|  Effect of ring stiffness and ambient pressure on the dynamical slowdown in
ring polymers †
---|---
|  Projesh Kumar Roy,a,b Pinaki Chaudhuri,a,b,∗ and Satyavani Vemparalaa,b,∗
|  Using extensive molecular dynamics simulations, we investigate the slowing
down of dynamics in a 3D system of ring polymers by varying the ambient
pressure and the stiffness of the rings. Our study demonstrates that the
stiffness of the rings determines the dynamics of the ring polymers, leading
to glassiness at lower pressures for stiffer rings. The threading of the ring
polymers, a unique feature that emerges only due to the topological nature of
such polymers in three dimensions, is shown to be the determinant feature of
dynamical slowing down, albeit only in a certain stiffness range. Our results
suggest a possible framework of exploring the phase space spanned by ring
stiffness and pressure to obtain spontaneously emerging topologically
constrained polymer glasses.

##

††footnotetext: a The Institute of Mathematical Sciences, C.I.T. Campus,
Taramani, Chennai 600113, India; Tel: +914422543293/+914422543257; E-mail:
pinakic<EMAIL_ADDRESS>b Homi Bhabha National
Institute, Training School Complex, Anushakti Nagar, Mumbai 400094,
India††footnotetext: † Electronic Supplementary Information (ESI) available:
See DOI: 10.1039/cXsm00000x/

##  1 Introduction

Entanglement in dense polymeric systems can arise due to many reasons and can
lead to interesting and complex rheological phenomena. For linear polymer
systems, entanglement can arise from spatial confinement via the presence of
other linear polymers in close contact. It is intuitive that behavior of a
single polymer in dilute vs dense conditions is very different. In the former
case, the dynamics that govern the polymer behavior are largely determined by
the interconnectivity of the polymer and the nature of the solvent surrounding
it, while in the latter, the presence of other polymer molecules within close
vicinity can significantly influence the dynamics of the polymers. Theoretical
models exist to describe these two different scenarios: Rouse-Zimm model 1, 2,
3, 4 for extreme dilute systems, reptation or tube models 5, 6, 7, 8 for the
dense linear polymer systems.

Entanglement can also arise from topological constraints via connecting the
ends of the linear polymers resulting in ring polymers and consequently
exhibiting different rheological properties 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19.The ring polymers have been considered as model systems for
understanding the compaction and organization of biologically relevant
polymers like chromatin 20, 21, 22, 23. In addition, recent studies 24, 25,
26, 27 have shown that systems composed of deformable particles such as
colloids exhibit glassy behavior as a function of packing fraction. The
introduction of deformability and softness in the hard-sphere particles can
induce novel packing behavior at high densities. The ring polymer systems thus
can be an excellent model platform to mimic such colloids, with the
possibility of soft and tuneable interactions, to understand emergence of
glassy behavior in such colloidal systems. The complex behavior of the
dynamics of the ring polymer systems poses an interesting challenge to the
theoretical understanding of polymer entanglement. In the last decade, several
attempts were made to understand their dynamical as well as statistical
properties using computer simulations 11, 12, 10. Simulations of ring polymers
show a sluggish decay in the stress relaxation modulus (G) at longer time-
scales when compared with theoretical models for such polymers 28, 29, 30. It
was postulated that such deviation is caused by ring-ring penetration, or
threading effect, which was not incorporated in the theoretical models like
self-similar lattice-animal model 31, 32 annealed-tree model 33, 34, or
fractal loopy globule model 35. Despite pure linear polymer dense systems not
exhibiting similar dynamical behavior, it was shown that the presence of even
a small amount of linear polymers in the ring polymer systems can dramatically
increase the relaxation times and make the ring polymer systems more sluggish
28, 36, 37, 21, 38, 39. Assuming a similarity of the conformations between the
ring and linear polymers at a length scale smaller than the radius of gyration
(Rg), some modified theories have been proposed to incorporate the threading
effect in pure ring polymer systems, e.g. Tsalikis et al 30, Dell et al 40.

