#  Topological transition in multicyclic chains with structural symmetry
inducing stress-overshoot phenomena in multicyclic/linear blends under biaxial
elongational flow

Takahiro Murashima<EMAIL_ADDRESS>Department of Physics,
Tohoku University, 6-3, Aramaki-aza-Aoba, Aoba-ku, Sendai, 980-8578, Japan
Katsumi Hagita  Department of Applied Physics, National Defense Academy,
1-10-20, Hashirimizu, Yokosuka, 239-8686, Japan  Toshihiro Kawakatsu
Department of Physics, Tohoku University, 6-3, Aramaki-aza-Aoba, Aoba-ku,
Sendai, 980-8578, Japan

###### Abstract

Blends of multicyclic and linear polymers under biaxial elongational flow were
analyzed using coarse-grained molecular dynamics simulations. The
multicyclic/linear blends displayed overshoot in the normal stress difference
at the start-up of biaxial elongational flow. This overshoot was steeper for
multicyclic/linear blends than for our previously reported monocyclic/linear
blends [T. Murashima, K. Hagita, and T. Kawakatsu, Macromolecules, 2021, 54,
7210]. Investigation of the origin of the overshoot in the multicyclic/linear
blends revealed a different mechanism than that previously observed in our
monocyclic/linear blends. Specifically, a “topological transition” mechanism
comprising a morphological change from the open- to closed-ring state was
observed in the multicyclic chains, but not in the monocyclic chains. This
topological transition drastically changes the stress of the rings. Although
the topological transition was also observed in asymmetric-multicyclic/linear
blends, no stress overshoot appeared, owing to the asymmetry in the
multicyclic chains. Therefore, we hypothesized that the structural symmetry in
multicyclic chains is indispensable for overshoot behavior to occur. We
determined that the topological transition in multicyclic chains with
structural symmetry induces stress-overshoot behavior in multicyclic/linear
blends under biaxial elongational flow.

###### keywords:

Multicyclic/linear blends, biaxial elongational flow, topology, coarse-grained
molecular dynamics simulation, Kremer–Grest model

###### keywords:

American Chemical Society, LaTeX

+81-22-795-5718 +81-22-795-6447 IR,NMR,UV

Graphical TOC Entry

(for Table of Contents use only)

##  1 Introduction

Advances in synthetic technology have led to the development of materials with
various topological structures that combine multiple rings 1, 2, 3. Such
topologies differ from those of conventional linear chains, branched chains,
and single rings, resulting in new material functionalities. For example,
Hossain et al. observed that the topologies of multicyclic polymers have a
marked influence on their glass transition temperature 4. Yan et al. reported
that multicyclic polymers display hierarchical inward relaxation, and that the
zero-shear viscosity decreased with an increasing number of constrained
segments on the coupling sites 5. Doi et al. investigated a dumbbell-shaped
polymer consisting of two small rings and a short linear chain. Their blend of
dumbbell-shaped and long linear polymers revealed that the dumbbell chain acts
as a pseudo-entanglement point with a longer characteristic time than that of
the host linear chain 6. However, while the understanding of monocyclic rings
has progressed rapidly in recent years, the basic understanding of multicyclic
polymers, except for these pioneering works, is still in its infancy. This is
because many studies have concentrated on monocyclic rings, especially those
with much larger molecular weights than the entanglement molecular weight of
linear polymers.

Single-ring polymers present different rheological properties than those of
linear chains because of the absence of free ends 7. Thus, the stress
relaxation of ring melts exhibits power-law behavior without an entanglement
plateau 8, 9. Moreover, the ratio of the zero-shear viscosity of a linear
polymer melt to that of its ring counterpart increases with the increasing
number of entanglements 10. The relaxation dynamics of rings surrounded by
obstacles is understood through three typical ring structures 11: (i)
obstacle-enclosing, (ii) non-obstacle-enclosing but ramified with double-
folded large loops, and (iii) non-obstacle-enclosing and non-ramified (or
ramified with small loops). The constraint release and reptation relaxation
mechanisms of the first and third structures are similar to those of linear
chains 12. On the other hand, the relaxation mechanism of the second structure
is difficult to understand in analogy with the conventional linear chain
dynamics. Instead, similar to an amoeba, such rings invade between the
surrounding obstacles while stretching and shrinking many branched double-
folded loops, a behavior described by the lattice animal model 13. Ge et al.
developed the fractal loopy globule model, implementing the self-similar
dynamics of entangled rings with self-consistent multi-ring dynamics, and
successfully reproduced the simulation and experimental results 14.

Rings are penetrated and interlocked by the surrounding rings 15, 16, 17, 18,
19, while in linear chain contaminants they are penetrated by linear chains
15, 20. Thus, the constraint release mechanism of rings is much slower than
theoretically expected. Moreover, since these penetrations lead to slow ring
polymer dynamics 21, 22, the rheological properties of ring/linear blends
differ from those of both the pure ring and linear melts. Specifically, a
small number of linear chains drastically increases the viscosity of the ring
melts 23, 24, whereas a low ring volume fraction increases the viscosity of
the linear melt up to the overlapping concentration of the rings 25. In a flow
field, rings represent nonlinear behaviors that are different from those of
linear chains owing to penetration. Zhou et al. observed transient stretching
of the DNA ring in planar elongational flow 26, 27, while Borger et al.
discovered a threading–unthreading transition in ring/linear blends under
uniaxial elongational flow 28. Parisi et al. investigated the steady-state
shear viscosity of pure ring melts and explained their shear thinning behavior
using a shear slit model 29. They also investigated the shear and elongational
viscosities of ring/linear blends 30.

Recently, we actively investigated ring/linear blend systems from the
viewpoint of penetration control for device application. By defining the
number of penetrations using the Gauss linking number 12, we established the
minimum ring size in which the ring has more than one penetration for
monocyclic/linear blends through coarse-grained molecular dynamics simulation
31. For multicyclic/linear blends, the number of penetrations is increased by
increasing the number of rings 32. Notably, these equilibrated simulation data
are publicly available for the development of a broad scientific community 33.
Following a mapping procedure 34 from the coarse-grained model to a real
material, we successfully reproduced the experimental observations 35.
Moreover, we confirmed that increasing the ring concentration reduces the ring
size and number of penetrations owing to the topological repulsion between the
rings 36. Cross-linking among the linear chains in a ring/linear blend can
create a material in which the rings are trapped in the networks. Interpreting
the number of penetrations as the trapping probability of the rings, we
confirmed that the trapping probabilities obtained from the experiments and
simulations were in good agreement 37. Moreover, the trapping probability of
multicyclic chains were evaluated, and the tricyclic chains in the network
presented different topologies depending on the penetration mode. These
topologies were expected to contribute to the physical characteristics of the
network. We also studied the role of rings trapped in a network under tensile
loading 38, 39.

To determine the nonlinear rheological properties, elongational and shear flow
simulations have been in high demand in recent years. Elongational flow
simulation with large strains, however, has remained difficult for many years,
except for planar elongational flow, in which the Kraynik–Reinelt (KR)
boundary conditions 40 are available to avoid the collapse of the simulation
box at large strains 41, 42. Indeed, the discovery of the generalized
Kraynik–Reinelt (gKR) boundary conditions 43, 44, 45 has seen the progress in
the simulation studies of uniaxial/biaxial elongational flows 46, 47, 48. In
particular, when studying ring polymers, O’Connor et al. discovered anomalous
thickening behaviors for pure ring melts under uniaxial elongational flow 49.
Further, they systematically investigated a threading–unthreading transition
28 for ring/linear blends with varying ring fractions 50. Moreover, our group
reported stress-overshoot phenomena under biaxial elongational flow for
ring/linear blends 51.

In our previous study 51, we investigated ring/linear blends using the
Kremer–Grest (KG)-type bead-spring model 52, where the ring-to-linear chain
blend weight ratio was fixed at 1:10. We observed stress-overshoot phenomena
under biaxial elongational flow at a ring size, $N_{\rm R}$, of 160 ($N_{\rm
R}>$80 — A ring with $N_{\rm R}=80$ has more than one penetration 31) and
linear chain length values, $N_{\rm L}$, of 20, 40, and 80. In contrast, no
overshoot was observed at $N_{\rm L}$ values of 10 and 160. The overshoot was
apparent at $N_{\rm L}=40$ under biaxial elongational flow, with a strain rate
$\dot{\varepsilon}$ of 0.001/$\tau$, where $\tau$ is the unit of time in the
Lennard-Jones (LJ) particle system. This strain rate is faster than
1/$\tau_{\rm e}$, where $\tau_{\rm e}$($\approx$ 2000$\tau$ 52) is the
entanglement time of the KG model. This overshoot behavior originated from
excessive ring elongation and slight shrinkage under fast elongational flow.
The excessive ring elongation was caused by the linear chains penetrating the
ring, whereby the linear chains spread inside the ring, and the ring was
pushed outward by the linear chains. When the linear chains penetrating the
ring were pulled out, the ring lost the force from the linear chains and
subsequently shrank. Excessive ring elongation was suppressed at $N_{\rm L}$
values $>$80, owing to entanglements among the linear chains. For $N_{\rm L}$
values $<$20, the linear chains penetrating the ring detached immediately
under the fast elongational flow, and they did not cause the excessive ring
elongation. Owing to our limited computational resources, we could not clarify
the $N_{\rm R}$-dependence of the stress-overshoot phenomena in our previous
work. Furthermore, multicyclic chains with $n$ rings exhibit more penetrations
than do monocyclic chains 32. Thus, we predicted that the stress-overshoot
phenomena would display the $n$-dependence in multicyclic/linear blends.

Our research group is also interested in the shape of multicyclic chains in
the steady state under biaxial elongational flow. As shown in our previous
work 51, the rings spread in the elongational plane, so that their steady
state was the open state. For multicyclic chains, the rings are also expected
to spread in the elongational plane, although the shape is thought to depend
on the connection of the rings. Both bond-connected and catenated multicyclic
chains can be considered 32. For a catenated multicyclic chain (catenane),
rings interpenetrate each other and do not have bond connections between them.
Because the rings in a catenane can slide and overlap with each other under
biaxial elongational flow, the steady-state shape will be close to that of
monocyclic chains, i.e., the open state. For a bond-connected multicyclic
chain, a junction part connects the rings. The steady state of such a chain
under biaxial elongational flow is significant, and the role of the junction
should be clarified. Therefore, in the present work, we focused on multicyclic
chains with bond connections between the rings.

