transition, there are small changes
in slope around $T_{A}$ and $T_{c}$.

Now if we define a corresponding dimensionless reduced variable for
$\Delta\kappa_{T}$, then we will be in a position to directly compare
$\kappa_{T}$ and $s_{c}$ in an appropriate form that recognizes the existence
of characteristic temperatures at which the thermodynamics and dynamics of GF
liquids change in a significant fashion. We then define a dimensionless
isothermal compressibility as,

$\delta\kappa_{T}\equiv(\kappa_{T,A}-\kappa_{T})/(\kappa_{T}-\kappa_{T,0}),$
(20)

where $\kappa_{T,A}$ is the value of $\kappa_{T}$ at $T_{A}$. This
dimensionless quantity vanishes at $T_{A}$, as in the case of $\delta s_{c}$.
However, it should be noted that our reduced variables are not well-defined as
$T$ approaches $T_{0}$, but this extremely low $T$ regime does not interest us
practically because the GET has limited interest below $T_{\mathrm{g}}$ where
equilibrium measurements on GF liquids are normally impossible. Accordingly,
we might expect $T_{\mathrm{g}}$ to somehow come into our reduced variable
description of $\delta\kappa_{T}$, since $T_{\mathrm{g}}$ is clearly an
important characteristic temperature of glass formation. We show the
comparison of $\delta\kappa_{T}$ to $\delta s_{c}$ for different $P$ in Figure
8a. These quantities are proportional to a high degree of approximation, if
not equality, and we may fix the prefactor by invoking a condition that the
reduced variable slopes should coincide by introducing $T_{\mathrm{g}}$ as a
reference point. In particular, we find the reduced variable relation,

$\delta\kappa_{T}=\mathcal{C}\delta s_{c},$ (21)

where the $P$-dependent proportionality factor may be given by
$\mathcal{C}=\delta\kappa_{T}(T_{\mathrm{g}})/\delta s_{c}(T_{\mathrm{g}})$ by
taking $T_{\mathrm{g}}$ as a reference point. We plot
$\delta\kappa_{T}/\mathcal{C}$ in Figure 8b, where we see that
$\delta\kappa_{T}/\mathcal{C}$ seems to scale perfectly linearly with $\delta
s_{c}$. The inset of Figure 8a indicates that $\delta\kappa_{T}/\mathcal{C}$
obeys thermodynamic scaling. It goes without saying that $\delta s_{c}$ obeys
thermodynamic scaling, which is demanded by mathematical consistency and was
checked explicitly. We now have a thermodynamic property exhibiting
thermodynamic scaling defined in terms of experimentally measurable
properties. As a separate matter, the inset of Figure 8b shows the $T$
dependence of $\delta s_{c}$, which is nearly quadratic in the reduced
temperature in the high $T$ regime near $T_{A}$, i.e., $\delta
s_{c}=C_{s}[(T_{A}-T)/T_{A}]^{2}$ with $C_{s}$ being a material specific
constant, 32 but in the low $T$ regime below $T_{c}$, the $T$ dependence of
$s_{c}$ changes to a variation in which the product of $s_{c}$ and $T$ is
linear, i.e., $Ts_{c}=K[(T-T_{0})/T_{0}]$, where $K$ is a constant defining
the thermodynamic fragility of glass formation. 32 The $T$ variation of
$s_{c}$ predicted by the GET has recently been reviewed in ref 69.

Quite apart from achieving our practically significant goal of transforming
our entropy theory to a model centered on the thermodynamically ‘natural’
variable of the isothermal compressibility, we may as well ask why
$\kappa_{T}$ itself does not exhibit thermodynamic scaling. Why do we need
such a transformation to restore the thermodynamic scaling property? As noted
before and recognized when developing the transformation, the situation here
is very similar to that arising in the second-order phase transition at which
a special critical temperature arises so that thermodynamic and dynamic
properties of the system do not depend on $T$ alone. Specifically, in the
$O(n)$ class of spin models describing the Ising model ($n=1$), X-Y model
($n=2$), Heisenberg model ($n=3$), …, spherical model ($n\rightarrow\infty$),
the thermodynamic and dynamic properties become universal functions of a
reduced temperature, $T=(T-T_{cr})/T$, with a non-universal prefactor near the
precisely defined critical temperature $T_{cr}$, where both the prefactor and
$T_{cr}$ depend on the interaction strength, material structure, order
parameter dimension $n$, spatial dimension, etc. However, thermodynamic
scaling is lost when the dimension is reduced below the lower critical
dimension at which $T_{cr}$ becomes zero so that the properties become only a
function of $T$, and no finite critical temperature and thus nontrivial
reduced critical temperature exists. This situation is realized, for example,
in the Ising model in one dimension. 185 It would appear that the breaking of
thermodynamic scaling in $\kappa_{T}$, and other properties for which virial-
potential scaling might have been anticipated, can be traced to a
thermodynamic transition phenomenon underlying glass formation that is
characterized by well-defined characteristic temperatures defined by the
equilibrium thermodynamics of the material. We have provided evidence for this
scenario in a previous work, 186 where we have emphasized that the transition
involves a particle supramolecular assembly transition responsible for the
emergence of dynamic heterogeneity in GF liquids, which is a type of rounded
transitions. It is characteristic of such self-assembly transitions that they
have both onset and ending transition temperatures analogous to $T_{A}$ and
$T_{\mathrm{g}}$ and a crossover transition in the middle analogous to $T_{c}$
so that the need for multiple characteristic temperatures rather than a single
$T_{c}$ is consistent with the proposed picture of the thermodynamic nature of
glass formation and conception of the origin of dynamic heterogeneity in such
transitions as a generic and dynamic phenomenon in condensed matter.
Importantly, this theoretical framework, which is embodied by the string model
of glass formation, 126, 128, 129, 127, 120, 130, 131, 93, 132, 133, 131, 141,
49, 50, 51 provides a framework for calculating and interpreting the dynamic
properties of glass formation.