Recently, it was shown that the dynamics of the ring polymer systems mimic a
glassy behavior when a randomly chosen minute fraction of the rings are pinned
in the simulation box 41, 42, 43. Further, it was estimated that at a large
chain-length of $\sim 3500$, glass formation of the ring polymer system
becomes spontaneous and independent of any external perturbations like pinning
41. This type of glass formation is unique as it does not require any cooling,
jamming, or modification of the interaction energies 44. Such glass-forming
systems are termed topological-glass 45 which linear polymers fail to exhibit
46. The phenomenon of topological glass formation as well as the slow
relaxation of G, was explained using the threading effect. This theory
suggests that the glassy phase originates as a result of the long-range
entanglements of the ring polymers in the form of threading, which creates
insurmountable energetic barriers for the reptation movement of the ring
polymers at larger timescale 31, 32. Formation of such barriers is controlled
by the topology (or conformations) and entropically driven mixing of the
rings. Several studies have focused on understanding the effects of threading
on the dynamics of the pure ring polymer systems using Monte-Carlo (MC) 45, 18
and Molecular-Dynamics (MD) 46, 30, 47, 48 methods and experiments 49, 50, 51.
The theoretical studies were performed on either flexible ring polymer
systems, or lattice models. Threading is a topological phenomenon, which
suggests that it should be correlated with the shape of the ring polymers.
However, a proper understanding of the correlation between shape and threading
in ring polymers is still missing in the literature. Recent studies have
started focusing on the effect of stiffness on threading parameter for systems
of knotted and unknotted polymers 52. However, there has not been much
attention on dense assemblies of ring polymers where there can be variations
in the shape of the polymers, and thereby affect threading and consequently
the dynamics under equilibrium conditions.

In this paper, we focus on emergence of dynamical slowing down in systems of
equilibrated ring polymer systems and identify the domain in phase space
spanned by pressure and ring stiffness, i.e. (P-$K_{\theta}$), in which the
glassy regime would emerge. Importantly, we observe that stiffer ring polymers
exhibit dynamical slowdown at smaller ambient pressures. We explicitly
calculate the total effect of threading in such systems, and show that the
threading effect is directly contributing to the observed slowdown at certain
P-$K_{\theta}$ state points. The pressure induced conformational changes of
the ring polymers and their effects on the threading properties in equilibrium
are also discussed.

The paper is organized as follows. After this introductory section, we provide
a section on methods where the model system that we study and the details of
the simulations that we are perform are outlined. Subsequently, there is an
extensive section discussing our results, where we detail the dynamical and
structural properties of the system, and the correlations between the two.
Finally, we have a concluding section where we summarize our findings and
provide some future perspectives.

##  2 Methods

###  2.1 Simulation details

Fig. 1: (Top) Snapshots of equilibrium configurations sampled at a pressure
value of $P=0.010$ and different ring stiffness, viz. $K_{\theta}$=1 (a),10
(b), and 20 (c). (Bottom) Conformation of a single polymer inside the bulk at
these P and $K_{\theta}$ values in (c),(d), and (e). Fig. 2: (a) Temporal
variation of the mean square displacement of the center-of-mass of ring
polymers ($g_{3}(t)$) for $K_{\theta}$=20 and different pressures ($P$) as
marked. Data are shown for regimes where equilibrium has been attained (in
solid lines) and where the dynamics is out-of-equilibrium within time-window
of observation (in dashed lines). Data is shown for $t\geq t_{d}$. (b)
Variation of center-of-mass diffusion constants (D) with pressure, calculated
from the equilibrium $g_{3}(t)$ data for different values of $K_{\theta}$ as
indicated. (c) Estimated iso-diffusivity contours in $P-K_{\theta}$ plane,
using data shown in (b), for different diffusion constant values, as marked.

A series of molecular dynamics (MD) simulations of ring polymer systems are
performed to probe the slowing down in dynamical behavior of such systems with
different values of ring stiffness and applied pressure. The pairwise
potential of the ring polymers is modeled using the Kremer-Grest (KG) bead-
spring model 53, 11, 12. In KG model, the repulsive part of a truncated and
shifted Lennard-Jones potential, aka Weeks-Chandler-Anderson (WCA) potential
54, is used to model the monomer-monomer non-bonded interactions:

$U_{\text{pair}}(r_{ij})=\begin{cases}4\epsilon\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}+\frac{1}{4}\right]&r_{ij}\leq
2^{1/6}\sigma\\\ 0&r_{ij}>2^{1/6}\sigma\end{cases}$ (1)

where $\sigma$ and $\epsilon$ are length and energy scales, respectively.
$\epsilon$ is measured in $k_{B}T$ units, where $k_{B}$ is the Boltzmann
constant, and $T$ is the temperature. $r_{ij}$ defines the distance between
any two monomers in the system. A finitely extensible non-linear elastic
(FENE) potential 53 is used to define the covalent bonds between the monomers:

$U_{\text{FENE}}(d_{ij})=\begin{cases}-0.5\kappa
R_{0}^{2}ln\left[1-\left(\frac{d_{ij}}{R_{0}}\right)^{2}\right]&d_{ij}<R_{0}\\\
\infty&d_{ij}\geq R_{0}\end{cases}$ (2)

where $\kappa=30$, $R_{0}=1.5$. Here, $d_{ij}$ denotes the distance between
two bonded monomers in a ring polymer. In our paper, we have used the Kratky-
Porod model 55, recently used in reference 52, for controlling the ring
stiffness,

$U_{\text{angle}}(\theta_{i})=K_{\theta}[1-cos(\theta_{i}-\pi)]$ (3)

where $K_{\theta}$ is the stiffness parameter, which is varied in our
simulations. $\theta_{i}$ defines the angle between three successive monomers
centered at $i^{th}$ monomer in a ring polymer. The temperature of the system
is kept fixed at 1.0 for all simulations. The mass of each monomer was taken
to be 1.0.

In this paper, we have simulated a ring polymer system of 100 rings, each with
100 monomers (total 10000 particles). The ring-size was kept smaller than used
in the previous simulations 42, 41, to understand if threading phenomenon can
be envisaged for small ring sizes as well. We randomly packed the ring
polymers in a cubic box of length 103 (in units of $\sigma$) with periodic
boundary conditions to create the initial low-density systems. Further, we
made sure that the rings remain non-concatenated during packing. The Nosé -
Hoover chains thermostat 56, 57 and barostat 58 were used to control the
temperature and the pressure of the system, respectively. We use an overdamped
barostat, which couples to the system over a timescale of $t_{d}=1.0$. The
molecular dynamics simulations were done using LAMMPS 59.

The pressure on the initial low density systems was iteratively increased for
100 steps with 1000 step intermediate relaxation periods using a time-step of
0.003 until the target pressure is reached. After the compression cycle is
completed, the systems were allowed to relax for $\sim 4\times 10^{7}$ steps.
A total of 108 production steps were subsequently ran with a larger time-step
of 0.012. This procedure was continued for different state-points from
$K_{\theta}=1$ to $20$, and $P=0.001$ to $20.0$. A few snapshots of the
simulation box and an arbitrarily chosen ring polymer at the end of the
production runs are included in Figure 1.

##  3 Results

###  3.1 Equilibrium dynamics

To monitor the dynamical behavior of the ring polymer systems, we study the
time evolution of the mean square deviation of the center-of-mass displacement
of the ring polymers, $g_{3}(t)$, which is defined as,

$g_{3}(t)=\langle[r_{\text{CM}}(t_{0})-r_{\text{CM}}(t_{0}+t)]^{2}\rangle$ (4)

where $t_{0}$ is the reference time origin and $\langle\rangle$ denotes the
average over all such time origins and all rings. The corresponding analysis
is discussed via Figure 2.

In Figure 2(a), we show the $g_{3}(t)$ data for a fixed value of ring
stiffness, viz. $K_{\theta}=20$, i.e. for the case where the rings are
relatively stiffer. We observe that for a range of pressure values, the long-
time behavior is linear, i.e. $g_{3}(t)\sim t$ (marked in solid lines). This
implies that the dynamics of the rings is diffusive and therefore indicates
that the equilibrium regime is being sampled for these ambient pressures. At
larger pressure values ($P\geq 1.0$), this diffusive regime is not attained
within the time window of our measurement. Rather, with increasing pressure, a
prolonged plateau-like regime is observed (see data shown in dashed lines in
Figure 2(a)), which would correspond to the caging of a ring by neighboring
rings. The observed caging behavior is reminiscent of the dynamics typical to
glass-forming liquids on the approach to dynamical arrest 60. With increasing
pressure and thus increased density, the localization of the rings
significantly increases, as is evident from the decreasing height of the
plateau-like regime in $g_{3}(t)$. Such dynamical behavior is also observed in
the rings with other $K_{\theta}$ values. In Supplementary Information (Figure
SI.3(a)), we show data for $K_{\theta}=1$ as an additional example.