In this study, we employed coarse-grained molecular dynamics simulations to
clarify the stress-overshoot phenomena in multicyclic/linear blends and
investigate the $N_{\rm R}$\- and $n$-dependences of these phenomena under
biaxial elongational flows. In particular, we considered symmetric bicyclic
and tricyclic chains with bond connections. Thus, we elucidated a new
mechanism that induces stress overshoot in multicyclic/linear blends, which is
significantly different from that observed in monocyclic/linear blends. This
observed topological transition mechanism comprises a morphological change in
multicyclic chains, which drastically changes the stresses on the rings in
multicyclic/linear blends. Herein, we describe the simulation methods and
results of our study. Consequently, we reveal that the stress-overshoot
phenomena in multicyclic/linear blends under biaxial elongational flow are
induced by topological transitions in multicyclic chains with structural
symmetry.

##  2 Methods

We investigated blends of multicyclic and linear chains. The polymer chains
considered in the present study are described using the KG model 52: A
multicyclic chain consists of $n$ rings, each ring composed of $N_{\rm R}$
particles and $N_{\rm R}$ bonds, while a linear chain consists of $N_{\rm L}$
particles and $N_{\rm L}-1$ bonds. Thus, a multicyclic chain of $n$ rings
comprises $nN_{\rm R}$ particles. For multicyclic chains with $n>$1, an extra
bond is required to connect the rings. Following our previous work 32, we
considered a bicyclic chain ($n=2$) to comprise one extra bond connecting two
rings and a tricyclic chain ($n=3$) to comprise three extra bonds connecting
three rings, as shown in Figure 1. For comparison, a monocyclic chain ($n=1$)
51 was also considered. When the numbers of rings and linear chains in the
system are $M_{\rm R}$ and $M_{\rm L}$, the total number of particles in the
system is $N_{\rm R}M_{\rm R}+N_{\rm L}M_{\rm L}$. To fix the number of rings
$M_{\rm R}$ in the system at the same $N_{\rm R}$, the number of multicyclic
chains was set to $M_{\rm R}/n$. Periodic boundary conditions were applied to
the LJ particle system with units of energy $\epsilon$, length $\sigma$, and
mass $m$. The time unit in this system was defined as
$\tau=\sigma(m/\epsilon)^{1/2}$. To realize uncrossable chains, the repulsive
LJ potential with the cutoff length $r_{\rm c}=2^{1/6}\sigma$ was applied to
each pair of particles, and the finite extensible nonlinear elastic (FENE)
potential, with spring constant $k=30\epsilon/\sigma^{2}$ and maximum bond
extent $R_{0}=1.5\sigma$, was applied between the bonded particles. The
density $\rho$ was set to 0.85$m/\sigma^{3}$. The temperature —
$T=1.0\epsilon/k_{\rm B}$, where $k_{\rm B}$ is the Boltzmann constant — was
controlled by the Langevin thermostat with friction coefficient
$\zeta=0.5m/\tau$. The positions and velocities of the particles were time-
integrated using the velocity-Verlet algorithm with the time step $\Delta
t=0.01\tau$. Hereafter, the unit symbols ($\epsilon$, $\sigma$, $m$, and
$\tau$) are not presented for simplicity if there is no confusion. Under a
deformation field with deformation rate tensor $\bm{\kappa}$, the particle
positions and velocities were evolved using the SLLOD algorithm 53. The
deformation rate tensor $\bm{\kappa}$ under biaxial elongational flow with a
constant Hencky strain rate $\dot{\varepsilon}$ is expressed as $\bm{\kappa}$
= diag($\dot{\varepsilon}$, $\dot{\varepsilon}$, $-2\dot{\varepsilon}$). On
applying an elongational deformation to the system for a long period, the
simulation box collapses under the conventional method, in which the
simulation box is placed parallel to the flow direction. Therefore, to avoid
such a collapse, we applied the gKR boundary conditions 40, 43, 44 to the
system under elongational flow.

Following our previous work 51, the multicyclic-to-linear chain blend weight
ratio was set to 1:10 ($(nN_{\rm R})(M_{\rm R}/n)$:$N_{\rm L}M_{\rm L}$ =
$N_{\rm R}M_{\rm R}$:$N_{\rm L}M_{\rm L}$), wherein the concentration of the
rings is less than the overlap concentration at equilibrium. Because the
$N_{\rm R}$-dependence was not considered in our previous work 51, we
reconsidered the monocyclic/linear blends and compared them with the
multicyclic/linear blends. In the present work, we considered 60 cases
originating from the combinations of $n$, $(N_{\rm R},M_{\rm R})$, and
$(N_{\rm L},M_{\rm L})$, where $n$ = {1, 2, 3}; $(N_{\rm R},M_{\rm R})$ =
{(40, 1536), (80, 768), (120, 512), (160, 384)}; and $(N_{\rm L},M_{\rm L})$ =
{(10, 61440), (20, 30720), (40, 15360), (80, 7680), (160, 3840)}. Because such
an exhaustive study requires vast computational resources, it is difficult to
investigate the various flow conditions. The present work focuses on stress-
overshoot phenomena under biaxial elongational flow with $\dot{\varepsilon}$ =
0.001$/\tau$, where monocyclic/linear blends exhibit overshoot behavior for
$\dot{\varepsilon}\tau_{\rm e}>$1 51. To prepare an equilibrium state for each
blend, we used HOOMD-blue 54 with GPU acceleration. Moreover, product runs
under biaxial elongational flow were performed using LAMMPS 55 with the UEFEX
package 46, which extends the UEF package 45 applicable to the Langevin
thermostat. Graphical analyses were performed using OVITO 56. Sufficiently
equilibrated states were produced in our previous studies 31, 32, 33, and
thus, we used these equilibrated data as the initial conditions.

Figure 1: Schematics of (a) monocyclic ($n$ = 1), (b) bicyclic ($n$ = 2), and
(c) tricyclic ($n$ = 3) chains. The black circles represent rings, red lines
represent extra bonds connecting the rings, and black points represent bond-
connected particles. Each black circle consists of $N_{\rm R}$ particles with
$N_{\rm R}$ bonds, whereas each red line represents a bond connecting two
black points.

##  3 Results and Discussion

###  3.1 Stress-overshoot behavior under biaxial elongational flow

This section summarizes the time-evolution behavior of the first normal stress
difference $(\sigma_{xx}+\sigma_{yy})/2-\sigma_{zz}$, which is a typical
characteristic variable under biaxial elongational flow. Here, $x$ and $y$ are
the elongational directions, and $z$ is the compressional direction. Because
the instantaneous stress values are noisy, owing to the thermal fluctuation,
the presented time-evolution data of the stress values were smoothed using the
Savitzky–Golay filter 57, which can smooth data without loss of the data
tendency.

Figure 2: Time evolution of the first normal stress difference on (a)
monocyclic/linear, (b) bicyclic/linear, and (c) tricyclic/linear blends under
biaxial elongational flow with a strain rate of 0.001$/\tau$. The blend ratio
of the cyclic/linear chains was 1/10. Blends with four different ring sizes
($N_{\rm R}$ = 40, 80, 120, and 160) were compared, while the linear chain
length $N_{\rm L}$ was fixed at 40. (d) Comparison of the stress-overshoot
behaviors in the three different blends with $N_{\rm R}$ = 160. For the
horizontal axis in (d) the Hencky strain $\varepsilon=t\dot{\varepsilon}$.

Figures 2a–c show the $N_{\rm R}$-dependences of stress-overshoot phenomena
under biaxial elongational flow with $\dot{\varepsilon}$ = 0.001$/\tau$ for
the monocyclic/linear, bicyclic/linear, and tricyclic/linear blends ($N_{\rm
L}$ = 40), respectively. In the monocyclic/linear blends, stress-overshoot
behavior is apparent for $N_{\rm R}$ = 160, as observed in our previous work
51. For $N_{\rm R}$ = 120, there is a small peak after the steep rise in the
first normal stress difference, while for $N_{\rm R}$ = 40 and 80, no stress
overshoot is observed. Because more than two chains penetrate a ring for
$N_{\rm R}$ = 120 and 160 31, the linear chains expand the ring outwardly
under biaxial elongational flow. On the other hand, for $N_{\rm R}$ = 40 and
80, some rings undergo single-chain penetration while many rings display no
penetration. In this situation, the force that pushes the ring outward is
weak. Therefore, there is no stress-overshoot behavior for $N_{\rm R}$ = 40
and 80 in the monocyclic/linear blends.

In the multicyclic/linear blends, for $n$ = 2 and 3, we observed the stress-
overshoot behaviors for $N_{\rm R}$ = 80, 120, and 160 but not for $N_{\rm R}$
= 40, reflecting whether the multicyclic chains have multiple-chain
penetration at the equilibrium state 32. Surprisingly, the stress overshoot
observed in the multicyclic/linear blends is much steeper than that in the
monocyclic/linear blends. Figure 2d compares the stress-overshoot behaviors,
at $N_{\rm R}$ = 160, of the monocyclic/linear and multicyclic/linear blends.
Here, the horizontal axis is replaced with the Hencky strain
($\varepsilon=t\dot{\varepsilon}$) for convenience in later discussion. The
stress-overshoot peak appears at $\varepsilon\sim$8 for the monocyclic/linear
blends and $\sim$4 for the multicyclic/linear blends. The peak strain values
were estimated from the least-squares fitting to a quadratic function as 8.5
for $n$ = 1, 4.6 for $n$ = 2, and 4.4 for $n$ = 3 (Figure 2d). Although the
tricyclic/linear blend presented a smaller peak strain value than that of the
bicyclic/linear blend, the difference is within the statistical error margin.
Moreover, Figure 2d shows that the steady-state value of the first normal
stress difference decreases with increasing $n$. These findings indicate the
different origins of the stress-overshoot behavior in the monocyclic/linear
and multicyclic/linear blends, the origin of which is discussed in Sections
3.2 and 3.3.

Figure 3: Decomposed stress evolution on (a) monocyclic/linear (1c-R160/L40),
(b) bicyclic/linear (2c-R160/L40), and (c) tricyclic/linear (3c-R160/L40)
blends under biaxial elongational flow with a strain rate of 0.001$/\tau$. The
total stress is decomposed into cyclic- and linear-chain stresses.

