Influence of Surface Grafted Polymers on the Polymer Dynamics in aSilica−Polystyrene Nanocomposite: A Coarse-Grained MolecularDynamics InvestigationAzadeh Ghanbari,†,* Mohammad Rahimi,‡ and Jaber Dehghany†
†Department of Systems Immunology, Helmholtz Centre for Infection Research, Inhoffenstr. 7, 38124 Braunschweig, Germany‡Eduard-Zintl-Institut fur Anorganische und Physikalische Chemie and Center of Smart Interfaces, Technische Universitat Darmstadt,Petersenstraße 20, D-64287 Darmstadt, Germany
*S Supporting Information
ABSTRACT: Performing coarse-grained molecular dynamics simulations, the localdynamics of free and grafted polystyrene chains surrounding a spherical silicananoparticle has been investigated, where the silica nanoparticle was either bare orgrafted with 80-monomer polystyrene chains. The effect of the free (matrix) chainmolecular weight and grafting density on the relaxation time of both the free andgrafted polystyrene chains has been investigated. Furthermore, we have analyzed thelocal mobility of the grafted chains at different separations from the nanoparticlesurface, as well as on the mean square displacement of the nanoparticles. Proximityto the surface, confinement by the surface, increased grafting density and increasedmatrix chain length were found to slow down the dynamics of the chain monomersand hence to increase the corresponding relaxation times. “Drying” of the graftednetwork of the nanoparticle via increasing the free chain lengths, which is known toshrink the brush-height, was found to slow down the relaxation of the brushes, too.The thickness of the interphase, beyond which the polymers showed bulklike behavior, was ∼2 nm for a bare nanoparticle,corresponding to four monomer layers, for all matrix chain lengths investigated. It increased to ∼3 nm for grafted nanoparticlesdepending on the grafting density and the matrix chain molecular weight.
1. INTRODUCTION
Nanocomposite materials (NCMs) exhibit novel propertieswhich give them the potential for a variety of applications inindustry, ranging from biomedical and electronic to automotiveand aerospace applications.1−3 Their high industrial potentialcauses attention from the scientific community to understandand to predict some of the features of NCMs. With this basicunderstanding, the development of new NCMs with desiredproperties should become easier. The structural properties ofpolystyrene (PS) chains around a spherical silica nanoparticle(NP), that was either bare or grafted with similar PS chains,were investigated performing either atomistic or coarse-grained(CG) molecular dynamics (MD) simulations4−6 of a modelNCM. These simulations revealed a layered polymer structurearound the nanoparticle, manifested as peaks of enhanceddensity. Moreover, the chain segments in the vicinity of the NPshowed tangential orientation along the surface for all chainlengths considered. In the current MD study, the same CGmodel has been employed to investigate how the dynamicalproperties of PS-silica NCMs are affected near the filler particle.In the presence of an individual filler particle, matrix chains
experience structural and dynamical perturbations by thesurface of the nanoparticle.4−8 The nature of theseperturbations, at a qualitative level, and also their magnitudeand spatial extent from the filler surface, at a quantitative level,
depends on the polymer−nanoparticle interaction as well as oncertain quantities of the filler (its shape, grafting state, particle/chain size ratio, etc.).8−11 The question of how deep into thepolymer phase the chain structure and dynamics is affected bythe presence of the nanoparticle, that is, how thick this so-calledinterphase is, is not entirely academic. Especially in nano-composites, the interphase contributes significantly to theproperties of the entire composite. Here the interstitial spacecan be of nanometer size and the polymer nowhere has theopportunity to develop its bulk behavior. In a CG model of ananocomposite, for example, Liu et al.10 reported a transitionfrom a tangential and to a perpendicular orientation of bonds,segments, and whole chains with respect to the particle surfacein the interfacial region when changing the filler−matrixinteraction type from strongly attractive to purely repulsive. Ithas also been shown that the NP-induced changes in theconformation of whole chains depend on the ratio of thechains’ radius of gyration to the nanoparticle size. Whereasshort chains become more stretched close to the filler, longerchains tend to compact, compared to bulk chains, and wraparound the NP.4,7 Recent MD studies of NCMs have
Received: July 17, 2013Revised: October 16, 2013Published: October 17, 2013
demonstrated that the interphase dimension does not onlydepend on the chosen NP and polymer but also on the quantitystudied for a given composite sample:12 Local structuralproperties (monomer density, monomer orientation) convergewithin a few monomer diameters, whereas anomalies in theglobal chain structure last for 1 or 2 radii of gyration.In order to predict and control the properties of NCMs
around individual particles quantitatively, it is essential tocharacterize the structure and properties of the interfacialregion. Whereas the changes in the local structural anddynamical properties of the matrix polymer confronting theNP surface have been investigated by experimental,13−15
