Monte Carlo simulations of a coarse grained model for an athermal all-polystyrene nanocomposite system
Georgios G. Vogiatzis, Evangelos Voyiatzis, Doros N. Theodorou
aa r X i v : . [ c ond - m a t . s o f t ] J a n Monte Carlo simulations of a coarse grained model for an athermal all-polystyrenenanocomposite system
Georgios G. Vogiatzis, Evangelos Voyiatzis, and Doros N. Theodorou ∗ School of Chemical Engineering, National Technical University of Athens, Zografou Campus, GR-15780 Athens, Greece
Received August 23, 2018; E-mail: [email protected]
Abstract:
The structure of a polystyrene matrix filled with tightlycross-linked polystyrene nanoparticles, forming an athermalnanocomposite system, is investigated by means of a Monte Carlosampling formalism. The polymer chains are represented as ran-dom walks and the system is described through a coarse grainedHamiltonian. This approach is related to self-consistent-field theorybut does not invoke a saddle point approximation and is suitablefor treating large three-dimensional systems. The local structure ofthe polymer matrix in the vicinity of the nanoparticles is found to bedifferent in many ways from that of the corresponding bulk, both atthe segment and the chain level. The local polymer density profilenear to the particle displays a maximum and the bonds developconsiderable orientation parallel to the nanoparticle surface. Thedepletion layer thickness is also analyzed. The chains orient withtheir longest dimension parallel to the surface of the particles. Theirintrinsic shape, as characterized by spans and principal momentsof inertia, is found to be a strong function of position relative to theinterface. The dispersion of many nanoparticles in the polymericmatrix leads to extension of the chains when their size is similar tothe radius of the dispersed particles.
Introduction
Nanocomposite polymeric materials consist of nanoparticles dis-persed in a polymeric matrix. This new family of composite mate-rials displays a variety of properties that attract great scientific andindustrial interest. The practice of adding nanoscale filler particlesto reinforce polymeric materials can be traced back to the earlyyears of the composite industry. The design of such conventionalcomposites has focused on maximizing the interaction betweenthe polymer matrix and the filler. This is commonly achieved byshrinking the filler particles in order to increase the surface areaavailable for interaction with the matrix. With the appearance ofsynthetic methods that can produce nanometer sized fillers, result-ing in an enormous increase of surface area, polymers reinforcedwith nanoscale particles should show vastly improved properties.Nanomaterials fabricated by dispersing nanoparticles in polymermelts have the potential for performances exceeding those of tradi-tional composites by far. Even though some property improvements have been achievedin nanocomposites, nanoparticle dispersion is difficult to control,with both thermodynamic and kinetic processes playing significantroles. It has been demonstrated that dispersed spherical nanopar-ticles can yield a range of multifunctional behaviour, includinga viscosity decrease, reduction of thermal degradation, increasedmechanical damping, enriched electrical and/or magnetic perfor-mance, and control of thermomechanical properties.
The tailor-made properties of these systems are very important to the manu-facturing procedure, as they fully overcome many of the existingoperational limitations. As a final product, a polymeric matrix en- riched with dispersed particles may have better properties than thenet polymeric material and can be used in more demanding andnovel applications. Therefore, an understanding and quantitativedescription of the physicochemical properties of these materials isof major importance for their successful production.The physical and chemical behavior of this kind of materials de-pends on the nature of the nanodispersed phase. The characteristicsof the surface of the particles significantly affect the behavior ofthe composite polymeric / particle system. In some cases, the dis-persed nanoparticles in the polymeric matrix may aggregate. Thepresence of particles with grafted chains improves wetting effectsin the system and suppresses aggregation, leading the material todramatically different performance.Extensive experimental work on all-polystyrene nanocompos-ites has been recently presented.
Regarding the conformationalproperties, the dimensions of matrix chains, measured by neutronscattering techniques, indicate swelling induced by the dispersedtightly cross-linked nanoparticles. This effect occurs only when theradius of gyration of the chain is larger than the nanoparticle radius.The all-polymer composite system appears to be a stable blend evenfor small interparticle distances, suggesting that chain shape in thepolymer matrix is highly distorted. As for the dynamic properties,the blend viscosity was found to demonstrate a non-Einstein-likedecrease with nanoparticle volume fraction. It was suggested thatthis phenomenon scales with the change in free volume introducedby the nanoparticles, while the entanglements seem to be totallyunaffected.The effect of incorporating a spherical nanoparticle on the con-formational properties of a polymer has been demonstrated bymeans of Monte Carlo (MC) simulations for dilute and semi-dilutesolutions.
These studies have been quite generic, employing abead-spring and hard-sphere representation for the polymer and thenanoparticle, respectively. The most significant changes in struc-tural properties and orientation occurred within the depletion layer.The effective interactions between two dispersed nanoparticles ina polymer solution have been quantified by calculating the po-tential of mean force through the expanded ensemble density ofstates technique. An oscillating long-range behaviour has beenrevealed.Efforts have also been made to predict the aggregation or thedispersion of polymer nanoparticles in polymer matrix by thermo-dynamic modelling. Recent advances are thoroughly reviewed in and include compressible regular solution free energy models, based on modification of theories for binary polymer blends, andgeneralizations of integral equation theories, such as the mi-croscopic Polymer Reference Interaction Site Model (PRISM). Theproposed models extend existing theories by incorporating spe-cific nanoparticle-nanoparticle and nanoparticle-polymer contribu-tions. The predicted phase diagrams indicate that such systemsexhibit a rich variety of behaviours, including upper critical solu-tion temperature-type, lower critical solution temperature-type, andhour-glass shape.A major challenge in simulating realistic nanocomposite materi- ls is that both the length and the time scales cannot be adequatelytreated by means of atomistic simulations because of the extensivecomputational load. This is why a variety of mesoscopic tech-niques have been developed for these particular systems. Amongthem, Self-Consistent Field Theory (SCFT) seems to be a well-founded simulation tool. This method adopts a field-theoretic de-scription of the polymeric fluids and makes a saddle-point (mean-field) approximation. Significant progress has been made in ap-plying the SCFT to nanoparticle-filled polymer matrices in the di-lute and semi-dilute region.