We note here that variation in the short-time dynamics is observed with
changing pressure at fixed $K_{\theta}$, even at constant temperature. This
occurs due to varying box fluctuations resulting from the coupling to a
barostat. However, and most importantly, we have verified that this has no
impact on the diffusive dynamics observed at long times (see Supplementary
Information Figure SI.2(a)), which is occurring much later vis-a-vis the
barostat’s coupling timescale $t_{d}$, in the range of pressures where
dynamical slowing down is happening. Relatedly, in all cases, we now display
dynamical data for timescales larger than the barostat coupling timescale
$t_{d}$.

It is evident in Figure 2(a) that the long-time diffusive dynamics shows a
slowing down with increasing pressure. This is illustrated via the plot
showing the dependence of the extracted center-of-mass diffusion coefficient
$D=\lim\limits_{t\to\infty}g_{3}(t)/6t$ versus pressure ($P$) as shown in
Figure 2(b), for a range of $K_{\theta}$ values. The data clearly suggests
that the onset of slow dynamics depends on both ring polymer stiffness,
$K_{\theta}$, and the pressure: rings with larger $K_{\theta}$ values clearly
display that the eventual onset of glassiness (i.e. $D=0$ or dynamical arrest)
would occur at lower pressures if one were to extrapolate the $D$ vs $P$
curves. This effect can be seen more clearly in Figure 2(c), where we plot the
isodiffusivity lines corresponding to $D=0.01,0.005,0.001$ in the
$K_{\theta},P$ plane. These iso-diffusivity lines act as guides to the shape
of the eventual line of glass transition in this phase-plane as we vary the
stiffness of the ring polymers and the ambient pressure.

###  3.2 Structural Characterization

Fig. 3: Structures of an arbitrarily chosen ring polymer at various
$K_{\theta}$ and pressures are shown in tabular form. All structures shown
here belong to the diffusive regime except for $K_{\theta}$=20 and P=1.0,
where we could not find a sample of an equilibrated trajectory.

In this section, we characterize the structural variation of the ring polymer
systems across the P-$K_{\theta}$ phase-diagram to understand the possible
connection between structure and emergence of slow dynamics. In Figure 4, the
variation of average density ($\langle\rho\rangle$), radius of gyration
($\langle{R_{g}}\rangle$), and eigenvalue ratio
($\langle\lambda_{\text{max}}/\lambda_{\text{min}}\rangle$, defined later)
with pressure for different ring polymer stiffness is shown. All data is
averaged over all trajectories within a time window of $0.5\times 10^{6}$
within the equilibrium regime.

Fig. 4: Average density ($\langle\rho\rangle$), radius of gyration
($\langle{R_{g}}\rangle$), and eigenvalue ratio
($\langle\lambda_{\text{max}}/\lambda_{\text{min}}\rangle$) as a function of
applied pressure values for different $K_{\theta}$ values, measured under
equilibrium conditions.

$R_{g}=\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}}$ (5)

where $\lambda_{1},\lambda_{2},\lambda_{3}$ are the principle moments of
inertia, calculated from the eigenvalues of the gyration tensor of each
polymer. In addition, we also compute the ratio of the maximum and minimum
eigenvalues to understand the shape of the ring polymers. For a globule shaped
polymer, this ratio remains close to 1 but increases as the structure becomes
more anisotropic. Results discussed below, for these two observables,
correspond to averages over all rings within configurations sampled in the
equilibrium regime.

As can be expected, the average density of the system increases sharply with
pressure for all values of $K_{\theta}$; see Figure 4(a). However, it should
be noted that the highest values of equilibrium density achieved in our
simulations, over the time window of our simulations, is for the most flexible
chains (low $K_{\theta}$). As the chain stiffness increases, the system cannot
access high density diffusive regimes, but rather falls out of equilibrium
within our observation time window. For example, for the highest and lowest
chain stiffness considered in the study ($K_{\theta}=20$, $K_{\theta}=1$), the
corresponding maximum equilibrium densities achieved are $\sim$ 0.3 and 1.13,
respectively. The $\langle{R_{g}}\rangle$ as a function of applied pressure,
exhibits varying behavior with different ring polymer stiffness seen in Figure
4(b). For low values of $K_{\theta}\lesssim 5$, with increasing pressure, the
value of $\langle{R_{g}}\rangle$ monotonically decreases, suggesting a more
collapsed conformation adopted by the ring polymers as the density of the
system goes up with applied pressure. However, as $K_{\theta}$ values
increase, the $\langle{R_{g}}\rangle$ values change with applied pressure in a
non-monotonic manner. The $\langle{R_{g}}\rangle$ values decrease upto a
certain pressure, and then increase again, strongly suggesting that the ring
polymers might be adopting more extended, plate/rod like structures at higher
density (see Figure 1). The increase of shape anisotropy at large pressure for
$K_{\theta}\gtrsim 5$ is reflected in the eigenvalue ratio plotted in Figure
4(c). Because of the plate or rod like shapes, the structure of the ring
polymers become more flat at larger $K_{\theta}$ values. With increasing
pressure, flat structures become more favorable as it helps ring polymers to
better pack in a dense environment.

