#  Is the Molecular Weight Dependence of the Glass Transition Temperature
Driven by a Chain End Effect?

William F. Drayer  [  David S. Simmons<EMAIL_ADDRESS>[

###### Abstract

The immense dependence of the glass transition temperature $T_{g}$ on
molecular weight $M$ is one of the most fundamentally and practically
important features of polymer glass formation. Here, we report on molecular
dynamics simulation of three model linear polymers of substantially different
complexity demonstrating that the 70-year-old canonical explanation of this
dependence (a simple chain end dilution effect) is likely incorrect at leading
order. Our data shows that end effects are present only in relatively stiff
polymers and, furthermore, that the magnitude of this end effect diminishes on
cooling. Instead, we find that $T_{g}(M)$ trends are instead dominated by
shifts in $T_{g}$ throughout the entire polymer chain rather than through a
chain end effect. We show that these data are consistent with a generic two-
barrier model of $T_{g}$ and its $M$-dependence, motivated by the Elastically
Collective Nonlinear Langevin Equation (ECNLE) theory. More broadly, this work
indicates both a need to reassess the canonical understanding of $T_{g}(M)$ in
linear polymers (and macromolecules at large) and an opportunity to reveal new
glass formation physics with renewed study of $M$ effects on $T_{g}$.

###### keywords:

American Chemical Society, LaTeX

Penn] Department of Materials Science and Engineering, University of
Pennsylvania, Philadelphia, PA 19104, USA USF] Department of Chemical,
Biological and Materials Engineering, University of South Florida, Tampa, FL
33620, USA IR,NMR,UV

##  1 Introduction

A diverse array of systems solidify on laboratory timescales through the glass
transition, a poorly understood phenomenon wherein relaxation times
dramatically grow on cooling over a finite range of temperature $T$ 1, 2, 3.
One central feature of this transition is a profound dependence on molecular
weight $M$ or polymer degree of polymerization $N$: the glass transition
temperature $T_{g}$ commonly differs over 200 K between the small molecule
(e.g., monomer) and the infinite $M$ polymer limits 4. Indeed, while this has
historically understood as an issue of polymer physics only, more recent work
has suggested a continuum of $T_{g}$ size dependence spanning from polymers
with large $M$ down to the genuine small molecule limit4.

The canonical textbook explanation 5, 6, 7, 8, 9, 10 for this trend was
established in the early 1950’s by Fox and Flory (FF) 11 and Ueberreiter and
Kanig (UK) 12. In the case of the former, specifically for polystyrene
polymers, end groups are suggested to “act like a foreign substance in
disrupting the local configurational order of the styrene units,” 11 which in
turn has been interpreted as a chain end free volume effect present in
polymers more generally.

Ueberreiter and Kanig are more explicit in their interpretation of chain ends
and their impact on $T_{g}$ at large. The original publication 12 in fact has
sections entitled “Polymers as Mixtures of End and Middle Groups” and “Chain
End Groups Acting as Plasticizers.” Their discussion of end groups states that
“[they] have a greater expansion coefficient according to an improved mobility
which is due to their privileged position” and later that “[i]t therefore
seems reasonable to treat the end groups as plasticizers.” Finally, they
remark in their summary that “[t]he end groups act as plasticizers and cause
the ‘self-plasticization’ of the polymer.”

These arguments are reasonable and intuitively appealing, with the ends
exhibiting some combination of faster dynamics or higher free volume due to
having one less bonded neighbor and the middle groups exhibiting slower
dynamics and/or lower free volume due to their extra bond relative to the end
groups. The effect on $T_{g}$ for growing chain length is then simply a
dilution of this chain end effect just like that of volume. One would expect
chain ends to exhibit enhanced mobility or free volume relative to other chain
segments, and that the infinite molecular weight limit is reached when the
majority of segments are beyond the dynamical or structural influence of chain
ends and thus exhibit mobility characteristic of an infinite chain. Indeed,
given that interactions cannot be infinite range, it _must_ follow that any
enhancement in mobility by the chain ends must radiate outward from the ends
via some gradient along the backbone or through space. However, this
underlying local mechanism has never been fully validated.

Despite the long-standing predominance of this perspective, and perhaps in
part due to the lack of a direct test to date, questions have emerged over
whether this represents the sole, or even the dominant, mechanism driving the
$M$ dependence of $T_{g}$. Early work by Cowie 13 argued for the presence of
three regimes of $T_{g}(M)$ behavior — a complexity not captured by the two-
parameter FF and UK forms. Novikov and Rössler have suggested that the
canonical scenario is missing a distinct mechanism that dominates in the low
$M$ limit 4. Indeed, their work emphasizes a continuity between the molecular
weight dependence of $T_{g}$ for polymers and that for small molecules, with
the former merging into the latter in the low molecular weight limit. In
addition to suggesting the need for an additional mechanism, this work
therefore also emphasizes the breadth of the importance of understanding how
molecular size impacts $T_{g}$ in both polymers and small molecules. Other
distinct scenarios have suggested that the $T_{g}(M)$ dependence is driven by
the growth of chain stiffness or intramolecular activation barriers with $M$,
neither directly mediated by any chain end effect 14, 15. Even the basic
physical rationale for the FF and UK perspectives has been reconsidered, with
Zaccone and Terentjev 16 showing that the the FF equation can be derived by a
chain connectivity rather than chain end dilution argument.

Data published by Miwa et al. are particularly interesting from the
perspective of this discussion 17. They reported that chain ends exhibit a
local drop in the temperature of a spin transition (which they argue is
proportional to $T_{g}$) for spin-labeled polystyrene, but that the chain end
spin transition was itself $M$-dependent, even at fairly high $M$. While the
enhancement in mobility at chain ends seems in accord with canonical chain end
dilution models, its local $M$ dependence is not; at least to leading order,
the polymer is modelled in FF and UK with chain ends that exhibit enhanced
mobility or lower $T_{g}$ for all $M$, with a chain of infinite length
infinitely diluting this effect wherein almost all polymer segments exhibit
$T_{g}$ of the infinite limit, $T_{g,\infty}$. The overall $M$ dependence is
instead expected to emerge at the mean chain level by averaging.

