# Kremer-Grest models for commodity polymer melts: Linking theory, experiment
and simulation at the Kuhn scale

Carsten Svaneborg<EMAIL_ADDRESS>University of Southern Denmark, Campusvej
55, DK-5230 Odense M, Denmark  Hossein Ali Karimi-Varzaneh  Nils Hojdis  Frank
Fleck  Continental, PO Box 169, D-30001 Hannover, Germany  Ralf Everaers  Univ
Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique and
Centre Blaise Pascal, F-69342 Lyon, France

###### Abstract

The Kremer-Grest (KG) polymer model is a standard model for studying generic
polymer properties in Molecular Dynamics simulations. It owes its popularity
to its simplicity and computational efficiency, rather than its ability to
represent specific polymers species and conditions. Here we show, that by
tuning the chain stiffness it is possible to adapt the KG model to model melts
of real polymers. In particular, we provide mapping relations from KG to SI
units for a wide range of commodity polymers. The connection between the
experimental and the KG melts is made at the Kuhn scale, i.e. at the crossover
from chemistry-specific small scale to the universal large scale behavior. We
expect the KG models to faithfully represent universal properties dominated by
the large scale conformational statistics and dynamics of flexible polymers.
In particular, we observe very good agreement between entanglement moduli of
our KG models and the experimental moduli of the target polymers.

##  I Introduction

Polymers are long chain molecules built by covalent linkage of a large numbers
of identical monomers Flory (1953); Flory and Volkenstein (1969). Some
properties of polymeric materials such as their density, their ability to form
semi-cristalline phases or their glass transition temperature depend on
specific chemical details at the monomer scale. Others, like the variation of
the melt viscosity with the molecular weight of the chains, are controlled by
the large scale conformational statistics and dynamics of long entangled
chains adopting interpenetrating random walk conformations Doi and Edwards
(1986). These latter properties, which are characteristic of polymeric
systems, are universal de Gennes (1979); Doi and Edwards (1986) in the sense
that a large number of chemically different systems have the same properties,
when expressed in suitable material-specific units.

The character of the target properties is crucial for making an intelligent
choice of which model to apply in a theoretical or computational
investigation. Universal properties can be predicted using simple,
analytically or numerically convenient lattice and off-lattice models, see
e.g. refs. Doi and Edwards (1986); Binder (1995); Kremer (2000, 2003); Kröger
(2004) for reviews. In contrast, predicting specific material properties for a
given chemical species often requires atom-scale modeling Müller-Plathe
(2002a). A growing body of work aims at developing such coarse-grained (CG)
polymer models Müller-Plathe (2002b); Sun and Faller (2005); Tzoumanekas and
Theodorou (2006); Peter and Kremer (2009); Li et al. (2013) designed for
specific polymer chemistries such as polyethylene Padding and Briels (2002);
Liu et al. (2013); Salerno et al. (2016), polyisoprene Faller and Müller-
Plathe (2002); Faller and Reith (2003); Li et al. (2011); Maurel et al.
(2015), polystyrene Harmandaris et al. (2006); Chen et al. (2007); Fritz et
al. (2011), polyamide Karimi-Varzaneh et al. (2008); Eslami et al. (2011),
polymethacrylate Chen et al. (2008), polydimethylsiloxane Maurel et al.
(2015), bisphenol-A polycarbonate Tschöp et al. (1998); Abrams and Kremer
(2003); Hess et al. (2006), polybutadiene Strauch et al. (2009), polyvinyl
Milano and Müller-Plathe (2005), and polyisobutylene Maurel et al. (2015).

Common to these approaches is the selected inclusion of specific chemical
details in the coarse-grained models. They offer insights into which atomistic
details of the chemical structure are relevant for particular non-universal
polymer properties. The inclusion of molecular details is supposed to preserve
a certain degree of transferability, i.e. models optimized to describe
materials at one state point are expected to remain approximately valid at
neighboring state points. Baschnagel et al. (2000); Carbone et al. (2008);
Peter and Kremer (2009) Similarly, careful coarse-graining is supposed to
assure representability, i.e. the ability of a model to predict properties
that it was not explicitly designed to reproduce. Johnson et al. (2007)

In the present paper, we use a minimal route to include specificity into a
generic polymer model, where non-trivial large scale properties emerge through
the same mechanisms as in the experimental target materials. With universality
assuring the strongest form of representability, we expect that by matching a
very limited number of conformational properties Sukumaran et al. (2005), we
can 1) design generic polymer models that match universal properties of real
chemical polymers, and 2) map simulation results for such models to real units
and hence make quantitative predictions for target polymer systems.

These generic polymer models display complex emergent phenomena. For instance,
static and dynamic entanglement effects including multichain mechanisms such
as constraint release Graessley (1982); Marrucci (1985); Rubinstein and Colby
(1988); Marrucci and Ianniruberto (2003); Ianniruberto and Marrucci (2014) and
crumpling Rosa and Everaers (2008); Halverson et al. (2012, 2011); Rosa and
Everaers (2014) as well as correlation hole effects Wittmer et al. (2004) will
naturally emerge in such models, without the description needing to be
accurate on the atomic scale. Polydispersity, branching Grest et al. (1989),
chemical cross-linking Grest and Kremer (1990); Duering et al. (1991);
Svaneborg et al. (2004), and network aging Svaneborg et al. (2008) are also
straightforward to include. Furthermore, such models can be used to study
effects of spatial confinement Aoyagi et al. (2001) in thin films Pierce et
al. (2009) and brushes Murat and Grest (1991, 1989); Grest (1996) or the
addition of filler particles in composite materials Yagyu and Utsumi (2009);
Sussman et al. (2014), or the welding dynamics at polymer interfaces Gersappe
(2002); Ge et al. (2014) to name a few examples.

Here we apply this philosophy to the most popular off-lattice polymer model
introduced by Kremer and Grest (KG) Grest and Kremer (1986); Kremer and Grest
(1990). In the KG model approximately hard sphere beads are connected by
strong non-linear springs generating the connectivity and the liquid-like
monomer packing characteristic of polymer melts. The spring potential is
chosen to energetically prevent two polymer chains from passing through each
other and thus to assure that the model reproduces the microscopic topological
constraints dominating the dynamics of long-chain polymers Kremer and Grest
(1990). The KG model is formulated in the natural units of the Lennard-Jones
potential describing the interaction potential between beads. The energy scale
$\epsilon$ together with the bead diameter, $\sigma$, and mass, $m_{b}$,
define the standard Lennard-Jones system of units including the time scale
$\tau=\sigma\sqrt{m_{b}/\epsilon}$.

The purpose of this paper is to establish KG simulations as a convenient tool
for exploring emergent universal properties of specific polymer materials. To
obtain valid coarse-grain descriptions for commodity polymer melts, we tune
the strength of a bending potential introduced into the model by Faller and
Müller-PlatheFaller et al. (1999, 2000); Faller and Müller-Plathe (2001). The
proposed mapping relies on results of a preceding paper, where we have studied
the dependence of the characteristic time and length scales in KG bead-spring
polymer melts on this parameter Svaneborg and Everaers (2018). Here we provide
tables specifying a one-parameter KG force field for a wide range of
experimental polymer melts, i.e. we 1) list which bending stiffness to use for
modelling a particular chemical polymer species, and 2) how to translate
simulation results expressed in KG units into predictions for the specific
polymer material expressed in SI units. Note that while we provide force-
fields for materials like polyethylene (PE), polyoxymethylene (POM),
polyethylene terephthalate (PET), polybutylene terephthalate (PBT),
polytetrafluoroethylene (PTFE) or isotactic polypropylene (i-PP), they cannot
be expected to reproduce the tendency to form semi-crystalline ordering. Such
KG models should thus be taken with a grain of salt or, maybe, as a reminder
that there is more to polymers than universal properties. Nonetheless, we note
that coarse-grain models have been used to study crystallizationMeyer and
Müller-Plathe (2001), and recently specialized KG models were developed and
optimizedHoy and Karayiannis (2013); Morthomas et al. (2017) to study
crystallization phenomena.

The paper is structured as follows; We introduce the Kuhn scale in Sec. II.
After characterizing experimental (II.2) and KG (II.3) melts in the
corresponding units, we derive mapping relations for static (II.4) and dynamic
(II.5) properties. Sec. III reviews theoretical insights into the emergence of
universal polymer behavior beyond the Kuhn scale in an attempts to outline the
types of problems to which Kuhn matched KG models may be profitably applied.
As a practical application, we compare plateau moduli inferred from KG models
to experimental values. The discussion in Sec. IV focuses on the place of Kuhn
scale matched KG models within the multiscale hierarchy of polymer polymers.
We propose to view Kuhn scale matching as a special case of structure based
coarse-graining, explain the advantages of matching the KG model to
experiments on the Kuhn rather than the entanglement scale, and discuss the
large effective time step of our models together with the expected speedup
relative to atomistic simulations. Finally, we briefly conclude in Sec. V.

##  II Matching at the Kuhn scale

The natural units of polymer physics de Gennes (1979); Doi and Edwards (1986);
Rubinstein and Colby (2003) are the mesoscopic Kuhn units: the Kuhn length,
$l_{K}$, the Kuhn time, $\tau_{K}$, and $k_{B}T$ as the natural energy scale
in entropy dominated systems. Kuhn’s seminal insight in the 1930s was to use
an $N_{K}$ step random walk of segment length $l_{K}$ to descripe the large
scale chain conformations Kuhn (1934). For the proper choice of $l_{K}$, the
Kuhn model reproduces both, the end-to-end distance at full extension,
$L=l_{K}N_{K}$, and the mean-square end-to-end distance, $\langle
R^{2}\rangle=N_{K}l_{K}^{2}$, of the target polymer. In particular, flexible
polymers exhibit universal behavior de Gennes (1979); Doi and Edwards (1986);
Rubinstein and Colby (2003) beyond the Kuhn scale defined by $l_{K}$ and the
corresponding time scale $\tau_{K}$. In contrast, behavior on smaller scales
is material specific and dependent on atomic details. For example, the large
scale flexibility has completely different microscopic origins in the wormlike
chain Kratky and Porod (1949) and in the rotational-isomeric-state Flory and
Volkenstein (1969) models. Similarly, there are well-documented exceptions
Ding and Sokolov (2006) to the strong form of time-temperature superposition
principle, which postulates identical temperature dependence for all
microscopic relaxation mechanisms down to the atomic scale. When it comes to
linking theory, experiments, atomistic and coarse-grain simulation, then we
believe that the natural approach is to match them at the Kuhn scale.

###  II.1 The Kuhn scale

The Kuhn length,

$l_{K}=\frac{\langle R^{2}\rangle}{L},$ (1)

characterizes the crossover from local rigid rod to random walk behavior.

It is not straightforward to infer the Kuhn length from the chemical structure
of a polymer in its melt state as it depends on intramolecular interactions,
chemistry-specific local packing, and universal long-range correlationsFlory
and Volkenstein (1969); Wittmer et al. (2004). However, a known Kuhn length
can be used to characterize the large scale structure of polymer melts via two
related dimensionless numbers.

The first such number is a dimensionless measure of chain length, the number
of Kuhn segments per chain:

$N_{K}=\frac{\langle R^{2}\rangle}{l_{K}^{2}}=\frac{L^{2}}{\langle
R^{2}\rangle},$ (2)

The second number, to which we refer as the Kuhn number, is a dimensionless
measure of density,

$n_{K}=\rho_{K}l_{K}^{3},$ (3)

defined as the number of Kuhn segments within the volume of a Kuhn length
cube.