Figure 9: Relationship between the isothermal compressibility and Debye-Waller
parameter in simulations. Relative isothermal compressibility
$(\kappa_{T}-\kappa_{T}^{o})$ versus $\left(\langle u^{2}\rangle/\langle
u_{A}^{2}\rangle\right)^{3/2}$ determined from simulations of polymer melts
having variable chain stiffness 49 and pressure. 50 (a) Observations for a
range of chain stiffness parameters $a_{\theta}$. (b) Observations for a range
of $P$. Note that $\kappa_{T}^{o}$ is the extrapolated value of $\kappa_{T}$
at the temperature at which $\langle u^{2}\rangle$ correspondingly
extrapolates to zero. Dashed lines indicates descriptions according to eq 22.

We are now in a position to test our starting hypothesis that $\langle
u^{2}\rangle^{3/2}$ corresponds to a measure of the ‘local compressibility’ by
directly comparing $\kappa_{T}$ to $\langle u^{2}\rangle^{3/2}$. This should
also provide insight into the interrelation between the entropy theory of
glass formation and the localization model, 169, 120, 182 which focuses on the
seemingly different property, $\langle u^{2}\rangle^{3/2}$. Historically,
$\langle u^{2}\rangle^{3/2}$ has been interpreted as a measure of dynamical
free volume, 174, 175, 187, 188, 189, 169 an interpretation that might
illuminate the interrelation between the entropy and free volume approaches to
the dynamics of GF liquids. The potential importance of this dynamical variant
of free volume can be traced back to some of the earliest scientific studies
of condensed matter, as illustrated below in our discussion of Sutherland’s
kinetic theory of solids. 190, 191, 192, 193 We point out that $\langle
u^{2}\rangle$ has also been interpreted in the theory of Leporini and
coworkers 168 to be a measure of reciprocal rigidity or ‘compliance’ in a
rheological terminology. Hence, $\langle u^{2}\rangle$ is clearly central to a
number of models of glass formation where a remarkable range of
interpretations have been given to this quantity.

Since we must determine $\langle u^{2}\rangle$ in simulations, we consider its
relation with $\kappa_{T}$ based on our simulation results. For generality, we
also analyze the simulation results based on a coarse-grained polymer melt
with variable chain stiffness parameter $a_{\theta}$. See ref 49 for details
about the polymer model. We first define $\kappa_{T}^{o}$ as the value of
$\kappa_{T}$ at the temperature at which $\langle u^{2}\rangle$ extrapolates
to zero, a condition that corresponds to the VFT temperature in the
localization model. 169, 120, 182 This correspondence has been confirmed in
previous simulation studies of polymeric 180, 183 and metallic GF liquids 182,
184 and grain boundary mobility. 181 However, we again emphasize that the
vanishing of $\langle u^{2}\rangle$ at a low $T$ should not be taken literally
just as the divergence of $\tau_{\alpha}$ at the VFT temperature should not be
taken literally. Taking the values of $\langle u^{2}\rangle$ at $T_{A}$ as
$\langle u_{A}^{2}\rangle$, our analysis based on polymer melts with variable
$a_{\theta}$ and $P$ in Figure 9 indicates that the relative isothermal
compressibility, $(\kappa_{T}-\kappa_{T}^{o})$, and the normalized quantity,
$\left(\langle u^{2}\rangle/\langle u_{A}^{2}\rangle\right)^{3/2}$, are
proportional to each other to a good approximation,

$(\kappa_{T}-\kappa_{T}^{o})=\mathcal{D}\left(\langle u^{2}\rangle/\langle
u_{A}^{2}\rangle\right)^{3/2},$ (22)

where the proportionality factor $\mathcal{D}$ may likewise be estimated by
taking $T_{\mathrm{g}}$ as a reference point, as in eq 21. This finding is
consistent with our hypothesis that $\langle u^{2}\rangle^{3/2}$ can be
interpreted as a local compressibility. The relative compressibility
$(\kappa_{T}-\kappa_{T}^{o})$ may be normalized to arrive at a dimensionless
reduction of the data for $\kappa_{T}$, but we found that this procedure
results in a plot with some scatter, given the large amount of data, so we do
not consider such a reduction in the present exploratory study. Note that
there is some curvature in some of the data for high $P$ and low $\langle
u^{2}\rangle$, which we do not currently understand, and we must admit that
the best fitted exponent could be a value somewhat different than the value of
$3/2$ assumed in Figure 9. These are refinements beyond our main point that
$\langle u^{2}\rangle$ can qualitatively be interpreted as a measure of local
fluid compressibility. We plan to study the interrelation between $\kappa_{T}$
and $\langle u^{2}\rangle$ more thoroughly in the future work.

