Monte Carlo simulations of a coarse grained model for an athermal all-polystyrene nanocomposite system

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by  Georgios G. Vogiatzis

MonteCarlosimulationsofa coarsegrainedmodelfor an athermalall-polystyrene
nanocompositesystem
GeorgiosG.Vogiatzis,EvangelosVoyiatzis,andDorosN.Theodorou∗
SchoolofChemicalEngineering,NationalTechnicalUniversityofAthens,ZografouCampus,GR-15780Athens,Greece
ReceivedJanuary16,2014;E-mail:[email protected]
richedwithdispersedparticlesmayhavebetterpropertiesthanthe
4 Abstract: The structure of a polystyrene matrix filled with tightly net polymeric material and can be used in more demanding and
1 cross-linked polystyrene nanoparticles, forming an athermal novel applications. Therefore, an understanding and quantitative
0
nanocompositesystem,isinvestigatedbymeansofaMonteCarlo descriptionofthephysicochemicalpropertiesofthesematerialsis
2
sampling formalism. The polymer chains are represented as ran- ofmajorimportancefortheirsuccessfulproduction.

n dom walksand the systemisdescribed through a coarsegrained Thephysicalandchemicalbehaviorofthiskindofmaterialsde-
a Hamiltonian. Thisapproachisrelatedtoself-consistent-fieldtheory pendsonthenatureofthenanodispersedphase.Thecharacteristics
J
but does not invoke a saddle point approximation and is suitable of the surface of the particles significantly affect the behavior of
4 fortreatinglargethree-dimensionalsystems. Thelocalstructureof thecompositepolymeric/particlesystem. Insomecases,thedis-
1 thepolymermatrixinthevicinityofthenanoparticlesisfoundtobe persednanoparticles inthepolymericmatrixmayaggregate. The

differentinmanywaysfromthatofthecorrespondingbulk,bothat presenceofparticleswithgraftedchainsimproveswettingeffects
]
t thesegmentandthechainlevel. Thelocalpolymerdensityprofile inthe system and suppresses aggregation, leading the material to
f
o near to the particle displays a maximum and the bonds develop dramaticallydifferentperformance.
s considerable orientation parallel to the nanoparticle surface. The Extensive experimental work on all-polystyrene nanocompos-
t. depletion layer thicknessis also analyzed. The chains orient with iteshasbeenrecentlypresented.8–10Regardingtheconformational
a theirlongestdimensionparalleltothesurfaceoftheparticles.Their properties, thedimensions ofmatrixchains, measuredbyneutron
m
intrinsic shape, as characterized by spans and principal moments scattering techniques, indicate swelling induced by the dispersed
- ofinertia,isfoundtobeastrongfunctionofpositionrelativetothe tightlycross-linkednanoparticles.Thiseffectoccursonlywhenthe
d
interface. The dispersion of many nanoparticles in the polymeric radiusofgyrationofthechainislargerthanthenanoparticleradius.
n
matrixleadstoextensionofthechainswhentheirsizeissimilarto Theall-polymercompositesystemappearstobeastableblendeven
o
theradiusofthedispersedparticles. forsmallinterparticledistances,suggestingthatchainshapeinthe
c
[ polymermatrixishighlydistorted. Asforthedynamicproperties,
the blend viscosity was found to demonstrate a non-Einstein-like

1 decreasewithnanoparticlevolumefraction. Itwassuggestedthat
v
Introduction thisphenomenonscaleswiththechangeinfreevolumeintroduced
4
by the nanoparticles, while the entanglements seem to be totally
6
Nanocomposite polymeric materials consist of nanoparticles dis- unaffected.
3
3 persedinapolymericmatrix. Thisnewfamilyofcompositemate- Theeffectofincorporatingasphericalnanoparticleonthecon-
. rialsdisplaysavarietyofpropertiesthatattractgreatscientificand formational properties of a polymer has been demonstrated by
1 industrialinterest. Thepracticeofaddingnanoscalefillerparticles meansofMonteCarlo(MC)simulationsfordiluteandsemi-dilute
0 to reinforce polymeric materials can be traced back to the early solutions.11,12Thesestudieshavebeenquitegeneric,employinga
4
yearsofthecompositeindustry. Thedesignofsuchconventional bead-springandhard-sphererepresentationforthepolymerandthe
1
: composites has focused on maximizing the interaction between nanoparticle, respectively. Themost significant changes instruc-
v thepolymermatrixandthefiller.1 Thisiscommonlyachievedby turalpropertiesandorientationoccurredwithinthedepletionlayer.
Xi shrinking the filler particles in order to increase the surface area Theeffective interactions between twodispersed nanoparticles in
available for interaction withthe matrix. Withthe appearance of a polymer solution have been quantified by calculating the po-
r
a syntheticmethodsthatcanproducenanometersizedfillers,result- tential of mean force through the expanded ensemble density of
ing inan enormous increase of surface area, polymers reinforced states technique.12 An oscillating long-range behaviour has been
with nanoscale particles should show vastly improved properties. revealed.
Nanomaterials fabricated by dispersing nanoparticles in polymer Efforts have also been made to predict the aggregation or the
meltshavethepotentialforperformancesexceedingthoseoftradi- dispersionofpolymernanoparticlesinpolymermatrixbythermo-
tionalcompositesbyfar.2 dynamicmodelling.Recentadvancesarethoroughlyreviewedin13
Even though some property improvements have been achieved andincludecompressibleregularsolutionfreeenergymodels,14,15
in nanocomposites, nanoparticle dispersion is difficult to control, based on modification of theories for binary polymer blends, and
withboththermodynamicandkineticprocessesplayingsignificant generalizations of integral equation theories,16,17 such as the mi-
roles. It hasbeen demonstrated that dispersed spherical nanopar- croscopicPolymerReferenceInteractionSiteModel(PRISM).The
ticles can yield a range of multifunctional behaviour, including proposed models extend existing theories by incorporating spe-
a viscosity decrease, reduction of thermal degradation, increased cificnanoparticle-nanoparticleandnanoparticle-polymercontribu-
mechanical damping, enriched electrical and/or magnetic perfor- tions. The predicted phase diagrams indicate that such systems
mance,andcontrolofthermomechanicalproperties.3–7Thetailor- exhibitarichvarietyof behaviours, includingupper criticalsolu-
madepropertiesofthesesystemsareveryimportanttothemanu- tiontemperature-type,lowercriticalsolutiontemperature-type,and
facturing procedure, as they fully overcome many of the existing hour-glassshape.