To assess whether the $T_{g}(M)$ dependence is predominantly driven by chain
end effects as anticipated by the FF and UK models, we measure local dynamics
in molecular dynamics (MD) simulations of three well-established polymers
models: a freely-jointed chain (FJC) 18, 19, a freely-rotating chain (FRC) 19,
and OPLS all-atom polystyrene (AAPS) 20, 21, 22. These models span a range of
complexity and strength of intramolecular correlations, which prior studies
23, 14, 15, 3, 24 have suggested play an important role in $M$ effects on
$T_{g}$.

## Methodology

### Model and Simulation Details

We study three models spanning from a fully-flexible bead-spring chain to a
chemically realistic polymer in this work. All simulations are performed in
LAMMPS 25. The simplest model, the freely-jointed chain (FJC), uses the
standard finite extensible nonlinear elastic (FENE) bond potential 18,

$E=-0.5KR_{0}^{2}\ln\left[1-\left(\frac{r}{R_{0}}\right)^{2}\right]+4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right]+\epsilon,$

with particle size $\sigma=1$, interaction strength $\epsilon=1$, FENE elastic
constant $K=30$, and maximum bond elongation $R_{0}=1.5$. To increase chain
stiffness, we employ an angle potential to model a freely-rotating chain
(FRC),

$E=K_{\theta}[\cos(\theta)-\cos(\theta_{0})]^{2}.$

We set the bending constant $K_{\theta}=25$ and bending equilibrium angle
$\cos\theta_{0}=-0.333$ (which corresponds to 109.5 degrees), as done in prior
work 19. Bead-spring simulations utilize the Stoermer-Verlet time integration
algorithm as implemented in LAMMPS with a timestep of $\tau=0.005$. Both the
FJC and FRC span chain lengths of $4\leq N\leq 60$ beads with total bead
counts of $N=30000$.

To analyze a model with realistic chemical structure, we perform additional
analysis of OPLS all-atom polystyrene (AAPS) simulations first published in
prior work 21, 22. Full details of those simulations can be found in those
prior publications. Degrees of polymerization for AAPS range from $3\leq N\leq
400$ chemical repeat units, with a total chemical repeat unit count per
simulation of approximately 800 (e.g., there are 160 chains for $N=5$ and two
chains for $N=400$).

We utilize the PreSQ simulation protocol 21, 22 wherein simulations begin with
a high temperature anneal before sequential linear quenches and further
isothermal annealing sufficient to yield equilibrium relaxation times at the
mean system level. These simulations are performed in the isothermal-isobaric
($NPT$) ensemble using the Nose-Hoover thermostat and barostat, with both
damping parameters set to $\tau=2$.

###  1.1 Analysis Details

Relaxation is determined using the self-part of the intermediate scattering
function,

$F_{s}(q,t)=\left\langle\frac{1}{S}\Sigma^{S}_{k}\frac{1}{N}\Sigma^{N}_{j}\left(\exp{\left(-i\mathbf{q}\cdot\left(\mathbf{r}_{j}(t+s_{k})-\mathbf{r}_{j}(s_{k})\right)\right)}\right)\right\rangle_{|\mathbf{q}|=q}$
(1)

choosing a wavenumber near the first peak in the structure factor:
7.07196/$\sigma_{LJ}$ (where $\sigma_{LJ}$ is the Lennard-Jones unit of length
and is of order 1 nm in real units) for both the freely-jointed and freely-
rotation chain models and 1.19952/Å for the all-atom polystyrene model. We
show representative data for $F_{s}$ as a function of time for each model with
a chain length of $N=10$ for all models, with the addition of $N=100$ for
AAPS, plotting both the mean system data and chain end data in the SI. The
slow relaxation process within these relaxation functions is then fit to a
stretched exponential and the alpha relaxation time is defined as the time at
which this function decays to 0.2, as done in several prior works 26, 27, 28,
21, 22.

We perform this analysis two ways. First, we compute a relaxation time for the
entire system by summing in Equation 1 over all segments in the system.
Second, we perform the analysis for particular repeat unit locations within
the chain. In the bead spring models, for example, we compute a relaxation
time for chain ends by summing over only end beads. We also do this for beads
bonded to chain ends, beads bonded to those beads, and so on, in general
computing a mean relaxation time at each position $i$ within the chain, where
$i=1$ is the chain end, $i=2$ denotes repeat units bonded to a chain end, and
so on. We perform a similar analysis for AAPS, but in this case for each
repeat unit location, we average Equation 1 over all atoms within that repeat
unit location in all chains.

After computing these relaxation times across a range of temperature, $T_{g}$
is then quantified by fitting these relaxation time $\tau$ and temperature $T$
data to the MYEGA functional form, suggested by Mauro et al. 29, which may be
written as

$\log{\tau}=\log{\tau_{\infty}}+\frac{A}{T}\exp{\frac{B}{T}}.,$ (2)

where $\tau_{\infty}$, $A$, and $B$ are the fitting parameters. We rewrite
this self-consistently in terms of the glass transition temperature, replacing
$A$ as a fitting parameter, as

$\log{\tau}=\log{\tau_{\infty}}+\left(\log\tau_{g}-\log{\tau_{\infty}}\right)\frac{T_{g}}{T}\exp{\left(B\left(\frac{1}{T}-\frac{1}{T_{g}}\right)\right)}$
(3)

in order to conveniently obtain standard errors on $T_{g}$ from a least-
squares regression upon choosing a timescale $\tau_{g}$ for $T_{g}$. We define
$T_{g}$ at an extrapolated experimental timescale of 100 seconds for AAPS, to
allow for experimental comparability (this comparison has been validated in
prior work at the mean system level 22). For the FJC and FRC, we instead
employ a computational timescale $T_{g}$ convention of $\tau_{g}=10^{4}$
dimensionless Lennard-Jones time units, due to the lack of experimental
analogue, as is standard for bead models 30, 31, 32.

The same process is used for both mean system (cf. Figure 1, filled markers)
and repeat unit specific (cf. Figure 1, crosses and open markers, and Figure
2, all markers) analysis by employing their respective relaxation times as
described above; we thus obtain $T_{g}$ at both a whole level and at the level
of particular repeat unit positions with the polymer (averaged over many
chains). We note that the definition of a local $\tau$ and thus $T_{g}$ in
this manner has extensive precedent, particularly from the perspective of
interfacial and nanoconfinement effects on dynamics in glass-formers 33, 34,
35, 36, 37, 38, 39, 40, 41. We especially highlight how Priestley and
coworkers have reported on the $T_{g}$ of individual segment locations as a
function of number of bonds from a block copolymer junction, which is very
similar to our segment-specific $T_{g}$ reported here42.