It is also suggested from a comparison of Figures 8 and 9 that a relationship
exists between $s_{c}$ and $\langle u^{2}\rangle^{3/2}$, a relationship that
was noticed, but not understood, in previous works. 180, 120 These
correspondences thus allow us to understand the occurrence of a common
thermodynamic scaling for $\delta s_{c}$, $\delta\kappa_{T}$, and $\langle
u^{2}\rangle$, thereby providing some understanding of the interrelations
between different models of glass formation that emphasize these thermodynamic
properties in relation to dynamics.

We may obtain some insight into the critical value of $\delta\kappa_{T}$ from
our previous study of a coarse-grained polymer melt under variable $P$
conditions, 133 which considers a dimensionless compressibility defined by the
ratio of $S(0)$ to the value of the static structure factor at its first peak
corresponding to an average intersegment distance, the so-called
‘hyperuniformity index’, $h$. All the $h$ curves intersect at a common
temperature that roughly equals the VFT temperature $T_{0}$ obtained from the
structural relaxation time to within simulation uncertainty, where $h$ is near
$10^{-3}$, the defining condition for effective hyperuniformity. 194, 9, 195,
196, 170 This intersection phenomenon of the isothermal compressibility curves
is also apparent in ref 194, where we estimated $h$ when the cohesive energy
parameter was varied at constant $P$ and then compared the results to those
for variable $T$ conditions but at constant $V$. The intersection point
appears to be about the same as in the variable pressure study, so it appears
to be some sort of invariant. This critical condition has great potential
relevance for defining an EOS for dense fluids that should have comparable
significance for gases and liquids near their other critical point associated
with the emergence of a hyperuniform state. 9

###  3.7 Quantification of Anharmonicity in the Non-Arrhenius Relaxation
Regime

Figure 10: Generalized entropy theory test of thermodynamic scaling of the
bulk modulus. (a) Bulk modulus $B$ versus $T$ for a range of $P$. The cross,
triangle, and square symbols indicate the positions of the onset $T_{A}$,
crossover $T_{c}$, and glass transition temperatures $T_{\mathrm{g}}$ of glass
formation, respectively. (b) $B$ versus $T/\phi^{\gamma}$ with $\gamma=7.0$.
Clearly, $B$ does not exhibit thermodynamic scaling. Figure 11: Simulation
test of thermodynamic scaling of the bulk modulus. (a) Bulk modulus $B$ versus
$k_{B}T/\varepsilon$ for a range of $P$. The cross symbols indicate the
positions of the onset temperature $T_{A}$ of glass formation. (b) $B$ versus
$(k_{B}T/\varepsilon)/(\rho\sigma^{3})^{\gamma}$ with $\gamma=7.0$. Clearly,
$B$ does not exhibit thermodynamic scaling, in accord with the predictions
from the GET. Figure 12: Pressure derivative of the bulk modulus in the
generalized entropy theory. (a) Bulk modulus $B$ versus $P$ for a range of
$T$. Lines are linear fits. (b) $dB/dP$ versus $T/T_{A}$. The cross symbols
indicate the position of the onset temperature $T_{A}$ of glass formation. The
dashed line indicates the value of $dB/dP$ at $T_{A}$, $dB/dP_{T_{A}}=8.7$.
The derivative of $B$ with respect to $P$ progressively deceases upon heating
to a value near the exponent of $7$ for thermodynamic scaling at $T_{A}$ for
Arrhenius relaxation. Figure 13: Pressure derivative of the bulk modulus in
simulations. (a) Bulk modulus $B$ versus $P$ for a range of $T$. Lines are
linear fits. (b) $dB/dP$ versus $T/T_{A}$. The cross symbol indicates the
position of the onset temperature $T_{A}$ of glass formation. The dashed line
indicates the value of $dB/dP$ at $T_{A}$, $dB/dP_{T_{A}}=8.8$. The derivative
of $B$ with respect to $P$ progressively deceases upon heating to a value near
the exponent of $7$ in the high $T$ Arrhenius regime. Evidently, $dB/dP$ is
somewhat larger than the scaling exponent at $T_{A}$, below which non-
Arrhenius relaxation starts to be observed. A dash-dotted line indicates that
the temperature where the observed thermodynamic scaling exponent is
comparable with our $dB/dP$ estimate lies in the Arrhenius regime where
$\tau_{\alpha}$ approaches its high frequency limit $\tau_{o}$. Figure 14:
Temperature dependence of the pressure derivative of the bulk modulus along
with the extent of stringlike cooperative motion. (a) $(dB/dP)/(dB/dP)_{A}$
and $s_{c}^{*}/s_{c}$ versus $T/T_{A}$ determined from the generalized entropy
theory. (b) $(dB/dP)/(dB/dP)_{A}$ and $L/L_{A}$ versus $T/T_{A}$ determined
from simulations.

