March 27, 2012 11:55 WSPC/245-JMM 00046 Journal of ... · MULTISCALE MODELING OF POLYMER/CLAY...

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Journal of Multiscale ModellingVol. 3, No. 3 (2011) 151–176c© Imperial College PressDOI: 10.1142/S1756973711000467

MULTISCALE MODELING OFPOLYMER/CLAY NANOCOMPOSITES

SIMAO PEDRO PEREIRA

IPC/I3N-Institute for Polymers and Composites,Department of Polymer Engineering,

University of Minho, Campus de Azurem,Guimaraes, 4800-058, Portugal

[email protected]

GIULIO SCOCCHI

Institute of Computer Integrated Manufacturingfor Sustainable Innovation (ICIMSI),

Department of Innovative Technologies (DTI),University of Applied Science of Southern Switzerland (SUPSI),

Via Cantonale — Galleria 2 Switzerland Manno, 6928, [email protected]

RADOVAN TOTH, PAOLA POSOCCO, DANIEL R. NIETO,SABRINA PRICL and MAURIZIO FERMEGLIA∗

Molecular Simulation Engineering (MOSE) Laboratory,Department of Industrial Engineering and Information Technology (DI3),

University of Trieste,Via Valerio 10 Trieste, 34127, Italy∗[email protected]

Received 21 April 2011Revised 14 November 2011

Multiscale molecular modeling (M3) is applied in many fields of material science, but itis particularly important in the polymer science, due to the wide range of phenomenaoccurring at different scales which influence the ultimate properties of the materials. Inthis context, M3 plays a crucial role in the design of new materials whose properties areinfiuenced by the structure at nanoscale. In this work we present the application of amultiscale molecular modeling procedure to characterize polymer/clay nanocompositesobtained with full/partial dispersion of nanofillers in a polymer. This approach relies ona step-by step message-passing technique from atomistic to mesoscale to finite elementlevel; thus, computer simulations at all scales are completely integrated and the calcu-lated results are compared to available experimental evidences. In details, nine polymernanocomposite systems have been studied by different molecular modeling methods,such as atomistic Molecular Mechanics and Molecular Dynamics, the mesoscale Dis-sipative Particles Dynamics and the macroscale Finite Element Method. The entirecomputational procedure has been applied to a number of diverse polymer nanocompos-ite systems based on montmorillonite as clay and different clay surface modifiers, and

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152 S. P. Pereira et al.

their mechanical, thermal and barrier properties have been predicted in agreement withthe available experimental data.

Keywords: Multiscale molecular modeling; polymer clay nanocomposites; mechanical,thermal and barrier properties prediction.

1. Introduction

In the vast field of nanotechnology, polymer materials reinforced with nanofillerssuch as layered silicates (clay), have become a prominent area of current researchand development. Generally speaking, nanocomposites are commonly defined asmaterials consisting of two or more dissimilar materials with well-defined inter-faces, at least one of the materials being nanostructured (having structural featuresranging in size from 1 to 100nm) in one, two, or three dimensions. The same refersto the spacing between the networks and layers formed by polymeric and inorganiccomponents. Depending on the strength of the interfacial tension between the poly-meric matrix and the layered silicate (modified or not), then resulting polymer/claynanocomposites (PCN)s can be categorized into two types, depending on the extentof the separation of the silicate layers: (i) intercalated nanocomposites, in which thepolymer chains are inserted between the layers of the clay such that the interlayerspacing is expanded, but the layers still bear a well-defined relationship to eachother, and (ii) exfoliated nanocomposites, in which the layers of the clay have beencompletely separated, and the individual mineral sheets are randomly distributedthroughout the polymeric matrix. Intuitively, the best performance of PCN-basematerials are achieved for system characterized by a high degree of clay exfoliationwithin the polymeric matrix.

PCNs have recently received tremendous attention from both scientific andindustrial communities due to their extraordinary enhanced properties.1–3 How-ever, from the experimental point of view, a thorough structural characterizationand a tailored fabrication of these hybrid nanostructure materials remain a grandchallenge. Nanomaterials are both exciting and puzzling at the same time, as theyinvolve components at “uncommon” characteristic scales at which conventionaltheories may fail. Understanding the behavior of materials at different scales isimportant both from the standpoint of basic science and future applications. Thedevelopment of such materials is still in its infancy and, as such, largely empiri-cal; thus, a fine degree of control of the resulting macroscopic properties cannotbe achieved so far. Moreover, as the ultimate properties of these hybrid systemscommonly depend on their structure at the nanoscale, it is of particular interest toestablish the mesoscopic morphology of the final composite and to link this charac-teristic to the material performance. To this purpose, the development of theoriesand the application of computer simulation techniques have opened avenues for thedesign of these materials, and the a priori prediction/optimization of their struc-tures and properties.4,5

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Multiscale Modeling of PCNs 153

The unique insights available through simulation of materials at a range ofscales, from the quantum and molecular, via the mesoscale, to the finite elementlevel, can produce a wealth of knowledge, significantly reduce wasted experiments,allow product and processes to be optimized, and permit large numbers of candidatematerials to be screened prior to production. Therefore, multiscale computationalapproach, which covers all methods (from atomistic/molecular through mesoscaleto macroscale) for each length and time scales can play an ever-increasing role inpredicting and designing material properties, and guiding such experimental workas synthesis and characterization.5