We also compute local values of the Debye-Waller factor, $\langle
u^{2}\rangle$, which is a measure of dynamical free volume 43, 44, 21, 45, 46,
47, 48, 49, 50, glassy elasticity 51, 52, and particle localization 53, 54,
55, 56, 57 that quantifies the space accessed by a segment within a cage of
its transient neighbors. Specifically, we choose the mean-square displacement
(MSD) at a time delta of 0.275 (or $\approx 10^{-0.55}$) dimensionless time
units for both the FJC and FRC and 0.711 ps (or $\approx 10^{-0.15}$ ps) for
AAPS (consistent with prior work21). As $\langle u^{2}\rangle$ is a (nearly
linear, see SI) function of temperature, we choose a consistent $T$ near the
lowest accessed by simulation for each model to assess trends with respect to
$N$. The chosen temperatures (0.4 for the FJC, 0.6 for the FRC, and 500 K for
AAPS) are at worst a slight extrapolation for the largest chain lengths (as
the longest chains exhibit the largest $T_{g}$ values) with the majority of
the data being interpolations. Plots of the MSD data for select systems and
$\langle u^{2}\rangle$ as a function of temperature are found in the SI (for
the same systems as done for the $F_{s}$ data).

##  2 Results

###  2.1 Quantifying Chain End Effects

Figure 1:  Mean system and chain end $T_{g}$, normalized by its value for the
highest $M$ simulated for a given system, plotted for the systems shown in the
legend. Error bars are standard errors on $T_{g}$ as a fit parameter for AAPS
(standard errors are smaller than the data points for the FJC and FRC).

We begin by considering how varying $M$ (by means of degree of polymerization
or chain length $N$) impacts $T_{g}$ for these systems. We initially define
$T_{g}$ for AAPS on the experimental timescale (100 seconds) by extrapolating
the fit to Equation 3. This extrapolation has has been robustly validated
against experiment for polystyrene based on this simulation model over a wide
range of chain lengths in prior work 22, such that we can reliably infer
experimental-timescale glass formation behavior from these simulations at much
shorter times. For the two coarse systems, consistent with prior work 31, 58,
59, 60 we define $T_{g}(M)$ on a computational timescale of $10^{4}\tau_{LJ}$
(Lennard-Jones time units). As shown in Fig. 1, all simulated systems exhibit
an appreciable dependence of their mean system $T_{g}$ on $N$, in a manner
comparable to experiment. This dependence is stronger in models with stronger
intramolecular correlations (AAPS is the most stiff while the FJC is the most
flexible), consistent with prior reasoning and common trends in experimental
polymers 23, 4, 14, 3, 15.

Figure 2:  $T_{g}$ (a-c), $\langle u^{2}\rangle$ (d-f), and $\log\tau$ (g-i)
plotted as a function of bead index $i$: chain ends are labeled $i=1$ (either
bead for FJC and FRC or monomer for AAPS), $i=2$ indicates repeat units bonded
to chain ends, and so on until $i=N/2$, which indicates pairs of middle-most
repeat units for even $N$ (for odd AAPS chain lengths, the last data point is
a single monomer). Each row corresponds to each model, as labeled inside each
panel. Error bars are standard errors on $T_{g}$ as a fit parameter (within
the data points for FRC). Note that error bars increase with chain length for
AAPS due to reduced statistical sampling.

Additionally shown in Fig. 1, all three models exhibit negligible or weak
local reduction in $T_{g}$ at chain ends, relative to the overall magnitude of
the trend of mean $T_{g}$ with $N$. Essentially no $T_{g}$ end effect is
observed in the FJC model and there is less than a 1% end reduction for the
FRC model ($\sim 4$ K in real units for typical glassy polymers). In AAPS the
end effect is of order 30K, relative to a shift of nearly 200K in mean $T_{g}$
with $M$ (consistent with Miwa et al.’s experimental findings 17). We expand
upon this in Figure 2a-c, which shows that local variations in $T_{g}$ along
the backbone near the chain end are weak or absent in all three models. Even
in AAPS, where a modest $T_{g}$ end effect is present, it does not appreciably
propagate to covalently connected segments. We instead observe a whole-chain
effect of increasing $T_{g}$ with respect to $N$ regardless of $i$, in
contrast to the expected chain-end effect. Evidently, a direct $T_{g}$ end
effect is vastly too small to account for the much larger variation in mean-
chain $T_{g}$ with $N$ observed in Fig. 1 for all three models.

Could a more pronounced end effect still be present in free volume, which is
the underlying proposition of the FF model? Interestingly, Fig. 2d-f indicate
that trends in $\langle u^{2}\rangle$ with $M$ and chain location are
nonuniversal. As with $T_{g}$, shifts in $\langle u^{2}\rangle$ with $M$ in
the FJC model are nearly uniform through the whole chain, with at most a very
weak chain end gradient. In contrast, AAPS exhibits a significant chain end
effect of enhanced $\langle u^{2}\rangle$. The FRC model exhibits a mix of
these two scenarios. This trend is perhaps as expected: intramolecular
correlations contribute to the cage scale in stiffer polymers, such that the
reduced bond connectivity near chain ends relieves caging more significantly
in these systems. However, as shown by the FJC, it is evidently possible for a
polymer to exhibit at least a 10% drop in $T_{g}$ without a significant chain
end effect in either $T_{g}$ or $\langle u^{2}\rangle$. This suggests that
chain end mobility or free volume effects cannot be the sole driver of the
$T_{g}(M)$ dependence. In contrast, data for FRC and AAPS indicate that even
the presence of an appreciable enhancement in chain-end $\langle u^{2}\rangle$
does not necessarily translate to a substantial suppression in chain end
$T_{g}$.

Figure 3:  $\tau$ vs $T$ for $N=100$ AAPS, for chain ends (open symbols) and
mean system (green symbols). Curves in corresponding colors are fits to the
MYEGA functional form 29. Green vertical lines (rightmost of each pair) denote
the $T$ at which the mean system $\tau$ equals $10^{-11}$s (dotted),
$10^{-9}$s (dot-dashed), and $10^{2}$s (dashed), respectively. Black vertical
lines (leftmost of each pair) denote the $T$ at which the chain ends exhibit
these same $\tau$. The $T$ difference, $\Delta T_{g}$, is reported for each of
the three timescales. Heavy red vertical segments highlight the chain end
$\tau$ reduction relative to the mean system at each timescale.  Figure 4:
Magnitude of chain end $T_{g}$ effect vs conventional timescale $\tau_{g}$ for
AAPS chain lengths indicated in the legend. Filled markers denote points for
which both mid-chain and end-chain $T_{g}$ values are interpolated from
simulation data; grey markers are points for which the mid-chain $T_{g}$ is
interpolated and end-chain values are mildly extrapolated; open markers denote
points for which $T_{g}$ values are extrapolated for all monomers.