A lattice-based simulation in thecontext of the Scheutjens-Fleer approximation has cast some lightonto the equilibrium dispersion of nanoparticles with grafted poly-mer chains into dense polymer matrices, where the matrix and thebrushes share the same chemical structure. An attempt to over-come the restrictions posed by the saddle-point approximation wasmade in. The coordinates of all particles in the system were ex-plicitly retained as degrees of freedom. It was crucial to update si-multaneously the coordinates and the chemical potential field vari-ables.In the same spirit of relaxing the mean field approximation, Bal-azs and coworkers have proposed a coupling of SCFT for the inho-mogeneous polymers and density functional theory (DFT) for thenanoparticles.
The extrema of the approximate energy func-tional created in this way correspond to the mean-field solution. Al-though the SCFT-DFT approach is quite promising, it is currentlyhindered by the difficulty in extending to systems for which reliabledensity functionals are not available.A novel way of incorporating fluctuation effects in self-consistent mean-field theories has been recently proposed.
The “Single Chain in Mean Field” simulation technique is particle-based and founded on an ensemble of independent chains in fluc-tuating, external fields. The fields mimic the effect of the instan-taneous interactions of a molecule with its neighborhood. Theyare frequently updated using the spatially inhomogeneous densitydistribution of the ensemble. The explicit conformations of themolecules evolve by Monte Carlo. This technique has been verysuccesfull in describing the self-assembly and the ordering of blockcopolymers on patterned substrates encountered in processes suchas nanolithography.
The present work examines the structure of a polymer matrix inthe vicinity of polymeric nanoparticles. Particular attention is givento capturing the orientational effects and the chain conformation ofthe matrix in a dense, polymeric nano-filled composite. The analy-sis is based on a coarse grained Monte Carlo simulation methodwhich can be readily applied to the complex three-dimensionalnanocomposite system. Although the level of description is quiteabstract (i.e. that of the freely-jointed chain for the matrix), thedeveloped model tries to predict the behaviour of a nanocompos-ite with specific chemistry, namely of an all-polystyrene material.A main characteristic of the method is that it treats in a differentmanner the interaction between polymer and between polymer andparticle: the former is accounted for through a suitable functionalof the local densities, while the latter is described directly by anexplicit interaction potential.
MethodPolymer Coarse Grained Model
The aim of the developed coarse grained model is to describethe properties of the nanocomposite material at mesoscopic lengthscales. The main parameters are the chain connectivity, the finitecompressibility of the melt and explicit pairwise potentials for thevan der Waals interactions between polymer segment - nanoparticle and nanoparticle - nanoparticle. The system is contained in volume V at temperature T . The matrix consists of n polymeric chainswhich are considered to obey the freely jointed chain model. Eachchain consists of N Kuhn segments. The bond length is kept fixedand is equal to the Kuhn length, b , of the specific polymer underconsideration. The n p nanoparticles are represented as impenetra-ble spheres. The interaction energy H includes only non-bondedcontributions: H ( ~ r ) = H pp { b r ( ~ r ) } + H pn + H nn (1)where H pp stands for the non-bonded interactions between thepolymeric chains, H pn for the interaction between polymeric seg-ments and nanoparticles and H nn for the interaction betweennanoparticles.An important difference in the treatment of polymer-polymer in-teractions compared to those arising because of the presence of thenanoparticle is that the former are taken into account through a suit-able functional of the instantaneous local density b r ( ~ r ) , while thelatter are accounted for by a pairwise interaction potential. The lo-cal density is computed directly from the segment positions by asmoothing procedure. The non-bonded interactions H pp are givenby: b H pp { b r ( ~ r ) } = r N Z V d ~ r (cid:18) k N [ b r ( ~ r ) − ] (cid:19) (2)where b = k B T with k B being the Boltzmann constant and r is theaverage bulk number density of segments. Deviations of the localdensity from the average bulk value are restricted by the quadraticterm which stems from Helfand’s approximation. Thus, the melthas a finite compressibility, which is inversely proportional to k : k = k B T k T r . (3)Although the coordinates of the particles, either polymeric seg-ments or nanoparticles, are the degrees of freedom in our model,the local densities that appear in ( ?? ) are not defined by the micro-scopic expression b r ( ~ r ) = (cid:229) nj = (cid:229) Ni = d (cid:0) ~ r − ~ r j , i (cid:1) . Rather, they areobtained by a particle-to-mesh (PM) assignment technique. A uni-form, rectangular grid with n sites sites is introduced. The local den-sities are defined on each site by applying a smoothing procedureto the bead’s position. As a consequence, a discretization param-eter D L has to be introduced; it is the grid spacing. Phenomenaand details of the system whose length scales are smaller than the D L parameter cannot be described accurately. Thus, this parametersets the resolution level of our observations. It also defines a sortof microscopic cutoff, since it determines the range of interactionbetween neighboring beads. It follows that D L must not be lessthan the average distance between two beads. The value at which D L should be set depends strongly on the system of interest and thepreferred smoothing procedure.The assignment procedure of a polymeric bead to the sites