Nano tools for macro problems: multiscale molecular modeling ofnanostructured polymer systems

Maurizio Fermegliaa,b*, Paola Posoccoa,b and Sabrina Pricla,b

aMolecular Simulation Engineering (MOSE) Laboratory, Department of Engineering andArchitecture, University of Trieste, Piazzale Europa 1, Trieste 34127, Italy; bNational

Interuniversity Consortium for Material Science and Technology (INSTM), Research UnitMOSE-DEA, University of Trieste, Trieste, Italy

(Received 1 March 2013; accepted 22 April 2013)

A current challenge of physical, chemical, and engineering sciences is to developtheoretical tools for predicting structure and properties of complex materials fromthe knowledge of a few input parameters. In this work, we present a general multi-scale molecular simulation protocol for predicting morphologies and properties ofnanostructured polymer systems and we apply it to three examples of industrial rele-vance. The first example is of general importance for the polymer industry and isrelated to the enhancement of mechanical and barrier properties, if a nanofiller is dis-persed into a polymer matrix: the role of multiscale modeling for the developmentof the material in the stage of screening, the best design is evidenced. The secondexample, important for the optoelectronic industry, is related to the prediction of thedispersion of gold nanoparticles into a diblock copolymer system forming differentnanostructures (lamellae, cylinders, …). In this case, it is relevant to understand howit is possible to influence the self-assembly of the nanoparticles in different regionsof the diblock copolymer structure. The third example is of interest to automotiveand polymer industries and involves inorganic nanoparticles grafted with organicside chains. The assembly is dispersed in a polymeric matrix and it is interesting topredict the effect of the chain length and grafting density on the nanostructure.

Keywords: nanotechnology; polymer nanocomposites; self-assembly; multiscalemolecular modeling

Introduction

The main goal of computational materials science is the rapid and accurate predictionof properties of new materials before their development and production. In order tobuild new materials and compositions with designed new properties, it is essential topredict these properties before preparation, processing, and characterization. Polymersare complex macromolecules that display structure ranging from the Å level of the indi-vidual backbone of a single chain to the scale of the radius of gyration, which canreach tens of nanometers. The corresponding time scales of the dynamic processes rele-vant for different material properties span a wider range, from femtoseconds to millisec-onds or even seconds or hours in glassy materials or for large-scale ordering processes,such as phase separation in blends. No single model or simulation algorithm can cover

*Corresponding author. Email: Maurizio.Fermeglia@di3.units.it

Composite Interfaces, 2013http://dx.doi.org/10.1080/15685543.2013.807131

� 2013 Taylor & Francis

such an interval of length and time scales. Consequently, in order to simulate apolymeric system one must consider models that range from those including quantumeffects and electronic degrees of freedom to chemically realistic, classical models. Thus,one of the most important issue in computational materials research, which holds partic-ular challenges for polymer materials, is multiscale simulation i.e. the bridging of lengthand time scales and the linking of computational methods to predict macroscopic prop-erties and behavior from fundamental molecular processes.[1–4] Despite the advancesmade in the modeling of the structural, thermal, mechanical, and transport properties ofmaterials at the macroscopic level (finite element analysis of complicated structures), atremendous uncertainty still remains about how to predict many properties of industrialinterest. The idea of performing simulations of materials across several characteristiclength and timescales, starting from fundamental physical principles and experimentaldata, has thus an obvious appeal as a tool of potentially great effect on technologicalinnovation and material design.[5]

Aim of this contribution is not a thorough presentation of all state-of-the-art currentmultiscale molecular modeling techniques available so far (detailed reviewed in [4,6–8]). Rather, this work illustrates the authors’ view on the possibilities that multiscalemodeling offers in understanding and controlling the properties of nanocomposite mate-rials and in tailoring them for specific applications. Examples presented in this overviewreflect authors’ own research interests, and are by no means exhaustive. Nevertheless, itis our hope that they can give a satisfactory cross-section of the current state of the fieldand can serve as inspiration for further developments in this exciting branch of science.