operationallimitations. Asafinalproduct,apolymericmatrixen- Amajorchallengeinsimulatingrealisticnanocompositemateri-
1
alsisthatboththelengthandthetimescalescannotbeadequately andnanoparticle-nanoparticle.Thesystemiscontainedinvolume
treatedbymeansofatomisticsimulationsbecauseoftheextensive V at temperature T. The matrix consists of n polymeric chains
computational load. This is why a variety of mesoscopic tech- whichareconsideredtoobeythefreelyjointedchainmodel. Each
niques havebeendeveloped for theseparticular systems. Among chainconsistsofN Kuhnsegments. Thebondlengthiskeptfixed
them, Self-Consistent Field Theory (SCFT) seems to be a well- andisequal totheKuhn length, b, of thespecificpolymer under
foundedsimulationtool.18Thismethodadoptsafield-theoreticde- consideration. Thenp nanoparticlesarerepresentedasimpenetra-
scriptionofthepolymericfluidsandmakesasaddle-point(mean- blespheres. TheinteractionenergyH includesonlynon-bonded
field) approximation. Significant progress has been made in ap- contributions:
plyingtheSCFTtonanoparticle-filledpolymermatricesinthedi-
luteandsemi-diluteregion.19–21 Alattice-basedsimulationinthe H (~r)=Hpp{r (~r)}+Hpn+Hnn (1)
contextoftheScheutjens-Fleerapproximationhascastsomelight where Hpp stands for the non-bonded interactions between the
ontotheequilibriumdispersionofnanoparticleswithgraftedpoly- polymericchains,Hpn forthebinteractionbetweenpolymericseg-
merchainsintodensepolymermatrices,wherethematrixandthe ments and nanoparticles and Hnn for the interaction between
brushes sharethesamechemical structure.22 Anattempt toover- nanoparticles.
cometherestrictionsposedbythesaddle-pointapproximationwas Animportantdifferenceinthetreatmentofpolymer-polymerin-
madein.23 Thecoordinatesofallparticlesinthesystemwereex- teractionscomparedtothosearisingbecauseofthepresenceofthe
plicitlyretainedasdegreesoffreedom. Itwascrucialtoupdatesi- nanoparticleisthattheformeraretakenintoaccountthroughasuit-
multaneouslythecoordinatesandthechemicalpotentialfieldvari- ablefunctional of the instantaneous local density r (~r), while the
ables. latterareaccountedforbyapairwiseinteractionpotential. Thelo-
Inthesamespiritofrelaxingthemeanfieldapproximation,Bal- cal density is computed directly from the segmentbpositions by a
azsandcoworkershaveproposedacouplingofSCFTfortheinho- smoothingprocedure. Thenon-bondedinteractionsHpparegiven
mogeneous polymers anddensityfunctional theory(DFT)forthe by:
nanoparticles.24–26 The extrema of the approximate energy func- b Hpp{r (~r)}= r 0 d3~r k 0N[r (~r)−1]2 (2)
tionalcreatedinthiswaycorrespondtothemean-fieldsolution.Al- N V 2
Z (cid:18) (cid:19)
hthionudgerhedthbeyStCheFdTi-fDfiFcuTltaypipnreoxatcehndisinqgutiotespyrsotemmissifnogr,withiischcurerrleianbtllye awvheerraegeb b=ulkkB1nTuwmibbtherkdBebnesiintygothfesBegomlteznmtsa.nnDbceovniasttiaonntsanodftrh0eilsotchael
densityfunctionalsarenotavailable. densityfromtheaveragebulkvaluearerestrictedbythequadratic
A novel way of incorporating fluctuation effects in self- termwhichstemsfromHelfand’sapproximation.32 Thus,themelt
consistent mean-field theories has been recently proposed.27,28 hasafinitecompressibility,whichisinverselyproportionaltok :
0
The“SingleChaininMeanField”simulationtechniqueisparticle-
1
btuaasteidnga,nedxtfeorunnadlefidelodns.anThenesfiemelbdlsemoifminicdethpeenedfefencttcohfaitnhseiinnsfltaunc-- k 0= kBTk Tr 0 . (3)
taneous interactions of a molecule with its neighborhood. They Althoughthecoordinatesoftheparticles,eitherpolymericseg-
arefrequentlyupdated usingthespatiallyinhomogeneous density ments or nanoparticles, arethe degrees of freedom in our model,
distribution of the ensemble. The explicit conformations of the thelocaldensitiesthatappearin(??)arenotdefinedbythemicro-
molecules evolve by Monte Carlo. Thistechnique has been very scopic expression r (~r)=(cid:229) nj=1(cid:229) Ni=0d ~r−~rj,i . Rather, they are
succesfullindescribingtheself-assemblyandtheorderingofblock obtainedbyaparticle-to-mesh(PM)assignmenttechnique. Auni-
(cid:0) (cid:1)
copolymersonpatternedsubstratesencounteredinprocessessuch form,rectangulargbridwithnsitessitesisintroduced.Thelocalden-
asnanolithography.22,29–31 sitiesaredefinedoneachsitebyapplyingasmoothingprocedure
Thepresentworkexaminesthestructureofapolymermatrixin tothebead’s position. Asaconsequence, adiscretizationparam-
thevicinityofpolymericnanoparticles.Particularattentionisgiven eter D L has to be introduced; it is the grid spacing. Phenomena
tocapturingtheorientationaleffectsandthechainconformationof anddetailsofthesystemwhoselengthscalesaresmallerthanthe
thematrixinadense,polymericnano-filledcomposite. Theanaly- D Lparametercannotbedescribedaccurately. Thus,thisparameter
sis is based on a coarse grained Monte Carlo simulation method setstheresolutionlevelofourobservations. Italsodefinesasort
which can be readily applied to the complex three-dimensional ofmicroscopiccutoff, sinceitdeterminestherangeofinteraction
nanocomposite system. Although thelevel ofdescription isquite between neighboring beads. It follows that D L must not be less
abstract (i.e. that of the freely-jointed chain for the matrix), the thantheaveragedistancebetweentwobeads. Thevalueatwhich
developed model triestopredict thebehaviour of a nanocompos- D Lshouldbesetdependsstronglyonthesystemofinterestandthe
itewithspecificchemistry, namelyofanall-polystyrenematerial. preferredsmoothingprocedure.