How does enhanced chain end $\langle u^{2}\rangle$ not directly lead to
suppressed chain end $T_{g}$? In Fig. 2g-i we report relaxation time $\tau$
gradients along the chain backbone at the same $T$ for which we reported
$\langle u^{2}\rangle$ end gradients. At least for AAPS, the chain end $\tau$
is indeed considerably reduced for $T$ well above that of $T_{g}$ when defined
on the experimental timescale. The failure of this increase to lead to a
chain-end $T_{g}$ enhancement lies in a subtle feature of its $T$ dependence,
which we demonstrate in Fig. 3. The chain-end mobility enhancement strengthens
on cooling on an _absolute_ basis; however, this strengthening is insufficient
to keep up with the growth in the overall activation barrier of relaxation on
cooling, such that it becomes _relatively_ weaker in its implications for
$T_{g}$. We further show this effect in Fig. 4, which emphasizes that this
effect occurs within computationally accessible timescales and is not merely a
result of extrapolation. Figures 3 and 4 demonstrate how the magnitude of the
$T_{g}$ end effect shrinks as the timescale $\tau_{g}$ that defines $T_{g}$ is
increased.

###  2.2 Theoretical Interpretations

Our data indicate that the chain end mobility effects intuited by FF and UK
are present at high $T$ in semiflexible chains (although negligible in quite
flexible chains) but diminish in significance upon cooling and become sub-
dominant by the experimental $T_{g}$ timescale. Within the context of many
classical theories of glass formation, this observation seems surprising. Many
of these theories, including free volume theory 61, 62 and classical entropy
theories 63, postulate the presence of a single dominant activation barrier to
relaxation in glass-forming liquids. This barrier is postulated to grow on
cooling in a manner that results from a _multiplicative product_ of the high
temperature barrier with a temperature-dependence amplification factor
associated with a growing cooperative lengthscale. As an example, within the
Adam-Gibbs theory of glass formation, $\log(\tau)$ goes as a high-$T$ barrier
times a cooperativity factor over $k_{B}T$ 63. A reduction in either high-$T$
activation barrier or cooperativity at the chain ends would thus not be
expected to diminish in importance on cooling. A similar intuition would seem
to hold for free volume approaches given the inverse proportionality of the
activation barrier to a single quantity (the free volume). It is this
intuition, that the alteration in activation barriers near the chain end
becomes highly important in the glass formation range, that drives the
classical FF and UK viewpoints.

More recently, a distinct alternative perspective has emerged that views
glassy super-Arrhenius behavior as emerging from an _additive_ , rather than
multiplicative, growth in the barrier on cooling 54, 55, 64. In particular,
the Elastically Collective Nonlinear Langevin Equation (ECNLE) theory of glass
formation formulates its activation barrier as a sum of a local barrier (which
grows relatively weakly on cooling) and a collective elastic barrier (which is
predicted to emerge and then grow relatively strongly on cooling towards
$T_{g}$) 54, 55.

We can understand our results within the context these newer perspectives.
Consider a generic two-barrier model wherein the total activation barrier
$F_{tot}^{mid}$ in the mid-chain is a sum of a local barrier $F_{loc}^{mid}$
and a collective barrier $F_{coll}^{mid}$:

$F_{tot}^{mid}\left(N,T\right)=F_{loc}^{mid}\left(N,T\right)+F_{coll}^{mid}\left(N,T\right).$
(4)

Alterations of this barrier at the chain end are rooted in alterations to
intramolecular correlations that are intrinsically present arbitrarily far
above $T_{g}$, and therefore most directly impact the local barrier (since the
collective barrier is absent far above $T_{g}$). We thus model the end effect
as a fractional reduction of the local barrier by a factor $\alpha^{end}$,
which we model as roughly temperature-invariant (but expect to be chemistry
dependent and larger for stiffer chains) since it reflects a truncation of
intramolecular barriers that are mainly steric and bonding in nature and
therefore relatively athermal:

$F_{tot}^{end}\left(N,T\right)={{\alpha}^{end}}F_{loc}^{mid}\left(N,T\right)+F_{coll}^{mid}\left(N,T\right).$
(5)

One can quantify the expected temperature dependence of the chain-end
relaxation time gradient within this perspective by employing the total
barrier forms above within a generalized activation law to compute the ratio
of chain-end to chain-mid relaxation times:

$\log\left(\frac{{{\tau}_{end}}\left(N,T\right)}{{{\tau}_{mid}}\left(N,T\right)}\right)=\frac{\left(1-{{\alpha}^{end}}\right){{F}_{loc}^{mid}}\left(N,T\right)}{kT}.$
(6)

This equation anticipates that the enhanced chain end mobility (relative to
the mid-chain) should grow on cooling, as seen above in Fig. 3 (the vertical
red bars, moving from right to left), simply because of the reduction in
temperature and any growth on cooling of $F_{loc}^{mid}$.

We can further combine Equations 4 and 5 with a generalized activation law as
above, rearrange them to solve for temperature, apply each at its
corresponding local $T_{g}$, and take their ratio. This gives

$\frac{T_{g}^{end}}{T_{g}^{mid}}=R\left[1-\left(1-{{\alpha}^{end}}\right)x_{loc}^{mid}\left(N,T_{g}^{end}\right)\right],$
(7)

in which we define two dimensionless ratios: a prefactor
$R={F_{tot}^{mid}\left(N,T_{g}^{end}\right)}/{F_{tot}^{mid}\left(N,T_{g}^{mid}\right)}$
and $x_{loc}^{mid}\left(N,T\right),$ which is the fraction of the total
barrier in the mid-chain that is contributed by the local barrier at $T$ such
that
$x_{loc}^{mid}\left(N,T_{g}^{end}\right)={F_{loc}^{mid}\left(N,T_{g}^{end}\right)}/{F_{tot}^{mid}\left(N,T_{g}^{end}\right)}\;$

For a weak $T_{g}$ end effect, we can approximate the relaxation process as
Arrhenius over the limited temperature range involved, giving $R\approx 1$.
Equation 7 then predicts that the $T_{g}$ end effect shrinks on cooling, even
as the $\tau$ end effect grows, because the fractional reduction from the
local barrier $x_{loc}^{mid}$ shrinks on cooling such that the end effects
become diluted within the faster-growing overall barrier. Indeed, this is the
behavior we see in Fig. 4, because the fractional reduction from the local
barrier $x_{loc}^{mid}$ shrinks on cooling 54 such that the end effects become
diluted within the faster-growing overall barrier.