of therectangular grid is shown in Figure 1. The i -th segment of the j -thchain at position ~ r j , i shown as a black dot, yields a contribution toa neighboring m -th site, shown as white dot, P (cid:2) ~ r j , i ,~ c m (cid:3) , where ~ c m is the position of the centre of the m th site and P the assignmentfunction. The resulting normalized density at site m is calculated asthe sum of the contributions of all segments in the system, givingthe instantaneous number of segments assigned to the site: b r m = r norm n (cid:229) j = ( N (cid:229) i = P (cid:2) ~ r j , i ,~ c m (cid:3))(cid:20) − (cid:0) d j , + d j , N (cid:1)(cid:21) (4)with d the Kronecker delta.The normalization constant r norm represents the segment num-ber density that the site m would have, provided that the melt wascompletely homogeneous without any density fluctuation, r norm = igure 1. Schematic of the PM technique used to estimate the local densi-ties. nN / n sites . The assignment function P could be chosen arbitrarily.In the present study, a zero-order (PM0) interpolation scheme isemployed. This choice corresponds to the case where the bead isassigned entirely to its closest site and does not contribute to anyother site. It follows that two beads interact only when they are as-signed to the same site. It is of great importance to use a dense gridwith many sites, so as to minimize the discretization effect on thedensity.Apart from the polymer-nanoparticle pairwise potential, themere existence of the nanoparticle, which is an impenetrable spher-ical object, defines a second, implicit way of interaction with thematrix. The available volume that the polymer can occupy in cells,that are in the vicinity of the nanoparticle, is less than the volumewhich can be occupied in cells far away from it. Since the volumeof the cells touching the nanoparticle is truncated, the normalizingdensity r norm of an arbitrary cell should be given by: r norm = r Z V accm d ~ r P (cid:2) ~ r j , i ,~ c m (cid:3) (5)where V accm is the volume of the cell space accessible to the poly-meric segments. For the case of the PM0 scheme, r norm is simplyproportional to the accessible (non nanoparticle-occupied) volumeassociated with site m , r norm = r V accm . The calculation of the ac-cessible free volume of each cell of the grid is based on a suitableadaptation of an analytical algorithm. The original algorithm de-termines in an analytical fashion the volume of a sphere delimitedby a set of arbitrary directed planes. It has been adapted to treatthe case of the volume of a sphere intersecting with a cubic boxand takes into account the periodic boundary conditions. The samestrategy can be applied to the case of multiple particles intersectingone box, since the former cannot intersect each other and the avail-able volume can be readily estimated by subtracting the volumeoccupied by each particle.An alternative to using a grid in order to estimate the local densi-ties is to resort to cloud-density methods. Such techniques call fora process that identifies the pair of beads which are close enough tointeract. A discretization parameter closely related to D L employedin the PM case, the Gaussian width, has to be defined. There is alsoa significant increase of computational load. Despite the demand-ing nature of the cloud methods, the accuracy of their predictionshas been shown to be the same as that of grid-mediated ones andboth methods give essentially the same results. Nanoparticle-polymer and nanoparticle-nanoparticleinteraction
The effect of dispersing bare nanoparticles in the polymer matrixis explicitly accounted. Pairwise potentials describing the effectiveinteraction between polymeric segment - nanoparticle, U e f fpn , andnanoparticle - nanoparticle, U e f fnn are introduced. The overall ex-plicit energetic effect is taken into account in the Hamiltonian bysumming up these interactions for all possible pairs of nanoparti-cles and polymeric segments: H pn = n p (cid:229) i n = nN (cid:229) i p = U e f fnp (cid:0) ~ r i n − ~ r i p (cid:1) (6)and H nn = n p − (cid:229) i n = n p (cid:229) j n = i n + U e f fnn (cid:0) ~ r i n − ~ r j n (cid:1) . (7)The reference energy level of the employed Hamiltonian ( ?? ) forthe homopolymer melt is that of a melt with uniform density profile.In this case, the effective interaction energy between two polymericbeads is taken as zero. The effective potentials introduced in ( ?? )and ( ?? ) should be such that, if the volume of the nanoparticleswas occupied by bulk homopolymer, then the energy of the sys-tem would be zero. The insertion of a nanoparticle in a uniformly-distributed melt can be thermodynamically accomplished in twosteps: a spherical volume of polymer, whose net interaction withthe rest of the polymer matrix is U equiv − ppp , is removed from themelt and a nanoparticle with equal volume is placed in its posi-tion, introducing a new net interaction U pn with the remaining bulkpolymer (Figure 2 from step A to step B). Thus, the net interac-tions between polymer-polymer should be subtracted from the netinteractions between polymer-nanoparticle: U e f fpn = U pn − U equiv − npp . (8)The treatment of the effective interactions between two nanopar-ticles is more sophisticated. Their insertion in the melt, followingthe above thermodynamic way, requires two more additional stepswith respect to the previous case. A second spherical volume ofpolymer is removed from the melt in the presence of one nanopar-ticle, whose net interaction with the rest of the system is U equiv − npn ,and the second particle is inserted in the created volume, introduc-ing new net interaction U nn with the composite system (Figure 2from step B to step C). U e f fnn = U nn − U equiv − npn + U equiv − npp (9)The coarse grained potentials describing the net interactions in-volved in ( ?? ) and ( ?? ) are derived on the grounds