The GET allows us to consider the molecular origin of configurational
anharmonicity quantitatively to better understand thermodynamic scaling since
we may calculate the bulk modulus $B$, which is just the reciprocal of the
isothermal compressibility, $\kappa_{T}$. We show our estimates of $B$ versus
$T$ in Figure 10 calculated from the GET for the same polymer model for a
range of fixed $P$, as described in Section 2.1. The $B$ estimates based on
the GET and their counterpart based on simulations in Figure 11 clearly do not
exhibit thermodynamic scaling, in consistency with the $S(0)$ data shown in
Figure 7, but we may estimate $B$ in the zero pressure limit, $B_{o}$, and
$(dB/dP)_{o}$, which provides a measure of anharmonicity given that
$(dB/dP)_{o}$ equals $\gamma_{M}$ in the Murnaghan EOS. In Figures 12 and 13,
we show how $dB/dP$ is estimated in the GET and simulations, along with our
predictions for how $(dB/dP)_{o}$ varies with $T$. As expected, this
anharmonicity measure increases progressively upon cooling below $T_{A}$. This
finding accords also with experimental observations of the Grüneisen constant
of PS as a function of $T$. 47 Regardless of the observed universal
thermodynamic scaling, the effective anharmonicity
$\gamma_{M}(T)\equiv(dB/dP)_{o}$ clearly becomes more prevalent in the $T$
regime below $T_{A}$. For some reason, thermodynamic scaling of relaxation and
diffusion remains dependent on $\gamma_{M}(T_{A})$. This finding is a bit of a
puzzle, but explains the existence of thermodynamic scaling in the $T$ regime
below $T_{A}$.

To gain a little further insight into this initially unexpected phenomenon, we
plot $dB/dP$ normalized by its value at $T_{A}$ and $s_{c}^{*}/s_{c}$ versus
$T/T_{A}$ determined from the GET in Figure 14a. The simulation counterpart is
provided in Figure 14b. Both quantities grow in a parallel fashion, suggesting
that the enhanced collective motion in cooled liquids tracks the enhanced
anharmonic interactions in the low $T$ regime where standard TST breaks down.
It would appear that $\gamma_{M}(T_{A})$ sets the rate of change of
$s_{c}^{*}/s_{c}$ and the corresponding extent of collective motion in cooled
liquids. We remark that the fragility of glass formation, which is determined
by the differential change of $s_{c}^{*}/s_{c}$ or the extent of collective
motion at a fixed cohesive interaction strength, is then determined by the
differential rate of change of the anharmonicity of intermolecular
interactions. This provides a novel viewpoint of the origin of fragility in
the dynamics of GF liquids and the origin of collective motion. The collective
motion is the physical expression of anharmonic interactions and fragility,
defined at a fixed cohesive interaction strength. We also note that estimates
of $\gamma_{M}$ over the $T$ range well below $T_{A}$ can be larger by a
factor as large as $3$. This phenomenon probably explains the discrepancy
between estimates of $\gamma_{M}$ from the $P$ derivative of the bulk modulus
in the observed thermodynamic scaling exponent in recent measurements by
Grzybowski and coworkers. 197

There is a generalization of the Murnaghan EOS that addresses the singular
repulsive and attractive contributions of the local interatomic interaction
potential, which we have largely ignored in our discussion because we focus on
‘simple’ materials with similar van der Waals interactions governed by an
intermolecular potential that has largely the same shape for this broad class
of materials. Notably, this is the same starting assumption of Pitzer in his
definition of ‘simple’ liquids 5, 6, 7, 8 and van der Waals in his theory of
non-ideal gases, 198 where the interplay of attractive and repulsive
interactions is essential in developing an EOS description of non-ideal gases
and condensed materials. Of course, the interparticle potential can be rather
different in inorganic ionic materials and metals than synthetic polymers, and
the potential shape can naturally have its own influence on anharmonicity.
Correspondingly, Gilvarry discusses an alternative isothermal EOS based on the
assumption that the pair potential can be described by a Mie potential, 199 in
which there is a repulsive power-law potential between interacting particles
scaling as a function of distance as $\mathcal{B}/r^{n}$ and an additive
attractive contribution of the opposite sign scaling as $-\mathcal{A}/r^{m}$,
where $\mathcal{A}$ and $\mathcal{B}$ are defined by lattice sums. The
original reference to the work by Grüneisen is a bit difficult to obtain, but
the essential results are reprised by Gilvarry. 28, 29 This alternative
generalization of a lattice theory of solids based on the cubic lattice with
this particular family of anharmonic potentials leads to an EOS that
generalizes the Murnaghan EOS,

