A Molecular Dynamics Study of Polymer/Graphene Nanocomposites

A Molecular Dynamics Study of Polymer/

Graphene Nanocomposites

Anastassia N. Rissanou,*1,2,3 Vagelis Harmandaris1,2,3

Summary: Graphene based polymer nanocomposites are hybrid materials with a

very broad range of technological applications. In this work, we study three hybrid

polymer/graphene interfacial systems (polystyrene/graphene, poly(methyl methac-

rylate)/graphene and polyethylene/graphene) through detailed atomistic molecular

dynamics (MD) simulations. Density profiles, structural characteristics and mobility

aspects are being examined at the molecular level for all model systems. In addition,

we compare the properties of the hybrid systems to the properties of the

corresponding bulk ones, as well as to theoretical predictions.

Keywords: graphene; molecular dynamics; nanocomposites; polymer

Introduction

Graphene is a very important material witha wide range of novel applications due to it’sexceptional physical properties[1]. For thisreason graphene can be thought as apromising candidate for the amplificationof polymer nanocomposites’ properties,taking the position of carbon nanotubes[2].. Graphene polymer nanocomposites arebased on the incorporation of graphene inpolymer matrices. The benefits which havebeen reported, for the hybrid system, arethe improvement of the electrical, thermal,mechanical and gas barrier properties ofgraphene-polymer composites[3].

Due to the above properties the study ofgraphene based polymer composite materi-als is a very intense research area. Forexample, many studies on the reinforce-ment of graphene – polymer nanocompo-sites have used either graphene oxide[4,5,6]

or a single atomic layer of exfoliatedgraphene. The former induces strongeradhesion between graphene and polymer.[7]

Besides experiments, simulation meth-odologies can also be used for the study ofpolymer/graphene hybrid nanostructuredsystems, providing a detailed investigationof the hybrid system, and especially of theinterface, at the molecular level. Forexample, Harmandaris et al. and Daoulaset al.[8], in a series of two papers, reportedon modeling of hybrid polymer/graphitesystems, through state-of-the-art atomisticMonte Carlo (MC) and Molecular Dynam-ics (MD) simulations. Furthermore, Awas-thi et al.[9], studied the graphene-polyethylene interfacial mechanical behav-ior using MD simulations. More recently,Lv et al.[10] performed a simulation studywhere the influence of the chemical func-tionalization of graphene on the interfacialbonding characteristics was presented.

In the current work we present results,which are a part of a general computationalapproach for the study of realistic polymer/graphene systems, with main goal thequantitative prediction of the macroscopicproperties of realistic nanocomposite sys-tems, especially at the interface. To achievethis, polymer/graphene hybrid systems arerequired to be examined at various lengthand time scales. Therefore, hierarchical

1 Department of Applied Mathematics, University ofCrete, GR-71409, Heraklion, Crete, GreeceFax: þ30 2810393701; E-mail: [email protected]

2 Archimedes Center for Analysis, Modeling & Com-putation, University of Crete, P.O. Box 2208 GR-71003, Heraklion, Greece

3 Institute of Applied and Computational Mathematics(IACM), Foundation for Research and TechnologyHellas (FORTH), GR-71110, Heraklion, Crete,Greece

Macromol. Symp. 2013, 331-332, 43–49 DOI: 10.1002/masy.201300070 | 43

� 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim wileyonlinelibrary.com

multi scale methodologies, which involvesystematic linking between simulationmethods from different levels of descrip-tion, are necessary[11,12]. Such simulationapproaches are described via typical length-time graphs presenting the length and timerange in which each simulation method isvalid, as the one shown in Figure 1. [13] Inmore detail, in Figure 1 representativesnapshots from polymer–solid interfacialsystems from different levels of description(from the quantum level, through classicalatomistic models, to coarse-grained modelsand up to continuummodels), as well as, thecorresponding simulation methods, arepresented. Note that a rigorous and sys-tematic procedure is required in order tolink the different levels of description. Theoverall methodology will allow us toprovide a fundamental study of the cou-pling between microstructure at the inter-face andmacroscopic properties (structural,mechanical, elastic and dynamical-rheolog-ical) of graphene/polymer nanocompositesystems.