Multiscale molecular modeling

Multiscale molecular modeling and simulation (M3S) combine methods that cover arange of time and length scales wide enough to study complex material systems. In thecontext of materials simulations, four characteristic time and length levels can be envis-aged before reaching the last step, i.e. engineering design (Figure 1):

(1) the quantum scale (�10�10 m and �10�12 s), in which nuclei and electrons arethe main players, and their quantum-mechanical state dictates the interactionsamong atoms. The possibility of obtaining data describing structural and electronfeatures of the system is being considered and the possibility of taking intoaccount effects associated with the break and formation of chemical bonds inmolecules, changes in electron configurations, and other similar phenomena arethe main advantage of methods working at quantum scale;

(2) the atomistic scale (�10�9m and 10�9–10�6 s). In atomistic simulations, allatoms are explicitly represented or in some cases, small groups of atoms are trea-ted by single sites (referred to as pseudo atoms). The potential energy in the sys-tem is estimated using a number of different classes of interactions, typically: (1)bonded interactions, including bond-length (stretch) potentials, bond-angle (bend)potentials, torsion (twist) potentials, and cross-terms, and (2) nonbonded interac-tions, which in general consist in Coulomb interactions and dispersion forces;

(3) the mesoscopic scale (�10�6m and 10�6–10�3 s). According to these methods,a molecule is usually treated with a field description or microscopic particles thatincorporate molecular details implicitly. Therefore, they are able to simulatephenomena on length and time scales, currently inaccessible by classical atomis-tic approach;

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(4) the macroscopic scale (�10�3m and 100–101 s), where a constitutive law gov-erns the behavior of the physical system, which is considered as effectivelyhomogeneous, ignoring discrete atomic and molecular structures and their influ-ence on overall system behavior. The final aim consists in representing a hetero-geneous material as an equivalent homogeneous one.

At each length and timescale, well established and efficient computationalapproaches have been developed over the years to handle the relevant phenomena. Totreat electrons explicitly and accurately at the lower scale, methods classified as quan-tum mechanics (QM) can be employed. QM methods have undergone enormousadvances in the past 10 years, enabling simulation of systems containing several hun-dred atoms with good accuracy.[9,10] For material properties at the atomic level, molec-ular dynamics (MD) and Monte Carlo (MC) simulations are usually performedemploying classical interatomic potential, which can often be derived from QM calcula-tions.[11,12] Although not as accurate as QM methods, classical MD and MC simula-tions are able to provide insight into atomic processes involving considerably largesystems.[13–15] At the mesoscopic scale, the atomic degrees of freedom are not explic-itly treated, and only large-scale entities are modeled (that is, agglomeration of atomscalled beads, obtained through a coarse-graining procedure). Mesoscale techniques areparticularly useful for studying the behavior of polymers and soft materials. They canmodel even larger molecular systems, but with the commensurate trade-off in accuracy.[16,17] Typical results of mesoscale simulation are the morphology and the structure ofthe matter at nanoscale level at the desired conditions of temperature, composition, andshear. Various simulation methods have been proposed to study the mesoscale structuresin polymer-based materials, the most common being Brownian Dynamics (BD),[18–20]

Figure 1. The diagram illustrates the concept of a unified message–passing, sequentialmultiscale approach able to reach length and time scales that individual methods fail to achieve.At the same time, this unified approach retains the accuracy that the individual techniquesprovide in their respective scales. The main objective of multiscale approaches is to predict theperformance and behavior of materials across all relevant length and timescales, striving toachieve a balance among accuracy, efficiency, and realistic description.

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Dissipative Particle Dynamics (DPD),[21–23] Lattice Boltzmann (LB),[24] time–depen-dent Ginzburg–Landau theory,[25] and Dynamic Density Functional Theory (DDFT).[26–29] Eventually, it is possible to transfer the simulated mesoscopic structure to finiteelements modeling (FEM) tools [30,31] to calculate macroscopic properties for the sys-tems of interest (Figure 1). In the multiscale protocol, it is important to be able to com-pare the calculated results with experimental evidences at each scale and a computationis performed. Fortunately, the experimental methods available now allows this compari-son along the entire multiscale procedure, as it is evidenced in Figure 2.

