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\DOI{10.5802/crphys.130}
\datereceived{2022-04-20}
\daterevised{2022-11-09}
\dateaccepted{2022-11-15}
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\dateposted{2023-02-08}
\datepublished{2024-04-26}
\begin{document}

\begin{noXML}

%\TopicEF{From everyday glass to unordered solids}{Du verre quotidien
%aux solides d\'esordonn\'es}

\title{Entropy of metallic glasses and the size effect on glass
transition}

\alttitle{Entropie des verres m\'etalliques et effet de taille sur la
transition vitreuse}

\author{\firstname{Yannick} \lastname{Champion}\CDRorcid{0000-0003-0147-6919}}
\address{Univ. Grenoble Alpes, CNRS, Grenoble-INP, \emph{SIMaP}, 38000
Grenoble, France.}
\email{yannick.champion@grenoble-inp.fr}

\keywords{\kwd{Metallic
glass}\kwd{Entropy}\kwd{Nano-indentation}\kwd{Mechanical
behavior}\kwd{Glass transition}}

\altkeywords{\kwd{Verre
m\'etallique}\kwd{Entropie}\kwd{Nano-indentation}\kwd{Comportement
m\'ecanique}\kwd{Transition vitreuse}}

\begin{abstract} 
A statistical description of metallic glasses is proposed based on
evaluation of an entropy and using the concept of potential energy
landscape (PEL). The PEL is probed using nano-indentation, assuming
that serrations observed are related to local energy levels. Serrations
sizes follow a Poisson distribution consistent with, that their
formation is a rare events. An entropy is derived revealing parameters
characterizing the glass structure. The relevance of the approach is
tested on the size effect on mechanical properties of metallic glass.
It is noticed that the relations explain the effect of film thickness
on the glass transition temperature observed for glass polymers.
\end{abstract} 

\begin{altabstract} 
Une description statistique des verres m\'etalliques est propos\'ee,
bas\'ee sur l'\'evaluation d'une entropie et utilisant le concept de
paysage du potentiel \'energ\'etique (PEL). Le PEL est sond\'e en
utilisant la nano-indentation, en supposant que les serrations
observ\'ees sont li\'ees aux niveaux d'\'energie locaux. La taille des
serrations suit une distribution de Poisson coh\'erente avec la
dynamique des \'ev\'enements rares. Une entropie est d\'eriv\'ee
r\'ev\'elant des param\`etres caract\'erisant la structure du verre. La
pertinence de l'approche est test\'ee sur l'effet de taille sur les
propri\'et\'es m\'ecaniques du verre m\'etallique. On constate que les
relations expliquent l'effet de l'\'epaisseur du film sur la
temp\'erature de transition vitreuse observ\'ee pour les polym\`eres
vitreux.
\end{altabstract} 

\maketitle

{\vspace*{-2pt}}

\end{noXML}

\section{Introduction}

There have been much efforts for decades to define relevant structural
descriptions of glasses, but the task is still very challenging and
absence of description a main obstacle to a better understanding and
modelling of properties. To contrast, crystalline solids are fully
described by Bravais lattice and pattern. From these reduced
quantities, defects are quantitatively defined, such as dislocation by
a simple lattice translation (the Burgers vector) and insight of
properties is well approached from symmetry. Even the complex
quasicrystals are described in a periodic 6-dimension space and
appropriate projections in the physical one. In this landscape of
symmetry and periodicity at the atomic scale, glass materials are
excluded. Even if local atomic \mbox{arrangements} and short range
order have been observed (the random distribution of
\mbox{near-icosahedra} in metallic alloys, octahedra in oxides, chains
of polymers) using high level characterizations technics and
simulations (X-rays absorption, NMR\,\ldots), no generic structural
rules are emerging.\looseness=-1

It is arising from disorder in glass that descriptions might be most
likely statistical in nature. In addition, disorder also implies that
none of the atoms or groups of atoms are able to define relevant
reference for spatial localization. This second requirement can be
faced by forgetting atoms and characterizing the glass using its
potential energy landscape (PEL) introduced by Stillinger~\cite{1} and
still a leading formalism for the analysis of glass forming
dynamics~\cite{2} or deformation~\cite{3}. The first requirement can be
fulfilled by evaluation from this PEL of an appropriate entropy for the
glass leading to characterization parameters or information parameters
with analogy to the Shannon entropy of information~\cite{4}. Besides,
it has been shown that the PEL can be divided following its various
characteristic domains (basin of attraction, minima\,\ldots) to form a
network, suggested to allow thermodynamics properties to be
calculated~\cite{5}.