Figures 3a–c show the contributions from the cyclic and linear chains to the
stress-overshoot behaviors at $N_{\rm R}$ = 160 and $N_{\rm L}$ = 40 for the
monocyclic/linear, bicyclic/linear, and tricyclic/linear blends, respectively,
under biaxial elongational flow with a strain rate of 0.001$/\tau$. The total
stress was decomposed into the partial stresses of the cyclic and linear
chains. To obtain the partial stresses, the stress on each particle was
calculated, and the obtained particle stresses were summed up separately for
the cyclic and linear chains. In the early stage for $t<$10${}^{3}\tau$, the
linear-chain stress is dominant, while when the Hencky strain
($\varepsilon=t\dot{\varepsilon}$) exceeds 1.0 ($t$ = 10${}^{3}\tau$ in the
case of $\dot{\varepsilon}$ = 0.001$/\tau$) the cyclic-chain stress exceeds
the linear-chain stress. Notably, stress-overshoot behavior is observed in
both the cyclic- and linear-chain stresses. The peaks for these two stresses
correspond to each other and appear at $\varepsilon\sim$8 in the
monocyclic/linear blend and $\sim$4 in the multicyclic/linear blends. These
results indicate that as expected, the stress-overshoot behavior is caused by
the cooperative dynamics between the cyclic and linear chains.

The rapid increase for $1<\varepsilon<4$ in the multicyclic/linear blends
originates from the difference in the end-to-end distance between the
monocyclic and multicyclic chains, where the end-to-end distance of a ring is
defined as the distance between two particles separated by $N_{\rm R}/2-1$
bonds along the chain. The end-to-end distances of the multicyclic chains for
$n$ = 2 and 3 are defined by two end particles on two different rings in a
multicyclic chain. Here, an end particle on a ring in a multicyclic chain is a
particle separated from a bond-connected particle by $N_{\rm R}/2-1$ bonds
along the chain (Figure 1). When the rings expand under biaxial elongational
flow, the end-to-end distance of the multicyclic chain is longer than that of
the monocyclic chain, owing to the extra ring. In addition, the multicyclic
chains have a branch point, while the monocyclic chain does not. In analogy
with the difference between stars and linear chains, the branch point slows
down the dynamics of multicyclic chains 12, 58, 59. Because of the longer end-
to-end distance and the presence of a branch point in the multicyclic chains,
the relaxation time is longer in the multicyclic chains than in the monocyclic
chain. Thus, at the same strain rate, the multicyclic chains have a higher
Rouse Weissenberg number ($Wi_{\rm R}=\dot{\varepsilon}\tau_{\rm R}$, where
$\tau_{\rm R}$ is the Rouse relaxation time) than that of the monocyclic
chain. The Rouse Weissenberg number indicates the strength of nonlinearity
under elongational flow, and thus, strain hardening is steeper in the
multicyclic/linear blends than in the monocyclic/linear blend.

The reason why the overshoot peak appears earlier for the multicyclic/linear
blends than for the monocyclic/linear blends remains unclear. However, a
possible reason was revealed by molecular shape analyses, as discussed later.
Notably, the slopes of the reduction process are near-identical for the
bicyclic/linear and tricyclic/linear blends (Figure 2d); however, the steady
state appears later for the latter blend. This implies that the
bicyclic/linear and tricyclic/linear blends share the same reduction
mechanism; however, the reduction time is longer for the latter blend. This
further reduction observed in the tricyclic linear blend is also discussed in
Section 3.3.

Figure 4: $N_{\rm L}$-dependence of the stress evolution in the (a)
monocyclic/linear, (b) bicyclic/linear, and (c) tricyclic/linear blends under
biaxial elongational flow with a strain rate of 0.001$/\tau$. Five different
linear chain lengths ($N_{\rm L}$ = 10, 20, 40, 80, and 160) were compared,
while the ring size $N_{\rm R}$ was fixed at 160.

Figures 4a–c show the $N_{\rm L}$-dependences of the time-evolution behaviors
of the first normal stress difference under biaxial elongational flow with
$\dot{\varepsilon}$ = 0.001$/\tau$ for the monocyclic/linear, bicyclic/linear,
and tricyclic/linear blends ($N_{\rm R}$ = 160), respectively. Although the
data of the monocyclic/linear blends shown in Figure 4a are the same as those
presented in our previous work 51, we replotted them for comparison with the
data of the multicyclic/linear blends. The data revealed that the $N_{\rm
L}$-dependencies of the monocyclic/linear and multicyclic/linear blends are
significantly different. Specifically, the stress overshoot is only
significant at $N_{\rm L}$ = 40 in the monocyclic/linear blends, whereas sharp
stress overshoot is observed for all $N_{\rm L}$ values in the
multicyclic/linear blends.

In the monocyclic/linear blends, strain hardening (nonlinear increase for
$\varepsilon>$1; $t>$10${}^{3}\tau$ for $\dot{\varepsilon}$ = 0.001$/\tau$)
occurs in both the long and short chains. The rings expand under biaxial
elongational flow with $\dot{\varepsilon}$ = 0.001$/\tau$, regardless of the
$N_{\rm L}$ value. The overshoot when $N_{\rm L}$ = 40 affords the most
notable behavior because the ring bond lengths, which are stretched rather
than in equilibrium, are maximized when $N_{\rm L}$ = 40. These findings,
which are reported in our previous work 51, occur because the short linear
chains ($N_{\rm L}<$20) are not sufficient to extend the bonds of the rings,
whereas the long linear chains ($N_{\rm L}>$80) disturb the extra ring
extension, owing to the entanglement network consisting of linear chains.

In contrast, the multicyclic linear blends exhibited stress-overshoot
behavior, regardless of the $N_{\rm L}$ value. Comparison of the
bicyclic/linear and tricyclic/linear blends revealed that the overshoot
becomes more significant as $n$ increases. Because in multicyclic/linear
blends this overshoot is observed even in short linear chains, we deduced that
the overshoot is not caused by bond elongation on the rings. Moreover, this
overshoot is not affected by the entanglements of the long linear chains.
Thus, evidently, the overshoot mechanism in multicyclic/linear blends is
different from that in monocyclic/linear blends, which origin is the extra
ring elongation followed by slight ring shrinkage. Notably, the overshoot
displays $N_{\rm R}$-dependency, as shown in Figure 2. The overshoot in
multicyclic/linear blends is caused by the linear-chain penetration, in
particular multiple-chain penetrations, similar to those observed in the
monocyclic/linear blends.

The above results revealed that the overshoot behaviors in multicyclic/linear
blends are phenomena in which multiple linear chains penetrate and expand the
ring; however, overshoot occurs without extra bond elongation and shrinkage of
the ring. It is considered that the origin of multicyclic/linear blends is
different from that of monocyclic/linear blends. In the case of
multicyclic/linear blends, we established that molecular morphology analyses
should be performed to investigate the cause of the overshoot behavior in more
detail. The next section focuses on bicyclic/linear blends for morphological
analysis.

###  3.2 Ring-shape analysis on bicyclic chains under biaxial elongational
flow

This section discusses the cause of overshoot arising at a strain
$\varepsilon$ of $\sim$4 in multicyclic/linear blends through molecular
morphology analysis. We first focus on the behavior of bicyclic chains in a
bicyclic/linear blend, with $N_{\rm R}$ = 160 and $N_{\rm L}$ = 40, under
biaxial elongational flow with $\dot{\varepsilon}$ = 0.001$/\tau$. Figure 5
shows superimposed snapshots of the bicyclic chains at each strain, which were
captured by moving the bicyclic chains in the system so that their mass
centers overlapped at the origin of the coordinates. The value of the gyration
radius $R_{\rm g}$ of the bicyclic chains, presented in each snapshot, is an
average value of all the bicyclic chains in the system. Each molecule is color
coded according to its identification number. The red thick molecule displayed
the closest $R_{\rm g}$ value to that of the mean shape in the system. Thus,
we concluded that the shape of the red thick molecule best approaches the mean
shape. When the strain is equal to zero, the mean shape of the bicyclic chain
is the equilibrium coiled state. As the strain increases, the coiled state of
the bicyclic chain is compressed in the $z$ direction and radially spreads in
the $xy$ plane. The two rings open and broaden up to $\varepsilon$ = 4.
Surprisingly, the mean shapes at $\varepsilon$ = 6, 8, and 10 represented a
tadpole-like shape of the bicyclic chain, where the hole in one of the rings
is tightly closed to form a double-folded string. On the other hand, for the
monocyclic chains, this hole remained open under biaxial elongational flow,
and such a closed ring was not observed in our previous work (see Figure 6 in
Ref. 51). Furthermore, because the morphological (open-to-closed) transition
of the ring occurs when the strain in the bicyclic chains is $>$4, we supposed
that this transition is strongly related to the overshoot behavior.

Figure 5: Strain-dependent mean shape of the bicyclic chains in the
bicyclic/linear blends (2c-R160/L40) under biaxial elongational flow with a
strain rate of 0.001$/\tau$. The bicyclic chains are presented and viewed from
the $z$-axis (compression direction). The mass center of each bicyclic chain
is shifted to the origin (0,0,0), and the average value of the gyration radius
$R_{\rm g}$ of the bicyclic chains is presented with the strain value. The red
thick bicyclic chain in each snapshot has the closest value of $R_{\rm g}$ to
its average value at each strain.

To observe in detail the changes in the ring morphology, we next introduced
two ring internal lengths, $l_{1}$ and $l_{2}$. Ring internal length $l_{1}$
is the distance between a bond-connected (first) particle and an end (($N_{\rm
R}$/2)-th) particle on the same ring, separated by $N_{\rm R}/2-1$ bonds along
the ring from the bond-connected particle, while $l_{2}$ is the distance
between the ($N_{\rm R}$/4)-th and (3$N_{\rm R}$/4)-th particles. Notably, the
gyration tensor is generally used to characterize the morphology of a
molecule. However, attempts to distinguish between the open and closed states
using the gyration tensor failed. This is because the gyration tensor
approximates a molecule as an ellipsoidal shape, and the eigenvalues of the
gyration tensor are therefore insensitive to the open and closed states of the
rings. Thus, we used $l_{1}$ and $l_{2}$ to capture the open and closed states
under biaxial elongational flow.