To characterise the dynamics, one can define the friction coefficient,
$\zeta_{K}$, of a Kuhn segment undergoing Brownian motion. Interpreting
$\zeta_{K}$ as a viscous Stokes drag, $\zeta_{K}\propto\eta_{K}l_{K}$, it is
convenient to define an effective viscosity at the Kuhn scale as

$\eta_{K}=\frac{1}{36}\frac{\zeta_{K}}{l_{K}}\ .$ (4)

The fundamental time scale of the dynamics of intrinsically flexible polymers
is set by the time that it takes a Kuhn segment to diffuse
($D_{K}=k_{B}T/\zeta_{K}$) over a distance comparable to its own size. Again
it turns out to be practical to incorporate some numerical prefactors into the
definition of the Kuhn time:

$\tau_{K}=\frac{1}{3\pi^{2}}\frac{\zeta_{K}l_{K}^{2}}{k_{B}T}=\frac{12}{\pi^{2}}\frac{\eta_{K}l_{K}^{3}}{k_{B}T}.$
(5)

###  II.2 Commodity polymer melts at the Kuhn scale

name | $\begin{array}[]{c}T_{ref}\\\ \mbox{[}K\mbox{]}\end{array}$ | $\begin{array}[]{c}\langle R^{2}\rangle/M_{c}\\\ \mbox{[}\mathring{A}^{2}mol/g\mbox{]}\end{array}$ | $\begin{array}[]{c}\rho_{bulk}\\\ \mbox{[}g/cm^{3}\mbox{]}\end{array}$ | $\begin{array}[]{c}G_{e}\\\ \mbox{[}MPa\mbox{]}\end{array}$ | $\begin{array}[]{c}p\\\ \mbox{[}\mathring{A}\mbox{]}\end{array}$ | $\begin{array}[]{c}d_{T}\\\ \mbox{[}\mathring{A}\mbox{]}\end{array}$ | $\begin{array}[]{c}n_{K}\\\ \mbox{ }\end{array}$ | $\begin{array}[]{c}l_{K}\\\ \mbox{[}\mathring{A}\mbox{]}\end{array}$ | $\begin{array}[]{c}M_{K}\\\ \mbox{[}g/mol\mbox{]}\end{array}$ | $\begin{array}[]{c}M_{K}/M_{m}\\\ \mbox{ }\end{array}$ | $\begin{array}[]{c}\rho_{K}\\\ \mbox{[}nm^{-3}\mbox{]}\end{array}$ | $\begin{array}[]{c}G_{e}l_{K}^{3}/k_{B}T\\\ \mbox{ }\end{array}$ | $\begin{array}[]{c}N_{eK}\\\ \mbox{ }\end{array}$ | $\begin{array}[]{c}\alpha\\\ \mbox{ }\end{array}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
PI-50 | 298 | 0.528 | 0.893 | 0.51 | 3.52 | 47.7 | 2.50 | 8.80 | 146.60 | 2.15 | 3.66 | 0.085 | 29.41 | 13.6
PI-7 | 298 | 0.596 | 0.900 | 0.44 | 3.10 | 55.1 | 2.72 | 8.44 | 119.60 | 1.76 | 4.52 | 0.064 | 42.55 | 17.8
PDMS∗ | 298 | 0.422 | 0.970 | 0.25 | 4.06 | 63.7 | 2.82 | 11.42 | 309.28 | 4.17 | 1.89 | 0.091 | 31.08 | 15.7
PI-20 | 298 | 0.591 | 0.898 | 0.44 | 3.13 | 54.8 | 2.86 | 8.98 | 136.50 | 2.00 | 3.95 | 0.077 | 37.17 | 17.5
PI-34 | 298 | 0.585 | 0.965 | 0.44 | 2.94 | 56.5 | 3.02 | 9.58 | 156.90 | 2.30 | 3.44 | 0.093 | 32.32 | 19.2
cis-PBd | 298 | 0.758 | 0.900 | 0.95 | 2.43 | 42.2 | 3.40 | 8.28 | 90.50 | 1.67 | 5.99 | 0.131 | 25.93 | 17.3
PIB(413) | 413 | 0.557 | 0.849 | 0.38 | 3.51 | 65.8 | 3.47 | 12.20 | 267.90 | 4.77 | 1.91 | 0.119 | 29.02 | 18.7
cis-PI | 298 | 0.679 | 0.910 | 0.72 | 2.69 | 46.0 | 3.47 | 9.34 | 128.60 | 1.89 | 4.26 | 0.144 | 24.15 | 17.1
a-PP(463) | 463 | 0.678 | 0.765 | 0.53 | 3.20 | 61.7 | 3.53 | 11.20 | 183.40 | 4.36 | 2.51 | 0.115 | 30.59 | 19.3
i-PP | 463 | 0.694 | 0.766 | 0.54 | 3.12 | 61.7 | 3.64 | 11.40 | 187.80 | 4.46 | 2.46 | 0.125 | 29.22 | 19.8
a-PP(413) | 413 | 0.678 | 0.791 | 0.59 | 3.10 | 56.0 | 3.65 | 11.20 | 183.40 | 4.36 | 2.60 | 0.145 | 25.21 | 18.1
a-PP(348) | 348 | 0.678 | 0.825 | 0.60 | 2.97 | 51.9 | 3.81 | 11.20 | 183.40 | 4.36 | 2.71 | 0.175 | 21.70 | 17.5
a-PP | 298 | 0.678 | 0.852 | 0.60 | 2.87 | 48.8 | 3.92 | 11.20 | 183.40 | 4.36 | 2.79 | 0.205 | 19.15 | 17.0
PIB | 298 | 0.570 | 0.918 | 0.43 | 3.17 | 55.2 | 3.94 | 12.50 | 274.20 | 4.89 | 2.02 | 0.202 | 19.52 | 17.4
a-PMMA | 413 | 0.390 | 1.130 | 0.39 | 3.77 | 62.5 | 4.07 | 15.30 | 598.00 | 5.97 | 1.14 | 0.243 | 16.72 | 16.6
i-PS∗ | 413 | 0.420 | 0.969 | 0.24 | 4.08 | 76.7 | 4.19 | 17.11 | 697.12 | 6.69 | 0.84 | 0.209 | 20.10 | 18.8
a-PMA | 298 | 0.436 | 1.110 | 0.31 | 3.43 | 61.9 | 4.29 | 14.70 | 494.60 | 5.75 | 1.35 | 0.241 | 17.79 | 18.1
PI-75 | 298 | 0.563 | 0.890 | 0.46 | 3.31 | 51.8 | 4.53 | 15.00 | 399.30 | 5.86 | 1.34 | 0.379 | 11.94 | 15.6
PBd-20 | 298 | 0.841 | 0.895 | 1.34 | 2.21 | 37.3 | 4.54 | 10.10 | 122.40 | 2.26 | 4.41 | 0.335 | 13.55 | 16.9
a-PS∗ | 413 | 0.437 | 0.969 | 0.25 | 3.92 | 76.3 | 4.54 | 17.80 | 725.34 | 6.96 | 0.80 | 0.247 | 18.35 | 19.4
PBd-98 | 300 | 0.661 | 0.890 | 0.71 | 2.82 | 45.4 | 4.83 | 13.70 | 284.80 | 5.27 | 1.88 | 0.442 | 10.93 | 16.1
PEO∗ | 353 | 0.805 | 1.060 | 2.25 | 1.95 | 33.4 | 4.99 | 9.71 | 117.12 | 2.66 | 5.45 | 0.423 | 11.81 | 17.1
POM∗ | 473 | 0.763 | 1.140 | 2.12 | 1.91 | 40.1 | 5.06 | 9.65 | 122.11 | 4.07 | 5.62 | 0.293 | 17.28 | 21.0
a-PHMA | 373 | 0.366 | 0.960 | 0.11 | 4.73 | 98.4 | 5.19 | 24.40 | 1622.00 | 9.53 | 0.36 | 0.317 | 16.35 | 20.8
a-PVA∗ | 333 | 0.490 | 1.080 | 0.44 | 3.14 | 57.9 | 5.26 | 16.50 | 555.70 | 6.45 | 1.17 | 0.428 | 12.30 | 18.4
SBR | 298 | 0.818 | 0.913 | 0.98 | 2.22 | 43.6 | 5.33 | 11.90 | 173.60 | 2.61 | 3.16 | 0.399 | 13.35 | 19.6
P6N∗ | 543 | 0.853 | 0.985 | 2.25 | 1.98 | 41.1 | 5.53 | 10.93 | 140.05 | 1.24 | 4.24 | 0.392 | 14.11 | 20.8
a-P$\alpha$MS∗ | 473 | 0.442 | 1.040 | 0.40 | 3.61 | 67.2 | 5.66 | 20.43 | 944.61 | 7.99 | 0.66 | 0.523 | 10.82 | 18.6
a-PEA | 298 | 0.463 | 1.130 | 0.45 | 3.17 | 53.7 | 5.70 | 18.10 | 710.10 | 7.09 | 0.96 | 0.649 | 8.79 | 16.9
PET∗ | 548 | 0.845 | 0.989 | 3.88 | 1.99 | 31.3 | 7.50 | 14.91 | 263.15 | 1.37 | 2.26 | 1.698 | 4.42 | 15.8
s-PP | 463 | 1.030 | 0.766 | 1.69 | 2.10 | 42.4 | 7.99 | 16.90 | 278.70 | 6.62 | 1.66 | 1.274 | 6.27 | 20.2
PE(413) | 413 | 1.250 | 0.785 | 3.25 | 1.69 | 32.2 | 8.09 | 13.70 | 150.40 | 5.36 | 3.14 | 1.466 | 5.52 | 19.0
a-POA | 298 | 0.442 | 0.980 | 0.20 | 3.83 | 73.3 | 8.34 | 31.90 | 2295.00 | 12.45 | 0.26 | 1.578 | 5.29 | 19.1
PC∗ | 473 | 0.864 | 1.140 | 3.38 | 1.69 | 33.9 | 10.93 | 18.43 | 393.25 | 1.55 | 1.75 | 3.237 | 3.38 | 20.1
PE | 298 | 1.400 | 0.851 | 4.38 | 1.39 | 26.0 | 11.10 | 15.40 | 168.30 | 6.00 | 3.04 | 3.884 | 2.86 | 18.6
PTFE∗ | 653 | 0.598 | 1.460 | 2.12 | 1.90 | 47.2 | 12.30 | 23.40 | 915.41 | 9.15 | 0.96 | 3.019 | 4.07 | 24.8
Table 1: Kuhn descriptions and entanglement properties of polymersFetters et
al. (2007) characterized by experimental temperature $T_{ref}$, mean-square
extension per molecular mass $\langle R^{2}\rangle/M_{c}$, bulk mass density
$\rho_{bulk}$, entanglement modulus $G_{e}$, and derived quantities such as
the packing length $p$ and tube diameter $d_{T}$. The polymers are also
characterized by their Kuhn number $n_{K}$, Kuhn length $l_{K}$, mass of a
Kuhn segment $M_{K}$, Kuhn density $\rho_{K}$, and the reduced entanglement
modulus $G_{e}l_{K}^{3}/[k_{B}T_{ref}]$, Kuhn segments between entanglements
$N_{eK}$, and number of entanglement strands per entanglement volume $\alpha$.
$*$ denotes polymers where we have derived the Kuhn scale descriptions, for
the rest we use the values from Ref. Fetters et al. (2007). To uniquely
identify polymers, we have added the reference temperature to some of the
polymer names.

At a given state point (temperature), a melt of monodisperse chains (with
molecular weight $M_{c}$) can be characterized by just a few experimental
observables: the mass density $\rho_{bulk}$, the average chain end-to-end
distance per unit mass $\langle R^{2}\rangle/M_{c}$, and the maximal chain
extension, $L$. Values for these observables for a large number of typical
polymers are collected in Ref. Fetters et al. (2007). We present data for a
selected subset of polymers expressed in Kuhn units in Tab. 1. The Kuhn
lengths are in the $1-2$ nanometer range, with a Kuhn segment mass
$M_{K}=M_{c}/N_{K}\sim 100-2000$ g/mol. The number of monomers in a Kuhn
segment varies in the range of $1-13$, and the number density of Kuhn
segments, $\rho_{K}$, varies in the range of $0.5-5nm^{-3}$.

A key characteristic of polymer species is their Kuhn number, which varies for
common, flexible commodity polymers in the range $2\leq n_{K}\leq 12$. For
comparison, $n_{K}\gg 10$ in gels of tightly entangled filamentous proteins
such as f-actin.Uchida et al. (2008a) In Tab. 1, we observe a strong and
systematic correlation between the Kuhn number and emergent properties such as
the entanglement modulus and the entanglement length measured in Kuhn units.
This observation suggests that the Kuhn number is a key molecular parameter in
predicting entanglement related phenomena. We will return to this point in
Sec. III.

###  II.3 Kremer-Grest model polymer melts at the Kuhn scale

The Kremer-Grest modelGrest and Kremer (1986); Kremer and Grest (1990) is a
defacto standard model in Molecular Dynamics investigations of generic polymer
properties. The KG model is a bead-spring model, where the mutual interactions
between all beads are given by the Weeks-Chandler-Anderson (WCA) potential
(the truncated and shifted repulsive part of the 12-6 Lennard-Jones
potential),

$U_{WCA}(r)=4\epsilon\left[\left(\frac{\sigma}{r}\right)^{-12}-\left(\frac{\sigma}{r}\right)^{-6}+\frac{1}{4}\right]\quad\mbox{for}\quad
r<2^{1/6}\sigma\ ,$ (6)

where $\epsilon$ defines the energy scale and $\sigma$ the bead diameter. The
standard choice for the temperature is $k_{B}T=\epsilon$. Bonded beads
interact through the finite-extensible-non-linear spring (FENE) potential
given by

$U_{FENE}(r)=-\frac{kR^{2}}{2}\ln\left[1-\left(\frac{r}{R}\right)^{2}\right]\
.$ (7)

Choosing a spring constant $k=30\epsilon\sigma^{-2}$ and a bond length,
$R=1.5\sigma$, where the FENE potential diverges, the bond length is
$l_{b}=0.965\sigma$. The standard choice for the bead density is
$\rho_{b}=0.85\sigma^{-3}$. Faller and Müller-Plathe Faller et al. (1999,
2000); Faller and Müller-Plathe (2001) augmented the standard KG model with a
bending potential,

$U_{bend}(\Theta)=\kappa\left(1-\cos\Theta\right),$ (8)

where $\Theta$ denotes the angle between subsequent bonds.