theoretical,16−18 and computer simulation methods,5,6,9,19−22 aconclusive understanding of the interphase region has still notbeen reached; especially its dynamical properties have been asubject of an ongoing debate. For example, several independentexperiments (including NMR, quasi-elastic neutron scattering,dielectric relaxation spectroscopy, differential scanning calorim-etry, and dynamical mechanical measurements)23−29 haveshown the existence of two distinct dynamic processes inNCMs, manifested by different relaxation times. One ischaracteristic for an immobile region (or at least one ofreduced mobility) caused by the adhesion of the polymerchains to the surface, which grows with the filler content.30 Thesecond refers to a free region which shows shorter relaxationtimes. It has been demonstrated that functionalization of thefiller surface by suitable polymer chains weakens the adhesionof matrix chains to the surface, whereas an increase of theinterfacial surface area and the polymer−surface attractionenhances it. There are also contradicting experimental studies(like dynamical mechanical spectroscopy, NMR, dielectricspectroscopy, and calorimetry) which conclude that the localsegmental dynamics of the chains in the vicinity of fillerparticles is basically the same as that of the unfilled system.31−34
Despite some stiffening of the elastomer in the vicinity of fillers,there is no appreciable difference in the segmental dynamics.35
There are even reports of increased mobility of the interfacialchains,36−39 which, together with the above-mentioned ones,shows the lasting complexity of the subject. Robertson andRoland40 have reviewed the literature dealing with the effects ofproximity to the filler surface on the local segmental mobility ofpolymer chains. They mentioned that some of thesediscrepancies can be ascribed to ambiguous methods of dataanalysis; others likely reflect changes in the filler−polymerinteraction between different systems.The interpretation of the reported empirical data can become
difficult, as alternative methods/models of data analysis canlead to completely different conclusions for the sameexperimental observation. In fact, some alternative interpreta-tions of the measurements on interfacial chain mobility havebeen proposed,40,41 showing that care must be taken in thisissue. On the other hand, there is an additional complication bythe necessary distinction between different regions of a sample(under study) contributing to the experimental measurements.For the interfacial dynamics of polymer chains, Robertson andRoland40 have emphasized that it is not always evident fromexperimental studies that a clear distinction has been madebetween chain segments immobilized by their spatial proximityto filler particles versus specific chemical units adsorbed atspecific sites on the filler surface. Such a precise distinction,being very difficult (if not impossible) in experiment, is ratherstraightforward in computer simulations.
Computer simulation methods have been employed to studythe interfacial region in NCMs in terms of structural5,7,42,43 anddynamical6,8,10,44−46 properties. For example, employing abead−spring polymer model, Star et al.44 showed that therelaxation of the monomers closest to the nanoparticle surfaceis slowest when the monomers and the nanoparticle have anattractive interaction. Such a reduced mobility in the presenceof an attractive filler−polymer interaction has been observed inother investigations, too.6,47−49 Conversely, Star and Glotzerobserved a significantly enhanced (compared to the bulk)relaxation of surface layer monomers when there was onlyrepulsion between the monomers and the nanoparticle.8
Borodin et al.50 reported the same reduced local polymermobility near the surface, accompanied by an enhancedpolymer density, for attractive interactions between the surfaceand the polymer chains. Moreover, the dynamic behavior of PSmatrix chains was found by Ndoro et al.6 to be affected both bythe grafting state of the silica nanoparticle (moderately) and itsdiameter or curvature (strongly). A higher NP radius (= lowersurface curvature) leads to a slower polymer dynamics and toan increased thickness of the “slow zone” around the particle.The common result of all computer simulations is that themere existence of an interface between the polymer and fillermodifies the polymer structure. Even though these perturba-tions can reach beyond the first molecular layer, they persistonly for a few nanometers from the filler surface.4−8,50,51
Beyond such a distance all structural and dynamical propertiesshow bulklike behavior. Investigating the influence of thegrafting state, as well as the grafting density and matrix-chainlength on the dynamical properties of the interphase region isthe main aim of the present coarse-grained MD study of PS-silica NCMs.The available results indicate that all modifications in the
properties of a NCM relative to an unfilled bulk polymer can beattributed to the altered behavior of the polymer in theinterphase. Since each filler particle contributes to theenormous volume of the interphase, its area is most efficientwhen nanosized particles are uniformly dispersed in the matrixpolymer. The stabilization of a dispersion of nanoparticles willbe addressed in future applications of the current NCM model.