$P/[3B_{o}/(n-m)]=(V_{o}/V)^{n/3+1}-(V_{o}/V)^{m/3+1}=(V_{o}/V)^{n/3+1}[1-(V_{o}/V)^{(m-n)/3}],$
(23)

where special potential index pairs have been investigated extensively in
applications to particular materials. Interestingly, the Grüneisen parameter
for this extension of the Debye model can be calculated exactly for this class
of model anharmonic potentials 28 and is independent of $\mathcal{A}$ and
$\mathcal{B}$, $\gamma_{G}(\mathrm{Mie})=(m+n+3)/6$. More singular attractive
and repulsive interactions increase the degree of aharmonicity derived from
the shape of the interparticle potential, more singular potential interactions
giving rise to larger values of $\gamma_{G}(\mathrm{Mie})$ and thus
$\gamma_{M}=2\gamma_{G}(\mathrm{Mie})+1/3$. For fluids interacting through van
der Waals interactions, this source of anhamonicity, i.e., the ‘potential
anharmonicity’, probably just makes an additive contribution that is
relatively constant in this broad class of materials so that anharmonicity
related to monomer shape, molecular stiffness, cohesive interaction strength,
etc., ‘configurational anharmonicity’, makes the predominant contribution to
the thermodynamics and dynamics of GF liquids arising from anharmonic
interactions. Anderson 200 has carefully discussed the assumptions underlying
the derivation of this Grüneisen-Mie EOS and further derived an often cited
analytic expression for the $T$ dependence of the bulk modulus $B$ in terms of
anharmonicity parameters based on the same theoretical framework. The
resulting equation for $B$ from this analysis is equivalent to the
phenomenological ‘Wachtman equation’, 200 which has been highly successful in
describing the $T$ dependence of the shear and Young’s moduli of diverse
inorganic materials.

This type of anharmonicity, which should not be confused with the various
forms of anharmonicity that arise in molecular fluids in association with
packing and interaction frustration, e.g., due to the presence of semiflexible
bonds, could possibly account for thermodynamic scaling if either the
attractive or repulsive interactions were formally neglected in comparison to
the other term. We anticipate that this approach should have some
applicability to certain atomic liquids where the intermolecular potential can
reasonably be described by a functional form of this idealized form, 201 but
we are doubtful that this approach would be very useful in molecular fluids
where the interparticle potentials are far more complex. In particular, we
expect this framework to provide useful insights into thermodynamic scaling
observed in ‘strongly correlating fluids’, 142, 143 where the potential
clearly gives rise to potential-virial correlations. Leporini and coworkers
149 have considered thermodynamic scaling in liquids described by the class of
Mie potentials, providing an attractive set of computational data to analyze
from this perspective. We plan to purse an investigation of this kind in the
future.

###  3.8 Implications of Thermodynamic Scaling

There are many implications of thermodynamic scaling, and we discuss a few
here. First, we note that a fixed value of $TV^{\gamma}$ defines a state of
fixed configurational entropy density ratio $s_{c}^{*}/s_{c}$ rather than a
structurally ‘isomorphic’ state defined by the pair correlation function or
its integral. We are not aware of any other thermodynamic property that allows
for the definition of such an isokinetic state for the fluid dynamics. The
existence of the fixed reduced isoconfigurational entropy state means that we
may determine $\gamma$ by the simple derivative relation,

$\gamma=-(\partial\log T_{\mathrm{g}}/\partial V_{\mathrm{g}})_{P},$ (24)

which has been utilized in some experimental studies to estimate the exponent
$\gamma$ for thermodynamic scaling. 202 We may deduce relaxation functions
from the GET in a reduced variable form, $TV/T_{\mathrm{g}}V_{\mathrm{g}}$, as
considered in recent experimental studies. 97, 153, 203 We plan to discuss
this representation based on the GET in a future study.

We may also make significant statements about the $P$ dependence of
$T_{\mathrm{g}}$ and $T_{m}$. The original derivation of a quantitative
‘Lindemann relation’ by Gilvarry 28, 29 (see also ref 40) for the melting
temperature $T_{m}$ of crystalline materials was based on the Gilvarry’s
quantitative, but nonetheless semi-empirical, estimate for crystal
instability, 63

$RT_{m}=\Omega_{c,m}B_{m}V_{m},$ (25)

where $R$ is the gas constant, $\Omega_{c,m}$ is a ‘critical constant’ that
depends on the Poisson ratio, and $B_{m}$ and $V_{m}$ are the bulk modulus and
molar volume at $T_{m}$, respectively. Gilvarry 63 attributed the above
melting point criterion to an earlier work by Einstein and Lindemann. Equation
25 offers a significant improvement over Lindemann’s original instability
condition for melting, based on the assumption that particles actually began
to touch, while Gilvarry’s criterion, supported by measurements, indicated a
critical displacement scale on the order of $1/10$th the interparticle
distance. 204, 205 Accordingly, many authors refer to the widely accepted
semi-empirical melting criterion given by ref 25 as the Lindemann-Gilvarry
melting criterion.

