Coarse-graining polymers with the MARTINI force-field: polystyrene as a benchmark case

Abstract

Rossi, Giulia Monticelli, Luca Puisto, Sakari R. Vattulainen, Ilpo Ala-Nissila, Tapio

Full text

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Coarse-graining polymers with the MARTINI force-field: polystyrene as
a benchmark case†
Giulia Rossi,*aLuca Monticelli,bcSakari R. Puisto,dIlpo Vattulainenaefand Tapio Ala-Nissilaag
Received 7th June 2010, Accepted 14th August 2010
DOI: 10.1039/c0sm00481b
We hereby introduce a new hybrid thermodynamic-structural approach to the coarse-graining of
polymers. The new model is developed within the framework of the MARTINI force-field (Marrink
et al., J. Phys. Chem. B, 2007, 111, 7812), which uses mainly thermodynamic properties as targets in the
parameterization. We refine the MARTINI procedure by including one additional target property
related to the structure of the polymer, namely the radius of gyration. The force-field optimization is
mainly based on experimental data. We test our procedure on polystyrene, a standard benchmark for
coarse-grained (CG) polymer force-fields. Our model preserves the backbone-ring structure of the
molecule, with each monomer represented by four CG beads. Structural properties in the melt are well
reproduced, and their scaling with chain length agrees with the available experimental data. The time
conversion factor between the CG and the atomistic simulations is nearly constant over a wide
temperature range, and the CG force-field shows reasonable transferability between 350 and 600 K.
The model is computationally efficient and polymer melts can be simulated over length scales of tens of
nanometres and time scales of tens of microseconds. Finally, we tested our model in dilute conditions.
The collapse of the polymer chains in a bad solvent and the swelling in a good solvent could be
reproduced.
1 Introduction
Optimization of polymer properties in industrial applications is
generally achieved by controlling the fine details of their chemical
composition, often through expensive and time-consuming trial-
and-error procedures. Ideally, one can speed up these procedures
via computer modeling, predicting changes in the material
properties as a function of chemical composition. Unfortunately,
despite ever increasing computational resources, simulations of
polymer melts from atomistic detail are subject to stringent
limitations in both the time and length scales of the phenomena
that can be observed. For example, simulating the mechanical
response of a polymeric material requires simulations of tens or
hundreds of microseconds and a length scale of tens of nano-
metres, currently not accessible with fully atomistic descriptions.
Coarse-graining strategies can help to overcome these limita-
tions. Coarse-graining involves grouping clusters of atoms into
super-atoms, or beads, thus reducing the total number of parti-
cles in the system. Generally, coarse-grained (CG) models are
computationally faster than atomistic ones thanks to a reduction
in the number of degrees of freedom and the use of smoother
interaction potentials, allowing for a longer time step in the
molecular dynamics (MD) simulations. Simple, chemically
aspecific CG models of polymer chains have been very useful in
improving our understanding of the general features of polymer
structure and dynamics.1Generic freely jointed, bead-spring and
self-avoiding model chains are still the subject of computational
and theoretical investigation. These models do not contain any
information on the specific chemistry of the polymer, therefore
they cannot be used to investigate the difference in material
properties of different polymers.
During the last decade, driven by the need for less generic
polymer models, several CG models have been developed with
some chemical specificity. These models aim to capture some of
the properties of polymer systems with a specific chemical
composition without studying the molecules at the atomic level
of detail. Most of the development of specific CG models has
so far been focused on typical benchmark chains like poly-
carbonates,2polystyrene3–6(PS) and polyamide.4
Different methodologies have been used in the development
of CG models for polymers including chemical specificity. In
structure-based CG models,3–5the CG interaction parameters are
tuned to reproduce accurately structural features of the system
(typically, radial distribution functions) generally derived from
all-atom MD simulations. Structure-based CG models also
suffer from some limitations. First of all, transferability of
interaction parameters to different temperatures is often poor, as
thoroughly discussed by Carbone et al.4For simple molecules,
such as ethylbenzene, the temperature dependence of the
aDepartment of Applied physics, Aalto University School of Science and
Technology, P.O. Box 11000, FI-00076 AALTO Helsinki, Finland.
E-mail: giulia.rossi@tkk.fi
bINSERM, UMR-S 665, DSIMB, 6 rue Alexandre Cabanel, 75015 Paris,
France
cUniversit? e Paris Diderot – Paris 7, UFR Life Sciences, Paris, France
dMatox Ltd, Pembroke House, 36–37 Pembroke Street, Oxford, OX1
1BP, United Kingdom
eDepartment of Physics, Tampere University of Technology, P.O. Box 692,
FI-33101 Tampere, Finland
fMEMPHYS - Center of Biomembrane Physics, University of Southern
Denmark, Campusvej 55, DK-5230 Odense M, Denmark
gDepartment of Physics,BrownUniversity,P.O. Box1843,Providence,RI,
02912-1843, USA
† Electronic supplementary information (ESI) available: Simulation
details, atomistic and CG distributions of bonded degrees of freedom
and radial distribution functions. See DOI: 10.1039/c0sm00481b
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structure-based coarse-grained potential can be corrected a pos-
teriori,7but this is not always the case with other polymers.
Ultimately, structure-based models of polymers require addi-
tional parameterization efforts if one wishes to use them in
combination with solvents7or other polymer species.8Indeed,
CG simulation studies of polymers in dilute conditions are rare.
Compared to structure-based CG models, thermodynamics-
based models rely on a very different coarse-graining strategy.