We used nano-indentation mechanical testing to probe the PEL in order
to reveal the glass zones where (atomic) rearrangements are
allowed~\cite{6} and evaluate the statistic of these events. From this
analysis an apparent entropy is derived and analyzed with the objective
of identifying a specific length scale characterizing the glass, as
cell parameter is characterizing the crystal or the Burgers vector the
dislocation. The work reported in this article originally focused on
the description of metallic glasses. But, we shall see that the results
and analyses can be generalized to interpret behaviors of oxides and
polymers glasses. With some hindsight, this is little surprising. The
various classes of glass exhibit different physical, chemical and
mechanical properties due to difference in their atomic bonds and local
atomic arrangement. But the common high degree of disorder, leads to
similar dynamics with only difference in the length and time scales.
The emblematic common characteristic is the glass transition
temperature, $T_{g}$ delimiting absence and large plastic yielding
behavior. The entropic description presented is tested on the
prediction of the size effect on the mechanical behavior (transition
from localized to homogenous deformation) observed for metallic and
oxide glass, as well as the on the $T_{g}$ variation with film
thickness in polymers.

\section{Derivation of entropy}

Among main features, metallic glasses exhibit nano-metric serrations
observed in nano-indentation~\cite{7,8} or nano-pillar compression
tests~\cite{9}. A serration is the results of atomic displacement under
stress effect and its magnitude is naturally assumed to be related to
local specific potential energy in the PEL. Hence, the mechanical test
should be able to probe the local structure of the glass, showing in
the displacement extent (serration size) the capacity for local atomic
rearrangement related to the local energy. It is well known that the
occurrence of serration events is strain rate dependent~\cite{8}.
Hence, the most relevant structural description obtained from the
mechanical probing, necessarily requires that the time scale of
experiment is lower than the timescale of the mechanisms controlling
the serration formation. To an experimental point of view, testing must
be performed in quasistatic conditions that is at the lower strain rate
as possible.\looseness=-1

Experiments were carried out on the
$\text{Mg}_{65}\text{Cu}_{12.5}\text{Ni}_{12.5}\text{MM}_{10}$ alloy,
prepared by electromagnetic melting of Mg, Cu, Ni and mischmetal
($\text{Ce}_{75}\text{La}_{25}$) in a cold crucible under controlled
atmosphere~\cite{10}. The metallic glass was then produced by twin roll
casting of the liquid alloy in He atmosphere into the form of a sheet
with a thickness of about $400~{\upmu}\text{m}$, a width of about 3~cm
and a length of about 30~cm. In the as cast conditions, the alloy
surface roughness is very low, controlled by the imaging mode of the
nano-indenter. An oxide is also formed at the surface with a thickness
of about 2~nm negligible at the experimental indent depth considered
for the analyses. Characterization of the amorphous alloys is reported
in~\cite{10}. As cast sample containing free volume was investigated.
Nanoindentation was performed with a TI 950 Hysitron Tribo Indenter
equipped with a Berkovich tip (threefold symmetry pyramid). For each
measurement, the load was applied at a loading rate of
$4~\text{mN}{\cdot}\text{s}^{-1}$ up to the maximum load of
$8000~{\upmu}\text{N}$. Strain rate, $\dot{\varepsilon} =
{\text{d}h}/{h\,\text{d}t}$ ($h$~is the indent depth and $t$ is the
time) is extremely high at the onset of the deformation and thus
serrations are too small to be detected due to high strain rate effect.
After about 100~nm indentation depth, the strain rate is nearly
constant at a sufficiently low value of about~1 to $0.3~\text{s}^{-1}$
(nearly quasistic conditions).