The ultimate goal of a multiscale modeling is, hence, to predict the macroscopicbehavior of an engineering process from first principles, i.e. starting from the quantumscale and passing information into molecular scales and eventually to process scales.One computes information at a smaller (finer) scale and passes it to a model at a larger(coarser) scale by leaving out (i.e. coarse graining) degrees of freedom.[32–34]

Two ingredients are required to construct a successful sequential multiscale model:first, it is necessary to have a priori and complete knowledge of the fundamental pro-cesses involved at the lowest scale. This knowledge or information can then beemployed for modeling the system at successively coarser level. Second, it is necessaryto acquire a reliable strategy for incorporating the lower-scale information into the coar-ser scales.

This is often accomplished by phenomenological theories, which contain few keyparameters, the value of which is determined from the information at the lower scale.This message-passing approach can be performed in sequence for multiple-lengthscales. The key attribute of the sequential approach is that the simulation at higher levelcritically depends on the completeness and the correctness of the information gatheredat the lower level, as well as the efficiency and reliability of the model at the coarser

Figure 2. Experimental and theoretical tools for characterization and modeling of polymer-basednanocomposites, plotted over their respective time and length scale domain of applicability.Experimental methods include X-ray diffraction, nuclear magnetic resonance (NMR), transmissionelectron microscopy (TEM), tomography, atomic force microscopy (AFM), small-angle neutronscattering (SANS), small-angle X-ray scattering (SAXS), mechanical testing. Theoretical toolsinclude QM, density functional theory (DFT), MD, MC simulations, DPD, BD, LB, DDFT.

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level. To obtain first principle-based results for macroscale systems, we must ensure thateach scale of simulation overlaps sufficiently with the finer description, so that all inputparameters and constitutive laws at each level of theory can be determined from morefundamental theory. It is equally important that we must ensure these relations areinvertible so that the results of coarse level simulations can be used to suggest the bestchoices for finer level parameters, which can be used to suggest new choices of compo-sition and structure. The problem for polymers is that the method of coarsening thedescription from atomistic to mesoscale or mesoscale to continuum is not as obvious asit is going from electrons to atoms.[2] In other words, the coarsening from QM to MDrelies on basic principles and can be easily generalized in a method and in a procedure,while the coarsening at higher scales is system specific. In this respect, for example,Doi [34] developed a suite of simulation tools that model polymers at the molecularand mesoscale level. Although each tool performs calculations using only one tech-nique, the output from one level can be used directly as input for another, allowing anoff-line bridging of length and time scales. To achieve what he and others refer to as“seamless zooming”, namely the ability to spawn higher resolution simulations usingmore detailed methods where needed, will require additional theoretical and computa-tional advances. Along similar lines, off-line multiscale simulations of nanofilled poly-mers using coarse-grained MD, mesoscopic time dependent Ginsburg–Landau theory,and macroscopic continuum finite element techniques have been carried out. Significantadvances in uniquely mapping atomistic models of polymers onto coarse-grained mod-els [35–37] have been made in recent years, in some cases providing nearly exact quan-titative agreement between the two models for certain quantities. Scale integration inspecific contexts in the field of polymer modeling can be done in different ways. Any‘recipe’ for passing information from one scale to another (upper) scale is based on thedefinition of multiscale modeling which consider ‘objects’ that are relevant at that par-ticular scale, disregard all degrees of freedom of smaller scales and summarize thosedegrees of freedom by some representative parameters. All approaches are initiallybased on the application of a force field (FF) that transfers information from quantumchemistry to atomistic simulation. From atomistic simulation to mesoscale model, essen-tial features of the system have to be maintained while reducing the degree of freedom.So far, the features chosen for the reproduction by coarse-grained models have beenmainly structural, thermodynamical or both, with structure prevailing.[38] This linkingthrough the mesoscale in which we can describe microstructure is probably the mostchallenging step toward the developing reliable first principles methods for practicalmaterial’s design applications.