In the present study, we report datathrough all-atom classical simulations ofhybrid systems consisted of graphene andthree well known and widely used polymers;the atactic Polystyrene (PS), the atactic

Poly(methyl-methacrylate) (PMMA) andPolyethylene (PE). We examine theirstructural and dynamical properties anddetect the effect of the graphene layer onthem, through a comparison with thecorresponding bulk systems’ properties.Moreover, we make a comparison amongthe three different hybrid systems.

Systems and Simulation Method

The systems studied in this work aredepicted in Table 1. These are the follow-ing polymer/graphene composites: (a) PS/Graphene, (b) PMMA/Graphene and (c)PE/Graphene. The corresponding bulkpolymer systems are studied as well. N isthe number of polymer chains in thesimulation box. For PS and PMMApolymer chains were 10-mers, while PEchains consist of 22-mers, (i.e. 22 CH2

groups). Note that the molecular backbonelength of all systems is very similar, sincePS and PMMA chains have 20 (CH2 orCH) groups in the backbone. The refer-ence bulk systems consist of 56 10-merchains for PS, 54 10-mer chains for PMMAand 420 22-mer chains for PE. The filmthickness (d), which is included in the third

Figure 1.

A schematic representation of the hierarchical simulation approach linking different levels of description[13].

Macromol. Symp. 2013, 331-332, 43–4944 |

� 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ms-journal.de

column of Table 1, is calculated from thebox length along the z-direction subtract-ing the thickness of the graphene layer,(0.34nm, i.e. about one van der Waalsradius), which is placed at z¼ 0. Thenumber and the length of the polymerchains in each system were chosen so thatthey form polymer films of almost equalthicknesses. The last three columns ofTable 1 will be discussed in the nextsection.

We performed atomistic NPT and NVTMolecular Dynamics (MD) simulationsusing the GROMACS code.[14] All simu-lations were carried out at constant tem-perature equal to T¼ 500K for PS andPMMA and T¼ 450K for PE, and pressureP¼ 1atm. We chose relatively high temper-ature values, in order to enhance themobility of polymers so as to study theirdynamics, but at the same time temper-atures where experiments can be per-formed. For both PS and PMMAT¼ 500K is an appropriate value, which isalmost equidistant from their glass transi-tion temperatures (i.e. Tg

PS ffi 360K, andTg

PMMA ffi 380K). For PE TgPE ffi 190K, is

substantially smaller than the Tg values ofthe other two polymers and an equidistanttemperature from Tg

PE would lead totemperature values where PE is crystallized(i.e., T� [310-330]K). Therefore, since themelting point of PE is about 410K, we choseT¼ 450K for this system.

An all atom representation model hasbeen used. For the description of theintermolecular and intramolecular interac-tions of PMMA and PS we have used theOPLS[15] atomistic force field, while for PEthe TRaPPE model[16] has been used. For

the interaction between polymer atoms andgraphene layers the geometric means eij ¼ðeiiejjÞ0:5 and sij ¼ ðsiisjjÞ0:5 were used with:ecc/kB¼ 28K and scc¼ 3.4Å. The set of e ands values for the graphene carbons are theones used by Steel[17], optimized for a basalplane of graphite. Graphene has beenrepresented as a set of LJ carbon atoms,centered at their crystallographic positions.The lattice constant which has been used isthat of graphite at 300K, equal to 2.462Å.Periodic boundary conditions have beenused in all three directions, so that thepolymer interacts with the graphene layer,which was placed at the bottom of thesimulation box, on the xy plane, and itsperiodic image at the top of the simulationbox simultaneously. This setup renders oursystems polymer films confined betweentwo graphene surfaces, i.e. assuming idealdispersion of graphene sheets. This iscertainly not the usual case for realisticnanocomposites[18], however we do notexpect this assumption to introduce arti-facts, since we are particular interested inlarge (compared to chain size) graphenelayers and short time dynamics in which themotion of the layers is not important. Thedimensions of the graphene sheets (i.e. xy-direction) in the simulation box are [5.5-7.5]times larger than the radius of gyration ofthe respective polymer chains. Neverthe-less, periodic boundary conditions ensure aninfinite graphene layer with no edges, whichis eclectically neutral. More details aboutthe all-atom force field, as well as the MDsimulations are given elsewhere[19]. TheNPT simulations of bulk systems provide adensity of 0.965g/cm3 for the PS, 1.054g/cm3