2. SYSTEMS AND METHODS
2.1. Mapping Scheme and Coarse-Grained Potentials.In the present study we have employed the coarse-grainingscheme proposed by Qian et al.52 for the PS matrix, where eachrepeat unit of PS is represented by one CG bead whoseinteraction center is located at the repeat unit’s center-of-mass.The same mapping scheme is used for the free and grafted PSchains. Two different beads (R and S) are used to account forthe chirality of atactic PS. The grafted chains are attached to theNP via a linker unit (H−[CH2CHCHCH2]3(CH3)2Si−) asused in experiments (see the Supporting Information of ref 5for its structure). The linker has been divided into four CGbeads of two kinds having approximately the same mass. Thefirst one corresponds to −CH2CHCHCH2− and the secondto −(CH3)2Si. For both, the CG interaction centers are locatedat the center of mass of the group of atoms. The spherical silicaNP has a diameter of ∼4 nm and is filled with 873 CG beads ofSiO2; their centers of mass are located at the position of the Siatoms. The NP beads have been distinguished as either surfaceor core beads which have different interactions with thepolymer, the surface beads contributing more. Detailed
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information on CG potentials has been given in a previousstudy.4
The CG interaction potentials for PS, which containnonbonded pairwise intermolecular interactions, bond inter-actions between neighbors as well as angle potentials betweenthree subsequent beads, are prepared using iterative Boltzmanninversion (IBI).53 The same is valid for all other intermolecularinteractions (PS−linker, PS−NP, linker−NP, and linker−linker). The aim of the IBI method is the reproduction ofthe atomistic structure by a CG model. In the present setup, thestructure is defined by radial distribution functions betweendifferent bead types as well as by bond-length and angledistributions. In our previous study of silica-PS composites, wehave shown a satisfactory agreement between CG and atomisticstructures.4 Furthermore, the reliability of the CG potentialswas confirmed by back-mapping the equilibrated CG structureto an atomistic resolution via a new back-mapping scheme.54
For more details of the potential development and the adoptedmapping scheme, the reader is referred to our previous study.4
2.2. Initial Configurations. Silica NPs of radius 2 nm wererandomly distributed in the simulation box; the PS chains wereadded to the system afterward. All initial configurations weregenerated by a method developed and described previously.4 Atfirst, the complete NPs with all their superatoms were copied totheir random (nonoverlapping) positions in the simulation box.Then the linker beads (connecting the grafted chains, if any, tothe NP) were placed next to the NP beads which they aresupposed to be bound to. These linkers constitute the firstbeads of each grafted chain. At the next step, a new bead wasadded to each grafted chain. The use of a self-avoiding randomwalk (SARW) procedure guaranteed that its distance from theprevious bead is equal to b (the average bond length betweenpolymer beads: 0.5 nm). At the same time, the shortest alloweddistance between two nonbonded beads (PS−PS or PS−NP)was chosen to be one-half of this length (dnb = 0.25 nm). Twobeads within this distance were considered as overlapping. Theposition of each new bead was chosen randomly on the surfaceof a sphere of radius b around the last bead of the parent chain.If there was no overlap between the new bead and any otherbead in the system, we placed the new bead at the chosenposition. Otherwise, another random point was chosen on thesurface of the same sphere. If, due to an overlap between the
new bead and the other beads of the system, the new positionwas not suitable, the algorithm stopped after a maximum of1000 attempts. At the end of such a step, each grafted chainreceived one additional bead. A simultaneous growth of allgrafted polymer chains was initiated when repeating thisprocedure until all chains of all NPs reached their desired chainlength. We then added free chains to the system. To add Nf freechains, we first placed Nf seeds (first monomers) randomly inthe simulation box, such that they overlapped neither withbeads of the NPs nor with beads of the grafted corona. Thenthey were grown in the same way as allowed for the graftedchains. To circumvent the possible overlap of the free andgrafted chains during the SARW process, we checked theoverlap of each newly added free bead to the beads of all otherfree and grafted chains, as well as NPs, using the above dnbcriterion between nonbonded beads. It should be mentionedthe use of larger dnb values generated configurations preventingthe addition of new beads as they experienced earlier theoverlap with other beads.