Mapping procedure from quantum to macroscale

Based on accurate QM calculations, a FF is determined, which includes charges, forceconstants, polarization, van der Waals interactions, and other quantities that accuratelyreproduce the QM calculations. With the FF, the dynamics is described with Newton’sequations (MD), instead of the Schrödinger equation. The MD level allows predictingthe structures and properties for systems much larger in terms of number of atoms thanfor QM, permitting direct simulations of the properties of many interesting systems.

For passing to atomistic to mesoscale simulations, one can use a traditionalapproach based on the estimation of the characteristic ratio, the Kuhn length, and theFlory-Huggins interaction parameter.[39] This approach for determining the inputparameters for mesoscale simulation is based on the following information: (1) the bead

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size and Gaussian chain architecture, (2) the bead mobility M, and (3) the effectiveFlory-Huggins χ interaction parameters. With this approach, the Flory-Huggins χparameters between two components of the coarse-grained molecular models in themesoscopic simulation are estimated through the atomistic simulation, and a mesoscopicstructure is predicted using these parameters (see Supporting information for details).Mesoscopic simulations are performed using a coarse-grained molecular model asshown in Figure 3: the particle in mesoscopic simulation is related to a group of severalatoms in the atomistic simulation. Mesodyn and DPD mesoscale theory and simulationprotocols are fully described in the literature.[16,17]

The traditional approach described above can be enhanced and improved by consid-ering the detailed structure at the polymer–nanofiller interface.[40] If one resorts to aparticle based method for describing the system at mesoscale, atomistic MD simulationgives the necessary details of the interface with a particular attention to the bindingenergies among components. Mapping of the binding energies on mesoscale beads bymeans of a combinatorial approach to repulsive parameter for particles is then carriedout and the system is simulated at mesoscale (see Supplemental material for details).

Typical mesoscale simulation results are the morphology and the structure of thematter at nanoscale level at the desired conditions of temperature, composition, andshear. For the description of flow of polymeric materials on a processing scale, onemust incorporate system hydrodynamics and other phenomena occurring on mesoscopicto macroscopic length and time scales. For example, to capture the nonNewtonian prop-erties of polymer flow behavior one can either use special models for the materialsstress tensor, or obtain it from a molecular simulation using the instantaneous flowproperties of the hydrodynamic fields as input. In the area of high-performancematerials and devices, polymer composites are finding a widespread application, andthe modeling of these materials was until recently done primarily through finite elementmethods (FEM), and are beyond the realm of application of molecular modelingapproaches. Nonetheless, a real problem in using FEM is the definition of the physicalproperty of a complex material such as a polymer blend with phase segregation and/ora polymer with microinclusion of nanosized platelets.[21–23,30,31,41,42] Mesoprop is amethod based on finite elements for estimating properties of a complex material startingfrom the density distribution at mesoscale. The method uses the results of a mesoscale

Figure 3. Coarse-grained modeling from atomistic model.

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simulation under the form of three-dimensional density maps, and transforms suchinformation into a fixed grid that is used for the integration of the equations to deter-mine macroscopic properties. Palmyra is a different method that allows the simulationat FEM level with a variable grid methodology that allows extending the size of thesystem.[31,41,42]

Figure 4 shows how the mapping from mesoscale to macroscale can be achieved.At FEM level each finite element corresponds to one phase, with property tensor Pi,while at mesoscale level (MesoDyn or DPD) each element contains a mixture ofphases, with concentration Ci. Accordingly, it is necessary to perform a geometrymapping by converting MesoDyn/DPD cubic elements to Palmyra tetrahedrons (seeSupplemental material for details). Once this is done, Laplace equation is solveddirectly for obtaining direct properties, such as electric conductance, diffusion,permeability, and so on. Local deformation allows the calculation of mechanicalproperties mesoscale (see Supplemental material for details).