A main characteristic of the method isthat it treatsin a different Theassignmentprocedureofapolymericbeadtothesitesofthe
mannertheinteractionbetweenpolymerandbetweenpolymerand rectangulargridisshowninFigure1. Thei-thsegmentofthe j-th
particle: theformerisaccountedforthroughasuitablefunctional chainatposition~rj,i shownasablackdot,yieldsacontributionto
of the local densities, while the latter is described directly by an aneighboringm-thsite,shownaswhitedot,P ~rj,i,~cm ,where~cm
explicitinteractionpotential. isthepositionof thecentreofthemthsiteandP theassignment
(cid:2) (cid:3)
function.Theresultingnormalizeddensityatsitemiscalculatedas
thesumofthecontributionsofallsegmentsinthesystem, giving
Method
theinstantaneousnumberofsegmentsassignedtothesite:
PolymerCoarseGrainedModel r m= r n1or (cid:229)n (cid:229)N P ~rj,i,~cm 1−21 d j,0+d j,N (4)
m j=1(i=0 )(cid:20) (cid:21)
The aim of the developed coarse grained model is to describe (cid:2) (cid:3) (cid:0) (cid:1)
thepropertiesofthenanocompositematerialatmesoscopiclength withbd theKroneckerdelta.
scales. Themainparametersarethechain connectivity, thefinite The normalization constant r mnor represents the segment num-
compressibilityofthemeltandexplicitpairwisepotentialsforthe berdensitythatthesitemwouldhave,providedthatthemeltwas
vanderWaalsinteractionsbetweenpolymersegment-nanoparticle completely homogeneous without anydensity fluctuation, r mnor =
2
Nanoparticle-polymerandnanoparticle-nanoparticle
interaction
Theeffect of dispersing bare nanoparticles inthepolymer matrix
isexplicitlyaccounted. Pairwisepotentialsdescribingtheeffective
eff
interaction between polymeric segment - nanoparticle, Upn , and
nanoparticle - nanoparticle,Ueff areintroduced. The overall ex-
nn
plicitenergeticeffect istakenintoaccount intheHamiltonianby
summing uptheseinteractionsfor all possiblepairsof nanoparti-
clesandpolymericsegments:
Hpn= (cid:229)np (cid:229)nN Unepff ~rin−~rip (6)
in=1ip=1
(cid:0) (cid:1)
and
Hnn=n(cid:229)p−1 (cid:229)np Unenff ~rin−~rjn . (7)
in=1 jn=in+1
(cid:0) (cid:1)
ThereferenceenergyleveloftheemployedHamiltonian(??)for
thehomopolymermeltisthatofameltwithuniformdensityprofile.
Figure1. SchematicofthePMtechniqueusedtoestimatethelocaldensi-
ties. Inthiscase,theeffectiveinteractionenergybetweentwopolymeric
beadsistakenaszero. Theeffectivepotentialsintroducedin(??)
and (??) should be such that, if the volume of the nanoparticles
nN/nsites. TheassignmentfunctionP couldbechosenarbitrarily. was occupied by bulk homopolymer, then the energy of the sys-
In the present study, a zero-order (PM0) interpolation scheme is temwouldbezero. Theinsertionofananoparticleinauniformly-
employed. Thischoicecorresponds tothecasewherethebeadis distributed melt can be thermodynamically accomplished in two
assigned entirelytoitsclosest siteanddoes not contributetoany steps: a spherical volume of polymer, whose net interaction with
othersite.Itfollowsthattwobeadsinteractonlywhentheyareas- the rest of the polymer matrix isUpeqpuiv−p, is removed from the
signedtothesamesite.Itisofgreatimportancetouseadensegrid melt and a nanoparticle with equal volume is placed in its posi-
withmanysites,soastominimizethediscretizationeffectonthe tion,introducinganewnetinteractionUpnwiththeremainingbulk
density. polymer (Figure 2 from step A to step B). Thus, the net interac-
Apart from the polymer-nanoparticle pairwise potential, the tionsbetweenpolymer-polymershouldbesubtractedfromthenet
mereexistenceofthenanoparticle,whichisanimpenetrablespher- interactionsbetweenpolymer-nanoparticle:35
ical object, defines a second, implicit way of interaction withthe Upenff =Upn−Upeqpuiv−n . (8)
matrix.Theavailablevolumethatthepolymercanoccupyincells,
thatareinthevicinityofthenanoparticle, islessthanthevolume Thetreatmentoftheeffectiveinteractionsbetweentwonanopar-
whichcanbeoccupiedincellsfarawayfromit. Sincethevolume ticlesismoresophisticated. Theirinsertioninthemelt,following
ofthecellstouchingthenanoparticleistruncated,thenormalizing theabovethermodynamicway,requirestwomoreadditionalsteps
densityr nor ofanarbitrarycellshouldbegivenby: with respect to the previous case. A second spherical volume of
m
polymerisremovedfromthemeltinthepresenceofonenanopar-
r mnor=r 0 d~rP ~rj,i,~cm (5) ticle,whosenetinteractionwiththerestofthesystemisUpenquiv−n,
ZVmacc andthesecondparticleisinsertedinthecreatedvolume,introduc-
whereVmacc isthevolumeofthecellsp(cid:2)aceacc(cid:3)essibletothepoly- ing new net interactionUnn with the composite system (Figure 2
mericsegments. ForthecaseofthePM0scheme, r mnor issimply fromstepBtostepC).
proportionaltotheaccessible(nonnanoparticle-occupied) volume
associatedwithsitem, r mnor =r 0Vmacc. Thecalculationof theac- Unenff =Unn−2Upenquiv−n+Upeqpuiv−n (9)
cessiblefreevolumeofeachcellofthegridisbasedonasuitable Thecoarsegrainedpotentialsdescribingthenetinteractionsin-
adaptationofananalyticalalgorithm.33Theoriginalalgorithmde- volvedin(??)and(??)arederivedonthegroundsofHamakerthe-
terminesinananalyticalfashionthevolumeofaspheredelimited ory.36Amajoradvantageisthattheexistenceofaccurateatomistic
by a set of arbitrary directed planes. It has been adapted to treat potentials describing these interactions is exploited. The need to
the case of the volume of a sphere intersecting with a cubic box introduceempiricallyadjustableparametersiskepttoaminimum.