Several multiscale computational approaches, spanning different length/timescale domains have been proposed in recent years for the characterization of poly-mer/clay nanocomposites (PCNs), including molecular mechanics (MM) and molec-ular dynamics (MD), mesoscale (MS), and finite element simulations (FEM).5

Multiscale modeling has been shown to be a valuable tool for the characteriza-tion and/or prediction of macroscopic properties of polymer nanocomposites withdifferent fillers, as layered clays,4 carbon and boron nanotubes,6–9 fibers,10,11 orspheres.12 In spite of these efforts devoted to the multiscale simulation of nanos-tructured systems, a thorough, systematic and comprehensive study dealing withmultiscale modeling and simulations of polymer nanocomposites covering all lengthscales with the final aim to study morphology phenomena and prediction of macro-scopic effective property still exists.

With the ambitious aim of filling this gap, our recent efforts in nanocompos-ites simulation were initially concerned with binding energy evaluations for well-characterized polymer-clay systems13–16 using atomistic molecular dynamics (MD)methods. At the same time, part of our activity was focused on the development andapplication of mesoscale simulation (MS) recipes to polymer blends and nanocom-posites morphology investigations on one side, and on the integration of these toolswith both lower scales (i.e., atomistic approaches10,17) and higher scales, such asmacroscale approaches through finite element method calculations.21,22

The main goal of this paper is to present the concept of multiscale molecularmodeling with emphasis on the detailed description of the procedure which is basedon the following ansatz: (i) data obtained from atomistic MD and/or Monte Carlo(MC) simulations are used to derive accurate input parameters for mesoscale sim-ulations, (ii) the morphological data obtained at the mesoscopic level are fed intothe micromechanical calculation, and (iii) the finite element modeling finally pro-vide quantitative information regarding the macroscopic properties of the simulatedmaterial.

Central to the above described multiscale modeling recipe is the mesoscopiclevel, alias, that material time-space spanning from nanometers to micrometers inlength scale and investigating relaxation phenomena up to microseconds in the timedomain. In mesoscale modeling, the familiar atomistic description of the moleculesis coarse grained, leading to beads of material (representing the collective degreeof freedom of many atoms). These beads interact through pair-potentials which

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capture the underlying interactions of the constituent atoms. The primary out-put of mesoscale modeling are material phase morphologies with size up to themicron level. These morphologies are of interest per se, although little predictionof the material properties can be obtained with the mesoscale tools. Finite Ele-ment (FE) modeling then comes into play, and the material properties of interestcan be calculated accordingly by mapping the material structures formed at thenano/micrometer scale onto the finite element grid and coupling this informationwith the properties of the pure components that comprise the complex system.21,22

Using standard solvers, the FE code can then calculate the properties of the realisticstructured material. Figure 1 presents the flow-chart of the multiscale simulationprotocol which is adopted in the present work and applied to predict a set ofthermophysical properties of several PCN systems based on poly(propylene) (PP),nylon6 (PA6) and a thermoplastic polyurethane (TPU) as the polymeric matrix,and organically modified montmorillonite (MMT) as filler.

According to the computational procedure outlined above, the interac-tions between all individual components (filler, polymer and surface modi-fier/compatibilizer) of each nanocomposites, which occur at a molecular level, werecalculated using atomistic MM/MD simulations as a first step. Similarly, usingother MM/MD-based protocols, the basal spacing of the MMT stacks in PCNswere derived. Secondly, the information obtained from the atomistic simulationswere expanded by employing mesoscale models for the prediction of density profilesand system morphologies. To this purpose, the resulting MD data were mappedonto the corresponding mesoscale models via the corresponding interaction param-eters, and the results generated at both length scales were compared for consistency.Lastly, the density profiles and the morphologies resulting from the MS simulationswere imported into a FE code, and some characteristic macroscopic properties ofthese systems — e.g. Young’s Modulus (E), thermal conductivity (k) and perme-ability (P ) — as functions of filler loading and/or a given degree of dispersion inPCNs were predicted and compared with the corresponding experimental valuesavailable in the current literature. The schema of the overall procedure adopted forthe multiscale simulation of polymer-clay nanocomposites is illustrated in Figs. 1and 2, and the entire set of PCN systems studied in this work are summarizedin Table 1.

2. Simulation Methods and Computational Details

2.1. Atomistic simulations of PCNs

The first, essential step in all atomistic MD simulation is the definition of a reli-able force field (FF) for the description of inter-and intra-molecular interactionswhich is based on the evaluation of structural parameters and physical propertiesfor the systems of interest. Since the calculated quantities are highly sensitive tothe nonbonded components of the force field employed (i.e., atomic charges andvan der Waals parameters), based on our previous experience in this field21 in this

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Fig. 1. Example of the proposed multiscale simulation protocol for polymer/clay nanocomposites.

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Table 1. Names and compositions of the polymer nanostructured sys-

tems studied in this work. MMT clays C15A and C20A have the samesurface modifier but a different cation exchange capacity (CEC).