To prepare the present study, we have investigated the dependence of the
characteristic time and length scales in KG bead-spring polymer melts on the
reduced bending energy, $x\equiv\beta\kappa=\kappa/k_{B}T$ Svaneborg and
Everaers (2018). In particular, we found for the Kuhn length:

$\displaystyle l_{K}(x)$ $\displaystyle=$ $\displaystyle l_{K}^{(0)}+\Delta
l_{K}$ (9) $\displaystyle l_{K}^{(0)}(x)/\sigma$ $\displaystyle=$
$\displaystyle\frac{l_{b}}{\sigma}\begin{cases}\frac{2x+e^{-2x}-1}{1-e^{-2x}(2x+1)}&\text{if
$x\neq 0$}\\\ 1&\text{if $x=0$}\end{cases}$ $\displaystyle\Delta
l_{K}(x)/\sigma$ $\displaystyle=$ $\displaystyle 0.769888+0.776514\times$
$\displaystyle\tanh\left(0.147244-0.405299x-0.0298337x^{2}\right)$

From this relation, we can directly infer the dimensionless Kuhn number, Eq.
(3), characterising KG melts:

$n_{K}(\beta\kappa)=\rho_{b}\frac{l_{b}}{l_{K}}l_{K}^{3}\ .$ (10)

Finally, the number of Kuhn segments between entanglements, the Kuhn friction
and Kuhn time of the KG model are given by

$\displaystyle N_{eK}(x)$ $\displaystyle=$ $\displaystyle
39.130-30.237x+4.2834x^{2}$ (11)
$\displaystyle+3.2065x^{3}-1.2879x^{4}+0.1372x^{5}$
$\displaystyle\zeta{}_{K}(x)/(m_{b}/\tau)$ $\displaystyle=$ $\displaystyle
36.021+10.149x+5.6341x^{2}$ (12) $\displaystyle\tau_{K}(x)/\tau$
$\displaystyle=$ $\displaystyle 4.2076+2.8451x+1.6485x^{2}$ (13)
$\displaystyle+0.6464x^{3}+0.6524x^{4}$

The parameterization of the Kuhn length we believe to be valid for arbitrary
values for stiffness $\beta\kappa$, while the other relations hold for bending
rigidities in the interval $-1<\beta\kappa<2.5$. A negative chain stiffness
partly counteracts the stiffness induced by excluded volume interactions
between next-nearest beads along a chain, and hence makes the chain more
flexible than the standard KG model.

###  II.4 A one parameter Kremer-Grest “force field” for commodity polymer
melts

Figure 1: KG chain stiffness vs. Kuhn number using eq. (10) (solid black line)
and our approximate inversion eq. (15) (green symbols). The inset shows the
error of our numerical inversion.

As a first step in defining a KG force field for a polymeric material we fix
the energy scale to the thermal excitation energy Kröger (2004),

$\epsilon=k_{B}T\ ,$ (14)

at the temperature of interest. The key step is the choice of the bending
stiffness: we match the dimensionless Kuhn numbers $n_{K}$ characterising the
experimental system and the model polymer melt. A priori, this requires the
numerical inversion of the combination of Eqs. (9) and (10). As shown in Fig.
1, the approximate relation

$\displaystyle\frac{\kappa(n_{K})}{k_{B}T}=0.822264\log(n_{K}-2.0)-0.000290834n_{K}^{3}$
(15) $\displaystyle+0.00871154n_{K}^{2}-0.0552417n_{K}+0.280663$

provides an excellent approximation over the experimentally relevant range,
$2\leq n_{K}\leq 15$. Through eq. (15), $l_{K}$, $N_{eK}$, $\zeta_{K}$, and
$\tau_{K}$ become functions of the Kuhn number. Note that the standard KG
model with $\beta\kappa=0$ essentially corresponds to the intrinsically most
flexible polymers such as PDMS or PI with $7-50\%$ 3,4 content.

name | $n_{K}$ | $\beta\kappa$ | $c_{b}$ | $l_{K}/\sigma$ | $M{}_{m}/M_{b}$ | $M_{b}/[g/mol]$ | $\epsilon/10^{-21}J$ | $\sigma/nm$ | $\epsilon\sigma^{-3}/MPa$
---|---|---|---|---|---|---|---|---|---
PI-50 | 2.50 | -0.378 | 1.81 | 1.74 | 0.84 | 81.13 | 4.11 | 0.50 | 32.0
PI-7 | 2.72 | -0.086 | 1.89 | 1.82 | 1.07 | 63.37 | 4.11 | 0.46 | 41.3
PDMS∗ | 2.82 | 0.013 | 1.92 | 1.85 | 0.46 | 161.05 | 4.11 | 0.62 | 17.6
PI-20 | 2.86 | 0.056 | 1.94 | 1.87 | 0.97 | 70.50 | 4.11 | 0.48 | 37.1
PI-34 | 3.02 | 0.191 | 1.99 | 1.92 | 0.86 | 78.86 | 4.11 | 0.50 | 33.1
cis-PBd | 3.40 | 0.445 | 2.11 | 2.04 | 1.26 | 42.87 | 4.11 | 0.41 | 61.3
PIB(413) | 3.47 | 0.483 | 2.13 | 2.06 | 0.45 | 125.69 | 5.70 | 0.59 | 27.3
cis-PI | 3.47 | 0.484 | 2.13 | 2.06 | 1.13 | 60.32 | 4.11 | 0.45 | 44.0
a-PP(463) | 3.53 | 0.518 | 2.15 | 2.07 | 0.49 | 85.25 | 6.39 | 0.54 | 40.7
i-PP | 3.64 | 0.575 | 2.18 | 2.11 | 0.49 | 85.96 | 6.39 | 0.54 | 40.4
a-PP(413) | 3.65 | 0.580 | 2.19 | 2.11 | 0.50 | 83.84 | 5.70 | 0.53 | 38.2
a-PP(348) | 3.81 | 0.656 | 2.23 | 2.16 | 0.51 | 82.08 | 4.80 | 0.52 | 34.3
a-PP | 3.92 | 0.708 | 2.27 | 2.19 | 0.52 | 80.84 | 4.11 | 0.51 | 30.7
PIB | 3.94 | 0.714 | 2.27 | 2.19 | 0.47 | 120.66 | 4.11 | 0.57 | 22.2
a-PMMA | 4.07 | 0.770 | 2.31 | 2.23 | 0.39 | 258.82 | 5.70 | 0.69 | 17.6
i-PS∗ | 4.19 | 0.819 | 2.35 | 2.26 | 0.35 | 297.22 | 5.70 | 0.76 | 13.2
a-PMA | 4.29 | 0.856 | 2.37 | 2.29 | 0.41 | 208.42 | 4.11 | 0.64 | 15.6
PI-75 | 4.53 | 0.941 | 2.44 | 2.35 | 0.42 | 163.78 | 4.11 | 0.64 | 15.9
PBd-20 | 4.54 | 0.944 | 2.44 | 2.35 | 1.08 | 50.16 | 4.11 | 0.43 | 52.2
a-PS∗ | 4.54 | 0.944 | 2.44 | 2.35 | 0.35 | 297.19 | 5.70 | 0.76 | 13.2
PBd-98 | 4.83 | 1.039 | 2.52 | 2.43 | 0.48 | 113.08 | 4.14 | 0.56 | 23.1
PEO∗ | 4.99 | 1.086 | 2.56 | 2.47 | 0.96 | 45.77 | 4.87 | 0.39 | 80.2
POM∗ | 5.06 | 1.105 | 2.58 | 2.48 | 0.63 | 47.40 | 6.53 | 0.39 | 111.5
a-PHMA | 5.19 | 1.143 | 2.61 | 2.52 | 0.27 | 621.59 | 5.15 | 0.97 | 5.7
a-PVA∗ | 5.26 | 1.162 | 2.63 | 2.53 | 0.41 | 211.52 | 4.60 | 0.65 | 16.7
SBR | 5.33 | 1.182 | 2.65 | 2.55 | 1.01 | 65.62 | 4.11 | 0.47 | 40.6
P6N∗ | 5.53 | 1.234 | 2.69 | 2.60 | 2.18 | 51.98 | 7.50 | 0.42 | 100.9
a-P$\alpha$MS∗ | 5.66 | 1.265 | 2.72 | 2.63 | 0.34 | 346.66 | 6.53 | 0.78 | 13.9
a-PEA | 5.70 | 1.276 | 2.74 | 2.64 | 0.39 | 259.56 | 4.11 | 0.69 | 12.8
PET∗ | 7.50 | 1.646 | 3.14 | 3.03 | 2.29 | 83.82 | 7.57 | 0.49 | 63.4
s-PP | 7.99 | 1.728 | 3.24 | 3.12 | 0.49 | 86.03 | 6.39 | 0.54 | 40.5
PE(413) | 8.09 | 1.744 | 3.26 | 3.14 | 0.61 | 46.15 | 5.70 | 0.44 | 69.0
a-POA | 8.34 | 1.785 | 3.31 | 3.19 | 0.27 | 693.22 | 4.11 | 1.00 | 4.1
PC∗ | 10.93 | 2.136 | 3.79 | 3.65 | 2.45 | 103.76 | 6.53 | 0.50 | 51.0
PE | 11.10 | 2.156 | 3.82 | 3.68 | 0.64 | 44.07 | 4.11 | 0.42 | 56.4
PTFE∗ | 12.30 | 2.291 | 4.02 | 3.87 | 0.44 | 227.70 | 9.02 | 0.60 | 41.1
Table 2: Kremer-Grest model parameters for the polymers shown in Tab. 1 in
terms of the Kuhn number, bending stiffness $\beta\kappa$, number of beads per
Kuhn segment $c_{b}$, Kuhn length expressed in KG units, number of beads per
monomer $M_{m}/m_{b}$, and finally the conversion relations from KG units for
energy $\epsilon$, length $\sigma$, and stress $\epsilon\sigma^{-3}$ to SI
units.

We can obtain the number of beads per Kuhn length as

$c_{b}(n_{K})\equiv\frac{l_{K}}{l_{b}}=\sqrt{\frac{n_{K}}{\rho_{b}l_{b}^{3}}}$
(16)

and hence the number of beads required to represent a chain of a given length
(molecular mass)

$N_{b}=c_{b}(n_{K})N_{K}\ .$ (17)

What remains is to fix the mapping relations for the simulation units of
length, mass, and time. Equating the model and experimental Kuhn lengths and
accounting for the small difference, $l_{b}=0.965\sigma$, between the bond
length and the bead diameter in the KG model, we obtain

$\sigma=\frac{l_{K}^{exp}}{0.965\times c_{b}(n_{K})}$ (18)

The bead mass is obtained along the same lines by equating the experimental
mass of a Kuhn segment to the mass of a Kuhn segment in the model:

$m_{b}=\frac{M_{K}^{exp}}{c_{b}(n_{K})}\ .$ (19)

In Table 2 we have listed the resulting Kremer-Grest model parameters and
mappings for the polymer species shown in Tab. 1. By construction, the number
of beads per Kuhn length is an increasing function of $n_{K}$ and varies
between 1.7 and 3.9. With $-0.4\epsilon\leq\kappa\leq 2.3\epsilon$ the
required stiffness parameters falls into the validity range of our empirical
relations for Kuhn length, entanglement length, and Kuhn friction. Bead
diameters vary between $4$ and $10\mathring{A}$, the energy scale is given by
the experimental temperature range and varies within a factor of two.
Nevertheless, the KG unit of stress, $4MPa<\epsilon\sigma^{-3}<112MPa$,
exhibits a much larger spread. As a rule of thumb, beads correspond to
monomers. But there are important variations. To cite some examples,

PI and PDMS

(polyisoprene and polydimethylsiloxane) are effectively the most flexible
chains, which map fairly well on the standard KG model with $\beta\kappa\equiv
0$. PI beads have a diameter of $5\mathring{A}$ and represent one monomer;
PDMS beads have a diameter of $6\mathring{A}$ and represent two monomers.

PS

(polystyrene) beads represent three monomers and have with $7.6\mathring{A}$ a
correspondingly larger diameter.

PE

(polyethylene) is among the effectively stiffest chains, which 4 beads per
Kuhn length and $\beta\kappa\approx 2$. PE beads represent 1.5 monomers; with
$4\mathring{A}$ they are relatively small.

PC

(polycarbonate) is comparable to PE in effective stiffness. With a diameter of
$5\mathring{A}$, PC beads are comparable to PI beads. However, in the case of
PC 2.5 beads are required to represent the more complex monomers, which is
remarkably similar to 2 bead / ellipsoid models per monomer used by Tschoeb et
al.Tschöp et al. (1998)

A priori, the parameters listed in Table 2 are only valid at the indicated
reference temperatures, where the chain dimensions were determined
experimentally. To model polymer melts at different temperatures, we have, in
principle, to account for changes in (i) the single chain statistics and (ii)
the overall density. It is straightforward to translate a temperature
dependent Kuhn length, $l_{K}=l_{K}(T)$, into a temperature dependent bending
free energy, $\kappa=\kappa(T)$, of our coarse-grain model. To a first
approximation, $\kappa(T)$ can be written in the form
$\kappa(T)=\kappa_{h}-\kappa_{s}T$, where $\kappa_{h}$ and $\kappa_{s}$ are
the enthalpic and entropic contributions to the stiffness.Meyer et al. (2013)
Since the Kuhn length of the KG model, Eq. (9), depends on the reduced bending
free energy, $x=\kappa(T)/k_{B}T$, a temperature-independent Kuhn length
corresponds to a bending free energy of entropic origin. Note, however, that
density changes and the corresponding temperature variations in $n_{K}$ result
in less intuitive shifts in bead diameters and weights with temperature. This
is a consequence of our choice to preserve the “canonical” KG bead density of
$\rho_{b}=0.85\sigma^{-3}$.