2.3. Systems and Simulation Details. We haveperformed simulations for different grafting states of thenanoparticle (bare, low = 0.5 chains/nm2; and high graftingdensity = 1.0 chains/nm2). We have put six NPs into thesimulation box of an initial volume ∼ (24.5 nm)3, with the NPvolume fraction initially being ∼1.4%. For comparison, we havealso simulated a pure unfilled PS system. Details of the systemsstudied in this part of our work are provided in Table 1.The prepared initial configurations of the systems are far
below the equilibrium density (∼660−680 kg/m3 at thebeginning) and still have some overlaps. Hence, they have tobe relaxed. In order to equilibrate the initial configuration, thesystems were simulated in the NPT ensemble for 12−20 ns,depending on the chain length, to reach the desired density(∼940−970 kg/m3). After this step, the simulation box has thevolume of ∼(22 nm)3 and a NP volume fraction of 2%. Thesystems were then simulated for 40 ns for data analysis using atime step of 4 fs. All MD runs were carried out at a pressure of101.3 kPa and a temperature of 590 K. Berendsen’s thermostat(with a coupling time of 0.2 ps) and barostat (with a couplingtime of 5 ps and an isothermal compressibility of 1.0 × 10−6
kPa) were used to control temperature and pressure. The cutofffor the nonbonded interactions was 1.5 nm, and the neighbor
Table 1. Diffusion Coefficient of Nanocomposite Systems with a Fixed Nanoparticle Radius of 2 nm (T = 590 K, P = 101.3kPa)a
grafting density(chains/nm2)
no. of graftedchains per NP
grafted chain length(monomers)
no. of freechains
free chain length(monomers)
diffusion coefficient of freechains (×10−6 nm2/ps)
diffusion coefficient, comparedto pure polystyrene
aThe values for 200-mer chains are not reported, because they did not reach the diffusion limit within 40 ns of production run. There are sixnanoparticles in the simulation box of (22 nm)3, and the nanoparticle volume fraction is 2%. The last column provides the diffusion coefficientnormalized by that of pure polystyrene chains of the same length.
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list cutoff was 1.6 nm. Configurations were sampled every 1000time steps (4 ps). All CG simulations were performed with theIBIsCO code.55
3. MONOMER DYNAMICS OF FREE CHAINSThe influence of added filler particles on the matrix chaindynamics is investigated in the following. Table 1 shows thecenter-of-mass diffusion coefficient of all free PS chains in thepresence of nanoparticles for different grafting densities andmatrix chain lengths. Even though the volume fraction of fillerparticles is low (2%), the introduction of nanoparticles (NPs)always leads to a lower diffusion coefficient, and hence to ahigher viscosity of the polymer chains in the nanocomposite.The diffusion of the polymer chains, however, provides only aglobal view of changes in their dynamics induced by the NPs. Inorder to elucidate the local dynamics of monomers around aNP in the presence of free chains of different length, we haveused the self-scattering-function, S(q,t). It shows the correlationof monomer positions at time t with their initial positions at t =0. The self-scattering-function is defined as S(q,t) = (1/N)⟨Σj = 1
N exp{−iq[rj(t) − rj(0)]}⟩ where q is the wave vector, Nis the total number of monomers, rj(t) is a vector pointing tothe position of monomer j at time t, and ⟨...⟩ is the ensembleaverage for all PS monomers in the system. The monomerdynamics can also be monitored by the mean squaredisplacement (MSD). The main advantage of the self-scattering-function over the MSD, however, is its experimentalaccessibility. Moreover, S(q,t) allows a simpler quantitativecomparison of the dynamics in different shells around the NPof the systems. Unlike the MSD it simply varies between 1(highly correlated) and 0 (totally uncorrelated). In thedefinition of S(q,t), which has been given above, the wavevector q determines the displacement range beyond which amonomer can be considered as out-of-phase which means thatit has lost the spatial correlation with its original position (at t =0). Choosing an appropriate value for q, which is in line withthe length scales in the system, gives us an additional degree offreedom to define and measure the degree of autocorrelation ofthe monomer position. We have chosen q = 0.78 nm−1 = (π/4nm) which leads to a phase difference of π when a monomer isdisplaced from its position by a distance equal to the NPdiameter. S(q,t) calculated in this way provides the spatialcorrelation of monomers with their initial position.We have considered concentric spherical shells of equal