In the thermodynamics-based approach, interaction parameters
are chosen in such a way as to reproduce the selected thermo-
dynamic properties of the system, and only a few structural
properties. Thermodynamics-based approaches have been often
used for modeling biological environments, and amphiphilic
molecules therein.9Nielsen et al.10developed a CG model for
n-alkanes based on the reproduction of experimental bulk
density and surface tension. The model could be later combined
with a structure-based CG description of hydrophilic groups,
resulting in a CG model of amphiphilic molecules such as
dimyristoylphosphatidylchloline
poly(ethyleneoxide)-poly(ethylethylene).11
Another example of thermodynamics-based CG models is the
MARTINI force field. The MARTINI CG force-field was
originally developed for lipids12,13and then extended to
proteins,14fullerenes15and carbohydrates.16In the MARTINI
approach, interaction parameters are determined by reproducing
densities and free energies of partitioning. With respect to other
approaches, this offers a number of advantages. (i) MARTINI
has been parameterized for a large number of chemical building-
blocks, which offers the possibility to build-up CG models of
a wide range of molecules. (ii) Its parameterization of non-
bonded (excluded-volume and inter-molecular) interactions
targets experimental data, and thus it does not inherit possible
failures of all-atom force fields. (iii) Its non-bonded interactions
are modeled by simple Lennard-Jones and Coulomb functions,
making the parameterization of new building blocks a relatively
easy task. On the other hand, since little structural input is taken
into account during the parameterization stage, MARTINI can
fail in reproducing some of the specific structural characteristics
of polymer systems.
One possible solution to overcome some of the typical limi-
tations of structure-based and thermodynamics-based models is
to adopt a more ‘‘hybrid’’ approach, in which both structural
and thermodynamic properties are used as a target during the
parameterization. In the present work we describe the applica-
tion of such a hybrid approach to the case of polystyrene. Since
most of the parameterization work is consistent with the
MARTINI philosophy, we can still refer to our polymer model
as a MARTINI polymer model. To our knowledge, MARTINI
has been applied so far to model only one polymer system,
namely polyethylene glycol.17Polystyrene has more complex
structural features, and should thus serve as an excellent
benchmark for the development of MARTINI-based models for
polymers.
This manuscript is divided in three main parts. In Section 2 we
will describe the force-field parameterization and optimization.
Then, in Section 3 the model will be validated by showing to what
extent we are able to reproduce structural and dynamic proper-
ties that were not used as input during the optimization stage.
Finally, in Section 4 we will discuss the performance of the model
andadiblock copolymer,
in terms of temperature and environment transferability, and
highlight the advantages and limitations of our model compared
to other models available in the literature.
2Development of the polystyrene models
Before considering the parameterization problem, two funda-
mental choices need to be made: how to map atoms onto coarse-
grained interaction sites and what properties should be used as
targets in the parameterization. The choice of interaction sites to
be mapped, from the atomistic to the CG description, is often
arbitrary. Recently, attempts to investigate the possibility to
develop automatic mapping procedures have been reported.18–20
We have considered two alternative mappings for polystyrene, as
described in Section 2.1. As for the target properties to be used in
the parameterization of the model, we chose to rely on both
structural and thermodynamic properties. Distributions of the
distances and angles from atomistic simulations were used as
a target in the parameterization of the bonded interactions.
Non-bonded interactions were based on reproducing the exper-
imental density and radius of gyration of the polymer chains in
the melt (at 500 K). Due to the lack of experimental data on
the radius of gyration for smaller molecular weights, we also
used results from atomistic simulations. Since the parameteri-
zation of both bonded and non-bonded interactions relies partly
on comparing atomistic and CG simulations, we first provide
a description of the setup and conditions in Section 2.2. An
explanation of the parameterization procedure is reported in
Sections 2.3 and 2.4.
2.1Mapping
In the MARTINI approach, mapping of CG beads onto atoms is
not uniquely defined, and usually a few alternatives are possible.
MARTINI beads generally represent a group of four heavy
atoms.12More recently smaller beads have been introduced13to
better match the structure of cyclic compounds, such as benzene
and cyclohexane, which can be represented by three inter-
connected small beads. In our PS model, each styrene unit is
represented by four small MARTINI beads, with three beads
representing the phenyl moiety and one bead for the aliphatic
part, as illustrated in Fig. 1. Thanks to the relatively low level of
coarse-graining, this approach allows us to preserve the planar
structure of the phenyl rings. This mapping can be realized in two
different ways that we name A-mapping and B-mapping (see
Fig. 1). In the A-mapping, the center of the backbone bead is
placed at the center of mass of Ca–Cb–Ca, in a position between
two consecutive rings. In calculating the center of mass position,
the external Ca are weighted one half of their mass, because they
are shared with the neighbouring beads. In the B-mapping the
backbone bead is placed at the center of mass of Cb–Ca–Cb, on
top of the ring. Here, the Cb atoms are shared.
For structure-based CG schemes, it has been shown5that the
choice of the bead position changes the target radial distribution
functions (RDFs), and can therefore have important conse-
quences on the model performances. Our CG approach, which is
not optimized against RDFs, should be less sensitive to these
details. Still, A- and B-mappings could exhibit different packing
and conformational preferences, and it is difficult to predict
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a priori which one is going to be more accurate in reproducing PS
properties. For this reason, both models have been tested and
their performances compared.
2.2 Simulation details
2.2.1
parameters partially relied on atomistic simulations of poly-
styrene in melt and dilute conditions. For simulations of the
polystyrene melt, we generated atactic chains of 30 styrene units
(PS30). A simulation box containing 24 PS30 chains was first
equilibrated at T ¼ 500 K for 20 ns. Polymer properties were
calculated from the subsequent production run of 1.5 ms at T ¼
500 K. For simulations in dilute conditions, a single atactic 100-
monomer chain (PS100) was inserted in a simulation box con-
taining 540 benzene molecules and equilibrated for 200 ns. Five
independent production runs were carried out at T ¼ 300 K,
amounting to a total simulation time of 1 ms.