Applications of multiscale modeling

In the following paragraphs, some applications of the message passing multiscale mod-eling described in the previous sections will be presented and discussed.

Polymer-organoclay nanocomposites

Polymer-organoclays are important materials since the addition of natural clay to apolymer enhances the properties of the polymer. To do this, the nanofiller must be com-pletely exfoliated in the polymer and this is normally achieved by modifying the surfaceof the nanofiller with a particular component that is responsible of the clay exfoliation.For designing the best surface modifier, the estimation of the binding energy and basalspacing of e.g. a montmorillonite–polymer–surface modifier system is necessary. This,in fact, will allow the calculation of the necessary parameters to be passed at the upperscale for the estimation of morphology at mesoscale and the mechanical properties atmacro scale as explained in the previous paragraphs. As an example, atomistic simula-tions allowed us to rank different quaternary ammonium salts (quats) for the ABS/mont-morillonite selected clay/polymer system for determining the best surface modifier.[43]Table 1 lists the binding energies for the best quat resulting from the screening, whichis the double-tailed ammonium salt Cloisite 20A®, and the relevant spacing between the

Figure 4. From mesoscale morphology to FEM analysis.

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clay platelets. By looking at the values of the Epol-quat term in Table 1, we see that mostpolar acrylonitrile (AN) homopolymer is more affine to quat than all other ABS homo-polymer chain components. Also, if one considers the mean binding energy of styrene-acrylonitrile (SAN) and acrylonitrile–styrene (ANS) and the binding energy for AN, thecorresponding values are, respectively, very similar. The same can be noticed if onecompares the mean values of the binding energy for polybutadiene (polyB)-SAN, andSAN or AN.

This reasoning allowed us to conclude that, in full accordance with mesoscale simu-lations [43] and experimental evidences [44] exfoliation takes place only partially. Onlythe SAN copolymers enter between MMT layers, creating stacks and the rubber phases,isolated in the shape of round islands, remain next to the stacks while SAN copolymeressentially constitute the matrix phase.

The interaction energies reported in Table 1 were next used to estimate mesoscalesimulation parameters and the morphologies obtained at mesoscale were ultimatelytransferred to FEM software according to the protocol described in the previousparagraphs.

Table 2 shows the overall results for ABS nanocomposite Young’s modulus as pre-dicted by FEM. The small discrepancy between the experimental and the calculateddata is due to the slightly different formulation of ABS used in the calculation, as realABS has a wide range of compositions. For this reason, a comparison of G/G° ratio(ratio of nanocomposite and pure ABS Young’s modulus) between simulations andexperiments are also reported, from which the agreement between real and virtual dataemerges more clearly.

Table 1. Binary binding energy (kcal/mol) and basal spacing (Å) for cloisite 20A® and MMT inhomopolymers acrylonitrile (PAN), styrene (PS), butadiene (PB), block copolymer (SAN)oriented with S next to MMT surface, block copolymer (ANS) oriented with AN next to MMT,branched copolymer (polyB-SAN) with a main butadiene (B) chain and lateral chains of SAN,branched copolymer (polyB–SANa) with a main chain of B oriented normal to MMT surface.

Polymer PS PB PAN SAN ANS polyB-SAN PolyB-SANa

EMMT-pol �24 �7 �41 �9 �15 �5 �19EMMT-quat �1216 �1266 �1248 �1165 �1102 �1386 �1230Equat-pol �32 �16 �100 �36 �146 �116 �71Basal spacing 3.108

Note: ‘Reprinted from Microporous and Mesoporous Materials, 107, Paolo Cosoli, Giulio Scocchi, SabrinaPricl, Maurizio Fermeglia, Many-scale molecular simulation for ABS–MMT nanocomposites: Upgrading ofindustrial scraps, Pages 169–179, Copyright 2008, with permission from Elsevier.’

Table 2. Young’s modulus for ABS–MMT nanocomposite and its comparison with experimentaldata [44].