System Matrix Filler Surface modifier

PP/C10A PP MMT (H3C)2 N(C18H37)(C2H4-Ph)PP/C15A PP MMT (H3C)2 N(C18H37)2PP/C20A PP MMT (H3C)2 N(C18H37)2PP/C30B PP MMT (H3C) N(C18H37)(C2H4-OH)2PP/ODA PP MMT H3 N(C18H37)PA6/C20A PA6 MMT (H3C)2 N(C18H37)2PA6/C30B PA6 MMT (H3C) N(C18H37)(C2H4-OH)2PA6/M3C18 PA6 MMT (H3C)3N(C18H37)TPUa/C30B TPU MMT (H3C) N(C18H37)(C2H4-OH)2

aElastollan 1185A, a commercial polyether polyurethane.

work we adopted the ad hoc force field developed by Heinz and coworkers.23,24 Asdemonstrated by the authors for sodium MMT and other phyllosilicate PCNs, thisaccurately derived FF is able to describe, among many other properties, the ther-modynamics of surface processes more reliably by reducing deviations of 50–500%in surface and interface energies to less than 10%, which constitutes a fundamentalstep towards quantitative modeling of interface processes involving layered silicates.The lattice of our MMT model is monoclinic, with space group C2/m, and charac-terized by the following lattice parameters: a = 5.20 A, b = 9.20 A, c = 10.13 A, andα = 90◦, β = 99◦, γ = 90◦, in excellent agreement with the available literature.24–26

All atomistic simulations were performed using Materials Studio (v.4.3, Accelrys,San Diego, USA).

Fig. 2. Multiscale molecular modeling approach for polymer/clay nanocomposites: general schemaof the protocol.

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The model structures of the MMT surface modifiers (i.e., quaternary ammo-nium salts or quats) were generated using the sketcher tool of Materials Studio.All molecules were subjected to an initial energy minimization using the Compassforce field (FF),27 the convergence criterion being set to 10−4 kcal/(mol A). Thegeneration of accurate model amorphous structures for polymers was conducted asfollows.15,16 First, the constitutive repeating unit (CRU) was built and its geom-etry optimized by energy minimization again using the Compass FF. Hence, theCRU was polymerized to a conventional degree of polymerization (DP) equal to 20.Although, this chain may be too short to capture the genuine response of a longpolymer molecule; in the case of polystyrene (PS) it has been verified that a polymerwith the same DP is longer than the average persistence length of PS in polymerclay nanocomposites.28 Further, polymers of similar lengths have been already suc-cessfully employed by us in similar studies.15,16 Explicit hydrogens were used inall model systems. The Rotational Isomeric State (RIS) algorithm,29 as modifiedby Theodorou and Suter,30 was used to create the initial polymer conformationat T = 300 K. The polymer structure was then relaxed to minimize energy andavoid atom overlaps using the conjugate gradient method. Such a straightforwardmolecular mechanics scheme is likely to trap the simulated system in metastablelocal high-energy minimum. To prevent the system from such entrapments, therelaxed structure was subjected to our well-validated combined molecular mechan-ics/molecular dynamics simulated annealing (MDSA) protocol.15,16 At the end ofeach annealing cycle, the structure was again relaxed via force field, using a root-mean-square (RMS) of the force < 0.1 kcal/(mol A). The same MDSA protocolwas applied to model of clay surface modifiers. After each component was mod-eled (MMT platelet, quats, and polymer), the overall systems for binding energycalculations and basal spacing determination, respectively were built (see Fig. 3).

To generate a proper surface in the model for binding energy calculations, thelattice constant c of the MMT cell with five quats molecules on one side wasextended to 125–150 A depending on the length of the surface modifiers and poly-mer molecules. The closest distance of a part of the polymer to the quat was around5 A. According to previous work,15,16 this ensures that polymer chain is attractedto the modified MMT surface by favorable nonbonded interactions, but is far awayenough so that its initial conformation hardly affects the equilibrium states of thesystem. Further, c = 125–150A assures that polymer molecules do not interactwith the K+ ions on the mineral surface of the nearest neighbor platelet. In otherwords, even if our model is 3D periodic, there are no interactions between theperiodic images in the z-direction, ultimately resulting in a pseudo 2D periodicsystem.31 Isothermal–isochoric (NVT) molecular dynamics experiments were runat T = 300 K. Each MD run was started by assigning initial velocity for atomsaccording to Boltzmann distribution at 2×T . It consisted of an equilibration phaseof 50 ps, during which system equilibration was monitored by recording the instan-taneous values of total, potential and nonbonded energy, and data collection phase,which was extended up to 300 ps.

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(a) (b)

Fig. 3. Three component model used for (a) binding energy calculations and (b) basal spacingdetermination. The clay is portrayed in atom-colored sticks (Si, orange; O, red; Al, green; Mg.magenta; K+, purple). The quat is depicted via its van der Waals surface as transparent bluespheres, while the polymer is shown in green solid sphere representation (Color online).

Further, molecular dynamics simulations for the determination of the basal spac-ing of nanocomposite resulting from the assembly of different polymers and differentsurface modifiers were performed. A typical three component model consisting oftwo layers of montmorillonite, several quat molecules, and a polymer chain is pre-sented in Fig. 3(b). The c values in the initial model of MMT crystal supercells wereprolonged according to a bi-layer arrangement of the quaternary ammonium salts.After system energy relaxation process, isothermal–isobaric (NPT) MD experimentswere run at 300K for a total time of 250ps (i.e., 25,0000 steps with integration timestep of 1 fs). The Ewald summation method was used to handle long range non-bonded interaction.32 During each MD, both montmorillonite layers were treatedas rigid bodies by fixing all cell dimensions except for c, and all atoms in the inter-layer space including K+ cations were allow to move without any constraint. Theprocedure used to calculate the interaction energies and, hence, the binding energyvalues Ebind between all system components, is described in details in our previouspapers.17–21 Accordingly, it will only briefly summarized below.