Force-field parameters for all atomistic simulations were taken
from OPLS-AA21force-field. A molecule-based cutoff of 1.3 nm
was used for the calculation of non-bonded interactions in all
simulations, without switch function or long-range dispersion
correction. Simulations were performed with a time step of 2 or
3 fs, with updates of the neighbour list every 5 steps (10 or 15 fs).
Simulations were carried out in the NpT ensemble using the
Nos? e–Hoover thermostat (sT ¼ 0.3 ps) and the Parrinello–
Rahman barostat (sP¼ 2 ps). These conditions allowed a faithful
reproduction of benzene properties as reported by Jorgensen and
Severance.21
Atomistic simulations. The optimization of interaction
2.2.2
bonded interactions was based on the comparison between all-
atom simulations of a single PS100 chain in benzene and CG
simulations of a PS10 chain in benzene. The atomistic chain has
been built by stacking monomers with random chirality. Thanks
to its considerable length, the average distributions of bonds and
angles were not influenced by local tacticity. The temperature
(300 K) and pressure (1 bar) in the atomistic and CG systems
were identical. In the CG simulations, a cutoff of 1.2 nm was
used in the calculation of the non-bonded interactions, with
a shift function starting from 0.9 nm for dispersion interactions.
A time step of 20 fs was used, and the neighbour list for non-
bonded interactions was updated every 200 fs. These conditions
correspond to the standard ones used in the parameterization of
MARTINI. Simulations in the NpT ensemble were carried out
Coarse-grainedsimulations.The optimizationof
with the Nos? e–Hoover thermostat (sT¼ 1 ps) and the Parrinello–
Rahman barostat (sP¼ 4 ps).
The parameterization of non-bonded interactions was based
on the density and radius of gyration of polystyrene chains with
30 and 100 monomers in the polymer melt at 500 K. Equilibrated
melts were prepared by (i) allowing a single fully stretched chain
to relax to a half-coiled conformation in a vacuum; (ii) stacking
npolidentical, half-coiled chains in an approximately cubic box,
(iii) allowing the system to achieve constant density through
a 100 ns run at temperature T ¼ 500 K and atmospheric pressure,
and finally (iv) letting the system equilibrate until the center of
mass of the chains diffused over a distance comparable to their
expected radius of gyration (details are reported in the ESI†).
After the equilibration stage, runs were performed as detailed in
Table 1. Simulation boxes were large enough to avoid any self-
interaction of the chains due to periodic boundary conditions.
Long (microsecond) time scales were required for proper
sampling of the density and of the radius of gyration of the
chains. All other simulation parameters (treatment of non-
bonded interactions, time step, barostat and thermostat) were
the same as in the simulations with the solvent (standard
MARTINI parameters). All the atomistic and CG simulations
were performed with the GROMACS 4 MD package.22
Fig. 1
position of their backbone bead.
The atomistic and coarse-grained descriptions of polystyrene superposed. Grey areas represent CG beads. A- and B-mapping differ in the
Table 1
are the average length of the simulation box edge (nm), the simulation
time (ms), the number of polymer chains and the number of solvent
molecules, respectively. Many independent simulations of PS10 in
benzene were used during the parameterization stage, to tune the bonded
interactions. Similarly, simulations of PS30 and PS100 in the melt were
used to tune the non-bonded interactions. The simulation of PS10 in
benzene and all the melt simulations were performed for both A- and
B-mappings, while the simulations of PS100 in the solvent environment
were performed for A-mapping only. All the simulations were used for
model validation
Details of the CG simulations performed. lbox, tsim, npoland nsol
lbox(nm)
tsim(ms)
npol
nsol
Melt
PS10
PS30
PS60
PS100
PS10 in solution
Benzene
PS100 in solution
Benzene
Water
Cyclohexane
5.2
5.6
5.8
6.9
1
4
5
9
75
32
18
18
—
—
—
—
3 0.11150
8.9
13
12
6
3
6
1
1
1
3880
18000
4370
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2.3 Bonded interactions
We used distributions of distances and angles from atomistic
simulations as target properties in the parameterization of the
bonded interactions. We first generated a long atomistic trajec-
tory for a single PS100 chain in benzene, and converted it into the
corresponding CG trajectory. This is the trajectory of the centres
of mass (COMs) of groups of atoms, according to the A- and
B-mapping schemes. From the CG trajectory we extracted
distributions of distances and angles between COMs. We chose
to use harmonic potentials with the appropriate equilibrium
lengths and angles, corresponding to the average distance and
angles between COMs. The force constants were chosen to
reproduce the width of the distributions. Following ref. 23, we
checked that this parameterization satisfied the uncrossability
conditions for the polymer chains. Our CG model explicitly
treats bonds and angles, but does not include any torsional
potential. The final choice of parameters is reported in Table 2.
We note here that the atomistic distributions, for both A- and
B-mappings, are often bimodal, while all the distributions of the
CG models obtained during the optimization procedure are
Gaussian-like. Bimodality arises from the choice of the mapping
scheme and from the atacticity of the PS100 sequence. Our model
reflects the average property of the atactic chains, and does not
retain any information about the local tacticity of the original
atomistic sequence. In the ESI†, the comparison between the
atomistic and CG distributions is shown.
2.4 Non-bonded interactions
Target properties for the optimization of A- and B-mapping
parameterizations are the density, r, and the radius of gyration,
Rg, of polystyrene in the melt. These were taken mostly from
experiments. In the case of low molecular weight PS, an estimate
for the radius of gyration was derived from atomistic simulations
(see Table 3 for details).
We initially assigned original MARTINI bead types to the A-
and B-mapping beads. The apolar type SC1 was chosen to
represent three consecutive C atoms in the PS backbone (see the
paper by Marrink et al.13for more details about the interaction
sites in MARTINI). Each SC1 bead shares the two external C
atoms with its adjacent beads. By analogy with the MARTINI
model of benzene, three interconnected apolar SC4 beads were
chosen to represent the hanging phenyl rings. This topology will
be referred to as TOP0.