andtakesintoaccounttheperiodicboundaryconditions.Thesame Theintegratedpotentialsarecomputationallyeffective. Eachpar-
strategycanbeappliedtothecaseofmultipleparticlesintersecting ticle is treated as a collection of atoms which interact only with
onebox,sincetheformercannotintersecteachotherandtheavail- atomsofdifferentparticles:
able volume can be readily estimated by subtracting the volume
occupiedbyeachparticle. U(r12)= r 1(~r1)r 2(~r2)ULJ(r12)dV1dV2 (10)
Analternativetousingagridinordertoestimatethelocaldensi- Zsphere1Zsphere2
tiesistoresorttocloud-densitymethods.34Suchtechniquescallfor wherer 1andr 2standforthedensityofinteractioncentresineach
aprocessthatidentifiesthepairofbeadswhicharecloseenoughto particles, r12 is the distance between the centres of the particles,
interact.AdiscretizationparametercloselyrelatedtoD Lemployed anddV1,dV2theelementaryvolumesofeachparticle.Thediscrete
inthePMcase,theGaussianwidth,hastobedefined.Thereisalso distributionoftheinteractingatomsineachparticleisassumedto
asignificantincreaseofcomputationalload. Despitethedemand- be continuous and uniform. The shape of the particles is spheri-
ingnatureofthecloudmethods, theaccuracyoftheirpredictions cal. Ananalyticalclosed-formsolutioncanberigorouslyderived,
hasbeenshowntobethesameasthatof grid-mediatedonesand
bothmethodsgiveessentiallythesameresults.29,34
3
sponding to a Kuhn segment and the experimental mass density
(approximately Rb=6.7 Å). The closest distance of approach is
equalorgreaterthantheradiusthatisusedfortheestimationofthe
Hamaker interaction between the nanoparticle and the Kuhn seg-
ment. TheatomisticparametersforCH(aromatic),CH (aliphatic),
2
CH(aliphatic),andCH (aliphatic)sitesinteractingwitheachother
3
aretakenfrom.38Thecross-interactionparametersareestimatedby
ageometricmeancombiningrule.39Bothnanoparticle-polymeric
segment and nanoparticle - nanoparticle potential are depicted in
Figure3.
3
20
0
2 −20
−40
U / k TB 1 −60 Nanoparticle − nanoparticle
−80
0 5 10 15 20 25 30 35 40
0
Polymeric Segment − Nanoparticle
−1
0 5 10 15 20 25 30 35 40
Surface to surface distance of interacting particles [Å]
Figure3. Pairwiseinteractionpotentialsusedforpolymericsegmentand
Figure2. Insertion oftwo nanoparticles inthe matrix, inthree discrete nanoparticles with the energy axis scaled by kBT. The density of the
steps. InstepAasphericalvolumeofpolymericmatrixisremovedfrom nanoparticleis1.15cmg3 anditsradiusisRn=36Å.Theradiusofthepoly-
themelt. InstepBthefirstsphericalparticleisplacedintheplaceofthe mericsegmentinteractionsiteRbis6.7Å.Thecentre-to-centredistanceis
removedvolumeofpolymericmatrix. FinallyinstepCanothervolumeof shiftedbyRn+Rbinthecaseofthepolymericsegment-nanoparticleinter-
polymericmatrixisreplacedbythesecondnanoparticle. action(mainfigure)andby2Rn inthecaseofnanoparticle -nanoparticle
interaction(inset).
providedthattheatomisticpotentialisLennard-Jones12-6:36,37
DetailsoftheMonteCarloSimulations
U(r12) = UA(r12)+UR(r12) (11)
A 2a a 2a a Allsimulationswerecarriedoutinthecanonicalstatisticalensem-
UA(r12) = − 612"r122−(a11+2 a 2)2+r122−(a11−2a 2)2 bculebi(cNaVnTd).itTsheedtgeemlpeenrgatthurweaosf6th0e0syÅs.teImtswhoausl4d0b0eKn.oTtehdetbhoaxt,wfoasr
+ ln rr112222−−((aa 11+−aa 22))22!# (12) athllecrhaadiinusleonfggthysra,ttihoeneodfgaeclehnaginth,siosagsretaotearvothidanfi4nRitegswiziteheRffgecbtesi.n1g1
Thesystemsstudiedwerestrictlymonodisperse, whilethedegree
A s 6
UR(r12) = 12 LJ ofpolymerizationofthechainsNwasvaried,withN=32,64,128
37800r12 and256beads(Figure4.Thecorrespondingmolecularweightsare
r122−7r12(a 1+a 2)+6 a 12+7a 1a 2+a 22 23, 47, 93 and 187 kg/mol, respectively). The number of chains
" (r12−a 1−(cid:0)a 2)7 (cid:1) idnenthsietysyosftethmespioslyvmareierdinacthceoradcicnegsstioblcehavionlulmenegtihs0s.o97thga/tcthme3m.Teahne
r122+7r12(a 1+a 2)+6 a 12+7a 1a 2+a 22 Kuhnsegmentlengthbis18.3Å,whichisassignedtosevenstyrene
+
(r12+a 1+(cid:0)a 2)7 (cid:1) monomers. Theradiusofthenanoparticleis36Å.Thedensityof
− r122+7r12(a 1−a 2)+6 a 12−7a 1a 2+a 22 athcetecrirsotsicslsinokfetdhepsoylystsetmyrsensetundaiendo,pia.ert.icthleeidseg1r.1ee5go/fcpmo3ly.mTehreizcahtiaorn-
(r12+a 1−(cid:0)a 2)7 (cid:1) of the chain, the radius and the mean density of the crosslinked
r122−7r12(a 1+a 2)+6 a 12−7a 1a 2+a 22 polystyrenenanoparticle,areincloseconnectionwithsystemsthat
− (r12−a 1+(cid:0)a 2)7 (cid:1)# havIneboeuerninsittuidailecdaelcxupleartiimonesn,taallsyinbgyleMnaacnKopaayrtaincdlecios-wploarckeedrsa.t8,t9he
(13)
centre of the simulation box. The polymer chains are built ran-
whereA12=4p 2eLJr 1r 2s L6J istheHamaker constant, a 1,a 2 the domly40withtheconstraintthatnoneofthepolymericbeadsover-
radiiofthesphericalparticles,r thedistancebetweentheircen- laps with the nanoparticle. When more than one particles had to
12
tresandeLJ,s LJ theLennard-Jones energy anddistance crossin- becontainedinaconfiguration,thenthenanoparticleswereplaced
teractionparametersforthepairofinteractionsites. Inthefollow- first at randomly selected positions, so that they did not overlap,
ing, the radius of the particle representing a polymeric segment, andthenthepolymerwasbuiltaroundthem. Theinitialconfigura-
Rb, is chosen such that (4/3)p R3b equals the volume per Kuhn tionscreatedbythismethodhavelargelocaldensityfluctuations.