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Starting from the concept that the total potential energy of a PCN ternary sys-tem is composed by the polymer chains (P ), the montmorillonite platelets (MMT ),and quat molecules (Q), its value E(P/MMT/Q) may be written as:

E(P/MMT/Q) = EP + EMMT + EQ + EP/MMT + EMMT/Q + EP/Q , (1)

where the first three terms represent the energy of polymer, MMT and quat (con-sisting of both valence and nonbonded energy terms), and the last three terms arethe interaction energies between each of two component pairs (made up of non-bonded terms only). By definition, the binding energy Ebind is the negative of theinteraction energy.

To calculate the binary binding energy Ebind(P/MMT ), for instance, we firstcreated a P–MMT system deleting the Q molecules from one of the equilibrated MDtrajectory frames, and then calculated the potential energy of the system EP/MMT .Next, we deleted the MMT platelets, leaving the P chains alone, and thus calculatedthe energy of the P molecules, EP . Finally, we deleted the P molecules from theP–MMT system, and calculated EMMT. Then, the binding energy Ebind(P/MMT )can be calculated from the following equation:

Ebind(P/MMT ) = EP + EMMT − EP/MMT . (2)

Similarly, the binding energies Ebind(P/Q) and Ebind(MMT /Q) can be computedas follows:

Ebind(P/Q) = EP + EQ − EP/Q , (3)

Ebind(MMT /Q) = EMMT + EQ − EMMT/Q . (4)

Importantly, these binding energies between the individual components of eachnanocomposite system also constitute the input parameters for the higher level,mesoscale simulations, as described in the next section.

2.2. Mesoscale models and simulations

In order to simulate the morphology of the systems of interest at a mesoscopiclevel, we used the DPD33,34 simulation tool as implemented in the Material Studiomodeling suite. In the framework of a multiscale approach to nanocomposites sim-ulation, the interaction parameters required as input for DPD calculations wereobtained by a mapping procedure of the binding energy values between differentspecies obtained from simulations at a lower (atomistic) scale.18,21

In the Dissipative Particle Dynamics simulation method, a set of particles movesaccording to Newton’s equation of motion, and interacts dissipatively through sim-plified force laws. Also, in the DPD model individual atoms or molecules are notrepresented directly, but are coarse-grained into beads. These beads, or particles,constitute local “fluid packages” able to move independently.

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The force acting on the beads, which is pairwise additive, can be decomposedinto three elements: a conservative (FC

ij), a dissipative (FDij ), and a random (FR

ij)force.35 Accordingly, the effective force Fi acting on a particle i is given by:

Fi =∑i�=j

(FCij + FD

ij + FRij) , (5)

where the sum extends over all particles within a given distance rc from the ithparticle. This distance practically constitutes the only length scale in the entiresystem. Therefore, it is convenient to set the cutoff radius rc as a unit of length(i.e., rc = 1), so that all lengths are measured relative to the particles radius.35

The conservative force is a soft repulsion, given by:

FCij = f(x) =

{aij(1 − rij)rij rij < 1 ,

0 rij ≥ 1 ,(6)

where aij is the maximum repulsion between particles i and j, rij is the magnitudeof the particle-particle vector rij = ri − rj (i.e., rij = |rij|), and rij = ri/rj is theunit vector joining particles i and j.

The first step necessary for the determination of the DPD input parametersgenerally consists in defining the DPD bead dimensions, thus implicitly definingthe characteristic length of the system (rc). As stated above, the interaction rangerc sets the basic length-scale of the system; in other terms, rc can be defined as theside of a cube containing an average number of ρ beads. Therefore:

rc = (ρVb)13 , (7)

where Vb is the volume of a bead.Thus, even in a heterogeneous system consisting of several different species

such as a nanocomposite a basic DPD assumption is that all bead-types (eachrepresenting a single species) are of the same volume Vb. In the case of a surfacemodifier made up by a strongly polar “head” and two almost apolar “tails” (typicalsituation for e.g., PCN), we considered them as made up by two different kind ofbeads, H and T , respectively, as schematically illustrated in Fig. 4.

The chosen bead volume has an average value of 130 A3, that roughly corre-sponds to that of a single monomer of polymer, as evaluated by the Connollyalgorithm.36–38 The corresponding polymer chain was chosen to be made up of 100beads of type P . Obviously, other numbers can be used in case of different polymersand organic modifiers, but the underlying concept is the same. Once we determinedthe bead size, and fixed the DPD system density to ρ = 3, we could calculate thecharacteristic dimension of the mesoscopic system (i.e., the cutoff distance rc) fromEq. (7), obtaining rc = 7.4 A. As said, this value represents the soft potential cutoffdistance, but is also the length of one of the unit cells in the DPD simulation box.Our DPD system was chosen to be constituted by 20× 20× 6 unit cells; therefore,the effective dimensions characterizing each single system can be calculated once

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Fig. 4. Schematic drawing of a surfactant (quat) mapping from atomistic representation to DPDbeads.

the corresponding intercalated basal spacing was obtained (see below). To representthe clay platelets, we made use of the available software option of including a repul-sive wall in the simulation box, perpendicular to the z-axis at the origin. Further,we introduced a bead M with no connectivity and no repulsion towards the wall,in order to fill the space between the two wall surfaces, thus representing a MMTplatelet.