Figure 6: Strain-dependent scatter plots of $l_{1}$ versus $l_{2}$ in the
bicyclic/linear blend (2c-R160/L40) under biaxial elongational flow with a
strain rate of 0.001$/\tau$. Ring internal length $l_{1}$ is the distance
between a bond-connected (first) particle and an end (($N_{\rm R}$/2)-th)
particle on the same ring, separated by $N_{\rm R}/2-1$ bonds along the ring
from the bond-connected particle, while $l_{2}$ is the distance between the
($N_{\rm R}$/4)-th and (3$N_{\rm R}$/4)-th particles.

Figure 6 shows scatter plots of $l_{1}$ versus $l_{2}$ for each strain: When
$\varepsilon$ = 0, the points are distributed at approximately $l_{1}$ =
$l_{2}$ = 10. As the strain increases ($\varepsilon<$2), the points are widely
dispersed, while at $\varepsilon$ = 3, a concentrated region at $60<l_{1}<80$
and $l_{2}<$3 is observed. With the further increase in the strain
($\varepsilon>$4), the distribution is divided into two regions, namely a
dense region ($60<l_{1}<80$ and $l_{2}<$3) comprising closed rings and a low-
density region, in which the open rings are widely distributed. Because the
dense region contains rings with a tightly constricted center, $l_{1}$ and
$l_{2}$ were not sufficient to determine the closed-ring state. Thus, to
determine the closed-ring state more precisely, we introduced auxiliary ring
internal lengths, $l_{3}$ and $l_{4}$, where $l_{3}$ is the distance between
the ($N_{\rm R}$/8)-th and (7$N_{\rm R}$/8)-th particles and $l_{4}$ is the
distance between the (3$N_{\rm R}$/8)-th and (5$N_{\rm R}$/8)-th particles.

Figure 7: Time-evolution behaviors of four ring internal lengths
($l_{1}$–$l_{4}$) for two rings on molecules with Mol.ID = 1 (a), 11 (b), and
37 (c) in the bicyclic/linear blend (2c-R160/L40) under biaxial elongational
flow with a strain rate of 0.001$/\tau$. (d) Schematic explanations of
$l_{1}$–$l_{4}$. The Mol.ID is the molecule identification number, while the
RingID represents one of the two rings in a bicyclic chain.

Figure 6 reveals the presence of two types of ring shapes, namely open- and
closed-ring states. To characterize the morphology of the bicyclic chains
under biaxial elongational flow, we analyzed four ring internal lengths
($l_{1}$–$l_{4}$) for each ring in the bicyclic/linear blend (2c-R160/L40)
under biaxial elongational flow. The time-evolution behaviors of
$l_{1}$–$l_{4}$ for molecule identification numbers (Mol.IDs) 1, 11, and 37
are presented in Figures 7a–c, respectively, wherein the RingID represents one
of the two rings in a bicyclic chain. In addition, the schematic explanation
of $l_{1}$–$l_{4}$ is presented in Figure 7d. The $l_{1}$ values for both
rings monotonically increase for $0<t<10,000$ and then represent steady states
for $t>$10,000. In Figures 7a–c, the $l_{1}$ behaviors are similar in all
three graphs, while at the steady state of $l_{1}$ ($t>$10,000) the behaviors
of $l_{2}$–$l_{4}$ in the three graphs exhibit some differences. Figure 7a
shows the most typical case in which, in the steady state, one of the rings
has small $l_{2}$–$l_{4}$ values indicating a slightly open ring, while the
other ring has large $l_{2}$–$l_{4}$ values indicating a fully open ring. A
total of 136 out of 192 samples exhibited similar behaviors. For $t>$10,000,
the respective estimated means and standard deviations are 1.43 and 0.39 for
$l_{2}$, 1.35 and 0.39 for $l_{3}$, and 1.57 and 0.5 for $l_{4}$ on the
slightly open ring and 40.6 and 8.9 for $l_{2}$, 29.5 and 5.6 for $l_{3}$, and
31.8 and 5.3 for $l_{4}$ on the fully open ring. Although the values of
$l_{2}$–$l_{4}$ for the slightly open ring sometimes reached a value of 3, as
shown in Figure 7a, the ring tended to be closed. This implies that the
slightly open ring was stable at $l_{2}$–$l_{4}$ values $<$3\. A transition
between the fully and slightly open rings was sometimes observed in the steady
state, as shown in Figure 7b. Moreover, a rare example of alternating the
fully and slightly open rings is shown in Figure 7c. From these transitions,
we could roughly estimate the lifetime of the steady-state morphology of the
bicyclic chain under biaxial elongational flow. When counting the number of
transitions, we ignored any instantaneously large fluctuations, which can be
interpreted as precursory phenomena before the transition. Thus, we determined
that one transition occurred when the values of $l_{2}$–$l_{4}$ changed from
$<$3 to $>$15 (or vice versa) and then stabilized for several time steps.
Values of 3 and 15 were selected by three standard deviations from the mean
value of $l_{4}$ for the slightly open ring and fully open ring, respectively.
Figure 7b illustrates one transition, while in Figure 7c the transition occurs
twice. For a total of 192 samples in the current system, the transition was
observed once in 38 samples and twice in 18 samples, while 136 samples
presented no transitions in the range $10,000<t<100,000$. Thus, the
probability of a transition being observed at $10,000<t<100,000$ was
approximately 0.39. The lifetime was approximately 230,000$\tau$ ($\approx$
90,000/0.39), which is sufficiently long under biaxial elongational flow with
$\dot{\varepsilon}$ = 0.001$/\tau$ (Hencky strain = 230). Hereafter, a ring is
defined as being in the closed-ring state when it satisfies the conditions
$l_{1}>$60, $l_{2}<$3, $l_{3}<$3, and $l_{4}<$3, labeled as the closed-ring
condition; otherwise, the ring is considered to be in an open-ring state.
According to this definition, the rings in equilibrium are the open-ring
state. A slightly open ring with a large $l_{1}$ is regarded as the closed-
ring state, accounting for a ring transiently opened by thermal fluctuations.
Notably, the condition $l_{1}>$60 depends on $N_{\rm R}$, whereas the other
conditions are independent of $N_{\rm R}$.

Figure 8: Appearance frequency versus strain in the bicyclic/linear blend
(2c-R160/L40) under biaxial elongational flow with a strain rate of
0.001$/\tau$. Three typical states are observed in the bicyclic chains: open-
open, open-closed, and closed-closed states.

In the case of bicyclic chains, three combinations (open-open, open-closed,
and closed-closed) were considered. Examining the strain dependency of the
frequency of these three combinations revealed some interesting facts, as
shown in Figure 8. As the strain increases, the number of open-open states
decreases, whereas those of open-closed and closed-closed states increase. At
$\varepsilon$ = 4, the magnitude relationship between the open-open and open-
closed is reversed. Regarding the contribution to the first normal stress
difference under biaxial elongational flow, that of the open ring is higher
than that of the closed ring. This is because the former has the force to push
the ring outward, owing to the linear chains penetrating it, whereas the
closed ring does not. Thus, the morphological change of the ring causes stress
overshoot.

Figure 9: Strain-dependent distribution of the average bond lengths on the
open and closed states of the rings in the bicyclic/linear blends
(2c-R160/L40) under biaxial elongational flow with a strain rate of
0.001$/\tau$. The bond lengths are averaged for one ring.

Figure 9 shows the relationship between the state of the ring and its bond
length for each strain; the bond lengths are averaged for each ring. At the
initial state ($\varepsilon$ = 0), all rings are in the open state, and the
average bond length in each ring is $<$1.0. This is because the equilibrium
bond length of the KG model is 0.97. As the strain increases, the average bond
length also increases. Moreover, at $\varepsilon$ = 2, closed states begin to
appear, until for $\varepsilon>$4, the closed states are distributed within
the range where the average bond length is $<$1.0. This implies that the bond
in the closed state is in a relaxed state. However, many bonds in the open
state have a bond length that is much larger than the equilibrium length.
Therefore, we supposed that the morphological transition, from the open to
closed state, causes bond relaxation and a decrease in tension at the joint
connecting two rings.

The reason why the closed state does not occur in a single ring but in a
bicycle can be understood from the morphological difference between the two.
In the former, the center of the molecule is the void of the ring, whereas in
the latter, the center is the joint connecting the two rings. Because the
molecules are expanded in the $xy$ plane under biaxial elongational flow,
single rings are isotropically expanded, whereas in a bicyclic chain, the
individual rings spread radially around the joint connecting two rings. In
addition, under elongational flow, the farther the molecule is from its
center, the greater is the deformation. Therefore, in the case of a bicyclic
chain, the holes of both rings tend to close, whereas the linear chains
penetrating the rings disturb ring closure. When the linear chains penetrating
one of the rings disappear, the ring without linear chain penetration
approaches a double-folded string shape, which is the characteristic (a non-
obstacle-enclosing and non-ramified) shape at equilibrium 11. A double-folded
string shape has also been considered in nonlinear shear flow in the shear
slit model 29, 30. Under biaxial elongational flow, the double-folded string
is confined by the surrounding chains and is highly stretched, whereas the
orientation is random in the elongational plane. Once linear chains
penetrating a ring are pulled out, the ring loses the tension spread outward
by the penetration of the linear chain, and then forms a double-folded string
shape. If either of the two rings forms a double-folded string first, the
balance between them is broken, and the joint relaxes. As the tension at the
joint weakens, the remaining ring behaves as a single ring and stabilizes in
an expanded state. From the above considerations, we concluded that a
morphological transition causes the stress-overshoot behavior of the
multicycle/linear blends, and the open-closed state in a bicycle is dominant
under biaxial elongational flow with $\dot{\varepsilon}$ = 0.001$/\tau$.

The morphological transition in bicyclic chains can be mathematically
explained in terms of the topology. Thus, a bicyclic chain with two open rings
(open-open state) is regarded as genus 2. Similarly, the open-closed state is
regarded as genus 1, and the closed-closed state as genus 0. Specifically, the
number of open rings in a multicyclic chain is regarded as the value of its
genus. This is not just wordplay but also a physically meaningful
interpretation. As previously discussed, an open ring allows chain
penetration, whereas a closed ring inhibits it. The open ring is pushed
outward by the chains penetrating it, whereas the closed ring does not
experience such a force. This implies that the topological transition from the
open- to closed-ring state drastically changes the stress of the ring. It is
therefore better to consider stress-overshoot behavior as a difference in the
topology rather than simply a morphological change. Notably, the maximum value
of the genus in a bicyclic chain is 2. Although an open ring can have several
holes depending on its morphology, the value of the genus was evaluated as 1
in this study. Therefore, a tricyclic chain can assume a state ranging from
genus 0 to genus 3. To further understand the topological transition,
tricyclic chains are analyzed in the next section.