Overall properties G0 (GPa) G (GPa) G/G0

% MMT 0 2 2Experimental 2.20 2.75 1.25Calculated 2.42 3.15 1.30

Note: ‘Reprinted from Microporous and Mesoporous Materials, 107, Paolo Cosoli, Giulio Scocchi, SabrinaPricl, Maurizio Fermeglia, Many-scale molecular simulation for ABS–MMT nanocomposites: Upgrading ofindustrial scraps, Pages 169–179, Copyright 2008, with permission from Elsevier.’

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Another example in the area of polymer clays is the polyamide-based system. Inthis case, atomistic simulation allowed us to calculate binding energies between thedifferent mesoscopic beads into which the entire system was coarse grained for nextscale simulations. In this case, however, we used mesoscale simulation to model thespace between two intercalated clay layers and subsequently imported the resultingdensity fields in a FEM code (Palmyra), in order to calculate mechanical and barrierproperties for small stacks of intercalated particles. Finally, models of the whole PCN(a box containing 36 single layer and stack clay particles) was modeled with Palmyrausing different degrees of exfoliation and different stack dimensions and properties,according to literature data [45,46] and DPD box analysis, respectively.[47] The interac-tion parameters needed as input for the mesoscale level DPD calculations were obtainedby a mapping procedure described in [40] (see also Supplemental material).

Calculations of the Young modulus yielded results which are comparable to experi-mental data if we consider the values along the x and y directions (i.e. parallel to thesimulated extrusion direction). Discrepancies between virtual and experimental datacould be due to a different degree of crystallinity or different crystalline forms of Nylon6 in presence of MMT nanofillers. As far as permeability is concerned, it is possible tonote that, as expected, O2 diffusion is hindered only in the z direction, i.e. perpendicularto platelet x–y plane. Permeability is anyway reduced only to a small extent (approxi-mately 10–15%), because of the low nanofillers content (less than 2% in volume). Dataare reported in Table 3.

The same procedure has been applied to a large number of systems, thus allowingthe selection of the best nanocomposite in terms of the desired property.[7,8] Figure 5shows Young modulus E, thermal conductivity k, and oxygen permeability P for a num-ber of different polymer-organoclay nanocomposites.

Mesoscale simulation for diblock copolymers with dispersion of nanoparticles

Mixing microphase-separating diblock copolymers and nanoparticles can lead to theself-assembly of organic/inorganic hybrid materials that are spatially organized on thenanometer scale. Controlling particle location and patterns within the polymeric matrixdomains remains, however, an unmet need. Computer simulation of such systemsconstitutes an interesting challenge, since an appropriate technique would require thecapturing of both the formation of the diblock mesophases and the copolymer–particleand particle–particle interactions, which can affect the ultimate structure of the material.In this example [48] we discuss the application of DPD to the study of the distributionof nanoparticles in lamellar and hexagonal A-B diblock copolymer matrices. The DPDparameters of the systems were calculated according to a multiscale modeling approach,i.e. from lower scale (atomistic) simulations as explained in the previous paragraphs(see also Supplemental material).

Table 3. Values of the Young modulus (E) and O2 permeability (P) in the x, y, and z directionsfor the 36 particles boxes of both PCNs considered.

Exx [GPa] Eyy [GPa] Ezzx [GPa] Pxx [barrer] Pyy [barrer] Pzz [barrer]

36 particle box for PCN with M2(C18)23.85� 102 3.76� 102 3.20� 100 1.52� 10�2 1.52� 10�2 1.37� 10�2

36 particle box for PCN with M3C18

4.04� 102 4.03� 102 3.23� 100 1.52� 10�2 1.54� 10�2 1.34� 10�2

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In agreement with some experimental evidence we found that, depending on thenature and type of nanoparticle covering (e.g. only A- or B-type covering), the particlescan segregate into the centers of the corresponding compatible domains (these beinglamellae or cylinders), forming nanowire-like structures that extend throughout thematerial (Figure 6). In effect, the interplay between micro phase separation and favor-able interactions do result in the self-assembly of spatially ordered nanocomposites.