Having set the DPD box dimensions, we proceeded by calculating the numberof quat molecules that had to be inserted in order to match the CEC of the MMTused in MD simulations. The procedure employed in this calculation is describedbelow for the 5-bead surfactant shown in Fig. 4 as a proof-of-concept. The claysurface in our atomistic molecular model has an area of approximately 277.3 A2

(i.e., 15.5 A × 17.9 A), and bears a charge of −2e accordingly, taking the case ofthe 5-bead surfactant shown in Fig. 4, two quat molecules can be placed on it. Eachsurface in the DPD has an area of roughly 21351 A2 (i.e., 146.1 A × 146.1 A), andhence is 77 times larger than the molecular model surface and bears a charge of−154e. Accordingly, the total number of quat molecules to be inserted in our DPDbox is 308 (154 on each platelet). The number of MMT beads to be inserted canin turn be calculated once the volume of MMT platelet molecular model (2387 A3)is known. This leads to a total of 1414 MMT beads. Finally, being 7200 the totalnumber of beads in the box, and the number of quat beads estimated to be 1540(= 308 × 5), the number of polymer beads is 4246 (= 7200 – (1540 + 1414)). Theresulting DPD nanocomposite model thus contains at least four species of differentbeads. Figure 5 illustrates the corresponding resulting mesoscale model employedin DPD simulations.

The next, important issue of a DPD simulation is the definition of the bead-beadinteraction parameters aij . The detailed procedure for obtaining these mesoscale

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Fig. 5. DPD model of PP/C20A nanocomposites. The color code for the DPD beads are as follows:orange, MMT; blue, quat head; green, quat tail; red, polymer.

interaction parameters from atomistic molecular dynamics binding energies isreported in details elsewhere.18,21 Adapting this recipe to the present system, wedetermined the bead-bead interaction parameters for the individual components inall systems, as shown in Table 2.

Table 2. DPD bead-bead interaction parameters aij forall PCN systems considered.

P T H M W

PP/C10AP 25 31 30.9 32 320T 31 31.3 35.3 30.4 304H 30.9 35.3 55.5 5.2 52

M 32 30.4 5.2 10 1

PP/C15AP 25 32.7 32.6 33 330T 32.7 32.8 36 30.7 307H 32.6 36 64.2 5 50M 33 30.7 4 10 1

PP/C20AP 25 32.7 32.6 33 330T 32.7 32.8 36 30.7 307H 32.6 36 64.2 5 50M 33 30.7 4 10 1

PP/C30BP 25 31.1 31 31 310T 31.1 30.3 32.9 30.1 301H 31 32.9 54.1 4.7 47M 31 30.1 4.7 10 1

PP/ODAP 25 31 30.9 32 320T 31 31.3 35.3 30.4 304H 30.9 35.3 55.5 5.2 52M 32 30.4 5.2 10 1

PA6/C20AP 25 33.56 34.06 33.09 330.9T 33.56 32.11 33.09 30.72 307.2H 34.06 33.09 500 5 50M 33.09 30.72 5 15 1

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Table 2. (Continued )

P T H M W

PA6/C30BP 25 32 30 29 290T 32 30.2 32.7 30.1 301H 30 32.7 500 5 50M 29 30.1 5 15 1

PA6/M3C18P 25 31.47 31.12 31.79 317.9T 31.47 31.78 30.95 30.6 306H 31.12 30.95 500 5 50M 31.79 30.6 5 15 1

TPU/C30BP 25 32.5 33.3 31 310T 32.5 31.1 35.8 30.8 308H 33.3 35.8 500 5 50M 31 30.8 5 15 1

2.3. Finite element models for the calculation of

PCN effective properties

The prediction some representative macroscopic properties of the polymernanocomposites considered in this work as a function of loading and degree ofexfoliation constitutes the final step of our multiscale modeling strategy. In partic-ular, the Young modulus E, the thermal conductivity k, and the gas permeabilityP were the properties of election, since it is in these performances that lies mostof the industrial interest towards these new materials. To this purpose, FE simu-lations were performed using the software MesoProp (Accelrys, San Diego, USA)and Palmyra (v. 2.5, MatSim, Zurich, CH), dedicated to the calculation of effec-tive properties for random composite and multiphase materials using Represen-tative Volume Elements (RVEs) of the microstructure. These software have beenvalidated on different composite material morphologies by several authors includ-ing us,18,21,39,40 yielding reliable results. FE calculations were performed in orderto analyze both platelet stacks and overall nanocomposite properties, using twodifferent methods for model generation and analysis. For MMT stacks, the den-sity profiles obtained from mesoscopic simulations were mapped to a fixed regulartetrahedral grid, whereas for the overall microstructure different RVEs were firstcreated according to different degrees of exfoliation and filler volume fractions andthen discretized using and adaptive tetrahedral mesh. In both cases, the resultingeffective properties were estimated using an energy minimization method.41