In practice, the static melt properties are relatively insensitive to changes
of temperature: the relative density expansion coefficient is
$d\ln\rho_{bulk}/dT\approx-6\times 10^{-4}K^{-1}$, while typical thermal chain
expansion coefficients $|d\ln\langle
R^{2}\rangle(T)/dT|<10^{-3}K^{-1}$.Fetters et al. (2007) Since
$n_{K}(T)\sim\rho_{bulk}\langle R^{2}\rangle^{2}$, we obtain $|d\ln
n_{K}(T)/dT|<3\times 10^{-3}K^{-1}$. In the one case where Ref. Fetters et al.
(2007) provides data, atactic polypropylene, the 50% increase in temperature
over the interval $298K\leq T\leq 463K$ causes a slight increase in density
while apparently leaving the chain dimensions unchanged. The corresponding
reduction of the Kuhn number from $n_{K}=3.92$ to $n_{K}=3.43$, suggests that
one changes the bead weights from $81$ to $85$ $g/mol$, the bead diameters
change from $5.1$ to $5.4$ $\mathring{A}$, while the required reduction of the
bending stiffness decreases the Kuhn length in LJ units from
$l_{K}=2.19\sigma$ to $l_{K}=2.07\sigma$. Compared to the dynamic effects
discussed in the following section, it thus seems safe to transfer the
$\beta\kappa$, $M_{b}$ and $\sigma$ values listed in Tab. 2 to other
temperatures. This suggests that the effective bending rigidity at the KG-
level appears is essentially entropic in origin, $\kappa_{h}\ll-
T_{ref}\kappa_{s}$, and related to rotations between isomeric states Flory and
Volkenstein (1969). Obviously, the relations provided above can be used to
obtain an improved parameterization, if there is information available about
the end-to-end distance and bulk density at the state point of interest.

###  II.5 Time mapping

name | $T_{g}[K]$ | $T_{ref}^{dyn}[K]$ | $\tau_{e}^{exp}[s]$ | $n_{K}$ | $N_{eK}$ | $\tau_{K}(T_{ref}^{dyn})[ps]$ | $\eta_{K}(T_{ref}^{dyn})[mPa\ s]$ | $\tau(T_{ref}^{dyn})[ps]$ | $\sigma\sqrt{M_{b}/k_{B}T_{ref}^{dyn}}[ps]$
---|---|---|---|---|---|---|---|---|---
PI-7 | $206$ | $298$ | $1.9\times 10^{-5}$ | $2.72$ | $42.55$ | $3.9$ | $61.5$ | $2800$ | $2.3$
PDMS | $150$ | $298$ | $1.1\times 10^{-7}$ | $2.82$ | $31.08$ | $4.2$ | $0.162$ | $17$ | $5.0$
cis-PBd | $174$ | $298$ | $8.8\times 10^{-8}$ | $3.40$ | $25.93$ | $5.8$ | $0.726$ | $21$ | $1.7$
cis-PI | $206$ | $298$ | $6.7\times 10^{-5}$ | $3.47$ | $17.1$ | $6.0$ | $424$ | $17000$ | $2.2$
a-PP | $262$ | $348$ | $1.9\times 10^{-6}$ | $3.81$ | $21.70$ | $7.1$ | $11.2$ | $560$ | $2.8$
PIB | $201$ | $298$ | $1.1\times 10^{-3}$ | $3.94$ | $19.52$ | $7.5$ | $4550$ | $350000$ | $4.0$
a-PS | $375$ | $453$ | $3.4\times 10^{-4}$ | $4.54$ | $18.35$ | $9.5$ | $1210$ | $140000$ | $6.8$
PEO | $210$ | $348$ | $1.5\times 10^{-8}$ | $4.99$ | $11.81$ | $11$ | $0.332$ | $7$ | $1.6$
Table 3:  Parameters and characteristic times for commodity polymer melts and
the corresponding KG models. The first set of numbers defines the experimental
input: the experimental glass transition temperature and the dynamic reference
temperature $T_{ref}^{dyn}$ for the experimental entanglement time $\tau_{e}$
(and/or more suitable VF parameters). Using the static mapping and, in
particular, the Kuhn number $n_{K}$, we can infer the number of Kuhn segments
per entanglement length, $N_{eK}$, from Eq. (26) or more refined estimates
Uchida et al. (2008b); Svaneborg and Everaers (2018). The Kuhn time,
$\tau_{K}(T_{ref}^{dyn})$, and the viscosity at the Kuhn scale,
$\eta_{K}(T_{ref}^{dyn})$, follow from Eqs. (5) and (30), the characteristic
time scale, $\tau(T_{ref}^{dyn})$, of the corresponding Kuhn mapped KG model
is given by Eq. (13). For comparison, we also list the estimate for $\tau$
that results from the static mapping. Finally, we can use Eqs. (20) and (21)
to estimate $\tau_{K}$, $\eta_{K}$, and $\tau$ over the entire TTS validity
range (Fig. 2). References for experimental data: PI-7Gotro and Graessley
(1984); Abdel-Goad et al. (2004), PDMSVaca Chávez and Saalwächter (2011), cis-
PDbVaca Chávez and Saalwächter (2011), cis-PIVaca Chávez and Saalwächter
(2011), cis-PIAuhl et al. (2008), a-PPvan Meerveld (2004), and PEO Niedzwiedz
et al. (2008).  Figure 2:  KG time unit $\tau$ in $ns$ as function of
temperature for a number of polymer species listed in the legend and
distinguished by color. Typical time steps in simulations are $\delta
t=10^{-2}\tau$. Solid lines: WLF-extrapolation over the temperature range
$[T_{g},T_{g}+100K]$, Thick dashed lines: WLF-extrapolation for
$T>T_{g}+100K$, Symbols: Estimate of $\tau$ derived from experimental data for
the dynamic reference temperature, $T_{ref}^{dyn}$, underlying the WLF
extrapolation. Thin dashed lines: standard estimation of the LJ time
$\tau=\sigma\sqrt{M_{b}/\epsilon}$ using the mapping values for bead diameter,
bass, and energy scale.

To reproduce not only static but also dynamic properties of target systems, we
require input on their Kuhn time, $\tau_{K}$, or their effective viscosity,
$\eta_{K}$, at the Kuhn scale. Equating with $\tau_{K}$ or $\eta_{K}$ of the
Kuhn mapped KG model, we can directly infer the value of the KG time unit
$\tau$ in SI units from Eq. (13), since the value of $x=\beta\kappa(n_{K})$ is
known via Eq. (15) for the Kuhn number of the experimental system. However, to
carry out this program, we needed to overcome two difficulties.

While conceptually useful, $\tau_{K}$ and $\eta_{K}$ are not straightforward
to observe directly. Typically, one can extrapolate down to the Kuhn scale
within a model, if there is information on (emergent) macroscopic behavior or
time scales at some dynamic reference temperature, $T_{ref}^{dyn}$. In Section
III we discuss suitable examples like the viscosity of unentangled chains,
$\eta=n_{K}N_{K}\ \eta_{K}$, the entanglement time,
$\tau_{e}=N_{eK}^{2}\tau_{K}$, the Rouse time, $\tau_{R}=N_{K}^{2}\tau_{K}$,
or the terminal relaxation time, $\tau_{max}$. Experimentally, the Kuhn time
or equivalently the Kuhn friction can be obtained from neutron spin echo data
Richter et al. (2005) by applying expressions from Rouse theory to analyse the
monomeric dynamics below the entanglement time scale as in our analysis of
simulation data Svaneborg and Everaers (2018). The entanglement time,
$\tau_{e}$, can be measured by oscillatory rheological experiments, dielectric
relaxation and transverse relaxation NMR measurements, see e.g. Refs. Bird et
al. (1977); Watanabe (1999); McLeish (2002); Saalwächter (2007). We note that
published estimates might be obtained by fitting data to expressions, which
define these times using conventions for prefactors, which differ from those
we have adopted here.

The second difficulty is the pronounced temperature dependence of the chain
dynamics in polymer melts. Most commodity polymer melts become glassy below a
temperature $T_{g}$ in or slightly below the experimentally relevant
temperature range. As a consequence, even a small change in temperature can
have a significant impact on the dynamics. Experimentally, time-temperature
superposition (TTS) Ngai and Plazek (2007) is used to explore polymer dynamics
over a much wider range of frequencies than those directly accessible to a
given measurement instrument. Here we use this approach to estimate the Kuhn
time at the temperature of interest, $T$, given a Kuhn time measured at the
reference temperature, $T_{ref}^{dyn}$:

$\frac{\tau_{K}(T)}{\tau_{K}(T_{ref}^{dyn})}=a_{T}(T,T_{ref}^{dyn})$ (20)

As discussed in appendix A, the shift factor can be written as

$\ln
a_{T}(T,T_{ref}^{dyn})=-\frac{C_{VF}(T-T_{ref}^{dyn})}{(T_{ref}^{dyn}-T_{VF})(T-T_{VF})}\
.$ (21)

Eq. (21) should be valid above the glass transition temperature in the
temperature range $[T_{g},T_{g}+100K]$. Here we use a “universal” Vogel-
Fulcher constant $C_{VF}=\ln(10)\times 17.44\times 51.6K=2072K$. Similarly, we
set $T_{VF}=T_{g}-51.6K$ for the Vogel-Fulcher temperature, where viscosities
and associated time scales formally diverge. More detailed information and
specific tables with fitted VF (or WLF, see appendix A) parameters can be
found in Ref. Ngai and Plazek (2007).

The mapping relations for the temperature dependent conversion of the KG time
$\tau$ resulting from Eq. (13) are shown in Tab. 3 and illustrated in Fig. 2.
The converted values vary over a much wider range than the static parameters
in Table 2:

PDMS, cis-PDb and PEO

are experimentally studied about $150K$ above $T_{g}$, resulting in KG time
scales in the $10ps$ range.

PI

melts at $100K$ above $T_{g}$ are represented by KG models with $\tau$ in the
$10ns$ range.

PIB

has a significantly higher $\tau\approx 350ns$ at a similar distance from the
glass transition temperature. Perhaps this can be explained by specific
intramolecular rotational barriers.Arbe et al. (2001).

a-PS

has a comparable $\tau\approx 140ns$ at $80K$ above $T_{g}$, while

a-PP

maps onto a KG model with $\tau\approx 0.6ns$ at a comparable distance from
$T_{g}$

##  III Predicting emergent universal behavior beyond the Kuhn scale

Having completed the matching of experimental systems and KG model polymer
melts at the Kuhn scale, an obvious question arises: do we have reasons to
believe, that this is “enough” to quantitatively describe the emergent
behavior on much larger scales?

In the following, we provide a brief outline of polymer theory Doi and Edwards
(1986); de Gennes (1979); Rubinstein and Colby (2003); Khokhlov et al. (1994)
to argue that this is indeed the case. The point of the exercise is to
illustrate, that two monodisperse polymer melts of chemically different
polymers are expected to show the same universal large scale properties,
provided (i) they are characterized by the same number of Kuhn segments per
chain, $N_{K}$, (ii) the same dimensionless Kuhn number, $n_{K}$, and (iii)
properties are measured in the “natural” Kuhn units. Our KG models can be
expected to have predictive power for emergent polymer properties, if we may
take it for granted, that this universality of properties of different
chemical species also extends to computational models that exhibit the key
features of polymer melts: chain connectivity, local liquid-like monomer
packing, and the impossibility of chain backbones to dynamically cross through
each other.

We begin our short tour d’horizon with the Rouse model Rouse (1953); Doi and
Edwards (1986), which describes the dynamics of short unentangled polymers.
Rouse considered the Langevin dynamics of a “Gaussian” chain composed of
beads, which experience local friction and which are connected by harmonic
springs representing the entropic elasticity of polymer sections beyond the
Kuhn scale. In this model, the maximal internal relaxation time of a chain is
given by the Rouse time

$\tau_{R}=N_{K}^{2}\,\tau_{K}\ .$ (22)

In the Rouse regime, the macroscopic melt viscosity can be written as

$\eta=n_{K}N_{K}\ \eta_{K}\ .$ (23)

The key for understanding the properties of polymer melts is the realisation,
that chains strongly interpenetrate. The Flory number, $n_{F}=\rho_{c}\langle
R^{2}\rangle^{3/2}$, is defined as the number of chains populating, on
average, the volume spanned by one chain. For our present purposes, it is
important to note that $n_{F}$ and hence the degree of this interpenetration
can again be simply expressed as a sole function of the two dimensionless
numbers characterising a melt on the Kuhn scale:

$n_{F}=n_{K}N_{K}^{1/2}\ $ (24)

or $n_{F}\sim 350$ for natural rubber.

That the Flory number is large, explains why chains behave nearly ideally in
dense meltsFlory (1949) and why such polymer systems can often be well
described by mean-field theories Schmid (1998); Müller and Schmid (2005).
Beyond the Kuhn scale, the universal aspects of mesoscale conformationsWittmer
et al. (2007); Glaser et al. (2014) and liquid structureMüller and Binder
(1995); Clark et al. (2013) in polymeric systems can be described by Gaussian
chain models. In this model, the notion of a chain contour length is lost.
This is particularly apparent in the continuum limit, which is frequently
employed in theoretical calculationsDoi and Edwards (1986); Freed (1987).
Systems are thus characterised by a single dimensionless number, which is
often referred to as the invariant degree of polymerization,

$\bar{N}=\rho_{c}^{2}\langle R^{2}\rangle^{3}=n_{K}^{2}N_{K}=n_{F}^{2}\ .$
(25)

This dimensionless measure plays a key role in more complex polymer systems
such as block-copolymers undergoing micro phase separationFredrickson and
Helfand (1987); Glaser et al. (2014), since it is related to the number of
intermolecular pair-interactions a given reference chain experiences. It is
evident from Eq. (25) that Kuhn matched KG melts can be expected to
automatically reproduce all emergent properties, which depend of the invariant
degree of polymerization, $\bar{N}$. For a more detailed discussion of these
aspects in the context of multi-scale modeling, we refer the reader to Refs.
Müller (2013); Zhang et al. (2015). In the following, we take a closer look a
dynamic properties.