for the PMMA and 0.663g/cm3 for the PE.

Table 1.Number of chains, film thickness, radius of gyration, end-to-end distance and segmental relaxation timefor the PS the PMMA and the PE systems.

System N d(nm) Rg(nm) Ree(nm) tseg(ns)

PMMA 135 13.35 0.697� 0.001 1.71� 0.02 250.55� 20.0Bulk PMMA 54 – 0.695� 0.001 1.70� 0.02 64.15� 10.0PS 120 13.67 0.695� 0.001 1.56� 0.01 26.30� 8.0Bulk PS 56 – 0.696� 0.001 1.54� 0.01 10.91� 3.0PE 420 13.91 0.631� 0.003 1.82� 0.01 0.0083� 0.002Bulk PE 420 – 0.620� 0.003 1.77� 0.01 0.0044� 0.001

Macromol. Symp. 2013, 331-332, 43–49 | 45


Results

We start the discussion of the simulationresults by calculating the chain dimensions.In Table 1 both the average radius ofgyration, <Rg> , and the end-to-end dis-tance,<Ree> , are presented for all systemsstudied here. The values of the correspond-ing bulk systems are also included. PS andPMMA have almost equal radius of gyra-tion, while the end-to-end distance of PS issmaller than the one of PMMA. For PE,<Rg> is lower, though <Ree> is higher,compared to the other two polymers. Aninteresting observation is that the values ofboth <Rg> and <Ree> are almost equalbetween the bulk and the confined systemsrespectively, except for PE, where the bulksystem has a rather lower<Rg> and<Ree>

value, compared to the confined one.The polymer arrangement, with respect

to the surface, is presented through thecalculation of the density profiles as afunction of the distance from the graphenelayer, r(r). In Figure 2 three density profiles

are depicted for the three hybrid polymer/graphene systems. Density profiles arebased on the monomer center of massand are averaged over time. There arecommon characteristic in all three cases: (a)An attraction of the polymer atoms fromthe graphene layer, which is denoted by thehigh peak (maximum) in the density curves.(b) The density profiles are, as expected,symmetrical with respect to the center ofthe film, due to the specific setup. (c) Allthree polymers attain their bulk density,almost in the middle of the polymer film.The bulk density value is presented with adashed horizontal line in each system.

Although the qualitative picture isalmost the same, for the three hybridsystems, a more careful observation bringsout quantitative differences. The highestattraction from the surface is exerted on PS,PMMA follows, while the lowest is the oneon PE, as it is obvious from the values ofthe peaks in the density profiles. This can beattributed to: (a) the existence of a sidegroup in the first two polymers (i.e., a

0 1 2 3 4 9 10 11 12 13 140

1

2

3

4 (c)

ρ (gr

/cm

3 )ρ(

gr/c

m3 )

PE

ρ(gr

/cm

3 )

r(nm)

0 1 2 3 4 9 10 11 12 13 140

1

2

3

4(b) PMMA

0 1 2 3 4 9 10 11 12 13 140

1

2

3

4 PS

(a)

Figure 2.

Monomer density profiles as a function of distance from graphene layers for a) PS, b) PMMA and c) PE, hybrid

polymer/graphene systems. Bulk system’s density is represented by a dashed horizontal line in all cases.