2.3.1. Calculation of clay stack effective properties

The prediction of effective properties for materials characterized by complexmicrostructural features at the nanoscale such as PCNs should ideally include

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information deriving from models developed at the molecular level. A possiblemethod for incorporating such details consists in mapping mesoscale density fieldsonto a fixed grid FE model, and assigning properties to each mesh element accord-ingly. This method has been implemented in the commercial software MesoProp,42

which allows the morphology of each phase present in a MS model to be exported,via density field mapping, to a cubic or orthorhombic cell featuring periodic bound-ary conditions and a fixed tetrahedral mesh. For each mesh element, pure compo-nent properties are averaged according to the imported density map (see Fig. 6), andthe effective properties of the entire macrostructure can thus be calculated basedon the morphological details of the MS model. As mentioned above, we applied thisprocedure to the prediction of the polymer-intercalated clay stacks whose phase den-sity maps were previously calculated by performing MS DPD simulations. Thesemesoscopic morphological information were then assigned to those inclusions repre-senting clay stacks in the overall nanocomposite RVE model described below and,using the properties of the single components listed in Table 3 as further input, theeffective properties of the composites could be finally estimated as schematicallyillustrated in Fig. 7.

2.3.2. Calculation of PCN effective properties

A suitable model for FE calculations of the effective properties of a composite mate-rial consists of a properly discretized representative volume element, i.e., a digitalsample of the material microstructure (either generated or obtained by experimental

Fig. 6. MesoProp mapping.

Table 3. Macroscopic properties of pure compo-nents: Young Modulus (E), Poisson ratio (ν),thermal conductivity (k), and permeability (P ).

Property PP PA6 TPU MMT

E (GPa) 1.55 2.8 0.02 178ν(−) 0.36 0.35 0.45 0.2k(W/mK) 0.25 0.28 0.2 1.14P (O2) (barrer) 9.6 4.7 2.1 0.0001

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Fig. 7. Calculation schema of MesoProp.

techniques) which could be descriptive of the overall morphology characterizing agiven complex material such as PCNs. The definition of a RVE is not trivial, anddepends on the specific characteristic of the material under investigation. In thiswork, all models were based in a RVE comprising 40 non-overlapping platelet-likeinclusions, allowing for reasonably reliable results on the base of previous work.22,43

Practically speaking, since most of the effects exerted by the filler addition isgenerally observed at low filler contents, we decided to adopt 2 standard values forclay of Φw = 0 2.0 and 4.6%w/w, respectively. MMT particles (both single parti-cles and stacks) were modeled as disks with a toroidal rim. Each platelet thicknesswas defined by the height of the corresponding symmetry axis h and diameter d,thus being characterized by an aspect ratio of a = d/h. By setting d = 120 nmand h = 1 nm for each single particle, the aspect ratio a was equal to 120, a valuein agreement with common literature data for layer silicates.44 Orientation to theplatelets was imparted by assigning the value of 0.06 to the eigenvalues of theorientation tensor 1 and 2; thus, the value of 0.88 was automatically assigned tothe eigenvalue 3. In order to evaluate and compare the influence of exfoliation onthe PCN ultimate properties we created models characterized by several, differ-ent degrees of exfoliation by substituting groups of single MMT platelets with anequivalent number of stacks. For example, a highly exfoliated system was definedwith 32 platelets having an aspect ratio of 120, and 8 stacks with two plateletseach, representing a 66.7% of exfoliated platelets. The aspect ratio of stacks rangesfrom 10 to 13 due to different basal spacing obtained from MD. Figure 8 showsan example of one these models for the PCN based on poly(propylene). A volumefraction Φv of 1.5% v/v was used for PP-based PCNs, while a value of Φv of 1.9%v/v was chosen for TPU- and PA6-based systems. This value of Φv corresponds tothe same value of the weight fraction Φw = 4.6% w/w for all PCNs.

As mentioned previously, density profiles from mesoscale simulations were usedas an input for macroscopic property calculation of the stacks, for which the cor-responding model is illustrated in Fig. 9. Finally, the bulk properties of the purecomponents of the diverse nanocomposite systems reported in Table 3 constitutedthe last necessary input of FEM calculations.

It is worth noticing here that a high-quality mesh is of fundamental importantin order to retrieve realistic values the macroscopic properties of interest for PCNsystems. In particular, an improper definition of inclusion surface grid nodes canresult in severe discrepancies between the theoretical volume fraction, defined by

March 27, 2012 11:55 WSPC/245-JMM 00046


(a) (b)

Fig. 8. Intercalated stacks and single platelets in the FE RVE model of the PP/C20A system: (a)global model configuration and (b) relative meshed volume.

Fig. 9. Intercalated stack in the FE RVE model of the PP/C20A nanocomposites.

platelets dimensions and number, and the effective volume fraction, resulting fromthe meshed RVE. For instance, surfaces of platelet-like inclusions in Palmyra arediscretized by assigning a given number of nodes to the toroidal rim (base lay-ers) and another number of nodes to the flat sides (extra layers). Thus, a correctbalance of these two quantities is fundamental for the creation of a grid whichcan accurately reproduce inclusions volume fraction and describe particle geome-try without increasing the total number of elements in the RVE and, hence, thecomputer memory requirements needed for computation. Figures 10 and 11 showthe different quality of the meshes obtained for the case of a single MMT plateletwith aspect ratio a = 120 occupying 0.40% of a cubic box by varying the numberof base and extra layers, whereas Tables 4 and 5 summarize the parameters usedto generate the corresponding meshes.