###  3.3 Topological transition in a tricyclic/linear blend under biaxial
elongational flow

As shown in the previous section, the morphological changes in bicyclic chains
induced stress-overshoot behavior. This phenomenon can be explained from the
topology of the bicyclic chains. Thus, in this section, we analyze
tricyclic/linear blends under biaxial elongational flows in terms of topology.
A tricyclic chain has four topologically different states: open-open-open,
open-open-closed, open-closed-closed, and closed-closed-closed. These four
states can be classified by their genus. In this study, we regarded the number
of open rings in a multicyclic chain as the value of the genus, and thus, a
tricyclic chain is expected to represent four genera ranging from 0 to 3.
Schematic images of the four genera are shown in Figure 10a.

Figure 10: (a) Schematic images of the four genera. (b) Appearance frequency
versus strain in the tricyclic/linear blend (3c-R160/L40) under biaxial
elongational flow with a strain rate of 0.001$/\tau$. Four typical states are
classified by four genera: genus 3 (open-open-open), genus 2 (open-open-
closed), genus 1 (open-closed-closed), and genus 0 (closed-closed-closed). (c,
d) Mean shapes of the tricyclic chains at $\varepsilon$ = 4 and 10.

We analyzed the morphologies of the tricyclic chains in the tricyclic/linear
blend 3c-R160/L40, under biaxial elongational flow with $\dot{\varepsilon}$ =
0.001$/\tau$. Based on the analyses in the previous section, the closed state
of the ring is specified by the closed-ring condition $l_{1}>$60, $l_{2}<$3,
$l_{3}<$3, and $l_{4}<$3\. Figure 10b shows the frequency of the four
topologies versus strain in the tricyclic/linear blend under biaxial
elongational flow. Similar to that in Figure 8, the genus 1 state displays the
highest percentage in the steady state ($\varepsilon>$10). This is because the
single open state is stabilized under biaxial elongational flow, as previously
discussed. Around the stress-overshoot peak ($\varepsilon$ = 4), the genus 2
state is the most populous. Reflecting this, the mean shape of the tricyclic
chains at $\varepsilon$ = 4 (Figure 10c) represents the genus 3 state;
however, with one of the rings closing. Because the state around $\varepsilon$
= 4 is transient, it is probable that a state in the middle of the transition
from genus 3 to genus 2 was observed in this case. For the tricyclic chain,
the genus 1 state is again a tadpole-like in shape, with the two closed rings
tightly stuck together, as shown in Figure 10d. Further reduction in the first
normal stress difference in Figure 2d is also understood from the topology.
Both the bicyclic and tricyclic chains prefer the steady state of genus 1
under biaxial elongational flow. Specifically, the bicyclic chain closes one
ring, whereas the tricyclic chain closes two rings in the steady state.
Therefore, the stress reduction in the tricyclic/linear blend by the open-to-
closed transition is twice that of the bicyclic/linear blend.

Figure 11: Four typical topological states of tricyclic chains at
$\varepsilon$ = 10 in the tricyclic/linear blend (3c-R160/L40) under biaxial
elongational flow with $\dot{\varepsilon}$ = 0.001$/\tau$: (a) genus 3, (b)
genus 2, (c) genus 1, and (d) genus 0. Molecules are color coded according to
the molecule identification number, ranging from 1 to 128. The gyration radius
$R_{\rm g}$ of each molecule is also presented. The particle diameter was set
to 3, accounting for the closed-ring condition for $l_{2}$ to $l_{4}$
($l_{2}<$3, $l_{3}<$3, and $l_{4}<$3).

Furthermore, the typical examples of the topological states on the tricyclic
chains (genera 3 to 0) at $\varepsilon$ = 10 are summarized in Figures 11a–d,
respectively. The diameter of the particles increased up to a value of 3,
corresponding to the closed-ring condition for $l_{2}$–$l_{4}$ ($l_{2}<$3,
$l_{3}<$3, and $l_{4}<$3). The figure for genus 1 in (c) corresponds to the
red thick molecule shown in Figure 10d. Notably, a double-tail tadpole shape,
in which two closed rings are unattached, is also observed in genus 1,
although the tadpole shape shown in (c) is the most populous. Topological
classification was successful, except for genus 3, where at $\varepsilon$ =
10, only one case was classified as belonging to genus 3. Although this case
is very close to genus 0, the three rings break the closed-ring condition
$l_{2}>$3, as evidenced by the small gap in each ring in Figure 11a. Genus 3
was considered unstable at $\varepsilon$ = 10. Genera 2 and 1, clearly
comprise opened and tightly closed rings. The genus 2 state is a dragonfly-
like in shape, while the genus 0 state is in the shape of a three-arm star. As
shown in Figure 10b, the populations of these topologies in the steady state
are of the order genus 1 $>$ genus 2 $>$ genus 0 $>$ genus 3. Although the
single open-ring state (genus 1) is the most stable under biaxial elongational
flow, as previously discussed, the state of genus 2 accounts for approximately
10% at the large strain region. It should be reminded that the state of genus
2 was suppressed in the bicyclic/linear blend. This indicates that the closure
of one ring contributes to the stabilization of genus 2. If multicyclic
chains, in which more than three rings are connected, were to be considered, a
stable genus 3 state might also be observed under biaxial elongational flow.
To verify this speculation, however, further investigations are necessary.

The present study found that stress overshoot of multicyclic/linear blends
under biaxial elongational flow is induced by the topological transition in
symmetric multicyclic chains; however, whether asymmetric multicyclic chains
exhibit stress overshoot remains an open query. Thus, we performed additional
computations on asymmetric-tricyclic chains and tricyclic chains with two
symmetric rings, presented in Sections 3.4 and 3.5, respectively. These
additional studies clarify that the symmetry of rings in a multicyclic chain
plays a key role to produce stress overshoot under biaxial elongational flow.

###  3.4 Asymmetric tricyclic chains

In the present study, we investigated symmetric-multicyclic/linear blends
under biaxial elongational flow and observed the stress overshoot induced by
the topological transition in multicyclic chains with structural symmetry. To
clarify the stress overshoot observed in asymmetric multicyclic chain, two
asymmetric tricyclic chains, each composed of three different rings are
discussed here, namely, R160-R120-R80 ($N_{\rm R}$ = 160, 120, and 80) and
R200-R160-R120 ($N_{\rm R}$ = 200, 160, and 120). After preparing the
asymmetric-tricyclic/linear blends with a 1:10 weight blend ratio, the blends
were sufficiently equilibrated for 1.0 $\times$ 109 steps. Subsequently, a
biaxial elongational flow with $\dot{\varepsilon}$ = 0.001$/\tau$ was applied.
The obtained stress–strain curves are presented in Figure 12a and reveal that
the asymmetric-tricyclic/linear blends do not exhibit stress overshoot. The
mean shapes of R160-R120-R80 and R200-R160-R120 at $\varepsilon$ = 10 are
presented in Figures 12b and 12c, respectively. In both cases, the largest
rings are in open-ring states, while the middle-sized and smallest rings are
in closed-ring states. The steady-state behaviors are similar to those
observed for the symmetric-tricyclic chains. Because the open rings (the
largest rings) mainly contributed to the steady-state stress, the steady-state
value of R160-R120-R80/L40 corresponds to that of 3c-R160/L40. On the other
hand, the steady-state value of R200-R160-R120/L40 is supposed to correspond
to that of 3c-R200/L40, although this confirmation is beyond the scope of this
study. For the asymmetric-tricyclic chains, the smaller rings preferentially
adopt the closed-ring state under biaxial elongational flow (Figure 12d)
because the number of linear chain penetrations is smaller. In contrast, in
the symmetric-multicyclic chains, a ring takes the closed-ring state only when
there is a difference in the number of linear chain penetrations between the
interconnected rings. Otherwise, the rings in the symmetric-multicyclic chain
elongate until a difference in the number of penetrations occurs.
Specifically, the stress increases until a topological transition occurs.
Structural symmetry in multicyclic chains is an important factor for inducing
stress-overshoot behavior. Indeed, we supposed that a tricyclic chain composed
of two symmetric large rings and a small ring, may exhibit stress-overshoot
behavior. This hypothesis is verified in the next section.

Figure 12: (a) Comparison of the stress–strain curves of R160-R120-R80/L40 and
R200-R160-R120/L40, under biaxial elongational flow with a strain rate of
0.001$/\tau$, with those of 1c-R160/L40 and 3c-R160/L40. (b, c) Mean shapes of
asymmetric tricyclic chains at $\varepsilon$ = 10 for R160-R120-R80 and
R200-R160-R120, respectively. (d) Plots of the number of closed rings against
strain for each ring. When a ring satisfied the conditions $l_{1}>$$l_{\rm
min}$, $l_{2}<$3, $l_{3}<$3, and $l_{4}<$3, it was deemed as being in the
closed-ring state. The value $l_{\rm min}$ was determined from the steady-
state behaviors of $l_{1}$, where $l_{\rm min}$ = 60 for $N_{\rm R}$ = 160, 40
for $N_{\rm R}$ = 120, and 20 for $N_{\rm R}$ = 80.

###  3.5 Tricyclic chains with two symmetric rings

The symmetric-tricyclic/linear blends exhibited stress-overshoot behaviors,
whereas the asymmetric-tricyclic/linear blends did not. If we consider a
tricyclic chain with two identical rings and one small ring, stress-overshoot
phenomena are expected. We investigated two tricyclic chains with two
symmetric rings, namely, R140-R140-R80 ($N_{\rm R}$ = 140, 140, and 80) and
R180-R180-R120 ($N_{\rm R}$ = 180, 180, and 120). The obtained stress–strain
curves of the two tricyclic chains under biaxial elongational flow with
$\dot{\varepsilon}$ = 0.001$/\tau$ are presented in Figure 13a. The tricyclic
chains comprising two symmetric rings presented stress-overshoot phenomena, as
expected. The mean shapes of R140-R140-R80 and R180-R180-R120 at $\varepsilon$
= 10 are shown in Figures 13b and 13c, respectively. The mean shapes represent
genus 1, and the symmetry of the two symmetric rings is broken. The steady-
state values of R140-R140-R80/L40 and R180-R180-R120/L40 are lower and higher,
respectively, than those of 3c-R160/L40, and depend on the $N_{\rm R}$ of an
open ring. Moreover, the value of the first normal stress difference at the
peak was larger for a larger $N_{\rm R}$, while for $N_{\rm R}<$160, the
strain at the peak was insensitive to $N_{\rm R}$. Further investigation is
required for large rings ($N_{\rm R}>$160). These results show that
multicyclic chains with symmetric rings exhibit stress-overshoot phenomena
induced by the topological transition in the symmetric multicyclic chains.
Thus, the symmetry of rings in a multicyclic chain is essential to produce
stress overshoot under biaxial elongational flow.