of Hamaker the-ory. A major advantage is that the existence of accurate atomisticpotentials describing these interactions is exploited. The need tointroduce empirically adjustable parameters is kept to a minimum.The integrated potentials are computationally effective. Each par-ticle is treated as a collection of atoms which interact only withatoms of different particles: U ( r ) = Z sphere Z sphere r ( ~ r ) r ( ~ r ) U LJ ( r ) dV dV (10)where r and r stand for the density of interaction centres in eachparticles, r is the distance between the centres of the particles,and dV , dV the elementary volumes of each particle. The discretedistribution of the interacting atoms in each particle is assumed tobe continuous and uniform. The shape of the particles is spheri-cal. An analytical closed-form solution can be rigorously derived, igure 2. Insertion of two nanoparticles in the matrix, in three discretesteps. In step A a spherical volume of polymeric matrix is removed fromthe melt. In step B the first spherical particle is placed in the place of theremoved volume of polymeric matrix. Finally in step C another volume ofpolymeric matrix is replaced by the second nanoparticle. provided that the atomistic potential is Lennard-Jones 12-6: U ( r ) = U A ( r ) + U R ( r ) (11) U A ( r ) = − A " a a r − ( a + a ) + a a r − ( a − a ) + ln r − ( a + a ) r − ( a − a ) ! (12) U R ( r ) = A s LJ r " r − r ( a + a ) + (cid:0) a + a a + a (cid:1) ( r − a − a ) + r + r ( a + a ) + (cid:0) a + a a + a (cid:1) ( r + a + a ) − r + r ( a − a ) + (cid:0) a − a a + a (cid:1) ( r + a − a ) − r − r ( a + a ) + (cid:0) a − a a + a (cid:1) ( r − a + a ) (13)where A = p e LJ r r s LJ is the Hamaker constant, a , a theradii of the spherical particles, r the distance between their cen-tres and e LJ , s LJ the Lennard-Jones energy and distance cross in-teraction parameters for the pair of interaction sites. In the follow-ing, the radius of the particle representing a polymeric segment, R b , is chosen such that ( / ) p R b equals the volume per Kuhnsegment in the bulk polymer, as calculated from the mass corre- sponding to a Kuhn segment and the experimental mass density(approximately R b = . (aliphatic),CH(aliphatic), and CH (aliphatic) sites interacting with each otherare taken from. The cross-interaction parameters are estimated bya geometric mean combining rule. Both nanoparticle - polymericsegment and nanoparticle - nanoparticle potential are depicted inFigure 3. −1 0 1 2 3 0 5 10 15 20 25 30 35 40 U / k B T Surface to surface distance of interacting particles [Å] Polymeric Segment − Nanoparticle−80−60−40−20 0 20 0 5 10 15 20 25 30 35 40Nanoparticle − nanoparticle
Figure 3.
Pairwise interaction potentials used for polymeric segment andnanoparticles with the energy axis scaled by k B T . The density of thenanoparticle is 1 . gcm and its radius is R n =
36 Å. The radius of the poly-meric segment interaction site R b is 6 . R n + R b in the case of the polymeric segment-nanoparticle inter-action (main figure) and by 2 R n in the case of nanoparticle - nanoparticleinteraction (inset). Details of the Monte Carlo Simulations
All simulations were carried out in the canonical statistical ensem-ble (NVT). The temperature of the system was 400 K. The box wascubic and its edge length was 600 Å. It should be noted that, forall chain lengths, the edge length is greater than 4 R g with R g beingthe radius of gyration of a chain, so as to avoid finite size effects. The systems studied were strictly monodisperse, while the degreeof polymerization of the chains N was varied, with N = , , .
97 g / cm . TheKuhn segment length b is 18 . . / cm . The char-acteristics of the systems studied, i.e. the degree of polymerizationof the chain, the radius and the mean density of the crosslinkedpolystyrene nanoparticle, are in close connection with systems thathave been studied experimentally by MacKay and co-workers. In our initial calculations, a single nanoparticle is placed at thecentre of the simulation box. The polymer chains are built ran-domly with the constraint that none of the polymeric beads over-laps with the nanoparticle. When more than one particles had tobe contained in a configuration, then the nanoparticles were placedfirst at randomly selected positions, so that they did not overlap,and then the polymer was built around them. The initial configura-tions created by this method have large local density fluctuations.These fluctuations increase with increasing chain length. A zerotemperature Monte Carlo optimization procedure took place in or- igure 4. Schematics of the relative size of nanoparticle and the four chainlengths under consideration. For each length the R g in pure melt is usedfor the comparison. The chain lengths are 32 (a), 64 (b), 128 (c) and 256 (d)Kuhn segments per chain respectively. der to reduce the density fluctuations. During this stage, all movesleading to more uniform density profile, thus decreasing the densityfluctuations, are accepted. In the opposite case, they are rejected. Five intermolecular moves were employed which treat the entirechains as rigid bodies: • Translation of individual chains in a random direction • Rotation of individual chains by random angles around ran-dom axes through their centres of mass • Reflection of individual chains at random mirror planes go-ing through the centre of mass • Inversion of individual chains at their centres of mass • Exchange of two chains preserving the centre of mass po-sitionsWhen the Monte Carlo proceeds, the internal shape of the chainsis restructured by four intramolecular moves: the flip, the end ro-tation, the reptation and the pivot moves. The exact mix of movesand acceptance rates of the moves are given in Table 1 for puremelt polystyrene system. The nanoparticles are allowed to translatein random directions every 500 MC steps.
Table 1.