###  2.3 What causes $T_{g}(M)$ if not chain end effects?

Across three models, chain ends evidently do not exhibit sufficiently reduced
$T_{g}$ values to account for the dependence of mean $T_{g}$ on chain length
on the basis of a chain end dilution effect. This would appear to demand a
reevaluation of the textbook understanding of the $T_{g}(M)$ dependence. Our
data indicate that $T_{g}(M)$ variations are primarily driven by the $M$
dependence of the mid-chain (or whole-chain) activation barrier
$F_{tot}^{mid}(N,T)$, which in turn may result from whole-chain trends in some
combination of the local and collective barriers. This type of scenario has
been predicted within the ECNLE theory, where both local and collective
elastic contributions to this activation barrier grow with increasing size of
the fundamental dynamical repeat unit. This unit is taken to be the entire
molecule in small rigid molecules 54, 55 and as the Kuhn segment in polymers
14, 24. In the small molecule case, this leads to the prediction that
$T_{g}\sim\sqrt{M}$ 54, 55, which is consistent with the small molecule limit
identified by Novikov and Rössler 4. In the polymer case, the $M$ dependence
follows from the growth of the Kuhn segment with increasing $N$ (as measured
by growth of the chain’s characteristic ratio $C_{N}$): a reflection of
increases in effective chain stiffness with $M$. 14, 24.

Figure 5:  $T_{g}$ plotted as a function of normalized $C_{N}$, each
normalized by their value for the longest chain of that type simulated, for
the FJC (blue circles) and FRC (orange diamonds). Lines in corresponding
colors are linear fits.

Indeed, prior studies have found that variation of $T_{g}$ with $M$ tracks
with variations in $C_{N}$ for at least polystyrene, poly(methyl
methacrylate), and polyethylene. This has also been reported in polydimethyl
siloxane, although this is in dispute 15. While we cannot add to the extant
data for PS $T_{g}$ vs $C_{N}$ correlations due to the limited number of
chains that can be simulated in an AA simulation accessing the glass formation
range, Fig. 5 illustrates that $T_{g}$ is proportional to $C_{N}$ for our two
bead-spring systems, adding to evidence that $T_{g}(M)$ closely tracks with
$C_{N}(M)$ as $M$ is varied for a given polymer, in line with the ECNLE
scenario 14, 24.

It is not clear whether this $C_{N}$ scenario alone fully accounts for the
$T_{g}(M)$ dependence given suggestions that $T_{g}(M)$ exhibits multiple
regimes, particularly in stiffer polymers. Baker et al. have argued that this
may result from nontrivial variations in chain conformational statistics,
combined with an intramolecular dynamical facilitation effect 15. While not
excluding that scenario, our results suggest a potential alternate scenario
given that we find chain end effects on $T_{g}$ to be present, if weak, in our
stiffer systems. It may be that the multiple regimes observed in some stiffer
polymers reflect a combination of a leading order stiffness effect with a
perhaps second-order end effect with parallels to FF and UK.

##  3 Conclusions

We have reported on local dynamics for three model polymers of varying degrees
of complexity and stiffness, ranging from a freely-jointed polymer chain, with
only excluded volume added relative to a random walk in three dimensions, to a
fully atomistic polystyrene chain which indeed agrees with experimental
$T_{g}(N)$ quite well 22. Surprisingly, we find that chain-end mobility
enhancements in dynamics or free volume are insufficient to account for the
shift in overall system $T_{g}$ with varying M. Indeed, the freely-jointed-
chain model exhibits an almost complete absence of chain end gradient in
$T_{g}$, relaxation time, or Debye-Waller factor $\langle u^{2}\rangle$ (one
measure of a dynamic free volume). Nevertheless, this model exhibits a 10%
reduction in $T_{g}$ from the highest to lowest molecular weight simulated.

The results for the freely-rotating-chain model and all-atom polystyrene are
more complex, exhibiting an enhancement in $\langle u^{2}\rangle$ at the chain
end. However, as noted above, this does not translate into appreciable $T_{g}$
gradients at chain ends. Within many classical theories of the glass
transition, this would be difficult to understand. However, we show that it
can be understood in terms of a simple two-barrier scenario of the glass
transition inspired by the Elastically Collective Nonlinear Langevin Equation
(ECNLE) theory of glass formation 54, 55, which predicts additive
contributions to the activation barrier for relaxation from a local barrier
and an longer-ranged collective barrier that grows more strongly on cooling.
If chain-end enhancements in $\langle u^{2}\rangle$ primarily alter the local
barrier contribution, we show that the they become less important at low
temperatures where the collective barrier dominates.

Does the absence of a locally-driven chain-end $T_{g}$ suppression
sufficiently large to account for the $T_{g}(M)$ dependence via spatial
averaging genuinely provide compelling evidence that some other mechanism must
be dominant? As we discuss in the introduction, any model in which faster
dynamics or lower $T_{g}$ are effectively _nucleated_ by a chain end effect
requires that this effect must emanate over a _finite_ distance from the chain
end, since correlations in supercooled liquids are not of infinite range. Some
gradient in these properties is therefore to be expected. Our longest chains
are 400 repeat units in the case of AAPS. Most segments in this system are not
within the first neighbor shell of an end; at this molecular weight, chain
ends are expected to be approximately 6 segmental diameters apart on average.
We nevertheless do not observe gradients remotely large enough to account for
$T_{g}(M)$.