Figure~\ref{fig1} is an example of a nano-indentation curve, from the
work of Thurieau {\etal}~\cite{10}, with the loading part showing
serrations of size $h$ and the waiting time $\updelta t$, between two
successive serrations (inset). The curve shows a plateau at the maximum
load; this is a depth-time relaxation experiment at a constant load,
used for evaluation of the activation volume of the deformation process
(published elsewhere~\cite{10}). The nano-indentation test is ending by
the unloading part. Serrations size and waiting times were
automatically detected and measured using a tailor-made automated
procedure based on the detection of slope discontinuities along the
loading part of the indentation curves. The resolution in depth
displacement is lower than 1~nm. 7980 serrations were measured from 320
indentation curves. The large dataset enables a relevant statistical
analysis, containing in particular large size events having very low
probability of occurrence.

\begin{figure}[t!]
\includegraphics{fig01}
{\vspace*{-2pt}}
\caption{\label{fig1}Nano-indentation curve showing serrations on the
loading branch, also shown in the inset with the serration size, $h$
and the waiting time between two successive serrations $\updelta t$
(data from~\cite{10}).}
{\vspace*{-4pt}}
\end{figure}

To connect the serrations to the PEL structure, the glass is
discretized in non-periodic regions of size~$l$, called ``clusters'',
having various potential energy in the PEL. Figure~\ref{fig2} is a 2D
scheme of a glass in such description, where black clusters are in
favorable energy configuration for rearrangement under an external
solicitation. The grey clusters depict the potential gradient around
favorable ones. Under the applied shear stress, $\tau$, atomic
rearrangement is expected over a distance $h \approx k \times l$,
corresponding to successive favorable and subsequent activated clusters
(black and grey) one calls ligament. In that picture, the total number
of clusters is very large ($n_{c} \gg 1$) and the probability to find a
ligament (favorable clusters) is very small ($p \ll 1$). Such rare
event configuration is described by a Poisson distribution, with $P_{k}
= {\lambda^{k}\text{e}^{-\lambda}}/{k!}$ the probability of finding
ligaments formed of $k$ clusters. $\lambda = p \times n_{c}$ is the
average number of clusters giving the most expected ligament size
$\overline{h} = \lambda \times l$. In this approach, the ligaments are
associated to experimental serrations observed in the mechanical
testing.

\begin{figure}[t!]
\includegraphics{fig02}
\caption{\label{fig2}2D scheme of the glass description in a PEL
approach. The glass is divided in ``clusters'' of size~$l$, formed in
average of the same number but different atoms. Clusters are
distributed over the PEL, where some (black) surrounded by various
energetic gradient of clusters, are in favorable energy configuration
for atomic rearrangement. An external shear stress solicitation $\tau$
produces the rearrangement in the stress direction over various number
of clusters, depending on the potential energy configuration.}
\end{figure}

The normalized distribution of experimental serrations, $P_{e}(h)$ is
plotted in the Figure~\ref{fig3} and compared to a normalized Poisson
distribution $P(h)$ with the fitting parameters: $\overline{h}=3.47 \pm
0.03$~nm, $l=0.64 \pm 0.03$~nm, which gives $\lambda \approx 5$. The
waiting times, $\updelta t$ distribution (inset of
\mbox{Figure}~\ref{fig3}) is also consistent with the Poisson statistic
and verifies $P(\updelta t) = A\text{e}^{-\lambda\updelta t}$. It was
reported that the activation volume controlling the shear band
formation in this Mg glass is of the order of 3 atoms~\cite{10}.
Similar value was reported by Schall and collaborators~\cite{11} and Ju
and collaborators~\cite{12} for different materials. The present
statistical analysis is consistent with that result, considering that
an elementary displacement of the order of~$l$, which is about two
interatomic distances, needs a rearrangement of at least 3~atoms. It is
much satisfactory to find analogy between cluster size used for the
discretization of the glass and the activation volume controlling shear
band formation, having a physical meaning.