TOP0 performs poorly in terms of density. The PS100 melt
density is 880 kg m?3, lower than the experimental value reported
for 10-monomer polystyrene chains, 895 kg m?3.24Such a low
density is inherited from the original MARTINI description of
benzene,13whose density is 10% lower than the experimental one.
Concerning the structural target, the values of Rgin the melt are
15–20% lower than the experimental value of Rgof PS100 in its
theta solvent.
In order to improve the performances of the model, we
introduced a new bead type, named SC4PS, to replace the SC4
beads of TOP0. The SC4PSbeads are connected using the same
bonded interactions as in TOP0, while non-bonded ring–ring
interaction parameters are modified. MARTINI non-bonded
interactions between the non-charged beads are described by
Lennard-Jones potentials. Table 4 reports the optimized values
of the s and 3 parameters of the Lennard-Jones backbone–
backbone (B–B), backbone–ring (B–R) and ring–ring (R–R)
interactions. With respect to the original TOP0 parameters, only
ring–ring interactions (i.e., interactions between SC4PSbeads)
are changed. All other interactions are identical to the original
MARTINI parameterization. Ring beads are made smaller by
reducing the value of sR–R. Ring–ring interactions are also made
weaker by reducing 3R–Rdown to 2.4 kJ mol?1. sR–Rand 3R–R
modifications are performedin such away as topreserve thetotal
energy density, giving better consistency with the rest of the
force-field. Polymer models containing the optimized SC4PS
parameters are named TOP1.
The performance of TOP1 in terms of density and radius of
gyration is reported in Table 3. A- and B-mapping do not differ
much in terms of density. Density values of our PS models are in
Table 2
backbone and ring beads, respectively. req(nm) and kb(kJ mol?1) are the
equilibrium bond length and the elastic constant of the harmonic bond
potential. Very narrow atomistic distributions induced us to constrain
some bonds. qeq(deg) and ka(kJ mol?1) are the equilibrium angle and
the elastic constant of the harmonic angle potential
Parameters of bonded interactions. B and R indicate the
Bond
req
kb
angle
qeq
ka
A-mapping
B–R
R–R
0.27
0.27
8000
constr.
B–R–B
R–B–R
B–R–R
52
120
136
550
25
100
B-mapping
B–R
R–R
B–B
0.217
0.27
0.25
constr.
constr.
8000
B–B–B
R–B–B
B–R–R
170
125
156
25
45
200
Table 3
PS in the melt (two top rows), and the same quantities resulting from
the optimization of our model (two bottom rows). All atomistic and CG
simulation data are from the present work, while superscripts a, b and c
refer to the experimental works by Zoller,24Hocker25and Konishi,26
respectively
Target values for the density and the radius of gyration of the
Source
PS30
r/kg m?3
PS100
r/kg m?3
PS30
Rg/nm
PS100
Rg/nm
Experimental 895a(PS10) 959b(PS500)
Atomistic 951
A-mapping
(TOP1)
B-mapping
(TOP1)
2.73c(cyclohex)
1.21 ? 0.03
1.38 ? 0.02 2.61 ? 0.02938 ? 5946 ? 5
944 ? 5951 ? 51.20 ? 0.03 2.32 ? 0.04
Table 4
TOP1 PS models. B and R subscripts stand for backbone and ring,
respectively. sB–B, 3B–B, sB–Rand sR–Rare the same as in TOP0, and
correspond to SC1–SC1 and SC1–SC4 (or, equivalently, SC1–SC4PS)
MARTINI type interactions. sR–Rand 3R–Rare the non-bonded self-
interaction parameters for the new SC4PStype
Optimized values of 3 and s Lennard-Jones parameters for the
Parameter
sB–B/
nm
3B–B/
kJ mol?1
sB–R/
nm
3B–R/
kJ mol?1
sR–R/
nm
3R–R/
kJ mol?1
PS model0.432.6250.432.3250.412.4
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good agreement with the experimental data, even if the depen-
dence of the density on the chain length is probably smoother
than reported experimentally. The radius of gyration of PS100 in
the melt is 4% (A-mapping) and 15% (B-mapping) lower than the
one measured for PS100 in a theta solvent, where the chain
configuration is expected to be the same as in the melt.1,27For
PS30, B-mapping provides the same Rgas the atomistic simula-
tions.
3 Validation of the model
For the purpose of validating our model, we compared a series of
CG simulations with the same all-atom simulations of the PS30
and PS100 melts, already used during the optimization stage.
Additional CG simulations were carried out for two more chain
lengths, namely 10 and 60 styrene units (PS10 and PS60). The
MD setup, equilibration and production protocols are the same
as described in Section 2.2. The time scales allowed for a proper
sampling of both structural (radius of gyration, end-to-end
distance) and dynamical (diffusion coefficient) properties of the
system, as detailed in Table 1.
3.1 Structural properties
3.1.1
size-dependent behavior of the A- and B-mapping models has
been analyzed by calculating the radius of gyration and the end-
to-end distance for PS chains of four different lengths, namely
PS10, PS30, PS60 and PS100. The results are shown in Fig. 2. In
the top panel, the values of Rgin the melt from the CG simula-
tions are plotted together with the small angle neutron scattering
(SANS) data given by Konishi et al.26Experiments concerned
oligomers of PS in cyclohexane, at the theta temperature, 307 K.