Chains undergoing Brownian motion can slide past each other, however, their
backbones cannot crossDoi and Edwards (1986). As a consequence, the motion of
long chains is subject to long-lived topological constraintsEdwards (1967a).
Modern theories of polymer dynamics Doi and Edwards (1986) are based on the
idea that molecular entanglements confine individual chain segments to a one-
dimensional, diffusive motion (reptation de Gennes (1971)) in tube-like
regions in space Edwards (1967b). The constraints become relevant at scales
beyond the entanglement (contour) lengthFetters et al. (1999a, b), $L_{e}$, or
the equivalent number of Kuhn units between entanglements,
$N_{eK}=L_{e}/l_{K}$. According to the packing argument for loosely entangled
polymers Lin (1987); Kavassalis and Noolandi (1987); Uchida et al. (2008a),
there are

$N_{eK}=\left(\frac{\alpha}{n_{K}}\right)^{2}$ (26)

Kuhn segments per entanglement length. The corresponding spatial scale is the
tube diameter which is given by

$\frac{d_{T}}{l_{K}}=\sqrt{\frac{\langle
R^{2}(N_{eK})\rangle}{l_{K}^{2}}}=\sqrt{N_{eK}}=\frac{\alpha}{n_{K}}$ (27)

To define a length scale characterizing the chain packing in a polymeric
material, one can consider a spherical region centered on a chosen monomer. If
the region is small, then most monomers found inside will belong to the same
chain as the chosen monomer. If the region is large, then most monomers inside
the region belong to other chains. The packing length Witten et al. (1989);
Fetters et al. (1999b) is related to the Kuhn number as

$\frac{p}{l_{K}}=\frac{V_{c}}{l_{K}\langle
R^{2}\rangle}=\left(l_{K}^{3}\rho_{K}\right)^{-1}=n_{K}^{-1},$ (28)

The number of entanglement strands per entanglement volume is

$\alpha=\frac{d_{T}}{p}=\frac{\rho_{K}}{N_{eK}}d_{T}^{3}=18\pm 2\ ,$ (29)

which appears to be a universal constant for all flexible polymers Lin (1987);
Kavassalis and Noolandi (1987); Fetters et al. (1994); Rosa and Everaers
(2014). Values of $\alpha$ for the present polymer species are shown in Tab.
1.

Just like the fundamental time scale $\tau_{K}$ is determined by Kuhn segment
diffusion coefficient and the Kuhn length, the diffusion of an entanglement
segment ($D_{e}=k_{B}T/[N_{eK}\zeta_{K}]$) and the tube diameter also define a
characteristic time scale $\tau_{e}$:

$\frac{\tau_{e}}{\tau_{K}}=N_{eK}^{2}=\left(\frac{\alpha}{n_{K}}\right)^{4}.$
(30)

Above the entanglement time, the Rouse model fails to describe dynamic
correlations in polymer melts. Again, the point to note is, that when defined
in Kuhn units, the spatial and temperal scales that emerges due to
entanglements only depend on the Kuhn number, $n_{K}$.

In the limit of long chains the maximal relaxation time is de Gennes (1971) is
$\tau_{max}=3Z^{3}\tau_{e}$, where $Z=N_{K}/N_{eK}$ denotes the number of
entanglements per chain. Using Eq. (26), we see that

$Z=\frac{n_{K}^{2}N_{K}}{\alpha^{2}}$ (31)

only depends on dimensionless constants, which are reproduced in KG melts
parameterized at the Kuhn scale. As a consequence, the correct number of
entanglements emerges from the microscopic topological constraints in the KG
melts, suggesting that they also exhibit the same asymptotic maximal
relaxation time as the experimental target systems:

$\frac{\tau_{max}}{\tau_{K}}=3Z^{3}\frac{\tau_{e}}{\tau_{K}}=3\frac{n_{K}^{2}N_{K}^{3}}{\alpha^{2}}\
.$ (32)

For chains of finite length, relaxation processes such as contour length
fluctuations and constraint release modify the maximal relaxation time Doi and
Edwards (1986); Likhtman and McLeish (2002a). But as such effects are
controlled through the effective chain length, $Z$, in entanglement units,
they can be also expected to be faithfully reproduced in our KG models Hou et
al. (2010).

Turning now to macroscopic material properties, in slowing down the chain
equilibration after a deformation, entanglements dominate the viscoelastic
behavior of high molecular weight polymeric liquids. For
$\tau_{e}<t<\tau_{max}$ the shear relaxation modulus, $G(t)$ exhibits a
rubber-elastic plateau, $G_{N}=\frac{4}{5}G_{e}$, where the entanglement
modulus isLin (1987); Kavassalis and Noolandi (1987),

$\frac{G_{e}l_{K}^{3}}{k_{B}T}=\frac{\rho_{K}l_{K}^{3}}{N_{eK}}=\frac{n_{K}^{3}}{\alpha^{2}}.$
(33)

From the time dependent shear relaxation modulus one can obtain the shear
compliance and the melt viscosity. The asymptotically expected result Doi and
Edwards (1986) for long entangled chains is given by

$\frac{\eta}{\eta_{K}}=\frac{\pi^{2}}{15}\frac{G_{e}\tau_{max}}{\eta_{K}}=\frac{12}{5}\frac{n_{K}^{5}N_{K}^{3}}{\alpha^{4}}\
.$ (34)

suggesting that this property should also be quantitatively reproduced by our
KG models.

Figure 3: Reduced entanglement moduli for the polymers in Tab. 1 compared to
the theoretical expectation for flexible polymers eq. (33) (red dashed line)
to our the semi-empirical prediction of eq. (42) in Ref. Svaneborg and
Everaers (2018)) (black dotted line) The symbols denote in order of increasing
Kuhn number: PI-50 (orange $\circ$), PI-7 (red $\times$), PDMS (orange
$\ast$), PI-20 (magenta $+$), PI-34 (green $\bigtriangleup$), cis-PBd (blue
$\Box$), PIB(413) (red $\circ$), cis-PI (blue $\ast$), a-PP(463) (orange $+$),
i-PP (orange $\Box$), a-PP(413) (green $\Box$), a-PP(348) (black $+$), a-PP
(red $\Box$), PIB (magenta $\ast$), a-PMMA (indigo $\bigtriangleup$), i-PS
(indigo $\diamond$), a-PMA (red $\diamond$), PI-75 (black $\Box$), PBd-20
(indigo $+$), a-PS (blue $\diamond$), PBd-98 (black $\bigtriangleup$), PEO
(green $\ast$), POM (black $\diamond$), a-PHMA (black $\ast$), a-PVA (blue
$\times$), SBR (blue $\circ$), P6N (black $\times$),h a-P$\alpha$MS (red
$\ast$), a-PEA (green $+$), PET (green $\diamond$), s-PP (indigo $\ast$),
PE(413) (orange $\bigtriangleup$), a-POA (magenta $\diamond$), PC (red
$\bigtriangleup$), PE (orange $\times$), and PTFE (black $\circ$).  Figure 4:
Reduced entanglement moduli for the experimental data in Fig. 3 compared to
the range of KG models for $-1\leq\beta\kappa\leq 2.5$ (blue solid line), with
indications of $\pm 25\%$ error (dashed black lines). Also shown are our the
semi-empirical prediction of eq. (42) in Ref. Svaneborg and Everaers (2018))
(black dotted line)

Figure 3 shows the reduced entanglement moduli as a function of Kuhn number.
The experimental data are in good agreement with eq. (33) for flexible
chains.Fetters et al. (1994) The scatter observed between the experimental
plateau moduli and the predicted plateau modulus line must be attributed
either to chemical details causing some small degree of non-universal
behaviourUnidad et al. (2015), such as a non-negligible crystalline fraction,
or to experimental uncertainties in accurately estimating the plateau modulus
which can be quite difficult.Likhtman and McLeish (2002b) For the very largest
Kuhn numbers, the experimental data points can not discriminate between the
packing argument and the predicted cross-over to the tightly entangled
regime.Uchida et al. (2008a); Svaneborg and Everaers (2018)

Figure 4 shows a comparison between experimental plateau moduli and
entanglement moduli of KG melts extracted from Primitive Path Analysis Hou et
al. (2010); Svaneborg and Everaers (2018). Most of the experimental values are
within the $25\%$ error interval around the line defined by the one parameter
KG models. This is concrete evidence, that the emergent entanglement
properties of our KG models agree with those of the targetted experimental
polymer systems. Interestingly, the stiffer KG models also seem to be in
excellent agreement with the predicted cross-over to tightly entangled
regime.Uchida et al. (2008a); Svaneborg and Everaers (2018)

To summarise: we have illustrated how static, dynamic, mesoscopic and
macroscopic properties of polymer melts depend on just two dimensionless
parameters: the Kuhn number and the number of Kuhn segments per chain length
when these properties are expressed in natural dimensionless form using the
Kuhn length, Kuhn time, Kuhn viscosity and the thermal excitation energy.
Hence we expect to obtain identical dimensionless results from experiments of
chemical polymer systems and from simulations of model polymer systems when
the Kuhn number and number of Kuhn units per chain used in the KG model
simulations match those of the polymer species.

##  IV Discussion

Polymeric systems exhibit a wide range of characteristic time and length
scales. This is readily illustrated for the example of natural rubber, i.e.
melts of cis-PI chains with a typical length of $N_{K}=10^{4}$ Kuhn segments.
Important characteristic length scales comprise (i) the Kuhn length,
$l_{K}\approx 1nm$, (ii) the tube diameter, $d_{T}\approx 5nm$, (iii) the coil
diameter, $\langle R^{2}\rangle\approx 100nm$, and (iv) the contour length,
$L\approx 10\mu m$. The spread is even larger between the characteristic time
scales. There are already almost three orders of magnitude between the Kuhn
time, $\tau_{K}\sim 1\times 10^{-7}s$, and the entanglement time,
$\tau_{K}\sim 7\times 10^{-5}s$. The Rouse time of $\tau_{R}\sim
10^{8}\tau_{K}\sim 10s$ governs fast processes such as the tension
equilibration inside the tube Doi and Edwards (1986), while the estimated
disentanglement time is $\tau_{max}\sim 4h$.

The slow dynamics has dramatic consequences for macroscopic properties such as
the viscosity. Our estimate of the effective viscosity at the Kuhn scale is
$\eta_{K}\approx 0.4Pa\,s$. The viscosity of a short chain melt at the
entanglement threshold, $N_{K}=N_{eK}$, is already two orders of magnitude
larger, $\eta_{e}\approx 40Pa\,s$, while for our strongly entangled
($Z=N_{K}/N_{eK}=400$) example, $\eta\approx 5\times 10^{9}Pa\,s$. In other
words, the long chain melt exhibits a macroscopic viscosity similar to glass
forming liquids close to $T_{g}$, even though locally the chains experience a
friction as if they were immersed in motor oil.

The wide range of relevant time and length scales in polymeric systems makes
them natural targets for multi-scale modelling.Faller (2007); Peter and Kremer
(2009, 2010) In particle-based models, the resolution ranges from the atom
scale to DPD-like descriptions, where entire chains are represented by one or
two soft spheres or ellipsoidsHahn et al. (2001); Müller (2013). What is the
natural place of KG-like models in this hierarchy? And how should they be
parameterized?

Figure 5:  Speed up of KG models due to increased stiffness. The left graph
shows the speed up relative to the standard KG model with $\beta\kappa=0$,
while the right graph shows the number of particle updates,
$N_{eb}\frac{\tau_{e}}{\tau}\frac{\tau}{\delta t}$, required to follow the
dynamics of one entanglement strand over the entanglement time.

###  IV.1 Entanglement vs. Kuhn scale as targets for KG models

Typically, the KG model is mapped to experiments Kremer and Grest (1990);
Kröger (2004) or simulations of more microscopic models Takahashi et al.
(2017) on the entanglement scale. Inspections of the formulas in Sec. III
suggests, that one can hope to reproduce the large scale dynamics by matching
the number of entanglements, $Z$, between target and model system and by
identifying the tube diameter, $d_{T}$, as the unit of spatial distance as
well as the entanglement time, $\tau_{e}$, as the unit of time. Conceptually,
this is what universality is all about and not different from using
experimental data for PDMS to predict universal aspects of the behavior of,
say, amorphous polystyrene.

As illustrated by Fig. 5, it is tempting to use the additional stiffness
parameter Faller et al. (2000) to reduce the CPU time required to reach the
entanglement scale Faller and Müller-Plathe (2001); Halverson et al. (2011).
But how far up the scales can one safely the push characteristic features of
the KG model like the well-defined, almost inextensible contour length and the
almost fully excluded molecular volume? These features are adequate for a
description on the Kuhn scale, but not for a generic model of loosely
entangled chains at the entanglement scale.