Macromol. Symp. 2013, 331-332, 43–4946 |


phenyl ring in PS, a carboxyl and a methylside group in PMMA), and (b): the fact thatthe polymer/graphene dispersion (van derWaals) forces are larger for PS and PMMAthan for PE, because of their larger bulkdensity, i.e. there are more polymer atomsper unit volume. Moreover, graphene layerseems to prefer the phenyl ring more thanthe side groups of PMMA, which isreflected in the higher first peak of r(r)for PS compared to PMMA. Anotherinteresting observation is that PE, due tothe absence of a side group, appears a morewell ordered layered structure close to thesurface, as it is evident from the second andthe third peak in the density profile. Thepeaks become gradually lower as thedistance from the surface increases (i.e.the attraction is smaller). On the contrary, avery small second peak is rather detectablein PS density profile, whereas the curve isstructureless after the first peak in PMMA.

In order to extract information for theconformations of the polymer chains closeto the surface we calculate the second rankbond order parameter, P2(cosu), defined as:P2 cosuð Þ ¼ 3

2 cos2u� �� 1

2 where, u is theangle between an arbitrary vector, whichis defined along the molecule, and oneCartesian axis. This quantity providesinformation about the orientation of indi-vidual parts of the polymer chain. In thiswork, we define a characteristic vectoralong the backbone of the polymer chain(vbb), for each polymer, which connects onecarbon atom to the next not consecutivecarbon atom (1-3) and we calculate its anglewith z-axis (normal to the surface).

In Figure 3 P2(cosu) has been analyzedas a function of the distance from thesurface. Note that the data have beenaveraged over the equal distances fromthe two graphene layers in order to improvestatistics, utilizing system’s symmetry.P2(cosu)¼ -0.5 signifies a vector withorientation parallel to surface, P2(cosu)¼ 1.0 normal to the surface, and P2(cosu)¼ 0.0 random orientation with respect tothe surface. P2(cosu) values denote anorientation almost parallel to the surfaceclose to the graphene layer, which is

gradually becomes random as the distancefrom the graphene increases. This behavioris common for the three systems, thoughmore pronounced for PE. This is directlyrelated to the layered structure, which hasbeen observed in the density profile of PE.

Finally, we present a brief discussion forthe dynamics of the hybrid interfacialsystems. Dynamics in the segmental levelcan be studied through the calculation ofthe second-order bond order parameter as afunction of time: P2 tð Þ ¼ 3

2 cos2uðtÞ� �� 12 for

the characteristic vbb vector, which has beendescribed previously. In this formula, u(t) isthe angle of the vector under considerationat time t relative to its position at t¼ 0. Weexamine the average dynamics over theentire film. In Figure 4 the time autocorre-lation functions of vbb characteristic vectoris presented, for the three systems. Aconsiderable difference in the relaxationtime of the PE compared to the other twopolymers is observed. vbb characteristicvector is fully decorrelated after just 1ns,whereas the time autocorrelation functionsfor the other two polymers have significantvalues at that time. Such differences in themobility of the different polymers are notsurprising if we consider the combinatoryeffect of both different interactions anddifferent Tg values on the local polymerrelaxation times. Moreover, the confined

0 1 2 3 4 5 6 7-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

PS PMMA PE

P 2z

r(nm)

Figure 3.

Bond order parameter P2(cosu) along z-axis as a

function of distance from graphene layer for a vector

defined along the backbone in each of the three

systems.

Macromol. Symp. 2013, 331-332, 43–49 | 47


systems have slower dynamics than therespective bulk ones.