Typically, at a constant temperature, size of the molecules determines the
average diffusion coefficient of a system. In case of ring polymers,
$\langle{R_{g}}\rangle$ values at large $K_{\theta}$ values ($\gtrsim 5$) seem
to have a non-monotonic dependence on pressure, however, the diffusion
coefficient does not follow the similar trend (see Figure 2). This behavior
suggests that underlying mechanism of the glass formation changes as the
system accesses higher P-$K_{\theta}$ state points in the phase diagram. In
this region, the polymers adopt more anisotropic extended structures (see
Figure 3) compared to the isotropic crumpled-like structures seen at lower
values of $K_{\theta}$. Hence, theories based on diffusion of spherical
objects would no longer be applicable in this regime. Here, one has to
consider the effect of entanglement to describe the dynamics of the system.
The increasing effect of the threading between the ring polymers, a
manifestation of entanglement, can describe the dynamical slow-down at higher
$K_{\theta}$ values, which we describe in next section.

###  3.3 Geometric analysis of the threading events

Fig. 5: A pictorial description of a threading event. The horizontal lines
inside each ring polymer surfaces depict the minimal surfaces detected by the
algorithm.

In this section, we correlate the threading of rings and emergence of slow
dynamics in ring polymer systems. We develop a threading-detection algorithm,
partly based on previously published triangulation algorithm 30, and
incorporating geometric analysis of individual rings in addition. The
geometrical analysis was used to identify loops along the contour of the ring
polymer which helped to calculate the area of the minimal surface with a
higher accuracy as compared to the contour reduction methods 30.

To identify threading between any two ring polymers, A and B, we compute all
the intersection points between bond vectors of ring polymer A and triangular
planes of ring polymer B which lie on the minimal surface of ring polymer B.
If number of such intersections points are greater than zero, we assume that A
is threaded into B (not the other way around). Due to geometric reasons, total
number of intersections will be an even number. The quality of threading is
identified by considering the length of the threaded sections 47, 48. A
threading event between ring polymers A and B is defined by the two successive
intersections found along the bond vectors of A. The extent of threading,
$N_{\text{th}}$ is defined as the total number of bond vectors of ring polymer
A present between the two successive intersections with ring polymer B. We
denote the two sides of the minimum plane of ring B as odd-side and even-side.
By construction,
$N_{\text{th}}^{\text{odd}}+N_{\text{th}}^{\text{even}}=N_{\text{ring}}$,
where $N_{\text{th}}^{\text{odd}}$ and $N_{\text{th}}^{\text{even}}$ denote
the extent of threading at odd- and even-side, respectively. To quantify the
degree of threading, we define a function,

$\Delta_{\text{th}}=1-\left|{\frac{2\sum\limits_{i=\text{odd}}N_{\text{th}}^{i}}{N_{\text{ring}}}-1}\right|$
(6)

where $\Delta_{\text{th}}$ is the degree of threading between two ring
polymers, who have at least one threading event between them. $\Delta_{th}=1$
for $\sum_{i=odd}N^{i}_{\text{th}}=N/2$, and $\Delta_{\text{th}}=0$ when
$\sum_{i=odd}N^{i}_{\text{th}}=0$. For a single ring polymer, the mean degree
of threading, $\langle\Delta_{th}\rangle$, is determined by averaging over all
$\Delta_{\text{th}}$ with all other ring polymers for all the sampled
trajectories. For more details of the threading detection algorithm, see
Supplementary Information, section IA.