segment in the bulk polymer, as calculated from the mass corre- These fluctuations increase with increasing chain length. A zero
temperatureMonteCarlooptimizationproceduretookplaceinor-
4
smallerchainsleadtodrasticvariationofdensityfluctuationswith-
outaffectingdramaticallytheoverallenergyofthegrid. Aslarger
moleculesareconsidered, moves ofthiskindbecome lessdrastic
becausetheytendtocreatelargerdensityfluctuations(sincealarge
number of segments change positions at the same time). On the
other hand, intramolecular moves and especially the pivot move
become very efficient for large chains, because of their ability to
rearrangeaspecificpartof thechain. Thetotallengthof simula-
tionsvariedbetween250and800millionsiterations,oneorderof
magnitudeabovethetimeneededfortheend-to-endvectororien-
tationalautocorrelationfunctiontodroptozero.
1.0
Figure4. Schematicsoftherelativesizeofnanoparticleandthefourchain 32 Kuhn segments/chain
64 Kuhn segments/chain
lengthsunderconsideration. ForeachlengththeRg0 inpuremeltisused 122586 KKuuhhnn sseeggmmeennttss//cchhaaiinn
forthecomparison.Thechainlengthsare32(a),64(b),128(c)and256(d) 0.8
Kuhnsegmentsperchainrespectively.
dertoreducethedensityfluctuations. Duringthisstage,allmoves < (t)·(0) >uu 00..46
leadingtomoreuniformdensityprofile,thusdecreasingthedensity
fluctuations,areaccepted.Intheoppositecase,theyarerejected.41
Five intermolecular moves were employed which treat the entire 0.2
chainsasrigidbodies:
0.0
• Translationofindividualchainsinarandomdirection 0 2 · 106 4 · M10o6nte Carlo ste6p ·s 106 8 · 106 1 · 107
• Rotationofindividualchainsbyrandomanglesaroundran- Figure5. Decayofthechainend-to-endvectororientationalautocorrela-
domaxesthroughtheircentresofmass tionfunctionh~u(t)·~u(0)iforfoursystemsofdifferentchainlengthatbulk
polymerdensity0.97 g .Theradiusofthenanoparticleis36Å.
cm3
• Reflectionofindividualchainsatrandommirrorplanesgo-
ingthroughthecentreofmass
Results
• Inversionofindividualchainsattheircentresofmass
Local polymer structure in the presence of a
• Exchangeoftwochainspreservingthecentreofmasspo-
nanoparticle
sitions
Thelocalstructureinthevicinityofthenanoparticle attheKuhn
WhentheMonte Carloproceeds, theinternal shape of thechains
segment level isfirst studied through the radial mass density dis-
isrestructuredbyfourintramolecular moves: theflip, theendro-
tributionfrom thesurface of thenanoparticle, which is displayed
tation,thereptationandthepivotmoves. Theexactmixofmoves
inFigure6. Thebeadsofthepolymerhavebeenclassifiedin1.5-
and acceptance rates of the moves are given in Table 1 for pure
Å-thicksphericalshellswhoseoriginisnanoparticle’scentre. The
meltpolystyrenesystem.Thenanoparticlesareallowedtotranslate
distanceoftheclosestsegmentfromthenanoparticlesurfaceisap-
inrandomdirectionsevery500MCsteps.
proximately equal to 7Å, which corresponds to theradius of the
Table1. MixtureofMCmovesusedforequilibrationofpuremeltofchains segment’sHamakerinteractionsite. Farawayfromthenanoparti-
with32polymericsegmentsperchain cle, the density assumes its bulk experimental value for all chain
Move %Attempted %Accepted lengths. Inthevicinityofthenanoparticle thereisastrong max-
Rigidtranslation 8% 62.3% imum that reaches approximately 1.8 g , located at the distance
cm3
Rigidrotation 8% 42.7% whereappearstheminimumoftheHamakerpotential. Thisvalue
Rigidreflection 8% 39.9% isobserved for all chain lengths. The enhancement of polymeric
Rigidinversion 8% 37.5% segmentdensityisverysimilarbetweenthefourchainlengthsun-
Rigidexchange 8% 35.5% derconsideration.Althoughbeingclosetothenanoparticleleadsto
Flip 15% 77.5% adecreaseofthechainentropy(allowedconformations arefewer
Endrotation 15% 70.4% thaninthebulk),attractiveenergeticcontributionsduetothehigher
Reptation 15% 61.8% densityofthenanoparticlesovercometheseentropiccontributions
Pivot 15% 53.9% andleadtoanincreaseinlocaldensity. Monomerpackingeffects
areimportant, sincemonomers formlayersaround thenanoparti-
TodemonstratetheefficiencyoftheMCmethodinequilibrating cle. A second peak is also observed in the radial density distri-
thesystemsunderstudy,theorientationalautocorrelationfunction butionofsegmentsandthepositionineachcasedoesnotdepend
h~u(t)·~u(0)i of a unit vector directed along the chain end-to-end onthenumber of Kuhnsegments ofthechain. Thissecond peak
vectorwasevaluatedasafunctionofnumber ofsimulationsteps. islocatedroughlyonesegmentdiameterawayfromthefirstpeak.