To further quantify the differences indynamics among the three systems wecalculate the segmental relaxation timesdefined as, tsegm: ¼ tKWW

bG 1

b

� �where tKWW

is the relaxation time, which is extractedfrom a fitting of P2(t), with a stretchexponential function[20] (Kohlrausch-Williams-Watts, (KWW)): P2ðtÞ ¼Aexp

�� t

tKWW

�b and b is the stretchexponent, which takes into account thedeviation from the ideal Debye behavior(b¼ 1). Segmental relaxation times tseg forall hybrid and bulk polymer model systemsstudied here, are shown in the last columnof Table 1. It is clear that for all systems thedynamics of the hybrid systems is slower,with a factor of 2 for PE, to about 3 for PSand up to 4 for PMMA, than the corre-sponding bulk ones. Note here, that theratios of tseg of the hybrid over the tseg ofthe corresponding bulk system reveal theeffect of different polymer/surface interac-tions on polymer dynamics. Furthermore,

the dynamics of PE is much faster (about 4-5 orders of magnitude) than the relaxationof both PS and PMMA, while the relaxationtime of PMMA chains is about 10 timesslower than the one of PS. Finally, we haveto mention that segmental relaxation timescan be extracted through different vectorswithin the polymer chain, something whichis also apparent among various experimen-tal techniques. Here we chose vbb charac-teristic vector, while a further analysis,where more vectors have been studied, anda comparison among different segmentalrelaxation times has been made, is pre-sented in our recent publications.19,[21]

Conclusion

The current work presents a comparativestudy of three different polymer/graphenenanocomposites through detailed atomisticsimulations. We performed atomistic molec-ular dynamics simulations on three differentpolymer/graphene interfacial systems, using

1E-4 1E-3 0,01 0,1 1 10 1000,00

0,25

0,50

0,75

1,00 PE PE-Bulk

AC

F P2

AC

F P2

(c)

AC

F P2

t(ns)

1E-3 0,01 0,1 1 10 1000,00

0,25

0,50

0,75

1,00 PMMA PMMA-Bulk

(a)

1E-3 0,01 0,1 1 10 1000,00

0,25

0,50

0,75

1,00 PS PS-Bulk

(b)

Figure 4.

The autocorrelation functions of P2 for the backbone characteristic vector (vbb) for the three hybrid polymer/

graphene systems.

Macromol. Symp. 2013, 331-332, 43–4948 |


polystyrene (PS), poly(methyl methacrylate)(PMMA) and polyethylene (PE) polymers.Our focus was on the structural conforma-tional and dynamical properties of thepolymer chains and the way that theydiversify form their corresponding bulkbehavior, as a function of the distance formthe graphene layer. The three polymer filmsare almost of the same thickness. The radiusof gyration, Rg and the end-to end vector,Ree, are measures of the mean size and theextension of the entire polymer chainrespectively. For both quantities, the valuesfor the confined systems are very close to thecorresponding bulk values. The densityprofiles of all three polymers highlight theattraction between the graphene layer andthe polymer chains and display a symmetricshape with respect to the center of the film.The bulk density value is attained away fromthe surface, for each polymer respectively.Furthermore, quantitative differences areobserved among the three polymers. Thehighest attraction from the graphene layer isobserved for PS, then for PMMA and finallyfor PE polymer chains. Moreover, PE formsa well ordered layered structure close to thesurface in contrast to the other two polymers.

The orientation of the segments of thepolymer chains close to the surface isquantified by the calculation of the secondrank bond order parameter, P2(cosu). Anorientation almost parallel to the surfaceclose to the graphene layer is observed,which is gradually becoming random as thedistance from the graphene increases. Thisbehavior is more pronounced for PEpolymer, something which is consistent withits layered structure.

Finally, information about the mobilityof the three different polymers, and the waythat the graphene layer affects them, isextracted from the calculation of theautocorrelation function of the second-order bond order parameter, P2(t), for acharacteristic vector at the monomer level.PE is substantially faster than the other twopolymers as it is revealed from a comparisonof the autocorrelation functions of Figure 4as well as from the calculation of thesegmental relaxation time, tsegm (Table 1).

PS presents the second faster dynamics,while the slowest one is PMMA. A moredetailed study on the structural, the confor-mational and the dynamical properties ofPMMA/graphene interfacial systems, ispresented in our recent publication19, whilea further study on dynamics of differentpolymer/graphene systems will be presentedin a forthcoming publication (Rissanou andHarmandaris to be submitted).

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A Molecular Dynamics Study of Polymer/Graphene Nanocomposites