Although Lindemann and Gilvarry are often recognized for the instability view
of melting in the recent scientific literature, we note that it was Sutherland
190, 191, 192, 193 who first formulated an instability of melting as part of
his pioneering and ambitious kinetic theory of solids. The theory of
Sutherland was the first theory of solids to recognize the significant role of
molecular kinetic energy in counterbalancing the strong attractive cohesive
interactions characteristic of condensed materials rather than just static
‘structure’ defined by density and molecular configurations in space. The
Sutherland theory explained the $T$ dependence of the rigidity of materials as
being caused by increasing repulsive intermolecular interactions caused by
molecular kinetic energy, as in a heated gas, and his theory of melting, or
more generally, fluidization, emphasized that the volume of molecular
‘domains’ explored by the molecules by virtue of their kinetic energy
increased in relation to the hard sore size of the molecules to a critical
point at which the molecules were no longer ‘imprisoned’ by their neighbors.
Sutherland also performed ‘simulations’ with marbles and further suggested
that the instability of the solid state might only require a relatively small
fraction of the molecules to be mobile in the technical sense. Sutherland
recognized that a paradox of this mode of instability is that the change in
the density and thus the static dimensions of the metallic materials that he
was preoccupied in studying experimentally and theoretically only allowed for
an increase of the average interparticle distance by a couple of percent upon
changing $T$ from absolute zero to $T_{m}$ so that the molecular size, as
measured by the interparticle distance, remains close to the hard core
molecular size. His struggle to understand the relatively small structural
change in terms of the density below $T_{m}$ continues to the present day. 124
In particular, the density measurements raise the question of how the
‘domains’ explored by the molecules, by virtue of their kinetic energy, could
be large enough to explain the existence of a melting instability. Much later
with the advent of MD simulation methods, Hoover and coworkers 174, 175 and
Reiss and Hammerich 187 illustrated the volumes explored by the centers of
particles in model liquids, providing insights into the intricate shapes of
the ‘rattle volume’ described by Sutherland, and these authors discussed the
direct relation of the geometry of this volume to the EOS, as discussed and
numerically estimated in more recent works by Sastry et al. 188 and Starr et
al. 180 for hard-sphere and LJ fluids, respectively.

Given the nearly fixed relation between the molecular size and interparticle
distance found in the condensed state, we may recognize Sutherland’s model of
melting as being basically equivalent to what is termed the Lindermann-
Gilvarry criterion of melting. However, the Sutherland criterion appears to
have some particular merit in the case of particles, such as polymers that are
not spherical, and the Sutherland picture of melting was more recently
advocated 206 without knowledge of Sutherland’s pioneering theory of solids
and melting. We remark that it is the strong interplay between the thermal
energy and interparticle interactions that lies at the core of thermodynamic
scaling, and the Sutherland model is the first theory of solids to address the
fundamental duality of kinetics and structure in the properties on condensed
materials without any reliance on the existence of a presumed lattice
structure of the solid. The Sutherland model of solids and its general view of
rigidity and mobility as being primarily kinetic rather than structural
phenomena have many merits, even from a modern perspective, and this model of
the condensed state deserves to be revisited with the aid of simulation
studies. Finally, we note the more recent interpretation by Starr and
coworkers 180 of the loss of ergodicity at the VFT temperature $T_{0}$ as
corresponding to a critical condition in which the cohesive interactions, and
corresponding increasing rigidity of cooled liquids, overwhelm the capacity of
thermal energy to displace molecules, resulting in particle localization and
‘structural arrest’. This interpretation of the fundamental nature of
solidification accords very well with the Sutherland model of solidification
as fundamentally arising from a cessation of capacity of molecules to explore
a domain large enough to escape their local environment because of
insufficient kinetic energies.

At any rate, the semi-empirical instability criterion by eq 25 appears to
describe $T_{m}$ for a wide range of materials where $\Omega_{c,m}$ is found
to depend on $\gamma_{G}$. For a fixed Poisson ratio, where the shear modulus
$G$ is simply proportional to $B$, we may simply replace $B_{m}$ in eq 25 by
$G_{m}$, after modifying $\Omega_{c,m}$ appropriately. The shift of $T_{m}$
with $P$ is then, 63

$T_{m}/T_{m,o}=(V_{m,o}/V_{m})^{\gamma-1}=(1+P/P^{*})^{(\gamma-1)/\gamma}.$
(26)

Note also that introducing the relation
$T_{m}/T_{m,o}=(V_{m,o}/V_{m})^{\gamma-1}$ into eq 1 leads to the empirically
successful Simon equation where $P$ is the melting pressure, $P_{m}$. 64, 63,
29, 207 A law of corresponding states analogous to that for gases may be
established for solid materials by taking the triple point as the reference
temperature in the Simon equation. 63

Arguments have also been given that the glass transition should be governed by
a Lindemann relation, 32, 124 and by extension of eq 25, we adopt the
corresponding relation for $T_{\mathrm{g}}$,

$RT_{\mathrm{g}}=\Omega_{c,\mathrm{g}}B_{\mathrm{g}}V_{\mathrm{g}}.$ (27)

Egami and coworkers derived an equation of the general form of eq 25 for
$T_{\mathrm{g}}$ with the specification of the Poisson ratio dependent
prefactor for metallic GF materials and observed good consistency with
experiment. 208 We also notice proportionalities between $T_{\mathrm{g}}$ and
$T_{m}$ and the bulk and shear moduli and with each other with a constant of
proportionality depending on the Poisson ratio. This has been verified in
experimental studies. 100, 101