\begin{figure}[t!]
\includegraphics{fig03}
\caption{\label{fig3}Normalized experimental distribution of serration
sizes $P_{e}(h)$ (open dots) compared with the Poisson distribution
$P(h)$. Distribution of waiting time, $\updelta t$ between successive
serrations is shown in the inset.}
\end{figure}

For derivation of thermodynamic quantities, the previous analysis is
linked to a statistical physic approach similar to that proposed by
Gibbs and Adam~\cite{13} to describe glass viscosity. The probability
to find a ligament formed of $k$ clusters writes, $P_{k} =
{g_{k}\text{e}^{{-u_{k}}/{k_{b}T}}}/{Z}$, where $u_{k}$ is the
energy of a ligament of $k$ clusters in the PEL, $g_{k}$ is a
degeneracy factor, $Z$ is the partition function, $k_{b}$ is the
Boltzmann constant and $T$ is the absolute temperature. For the most
probable ligament of $\lambda$ clusters, it is arbitrarily chosen that
$g_{\lambda}=1$. Combining with the Poisson distribution, the free
energy is obtained: $f = - k_{b}T\ln Z = u_{\lambda} - k_{b}
T\ln({\lambda!}/{\lambda^{\lambda}\text{e}^{-\lambda}})$, where the
energy of the ligament, $u_{\lambda}$ is unknown. From this quantity, a
simple apparent entropy per ligament is derived:

{\begin{equation}\label{eq1}
s = k_{b}\ln\left(\frac{\lambda!}{\lambda^{\lambda}
\text{e}^{-\lambda}}\right).
\end{equation}}\unskip
With analogy to the Shannon measure of information (information
entropy), the value of relation~(\ref{eq1}) is a first step in a
statistical description of the glass, which depends on the unique
structural parameter $\lambda$ related to the glass PEL.

\section{The size dependence of the mechanical behavior in metallic glass}

As a first attempt for validation, the entropic description is tested
in regards to the size dependence of the mechanical behavior. Glasses
(metallic, oxide, polymers) flows at high temperature allowing easy
forming, but are brittle below the glass transition temperature. Of
course, fundamental mechanisms for absence of macroscopic plastic
deformation are distinct between these phases due to their difference
in atomic bonds. However, in both cases the brittleness stems from the
stress localization leading to formation of cracks in oxides and a
shear bands in metals and polymers. It is then very intriguing to
observe homogenous plastic deformation when glass sample is getting
sufficiently small. The first evidence was reported by Taylor in 1949
using indentation on a borosilicate optical glass~\cite{14}. Since,
mechanical tests on micro pillars reveal plasticity for diameter of
sample of few micrometers~\cite{15}. For metallic glasses, the first
evidence was reported by Volkert and collaborators on a PdSi alloy
nano-pillars with diameter of 440~nm~\cite{16}. So far, an explanation
has been based on a necessary critical volume for crack or shear band
formation and propagation~\cite{16,17}. With analogy to the Griffith
criterion, the critical size is estimated from the balance between
elastic energy stored under the applied stress $\sigma$ and the crack
or shear band energy, $\Delta \propto {\Gamma E}/{\sigma^{2}}$ where
$\Gamma$ is the energy per unit area of crack or shear band and $E$ is
the Young modulus. Origin and mechanism for the change in the
deformation mode from brittle to ductile in glasses is still debating.
In their work, Guo and collaborators~\cite{18} observed in situ at the
microscopic scale, using transmission electron microscopy, the
homogenous deformation of a metallic glass followed by instability
leading to necking similar to deformation in crystals. \mbox{Evidence}
of shear band inhibition was interpreted through several angles. With
the ``Griffith criterion'', they proposed the interesting aspect of
necessary distance for shear band to reach a mature stage. In addition,
the authors emphasized the fact that the sample size is most likely
below the range of shear band spacing. They suggest that the sample
volume is less likely to contain ``fertile'' sites for the initiation
of shear bands (those favorable to form shear transformation zones).