Mixture of MC moves used for equilibration of pure melt of chainswith polymeric segments per chain Move % Attempted % AcceptedRigid translation 8 % 62.3 %Rigid rotation 8 % 42.7 %Rigid reflection 8 % 39.9 %Rigid inversion 8 % 37.5 %Rigid exchange 8 % 35.5 %Flip 15 % 77.5 %End rotation 15 % 70.4 %Reptation 15 % 61.8 %Pivot 15 % 53.9 %To demonstrate the efficiency of the MC method in equilibratingthe systems under study, the orientational autocorrelation function h ~ u ( t ) · ~ u ( ) i of a unit vector directed along the chain end-to-endvector was evaluated as a function of number of simulation steps.Figure 5 shows the results for the four chain lengths consideredhere. The shorter chain systems exhibit faster equilibration ratedue to the increased acceptance of rigid moves. Rigid moves of smaller chains lead to drastic variation of density fluctuations with-out affecting dramatically the overall energy of the grid. As largermolecules are considered, moves of this kind become less drasticbecause they tend to create larger density fluctuations (since a largenumber of segments change positions at the same time). On theother hand, intramolecular moves and especially the pivot movebecome very efficient for large chains, because of their ability torearrange a specific part of the chain. The total length of simula-tions varied between 250 and 800 millions iterations, one order ofmagnitude above the time needed for the end-to-end vector orien-tational autocorrelation function to drop to zero. < u ( t ) · u ( ) > Monte Carlo steps32 Kuhn segments/chain64 Kuhn segments/chain128 Kuhn segments/chain256 Kuhn segments/chain
Figure 5.
Decay of the chain end-to-end vector orientational autocorrela-tion function h ~ u ( t ) · ~ u ( ) i for four systems of different chain length at bulkpolymer density 0 . gcm . The radius of the nanoparticle is 36 Å. ResultsLocal polymer structure in the presence of ananoparticle
The local structure in the vicinity of the nanoparticle at the Kuhnsegment level is first studied through the radial mass density dis-tribution from the surface of the nanoparticle, which is displayedin Figure 6. The beads of the polymer have been classified in 1 . . gcm , located at the distancewhere appears the minimum of the Hamaker potential . This valueis observed for all chain lengths. The enhancement of polymericsegment density is very similar between the four chain lengths un-der consideration. Although being close to the nanoparticle leads toa decrease of the chain entropy (allowed conformations are fewerthan in the bulk), attractive energetic contributions due to the higherdensity of the nanoparticles overcome these entropic contributionsand lead to an increase in local density. Monomer packing effectsare important, since monomers form layers around the nanoparti-cle. A second peak is also observed in the radial density distri-bution of segments and the position in each case does not dependon the number of Kuhn segments of the chain. This second peakis located roughly one segment diameter away from the first peak.Similar studies in the semi-dilute region have also shown that,as the polymer density was increased, the initially chain length-dependent peaks become independent. On the basis of the pertur- ations in the total density profile from its bulk value, the thicknessof the interface could be estimated as less than 40 Å. The typicallength scale characterizing the density distribution in the vicinity ofthe nanoparticle is the depletion layer thickness d , which is shownin the inset of Figure 6. The depletion layer can be defined as thezone next to the particle where there are no polymer monomers. It is an equimolar dividing surface. The number of segments de-pleted is estimated by an appropriate mass balance and leads to anexpression of the form: ( R n + d ) − R n = Z ¥ R n r · ( − g ( r )) dr (14)where R n is the radius of the nanoparticle and g ( r ) = r ( r ) / r is thepair distribution function between the nanoparticle and the poly-meric segments around it. As shown in the inset to Figure 6, thethickness of the depletion layer increases slightly with increasingthe number of Kuhn segments in a chain. The value is close to theradius calculated for the segment’s spherical interaction site. D en s i t y [ g / c m ] r − R n [Å] 32 Kuhn segments/chain64 Kuhn segments/chain128 Kuhn segments/chain256 Kuhn segments/chainBulk polymer density at 400K 6 6.1 6.2 6.3 6.4 0 50 100 150 200 250 D ep l e t i on t h i ck ne ss [ Å ] Segments/chain
Figure 6.
Radial mass density distribution from the surface of the nanopar-ticle accumulated in 1 . N =
32 (continuous line), N = N =
128 (short-dashed line) and N =
256 (dotted line).The radius and the mass of the nanoparticle are 36 Å and 135 kDa, respec-tively. The straight dot-dashed line represents the bulk polymer density at400 K. In the inset, the variation of the depletion layer thickness d withchain length is shown. The local orientation of chain segments induced by the nanopar-ticle can be quantified by the second order Legendre polynomial P .An angle q may be defined by the bond vector of two consecutiveKuhn segments i and i + i . This angle is used to de-fine the P function: P = (cid:0) h cos q i − (cid:1) . P would assume val-ues of − .
5, 0 .
0, and 1 . P from the centre of a nanoparticle, accumulated in 5-Å-thick bins.A given bond is assigned to the bin in which its i Kuhn segment re-sides. The bonds lying in the vicinity of the particle are structuredand tend to be oriented tangentially to the interface. Chain lengtheffects on the orientational distribution of bonds are weak. Thetangential orientation tends to be stronger for the shortest chainsexamined. The shorter the chain, the more intense the orientationaleffects are. Kuhn segment orientation effects are weak beyond adistance of two Kuhn segment lengths from the particle surface.Similar behavior has been observed in molecular dynamics simu-lations of silica particles dispersed in a polyethylene-like matrix. One of the most important aspects of structure in polymer meltsis the so-called “correlation hole” effect . It arises in dense poly-meric fluids because of the melt’s incompressibility. Figure 8 shows −0.2−0.15−0.1−0.05 0 0.05 0 20 40 60 80 100 120 < P > r − R n [Å] 32 Kuhn segments/chain64 Kuhn segments/chain128 Kuhn segments/chain256 Kuhn segments/chain Figure 7.