By the same argument as above for $T_{m}$, we may expect $T_{\mathrm{g}}$ to
exhibit the same type of $P$ dependence as for $T_{m}$,

$T_{\mathrm{g}}/T_{\mathrm{g},o}=(V_{m,o}/V_{m})^{\gamma-1}=(1+P/P^{*})^{(\gamma-1)/\gamma}.$
(28)

Following Andersson and Andersson, 209 we have fitted the $P$ dependence of
all the characteristic temperatures ($T_{A}$, $T_{c}$, $T_{\mathrm{g}}$, and
$T_{0}$) of glass formation to this general functional form in our past works,
132, 133 along with $\Delta H_{A}$. We can now see the theoretical rationale
for this type of relationship.

Finally, we may gain physical insight into the $P$ dependence of the thermal
expansion coefficient $\alpha_{P}$. We can evidently calculate $\alpha_{P}$
from the GET, but the physical meaning of the result is not always clear. For
temperatures above the Debye temperature, a reasonable assumption for many
fluids is that the vibrational degrees of freedom are activated. We may invoke
the approximation of Delong and Petit, 210, 211 $C_{V}\approx 3R$, so that eq
25 implies, 212

$\alpha_{P}\approx\gamma_{G}(3R)\kappa_{T}/V.$ (29)

This indicates that we may estimate the change of $\alpha_{P}$ with $P$ as,

$\alpha_{P}/\alpha_{P,o}\approx(1+P/P^{*})^{-\delta_{T}},$ (30)

where the exponent $\delta_{T}$ has been termed the Anderson-Grünesisen
parameter. 213, 214, 215 This relation, with $\delta_{T}$ being an adjustable
parameter, has been used in applications in geophysical science. 216 Given eqs
26, 28, and 30, we can then readily understand the near constancy of
$\alpha_{P}(T_{m})T_{m}$ and $\alpha_{P}(T_{\mathrm{g}})T_{\mathrm{g}}$ in
both crystalline and GF materials. 124, 217 This is just the result of a near
cancellation of anharmonic interaction effects in $\alpha_{P}$ and the
characteristic temperatures, $T_{m}$ and $T_{\mathrm{g}}$. As a remarkable
general property of condensed materials, we note that there is a general
approximate inverse relation between the material stiffness (tensile modulus)
of diverse condensed materials and the square of $\alpha_{P}$, 47 although
there are exceptions of this general trend in some polymeric materials. We
also mention that Ledbetter 41 has derived a very interesting interrelation
between the thermal expansion coefficient and the bulk modulus and it
derivative with respect to $T$.

The ‘anomalous’ character of thermal expansion in polymer materials can be
rationalized, at least in part, by the observation that approximating $C_{V}$
by a constant is often not suitable in polymer materials because the potential
energy has large contributions arising from the intramolecular bonds, in
addition to the intermolecular interactions similar to those dominating the
potential energy in atomic and small-molecule liquids. 42, 47 Correspondingly,
the most effective models of the specific heat of polymer materials have been
modeled by taking the polymer chains to be one-dimensional crystals whose
interchain interactions are modeled by the Debye model or its extension to
include anharmonicity. 42, 47 The application of this type of model to many
experimental systems has indicated that the Debye temperature associated with
chain bonding tends to be quite large so that the $T$ regime where $C_{V}$ can
be taken as just a constant as in eq 29 is limited in these materials. We have
recently discussed these issues regarding the complexity of the $T$ dependence
of $C_{V}$ in polymers from a computational standpoint. 194

Thermodynamic scaling thus provides quantitative and qualitative insights into
widely observed scaling relations for how the melting temperature and the
characteristic temperatures of glass formation depend on pressure.

##  4 Summary

We have shown that thermodynamic scaling can be derived within a transition
state theory framework in conjunction with the well-validated Murnaghan
equation of state 23, 30, 31 relating changes of pressure to changes in volume
in condensed materials and the Gilvarry interpretation of this equation of
state 28, 29 in terms of an anharmonic extension of the Debye model of solid
materials that incorporates Grüneisen’s quantification of material
anharmonicity in interatomic interactions. 27 The strict applicability of
these calculations is limited to the Arrhenius regime where the fluid may be
considered dynamically homogeneous so that relaxation and diffusion can be
appropriately described by transition state theory. We have addressed the
important extension to the temperature regime below the onset temperature
$T_{A}$ for non-Arrhenius relaxation and diffusion through the vehicles of the
generalized entropy theory and molecular dynamics simulation, where the same
thermodynamic scaling has been shown to hold below $T_{A}$ as in the Arrhenius
regime. This fact has been rationalized by the observation of thermodynamic
scaling of the configurational entropy density $s_{c}$, normalized by its high
temperature value $s_{c}^{*}$, in explicit calculations based on the
generalized entropy theory. We have also shown that the extent $L$ of
stringlike collective motion normalized by its value $L_{A}$ at $T_{A}$, which
determines the temperature dependence of the activation energy below $T_{A}$
in the string model, obeys thermodynamic scaling, consistent with the
identification of the strings with the cooperatively rearranging regions of
Adam and Gibbs. 72