Could a gradient emanating from chain ends simply be so long-ranged as to
appear flat even over this spacing? There are several reasons to conclude that
this is not the case. First, prior simulations probing the range of
correlation length scales in glass-forming liquids have suggested that they
are of range only a few segmental diameters on the timescales we access; they
should therefore be readily observable within Figure 2 were they present. As a
possibly even stronger argument, let us consider what it would imply regarding
the behavior of even longer chains if some dramatically longer range were in
fact concealing the gradient within our simulations. This would require that a
$T_{g}$ gradient of magnitude comparable to the overall $T_{g}(M)$ trend would
have to emerge in Figure 2 for PS chains considerably longer than 400 repeat
units (about 40,000 g/mol). Since it would be incoherent for the $T_{g}$ at
any location in the chain to drop with increasing molecular weight, this would
imply that the high-molecular weight mid-chain $T_{g}$ would have to grow by
many tens of K beyond that observed here. This is not plausible, since the
mean $T_{g}$ value here has already plateaued in accord with high-molecular
weight $T_{g}$ observed for PS in experiment.

Collectively, these findings suggest that chain end effects are not the
dominant origin of the $T_{g}(M)$ dependence. They are entirely absent in at
least one system exhibiting a substantial $T_{g}(M)$ dependence, and even when
present, their effects grow weaker on cooling and play little role in $T_{g}$
at experimental timescales. Indeed, we note that even the modest chain end
$T_{g}$ suppression experimentally inferred by Miwa et al. 17 was based on a
shorter-timescale spin transition that occurs at higher temperatures where
this analysis would expect chain end effects to be modestly more important in
some systems than they are at experimental timescales. Perhaps the most
remarkable conclusion is that for the fully flexible chain, chain end effects
for _both_ $\tau$ and $\langle u^{2}\rangle$ are negligible, from which one
must conclude that _at least_ another mechanism must be responsible for the
observed 10% reduction in $T_{g}$ for this model. Consistent with prior work,
we report that $T_{g}$ in our freely-jointed and freely-rotating chain models
tracks linearly with the characteristic ratio is the molecular weight is
varied. This may be consistent with a scenario encoded within the ECNLE theory
wherein $T_{g}(M)$ is driven by stiffness variations with molecular weight, as
indicated by the growth in the characteristic ratio in longer chains. More
broadly, this may align with the proposition that intramolecular activation
barriers, which play a central role in polymer glass formation 65, may
qualitatively vary with molecular weight 15.

Overall, these findings indicate a need to reopen the study of $M$ effects on
$T_{g}$, with a focus on more recent theories wherein this trend is dominated
by whole-chain effects rather than end effects. Indeed, the finding that the
$T$-dependence of dynamical chain end effects can be understood based on a
two-barrier model of the glass transition suggests that a renewed focus on
this problem may have the potential to yield broader insights into the nature
of glass formation itself.

This material is based upon work supported by the National Science Foundation
under Grant No. DMR - 1849594. The authors acknowledge helpful discussions
with Dr. Kenneth Schweizer.

The supporting information contains curves for decay of the self-part of the
intermediate scattering function, the time evolution of the mean-square
displacement and the temperature dependence of the Debye-Waller factor for
representative chains of each model, along with temperature dependence of mean
relaxation time for each model and all chain length.