Average second Legendre polynomial P plotted against the dis-tance of segment i from the surface of the nanoparticle in 5-Å-thick bins.The P polynomial is calculated between the vector connecting the cen-tre of the nanoparticle and a Kuhn segment i and the bond vector of Kuhnsegments i and i +
1. The radius of the nanoparticle is 36 Å. A positive(negative) order parameter indicates an orientational tendency that is per-pendicular (parallel) to the nanoparticle’s surface. the variation of the total and the intermolecular pair distributionfunctions between polymeric segments. The correlation hole ef-fect is evident in the intermolecular distribution functions. On veryshort length scales, the total distribution function diverges as 1 / r because of the intramolecular contribution dictated by the freelyjointed model. This is an artificial effect, caused by the fact thatchain self-intersection is not prevented in the freely jointed chainsused in our calculations. In the regime of small distances, segmentsof other molecules are expelled from the volume of a referencechain and this gives rise to a correlation hole in the intermolecu-lar pair distribution function. The spike in the total pair distributionfunction is an intramolecular feature, caused by pairs of segmentsconnected directly by a Kuhn segment. If an atomistic descriptionwere used, this spike would be replaced by a series of intramolecu-lar peaks reflecting the bonded geometry and conformational pref-erences of polystyrene molecules. On the other hand, for largelength scales, the intermolecular distribution function approachesunity, as the intramolecular distribution function approaches zero.As shown in the inset of Figure 8, longer chains exhibit “correlationhole” effect for slightly longer distances. S eg m en t − s eg m en t pa i r d i s t r i bu t i on f un c t i on g (r) Distance between two polymeric segments [Å]Intermolecular g(r) − 32 segments/chainTotal g(r) − 32 segments/chain 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 0 20 40 60 80 100N=32N=64N=128N=256
Figure 8.
Polymeric segments, total and intermolecular pair distributionfunctions, accumulated in 1-Å-thick bins. One chain length is depicted, N =
32 (main figure). The intermolecular pair distribution function, in thepresence of a nanoparticle of radius 36 Å, is included in the inset. hain structure in the presence of a nanoparticle Complementary to the study of local structure at the segment level,long range structural features may be revealed by probing proper-ties at the level of entire chains. The distribution of the centre-of-mass (COM) of the entire chains around the nanoparticle, which isshown in Figure 9, is such an example. The COMs of the chainsare classified in 4 . N anopa r t i c l e − c ha i n c en t r e o f m a ss pa i r d i s t r i bu t i on f un c t i on r − R n [Å] 32 Kuhn segments/chain64 Kuhn segments/chain128 Kuhn segments/chain256 Kuhn segments/chain Figure 9.
Chain centre of mass distribution, accumulated in 4 . N =
32 (continuous line), N =
64 (long-dashed line), N = N =
256 (dotted line).
The overall shape of the chains is explored by determining theirspans and the eigenvalues of their radius of gyration tensors. Spansare defined according to Rubin and Mazur: they are the dimen-sions of the smallest orthorhombic box that can completely en-close the chain segment cloud. Our analysis, presented for spansin Figure 10(a), leads to the conclusion that the overall shape of thesegment cloud is strongly affected by the presence of the spheri-cal nanoparticle. Near the surface of the nanoparticle, chains tendto expand along their main axis (largest span, W , increases) andshrink along the smaller ones (spans W and W decrease). Thesame tendency is revealed through an examination of the eigenval-ues of the chain radius of gyration tensor. In the polymer melt,polymers have the shape of flattened ellipsoids. For an ellipsoidformed by a random walk model, the ratio of the three eigenvaluesis reported to be 12 : 2 . In our simulations, even the small-est chains are in excellent agreement with this prediction, as faras the small and the medium ellipsoidal semiaxes ( L and L ) areconcerned. At shorter distances from the nanoparticle centre thanthe mean size of the chain, an expansion of the chains across theirprincipal semiaxis ( L ) is found. This leads to an increase of ra-dius of gyration R g = L + L + L near the nanoparticle. Thedeformation of the molecules is smaller for longer chains (whosedimensions exceed by far the radius of the nanoparticle). Far fromthe surface of the nanoparticle, the chain dimensions, estimated ei-ther as spans or as radius of gyration tensor eigenvalues, reach theirbulk average values, since the molecules are not affected by thepresence of the nanoparticle.The orientation of chain segment clouds with respect to the sur-face of the nanoparticle is explored by computing order parametersfor chain spans and radius of gyration tensor eigenvectors. Thedefinition of the order parameter for spans is analogous to that used S pan s o f c ha i n s ( no r m a li z ed b y t he i r s u m ) r − R n [Å] W /W W /W W /W (a) R g T en s o r E i gen v a l ue s [ Å ] r − R n [Å] L L L L +L +L Random walk expected values (b)
Figure 10.
Overall shape of chains as a function of centre of mass position.(a,top) Spans of chains, normalized by their sum W = W + W + W . (b,bottom) Eigenvalues of the chain radius of gyration tensor L , L , L .These plots reveal that the overall shape of chain segment cloud is dependenton position relative to the nanoparticle. The systems consists of chains with32 Kuhn segments per chain and one nanoparticle of radius 36 Å. ere for the segments S W = ( / ) (cid:2) h cos ( q ) i − (cid:3) . The angle q isformed between the larger span W and the vector connecting thecentre of the nanoparticle and the centre of mass of the chain. Thesame is done for the smallest span, W . Plotted in Figure 11(a) arethe order parameters for the shortest and longest span as functionsof the chain centre of mass position. In the same fashion, S L is theorder parameter of the eigenvectors corresponding to the greaterand smaller eigenvalues (Figure 11(b)). In the interfacial region,we observe a strong tendency for the longest span to orient paral-lel to the surface and the shortest span to orient perpendicular tothe surface of the nanoparticle. This tendency remains unchangedif, instead of spans, the principal axes of inertia (i.e., the eigenvec-tors of the radius of gyration tensor) are used as measures of shape(Figure 11(b)). The reorientation of chains is confined to a regionwhose width is commensurate with R g . −0.5−0.25 0 0.25 0.5 0.75 1−40 −20 0 20 40 60 80 100 S W r − R n [Å] Longest spanShortest span (a) −0.5−0.25 0 0.25 0.5 0.75 1−40 −20 0 20 40 60 80 100 S L r − R n [Å] Longest axisShortest axis (b) Figure 11. (a,top) Order parameters for the longest and shortest spans ofchains as functions of centre of mass position. (b,bottom) Order parametersfor the eigenvectors corresponding to the largest and the smallest eigenval-ues of the chain radius of gyration tensor as functions on the centre of massposition. The systems consists of chains with 32 Kuhn segments per chainand one nanoparticle of radius 36 Å.