###  3.4 Threading of ring polymers and emergence of spontaneous pinning

In Figure 6(a), the $\langle\Delta_{\text{th}}\rangle$ for the ring polymer
systems is plotted as a function of pressure for various values of ring
polymer stiffness. For all values of ring polymer stiffness,
$\langle\Delta_{\text{th}}\rangle$ increases monotonically with pressure
suggesting the increase in threading events with the increasing density of the
ring polymer systems. While it is possible to calculate
$\langle\Delta_{\text{th}}\rangle$ from the simulation trajectories, it is
difficult to obtain experimentally 50. On the other hand,
$\langle{R_{g}}\rangle$, D, and $\langle\rho\rangle$ are readily measurable
via experiments. Here, we attempt to find some correlation between the
experimentally available structural quantities and
$\langle\Delta_{\text{th}}\rangle$ calculated from simulations.

We define an overlap concentration of the ring polymer system, $c^{*}$, to
capture the essence of possible overlap of rings as pressure on the system is
increased. The effective shape of the ring polymers can be approximated as
spheres with volume $(4\pi/3)\langle{R_{g}}\rangle^{3}$. Further, from
emergent density ($\langle\rho\rangle$) due to pressure applied on the system,
an effective volume per ring polymer can be defined as
$\langle{V_{m}}\rangle=N/\langle\rho\rangle$, where $N$ is the total number of
ring polymers. Using these two parameters, the overlap concentration $c^{*}$
is defined as:

$1/c^{*}=\frac{4\pi\langle{R_{g}}\rangle^{3}}{3\langle{V_{m}}\rangle}$ (7)

A higher value of $1/c^{*}$ can come from either large $R_{g}$, suggesting
possible non-collapse and anisotropic structures or smaller $V_{m}$ indicating
threading. In Figure 6(a) and 6(b), it can be seen that there is a strong
correlation in the trend of $(1/c^{*})$ and
$\langle\Delta_{\text{th}}\rangle$. This correlation suggests that it is
possible to estimate the average amount of threading in the ring polymer
systems from the inverse of $c^{*}$. Overlap concentration is measured in
dilute polymer solutions 61 via viscosity analysis. Due a strong correlation
between the threading and dynamics, it should be possible to define $c^{*}$ in
a pure ring polymer system in a similar manner. However, an extrapolation of
the data from dilute solution to dense systems is not straightforward and
requires further investigation, which is beyond scope of this paper.

Fig. 6: Variation of (a) average threading factor, (b) inverse of the overlap
concentration, and (c) average fraction of threaded rings with different
pressure values and ring stiffness, measured within the accessible equilibrium
regime. The horizontal dashed line in (b) and (c) cuts the y-axis at 1. Fig.
7: (a) Variation of threading correlation function $C(t)$ (see text for
definition) with pressure values at $K_{\theta}=10$. Filled symbols correspond
to pressure values where equilibrium has been attained, and open symbols
correspond to the case where the dynamics is not in equilibrium over the time
window of observation. Lines correspond to fits to equilibrium data with
stretched exponential function (see text) to obtain threading timescale
$\tau_{\text{th}}$. (b) Variation of $\tau_{\text{th}}$ with pressure, for
different $K_{\theta}$. (c) Correlation plot between different timescales,
i.e. $1/D$ (a diffusive timescale) and $\tau_{\text{th}}$, for different
P-$K_{\theta}$ state points.

The data in Figure 6(b) shows that the values of $(1/c^{*})$ become much
larger than 1 with increasing pressure for $K_{\theta}\gtrsim 5$. As discussed
earlier, such large increase is possible only if the polymers are highly
anisotropic in nature or they form highly entangled conformations. From the
structural characterization of previous section, we deduce the following. For
intermediate values of stiffness, $5\lesssim K_{\theta}\lesssim 10$, the
structures suggest that the anisotropy is not significant and the large value
of $(1/c^{*})$ primarily can be attributed to threading. To verify this, we
calculate the change in the average fraction of threaded polymers in the bulk
as seen in Figure 6(c). We found that as the bulk system approaches the regime
of increasingly slow dynamics, the fraction of threaded rings,
$f_{\text{th}}$, sharply increases and saturates to 1. We have further
verified that the total fraction of ring polymers participating in extensive
inter-ring threading is quite high ($\Delta_{\text{th}}$ > 0.75, see
Supplementary Information Figure SI.7(a)(d)). This further confirms that with
increasing pressures, the dense ring polymer systems prefer a well-threaded
environment. Therefore, in the glass phase, we predict that the system must
contain a vast network of threaded ring polymers. For stiffer rings, i.e.
$K_{\theta}>10$, threading alone perhaps does not account for the increase in
$1/c^{*}$ values. In this regime, the high values of $(1/c^{*})$ can also be
contributed by the anisotropical, plate/rod like structures at high density.
Similar emergence of anisotropy in the form of cluster-glass phase for ring
polymers at high $K_{\theta}$ values has also been seen previously 62, 63, 64.