## References

  * Debenedetti and Stillinger 2001 Debenedetti, P. G.; Stillinger, F. H. Supercooled liquids and the glass transition. _Nature_ 2001, _410_ , 259–267
  * Cavagna 2009 Cavagna, A. Supercooled liquids for pedestrians. _Physics Reports-Review Section of Physics Letters_ 2009, _476_ , 51–124, WOS:000267351200001
  * Novikov and Sokolov 2022 Novikov, V. N.; Sokolov, A. P. Temperature Dependence of Structural Relaxation in Glass-Forming Liquids and Polymers. _Entropy_ 2022, _24_ , 1101, Number: 8 Publisher: Multidisciplinary Digital Publishing Institute
  * Novikov and Rössler 2013 Novikov, V.; Rössler, E. Correlation between glass transition temperature and molecular mass in non-polymeric and polymer glass formers. _Polymer_ 2013, _54_ , 6987–6991
  * Hiemenz and Lodge 2007 Hiemenz, P. C.; Lodge, T. _Polymer Chemistry_ , 2nd ed.; CRC Press, 2007
  * Rubinstein and Colby 2003 Rubinstein, M.; Colby, R. H. _Polymer Physics_ ; OUP Oxford, 2003
  * Coleman and Painter 1998 Coleman, M. M.; Painter, P. C. _Fundamentals of Polymer Science: An Introductory Text_ , 2nd ed.; CRC Press: Lancaster, Pa, 1998
  * Rudin and P.Eng 2012 Rudin, A.; P.Eng, P. C. P. D. _The Elements of Polymer Science and Engineering: An Introductory Text and Reference for Engineers and Chemists_ , 3rd ed.; Academic Press: San Diego, 2012
  * Mathot and Benoist 1994 Mathot, V. B. F., Benoist, L., Eds. _Calorimetry and Thermal Analysis of Polymers_ ; Hanser Pub Inc: Munich ; New York : Cincinnati, 1994
  * Rosen 1993 Rosen, S. L. _Fundamental Principles of Polymeric Materials_ , 2nd ed.; Wiley-Interscience: New York, 1993
  * Fox and Flory 1950 Fox, T. G.; Flory, P. J. Second‐Order Transition Temperatures and Related Properties of Polystyrene. I. Influence of Molecular Weight. _Journal of Applied Physics_ 1950, _21_ , 581–591, Publisher: American Institute of Physics
  * Ueberreiter and Kanig 1952 Ueberreiter, K.; Kanig, G. Self-plasticization of polymers. _Journal of Colloid Science_ 1952, _7_ , 569–583
  * Cown 1975 Cown, J. M. G. Some general features of Tg-M relations for oligomers and amorphous polymers. _European Polymer Journal_ 1975, _11_ , 297–300
  * Mirigian and Schweizer 2015 Mirigian, S.; Schweizer, K. S. Dynamical Theory of Segmental Relaxation and Emergent Elasticity in Supercooled Polymer Melts. _Macromolecules_ 2015, _48_ , 1901–1913
  * Baker et al. 2022 Baker, D. L.; Reynolds, M.; Masurel, R.; Olmsted, P. D.; Mattsson, J. Cooperative Intramolecular Dynamics Control the Chain-Length-Dependent Glass Transition in Polymers. _Physical Review X_ 2022, _12_ , 021047
  * Zaccone and Terentjev 2013-04-26 Zaccone, A.; Terentjev, E. M. Disorder-Assisted Melting and the Glass Transition in Amorphous Solids. _Physical Review Letters_ 2013-04-26, _110_ , 178002, Publisher: American Physical Society
  * Miwa et al. 2003 Miwa, Y.; Tanase, T.; Yamamoto, K.; Sakaguchi, M.; Sakai, M.; Shimada, S. Influence of Chain End and Molecular Weight on Molecular Motion of Polystyrene, Revealed by the ESR Selective Spin-Label Method. _Macromolecules_ 2003, _36_ , 3235–3239, Publisher: American Chemical Society
  * Kremer and Grest 1990 Kremer, K.; Grest, G. S. Dynamics of entangled linear polymer melts: A molecular-dynamics simulation. _The Journal of Chemical Physics_ 1990, _92_ , 5057–5086
  * Bulacu and van der Giessen 2007 Bulacu, M.; van der Giessen, E. Molecular-dynamics simulation study of the glass transition in amorphous polymers with controlled chain stiffness. _Physical Review E_ 2007, _76_ , 011807
  * Jorgensen et al. 1996 Jorgensen, W. L.; Maxwell, D. S.; Tirado-Rives, J. Development and Testing of the OPLS All-Atom Force Field on Conformational Energetics and Properties of Organic Liquids. _Journal of the American Chemical Society_ 1996, _118_ , 11225–11236
  * Hung et al. 2019 Hung, J.-H.; Patra, T. K.; Meenakshisundaram, V.; Mangalara, J. H.; Simmons, D. S. Universal localization transition accompanying glass formation: insights from efficient molecular dynamics simulations of diverse supercooled liquids. _Soft Matter_ 2019, _15_ , 1223–1242
  * Hung et al. 2020 Hung, J.-H.; Patra, T. K.; Simmons, D. S. Forecasting the experimental glass transition from short time relaxation data. _Journal of Non-Crystalline Solids_ 2020, _544_ , 120205
  * Sokolov et al. 2007 Sokolov, A. P.; Novikov, V. N.; Ding, Y. Why many polymers are so fragile. _Journal of Physics: Condensed Matter_ 2007, _19_ , 205116
  * Zhou et al. 2022 Zhou, Y.; Mei, B.; Schweizer, K. S. Activated relaxation in supercooled monodisperse atomic and polymeric WCA fluids: Simulation and ECNLE theory. _The Journal of Chemical Physics_ 2022, _156_ , 114901, Publisher: American Institute of Physics
  * Thompson et al. 2022 Thompson, A. P.; Aktulga, H. M.; Berger, R.; Bolintineanu, D. S.; Brown, W. M.; Crozier, P. S.; in ’t Veld, P. J.; Kohlmeyer, A.; Moore, S. G.; Nguyen, T. D.; Shan, R.; Stevens, M. J.; Tranchida, J.; Trott, C.; Plimpton, S. J. LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales. _Computer Physics Communications_ 2022, _271_ , 108171
  * 26 Hanakata, P. Z.; Douglas, J. F.; Starr, F. W. Local variation of fragility and glass transition temperature of ultra-thin supported polymer films. _137_ , 244901
  * 27 Lang, R. J.; Merling, W. L.; Simmons, D. S. Combined Dependence of Nanoconfined Tg on Interfacial Energy and Softness of Confinement. _3_ , 758–762
  * 28 Hung, J.-H.; Mangalara, J. H.; Simmons, D. S. Heterogeneous Rouse Model Predicts Polymer Chain Translational Normal Mode Decoupling. _51_ , 2887–2898
  * Mauro et al. 2009 Mauro, J. C.; Yue, Y.; Ellison, A. J.; Gupta, P. K.; Allan, D. C. Viscosity of glass-forming liquids. _Proceedings of the National Academy of Sciences_ 2009, _106_ , 19780–19784
  * 30 Marvin, M. D.; Lang, R. J.; Simmons, D. S. Nanoconfinement effects on the fragility of glass formation of a model freestanding polymer film. _10_ , 3166–3170
  * 31 Hsu, D. D.; Xia, W.; Song, J.; Keten, S. Glass-Transition and Side-Chain Dynamics in Thin Films: Explaining Dissimilar Free Surface Effects for Polystyrene vs Poly(methyl methacrylate). _5_ , 481–486, Publisher: American Chemical Society
  * 32 Cheng, Y.; Yang, J.; Hung, J.-H.; Patra, T. K.; Simmons, D. S. Design Rules for Highly Conductive Polymeric Ionic Liquids from Molecular Dynamics Simulations. _51_ , 6630–6644
  * 33 Scheidler, P.; Kob, W.; Binder, K. Cooperative motion and growing length scales in supercooled confined liquids. _59_ , 701, Publisher: IOP Publishing
  * 34 Scheidler, P.; Kob, W.; Binder, K. The relaxation dynamics of a confined glassy simple liquid. _12_ , 5–9
  * 35 Scheidler, P.; Kob, W.; Binder, K. The Relaxation Dynamics of a Supercooled Liquid Confined by Rough Walls. _108_ , 6673–6686, Publisher: American Chemical Society