Chain conformation in the presence of manynanoparticles
The values of radius of gyration R g relative to the value for thepure polymer melt R g , are shown in Figure 12 as a function of thenanoparticle volume fraction for the four different chain lengthsused in this work. In general, an expansion of polymeric chains with increasing nanoparticle volume fraction is observed for allchain lengths. This expansion is maximal for 32 Kuhn segmentsper chain, where the radius of gyration R g =
42 Å is comparableto the radius of the nanoparticle R n =
36 Å. It seems that there isa tendency of chains to swell when their dimension is equal to orapproaches the dimension of the nanoparticle. This observation isin very good quantitive agreement with experimental data reportedfor the same system. In all other cases, the swelling due to thepresence of the nanoparticles is hardly distinguished. g0 = 42 Å 1 1.005 1.01 1.015 1.02 1.025 0 0.05 0.1 0.15Volume fraction of nanoparticlesbR g0 = 59 Å 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04 0 0.05 0.1 0.15Volume fraction of nanoparticlescR g0 = 84 Å 1 1.01 1.02 1.03 1.04 1.05 1.06 0 0.05 0.1 0.15Volume fraction of nanoparticlesdR g0 = 120 Å Figure 12.
Deviation of radius of gyration R g from its reference value(melt without nanoparticles) R g for the four chain lengths under consid-eration (a,b,c,d for 32, 64, 128 and 256 Kuhn segments per polymeric chainrespectively). In the presence of many nanoparticles, structural features maybe revealed by probing the pair distribution function between thenanoparticles and the polymer chain centres of mass. The COMof the chains are accumulated in 4 . N =
32 exhibit a tendency to surround the nanoparticles, having aclear maximum in their COM distribution inside the nanoparticle,very close to its centre, also outside the nanoparticle, some distancefrom its surface, with a minimum in between. Interestingly, with in-creasing distance the nanoparticle-chain centre of mass distributiongoes through a minimum and then rises again a short distance awayfrom the nanoparticle surface, because of the contribution of chainswhich are adsorbed to, but do not engulf the nanoparticle.
Conclusions
The structure of a polymer matrix surrounding an immersed spher-ical nanoparticle was studied in detail both at the segment and thechain levels. The simulation method employed relies on a hybridMonte Carlo formalism. It is a field theoretic approach, based onthe grounds of the self-consistent field, which can treat efficientlylarge systems. The system investigated bears great resemblancewith an athermal, all-polystyrene nanocomposite material that hasbeen studied experimentally. Several conformational properties ofthe system have been examined. At the segment level, an increaseof the local polymer density around the nanoparticle is found. Den-sity profiles are weakly dependent on the chain length. An increaseof the degree of polymerization of the chain results in slightly in-creased thickness of the depletion layer. The bonds tend to ori- N anopa r t i c l e − C ha i n c en t r e o f m a ss pa i r d i s t r i bu t i on f un c t i on r − R n [Å] 32 segments/chain64 segments/chain128 segments/chain256 segments/chain Figure 13.
Chain centre of mass distribution, accumulated in 4 . ent parallel to the nanoparticle surface, as revealed by the secondLegendre polynomial. All these effects have a characteristic lengthscale commensurate with the chain segment size.At the chain level, chain centres of mass are able to penetratethe interior of the particle, as chains engulf the particle. This phe-nomenon is more pronounced when the radius of gyration of chainsis comparable in size to the nanoparticle. The formation of chainlayers around the particle is suggested by the particle-polymer cen-tre of mass pair distribution function. The size and the shape of thechains are also affected. A flattening of polymer molecules whentheir centre of mass is close to the nanoparticle centre is revealed byboth the chain spans and the eigenvalues of the radius of gyrationtensor. This deformation of the polymer molecules is small for longchains.In the case of the dispersion of many nanoparticles, deformationof the chain shape is observed. The chains which share the samelength scale with the dispersed particles tend to swell. This is inagreement with recent neutron scattering measurements. The radiusof gyration increases with increased nanoparticle volume fraction. Acknowledgments
This work was funded by the European Union under the FP7-NMP-2007 program, Nanomodel Grant Agreement SL-208-211778.Computational work was carried out under the HPC-EUROPA2project (project number: 228398) with the support of the Euro-pean Commission Capacities Area - Research Infrastructures Ini-tiative. Fruitful discussions with Dr. Kostas Daoulas are gratefullyacknowledged.
References
References (1) Gersappe, D.