Our models were finally built using 5 base and 35 extra layers for each MMTplatelet. A volume mesh which could guarantee reliable results was then created byadding further nodes, defining a variable number of tetrahedra, and assessing theoverall mesh quality with respect to the volume of a regular tetrahedron. By usingan average mesh quality parameter d of 0.3–0.4, and eliminating all elements with

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Fig. 10. Different quality of surface meshes generated for one MMT platelet obtained with 5 (left),7 (middle), and 9 (right) extra layers, respectively.

Fig. 11. Different quality of surface meshes for MMT platelets obtained by using 5, 7, 9, and 35extra layers (from left to right), respectively.

Table 4. Parameters for generation of the surface meshesshown in Fig. 10.

Base layers 5 7 9

Extra layers 5 7 9Volume fraction 0.40% 0.40% 0.40%Effective volume fraction 0.090% 0.399% 0.40%Surface nodes 3073 22015 39617

Table 5. Parameters for generation of the surface meshes presented in Fig. 11.

Base layers 5 5 5 5

Extra layers 5 7 9 35Volume fraction 0.40% 0.40% 0.40% 0.40%Effective volume fraction 0.090% 0.121% 0.146% 0.399%Surface nodes 3073 3237 3454 10829

d < 0.1 by adding more nodes, the resulting volume meshes were characterized byan average number of volume elements in the range 2×106 −4×106 and were founddevoid of bridging elements and inclusion shape anomalies. Accordingly, they wereused for the calculations of the effective properties of all PCNs considered.

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3. Results and Discussion

3.1. Atomistic simulations of PCNs

Interaction energies, basal spacings, and density profiles obtained from atomisticsimulations are the main information required for both checking the correctnessof the calculations and setting the system for further mesoscale simulations. Infact, the repulsive DPD parameters and size of the DPD cell box are based onthe atomistic interactions energies and basal spacing values, respectively. Accord-ingly, Table 6 presents the basal spacing values obtained from the atomistic MDsimulations for all PCN systems considered in this work, which are in a good agree-ment with the corresponding experimental results, also shown in parentheses forcomparison.18,45,46

3.2. Mesoscale simulation of nanocomposites

As mentioned in the Simulation Methods and Computational Details section, themesoscale level simulations, performed using parameters generated via a mappingprocedure based on the information stemming from the lower scale MD simula-tions, requires an initial validation. In the case of an polymer-intercalated MMTstack and, in particular, the arrangement of all organic material (MMT surfacemodifiers and polymer chains) within the MMT layers, we compared the densityprofiles of the total organic material derived from the corresponding MD trajecto-ries with those obtained from the mesoscale DPD runs. As already observed by usand others18,21,47,48 there is a general trend in the arrangement of organic specieswithin the mineral gallery. Intercalated species usually exhibit alternating high andlow density layers in the proximity of the clay surface, whereas bulk polymer densityis reached in the center of the clay gallery.

This arrangement is re-confirmed by the distribution of the organic densityobtained from the present MD and DPD simulations of the system PP/C20A, asshown and compared in Fig. 12. Given the excellent agreement between MD andDPD density profiles, we can safely assume that a reasonable conformational link

Table 6. Basal spacing values of PCNstacks calculated by MM/MD simulations.

Stack Basal spacing (nm)

PP/C10A 3.73 (3.745)PP/C15A 4.90PP/C20A 4.10PP/C30B 3.68PP/ODA 3.73PA6/C20A 4.22 (4.2246)PA6/M3C18 4.00PA6/C30B 4.22TPU/C30B 4.22

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Fig. 12. Comparison between interlayer densities of organic species (compatibilizer and polymer) inthe interlayer space of the PP/C20A PCN as obtained from MD and DPD simulations, respectively.

between the atomistic and mesoscopic scale for these clay-based PCN has beensuccessfully established.

3.3. Effective properties of PCN stacks

The mesoscopic density profiles obtained as an output of DPD simulations werethen as input data for the FE calculations (i.e., using MesoProp) to predict agiven set of macroscopic properties of the stack such as its elastic modulus E,thermal conductivity k, and gas permeability P . In this last case, a value of 0.0001was chosen for the MMT platelet permeability, as we considered the stacks non-permeable to gas. An example of the fixed grid created by MesoProp is shown inFig. 13.

As mentioned above, the values of the Young modulus and the thermal con-ductivity for each stack of the NC were then evaluated. All results obtained arepresented in Table 7, and are expressed as an average value in the x and y direc-tion with respect to the orientation of the MMT platelets. From Table 7 it isclear that PA6 offers the best mechanical properties of the stacks, which can be

Fig. 13. Cut plane of the microFEM fix grid. The color code is as follows: blue, MMT; green, quathead; yellow, quat tail; red, polymer (Color online).

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Table 7. Properties of PCN stacks estimated using micro-

FEM calculations performed exploiting the density profilesderived from mesoscale simulations.