  * 36 Baschnagel, J.; Varnik, F. Computer simulations of supercooled polymer melts in the bulk and in confined geometry. _17_ , R851
  * 37 Peter, S.; Meyer, H.; Baschnagel, J. Thickness-dependent reduction of the glass-transition temperature in thin polymer films with a free surface. _44_ , 2951–2967, _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/polb.20924
  * 38 Kob, W.; Roldán-Vargas, S.; Berthier, L. Non-monotonic temperature evolution of dynamic correlations in glass-forming liquids. _8_ , 164–167, Number: 2 Publisher: Nature Publishing Group
  * 39 Hocky, G. M.; Berthier, L.; Kob, W.; Reichman, D. R. Crossovers in the dynamics of supercooled liquids probed by an amorphous wall. _89_ , 052311, Publisher: American Physical Society
  * 40 Kob, W.; Coslovich, D. Nonlinear dynamic response of glass-forming liquids to random pinning. _90_ , 052305, Publisher: American Physical Society
  * 41 Hao, Z.; Ghanekarade, A.; Zhu, N.; Randazzo, K.; Kawaguchi, D.; Tanaka, K.; Wang, X.; Simmons, D. S.; Priestley, R. D.; Zuo, B. Mobility gradients yield rubbery surfaces on top of polymer glasses. _596_ , 372–376, Number: 7872 Publisher: Nature Publishing Group
  * Christie et al. 2018 Christie, D.; Register, R. A.; Priestley, R. D. Direct Measurement of the Local Glass Transition in Self-Assembled Copolymers with Nanometer Resolution. _ACS Central Science_ 2018, _4_ , 504–511
  * McKenzie-Smith et al. 2022 McKenzie-Smith, T.; Douglas, J. F.; Starr, F. W. Relating dynamic free volume to cooperative relaxation in a glass-forming polymer composite. _The Journal of Chemical Physics_ 2022, _157_ , 131101, Publisher: American Institute of Physics
  * McKenzie-Smith et al. 2021 McKenzie-Smith, T. Q.; Douglas, J. F.; Starr, F. W. Explaining the Sensitivity of Polymer Segmental Relaxation to Additive Size Based on the Localization Model. _Physical Review Letters_ 2021, _127_ , 277802, Publisher: American Physical Society
  * Puosi et al. 2019 Puosi, F.; Tripodo, A.; Leporini, D. Fast Vibrational Modes and Slow Heterogeneous Dynamics in Polymers and Viscous Liquids. _International Journal of Molecular Sciences_ 2019, _20_ , 5708, Number: 22 Publisher: Multidisciplinary Digital Publishing Institute
  * Pazmiño Betancourt et al. 2015 Pazmiño Betancourt, B. A.; Hanakata, P. Z.; Starr, F. W.; Douglas, J. F. Quantitative relations between cooperative motion, emergent elasticity, and free volume in model glass-forming polymer materials. _Proceedings of the National Academy of Sciences_ 2015, _112_ , 2966–2971, Publisher: Proceedings of the National Academy of Sciences
  * Ottochian and Leporini 2011 Ottochian, A.; Leporini, D. Universal scaling between structural relaxation and caged dynamics in glass-forming systems: Free volume and time scales. _Journal of Non-Crystalline Solids_ 2011, _357_ , 298–301
  * Ngai 2004 Ngai, K. L. Why the fast relaxation in the picosecond to nanosecond time range can sense the glass transition. _Philosophical Magazine_ 2004, _84_ , 1341–1353, Publisher: Taylor & Francis _eprint: https://doi.org/10.1080/14786430310001644080
  * Starr et al. 2002 Starr, F. W.; Sastry, S.; Douglas, J. F.; Glotzer, S. C. What Do We Learn from the Local Geometry of Glass-Forming Liquids? _Physical Review Letters_ 2002, _89_ , 125501
  * Ngai et al. 2001 Ngai, K. L.; Bao, L.-R.; Yee, A. F.; Soles, C. L. Correlation of Positron Annihilation and Other Dynamic Properties in Small Molecule Glass-Forming Substances. _Physical Review Letters_ 2001, _87_ , 215901, Publisher: American Physical Society
  * van Zanten and Rufener 2000 van Zanten, J. H.; Rufener, K. P. Brownian motion in a single relaxation time Maxwell fluid. _Physical Review E_ 2000, _62_ , 5389–5396
  * Yang and Schweizer 2011 Yang, J.; Schweizer, K. S. Glassy dynamics and mechanical response in dense fluids of soft repulsive spheres. II. Shear modulus, relaxation-elasticity connections, and rheology. _Journal of Chemical Physics_ 2011, _134_ , 204909
  * Dyre 2006 Dyre, J. C. Colloquium: The glass transition and elastic models of glass-forming liquids. _Reviews of Modern Physics_ 2006, _78_ , 953–972
  * Mirigian and Schweizer 2014 Mirigian, S.; Schweizer, K. S. Elastically cooperative activated barrier hopping theory of relaxation in viscous fluids. I. General formulation and application to hard sphere fluids. _Journal of Chemical Physics_ 2014, _140_ , 194506, WOS:000336832700028
  * Mirigian and Schweizer 2014 Mirigian, S.; Schweizer, K. S. Elastically cooperative activated barrier hopping theory of relaxation in viscous fluids. II. Thermal liquids. _The Journal of Chemical Physics_ 2014, _140_ , 194507, Publisher: American Institute of Physics
  * Hall and Wolynes 1987 Hall, R. W.; Wolynes, P. G. The aperiodic crystal picture and free-energy barriers in glasses. _Journal of Chemical Physics_ 1987, _86_ , 2943–2948
  * Buchenau and Zorn 1992 Buchenau, U.; Zorn, R. A RELATION BETWEEN FAST AND SLOW MOTIONS IN GLASSY AND LIQUID SELENIUM. _Europhysics Letters_ 1992, _18_ , 523–528
  * Xia et al. 2015 Xia, W.; Hsu, D. D.; Keten, S. Molecular Weight Effects on the Glass Transition and Confinement Behavior of Polymer Thin Films. _Macromolecular Rapid Communications_ 2015, _36_ , 1422–1427, WOS:000359791700006
  * Mangalara and Simmons 2015 Mangalara, J. H.; Simmons, D. S. Tuning Polymer Glass Formation Behavior and Mechanical Properties with Oligomeric Diluents of Varying Stiffness. _ACS Macro Letters_ 2015, _4_ , 1134–1138
  * Ghanekarade et al. 2023 Ghanekarade, A.; Phan, A. D.; Schweizer, K. S.; Simmons, D. S. Signature of collective elastic glass physics in surface-induced long-range tails in dynamical gradients. _Nature Physics_ 2023, 1–7, Publisher: Nature Publishing Group
  * Doolittle 1951 Doolittle, A. K. Studies in Newtonian Flow. II. The Dependence of the Viscosity of Liquids on Free‐Space. _Journal of Applied Physics_ 1951, _22_ , 1471–1475
  * White and Lipson 2017 White, R. P.; Lipson, J. E. G. Explaining the T,V-dependent dynamics of glass forming liquids: The cooperative free volume model tested against new simulation results. _The Journal of Chemical Physics_ 2017, _147_ , 184503
  * Adam and Gibbs 1965 Adam, G.; Gibbs, J. H. On the Temperature Dependence of Cooperative Relaxation Properties in Glass‐Forming Liquids. _The Journal of Chemical Physics_ 1965, _43_ , 139–146
  * Schmidtke et al. 2015 Schmidtke, B.; Hofmann, M.; Lichtinger, A.; Rössler, E. A. Temperature Dependence of the Segmental Relaxation Time of Polymers Revisited. _Macromolecules_ 2015, _48_ , 3005–3013
  * 65 Colmenero, J. Are polymers standard glass-forming systems? The role of intramolecular barriers on the glass-transition phenomena of glass-forming polymers. _27_ , 103101, Publisher: IOP Publishing