Figure 13: (a) Comparison of the stress–strain curves of R140-R140-R80/L40 and
R180-R180-R120/L40 under biaxial elongational flow with a strain rate of
0.001$/\tau$ with those of 1c-R160/L40 and 3c-R160/L40. (b, c) Mean shapes of
asymmetric tricyclic chains at $\varepsilon$ = 10 for R140-R140-R80 and
R180-R180-R120, respectively.

##  4 Summary and Future Outlook

We investigated multicyclic/linear blends under biaxial elongational flow
using coarse-grained molecular dynamics simulations. Our results revealed
stress-overshoot phenomena in the multicyclic/linear blends, which were
steeper than that previously reported for monocyclic/linear blends 51.
Molecular morphology analyses revealed a topological transition mechanism in
the symmetric-multicyclic chain. This transition is a morphological change
from the open- to closed-ring state. We observed that in the closed-ring
state, one ring was tightly closed to form a double-folded string. This
double-folded string state, which was in a relaxed state under biaxial
elongational flow, was not observed in the monocyclic chains. Because the
closed-ring state inhibits chain penetration, the topological transition
decreases the first normal stress difference under biaxial elongational flow.
We therefore concluded that topological transition induces stress-overshoot
phenomena. Comparison of the bicyclic/linear and tricyclic/linear blends
revealed that the steady-state value of the first normal stress difference in
the tricyclic/linear blends was smaller than that in the bicyclic/linear
blends. This difference was also explained in terms of the topological
differences between the bicyclic and tricyclic chains: Because the state of
genus 1 (single open-ring state) is the most stable and populous state, the
reduction in the first normal stress difference increases with the increasing
number of rings in the multicyclic chain. In the case of tricyclic/linear
blends, the state of genus 2 remains within a certain proportion, suggesting
that the closure of one ring stabilizes this state. A multicyclic chain with
more than three rings may represent a stable of genus 3 under biaxial
elongational flow. Whether topological transitions occur under different flow
conditions (e.g., shear flow and uniaxial elongational flow) remains an open
query. Further investigations are currently underway to address these
problems.

Stress overshoots were observed in the symmetric-multicyclic/linear blends. We
also investigated asymmetric tricyclic chains and tricyclic chains composed of
two symmetric rings and one small ring. The asymmetric tricyclic chains also
showed morphological changes between the open- and closed-ring states,
although the stress-overshoot behavior was inhibited in the asymmetric-
multicyclic/linear blends. This is because smaller rings preferentially adopt
closed-ring states. The blends of tricyclic chains with two symmetric rings
and linear chains, however, displayed the stress-overshoot behavior. When a
multicyclic chain has symmetric rings, there is no fixed order for the closed-
ring states to occur, and thus, structural symmetry in a multicyclic chain
causes the dilemma of deciding which ring to close. The strain on the rings
increases until one ring is closed. This closed ring then relaxes, so that the
ring stress is drastically decreased. Therefore, topological transition in
symmetric-multicyclic chains induces the stress-overshoot behavior.

In the present study, we solely focused on the single strain rate value
($\dot{\varepsilon}=0.001[1/\tau]$). The strain rate dependence is beyond the
scope of this study and will be investigated in the future. Furthermore, we
expect that the multicyclic/linear blends will present interesting phenomena
under the different flow patterns (uniaxial, planar, and shear flows). Because
the topological transition was caused by pulling out the linear chains
penetrating the rings, we can regard this transition as a variant of the
threading–unthreading transition 28. The closed-ring state is apparently
unthreaded by linear chains, and therefore, we did not investigate the number
of penetrations 31, 32, 51 in the present study. We expect that the
multicyclic/linear blends also show such a threading–unthreading transition
under uniaxial elongational flow. The topological transition, however, will be
difficult to observe under uniaxial elongational flow, because both the
threaded and unthreaded states will be tightly closed under such conditions.
Recently, O’Connor et al. discussed the blend ratio dependence of
monocyclic/linear blends under uniaxial elongational flow and revealed that
the overshoot was remarkable at a certain ring volume fraction where the
topological constraints of the composite entanglement network are maximized
50. Indeed, the blend ratio dependence of the multicyclic/linear blends is
also important; however, this will also be investigated in a future study.
Notably, the open-closed state in a bicyclic/linear blend and open-closed-
closed state in a tricyclic/linear blend under biaxial elongational flow
correspond to the tadpole-shaped polymers 60. Comparison with the tadpole-
shaped polymers is also a subject for future study.

In the present study, we considered a multicyclic chain in which multiple
rings are connected by a bond for a bicyclic chain and three bonds for a
tricyclic chain. As similar multicyclic chains, catenanes are also highly
interesting 61, 62, 63. In catenanes, interpenetration amongst multiple rings
disrupt ring closure under biaxial elongational flow. Therefore, in the case
of catenanes, the closed-ring state observed in the present work would be
suppressed. We supposed that the catenane system does not present a
topological transition. Recently, we discovered that a fluid-solid phase
transition occurs as the catenane ring size is reduced 64. It has also been
found that reducing the ring size increases the porosity. Thus, it is more
difficult for a catenane to adopt the closed-ring state if the ring size is
smaller. It is therefore expected that bond-connected multicyclic chains and
catenanes display different characteristics under biaxial elongational flow. A
comparison of the two materials is a topic for future research.

The present work suggests the possibility of tuning the physical properties by
controlling the topology. The phenomenon observed in this work comprised a
transition in which an open ring closes; however, the opposite process is also
possible. For example, a ring closed by assuming an associative interaction 65
can be opened by an external stimulus. Thus, a strategy to freely control the
open and closed states of a ring would be useful for energy conservation, and
the viscosity would be controlled by the topological transition. Moreover, we
expect that this strategy will be used as a new method to recycle materials by
tuning their topology. This work is just the beginning of the study of
topological transitions. Future research will broaden our knowledge of
topological transitions and bridge topological transitions to innovative
applications.

Finally, we address the experimental verification possibilities in the present
work. For the same ring size, the bicyclic/linear blends and the
tricyclic/linear blends exhibited steeper strain hardening than did the
monocyclic/linear blends. This behavior can be immediately useful as an
indicator of multicyclization methodology. Figure 7 shows that the open-closed
states account for approximately 10% at $\varepsilon$ = 2. This suggests that
the open-to-closed transition (or its transition process) can be observed by
neutron scattering 66. Under biaxial elongational flow, multicyclic chains
orient randomly in the $xy$ plane (elongational plane). The closed-ring state
has a rod-like shape, while the open-ring state displays a ring shape; thus,
the form factors between the two states are different. Neutron scattering can
therefore be used to detect and distinguish between the two states. The
overshoot peak appeared at $\varepsilon\sim$4 and $\sim$8 in the
multicyclic/linear and monocyclic/linear blends, respectively. Unfortunately,
these overshoot behaviors under biaxial elongational flow are not observable
with the current experimental technique, which is limited to $\varepsilon$ =
2.5 67, 68. However, we predict that future technological advances will
eventually allow us to observe the overshoot behaviors under biaxial
elongational flow, and thus, the overshoot of the multicyclic/linear blends
can be further studied. The present work suggests that overshoot behavior
appears at lower strain values when designing a method to connect the rings.
Topological transitions in multicyclic chains with structural symmetry will
play a key role in observing overshoot behavior under biaxial elongational
flow in experiments.

TM thanks Prof. T. Taniguchi, Prof. M. Sugimoto, Prof. J.-I. Takimoto, Prof.
S. K. Sukumaran, Prof. T. Uneyama, Prof. T. Honda, and Dr. Y. Tomiyoshi for
their fruitful discussions, comments, and encouragement. The authors thank
Prof. H. Jinnai, Prof. T. Satoh, and Prof. T. Deguchi for their support and
encouragement. TM and TK thank the collaboration program of the Advanced
Imaging and Modeling Center for Soft-materials (AIMcS) of Tohoku University.
For the computations in this work, the authors were partially supported by the
Supercomputer Center, Institute for Solid State Physics, University of Tokyo;
MASAMUNE-IMR at the Center for Computational Materials Science, Institute for
Materials Research, Tohoku University; Grand Chariot and Polaire at Hokkaido
University Information Initiative Center; Flow at Nagoya University
Information Technology Center; SQUID at Cybermedia Center, Osaka University;
Fugaku at RIKEN Center for Computational Science; the Joint Usage/ Research
Center for Interdisciplinary Large-scale Information Infrastructures (JHPCN);
and the High-Performance Computing Infrastructure (HPCI) in Japan: hp200048,
hp200168, hp210102, hp210132, hp220019, hp220104, hp220113, hp220114,
jh210035, and jh220038. This work was partly financially supported by JSPS
KAKENHI, Japan, Grant numbers JP18H04494, JP19H00905, JP20K03875, JP20H04649,
and JP21H00111 and JST CREST, Japan, Grant numbers JPMJCR1993 and JPMJCR19T4.
The authors thank Editage (www.editage.com) for the English language editing.