thickness of 1 nm around each nanoparticle. This has enabledus to study how and to which extent the mobility of themonomers at different distances from surface (i.e., in thechosen shells) is affected by the presence of the NP. Bydefinition, a monomer belongs to that shell of a NP and thus tothat NP where it has spent most (at least 50%) of its time. Thesame criterion had been adopted already in our recent MDstudies of the present nanocomposite.4−6 The self-scattering-function of monomers of a given (nth) shell is defined as Sn(q,t)= (1/Nn,p)⟨Σj = 1
Nn,p exp{−iq[rj(t) − rj(0)]}⟩ where Nn,p is totalnumber of monomers in shell n around NP p and ⟨...⟩ denotesthe ensemble average over PS monomers belonging to nth shellof all NPs in the system.The way of assignment of monomers to the NPs and their
surrounding shells, as described above, excludes thosemonomers which have passed from one NP to the other andstayed less than 50% of their residence time around any one ofthem. If a monomer has regularly visited different shells of agiven NP and has stayed most of its time in one of the shells
(even not persistently), it is assigned to that particular shell.Consideration of larger presence times did not lead to changesin the graduation of dynamic properties.The influence of the grafting density on the dynamics of
monomers of matrix chains of different lengths in the graftedcorona has been investigated. In Figure 1, S(q,t) is shown for
monomers of free chains in the second shell around the NP (1to 2 nm from its surface) for different grafting densities. It isalso compared with that of bulk PS (black curve). Differentpanels show different free chain lengths. Independent of thegrafting state and matrix chain length, the presence of a NPslows down the dynamics of the free polymer compared to thebulk, albeit to different extent. As could be expected, anincrease of the free-chain length leads to a decrease in thedynamics for both the bulk and the free polymer (Figure 1).The overall effect of the grafting density on the dynamics ofmatrix chains is as follows: the larger the grafting density, theslower is the dynamics of free monomers at any given distance.Figure 1A shows that when the radius of gyration Rg of
matrix chains (1.01 nm in the case of the 20-mere bulk) issmaller than the average distance of the shell from the surface(1.5 nm), the monomer dynamics of the matrix chains does notdeviate considerably from that of the bulk polymer,independent of the particle’s grafting state. This happens inspite of the presence of the grafted corona which ischaracterized by a radius of gyration and a brush height ofthe grafted chains of 2.44 and 3.29 nm for high grafting densityas well as of 2.38 and 2.85 nm for low grafting density. Thebrush height is defined as the average radial distance of thegrafted chain monomers from the surface. When Rg of thematrix chains is 1.51 nm (40 monomers, Figure 1B), theirmonomer dynamics slows down, at least in the presence ofgrafted NPs. This indicates that the thickness of the region withreduced mobility, called the interphase, depends on the radiusof gyration of the matrix chains. The dynamics of the matrix
Figure 1. Self-scattering-function of free-chain monomers in thesecond shell (1−2 nm from the surface), Sn=2(q,t), around thenanoparticle as a function of the density of grafted chains (length 80monomers) and the length of the free chains (20, 40, 80, and 200monomers in panels A−D, respectively) for a constant q = 0.78 nm−1.The scattering function for an unfilled polymer bulk with the samechain lengths is shown for comparison.