Figure 5 shows the results for the four chain lengths considered Similar studies in the semi-dilute region11 have also shown that,
here. The shorter chain systems exhibit faster equilibration rate as the polymer density was increased, the initially chain length-
due to the increased acceptance of rigid moves. Rigid moves of dependent peaksbecomeindependent. Onthebasisofthepertur-
5
bationsinthetotaldensityprofilefromitsbulkvalue,thethickness
oftheinterfacecouldbeestimatedaslessthan40Å.Thetypical
0.05
lengthscalecharacterizingthedensitydistributioninthevicinityof
thenanoparticleisthedepletionlayerthicknessd ,whichisshown
0
intheinsetofFigure6. Thedepletionlayercanbedefinedasthe
zonenexttotheparticlewheretherearenopolymermonomers.42
It isan equimolar dividing surface. The number of segments de- −0.05
pletedisestimatedbyanappropriatemassbalanceandleadstoan P >2
<
expressionoftheform: −0.1
¥
(Rn+d )3−R3n=3ZRn r2·(1−g(r))dr (14) −0.15 32 Kuhn segments/chain
whereRnistheradiusofthenanoparticleandg(r)=r (r)/r 0isthe 12625486 KKKuuuhhhnnn ssseeegggmmmeeennntttsss///ccchhhaaaiiinnn
pair distribution function between the nanoparticle and the poly- −0.2
0 20 40 60 80 100 120
mericsegments around it. Asshown intheinset toFigure6, the r − Rn [Å]
thickness of thedepletion layer increases slightlywithincreasing
thenumberofKuhnsegmentsinachain. Thevalueisclosetothe Figure7. AveragesecondLegendrepolynomialP2plottedagainstthedis-
tanceofsegmentifromthesurfaceofthenanoparticle in5-Å-thickbins.
radiuscalculatedforthesegment’ssphericalinteractionsite.
The P2 polynomial is calculated between the vector connecting the cen-
treofthenanoparticleandaKuhnsegmentiandthebondvectorofKuhn
segments iandi+1. Theradius ofthenanoparticle is36Å.Apositive
(negative) orderparameterindicates anorientational tendencythatisper-
2 Å] pendicular(parallel)tothenanoparticle’ssurface.
3m] 1111....2468 Depletion thickness [ 6666 ....61234 0 50 100 150 200 250 fthuencvtiaorniastiboentwofeetnhepotolytamlearnicdstehgeminetnetrsm. oTlehceulcaorrrpealairtiodinsthriobluetieofn-
Density [g/c 0 .18 Segments/chain fsehcotrtisleenvgidthenstcianleths,etihnetetromtaolledciustlrairbduitsiotrnibfuutniocntiofunndcitvioenrgs.esOansv1e/ryr
because of the intramolecular contribution dictated by the freely
0.6
jointed model. Thisis an artificial effect, caused by the fact that
0.4 32 Kuhn segments/chain
64 Kuhn segments/chain chainself-intersectionisnotpreventedinthefreelyjointedchains
0.2 128 Kuhn segments/chain
256 Kuhn segments/chain usedinourcalculations.Intheregimeofsmalldistances,segments
Bulk polymer density at 400K
0 0 20 40 60 80 100 120 of other molecules are expelled from the volume of a reference
r − Rn [Å] chain and this gives rise to a correlation hole in the intermolecu-
larpairdistributionfunction.Thespikeinthetotalpairdistribution
Figure6. Radialmassdensitydistributionfromthesurfaceofthenanopar-
functionisanintramolecularfeature,causedbypairsofsegments
ticleaccumulatedin1.5-Å-thickbins,forN=32(continuousline),N=64
(long-dashedline),N=128(short-dashedline)andN=256(dottedline). connecteddirectlybyaKuhnsegment. Ifanatomisticdescription
Theradiusandthemassofthenanoparticleare36Åand135kDa,respec- wereused,thisspikewouldbereplacedbyaseriesofintramolecu-
tively. Thestraightdot-dashedlinerepresentsthebulkpolymerdensityat larpeaksreflectingthebondedgeometryandconformationalpref-
400K. Intheinset, thevariation ofthedepletion layer thickness d with
erences of polystyrene molecules. On the other hand, for large
chainlengthisshown.
length scales, the intermolecular distribution function approaches
Thelocalorientationofchainsegmentsinducedbythenanopar- unity, astheintramoleculardistributionfunctionapproaches zero.
ticlecanbequantifiedbythesecondorderLegendrepolynomialP. AsshownintheinsetofFigure8,longerchainsexhibit“correlation
2
Anangleq maybedefinedbythebondvectoroftwoconsecutive hole”effectforslightlylongerdistances.
Kuhnsegmentsiandi+1andthevectorconnectingthecentreof
thenanoparticlewiththeKuhnsegmenti.Thisangleisusedtode-
finetheP2function: P2= 12 3hcos2q i−1 .P2wouldassumeval- 1.4
uesof−0.5,0.0,and1.0forbondscharacterizedbyperfectlyparal- 1
ltPAetasehaifn2dlnfedg,efegpirrcsvetoaat.enesnmrnnTttdidioahoctbnhllemtoeoeon,tb,hcrbdroaieeeeennnisnsddoptotrsareraepiisctleeesioytnonriifivpgtntnaaeeengtdlnteniyeoddia.tnnnianFctdanootusilhgpglt(cid:0)eeatdauhonrrirevstetoiiibtbccarr7eiiililnnbeeldys,iuniittsanttyrtiapcooowtolcninatuofghhymneitoeschrf(cid:1)ruhteiehlfnblaioeptaotterasternriradvitfdtdhaieKsiciecnaltueaeols5.hrdh-aetnCÅhiorseserwht-tertaesseightisubamirnturckufktec.lcaienhtocbnutaTenigrinrhenooteshdesff-. ment − segment pair distribution function g(r) 0111.... 91231 00000000 ........0088889999..2468246889 0 20 40 60NNNN====1225368624 80 100
examined.Theshorterthechain,themoreintensetheorientational Seg Intermolecular g(r) − 32 segments/chain
effects are. Kuhn segment orientation effects are weak beyond a Total g(r) − 32 segments/chain
0.8
distance of two Kuhn segment lengths from the particle surface. 0 20 40 60 80 100 120 140
Distance between two polymeric segments [Å]
Similarbehavior hasbeenobserved inmolecular dynamics simu-
lationsofsilicaparticlesdispersedinapolyethylene-likematrix.43 Figure8. Polymericsegments, totalandintermolecular pairdistribution
functions, accumulated in 1-Å-thick bins. One chain length is depicted,
Oneofthemostimportantaspectsofstructureinpolymermelts N=32(mainfigure). Theintermolecularpairdistributionfunction,inthe
presenceofananoparticleofradius36Å,isincludedintheinset.