## References

  * Tezuka 2012 Tezuka, Y. Topological polymer chemistry for designing multicyclic macromolecular architectures. _Polymer Journal_ 2012, _44_ , 1159–1169, DOI: 10.1038/pj.2012.92
  * Jia and Monteiro 2012 Jia, Z.; Monteiro, M. J. Cyclic Polymers: Methods and Strategies. _Journal of Polymer Science Part A: Polymer Chemistry_ 2012, _50_ , 2085–2097, DOI: 10.1002/pola.25999
  * Isono et al. 2018 Isono, T.; Sasamori, T.; Honda, K.; Mato, Y.; Yamamoto, T.; Tajima, K.; Satoh, T. Multicyclic Polymer Synthesis through Controlled/Living Cyclopolymerization of $\alpha$,$\omega$-Dinorbornenyl-Functionalized Macromonomers. _Macromolecules_ 2018, _51_ , 3855–3864, DOI: 10.1021/acs.macromol.8b00355
  * Hossain et al. 2014 Hossain, M. D.; Lu, D.; Jia, Z.; Monteiro, M. J. Glass Transition Temperature of Cyclic Stars. _ACS Macro Letters_ 2014, _3_ , 1254–1257, DOI: 10.1021/mz500684v
  * Yan et al. 2018 Yan, Z.-C.; Hossain, M. D.; Monteiro, M. J.; Vlassopoulos, D. Viscoelastic Properties of Unentangled Multicyclic Polystyrenes. _Polymers_ 2018, _10_ , 973, DOI: 10.3390/polym10090973
  * Doi et al. 2022 Doi, Y.; Takano, A.; Takahashi, Y.; Matsushita, Y. Terminal Relaxation Behavior of Entangled Linear Polymers Blended with Ring and Dumbbell Shaped Polymers in Melts. _Rheological Acta_ 2022, DOI: 10.1007/s00397-022-01355-y, (published online)
  * McLeish 2002 McLeish, T. Polymer Without Beginning or End. _Science_ 2002, _297_ , 2005–2006, DOI: 10.1126/science.1076810
  * Kapnistos et al. 2008 Kapnistos, M.; Lang, M.; Vlassopoulos, D.; Pyckhout-Hintzen, W.; Richter, D.; Cho, D.; Chang, T.; Rubinstein, M. Unexpected Power-Law Stress Relaxation of Entangled Ring Polymers. _Nature Materials_ 2008, _7_ , 997–1002, DOI: 10.1038/nmat2292
  * Halverson et al. 2011 Halverson, J. D.; Lee, W. B.; Grest, G. S.; Grosberg, A. Y.; Kremer, K. Molecular Dynamics Simulation Study of Nonconcatenated Ring Polymers in a Melt. II. Dynamics. _The Journal of Chemical Physics_ 2011, _134_ , 204905, DOI: 10.1063/1.3587138
  * Pasquino et al. 2013 Pasquino, R. et al. Viscosity of Ring Polymer Melts. _ACS Macro Letters_ 2013, _2_ , 874–878, DOI: 10.1021/mz400344e
  * Klein 1986 Klein, J. Dynamics of Entangled Linear, Branched and Cyclic Polymers. _Macromolecules_ 1986, _19_ , 105–118, DOI: 10.1021/ma00155a018
  * Doi and Edwards 1986 Doi, M.; Edwards, S. F. _The Theory of Polymer Dynamics_ ; Clarendon press: Oxford, 1986
  * Obukhov et al. 1994 Obukhov, S. P.; Rubinstein, M.; Duke, T. Dynamics of a Ring Polymer in a Gel. _Physical Review Letters_ 1994, _73_ , 1263–1266, DOI: 10.1103/PhysRevLett.73.1263
  * Ge et al. 2016 Ge, T.; Panyukov, S.; Rubinstein, M. Self-Similar Conformations and Dynamics in Entangled Melts and Solutions of Nonconcatenated Ring Polymers. _Macromolecules_ 2016, _49_ , 708–722, DOI: 10.1021/acs.macromol.5b02319
  * Subramanian and Shanbhag 2008 Subramanian, G.; Shanbhag, S. Conformational Properties of Blends of Cyclic and Linear Polymer Melts. _Physical Review E_ 2008, _77_ , 011801, DOI: 10.1103/PhysRevE.77.011801
  * Lo and Turner 2013 Lo, W.-C.; Turner, M. The Topological Glass in Ring Polymers. _EPL_ 2013, _102_ , 58005, DOI: 10.1209/0295-5075/102/58005
  * Bernabei et al. 2013 Bernabei, M.; Bacova, P.; Moreno, A. J.; Narros, A.; Likos, C. N. Fluids of Semiflexible Ring Polymers: Effective Potentials and Clustering. _Soft Matter_ 2013, _9_ , 1287–1300, DOI: 10.1039/c2sm27199k
  * Tsalikis et al. 2016 Tsalikis, D. G.; Mavrantzas, V. G.; Vlassopoulos, D. Analysis of Slow Modes in Ring Polymers: Threading of Rings Controls Long-Time Relaxation. _ACS Macro Letters_ 2016, _5_ , 755–760, DOI: 10.1021/acsmacrolett.6b00259
  * Smrek et al. 2019 Smrek, J.; Kremer, K.; Rosa, A. Threading of Unconcatenated Ring Polymers at High Concentrations: Double-Folded vs Time-Equilibrated Structures. _ACS Macro Letters_ 2019, _8_ , 155–160, DOI: 10.1021/acsmacrolett.8b00828
  * Tsalikis and Mavrantzas 2014 Tsalikis, D. G.; Mavrantzas, V. G. Threading of Ring Poly(Ethylene Oxide) Molecules by Linear Chains in the Melt. _ACS Macro Letters_ 2014, _3_ , 763–766, DOI: 10.1021/mz5002096
  * Tsalikis and Mavrantzas 2020 Tsalikis, D. G.; Mavrantzas, V. G. Size and Diffusivity of Polymer Rings in Linear Polymer Matrices: The Key Role of Threading Events. _Macromolecules_ 2020, _53_ , 803–820, DOI: 10.1021/acs.macromol.9b02099
  * Mo et al. 2022 Mo, J.; Wang, J.; Wang, Z.; Lu, Y.; An, L. Size and Dynamics of a Tracer Ring Polymer Embedded in a Linear Polymer Chain Melt Matrix. _Macromolecules_ 2022, _55_ , 1505–1514, DOI: 10.1021/acs.macromol.1c02388
  * Roovers 1988 Roovers, J. Viscoelastic Properties of Polybutadiene Rings. _Macromolecules_ 1988, _21_ , 1517–1521, DOI: 10.1021/ma00183a049
  * Halverson et al. 2012 Halverson, J. D.; Grest, G. S.; Grosberg, A. Y.; Kremer, K. Rheology of Ring Polymer Melts: From Linear Contaminants to Ring-Linear Blends. _Physical Review Letters_ 2012, _108_ , 038301, DOI: 10.1103/PhysRevLett.108.038301
  * Parisi et al. 2020 Parisi, D.; Ahn, J.; Chang, T.; Vlassopoulos, D.; Rubinstein, M. Stress Relaxation in Symmetric Ring-Linear Polymer Blends at Low Ring Fractions. _Macromolecules_ 2020, _53_ , 1685–1693, DOI: 10.1021/acs.macromol.9b02536
  * Zhou et al. 2019 Zhou, Y.; Hsiao, K.-W.; Regan, K. E.; Kong, D.; McKenna, G. B.; Robertson-Anderson, R. M.; Schroeder, C. M. Effect of Molecular Architecture on Ring Polymer Dynamics in Semidilute Linear Polymer Solutions. _Nature Communications_ 2019, _10_ , 1753, DOI: 10.1038/s41467-019-09627-7
  * Zhou et al. 2021 Zhou, Y.; Young, C. D.; Lee, M.; Banik, S.; Kong, D.; McKenna, G. B.; Robertson-Anderson, R. M.; Sing, C. E.; Schroeder, C. M. Dynamics and Rheology of Ring-Linear Blend Semidilute Solutions in Extensional Flow: Single Molecule Experiments. _Journal of Rheology_ 2021, _65_ , 729–744, DOI: 10.1122/8.0000219
  * Borger et al. 2020 Borger, A.; Wang, W.; O’Connor, T. C.; Ge, T.; Grest, G. S.; Jensen, G. V.; Ahn, J.; Chang, T.; Hassager, O.; Mortensen, K.; Vlassopoulos, D.; Huang, Q. Threading–Unthreading Transition of Linear-Ring Polymer Blends in Extensional Flow. _ACS Macro Letters_ 2020, _9_ , 1452–1457, DOI: 10.1021/acsmacrolett.0c00607
  * Parisi et al. 2021 Parisi, D.; Costanzo, S.; Jeong, Y.; Ahn, J.; Chang, T.; Vlassopoulos, D.; Halverson, J. D.; Kremer, K.; Ge, T.; Rubinstein, M.; Grest, G. S.; Srinin, W.; Grosberg, A. Y. Nonlinear Shear Rheology of Entangled Polymer Rings. _Macromolecules_ 2021, _54_ , 2811–2827, DOI: 10.1021/acs.macromol.0c02839
  * Parisi et al. 2021 Parisi, D.; Kaliva, M.; Costanzo, S.; Huang, Q.; Lutz, P. J.; Ahn, J.; Chang, T.; Rubinstein, M.; Vlassopoulos, D. Nonlinear Rheometry of Entangled Polymeric Rings and Ring-Linear Blends. _Journal of Rheology_ 2021, _65_ , 695–711, DOI: 10.1122/8.0000186
  * Hagita and Murashima 2021 Hagita, K.; Murashima, T. Effect of Chain-Penetration on Ring Shape for Mixtures of Rings and Linear Polymers. _Polymer_ 2021, _218_ , 123493, DOI: 10.1016/j.polymer.2021.123493
  * Hagita and Murashima 2021 Hagita, K.; Murashima, T. Multi-Ring Configurations and Penetration of Linear Chains into Rings on Bonded Ring Systems and Polycatenanes in Linear Chain Matrices. _Polymer_ 2021, _223_ , 123705, DOI: 10.1016/j.polymer.2021.123705
  * Hagita et al. 2022 Hagita, K.; Murashima, T.; Ogino, M.; Omiya, M.; Ono, K.; Deguchi, T.; Jinnai, H.; Kawakatsu, T. Efficient Compressed Database of Equilibrated Configurations of Ring-Linear Polymer Blends for MD Simulations. _Scientific Data_ 2022, _9_ , 40, DOI: 10.1038/s41597-022-01138-3
  * Everaers et al. 2020 Everaers, R.; Karimi-Varzaneh, H. A.; Fleck, F.; Hojdis, N.; Svaneborg, C. Kremer-Grest Models for Commodity Polymer Melts: Linking Theory, Experiment, and Simulation at the Kuhn Scale. _Macromolecules_ 2020, _53_ , 1901–1916, DOI: 10.1021/acs.macromol.9b02428