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chains around a bare NP is still not affected, which shows thatthe interphase thickness depends on the grafting density, too.Further increasing the radius of gyration of the matrix chains to2.2 and 3.4 nm (80 and 200 monomers, Figure 1C and D,respectively) results in further slowing-down, though small,around the bare NP. Moreover, the dynamics of the matrixchains for high grafting density shows a qualitatively differenttrend when chains with Rg > 1 nm are used. It manifests itself asslower dynamics accompanied by a plateau at long times.Considering our preceding study on the structural properties ofthe same nanocomposite,4 this can be attributed to the “dryingeffect” of the grafted corona. Grafted chains contract fromfarther (mainly beyond 3 nm from the surface) to closerdistances, which leads to ∼25% increase in the density ofgrafted monomers, together with ∼50% reduction of thedensity of matrix chains, at 1.5 nm from the surface (center ofthe current shell). The ultimate effect of such a change in thecomposition of the monomers at this distance is a suppresseddynamics of the remaining free chains because of theirincreased involvement with grafted ones. The fact that thesame magnitude of slowing-down does not happen for a lowgrafting density is in line with the observation that a wet-to-drytransition in the grafted corona is much weaker in this case.4
When the matrix chain Rg becomes >2 nm (i.e., about twice itsdistances from the surface), the difference between the matrixmonomer dynamics around bare and low-grafted NPsdisappears.In the next shell, which is 2−3 nm from the surface
(Supporting Information, Figure SI-1), the dynamics aroundbare and low-grafted NPs is bulklike for all free-chain lengths.This means that the NP, whether bare or low-grafted, does notaffect significantly the monomer mobility of the matrix chainsbeyond 2 nm from its surface. Should the changes in thedynamics of the free chains be interpreted as a manifestation ofan interphase, its thickness would be ∼2 nm in this case. Such alength scale is in agreement with the interphase thicknessdefined by the extent to which the structure of the chains isaltered by the NP.4 In the presence of high-grafted NPs, on theother hand, the dynamics is still perceptibly slower than forbulk polymers, showing that the interphase thickness for thesame NP can be extended to ∼3 nm by increasing the graftingdensity. Such a behavior is again in line with the grafting-density dependence of the interphase thickness when measuredby structural properties (radius of gyration) of the chains.4 Thisfurther shows that the monomer dynamics is perturbed withinthe same interval as realized for the chain structure. Beyond 3nm, the chain dynamics is bulklike. It depends neither on thegrafting density nor on the length of the free chains (data notshown here).
4. MONOMER DYNAMICS OF GRAFTED CHAINSThe influence of the lengths of the free chains and graftingdensity on the mobility of the graf ted chains has been alsoinvestigated. To this end, the appropriate S(q,t) values arecalculated in different shells. An example (low grafting density)is given in Figure SI-2 (Supporting Information). It shows aclear attenuation in the dynamics of grafted chains in thepresence of longer free chains. The dynamics can, however, becondensed into a more compact form by using the relaxationtime τ of the scattering-function S(q,τ), Figure 2. The τparameter is the time where S(q,τ) = 0.5. Alternative definitionsof τ probably would not lead to qualitative changes in ourinterpretation. Figure 2 shows relaxation times of grafted
monomers in four different shells around a NP of high and lowgrafting density for different free chain lengths. Graftedmonomers in the first layer (0−1 nm) are mainly linkerunits. They are only slightly displaced as they are attached tothe NP surface. Their S(q,t) do not meet the chosen criterionfor a relaxation time; in most cases they do not decrease below0.6 (see Figure SI-2 in the Supporting Information). So weshow relaxation times only for shells 2−5 around the NP, whichcover distances of 1−5 nm from the surface. It is worth notingthat S(q,t) = 0.5 is an arbitrary criterion and not necessarilyequivalent to what is referred to as system “relaxation” or“equilibration”. The latter is measured by some decayingcorrelation functions which reach near-zero values at longenough times. However the S(q,t) of grafted monomers inFigures SI-2 and SI-3, near the NP surface, show nonzeroplateaus, because their degree of freedom is reduced by beingattached to the surface.Figure 2A demonstrates, for all free chain lengths, the
expected faster relaxation of grafted chains as the distance fromthe surface of highly grafted NP increases. The same trend isalso observed for the low grafting density (Figure 2B). Theconcentration of the grafted monomers decreases with distancefrom the NP. Due to its spherical geometry this leads to ahigher contribution of the more mobile free chains. In eachshell, the relaxation of the grafted monomers is faster when thefree chains are shorter. This behavior is observed for all shells atboth grafting densities, except for the 1−2 nm shell of thehighly grafted NP where the relaxation times are typically verylarge and not that much different. Close to the NP very largerelaxation times are found because of the attachment to thesurface together with high grafting density, which strongly
Figure 2. Relaxation time τ of grafted monomers at different distances,that is, in different shells for high (σ = 1.0 chains/nm2) and low (σ =0.5 chains/nm2) grafting densities (panels A and B); τ is defined as thetime when S(q,t), q = 0.78 nm−1, has decreased to 0.5. The free chainshave 20, 40, 80, and 200 monomers; the grafted chains contain 80monomers. Data for the first shell (0−1 nm) are not shown, as theyare mainly caused by linker molecules which relax too slow to estimatetheir relaxation time.