The values of $\langle\Delta_{\text{th}}\rangle$ respond in a non-monotonic
fashion with respect to ring polymer stiffness $K_{\theta}$ at some pressure
values (see Supplementary Information, Figure SI.5(a)), similar to a recent
work 52. This behavior can be attributed to the emergence of hollow disk
phase, where rings have lower tendency to thread into each other due to higher
stiffness 52. However, a direct correlation between
$\langle\Delta_{\text{th}}\rangle$ and diffusion is not straightforward. Some
special form of threading between two rings, like square knots, may lead to a
slow dynamics despite having a low $\Delta_{\text{th}}$ value 51, 65. Hence,
the lifetime of a threading event is more important quantity to understand the
effect of threading on dynamics of the ring polymers. To understand such
temporal evolution of threading event, a threading related correlation
function ($C(t)$) for a pair of initially threaded rings is calculated as,

$C(t)=\frac{\langle\Delta_{\text{th}}(t_{0})\Delta_{\text{th}}(t_{0}+t)\rangle}{\langle\Delta_{\text{th}}^{2}\rangle}$
(8)

By definition, $C(t)$ varies from 1 to 0 as the threading correlation
decreases between two rings. Once un-threading of the two rings occurs, the
threading event ends and thereby the computation of $C(t)$ corresponding to
that pair of rings also ends. Any re-threading events between two rings at a
later time will subsequently be treated as a new event. Hence $C(t)$ is
similar to a first passage correlator corresponding to an un-threading event
between two rings that were initially entangled. This definition of $C(t)$
makes it a continuous correlation function.

In Figure 7(a), we show the variation of $C(t)$ at different pressures with
time for a particular stiffness of the ring ($K_{\theta}=10$). We observe
that, for this value of $K_{\theta}$, in the regime of pressure where
equilibrium dynamics is observed (see Fig. 2), $C(t)$ decreases with time and
eventually decays to zero i.e. complete decorrelation of threaded polymers
occur. Further, it is evident that such unthreading processes occur over
longer timescales at larger pressures. In the pressure regime where we do not
observe equilibrium diffusion dynamics within time window of measurement,
$C(t)$ does not completely decorrelate, i.e, threadings continue to persist
over the observation time window.

In the equilibrium regime, it is possible to fit the eventual decay of $C(t)$
decay with a stretched exponential function, viz. $C(t)\sim
A\exp{[-(t/\tau_{\text{th}})^{p}]}$, where $A$ is an empirical constant,
$\tau_{\text{th}}$ is the threading timescale extracted via the stretched
exponential fit, and $p$ is the stretched exponent. In Figure 6(b), we show
that $\tau_{\text{th}}$ increases with pressure, for all the different
$K_{\theta}$ values that we have explored. Together with Figure 6(b) and
Figure 7(b), this result indicates that threading leads to formation of a
stable network among the ring polymers, whose lifetime increases with
increasing pressure. It is also evident that for any given pressure, the
persistence of threading events increases with the stiffness of the ring.

A relevant question to ask is how the threading lifetimes can be correlated
with the observed dynamical behavior. In Figure 7(c), we illustrate a positive
correlation between $\tau_{\text{th}}$ and a diffusive timescale corresponding
to the inverse of the diffusion coefficient demonstrating that threading is a
contributing factor for dynamical slowing down with increasing pressure at all
$K_{\theta}$ values. However, the extent of contribution of $\tau_{\text{th}}$
on dynamical behavior can not be extracted directly form Figure 7(c). In fact,
the data shows that the slowing down in diffusion happens slower than the
growth in threading lifetime, with increasing pressure. Hence there are likely
to be effects from other relevant relaxation processes on the dynamics of the
ring polymers. Possibly, such effects are originating from different forms of
inter-ring entanglements other than threading. For example, at small K,
$\tau_{\text{th}}$ is much smaller and threading is not expected to be the
dominant factor in the dynamical slowdown with increasing pressure. Rather,
excluded volume effects due to crowding among the adjacent globule-shaped ring
polymers are more contributing to its dynamics. With increasing K, rings are
no longer globular and such excluded volume effect is expected to diminish but
still remain relevant. However, in this regime, threading could become the
more contributing factor to the dynamical slowdown with increasing stiffness,
which is realized by order of magnitude increase of the $\tau_{\text{th}}$
data with pressure in Figure 7(c).