Targeting the Kuhn scale, as we advocate here, provides a simple physical
motivation for the choice of the stiffness parameter and should help to reduce
“gaps” Takahashi et al. (2017) relative to predictions of more microscopic
models for the local behavior. Since $\beta\kappa>0$ for most Kuhn matched KG
models of commodity polymers, they are more efficient than the original KG
model in reaching the entanglement scale, even though by targeting the Kuhn
scale the models are nominally more microscopic. Typical speedups are of the
order of 4, in the case of polycarbonate they reach a factor of 30 (Fig. 5).

###  IV.2 Linear vs. nonlinear universality in the rheology of polymer melts

Crucially, we can hope to extend the validity range of the KG model by, to
paraphrase Einstein, making the chains “as stiff as possible, but not
stiffer.” By reproducing the number of Kuhn segments per entanglement length,
$N_{eK}$, the models account for the maximal chain extension, $\sqrt{N_{eK}}$,
under strong deformations. Furthermore, KG melts parameterized at the Kuhn
scale plausibly exhibit friction reduction in fast elongational flows, in so
far as the effect can be attributed to the alignment of the Kuhn segments to
the stretching directionIanniruberto et al. (2012); Yaoita et al. (2012).
There are thus good reasons to expect, that the models discussed in the
present article fulfil all three conditions for non-linear universality in the
rheology of polymer melts Wingstrand et al. (2015).

###  IV.3 Computational performance of Kuhn matched KG models compared to
descriptions on neighboring scales

Kuhn matched KG models are computationally much less demanding than atomistic
simulations. This is due to two factors: (i) There is a considerable reduction
in the number of degrees of freedom. We have not counted atoms, but assuming
carbon and hydrogen atoms as the dominant components, molecular bead weights
between $40g/mol$ and $700g/mol$ translate to $3$ to $50$ united atoms that
are being represented by one KG bead. If hydrogen atoms are represented
explicitly, then these numbers increase by a an additional factor of two or
three. (ii) At the reference temperature, $T_{ref}^{dyn}$, of the rheological
experiments, our estimates for the physical meaning of the KG unit of time,
$\tau$, vary in the range $5ps<1\tau<0.35\mu s$. The corresponding time step
of $50fs<\delta t=10^{-2}\tau<3.5ns$ is thus $50$ to $3.5\times 10^{6}$ times
larger than the $1fs$ time step in atomistic simulations. The time step in
atomistic simulations is dictated by typical frequency of bond vibrations.
Whereas if bond lengths are constrained, then the typical time scale is that
of bond angle vibrations which occurs on time scales of tens of $fs$. Faller
(2007); Ciccotti et al. (1982) For systems closer to the glass transition, the
speedup in modelling the large scale behaviour along the present lines would
be exponentially larger. Compared to an atomistic model, this obviously comes
at the price of loosing the ability to predict any of the glassy behaviour.

With at least $10^{6}$ particle updates per entanglement strand and time
(inset Fig. 5), Kuhn matched KG models are bound to be slower than PPA-
parameterized slip-link augmented DPD-models Hua and Schieber (1998);
Masubuchi et al. (2001); Likhtman (2005); Chappa et al. (2012); Vogiatzis et
al. (2017). Again, the more coarse-grain description benefits from a reduction
of the number of degrees of freedom by a factor of the order of $1/N_{eK}$ as
well as a corresponding reduction of the number of time steps by a factor of
$\tau_{K}/\tau_{e}\sim 1/N_{eK}^{2}$. For the intrinsically most flexible
polymers in Table 1, the speedups may be as large as a factor of $10^{4}$ or
even $10^{5}$ in rare cases. While this approach is clearly successful, there
is nevertheless a price to be paid: effects of topological constraints do not
emerge through the same mechanisms as in the target systems, but are described
within a phenomenological model. While the tube/slip-link model is in general
well understood Likhtman and McLeish (2002b), we suspect that non-linear
universality Wingstrand et al. (2015) or the emergence of crumpling in non-
concatenated ring melts Halverson et al. (2011); Rosa and Everaers (2014)
remain a challenge.

###  IV.4 Kuhn scale matching as a special case of structure based coarse-
graining

The construction of coarse-grain models requires choices and the definition of
(subjective) priorities. A classic example is the tension between structure-
based approachesKremer (2000, 2003); Peter and Kremer (2009) and schemes
focused on preserving thermodynamic propertiesMarrink et al. (2007).

Kuhn scale matching can be viewed as a special case of structure-based coarse-
graining. It is guided by theoretical considerations, which identify the Kuhn
scale as controlling the emergent, universal behavior at larger time and
length scales. Consequently, no particular effort is made to reproduce the
local behavior. The resulting “one parameter force-field” for the KG model is
remarkably simple, but this simplicity obviously comes at the price of loosing
the ability to predict (or to understand) the behaviour of experimental target
systems below the Kuhn scale. In particular, this holds on the bead scale,
where we employ a computationally convenient, generic model without any
particular relation to the properties (or the structure) of the target system.

The techniques for structure-based coarse-graining are well
understoodLyubartsev and Laaksonen (1995); Müller-Plathe (2002b); Izvekov and
Voth (2005); R¨hle et al. (2009). If applied on a similar level of coarse-
graining as our KG models (i.e. retaining a comparable number of degrees of
freedom), the resulting models can be expected to offer a locally more
faithful representation. The differences are probably minor for polymers like
isotactic polystyrene, where our KG beads represent three polystyrene
monomers. The situation is different for polymers like polycarbonate, whose
monomers are represented by several KG beads. There is no reason whatsoever,
why the “beads” arising from systematic coarse-graining ought to be of equal
size, spherical or be joined in a straight line like those of our KG
models.Tschöp et al. (1998); Hahn et al. (2001); Müller-Plathe (2002b) Such
models may provide insight into the relation between structure, local
dynamics, and the dissipation mechanisms responsible for the glassy dynamics,
which is lost in our approach. In terms of computational performance, they
should fall in between atomistic descriptions and Kuhn matched KG models,
since they need to resolve motion on smaller time scales.

###  IV.5 Time scales in coarse-grain models

There is a persistent idea in the literature Kröger (2004) that the time scale
in simulations of coarse-grain models can be inferred by standard dimensional
analysis. The difficulty becomes clear, if we try to follow this approach on
the Kuhn scale. The time scale $l_{K}\sqrt{M_{K}/k_{B}T}$ can be understood as
the time required by a Kuhn segment to ballistically cover a distance
comparable to its size, $l_{K}$, if it moves at its thermal velocity,
$v_{K}^{th}=\sqrt{k_{B}T/M_{K}}$. In contrast, the physically relevant Kuhn
time, $\tau_{K}$, is controlled by the local viscosity, Eq. (4), which emerges
from microscopic interactions below the Kuhn scale and which is expected to
display an exponential WLF temperature dependence, Eqs. (20) and (21).

The systematic linking of times scales on different levels of spatial and
temporal resolution remains a challenge. A conceptual framework is provided by
the Mori-Zwanzig projector formalism.Mori (1965); Nordholm and Zwanzig (1975)
Here the projection operator is defined by the choice of "slow" CG variables.
The formalism provides a generalized Langevin equation (GLE) for the time
evolution of the CG variables, where the effect of the "fast" variables is
described by the GLE memory kernel giving rise to friction and stochastic
forces applied to the slow variables. In practice, sampling such GLE memory
kernels requires simulations of the fast dynamics for fixed slow variables,
which is complicated and has only been achieved relatively recently.Hijón et
al. (2010); Carof et al. (2014); Jung et al. (2017)

In practice Fritz et al. (2011), one often uses a mapping approach, where the
time scale of the coarse-grain simulations is determined by the condition,
that the coarse-grain and the microscopic model predict identical dynamics on
the largest time scales accessible to the microscopic approach. In the present
case, we have used a mapping on the Kuhn scale to estimate the physical
meaning of the KG time scale $\tau$. As shown in Table 3 and Figure 2, these
estimates exceed by orders of magnitude the time scale arising from the
standard combination $\sigma\sqrt{M_{b}/\epsilon}$ of the diameter and mass
$M_{b}$ of the KG beads with the energy scale of the model.

This mismatch strikes us as a natural and highly desired consequence of the
elimination of microscopic degrees of freedom and of the associated
dissipation mechanisms. In principle, it is possible to preserve
$\sigma\sqrt{M_{b}/\epsilon}$ as the definition of time by tuning the friction
of a Langevin (or preferentially, DPDSoddemann et al. (2003)) thermostat such
that the resulting $\tau_{K}$ matches the experimental target value. However,
this would make the simulations orders of magnitude more expensive in terms of
computer time without providing additional physical insight.

###  IV.6 Kuhn scale matched KG models as part of a multiscale hierarchy of
polymer polymers

In our opinion, the Kuhn scale merits to be systematically included in the
hierarchy of multi-scale models of polymeric systems. Omitting it risks to
mask a remarkable simplicity, which emerges from the universality of polymeric
behavior.

We have focused on the KG model with bending rigidity, because it has been
used in a vast number of publications as a basis for studying generic polymer
and materials physics, see e.g. Kremer (2000, 2003); Peter and Kremer (2009)
for reviews. Furthermore, there are several fast equilibration procedures
Zhang et al. (2015); Moreira et al. (2015); Svaneborg et al. (2016), which
allow to build very well equilibrated, highly entangled melt configurations at
relatively low computational cost. Obviously, one could apply the same logic
to bead-spring models with variable density, to models based on chains of rods
rather than beadsSirk et al. (2012) or to lattice modelsCarmesin and Kremer
(1988); Tries et al. (1997); Baiesi et al. (2010); Schram and Barkema (2018)
as long as these capture the relevant physics of polymers.

Kuhn matched polymer models are easy to connect to neighboring scales. In the
“up”-direction, the primitive path analysis Everaers et al. (2004) provides a
systematic link to phenomenological models describing polymers on the
entanglement scaleHua and Schieber (1998); Masubuchi et al. (2001); Likhtman
(2005); Chappa et al. (2012); Vogiatzis et al. (2017). In the “down”
direction, they enable the generation of well equilibrated atomistic material
models through fine-graining of melt configurations of a chemistry-specific KG
model.Carbone et al. (2010)

The information we used here to parameterize the KG model was obtained top-
down from experimentFetters et al. (2007). Our aim was to provide reasonable
estimates of these parameters for a wide variety of polymer species and over
the entire experimentally relevant temperature range. Alternatively, one could
analyse simulations of atomistic Padding and Briels (2002); Liu et al. (2013);
Salerno et al. (2016); Faller and Müller-Plathe (2002); Faller and Reith
(2003); Li et al. (2011); Maurel et al. (2015); Harmandaris et al. (2006);
Chen et al. (2007); Fritz et al. (2011); Karimi-Varzaneh et al. (2008); Eslami
et al. (2011); Chen et al. (2008); Maurel et al. (2015); Strauch et al.
(2009); Milano and Müller-Plathe (2005) or mildly coarse-grain Tschöp et al.
(1998); Abrams and Kremer (2003); Hess et al. (2006) models of target polymers
at specific state points. If the purpose is solely to parameterize the present
model, then it suffices to analyze the simulations along the lines of the
accompanying paper Svaneborg and Everaers (2018). The inferred Kuhn length,
density and time are straightforward to convert into a bottom-up
parameterization of the KG model, which then provides access to much larger
time and length scales than the original, more microscopic model.

##  V Conclusion

We have argued that the Kuhn scale is a natural scale (i) to link theories,
experiments and simulations of amorphous polymer melts and (ii) to target in
building computational polymer models. Omitting the Kuhn scale from the
hierarchy of multi-scale models risks to mask a remarkable simplicity, which
emerges from the universality of polymeric behavior.

In practical terms, we have shown how to model homopolymer melts of a large
variety of polymer species with an extension of the Kremer-Grest model Grest
and Kremer (1986); Kremer and Grest (1990), which was originally introduced by
Faller and Müller-Plathe Faller et al. (1999). The force field has a single
adjustable parameter, the chain stiffness. We determine this parameter by
matching the (Kuhn) number of Kuhn segments per Kuhn volume,
$n_{K}=\rho_{K}l_{K}^{3}$, of the target polymer species and the KG polymer
model. No attempt is made to reproduce smaller scale features. Besides
expressions for estimating the model parameters from experimental input, we
have provided tables listing which bending stiffness to use for particular
polymer species and how to translate KG into SI units. Our estimates for the
mapping from simulation to physical time are based on time-temperature
superposition.

Conceptually, Kuhn scale matching can be seen as a special case of structure
based coarse-graining. The choice of the structural features to be preserved
is guided by theoretical considerations, which identify the Kuhn scale as
controlling the emergent universal polymer behavior at larger time and length
scales.

The resulting coarse-graining level is about one bead per chemical monomer or
two to three beads per Kuhn segment. Kuhn matched KG models thus fall in
between atomistic or mildly coarse-grain models and descriptions on the
entanglement scale. Both coarse-graining steps, from the atom to the Kuhn and
from the Kuhn to the entanglement scale, are associated with performance gains
of several orders of magnitude. For systems close to the glass transition, the
speedup in modelling the large scale behaviour is even exponentially larger.
Compared to atomistic descriptions, Kuhn matched KG models loose the ability
to predict the microscopic (glassy) dynamics or to reproduce semi-crystalline
ordering. Compared to phenomenological entanglement models, Kuhn matched KG
models preserve the emergence of the full spectrum of universal amorphous
polymer properties through the same mechanisms as in the experimental target
systems. In particular, we expect them to automatically fulfil all three
conditions for non-linear universality in the rheology of polymer melts
Wingstrand et al. (2015).