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restricts the dynamics of the grafted chains in those shells. Veryslowly decaying S(q,t) together with statistical fluctuations ledto the observed reversal in the order of the monomersdynamics. The general decrease of the relaxation speed with thefree-chain length can be understood noting the faster dynamicsand lower viscosity of the shorter matrix chains on the onehand, and more infiltration of the shorter free chains into thegrafted corona on the other hand. The latter, leading to the so-called wetting of the grafted corona, has been reported in ourrecent study of the structure in the same nanocomposite.4
Table 2 shows that the dependence of the brush height of the
grafted chains (defined as the average radial distance of theirmonomers from the surface) as a function of the length of thefree chains supports this view. The data confirm a deswelling ofthe corona chains for large matrix-chain lengths. In agreementwith the previous theoretical calculations;56,57 however, thisbehavior is saturated at large chain lengths. Whereas thedependence of the dynamics of the grafted monomers on thelength of the free chains is stronger in the vicinity of the NP, itbecomes less pronounced at farther distances, for both graftingdensities. Moreover, for a given free chain length, graftedmonomers relax faster for smaller grafting density (Figure 2,compare panels A and B). This can be attributed to the largerinfiltration of the free chains in the low density corona,reflected by their higher contribution in the density profile.4
The chains have generally a faster dynamics when being “free”.At larger distances from the NP (Figure 2, 4−5 nm from
surface), the relaxation times converge to ∼2500 ps for bothgrafting densities. This shows that the grafting density plays nosignificant role in the determination of the dynamics of graftedmonomers at large distances. This is possibly an effect of theconvex geometry of the nanoparticle which necessarily leads toa reduction of the grafted-chain density with increasingseparation from the surface.At high grafting densities a closer look at the S(q,t), shown in
Figure SI-3, reveals a difference in the relaxation process ofgrafted chains in the presence of short and long matrix chains,moving from inner (panels A and B) to outer shells (panels Cand D). While the S(q,t) of grafted monomers in the presenceof different free chain lengths are ordered as 20-mer = 40-mer <80-mer < 200-mer in the inner shells (0−2 nm from thesurface), they follow a 20-mer = 40-mer < 80-mer = 200-mer inthe outer shells (2−4 nm from surface). Thus, we can deducethat short and long matrix chains induce a qualitatively differentdynamics in the grafted corona. Again, the explanation isprovided by the wet-to-dry transition of the grafted coronawhen the matrix chain length varies from shorter to longerrelative to that of the grafted chains (Table 2). The similarity ofthe corona relaxation in the presence of short and long free
chains is accompanied by the same similarity in its monomercomposition at separations of 2−4 nm from the surface. Thelatter quantity has been extracted from the density profile (seeFigure 9 of ref 4). Such a difference, however, is not assured atlow grafting densities. It is in line with a much weaker wet-to-dry transition effect in this case.The restriction of the motion of grafted monomers, however,
also shows up in their MSD. The MSD of grafted monomers,defined by MSD(t) = ⟨(R(0) − R(t))2⟩, is calculated for thedifferent shells; R(0) and R(t) are position vectors of themonomers in a particular shell at time arguments 0 and t,respectively, in the coordinate frame of the center of mass oftheir parent NP. Angle brackets denote the ensemble averageover all monomers in that particular shell, as well as an averageover all NPs. The assignment of the monomers to differentshells has been done by the 50% residence time criteriondescribed above. Figure 3 shows the MSDs of grafted
monomers in different shells versus time, in the presence offree chains of 20 and 200 monomers. As expected, the mobilityof grafted monomers increases with the distance from the NP;it decreases with the length of the matrix chains according totheir increased viscosity. All MSDs saturate, as the chains aretethered to the NPs.