##  4 Conclusion and perspectives

Emergence of dynamically arrested states in topologically constrained polymer
systems is an exciting area which can serve as model platform to explore the
dynamics of deformable entities and emergent new phases. Earlier simulations
have demonstrated the occurrence of glassy dynamics in ring polymers via
artificial pinning of the ring polymers, motivated by similar explorations of
random pinning in supercooled atomistic liquids 66. It is not very clear what
is the role played by both the ring polymer stiffness and ambient pressure
conditions in the formation of such novel phases and whether the presence of
quenched disorder is necessarily required for facilitating the path towards
the formation of topological glasses.

In this work, we explore the phase space spanned by ring polymer stiffness and
ambient pressure to elucidate the emergence of glassy states via large scale
simulations of a thermal assembly of ring polymers formed by hundred monomers.
Our work evidences that large ring size is not a prerequisite for observing
threading in the equilibrated systems of semi-stiff rings. We show that
dynamical slowing down and thereby the tendency towards glassiness emerges as
we scan across the phase space. We demonstrate that stiffer rings are likely
to get dynamically arrested at smaller ambient pressures, and correspondingly
smaller densities. By calculating various structural properties of ring
polymers in the P-$K_{\theta}$, we show that the ring polymers show a variety
of conformations from completely crumpled structures (flexible ring polymers)
to elongated and stiff plate like conformations (stiff ring polymers). These
structural results strongly suggest that the ring polymers considered here
encounter significant dynamical constraints and thus perhaps achieve better
close packing via clustering and shape modifications, some of which have been
see earlier 67, 68, 68, 63, 64 . As shown in Figure 3, at low $K_{\theta}$,
the system becomes more closed and globular with increasing pressure. At the
medium $K_{\theta}$ range, the rings take a rod like shape at high pressure.
At large $K_{\theta}$, rings take a circular shape which minimizes its energy
and helps packing via clustering.

To better understand the emergent slow dynamics with varying ring stiffness
and pressure, we explored the possibility of threading among the ring
polymers, which has been proposed as a mechanism for the formation of the
topological glass that is unique to these objects. For identifying such
conformations, we develop an improved threading algorithm with specific inputs
from the geometry of the ring polymers which gives an excellent quantification
of the extent of threading. Our threading analyses strongly suggests that for
any specific ring stiffness with increasing pressure values, the average
threading factor increases and contributes to the slow dynamics. Further, the
occurrence of such threading events increase with increasing ring stiffness,
at any particular ambient pressure condition. We define an overlap
concentration for the polymer systems – usually measured experimentally in
dilute polymer solutions 61 – which strongly correlates with the average
threading factor measured from the simulation. This result has the potential
to promote experimental studies on measurement of threading in dense ring
polymer systems. We also measure the life time of a threaded event and find
that as the pressure increases, the ring polymers show more long-lived
threaded events, in sync with increasing slow down in the dynamics. Taken
together all the structural and dynamical properties measured in our
simulations, we clearly demonstrate the spontaneous emergence of slow, glassy
dynamics in ring polymer systems – by suitable selection of ring polymer
stiffness and pressures applied, slow dynamics can be induced in ring polymers
at very low pressure values. We also note that we are observing the formation
of such topologically constrained states for relatively smaller ring sizes
compared to what has been explored earlier, indicating that the access to such
states can be obtained by tuning the properties of the ring. We hope that our
work will motivate experimental studies to validate our findings.

In future, interesting avenues of study could include the possibility of
introducing crowder particles into the ring polymer systems or adding
differential activities 69, 65, 70, 71 to the ring polymer via temperature
changes to further probe glass formation in such systems.

## Author Contributions

PC and SV conceived the study. PR carried out the simulations. All authors
contributed to writing of the manuscript.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgments

The authors thank the HPC facility at IMSc, Chennai, for providing the
necessary computational platform for this research.

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