Chain statistics in the melt and in theta solvent is expected to be
described by a Markovian random walk,1,27and almost identical
values for the radius of gyration of PS in the bulk and at the theta
conditions have been measured experimentally.28Our data show
a good overall agreement with the experimental results. The A-
mapping model, in particular, reproduces all the experimental
values within statistical error. B-mapping performs as well as the
A-mapping at small molecular weights, but shows some tendency
to overestimate chain contraction at the larger sizes. The size
dependence of the end-to-end distance of the CG models is
shown in the bottom panel of Fig. 2. By fitting to a power-law f
Nmfunction, the A- and B-mapping data gave values of m ¼ 0.49
and 0.63, respectively. The ratio between the mean-squared end-
to-end distance and the mean-squared radius of gyration, which
is equal to 6 in random-walk statistics, oscillates between 5.2 and
6.7 for A-mapping, and between 4.0 and 5.9 for B-mapping.
Scaling of the structural metrics with chain length. The
3.1.2
polystyrene is characterized by two peaks in the low-q region, as
shown in Fig. 1 of ref. 29. Based on atomistic MD simulations by
Chakravarthy et al.,29the peak at q ¼ 7.5 nm?1can be ascribed to
backbone–backbone interactions. Phenyl–phenyl and phenyl–
backbone interactions give rise to the peak at q ¼ 14 nm?1. The
structure factor can also be calculated directly from CG simu-
lations. The top panel of Fig. 3 shows the structure factor30S(q)
for PS100 melts at 500 K, for A- and B-mappings. Two peaks can
Structure factor. The X-ray scattering pattern of atactic
clearly be distinguished for both models. A- and B-mappings
agree in the positioning of the large-q peak at 14.5 nm?1, which is
in good agreement with the experimental value. The low-q peak
is found at q ¼ 6 nm?1for the A-mapping and at q ¼ 4.5 nm?1for
the B-mapping. The attribution of the two peaks to different pair
interactions coincides with the atomistic simulations reported by
Chakravarthy et al., as shown in the bottom panel of Fig. 3.
As the atomistic RDFs have not been used as targets during
the optimization stage, the models are not expected to reproduce
the short-range structural order of B and R beads with much
detail. In the ESI† the RDFs obtained from the atomistic
simulations are compared to those obtained with the A-mapping
model, which shows better structure factor agreement with the
experimental data. The backbone–backbone RDF from A-
mapping simulations of a PS30 melt overlaps satisfactorily with
the RDF obtained from the atomistic simulations. On the
contrary, ring–ring and backbone–ring RDF differ significantly
at short distances, in the CG and atomistic descriptions.
3.2Dynamic properties
In order to interpret the time scale of our CG simulations, we can
compare the chain self-diffusion coefficient, D, to the experi-
mental data and values from atomistic simulations. D is
Fig. 2
in the melt, as a function of chain length. In the lower panel, the lines
indicate power-law fits to the data (see text for details).
Radius of gyration (top) and end-to-end distance (bottom) of PS
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calculated from the mean square displacement h|DR|2i ¼ h|R(t) ?
R(0)|2i of the center of mass of each chain, averaged over time
and over all the chains (6D ¼ limt/Nh|DR|2i/t). Our atomistic
simulations of the PS30 melt at T ¼ 500 K yielded a diffusion
coefficient of (6.0 ? 0.1) ? 10?9cm2s?1. This value is 2.5 and 6.7
times smaller than that obtained from the A-mapping and B-
mapping CG simulations, respectively. A more precise time
scaling would necessitate an estimate of the glass transition
temperature (Tg) for both the atomistic and the CG PS30 melt, so
that the D values could be compared at the same relative distance
from Tg. Still, it is worth noting that this scaling factor is close to
the standard time conversion factor of MARTINI,13where the
diffusional dynamics of CG water, compared to atomistic water,
is sped up by a factor of four.
According to the Rouse model,1the diffusion coefficient of PS
oligomers is expected to depend on the number of monomers, N,
as D ? N?1. The Rouse model does not take into account the
mass dependence of the free volume or, equally, the mass
dependence of the friction coefficient, which accelerates polymer
diffusion at smaller sizes.31Experimentally, the measured power-
law dependence of D on the size of the PS molecule has a larger
exponent.32–34Fig. 4 shows D values versus N, comparing the
results from our A- and B-mappings to two experimental
measurements performed at the same temperature, 500 K. D
values are fitted to a power-law function D ? N?a. Antonietti
et al.,35reports an exponent a ¼ 2.4 in the range 60 < N < 700.
Experimental values from Fleischer32are fitted by a ¼ 2.2, those
from Bachus and Kimmich33by a ¼ 2.5 (even if, in the original
paper, the inclusion of larger molecular weights leads to a ¼ 2.0).
Our A- and B-mappings are fitted by a ¼ 1.8 and a ¼ 2.0,
respectively. Similar results are reported for the CG model of
polystyrene developed by Fritz et al.6and Harmandaris et al.31
3.3Temperature transferability
3.3.1
mapping models for polystyrene by comparing the temperature
dependence of the density in CG and atomistic simulations.
Fig. 5 shows the density of PS30 melts at different temperatures
between 350 and 500 K. Our coarse-grained models present the
same thermal expansion coefficient as the atomistic model across
the whole temperature range. No major differences are registered
between A- and B- mapping.
Density. We tested the transferability of our A- and B-
3.3.2
PS30 melt at different temperatures, namely 460, 500 and 540 K
in order to assess the transferability of the CG-to-atomistic time
conversion factor, d ¼ DCG/Datomistic. The values of d are
reported in Table 5. A-mapping appears to be more transferable,
Dynamics. We performed atomistic simulations of the
Fig. 3
mapping. In the bottom panel, the contributions from different pair
interactions are separated. The presence of two peaks and their attribu-
tion to the different pair interactions do not depend on the chain size or
on the model considered. Top panel refers to a PS100 melt, bottom panel
to a PS30melt (A-mapping only).The experimental and atomistic spectra
are shown in Fig. 1 of ref. 29.
In the top panel, the structure factor of a PS melt, A- and B-
Fig. 4
PS melt is shown. Experimental data from Fleischer32and Bachus and
Kimmich33are compared to the D values obtained from the A- and B-
mapping CG models. The lines correspond to the best fit of the data to
a power-law function D ? N?a.