5. DYNAMICS OF NANOPARTICLESThe center-of-mass mean square displacements of the nano-particles are shown in Figure 4, averaged over the six NPsconsidered. Free chains of different lengths (from 20 to 200monomers) are examined for each grafting density. An increaseof the grafting density from 0 to 0.5 and 1 chains/nm2 iscoupled to a decrease in the MSD of the NP. As the Einsteindiffusion limit is not reached for most of the systems, we takethe mean-square displacement of the nanoparticles after 10 ns
Table 2. Average Brush Height of Nanoparticle-GraftedPolymer Chains for High (σ = 1 chains/nm2) and Low (σ =0.5 chains/nm2) Grafting Density as a Function of the FreeChain Length; the Grafted Chain Length Is Constant (80monomers)
Figure 3. Mean-square displacement (MSD) of monomers of graftedchains at different distances from a high-grafted (σ =1.0 chains/nm2)nanoparticle, measured in the coordinate frame of the nanoparticlecenter-of-mass. The MSDs in different shells (each with 1 nmthickness, where the first one is found between 0 and 1 nm from thesurface) are shown by different colors, during 25 ns. Solid and dashedlines show the MSD in the presence of short (20 monomers) and long(200 monomers) free chains. To estimate the accuracy of thesimulations, representative error bars are shown.
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to compare their relative mobilities. This value is universallysmaller for low-grafted NPs than that for ungrafted NPs; theydiffer by a factor between 1.4 and 10, depending on the matrix-chain length. For high-grafted NPs it becomes even smaller, atleast by a factor of 2, for all chain lengths examined. This shows
that the motion of a bare nanoparticle is much easier than thatof grafted ones, due to its smaller hydrodynamic radius. Foreach grafting density, shorter matrix chains lead to larger valuesof the MSD of the NP in comparison to longer chains. Themobility of the NP in the polymer matrix depends also on thelength of the free chains. In the case of a bare NP, there is adifference of more than 1 order of magnitude in the MSD ofthe NP after 40 ns between matrix polymers with chain lengthsof 20 and of 200 monomers. Whereas the MSD decreasessubstantially between the free chain lengths of 20 and 80monomers, the change between 80 and 200 monomers is muchsmaller. Note that, only for a bare NP surrounded by theshortest free chains, the regime of anomalous diffusion is leftand Einstein diffusion is observed (Figure 4A) during thesimulation time considered here. Compared to bare NPs, theMSD of grafted NPs shows a quantitatively weaker dependenceon the free-chain length. In particular, the MSD is roughly thesame for free chains of 80 and 200 monomers.
6. SUMMARY AND CONCLUSIONSWe have investigated the effect of the matrix chain length andgrafting density on the dynamics of the matrix and graftedchains in the interphase region around bare and grafted silicaNPs of radius 2 nm embedded in atactic polystyrene, usingcoarse-grained MD simulations. Our results indicate that thepresence of filler particles attenuates the polymer dynamics inthe interfacial region. This effect smoothly decreases with theseparation from the surface. At any given distance from thesurface, the dynamics is further influenced by the matrix chainlength and grafting density. The matrix chain length controlsthe dynamics in the grafted corona as well as the diffusion ofthe nanoparticle. Shorter matrix chains, in comparison withlonger matrix chains, are not only able to swell the graftedcorona more (known as wet grafted corona), but also inducefaster relaxation of the grafted chains. This effect is stronger forlower grafting densities. For a given matrix chain length, a highgrafting density slows the polymer dynamics and expands theinterphase thickness, in addition to decreasing the nano-particle’s mobility. Whereas the interphase region around a barenanoparticle, measured by the local dynamics, persists only upto ∼2 nm (corresponding to four monomer layers) from thesurface, surface grafting increases it up to ∼3 nm.
■ ASSOCIATED CONTENT*S Supporting InformationS(q,t) of free monomers in the third shell around bare andgrafted NPs, and S(q,t) of grafted monomers in four shellsaround σ = 0.5 and σ = 1 chains/nm2 grafted NPs. Thismaterial is available free of charge via the Internet at http://pubs.acs.org.
■ ACKNOWLEDGMENTSSpecial thanks to Prof. F. Muller-Plathe and Prof. M. C. Bohmfor many fruitful discussions and comments. AG thanks Prof.M. Meyer-Hermann for his valuable support. This work has
Figure 4. Mean square displacement of (A) bare, (B) low-grafted (σ =0.5 chains/nm2), and (C) high-grafted (σ = 1.0 chains/nm2)nanoparticles in the presence of free chains with a length of 20(red), 40 (blue), 80 (green), and 200 monomers (black). The graftedchains have 80 monomers. The slopes 1 and 0.5 (dotted lines) havebeen shown here for better comparison with theoretical diffusionlimits.
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been funded by the EU project NanoModel (211778) as well asby the Deutsche Forschungsgemeinschaft through the PriorityProgramme 1369 “Polymer-Solid Contacts: Interfaces andInterphases”.
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