An alternative to the mechanical approach, necessarily based on stress
localization and shear band properties, is to assume that the glass
flows at a critical size as it does at the glass transition
temperature. Hence, a ``sample size---glass transition temperature''
equivalence is envisaged similar to the ``time-temperature''
equivalence in thermally activated process. The glass transition is the
temperature below which elementary species mobility is too low to
observe relaxation at the time scale of experiment. In their
statistical mechanics theory, Gibbs and DiMarzio, showed that the
increase in relaxation time at the glass transition is related to a
dramatic decrease towards very small value of the configurational
entropy~\cite{19}. Later, using this idea Gibbs and Adam introduced the
configurational entropy in the description of viscosity~\cite{13}.
Based on this approach, an interpretation of the size dependence would
be that an increase in specific configurational entropy happens with
decreasing the sample size. Homogenous deformation should be then
allowed when the value of the size dependent glass transition is
reached at room temperature.

A change in entropy with size was first reported in~\cite{20}. Before,
Stillinger suggested that minima in the PEL varies exponentially with
size~\cite{21}. From the relation~(\ref{eq1}), rewriting
$\text{e}^{-\lambda} = \lim_{n_{c} \rightarrow \infty}(1 -
({\lambda}/{n_{c}}))^{n_{c}}$ and identifying that ${{\lim_{n_{c}
\rightarrow \infty} (1 - ({\lambda}/{n_{c}}))}^{n_{c}} \approx \lim_{n
\rightarrow \infty} (1 - ({\lambda}/{n}))}^{n}$, $n$ is the total
number of ligaments, one obtains after Taylor series development, a
size dependent specific entropy following:
{\begin{equation}\label{eq2}
s = k_{b}\ln\left(\frac{\lambda!}{\lambda^{\lambda}
\text{e}^{-\lambda}}\right) + k_{b}\frac{\lambda^{2}}{2n}.
\end{equation}}\unskip
Reducing $n$ means that the statistic is changing from Poisson to
binomial, a well-known property for these distributions. In other
words, the way events (serrations) are ``drawn'' (probed) changes with
the reduction of the sample size. From this result and considering the
Gibbs and Adam analysis~\cite{13}, a ``sample size---glass transition
temperature'' equivalence is defined writing that the entropy of an
infinite sample at the glass transition temperature is equal to the
entropy of the small size specimen at room temperature.

This simple equality needs however, some special care. It is emphasized
that entropy of the small size sample is calculated with data obtained
from the mechanical testing, then considering only the ligaments probed
by the mechanical solicitation. Consequently, the entropy from room to
glass temperature must be rescaled by the ligaments fraction probed in
the mechanical testing. Heat capacity $c_{p}$ is rescaled in number of
ligaments (or mole of ligaments), that is by ${\sim}3\lambda$, assuming
cluster formed of about 3~atoms. Then the balance between the entropy
of the small size specimen and the entropy of infinite specimen at the
glass transition writes, with $\Omega$ the atomic volume and
${n_{l}}/{n}$ the fraction of ligaments probed:
{\begin{equation}\label{eq3}
k_{b}\ln\left(\frac{\lambda!}{\lambda^{\lambda}\text{e}^{-\lambda}}
\right) + k_{b}\frac{\lambda^{2}}{2n} = k_{b}\ln\left(\frac{\lambda!}
{\lambda^{\lambda}\text{e}^{- \lambda}}\right) + \frac{n_{l}}{n}
\int_{T_{\mathrm{RT}}}^{T_{g}}{c_{p}\frac{\text{d}T}{T}}.
\end{equation}}\unskip
The volume of the specimen is $V = n3\lambda\Omega$ and the volume
density of ligament is $\Psi = {n_{l}}/{V} =
{n_{l}}/{n3\lambda\Omega}$.