An interesting challenge for future work would be the parameterization of a
corresponding force-field for co-polymer systems, using for example the
technique from Ref. Zhang et al. (2017). While this should, in principle, be
possible at least for static properties, modelling the dynamics might no
longer be as simple as adjusting a single time scale. Similarly, it might be
possible to parameterise minimal models of glassyBennemann et al. (1999);
Baschnagel and Varnik (2005) or semi-crystallineMeyer and Müller-Plathe (2001)
polymers along the present lines.

##  Appendix A Time-temperature superposition

For a TTS reference temperature $T_{0}$ the empirical Williams-Landel-Ferry
(WLF)Williams et al. (1955) shift factor has the form

$\log_{10}a_{T}(T;T_{0})=-\frac{C_{1}(T_{0})(T-T_{0})}{C_{2}(T_{0})+(T-T_{0})}\
.$ (35)

Eq. (35) is valid above the glass transition temperature in the temperature
range $[T_{g},T_{g}+100K]$. Using $T_{g}$ as reference temperature, the
constants adopt “universal” values $C_{1}^{g}\approx 15$ and $C_{2}^{g}\approx
50K$.Williams et al. (1955) Other choices require suitably adjusted parameters
$C_{1}(T_{0})$ and $C_{2}(T_{0})$.

The conversion can be avoided by writing the shift factor in a form derived
from the equivalent Vogel-Fulcher-Tammann-Hesse equation [Refs. 14-16 from
Ngai]

$\ln a_{T}(T;T_{0})=-\frac{C_{VF}(T-T_{0})}{(T_{0}-T_{VF})(T-T_{VF})}\ .$ (36)

The relations

$\displaystyle T_{VF}=T_{0}-C_{2}(T_{0})$ $\displaystyle\Leftrightarrow
C_{2}(T_{0})=T_{0}-T_{VF}$ $\displaystyle
C_{VF}=\ln(10)C_{1}(T_{0})C_{2}(T_{0})$ $\displaystyle\Leftrightarrow
C_{1}(T_{0})=\frac{C_{VF}}{\ln(10)(T_{0}-T_{VF})}$

with $C_{VF}\approx 2000K$ allow to pass between the two representations. In
particular, Eqs. (35) and (36) suggests (i) that the viscosity diverges at the
Vogel-Fulcher temperature $T_{VF}=T_{g}-C_{2}^{g}$ located $\sim 50K$ below
the glass transition temperature and (ii) that the orders of magnitude by
which the viscosity drops in the opposite limit of $T\rightarrow\infty$ are
given by $C_{1}(T_{0})$ and are hence inversely proportional to the distance
of the reference from the Vogel-Fulcher temperature. More detailed information
and specific tables can be found in Ref. Ngai and Plazek (2007). Note,
however, that their Eq. (26.3) ought to read

$\log_{10}a_{T}(T;T_{0})=\frac{C_{VF}/\ln(10)}{T-T_{VF}}-\frac{C_{VF}/\ln(10)}{T_{0}-T_{VF}}\
.$ (39)