  * Iwamoto et al. 2018 Iwamoto, T.; Doi, Y.; Kinoshita, K.; Takano, A.; Takahashi, Y.; Kim, E.; Kim, T.-H.; Takata, S.; Nagao, M.; Matsushita, Y. Conformations of Ring Polystyrenes in Semidilute Solutions and in Linear Polymer Matrices Studied by SANS. _Macromolecules_ 2018, _51_ , 6836–6847, DOI: 10.1021/acs.macromol.8b00934
  * Hagita and Murashima 2021 Hagita, K.; Murashima, T. Molecular Dynamics Simulations of Ring Shapes on a Ring Fraction in Ring-Linear Polymer Blends. _Macromolecules_ 2021, _54_ , 8043–8051, DOI: 10.1021/acs.macromol.1c00656
  * Hagita et al. 2022 Hagita, K.; Murashima, T.; Ebe, M.; Isono, T.; Satoh, T. Trapping Probabilities of Multiple Rings in End-Linked Gels. _Polymer_ 2022, _245_ , 124683, DOI: 10.1016/j.polymer.2022.124683
  * Hagita et al. 2022 Hagita, K.; Murashima, T.; Jinnai, H. Demonstration of Reinforcement in Polymer Composite with Rings Penetrating the Diamond-Lattice Network. _Polymer_ 2022, _243_ , 124637, DOI: 10.1016/j.polymer.2022.124637
  * Hagita et al. 2022 Hagita, K.; Murashima, T.; Ohkuma, T.; Jinnai, H. Ring-Filling Effect on Stress-Strain Curves of Randomly End-Linked Tetra-Arm Prepolymers. _Macromolecules_ 2022, _55_ , 6547–6561, DOI: 10.1021/acs.macromol.2c00451
  * Kraynik and Reinelt 1992 Kraynik, A. M.; Reinelt, D. A. Extensional Motions of Spatially Periodic Lattices. _International Journal of Multiphase Flow_ 1992, _18_ , 1045–1059, DOI: 10.1016/0301-9322(92)90074-Q
  * Baranyai and Cummings 1999 Baranyai, A.; Cummings, P. T. Steady State Simulation of Planar Elongation Flow by Nonequilibrium Molecular Dynamics. _The Journal of Chemical Physics_ 1999, _110_ , 42–45, DOI: 10.1063/1.478082
  * Matin et al. 2000 Matin, M. L.; Daivis, P. J.; Todd, B. D. Comparison of Planar Shear Flow and Planar Elongational Flow for Systems of Small Molecules. _The Journal of Chemical Physics_ 2000, _113_ , 9122–9131, DOI: 10.1063/1.1319379
  * Dobson 2014 Dobson, M. Periodic Boundary Conditions for Long-Time Nonequilibrium Molecular Dynamics Simulations of Incompressible Flows. _The Journal of Chemical Physics_ 2014, _141_ , 184103, DOI: 10.1063/1.4901276
  * Hunt 2015 Hunt, T. A. Periodic Boundary Conditions for the Simulation of Uniaxial Extensional Flow of Arbitrary Duration. _Molecular Simulation_ 2015, _42_ , 347–352, DOI: 10.1080/08927022.2015.1051043
  * Nicholson and Rutledge 2016 Nicholson, D. A.; Rutledge, G. C. Molecular Simulation of Flow-Enhanced Nucleation in N-Eicosane Melts under Steady Shear and Uniaxial Extension. _The Journal of Chemical Physics_ 2016, _145_ , 244903, DOI: 10.1063/1.4972894
  * Murashima et al. 2018 Murashima, T.; Hagita, K.; Kawakatsu, T. Elongational Viscosity of Weakly Entangled Polymer Melt via Coarse-Grained Molecular Dynamics Simulation. _Nihon Reoroji Gakkaishi_ 2018, _46_ , 207–220, DOI: 10.1678/rheology.46.207, https://github.com/t-murash/LAMMPS-UEFEX
  * O’Connor et al. 2018 O’Connor, T. C.; Alvarez, N. J.; Robbins, M. O. Relating Chain Conformations to Extensional Stress in Entangled Polymer Melts. _Physical Review Letters_ 2018, _121_ , 047801, DOI: 10.1103/PhysRevLett.121.047801
  * O’Connor et al. 2019 O’Connor, T. C.; Hopkins, A.; Robbins, M. O. Stress Relaxation in Highly Oriented Melts of Entangled Polymers. _Macromolecules_ 2019, _52_ , 8540–8550, DOI: 10.1021/acs.macromol.9b01161
  * O’Connor et al. 2020 O’Connor, T. C.; T. Ge, M. R.; Grest, G. S. Topological Linking Drives Anomalous Thickening of Ring Polymers in Weak Extensional Flows. _Physical Review Letters_ 2020, _124_ , 027801, DOI: 10.1103/PhysRevLett.124.027801
  * T. C. O’Connor 2022 T. C. O’Connor, G. S. G., T. Ge Composite Entanglement Topology and Extensional Rheology of Symmetric Ring-Linear Polymer Blends. _Journal of Rheology_ 2022, _66_ , 49–65, DOI: 10.1122/8.0000319
  * Murashima et al. 2021 Murashima, T.; Hagita, K.; Kawakatsu, T. Viscosity Overshoot in Biaxial Elongational Flow: Coarse-Grained Molecular Dynamics Simulation of Ring-Linear Polymer Mixtures. _Macromolecules_ 2021, _54_ , 7210–7225, DOI: 10.1021/acs.macromol.1c00267
  * 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, DOI: 10.1063/1.458541
  * Evans and Morriss 2008 Evans, D. J.; Morriss, G. P. _Statistical Mechanics of Nonequilibrium Liquids_ , 2nd ed.; Cambridge University Press: New York, 2008
  * Anderson et al. 2020 Anderson, J. A.; Glaser, J.; Glotzer, S. C. HOOMD-blue: A Python Package for High-Performance Molecular Dynamics and Hard Particle Monte Carlo Simulations. _Computational Materials Science_ 2020, _173_ , 109363, DOI: 10.1016/j.commatsci.2019.109363, https://glotzerlab.engin.umich.edu/hoomd-blue/
  * Plimpton 1995 Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. _Journal of Computational Physics_ 1995, _117_ , 1–19, DOI: 10.1006/jcph.1995.1039, https://lammps.org/
  * Stukowski 2010 Stukowski, A. Visualization and Analysis of Atomistic Simulation Data with OVITO-the Open Visualization Tool. _Modelling and Simulation in Materials Science and Engineering_ 2010, _18_ , 015012, DOI: 10.1088/0965-0393/18/1/015012, https://www.ovito.org/
  * Savitzky and Golay 1964 Savitzky, A.; Golay, M. J. E. Smoothing and Differentiation of Data by Simplified Least Squares Procedures. _Analytical Chemistry_ 1964, _36_ , 1627–1639, DOI: 10.1021/ac60214a047
  * Milner and McLeish 1997 Milner, S. T.; McLeish, T. C. B. Parameter-Free Theory for Stress Relaxation in Star Polymer Melts. _Macromolecules_ 1997, _30_ , 2159–2166, DOI: 10.1021/ma961559f
  * Huang et al. 2016 Huang, Q.; Agostini, S.; Hengeller, L.; Shivokhin, M.; Alvarez, N. J.; Hutchings, L. R.; Hassager, O. Dynamics of Star Polymers in Fast Extensional Flow and Stress Relaxation. _Macromolecules_ 2016, _49_ , 6694–6699, DOI: 10.1021/acs.macromol.6b01348
  * Rosa et al. 2020 Rosa, A.; Smrek, J.; Turner, M. S.; Michieletto, D. Threading-Induced Dynamical Transition in Tadpole-Shaped Polymers. _ACS Macro Letters_ 2020, _9_ , 743–748, DOI: 10.1021/acsmacrolett.0c00197
  * Rauscher et al. 2018 Rauscher, P. M.; Rowan, S. J.; de Pablo, J. J. Topological Effects in Isolated Poly[n]catenanes: Molecular Dynamics Simulations and Rouse Mode Analysis. _ACS Macro Letters_ 2018, _7_ , 938–943, DOI: 10.1021/acsmacrolett.8b00393
  * Sawada et al. 2019 Sawada, T.; Inomata, Y.; Shimokawa, K.; Fujita, M. A Metal-Peptide Capsule by Multiple Ring Threading. _Nature Communications_ 2019, _10_ , 5687, DOI: 10.1038/s41467-019-13594-4
  * Sawada et al. 2019 Sawada, T.; Saito, A.; Tamiya, K.; Shimokawa, K.; Hisada, Y.; Fujita, M. Metal-Peptide Rings Form Highly Entangled Topologically Inequivalent Frameworks with the Same Ring- and Crossing-Numbers. _Nature Communications_ 2019, _10_ , 921, DOI: 10.1038/s41467-019-08879-7
  * Hagita et al. 2022 Hagita, K.; Murashima, T.; Sakata, N. Mathematical Classification and Rheological Properties of Ring Catenane Structures. _Macromolecules_ 2022, _55_ , 166–177, DOI: 10.1021/acs.macromol.1c01705
  * Wu and Chen 2022 Wu, S.; Chen, Q. Advances and New Opportunities in the Rheology of Physically and Chemically Reversible Polymers. _Macromolecules_ 2022, _55_ , 697–714, DOI: 10.1021/acs.macromol.1c01605
  * Shibayama 1998 Shibayama, M. Spatial Inhomogeneity and Dynamic Fluctuations of Polymer Gels. _Macromolecular Chemistry and Physics_ 1998, _199_ , 1–30, DOI: 10.1002/(SICI)1521-3935(19980101)199:1%3C1::AID-MACP1%3E3.0.CO;2-M
  * Venerus et al. 2010 Venerus, D. C.; Shiu, T.-Y.; Kashyap, T.; Hosttetler, J. Continuous Lubricated Squeezing Flow: A Novel Technique for Equibiaxial Elongational Viscosity Measurements on Polymer Melts. _Journal of Rheology_ 2010, _54_ , 1083–1095, DOI: 10.1122/1.3474599
  * Venerus et al. 2019 Venerus, D. C.; Mick, R. M.; Kashyap, T. Equibiaxial Elongational Rheology of Entangled Polystyrene Melts. _Journal of Rheology_ 2019, _63_ , 157–165, DOI: 10.1122/1.5062161