After rearrangement, a critical size for the transition in the specimen
size effect on the mechanical behavior is obtained:
{\begin{equation}\label{eq4}
\Delta=\sqrt[3]{\frac{k_{b}\lambda}{6\Psi\int_{T_{\mathrm{RT}}}^{T_g}
{c_{p}\frac{\text{d}T}{T}}}}.
\end{equation}}\unskip
For the Mg based metallic glass used in our nano-indentation
experiments, $\lambda = 5$ is determined from a robust statistical
distribution. The ligaments density is estimated of
$10^{-8}~\text{nm}^{-3}$, assuming a hemispheric zone, $v \approx
60h_{\max}^{3}$, where $h_{\max}$ is the maximum indent depth. 
An~\mbox{entropy} from room temperature to glass transition was numerically
evaluated from~\cite{22} of about $10~\text{J}{\cdot}\text{mol}^{-1}
{\cdot}\text{K}^{-1}$. Then the critical size for the transition from
brittle to ductile is calculated of about 400~nm, consistent with
observations by Lee and collaborators on pillars with diameter smaller
than 1000~nm~\cite{23}. The approach was applied successfully for
various metallic glasses tested on nanopillars reported in the
literature~\cite{20}. In particular, the first observation by
Volkert~\cite{16} reports on critical size of 440~nm, whilst the
relation~(\ref{eq4}) predicts 500~nm.

\section{Size dependence of the glass transition in glass polymer}

In the previous section, the motivation was to evaluate the critical
size for glass transition at room temperature. However, one notes that
this relation is valid at any temperature below $T_{g}$ and an apparent
size dependent glass transition temperature can be naturally derived,
substituting $T_{\mathrm{RT}}$ by $T_{g\Delta}$ in~(\ref{eq4}).
Assuming that the $c_{p}$ variation with temperature is little ($c_{p}$
is from 25 to $50~\text{J}{\cdot}\text{K}^{-1}{\cdot}\text{mol}^{-1}$
for metallic glass), a size dependent $T_{g}$ is obtained from
relation~(\ref{eq4}) following:
{\begin{equation}\label{eq5}
T_{g\Delta} \cong \frac{T_{g}}{\left\lbrack 1 + \frac{k_{b}\lambda}
{6\Psi C_{p}\Delta^{3}}\right\rbrack} = \frac{T_{g}}{\left\lbrack 1
+ \left(\frac{\Delta^{*}}{\Delta}\right)^{3}\right\rbrack}
\end{equation}}\unskip
$\Delta^{*} = \sqrt[3]{{k_{b}\lambda}/{6\Psi C_{p}}}$ is a
characteristic length scale of the glass.
{\vspace*{3pt}}

Substituting the exponent 3 by a free exponent $\delta$, and verifying
that $\Delta^{*} \ll \Delta$, a Taylor development of the
relation~(\ref{eq5}) gives:
{\begin{equation}\label{eq6}
T_{g\Delta} = T_{g}\left\lbrack 1 - \left(\frac{\Delta^{*}}
{\Delta}\right)^{\delta}\right\rbrack
\end{equation}}\unskip
which is the empirical relation proposed by Keddie and collaborators to
fit the experimental variation of $T_{g}$ with polystyrene film
thickness~\cite{24}.

In their work, $T_{g}$ is obtained by detecting the discontinuity in
expansivity using ellipsometry as the films are heated. These authors
found the fitting parameters of the relation~(\ref{eq6}) of $\delta=1.8
\pm 0.7$, a bulk $T_{g} = 373.8 \pm 0.7$~K and $\Delta^{*} = 3.2 \pm
0.6$~nm. Their data were fitted using the relation~(\ref{eq5}) with the
exponent $\delta$ instead of~3, giving the fitting parameters: $\delta
= 1.2 \pm 0.1$, a bulk $T_{g} = 375 \pm 1$~K and $\Delta^{*} = 1.4 \pm
0.4$~nm. Figure~\ref{fig4} shows the fitting of Keddie's data with good
agreement using the relation~(\ref{eq5}) derived from the entropic
description.

\begin{figure}[t!]
{\vspace*{-.15pc}}
\includegraphics{fig04}
{\vspace*{-.3pc}}
\caption{\label{fig4}Replot of the data from Keddie {\etal}~\cite{24},
$T_{g\Delta}$ as a function of the polystyrene film thickness $\Delta$
fitted with the relation~(\ref{eq6}) with exponent $\delta$ instead
of~3.}
{\vspace*{.3pc}}
\end{figure}