The scaling behavior of the chain self-diffusion coefficient in the
Table 5
coefficients (d ¼ DCG/Datomistic) for a PS30 melt at different temperatures
Ratio between coarse-grained and atomistic self-diffusion
d ¼ DCG/Datomistic
460 K500 K540 K
A-mapping
B-mapping
3.0
21.0
2.5
6.7
1.5
3.9
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with d values showing no major changes over the temperature
range considered. For the B-mapping model, diffusion is overall
faster than for A-mapping, and the value of d shows almost an
order of magnitude reduction going from 460 up to 540 K.
The effect of temperature on the viscoelastic properties of
polymers is commonly expressed in terms of a shift factor, aT,
which allows for the time–temperature superposition of visco-
elastic functions.36The shift factor is the ratio between the
relaxation time at temperature T and that at a reference
temperature, T0. In the framework of the free volume theory,37
the temperature dependence of the shift factor is
lnaT¼?C1ðT ? T0Þ
C2þ T ? T0
(1)
which translates into the temperature dependence of the self
diffusion-coefficient
D f eC1(T?T0)/(C2+T?T0).
(2)
This trend was first derived, empirically, by Williams, Landel and
Ferry (WLF).38The product C1C2is expected to be independent
of the choice of reference temperature and on the molar mass of
the melt,39being a characteristic of the polymer species. Exper-
iments39–41confirmed that the viscoelastic properties of poly-
styrene are well described by WLF behavior and suggest values
of C1C2for PS in the 570–600 K range. In Fig. 6 the values of
D vs. temperature are reported for our CG PS30 melt. Data are
best fit by the WLF function with C1C2¼ 558 and 540 K for A-
and B-mappings, respectively.
3.4 Transferability between different chemical environments
Solvents can significantly affect polymer behavior, and are
frequently used during polymerization and fabrication processes.
Standard characterization techniques, such as light scattering or
gel-permeation chromatography, are performed on dilute poly-
mer solutions. Solvents are usually classified as bad or good
solvents, depending on their tendency to make the polymer
collapse to a compact globule or swell, respectively. By varying
the temperature, a transition between the extended and collapsed
configuration of the solvated polymer is expected. The theta-
condition is realized when the polymer coil acts as an ideal chain,
with the solvent cancelling the effects of the excluded volume
expansion.
One of the advantages of using MARTINI to model a polymer
system is represented by the availability of a number of solvents
for which a MARTINI parametrization has been already
derived. Here we aim at describing the compatibility of our
PS model with three different MARTINI solvents, namely
benzene (good solvent), cyclohexane (theta solvent at 307 K) and
water (bad solvent). Solvent topologies and interactions are as
Fig.5
B-mapping) and atomistic simulations.
Temperature dependence of the densityfor the CG model(A- and
Fig. 6
PS30 melt. The lines correspond to the best fit of the simulation data to
the WLF behavior.
Self diffusion coefficient vs. temperature for a coarse-grained
Fig. 7
melt (b) and in water (c). Backbone beads are dark grey, phenyl ring
beads are white. Solvents and other PS molecules are not shown to better
appreciate the polymer shape.
Typical configurations for a PS100 chain in benzene (a), in the
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in the original MARTINI parameterization.13Benzene and
cyclohexane are modeled by a set of three small MARTINI
beads, of type SC4 and SC1, respectively. MARTINI water is
modeled by P4 beads, each of them representing four all-atom
water molecules. At this stage we consider A-mapping only, due
to its better performance in terms of reproduction of structural
properties in the melt and of temperature transferability. Simu-
lation details are reported in Table 1.
We monitored the radius of gyration of a single PS100 chain
diluted in each of these three solvents, over a time scale of
several microseconds. In water, at 300 K, the partially swollen
initial configuration collapses down to a disordered globule
whose radius of gyration is 1.28 ? 0.02 nm. The collapse takes
place within the first 150 ps, and the globule conformation
remains compact over the rest of the simulation, amounting to
2 microseconds. In benzene, the chain swells and its radius of
gyration undergoes large fluctuations during the simulation. A
run of 6 microseconds lets us estimate a radius of gyration of
3.40 ? 0.07 nm, which is in reasonable agreement with the
experimental values reported by Cotton et al.,42where Rgfor
a chain of 200 monomers is 5 nm. Typical configurations of
a PS100 chain in the melt, in benzene and in water are shown in
Fig. 7.
Cyclohexane is a theta solvent for polystyrene at 307 K.
The coil-to-globule transition of PS in cyclohexane whilst
varying the temperature of the solution has been experimentally
monitored for large molecular weight PS.43Since the theta-
condition is satisfied only for a narrow range of temperatures, it
is interesting to investigate to what extent our CG model is able
to capture it. During the simulations of PS100 in cyclohexane
we have observed the collapse of PS100 into a globular
configuration, no matter the temperature of the solution in the
300–400 K range. This led us to refine the parameterization of
the SC4PS–SC1 interactions. According to the original
MARTINI parameterization, the Lennard-Jones SC4–SC1
interaction is parameterized by 3 ¼ 0.75 ? 3.1 kJ mol?1and s ¼
0.43 nm. Modifying the strength of the interaction up to 3 ¼
0.75 ? 3.3 kJ mol?1leads to a satisfactory description of the
theta condition. At 300 and 320 K, we observe the temporary
stabilization of ordered structures, such as helical or planar
zigzag conformations. Their radius of gyration varies in the
2.2–3.0 nm range. The appearance of these metastable config-
urations in cyclohexane becomes less common as the temper-
ature increases. At 330 and 340 K planar configurations are
stable only for very short time spans (a few hundred ps) during
a 6 microsecond run, and completely disappear at 360 K. The
radius of gyration of PS in cyclohexane as a function of
temperature is reported in Table 6. In the 330–360 K range our
Rgvalues are in good agreement with the value of 2.73 nm
reported by Konishi et al.26for PS in cyclohexane at the theta
temperature of 307 K.