Phys. Rev. Lett. , , 058301.(2) Balazs, A. C.; Emrick, T.; Russel, T. P. Science , , 1107.(3) Bansal, A.; Yang, H.; Li, C.; Benicewiz, B. C.; Kumar, S. K.;Schadler, L. S. J. Polym. Sci., Part B: Polym. Phys. , , 2944–2950.(4) Lee, J. Y.; Buxton, G. A.; Balazs, A. C. J. Chem. Phys. , , 5531–5540.(5) Bockstaller, M. R.; Thomas, E. L. Phys. Rev. Lett. , , 166106.(6) Si, M.; Araki, T.; Ade, H.; Kilcoyne, A. L. D.; Fisher, R.; Sokolov, J. C.;Rafailovich, M. H. Macromolecules , .(7) Stratford, K.; Adhikari, R.; Pagonabarraga, I.; Desplat, J.-C.; Cates, M. E. Science , , 2198–2201.(8) Tuteja, A.; Duxbury, P. M.; Mackay, M. E. Phys. Rev. Lett. , ,077801. (9) Mackay, M. E.; Tuteja, A.; Duxbury, P. M.; Hawker, C. J.; Horn, B. V.;Guan, Z.; Chen, G.; Krishman, R. Science , , 1740.(10) Mackay, M. E.; Dao, T. T.; Tuteja, A.; Ho, D. L.; Van Horn, B.;Kim, C. J., Ho-Cheol amd Hawker Nat. Mater. , , 762–766.(11) Doxastakis, M.; Chen, Y.-L.; Guzman, O.; de Pablo, J. J. J. Chem. Phys. , , 9335–9342.(12) Doxastakis, M.; Chen, Y.-L.; de Pablo, J. J. J. Chem. Phys. , ,034901.(13) Hall, L. M.; Jayaraman, A.; Schweizer, K. S. Curr. Opin. Solid State Mater.Sci. , , 38 – 48, Polymers.(14) de Luzuriaga, A. R.; Grande, H. J.; Pomposo, J. A. J. Chem. Phys. , , 084905.(15) Pomposo, J. A.; de Luzuriaga, A. R.; Etxeberria, A.; Rodrà guez, J. Phys.Chem. Chem. Phys. , , 650.(16) Arthi Jayaraman, K. S. S. Macromolecules , , 9430–9438.(17) Hall, L. M.; Schweizer, K. S. J. Chem. Phys. , , 234901.(18) Fredrickson, G. H. The equilibrium theory of inhomogeneous polymers ;Clarendon Press, 2006.(19) Surve, M.; Pryamitsyn, V.; Ganesan, V.
Langmuir , , 969–981.(20) Surve, M.; Pryamitsyn, V.; Ganesan, V. Macromolecules , , 344–354.(21) Ganesan, V.; Khounlavong, L.; Pryamitsyn, V. Phys. Rev. E , ,051804.(22) Harton, S. E.; Kumar, S. K. J. Polym. Sci., Part B: Polym. Phys. , ,351–358.(23) Sides, S. W.; Kim, B. J.; Kramer, E. J.; Fredrickson, G. H. Phys. Rev. Lett. , , 250601.(24) Thompson, R. B.; Ginzburg, V. V.; Matsen, M. W.; Balazs, A. C. Macro-molecules , , 1060–1071.(25) Buxton, G. A.; Lee, J. Y.; Balazs, A. C. Macromolecules , , 9631–9637.(26) Lee, J. Y.; Shou, Z.; Balazs, A. C. Phys. Rev. Lett. , , 136103.(27) Daoulas, K. C.; Müller, M. J.Chem.Phys. , , 184904.(28) Daoulas, K. C.; Müller, M.; Pablo, J.; Nealey, P.; Smith, G. Soft Matter , , 573–583.(29) Detcheverry, F. A.; Pike, D. Q.; Nealey, P. F.; Müller, M.; de Pablo, J. J. Phys. Rev. Lett. , , 197801.(30) Detcheverry, F. A.; Kang, H.; Daoulas, K. C.; Müller, M.; Nealey, P. F.;de Pablo, J. J. Macromolecules , , 4989–5001.(31) Stoykovich, M. P.; Daoulas, K. C.; Müller, M.; Kang, H.; de Pablo, J. J.;Nealey, P. F. Macromolecules , , 2334–2342.(32) Helfand, E.; Tagami, Y. J.Chem.Phys. , , 3592.(33) Dodd, L. R.; Theodorou, D. N. Mol.Phys. , , 1313.(34) Laradji, M.; Guo, H.; Zuckermann, M. J. Phys. Rev. E , , 3199–3206.(35) Hiemenz, P. C. Principles of colloid and surface chemistry , 2nd ed.;Dekker, 1977.(36) Hamaker, H.
Physica(Amsterdam) , IV , 1058.(37) Everaers, R.; Ejtehadi, M. Phys.Rev.E , , 041710.(38) Spyriouni, T.; Tzoumanekas, C.; Theodorou, D.; MÃijller-Plathe, F.; Mi-lano, G. Macromolecules , , 3876–3885.(39) Good, R. J.; Hope, C. J. J.Chem.Phys. , , 111.(40) Marsaglia, G. Ann. Math. Stat. , , 645–646.(41) Auhl, R.; Everaers, R.; Grest, G. S.; Kremer, K.; Plimpton, S. J. J. Chem.Phys. , , 12718.(42) Aarts, D. G. L.; Tuinier, R.; Lekkerkerker, H. N. W. J. Phys.: Condens.Matter , , 7551–7561.(43) Brown, D.; Marcadon, V.; MÃl’lÃl’, P.; AlbÃl’rola, N. D. Macromolecules , , 1499–1511.(44) Rubin, J.; J.Mazur, Macromolecules , , 139–149.(45) Theodorou, D. N.; Suter, U. W. Macromolecules , , 1206–1214.(46) Picu, R. C.; Ozmusul, M. S. J. Chem. Phys. , , 11239–11248.(47) Mansfield, K. F.; Theodorou, D. N. Macromolecules , , 4430–4445.(48) Mansfield, K. F.; Theodorou, D. N. Macromolecules , , 4295–4309., 4295–4309.