On the other hand, for a different covering type (e.g. A6B6), the particles segregateat the interfaces instead of the centers of the lamellae (or cylinders), as shown inFigure 7. The results also indicate that the morphologies of the organic/inorganic hybridmaterials can be tailored by adding particles of specific size and chemistry.

Figure 5. Predicted enhancement ratio of the macroscopic properties (Young modulus E, thermalconductivity k and oxygen permeability P) of different polypropylene (PP), nylon 6 (PA6 andthermoplastic polyurethane (TPU)-clay nanocomposites. By definition, Ef=Pc/Pm, where Pc andPm are the permeability of the composite and the pristine polymer matrix, respectively. Thenanofiller concentration in all system corresponds to a weight fraction Φw of 4.6% w/w. [Toth R,Santese F, Pereira SP, Nieto DR, Pricl S, Fermeglia M, Posocco P. Size and shape matter! Amultiscale molecular simulation approach to polymer nanocomposites. J. Mater. Chem. 2012; 22:5398–5409. Reproduced by permission of The Royal Society of Chemistry. http://pubs.rsc.org/en/content/articlelanding/2012/jm/c2jm15763b]

Figure 6. Distribution of A (or B) covered nanoparticles in diblock copolymers: nanoparticlesare located in the center of each domain. Left: distribution of the nanoparticles with respect to thecopolymer domains; right: position of the center of mass of the nanoparticles. [Reprinted withpermission from Maly M, Posocco P, Pricl S, Fermeglia M. Self-assembly of nanoparticlemixtures in diblock copolymers: multiscale molecular modeling. Ind. Eng. Chem. Res. 2008; 47:5023–5038. Copyright (2013) American Chemical Society.]

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These findings highlight the fact that, in such complex mixtures, it is not simply theordering of the copolymers that templates the spatial organization of the particles: onthe contrary, the particles play an active role and affect the self-assembly of the poly-meric chains. In fact, we detected a phase transition from the hexagonal to lamellarmorphology induced by a nonselective block-particle interaction, indication that theparticles actively contribute to the determination of the system structure.[49,50]

Figure 7. Distribution of A and B equal coverage nanoparticles in diblock copolymers:nanoparticles are located at the interface of the domains; left: distribution of the nanoparticleswith respect to the copolymer domains; right: 3-D representation of the nanoparticles at theinterface. [Reprinted with permission from Maly M, Posocco P, Pricl S, Fermeglia M.Self-assembly of nanoparticle mixtures in diblock copolymers: multiscale molecular modeling.Ind. Eng. Chem. Res. 2008; 47: 5023–5038. Copyright (2013) American Chemical Society.]

Figure 8. Mapping of the atomistic nanocomposite silica/linker-PS grafted chains/PS matrixchains model system onto the corresponding DPD model.

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Dispersion of grafted silica nanoparticles into polystyrene

The systems considered in the next example are dispersions of silica nanoparticles withgrafted chains on the surface. The average diameter of the nanoparticle is 17 nm. Thegrafted chains are made of polystyrene (PS) and the nanoparticles are considered dis-persed in an atactic PS homopolymer matrix polymer with a well-defined molecularweight at a constant temperature of 300K. A multiscale simulation methodology wasalso applied to such system. In particular, atomistic simulations were performed for esti-mating interaction energies and density distributions around the silica nanoparticles, afunction of the grafted chain length and grafted chian density, and then used fordetermining the DPD energy interaction parameters aij [40] (see also Supplementalmaterial).

The specific DPD nanocomposite model contains four species of different beads:one for the polymer chain (PM) and two for the grafted PS chain-linker moiety (one forthe linker head (L), and one for the PS tail (PL)). The last one is for the silica surface(S). This is schematically illustrated in Figure 8.

The resulting DPD models and parameters have been validated by comparing thedensity profiles of the organic constituents as obtained from atomistic MD simulationsand mesoscale DPD simulations as it is reported in Figure 9.