istheso-called“correlationhole”effect. Itarisesindense poly-
mericfluidsbecauseofthemelt’sincompressibility.Figure8shows
6
Chainstructureinthepresenceofananoparticle
Complementarytothestudyoflocalstructureatthesegmentlevel,
longrangestructuralfeaturesmayberevealedbyprobingproper-
tiesatthelevelofentirechains. Thedistributionofthecentre-of-
mass(COM)oftheentirechainsaroundthenanoparticle,whichis
showninFigure9, issuchanexample. TheCOMsof thechains
areclassifiedin4.5-Å-thickbins.Itshouldbenotedthattheresults
attheleveloftheentirechainsaremorepronetostatisticalnoise,
becausethenumber ofbinnedchainsisconsiderablesmallerthan
thenumberofpolymericbeads. Therearepolymericchainswhose
COMpositionliesinsidethenanoparticle.Thesechainsengulfthe
nanoparticles. Theshorterthepolymericchains, themoreintense
thepenetrationofthenanoparticlesintothechains.
0.55
Nanoparticle − chain centre of mass pair distribution function 00001111 ........01224682468 1263254286 KKKKuuuuhhhhnnnn sssseeeeggggmmmmeeeennnnttttssss////cccchhhhaaaaiiiinnnn Spans of chains (normalized by their sum) 000 0000...234....2345555−40 −20 0 20 r − Rn [4Å0] 60 80WWW231///WWW 100
−40 −20 0 20 40 60 80 100
(a)
r − Rn [Å]
Figure9. Chaincentreofmassdistribution, accumulated in4.5-Å-thick
bins, for N=32 (continuous line), N=64(long-dashed line), N=128 6000
(short-dashedline)andN=256(dottedline). L12
L22
Theoverallshapeofthechainsisexploredbydeterminingtheir 5000 L32
cissasnieplrooegaFnsmndiessgeeutaofinhrnftneedect1cdhlt0hohea(aueacidsne)cm,ioigslsareeedlasglniedtnvmrssagoteltnunotogoetrltstyhcRhoelouoafrcbufhtfoidheonn.ecmcitarlOebnudridsuacirdboMiybnauonasttxazhholuaeyftrths:pgtia4hsry4te,ersactpoehtarvineneoecysnreecaantoloerltmenfesdsphttohhlafreeeposte.erdslpSiosymhppfeaaeetrhnnnniess--- 2R Tensor Eigenvalues [Å ]g 234000000000 Random walk expeLc1t2e+dL 2v2a+luLe3s2
calnanoparticle. Nearthesurfaceofthenanoparticle,chainstend 1000
to expand along their main axis (largest span,W , increases) and
3
shrink along the smaller ones (spansW2 andW1 decrease). The 0−40 −20 0 20 40 60 80 100
sametendencyisrevealedthroughanexaminationoftheeigenval- r − Rn [Å]
ues of the chain radius of gyration tensor. In the polymer melt,
polymershavetheshapeofflattenedellipsoids.45 Foranellipsoid (b)
formedbyarandomwalkmodel,theratioofthethreeeigenvalues Figure10. Overallshapeofchainsasafunctionofcentreofmassposition.
isreportedtobe12:2.7:1.46 Inoursimulations,eventhesmall- (a,top)Spansofchains,normalizedbytheirsumW=W1+W2+W3. (b,
est chains are in excellent agreement with this prediction, as far bottom) Eigenvalues of the chain radius of gyration tensor L12,L22,L32.
Theseplotsrevealthattheoverallshapeofchainsegmentcloudisdependent
asthesmallandthemediumellipsoidalsemiaxes(L andL )are
1 2 onpositionrelativetothenanoparticle.Thesystemsconsistsofchainswith
concerned. Atshorterdistancesfromthenanoparticlecentrethan 32Kuhnsegmentsperchainandonenanoparticleofradius36Å.
themeansizeofthechain,anexpansionofthechainsacrosstheir
principal semiaxis (L ) isfound. This leads toan increase of ra-
3
diusofgyrationR2g=L12+L22+L32 nearthenanoparticle. The
deformation of themoleculesissmaller forlonger chains (whose
dimensionsexceedbyfartheradiusofthenanoparticle). Farfrom
thesurfaceofthenanoparticle,thechaindimensions,estimatedei-
therasspansorasradiusofgyrationtensoreigenvalues,reachtheir
bulk average values, since the molecules are not affected by the
presenceofthenanoparticle.
Theorientationofchainsegmentcloudswithrespecttothesur-
faceofthenanoparticleisexploredbycomputingorderparameters
forchainspansandradiusofgyrationtensoreigenvectors.47,48The
definitionoftheorderparameterforspansisanalogoustothatused
7
hereforthesegmentsSW =(1/2) 3hcos2(q )i−1 .Theangleq is with increasing nanoparticle volume fraction is observed for all
formedbetweenthelargerspanW andthevector connecting the chain lengths. This expansion is maximal for 32 Kuhn segments
3
(cid:2) (cid:3)
centreofthenanoparticleandthecentreofmassofthechain. The perchain, wheretheradiusofgyrationRg0=42Åiscomparable
sameisdoneforthesmallestspan,W1. PlottedinFigure11(a)are totheradiusofthenanoparticleRn=36Å.Itseemsthatthereis
theorderparametersfortheshortestandlongestspanasfunctions atendencyofchainstoswellwhentheirdimensionisequal toor
ofthechaincentreofmassposition. Inthesamefashion,SListhe approachesthedimensionofthenanoparticle. Thisobservationis
order parameter of the eigenvectors corresponding to the greater inverygoodquantitiveagreementwithexperimentaldatareported
and smaller eigenvalues (Figure 11(b)). In the interfacial region, for the same system.8 In all other cases, the swelling due to the
weobserveastrongtendencyforthelongestspantoorientparal- presenceofthenanoparticlesishardlydistinguished.