4 Discussion
In the present work, we have described the development of
a hybrid thermodynamic-structural approach to polymer coarse-
graining and applied it to the case of polystyrene. Our procedure
is largely based on the MARTINI approach.12–16The parame-
terization relies on atomistic simulations for bonded interactions,
while non-bonded interactions are tuned to reproduce experi-
mental densities and free energies of transfer of the target
molecules between polar and non-polar phases. We have refined
the MARTINI parameterization by using one additional target
property, namely the radius of gyration of PS in the melt. Two
alternative mappings, A- and B-mappings, have been proposed
and tested.
Polystyrene is one of the most important commercial plastics.
A number of different methodologies has been used for the
development of CG models of polystyrene, and this polymer
has become one of the most popular benchmark systems.
Milano et al.3applied an iterative Boltzmann inversion (IBI)
CG procedure (initially developed by Reith et al.44) to poly-
styrene melts. In that model, each super-atom represented
a monomer, belonging to either a meso- or a racemo-diad of
atactic polystyrene. The bonded interactions were modeled by
multi-centered Gaussian-based potentials fit to atomistic bond
length, angle and dihedral distributions. The non-bonded
interactions were obtained through the IBI procedure, aiming
to reproduce the radial distribution functions calculated from
the all-atom simulations. Sun and Faller5applied the same
approach to polystyrene in the melt45and in other environ-
ments.8More recently, Fritz et al.6proposed a finer scale model,
where each PS monomer is represented by two beads, one
associated with the backbone and one with the phenyl ring. The
model takes into account and reproduces correlations between
adjacent bonded degrees of freedom. The bonded interactions
extend up to the beads separated by 4 bonds (1–5 pairs), and the
non-bonded interactions are derived from the all-atom simu-
lations of oligomers within a force-matching scheme. The
model can describe different PS tacticities.
Both the IBI-based and the force-matching procedures can
very accurately reproduce the structural features of polystyrene
at the optimization temperature. Our model is not expected to
reproduce accurately the short-range structural properties of
the melt, since they have not been used as targets during the
optimization of the force-field parameters. Nevertheless, our
model has a finer degree of coarse-graining than the models
developed by Milano et al.,3Sun and Faller,5Qian et al.7and
Fritz et al.6The more detailed description of PS molecular
structure allows us, in principle, to distinguish between ring–
ring, backbone–ring and backbone–backbone interactions. At
short distances, the model does reproduce the backbone–
backbone radial distribution function correctly, while the
centers of mass of the phenyl rings are predicted to be closer to
each other than in atomistic simulations. As for global struc-
tural properties of the chains in the melt, both A- and
B-mapping yield reasonable results. This is highlighted by the
good overlap between CG, experimental and atomistic struc-
ture factors.29Furthermore, there is good agreement between
model prediction and experimental data about the scaling of
Rgwith size.
Table 6
hexane at different temperatures
The Rgvalues for a single A-mapping PS100 chain in cyclo-
330 K340 K 360 K
Rg[nm] 2.79 ? 0.15 2.80 ? 0.202.62 ? 0.06
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In terms of computational efficiency, our model has a larger
number of beads per monomer compared to the other models
mentioned above. The additional cost is compensated for by the
use of simple analytical interaction potentials, which allow us to
use a time step of 20 fs in MD simulations. Such a time step is at
least 4 times larger than previously reported for other CG
models, and about 10 times larger than time steps commonly
used in atomistic simulations. Simulations of polymer melts on
a time scale of tens of microseconds are within reach. For
instance, the simulation of a melt of 18 PS100 chains on 32 cores
of a HP CP4000 BL ProLiant supercluster produces 1 ms per day.
The CG-to-atomistic time conversion factor, d ¼ DCG/
Datomistic, is not very large (2.5 and 6.7 for PS100, A-mapping and
B-mapping, at 500 K, see Table 5), compared to the 102–103time
conversion factors of the above mentioned models. For example,
a CG simulation spanning 1 microsecond would correspond to
2.5–6.7 microseconds in real time for our model, but 0.1 to
1 milliseconds with coarser models. This bears the obvious
disadvantage that describing transformations in the system
requires longer simulations with our MARTINI model. On the
other hand, our model is expected to describe PS dynamics in
a more realistic way, including the contribution to friction derived
from the presence of the pendant phenyl rings. This is a desirable
property, especially in view of the application of the model to the
study of rheological properties of all-polymer nanocomposites.46
An important criterion to assess the properties of CG force-
fields is the quality of temperature transferability. Decreased
transferability is intrinsic in the coarse-graining procedure, which
is performed at one particular thermodynamic state point.
Theoretical studies are currently ongoing in different research
groups to develop thermal rescaling approaches, which would
allow for the transfer of CG force-field parameters across
different temperatures.47,48
Several factors can affect the
temperature transferability of the force-field. The most impor-
tant one is the degree of coarse-graining: the coarser the
mapping, the less transferable the model. Another cause for poor
transferability can be related to the chemical properties of the
molecules themselves. Carbone et al.4reported that the hydrogen
bonding in polyamide-6,6 makes the atomistic dynamics at low
temperatures much slower than predicted by their coarse-grained
model, which is not able to incorporate hydrogen bonds. For
PS, as the temperature decreases the geometry of phenyl ring
packing becomes more and more important, affecting the density
of the melt. For this reason, CG models of PS where each CG
bead represents a whole monomer, generally present poor trans-
ferability.4Using two CG beads per monomer, one of them being
centered on the phenyl ring, significantly improves transferability.6
It is also possible that different mapping schemes with the same
number of interaction sites have a different degree of trans-
ferability. For example, Quian et al.7has observed that CG
parameters tend to be more tranferable when CG beads are map-
ped onto the center of mass of the cluster of atoms they represent.