The value of $\delta$ (non integer) from these measurements is rather
intriguing; thickness variation would have made more intuitive to
find~1. However, from our analysis it is pointed out that the
exponent~3 is not related to the 3 dimensions of the sample, but to
volume (or spacial) distribution of ligaments which should be still
valid for films. As a result, the reduction of one dimension or more,
should affect the ligament based entropy in the same way and the
$T_{g\Delta}$ variation with thickness should be intuitively expected
to follow the relation~(\ref{eq5}) with thickness exponent of~3.
Forrest and collaborators, proposed another model to explain the
thickness effect on $T_{g}$~\cite{25}. It is not discussed in the
present paper, but is is emphasized by these authors that the
polystyrene thin film studied by Keddie and collaborators were
supported on a (111) oriented silica wafer most likely affecting the
size dependence. In their work, Forrest and collaborators measured the
variations of $T_{g\Delta}$ with the thickness of free standing
polystyrene films, with molecular weight ranging from 116~K to 347~K.
Their data were replotted and fitted with the relation~(\ref{eq5}) with
still exponent $\delta$ instead of~3 (Figure~\ref{fig5}).
The fitting parameters are
obtained as $\delta = 3.2 \pm 0.5$, a bulk $T_{g} = 366 \pm 3$~K and
$\Delta^{*}=13 \pm 1$~nm. The exponent $\delta$ close to 3 confirms the
entropic description well. The bulk value of $T_{g}$ is corresponding
to polystyrene with average molecular weight of 120~K, consistent with
the values reported by Forrest {\etal}~\cite{25}.
As pointed out by these authors
influence of the substrate in the work of Keddie is understood from the
relation~(\ref{eq5}) as a ``bulk'' glass transition dependence with the
film thickness related to a gradient of material properties induced by
the substrate.

\begin{figure}[!h]
\includegraphics{fig05}
%{\vspace*{-.3pc}}
\caption{\label{fig5}Replot of the data from Forrest {\etal}~\cite{25},
$T_{g\Delta}$ as a function of the polystyrene film thickness $\Delta$
fitted with the relation~(\ref{eq6}) with exponent $\delta$ instead
of~3.}
%{\vspace*{-2.1pc}}
\end{figure}

\section{Concluding remarks}

An analysis based on a statistical physics evaluation of entropy also
from mechanical testing with a strong hypothesis that serrations are
related to the PEL, leads to identification of two main structural
parameters: the average number of successive ``clusters'' $\lambda$
corresponds to easy displacement domain (favorable energy domain for
rearrangement or ``fertile'' sites for the initiation of shear bands)
and the volume density $\Psi$ is the number of these domain per unit
volume. The cluster size which is the unit domain in the statistical
analysis is seemingly corresponding to the activation volume, $v_{a}$
controlling deformation in the glass, then this value has to be
associated to the two others. If our original objective was to identify
structural parameters, it must be emphasized that $\lambda$, $v_{a}$
and $\Psi$ are derived from mechanical testing, which means that other
properties may reveals other parameters. A second aspect is that $\Psi$
is a hidden parameter in the entropy definition and is revealed by the
size effect on the mechanical behavior of the glass at the glass
transition temperature. Eventually the analysis is based on the
investigation of a metallic glass for which serrations are visible with
the spatial resolution of the nano-indentation technique. Most
positively, is that derivation from the size effect on the glass
transition, leads to a unique characteristic length scale which gathers
all the structural parameters and which writes with inserting the
activation volume, $\Delta^{*} = \sqrt[3]{{k_{b}\lambda}/{3v_{a}\Psi
C_{p}}}$. This value gives a first insight on the understanding
difference in the size effect on the mechanical properties, in
particular the fact that oxide glass shows the largest size, in the
range of micron, compare to metallic glasses, in the range of 50 to
500~nm, and polystyrene in the range of 20~nm. The characteristic
length scale shows the dominant role of~$\Psi$, which can be assumed to
be the lowest in silica glass characterized by strong atomic bonds and
local well defined polyhedra, compared to softer bonds and larger
disorder in metallic glass and polymers.

\section*{Conflicts of interest}

The author has no conflict of interest to declare.

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