lel to the surface and the shortest span to orient perpendicular to
thesurfaceofthenanoparticle. Thistendencyremainsunchanged a b
1.25 1.025
if,insteadofspans,theprincipalaxesofinertia(i.e.,theeigenvec-
1.2 1.02
torsoftheradiusofgyrationtensor)areusedasmeasuresofshape
1.15 1.015
(Figure11(b)). Thereorientationofchainsisconfinedtoaregion
1.1 1.01
whosewidthiscommensuratewithRg. 1.05 1.005
1 Rg0 = 42 Å 1 Rg0 = 59 Å
0 0.05 0.1 0.15 0 0.05 0.1 0.15
Volume fraction of nanoparticles Volume fraction of nanoparticles
1 c d
Longest span
Shortest span 1.04 1.06
1.035 1.05
0.75 1.03
1.025 1.04
1.02 1.03
0.5 1.015 1.02
1.01
SW 0.25 1.00 15 Rg0 = 84 Å 1.0 11 Rg0 = 120 Å
0 0.05 0.1 0.15 0 0.05 0.1 0.15
Volume fraction of nanoparticles Volume fraction of nanoparticles
0
Figure12. Deviation of radius of gyration Rg from its reference value
(meltwithoutnanoparticles) Rg0 forthefourchainlengths underconsid-
−0.25 eration(a,b,c,dfor32,64,128and256Kuhnsegmentsperpolymericchain
respectively).
−0.5
−40 −20 0 20 40 60 80 100 In the presence of many nanoparticles, structural features may
r − Rn [Å]
be revealed by probing the pair distribution function between the
(a) nanoparticles and the polymer chain centres of mass. The COM
ofthechainsareaccumulatedin4.5-Å-thickbins(Figure13). The
pairdistributionfunctiondepicted, isindicativeoftheengulfment
1 of nanoparticles by the polymeric chains. There are polymeric
Longest axis
Shortest axis chainsofeverychainlength, whoseCOMpositionliesinsidethe
0.75 nanoparticle. ThereisaremarkabledifferencebetweenFigure13
(many nanoparticles) and Figure 9 (one nanoparticle). In Figure
0.5 9, theCOMsof longer chains aremoredepleted inthespace oc-
cupied by the nanoparticle. In Figure 13, the shortest chains of
SL 0.25 N=32exhibitatendencytosurroundthenanoparticles,havinga
clearmaximumintheirCOMdistributioninsidethenanoparticle,
0
veryclosetoitscentre,alsooutsidethenanoparticle,somedistance
fromitssurface,withaminimuminbetween.Interestingly,within-
−0.25
creasingdistancethenanoparticle-chaincentreofmassdistribution
goesthroughaminimumandthenrisesagainashortdistanceaway
−0.5
−40 −20 0 20 40 60 80 100
fromthenanoparticlesurface,becauseofthecontributionofchains
r − Rn [Å]
whichareadsorbedto,butdonotengulfthenanoparticle.
(b)
Conclusions
Figure11. (a,top)Orderparametersforthelongestandshortestspansof
chainsasfunctionsofcentreofmassposition.(b,bottom)Orderparameters
fortheeigenvectorscorrespondingtothelargestandthesmallesteigenval- Thestructureofapolymermatrixsurroundinganimmersedspher-
uesofthechainradiusofgyrationtensorasfunctionsonthecentreofmass icalnanoparticlewasstudiedindetailbothatthesegmentandthe
position. Thesystemsconsistsofchainswith32Kuhnsegmentsperchain
chainlevels. Thesimulationmethodemployed reliesonahybrid
andonenanoparticleofradius36Å.
MonteCarloformalism. Itisafieldtheoreticapproach, basedon
thegroundsoftheself-consistentfield,whichcantreatefficiently
large systems. The system investigated bears great resemblance
Chain conformation in the presence of many
withanathermal,all-polystyrenenanocompositematerialthathas
nanoparticles
beenstudiedexperimentally. Severalconformationalpropertiesof
thesystemhavebeenexamined. Atthesegmentlevel,anincrease
The values of radius of gyration Rg relative to the value for the ofthelocalpolymerdensityaroundthenanoparticleisfound.Den-
purepolymermeltR ,areshowninFigure12asafunctionofthe
g0 sityprofilesareweaklydependentonthechainlength.Anincrease
nanoparticle volume fraction for the four different chain lengths
ofthedegreeofpolymerizationofthechainresultsinslightlyin-
used in this work. In general, an expansion of polymeric chains
creased thickness of the depletion layer. The bonds tend to ori-
8
2 32 segments/chain (9) Mackay,M.E.;Tuteja,A.;Duxbury,P.M.;Hawker,C.J.;Horn,B.V.;
Chain centre of mass pair distribution function 001111 ......1682468 12625486 ssseeegggmmmeeennntttsss///ccchhhaaaiiinnn ((((((111111542310)))))) 1SPd20DHCGDKM3ce03ohoauoiai004mmle.xaxLcl4,9mn2,aa,kpu0,0s,sL0Caoz.181ttZ1yu.saaC..42,0oMk.krJ90h;i,,ii.aM0,essC,.J1g;,,m59.H4h.aJ.3MAM.,ea,o3EPn3y.-A..5;;C,8a;.h–;d.rGCyhC–9aeRse.hDm3h.;4L.oe4;2eaK8alun2n0Gon,az,r.,0,,PimurYs8AaYorhTd.,in.l-.am.-y;1dLHgLmS0eaT.a.;a,,nc;.,ew;GHh6,Ardw5sRku.eT..0e.zeJRur.P.miS;tz.Ncae;eaPibjaErneaol,tn,,to.mKxc,OMeA.epJb.Soa.;.2e;.st0drJeoCr.0erH,i.u6aJJPo2r,.,.,r0aA3.CAb01OD.lh3;.1ope,.RJ,,mi2.1noJL,C..7.d.7;4SJrPh6Ã.0oeh2V.Jlmyi–.dasg.7Cn.uSP6h2et6hHaze0.yt,m0oesJ5r...Mn,2PP,01ahh02tByye93ssr.,.,..;
Nanoparticle − 00..24 (((111768))) CHAFrlareatlhdrl,eirniLJcda.koyMsnaor.nP;a,mrSeGcash.snw,,H2Ke.0i.z0TSe6hr..e,SK.e.qMSua.ilcJibr.orCmiuhomelmetc.huPeloehsryys2.0o20f08i0,n84h,1o1,m29o84g,3e20n3–e49o94u03s18.p.olymers;
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project (project number: 228398) with the support of the Euro-
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9