As noted by Dyre and coworkers. 143 the existence of thermodynamic scaling
offers a ‘filter’ for assessing models of glass formation. Accordingly, we
found that the static structure factor in the long wavelength limit, $S(0)$,
and the isothermal compressibility, do not exhibit thermodynamic scaling,
raising questions about theories that purport that the structural relaxation
time and diffusion are directly related to $S(0)$. 171, 161, 162, 163, 164,
165, 166, 167 On the other hand, we found that the mean square displacement at
a caging time, the Debye-Waller parameter $\langle u^{2}\rangle$, follows
thermodynamic scaling to a high approximation, a point made by Leporini and
coworkers 168 and others. 151 This means that the localization model of glass
formation, 169, 120, 182 and the somewhat related model of Leporini and
coworkers, 168 passes this basic consistency test associated with
thermodynamic scaling. Thermodynamic scaling then offers a powerful tool for
testing models of glass formation, although this scaling property in itself
does not establish any such theory.

Thermodynamic scaling also has great potential for providing new research
directions on glass-forming liquids. The finding that $\langle u^{2}\rangle$
and the reduced configurational entropy obey a common thermodynamic scaling
has naturally led us to consider whether these properties might have some
direct relationship. Based on the further plausible assumption that $\langle
u^{2}\rangle^{3/2}$ might be interpreted as local ‘compressibility’, we have
found based on the GET that there is indeed a direct relation between the
fluid isothermal compressibility and configurational entropy once both
quantities are properly reduced. This scaling relation provides insight into
why the isothermal compressibility and $S(0)$ do not obey thermodynamic
scaling, and more importantly, it allows the entropy theory to be recast in
terms of a reduced isothermal compressibility, a quantity that is directly
measurable. This result provides conceptual insight into the interrelation
between changes in the material rigidity and configurational entropy that lie
at the heart of ‘shoving’ or elastic models, 99, 100, 101 and the localization
model, 169, 120, 182 a type of ‘dynamic free volume’ model of glass formation
that emphasizes the volume explored by particles in their condensed state
rather than the local density as in Sutherland’s kinetic theory of solids.
190, 191, 192, 193

We have considered the implications of our newly found capacity to quantify
the aharmonicity in condensed materials and found that there are different
types of anharmonicity, namely, potential and configurational anharmonicities,
derived from the shape of the pair potential and from the monomer shape,
rigidity, connectivity, and other factors that influence molecular packing,
respectively. This allows us to understand why the thermodynamic scaling
exponent varies greatly in polymeric and other molecular materials in which
molecules interact through a similar local pair potential describing their van
der Waals interactions. The generalized entropy theory allows for a direct
calculation of how the thermodynamic scaling exponent $\gamma$ relates to the
fragility of polymeric glass-forming materials, 81 along with other
thermodynamic and dynamic properties of these materials within a consistent
theoretical framework. We have also found the unexpected result that the
Murnaghan exponent $\gamma_{M}$, our anharmonicity measure, can change
appreciably with temperature and that these changes track the extent of
stringlike collective motion below the onset temperature $T_{A}$ for non-
Arrhenius relaxation. This phenomenon, which certainly requires further
investigation, seems to suggest that the collective motion below $T_{A}$ is a
direct physical expression of a growing anharmonicity of intermolecular
interactions in cooled liquids. Anharmonicity is either the cause or the
effect of this form of dynamic heterogeneity.

Finally, after establishing thermodynamic scaling as an attribute of our
simulation observations on glass-forming polymer melts and the generalized
entropy theory and the string model of glass formation, we have considered
some of the many ramifications of thermodynamic scaling, such as the shift of
the glass transition and melting temperatures with pressure and other material
properties (e.g., thermal expansion coefficient and isothermal
compressibility) that depend strongly on anharmonic interactions. We have
shown that many phenomenological relationships, such as the Simon relation 63
for the pressure dependence of the melting temperature and its analog for the
glass transition temperature of glass-forming materials, the Anderson relation
213, 214, 215 for the thermal expansion coefficient, the Tait equation, 52 and
the Bridgeman equation, 27, 43, 33 can be understood within the unifying
framework of the Murnaghan equation of state. We have also obtained a better
understanding of the near constancy of the product of the thermal expansion
coefficient and the melting or glass transition temperatures in materials that
solidify through crystallization or glass formation.

W.-S.X. acknowledges the support from the National Natural Science Foundation
of China (No. 21973089). A portion of the simulations was conducted at the Oak
Ridge National Laboratory’s Center for Nanophase Materials Sciences, which is
a DOE Office of Science User Facility. This research used resources of the
Compute and Data Environment for Science (CADES) at the Oak Ridge National
Laboratory, which is supported by the Office of Science of the U.S. Department
of Energy under Contract No. DE-AC05-00OR22725. This research also used
resources of the Network and Computing Center at Changchun Institute of
Applied Chemistry, Chinese Academy of Sciences.

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