In our study, both A- and B-mappings are promising with
respect to transferability-related issues. (i) They are relatively
finely coarse-grained. (ii) They both preserve the characteristic
planarity of the pendant groups, which plays a role in PS packing
when decreasing the temperature. (iii) The CG beads are centered
on the real center of mass, as for the transferable force-fields of
Fritz et al.6and Qian et al.7
Differences in transferability between A- and B-mapping
could not be excluded a priori. In terms of density vs. tempera-
ture, A- and B-mapping are both satisfactory. Both the coarse-
grained models appear to capture the correct qualitative
behavior for the temperature dependence of the diffusion coef-
ficient. A-mapping performance is in better quantitative agree-
ment with experimental data, being transferable over a
temperature range extending from 420 K up to roughly 600 K.
The vanishing of the self-diffusion coefficient that we can observe
in CG simulations is consistent with the experimentally reported
glass transition temperature of polystyrene, Tg¼ 373 K. When
compared to atomistic simulations, the CG-to-atomistic time
conversion factor is rather constant with temperature only for
A-mapping, while for B-mapping it varies over a wider range.
Beside transferability issues, CG force-fields are known to
suffer from representability issues, as pointed out by Louis and
coworkers.49,50At the state point, where the CG model has been
parameterized, the model is not expected to simultaneously
reproduce multiple physical properties. Users should be cautious
in modeling situations where an important role is played by
thermodynamical quantities that are different from those used
for parameterization or validation. For the particular case of our
polystyrene model, and more generally for the MARTINI force-
field, problems may arise when an accurate reproduction of
surface tension is needed.51,52Application of our model to
systems with gas–liquid interfaces has not been tested, and is
expected to require special caution.
Due to the excellent performance of A-mapping during our
validation stage, we have tested the behavior of our A-mapping
CG polystyrene in three different solvent environments. The
radius of gyration of PS100 in benzene, a good solvent for PS, is
in quantitative agreement with the available experimental data.
It is worth noticing that parameters for benzene were taken from
the original MARTINI parameterization with no modification.
PS behaves as expected in water, where it quickly collapses to
a compact globular configuration. The reproduction of the PS
behavior at the theta condition turned out to be a more delicate
task. The MARTINI parameterization for cyclohexane, which is
a theta solvent for PS at 307 K, had to be refined by increasing
the strength of the phenyl–cyclohexane interactions, in order to
avoid the collapse of the polymer to a globule. After refinement,
the theta temperature results are in the 330–360 K range, 20–
50 K higher than the experimental one.
The optimization of the phenyl–cyclohexane interactions has
been complicated by the stabilization of solid-like intra-chain
arrangements, mainly planar zigzag configurations, during the
simulation of PS100 in cyclohexane at 300–320 K. Experimen-
tally, similar configurations have been observed in syndiotactic
polystyrene53and mixtures of syndiotactic and atactic poly-
styrene54upon precipitation and solidification. It is worth
recalling here that the bonded interactions of our CG model were
obtained by averaging out the distributions of bonded interac-
tions of an atomistic atactic PS100 chain. In many cases those
distributions were bimodal or multi-modal, due to the presence
of stretches with different tacticity along the chain. Our CG
model does not retain the tacticity of the original model, differing
from simulations with other CG models of PS.3,6It is possible
that, during the simulations, phenyl rings rearrange forming
stacks, similar to syndiotactic polystyrene. This problem can in
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principle be solved by adding specific terms to the CG potential
energy function (e.g., dihedral potentials) to ‘‘discourage’’ the
formation of stacks. Dihedral potentials could also be used to
include the information on the tacticity of the polymer chain into
the model. We notice that a more complex potential energy
function would certainly lead to more realistic structural features
of the polymer chain, but it would also require a shorter inte-
gration time step. Since our simple model describes both struc-
tural and thermodynamic properties of atactic polystyrene
reasonably well, we believe the level of complexity used here
represents a good compromise between speed and accuracy.
5 Conclusions
In the present work we have introduced the use of a hybrid
structural–thermodynamic approach to coarse-grain polymers.
Such strategy has proved to be successful for the benchmark
case of polystyrene, which we have modeled in two alternative
ways. After having validated both A- and B-mapping models,
we can conclude that A-mapping offers the best performance.
On the whole, A-mapping performs slightly better than B-
mapping in terms of structural properties (Rgvalues, end-to-
end distance, scaling with size, structure factor), especially
when considering experimental data as the main reference
target. Moreover, A-mapping shows better temperature trans-
ferability, as demonstrated by the good agreement between
CG model and experimental WLF coefficients, and by a nearly
constant CG-to-atomistic time conversion factor. The compati-
bility of our polymer model with other MARTINI molecules has
been discussed for three different cases: benzene (good solvent),
cyclohexane (theta solvent at room temperature) and water (bad
solvent). The guidelines drawn here for model development and
validation can be applied to a variety of polymer systems,
provided that the CG-to-atomistic mapping scheme is consistent
with the MARTINI framework. The compatibility of our poly-
mer model with a well-established coarse-grained force-field
model for biological macromolecules is extremely promising,
especially in view of possible applications of polymer systems in
medicine (i.e., drug delivery).
Acknowledgements
GR thanks Paola Carbone for useful discussions. This work has
been supported in part by the Academy of Finland through its
Center of Excellence COMP grant and by the research pro-
gramme TransPoly. MatOx Ltd acknowledges support from
Becker Industrial Coatings Ltd. GR acknowledges HPC-
EUROPA2 for funding a visiting period to INSERM and
granting access to CINES supercomputing facilities. CSC IT
Center for Science Ltd. is acknowledged for the allocation of
other computational resources.
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