The DPD theory was hence used to predict nanoscale morphologies of the silica-PSnanocomposite according to the following procedure. Each DPD grafted silica nanoparti-cle was modeled by an icosahedral structure, devised as being constituted by a centralDPD bead, connected to 12 other DPD beads on each vertex of the icosahedron, asshown in Figure 10. To simulate the effect of different grafted chain length/density, eachvertex of the icosahedron was considered either as a bare silica portion or a lumpedPS-L grafted chain. The average diameter of the icosahedral nanoparticle is 5 nm.

According to this model, different length of the grafted PS-L chains can bemodulated by averaging the contribution of the corresponding aij parameters of allbeads making up the PS-L chain. At the same time, different degrees of grafting can besimulated (from 0 to 100%) by considering the 12 beads of the icosahedral particle asrepresenting bare silica or the grafted chains, respectively.

Finally, we used the morphologies predicted at mesoscale as input for the microFEM simulation as described previously. The prediction of the mechanical properties(Young modulus) estimated by the described multiscale simulation protocol is reported

Figure 9. Comparison of the density profiles as obtained from MD and DPD simulations for theorganic components of the PS/silica nanocomposites.

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in Table 4 and compared with experimental values (in parenthesis) for different graftingdensity and different nanoparticles loading. We see from this Table that the agreementbetween the two data-set is quite good.

Conclusions

In this paper we have illustrated the concept of multiscale molecular modeling and dis-cussed some general guidelines for its implementation. M3S is applied in many fieldsof material science, but it is of paramount importance in polymer science, due to thewide range of phenomena accruing at different scales (from quantum chemistry to themesoscale) influencing the final property of the materials. In this context, M3S can playa crucial role in the design of new materials whose properties are influenced by thestructure at nanoscale.

Several examples have been reported in this paper showing the methodology anddescribing the simulation protocols. A general good agreement in the comparison with

Table 4. Young modulus estimated with multiscale molecular modeling as a function ofnanoparticle loading (%NP) and grafting chain density. EC is the Young module of thenanocomposite, EM, is that of the matrix. Experimental values are given in parenthesis.

%NP EC (GPa) EC/EM

Grafting density 0.5 chains/nm2

1 2.59 (2.60±0.1) 1.035 2.94 (2.81±0.2) 1.1710 3.45 1.37

Grafting density 0.7 chains/nm2

1 2.62 1.045 3.07 1.2210 3.55 1.41

Grafting density 1 chains/nm2

1 2.62 (2.75±0.2) 1.045 3.34 (2.87±0.1) 1.33

Figure 10. Icosahedral nanoparticle model representing the PS-linker grafted silica nanoparticlein our DPD simulations. Each vertex of the icosahedron represents a lumped PS-L grafted chain.

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experimental literature data of mechanical properties and morphologies is obtained, thusshowing that the multiscale molecular modeling is a mature tool that may be used inthe design and development of new nanostructured polymeric materials. Advances incomputational material science, in general, will continue to facilitate the understandingof materials and materials processing, the prediction of properties and behavior, and thedesign of new materials and new materials phases, thus facilitating the application ofprocess system engineering to more sophisticated and innovative processes.

Supplemental material

The supplemental material for this paper is available online at http://dx.doi.10.1080/15685543.2013.807131

AcknowledgmentsAuthors acknowledge financial support within the following projects: FP6 EU Project:MULTIPRO – Design Development and Application of Multifunctional Optoelectronic Materials;FP6 EU Project 2007: MULTIHYBRIDS – Innovative sensor-based processing technology ofnanostructured multifunctional hybrids and composites; and FP7 EU Project 2008:NANOMODEL – Multi-scale modeling of nano-structured polymeric materials: from chemistryto materials performance. Most of the calculations presented in this work were performed at theCINECA supercomputer center (Bologna, Italy) in the framework of the granted project MOMA– Multiscale modeling of nanocomposite Materials.

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