## References

  * Flory (1953) P. J. Flory, _Principles of polymer chemistry_ (Cornell University Press, Ithaca N.Y, 1953).
  * Flory and Volkenstein (1969) P. Flory and M. Volkenstein, _Statistical mechanics of chain molecules_ (1969).
  * Doi and Edwards (1986) M. Doi and S. F. Edwards, _The Theory of Polymer Dynamics_ (Clarendon, Oxford, 1986).
  * de Gennes (1979) P. G. de Gennes, _Scaling Concepts in Polymer Physics_ (Cornell University Press, Ithaca NY, 1979).
  * Binder (1995) K. Binder, _Monte Carlo and Molecular Dynamics simulations in polymer science_ (Oxford University Press: New York, 1995).
  * Kremer (2000) K. Kremer, in _Soft and fragile matter: Non equilibrium dynamics, metastability and flow_ , edited by M. E. Cates and M. R. Evans (J. W. Arrowsmith Ltd., Bristol, U.K., 2000), p. 145\.
  * Kremer (2003) K. Kremer, Macromol. Chem. Phys. 204, 257 (2003).
  * Kröger (2004) M. Kröger, Phys. Rep. 390, 453 (2004).
  * Müller-Plathe (2002a) F. Müller-Plathe, Soft Mater. 1, 1 (2002a).
  * Müller-Plathe (2002b) F. Müller-Plathe, Chem. Phys. Chem. 3, 754 (2002b).
  * Sun and Faller (2005) Q. Sun and R. Faller, Comput. Chem. Eng. 29, 2380 (2005).
  * Tzoumanekas and Theodorou (2006) C. Tzoumanekas and D. N. Theodorou, Curr. Opin. Solid State Mater. Sci. 10, 61 (2006).
  * Peter and Kremer (2009) C. Peter and K. Kremer, Soft Matter 5, 4357 (2009).
  * Li et al. (2013) Y. Li, B. C. Abberton, M. Kröger, and W. K. Liu, Polymers 5, 751 (2013).
  * Padding and Briels (2002) J. Padding and W. Briels, J. Chem. Phys. 117, 925 (2002).
  * Liu et al. (2013) L. Liu, J. Padding, W. den Otter, and W. Briels, J. Chem. Phys. 138, 244912 (2013).
  * Salerno et al. (2016) K. M. Salerno, A. Agrawal, D. Perahia, and G. S. Grest, Phys. Rev. Lett. 116, 058302 (2016).
  * Faller and Müller-Plathe (2002) R. Faller and F. Müller-Plathe, Polymer 43, 621 (2002).
  * Faller and Reith (2003) R. Faller and D. Reith, Macromolecules 36, 5406 (2003).
  * Li et al. (2011) Y. Li, M. Kröger, and W. K. Liu, Polymer 52, 5867 (2011).
  * Maurel et al. (2015) G. Maurel, F. Goujon, B. Schnell, and P. Malfreyt, RSC Adv. 5, 14065 (2015).
  * Harmandaris et al. (2006) V. Harmandaris, N. Adhikari, N. F. van der Vegt, and K. Kremer, Macromolecules 39, 6708 (2006).
  * Chen et al. (2007) X. Chen, P. Carbone, W. L. Cavalcanti, G. Milano, and F. Müller-Plathe, Macromolecules 40, 8087 (2007).
  * Fritz et al. (2011) D. Fritz, K. Koschke, V. A. Harmandaris, N. F. van der Vegt, and K. Kremer, Phys. Chem. Chem. Phys. 13, 10412 (2011).
  * Karimi-Varzaneh et al. (2008) H. A. Karimi-Varzaneh, P. Carbone, and F. Müller-Plathe, J. Chem. Phys. 129, 154904 (2008).
  * Eslami et al. (2011) H. Eslami, H. A. Karimi-Varzaneh, and F. Müller-Plathe, Macromolecules 44, 3117 (2011).
  * Chen et al. (2008) C. Chen, P. Depa, J. K. Maranas, and V. G. Sakai, J. Chem. Phys. 128, 124906 (2008).
  * Tschöp et al. (1998) W. Tschöp, K. Kremer, J. Batoulis, T. Bürger, and O. Hahn, Acta. Polym. 49, 61 (1998).
  * Abrams and Kremer (2003) C. F. Abrams and K. Kremer, Macromolecules 36, 260 (2003).
  * Hess et al. (2006) B. Hess, S. León, N. Van Der Vegt, and K. Kremer, Soft Matter 2, 409 (2006).
  * Strauch et al. (2009) T. Strauch, L. Yelash, and W. Paul, Phys. Chem. Chem. Phys. 11, 1942 (2009).
  * Milano and Müller-Plathe (2005) G. Milano and F. Müller-Plathe, J. Phys. Chem. B. 109, 18609 (2005).
  * Baschnagel et al. (2000) J. Baschnagel, K. Binder, P. Doruker, A. A. Gusev, O. Hahn, K. Kremer, W. L. Mattice, F. Müller-Plathe, M. Murat, and W. Paul, in _Viscoelasticity, atomistic models, statistical chemistry_ (Springer, 2000), p. 41.
  * Carbone et al. (2008) P. Carbone, H. A. K. Varzaneh, X. Chen, and F. Müller-Plathe, J. Chem. Phys. 128, 064904 (2008).
  * Johnson et al. (2007) M. E. Johnson, T. Head-Gordon, and A. A. Louis, J. Chem. Phys. 126, 144509 (2007).
  * Sukumaran et al. (2005) S. K. Sukumaran, G. S. Grest, K. Kremer, and R. Everaers, J. Polym. Sci., Part B: Polym. Phys. 43, 917 (2005).
  * Graessley (1982) W. W. Graessley, Adv. Polym. Sci. 47, 67 (1982).
  * Marrucci (1985) G. Marrucci, J. Polym. Sci, Polym. Phys. Ed. 23, 159 (1985).
  * Rubinstein and Colby (1988) M. Rubinstein and R. H. Colby, J. Chem. Phys. 89, 5291 (1988).
  * Marrucci and Ianniruberto (2003) G. Marrucci and G. Ianniruberto, Phil. Trans. R. Soc. Lond. A 361, 677 (2003).
  * Ianniruberto and Marrucci (2014) G. Ianniruberto and G. Marrucci, J. Rheol. 58, 89 (2014).
  * Rosa and Everaers (2008) A. Rosa and R. Everaers, PLoS. Comput. Biol. 4, e1000153 (2008).
  * Halverson et al. (2012) J. D. Halverson, G. S. Grest, A. Y. Grosberg, and K. Kremer, Phys. Rev. Lett. 108, 038301 (2012).
  * Halverson et al. (2011) J. D. Halverson, W. B. Lee, G. S. Grest, A. Y. Grosberg, and K. Kremer, J. Chem. Phys. 134, 204904 (2011).
  * Rosa and Everaers (2014) A. Rosa and R. Everaers, Phys. Rev. Lett. 112, 118302 (2014).
  * Wittmer et al. (2004) J. P. Wittmer, H. Meyer, J. Baschnagel, A. Johner, S. Obukhov, L. Mattioni, M. Müller, and A. N. Semenov, Phys. Rev. Lett. 93, 147801 (2004).
  * Grest et al. (1989) G. S. Grest, K. Kremer, S. T. Milner, and T. A. Witten, Macromolecules 22, 1904 (1989).
  * Grest and Kremer (1990) G. S. Grest and K. Kremer, Macromolecules 23, 4994 (1990).
  * Duering et al. (1991) E. R. Duering, K. Kremer, and G. S. Grest, Phys. Rev. Lett. 67, 3531 (1991).
  * Svaneborg et al. (2004) C. Svaneborg, G. S. Grest, and R. Everaers, Phys. Rev. Lett. 93, 257801 (2004).
  * Svaneborg et al. (2008) C. Svaneborg, R. Everaers, G. S. Grest, and J. G. Curro, Macromolecules 41, 4920 (2008).
  * Aoyagi et al. (2001) T. Aoyagi, J.-I. Takimoto, and M. Doi, J. Chem. Phys. 115, 552 (2001).
  * Pierce et al. (2009) F. Pierce, D. Perahia, and G. S. Grest, Macromolecules 42, 7969 (2009).
  * Murat and Grest (1991) M. Murat and G. S. Grest, Macromolecules 24, 704 (1991).
  * Murat and Grest (1989) M. Murat and G. S. Grest, Phys. Rev. Lett. 63, 1074 (1989).
  * Grest (1996) G. S. Grest, Phys. Rev. Lett. 76, 4979 (1996).
  * Yagyu and Utsumi (2009) H. Yagyu and T. Utsumi, Comput. Mater. Sci. 46, 286 (2009).
  * Sussman et al. (2014) D. M. Sussman, W.-S. Tung, K. I. Winey, K. S. Schweizer, and R. A. Riggleman, Macromolecules 47, 6462 (2014).
  * Gersappe (2002) D. Gersappe, Phys. Rev. Lett. 89, 058301 (2002).
  * Ge et al. (2014) T. Ge, M. O. Robbins, D. Perahia, and G. S. Grest, Phys. Rev. E. 90, 012602 (2014).
  * Grest and Kremer (1986) G. S. Grest and K. Kremer, Phys. Rev. A 33, 3628 (1986).
  * Kremer and Grest (1990) K. Kremer and G. S. Grest, J. Chem. Phys. 92, 5057 (1990).
  * Faller et al. (1999) R. Faller, A. Kolb, and F. Müller-Plathe, Phys. Chem. Chem. Phys. 1, 2071 (1999).
  * Faller et al. (2000) R. Faller, F. Müller-Plathe, and A. Heuer, Macromolecules 33, 6602 (2000).
  * Faller and Müller-Plathe (2001) R. Faller and F. Müller-Plathe, Chem. Phys. Chem. 2, 180 (2001).
  * Svaneborg and Everaers (2018) C. Svaneborg and R. Everaers, _Characteristic time and length scales in kremer-grest bead-spring polymer melts as a function of chain stiffness. Manuscript submitted to Macromolecules._ (2018).
  * Meyer and Müller-Plathe (2001) H. Meyer and F. Müller-Plathe, The Journal of Chemical Physics 115, 7807 (2001).
  * Hoy and Karayiannis (2013) R. S. Hoy and N. C. Karayiannis, Phys. Rev. E 88, 012601 (2013).
  * Morthomas et al. (2017) J. Morthomas, C. Fusco, Z. Zhai, O. Lame, and M. Perez, Phys. Rev. E. 96, 052502 (2017).
  * Rubinstein and Colby (2003) M. Rubinstein and R. H. Colby, _Polymer physics_ (Oxford University, New York, 2003).
  * Kuhn (1934) W. Kuhn, Kolloid. 1, 2 (1934).
  * Kratky and Porod (1949) O. Kratky and G. Porod, Rec. Trav. Chim. 1106, 2211 (1949).
  * Ding and Sokolov (2006) Y. Ding and A. P. Sokolov, Macromolecules 39, 3322 (2006).
  * Fetters et al. (2007) L. J. Fetters, D. J. Lohse, and R. H. Colby, in _Physical properties of polymers handbook_ , edited by J. Mark (Springer, 2007), p. 447\.
  * Uchida et al. (2008a) N. Uchida, G. S. Grest, and R. Everaers, J. Chem. Phys. 128, 044902 (2008a).
  * Meyer et al. (2013) S. Meyer, D. Jost, N. Theodorakopoulos, M. Peyrard, R. Lavery, and R. Everaers", Biophys. J. 105, 1904 (2013).
  * Uchida et al. (2008b) N. Uchida, G. S. Grest, and R. Everaers, J. Chem. Phys. 128, 044902 (2008b).
  * Gotro and Graessley (1984) J. T. Gotro and W. W. Graessley, Macromolecules 17, 2767 (1984).
  * Abdel-Goad et al. (2004) M. Abdel-Goad, W. Pyckhout-Hintzen, S. Kahle, J. Allgaier, D. Richter, and L. J. Fetters, Macromolecules 37, 8135 (2004).
  * Vaca Chávez and Saalwächter (2011) F. Vaca Chávez and K. Saalwächter, Macromolecules 44, 1549 (2011).
  * Auhl et al. (2008) D. Auhl, J. Ramirez, A. E. Likhtman, P. Chambon, and C. Fernyhough, J. Rheol. 52, 801 (2008).
  * van Meerveld (2004) J. van Meerveld, Rheol. Acta. 43, 615 (2004).
  * Niedzwiedz et al. (2008) K. Niedzwiedz, A. Wischnewski, W. Pyckhout-Hintzen, J. Allgaier, D. Richter, and A. Faraone, Macromolecules 41, 4866 (2008).
  * Richter et al. (2005) D. Richter, M. Monkenbusch, A. Arbe, and J. Colmenero, _Neutron spin echo in polymer systems_ (Springer, 2005).
  * Bird et al. (1977) R. B. Bird, R. C. Armstrong, and O. Hassager, _Dynamics of Polymeric Liquids_ , vol. 1 (Wiley, New York, 1977).
  * Watanabe (1999) H. Watanabe, Prog. Polym. Sci. 24, 1253 (1999).
  * McLeish (2002) T. C. B. McLeish, Adv. Phys. 5, 1379 (2002).
  * Saalwächter (2007) K. Saalwächter, Prog. Nucl. Magn. Reson. Spectrosc. 51, 1 (2007).
  * Ngai and Plazek (2007) K. L. Ngai and D. J. Plazek, in _Physical Properties of Polymers Handbook_ (Springer, 2007), p. 455\.
  * Arbe et al. (2001) A. Arbe, M. Monkenbusch, J. Stellbrink, D. Richter, B. Farago, K. Almdal, and R. Faust, Macromolecules 34, 1281 (2001).
  * Khokhlov et al. (1994) A. R. Khokhlov, A. Y. Grosberg, and V. S. Pande, _Statistical Physics of Macromolecules_ (AIP-Press, Jericho NY, 1994).
  * Rouse (1953) P. E. Rouse, J. Chem. Phys. 21, 1272 (1953).
  * Flory (1949) P. J. Flory, J. Chem. Phys. 17, 303 (1949).
  * Schmid (1998) F. Schmid, J. Phys. Cond. Matter 10, 8105 (1998).
  * Müller and Schmid (2005) M. Müller and F. Schmid, in _Advanced Computer Simulation Approaches for Soft Matter Sciences II_ (Springer, 2005), p. 1.
  * Wittmer et al. (2007) J. P. Wittmer, P. Beckrich, H. Meyer, A. Cavallo, A. Johner, and J. Baschnagel, Phys. Rev. E. 76, 011803 (2007).
  * Glaser et al. (2014) J. Glaser, J. Qin, P. Medapuram, and D. C. Morse, Macromolecules 47, 851 (2014).
  * Müller and Binder (1995) M. Müller and K. Binder, Macromolecules 28, 1825 (1995).
  * Clark et al. (2013) A. Clark, J. McCarty, and M. Guenza, J. Chem. Phys. 139, 124906 (2013).
  * Freed (1987) K. F. Freed, _"Renormalization Group Theory of Macromolecules."_ (Wiley Interscience, New York, 1987).
  * Fredrickson and Helfand (1987) G. H. Fredrickson and E. Helfand, J. Chem. Phys 87, 697 (1987).
  * Müller (2013) M. Müller, in _Hybrid Particle-Continuum Methods in Computational Materials Physics_ , edited by M. H. Müser, G. Sutmann, and R. G. Winkler (Forschungszentrum Jülich, 2013), vol. 46, p. 127.
  * Zhang et al. (2015) G. Zhang, T. Stuehn, K. C. Daoulas, and K. Kremer, J. Chem. Phys. 142, 221102 (2015).
  * Edwards (1967a) S. F. Edwards, Proc. Phys. Soc. 91, 513 (1967a).
  * de Gennes (1971) P. G. de Gennes, J. Chem. Phys. 55, 572 (1971).
  * Edwards (1967b) S. F. Edwards, Proc. Phys. Soc. 92, 9 (1967b).
  * Fetters et al. (1999a) L. J. Fetters, D. J. Lohse, and W. W. Graessley, J. Polym. Sci., Part B: Polym. Phys. 37, 1023 (1999a).
  * Fetters et al. (1999b) L. J. Fetters, D. J. Lohse, S. T. Milner, and W. W. Graessley, Macromolecules 32, 6847 (1999b).
  * Lin (1987) Y.-H. Lin, Macromolecules 20, 3080 (1987).
  * Kavassalis and Noolandi (1987) T. A. Kavassalis and J. Noolandi, Phys. Rev. Lett. 59, 2674 (1987).
  * Witten et al. (1989) T. A. Witten, S. T. Milner, and Z.-G. Wang, in _Multiphase macromolecular systems_ , edited by B. Culbertson (Plenum Press, New York, 1989), p. 655\.
  * Fetters et al. (1994) L. J. Fetters, D. J. Lohse, D. Richter, T. A. Witten, and A. Zirkel, Macromolecules 27, 4639 (1994).
  * Likhtman and McLeish (2002a) A. E. Likhtman and T. C. McLeish, Macromolecules 35, 6332 (2002a).
  * Hou et al. (2010) J.-X. Hou, C. Svaneborg, R. Everaers, and G. S. Grest, Phys. Rev. Lett. 105, 068301 (2010).
  * Unidad et al. (2015) H. J. Unidad, M. A. Goad, A. R. Bras, M. Zamponi, R. Faust, J. Allgaier, W. Pyckhout-Hintzen, A. Wischnewski, D. Richter, and L. J. Fetters, Macromolecules 48, 6638 (2015).
  * Likhtman and McLeish (2002b) A. Likhtman and T. McLeish, Macromolecules 35, 6332 (2002b).
  * Faller (2007) R. Faller, Rev. Comp. Ch. 23, 233 (2007).
  * Peter and Kremer (2010) C. Peter and K. Kremer, Faraday Discuss. 144, 9 (2010).
  * Hahn et al. (2001) O. Hahn, L. D. Site, and K. Kremer, Macromol. Theory Simul. 10, 288 (2001).
  * Takahashi et al. (2017) K. Z. Takahashi, R. Nishimura, N. Yamato, K. Yasuoka, and Y. Masubuchi, Nature 7, 12379 (2017).
  * Ianniruberto et al. (2012) G. Ianniruberto, A. Brasiello, and G. Marrucci, Macromolecules 45, 8058 (2012).
  * Yaoita et al. (2012) T. Yaoita, T. Isaki, Y. Masubuchi, H. Watanabe, G. Ianniruberto, and G. Marrucci, Macromolecules 45, 2773 (2012).
  * Wingstrand et al. (2015) S. L. Wingstrand, N. J. Alvarez, Q. Huang, and O. Hassager, Phys. Rev. Lett. 115, 078302 (2015).
  * Ciccotti et al. (1982) G. Ciccotti, M. Ferrario, and J.-P. Ryckaert, Mol. Phys. 47, 1253 (1982).
  * Hua and Schieber (1998) C. C. Hua and J. D. Schieber, J. Chem. Phys. 109, 10018 (1998).
  * Masubuchi et al. (2001) Y. Masubuchi, J.-I. Takimoto, K. Koyama, G. Ianniruberto, G. Marrucci, and F. Greco, J. Chem. Phys. 115, 4387 (2001).
  * Likhtman (2005) A. E. Likhtman, Macromolecules 38, 6128 (2005).
  * Chappa et al. (2012) V. C. Chappa, D. C. Morse, A. Zippelius, and M. Müller, Phys. Rev. Lett. 109, 148302 (2012).
  * Vogiatzis et al. (2017) G. G. Vogiatzis, G. Megariotis, and D. N. Theodorou, Macromolecules 50, 3004 (2017).
  * Marrink et al. (2007) S. J. Marrink, H. J. Risselada, S. Yefimov, D. P. Tieleman, and A. H. De Vries, J. Phys. Chem. B. 111, 7812 (2007).
  * Lyubartsev and Laaksonen (1995) A. P. Lyubartsev and A. Laaksonen, Phys. Rev. E. 52, 3730 (1995).
  * Izvekov and Voth (2005) S. Izvekov and G. A. Voth, J. Chem. Phys. 123, 134105 (2005).
  * R¨hle et al. (2009) V. R¨hle, C. Junghans, A. Lukyanov, K. Kremer, and D. Andrienko, J. Chem. Theory Comput. 5, 3211 (2009).
  * Mori (1965) H. Mori, Prog. Theor. Phys. 33, 423 (1965).
  * Nordholm and Zwanzig (1975) S. Nordholm and R. Zwanzig, J. Stat. Phys. 13, 347 (1975).
  * Hijón et al. (2010) C. Hijón, P. Español, E. Vanden-Eijnden, and R. Delgado-Buscalioni, Faraday discuss. 144, 301 (2010).
  * Carof et al. (2014) A. Carof, R. Vuilleumier, and B. Rotenberg, J. Chem. Phys. 140, 124103 (2014).
  * Jung et al. (2017) G. Jung, M. Hanke, and F. Schmid, J. Chem. Theory Comput. 13, 2481 (2017).
  * Soddemann et al. (2003) T. Soddemann, B. Dünweg, and K. Kremer, Phys. Rev. E. 68, 046702 (2003).
  * Moreira et al. (2015) L. A. Moreira, G. Zhang, F. Müller, T. Stuehn, and K. Kremer, Macromol. Theory Simul. 24, 419 (2015).
  * Svaneborg et al. (2016) C. Svaneborg, H. A. Karimi-Varzaneh, N. Hojdis, F. Fleck, and R. Everaers, Phys. Rev. E 94, 032502 (2016).
  * Sirk et al. (2012) T. W. Sirk, Y. R. Slizoberg, J. K. Brennan, M. Lisal, and J. W. Andzelm, J. Chem. Phys. 136, 134903 (2012).
  * Carmesin and Kremer (1988) I. Carmesin and K. Kremer, Macromolecules 21, 2819 (1988).
  * Tries et al. (1997) V. Tries, W. Paul, J. Baschnagel, and K. Binder, J. Chem. Phys 106, 738 (1997).
  * Baiesi et al. (2010) M. Baiesi, G. T. Barkema, and E. Carlon, Phys. Rev. E 81, 061801 (2010).
  * Schram and Barkema (2018) R. Schram and G. Barkema, J. Comput. Phys. 363, 128 (2018).
  * Everaers et al. (2004) R. Everaers, S. K. Sukumaran, G. S. Grest, C. Svaneborg, A. Sivasubramanian, and K. Kremer, Science 303, 823 (2004).
  * Carbone et al. (2010) P. Carbone, H. A. Karimi-Varzaneh, and F. Müller-Plathe, Faraday Discuss. 144, 25 (2010).
  * Zhang et al. (2017) W. Zhang, E. D. Gomez, and S. T. Milner, Phys. Rev. Lett. 119, 017801 (2017).
  * Bennemann et al. (1999) C. Bennemann, J. Baschnagel, W. Paul, and K. Binder, Comput. Theor. Polym. Sci. 9, 217 (1999).
  * Baschnagel and Varnik (2005) J. Baschnagel and F. Varnik, J. Phys.: Condens. Matter 17, R851 (2005).
  * Williams et al. (1955) M. L. Williams, R. F. Landel, and J. D. Ferry, J. Am. Chem. Soc. 77, 3701 (1955).

Table of content graphic

Kremer-Grest models for commodity polymer melts: Linking theory, experiment
and simulation at the Kuhn scale.

Carsten Svaneborg, Hossein Ali Karimi-Varzaneh, Nils Hojdis, Frank Fleck, and
Ralf Everaers.