Stack E (GPa) StDev k (W/mK) StDev

PP/C10A 70.06 0.19 1.070 0.076PP/C15A 75.44 0.43 1.092 0.079PP/C20A 72.36 0.70 1.078 0.078PP/C30B 70.34 0.57 1.070 0.075PP/ODA 88.22 0.46 1.186 0.090

PA6/C20A 110.30 0.07 1.298 0.080PA6/M3C18 110.39 0.15 1.295 0.083PA6/C30B 110.30 0.12 1.295 0.082TPU/C30B 100.05 0.11 1.088 0.112

attributed to the higher polarity of the nylon chains with respect to the polypropy-lene macromolecules.

3.4. Macroscopic properties of PCNs

The prediction of the macroscopic properties of the entire PCN systems — i.e., thosein which both stacks and isolated platelets were present — was achieved with theFEM software Palmyra44 as described above. Basically, for each system a cell wascreated and objects as platelets and stacks were added until the required volumefraction was achieved. The properties of pure components (polymer and MMT)were inserted in materials database together with the properties of the stacks asobtained using the fix grid calculations described above. The set of properties andtheir values calculated for different PCN systems (see Table 1) are shown in Fig. 14in terms of enhancement factor Ef = Pc/Pm, i.e., the ratio between the given

Fig. 14. Predicted enhancement ratio Ef = Pc/Pm of the macroscopic properties of differentPP-clay nanocomposites.

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property value for the PCN and the corresponding value for the pristine polymermatrix.

An increase of 60% on mechanical properties is observed when 4.6% w/w ofMMT is added to PP or PA6 matrices. In principle, a higher Ef could be expectedfor PP/MMT systems due to lower elastic modulus of neat PP. However, FEMcalculations show that the elastic modulus of the PA6/MMT nanocomposites isslightly higher than that of PP/MMT systems (Fig. 14). This can be attributed tothe mechanical properties of PA6/MMT stacks, which are significantly higher thanthe corresponding PP/MMT counterparts (Table 7). On the other hand, in thecase of the TPU-based PCN, a huge increase in Young modulus is observed, whichconstitutes a somewhat expected result by virtue of the significantly lower elasticmodulus of the neat TPU. Concerning the other properties, an increase of 10% inthermal conductivity and a decrease of 25% (PP) and 36% (PA6) in permeabilityis predicted. The permeability depends mainly on orientation, aspect ratio andvolume fraction of the filler in the matrix. As the volume fraction of MMT in thePA6 matrix is higher, and also the density of PA6 is higher than that of PP, ahigher barrier to the diffusion of gas/liquid in the PA6/MMT PCN with respectto the PP-based nanocomposites predicted (Fig. 14). Other differences might benoted depending on the surface modifiers used, resulting from stack properties anddifferent aspect ratios.

As concerns a direct comparison of PCNs macroscopic properties predicted viathe present multiscale simulation and the corresponding experimental data, a seriesof different measurements and morphological characterizations were reported byFornes et al.49 in a complex and elegant study where sodium montmorillonite wasmodified with a series of organic amine salts to evaluate the effect on quat mor-phology and physical properties on the resulting PCNs. Within their set of results,three compare well with the calculated values reported in the present work. In par-ticular, Fornes and Paul49 observed that, in the case of systems C30B and M3C18

(see Table 1), the corresponding PCN systems were exfoliated and well dispersed;this morphological evidence clearly accounts for the higher values of the measuredmacroscopic quantities compared to the simulated ones. However, it is also clearthat the system with M3C18 yields higher enhancement in the young modulus thanthe C30B PCN according to both simulated and experimental results. Further, thepresent simulations show that C20A offers the highest increase in Young modulus,although experimentally the Ef is somewhat less pronounced, due to partial inter-calation that takes place by using this kind of surfactant.49 Interestingly, however,a good agreement is found between experimental and calculated results in a directcomparison for PA6-based PCN at Φw = 4.5% w/w.50 In fact, the experimentalenhancement factors Ef of 1.57, 1.62, and 1.69 for the Young modulus E of thesystems PA6/C20A, PA6/C30B, and PA6/M3C18 nicely match the correspondingcalculated values of 1.61, 1.57, and 1.60, respectively.

In the case of TPU-based nanocomposites, a good agreement of the predictedYoung modulus was found with the experimental results.50 Indeed, this group

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reported a value of E = 28GPa for the pristine TPU matrix, and E = 83GPafor TPU loaded with 5% w/w of C30B, corresponding to Ef = 2.96, in agreementsthe predicted value of 2.76. For this system, also the Ef for the thermal conductiv-ity = 1.1851 is found in agreement with the estimated value of 1.14. It is importantto note here that the value for thermal conductivity of a composite corresponds toa value averaged over the three main directions. It was observed that the increase inthe conductivity value along the x and y directions is substantially higher than thatin the z direction, due to the alignment of the platelets on those directions. Lastly,given the small difference between calculated and experimental k values (approxi-mately 3.5%), we can state that the predictive power of the multiscale procedureis validated also for this property.

The last property considered was the gas permeability. Considering the systembased on PP and characterized by a Φw = 4.6% w/w loading of MMT as a proof-of-concept, the permeability of the system to oxygen is predicted to decrease by 25%.The main factors that account for the decrease in the barrier properties are the clayloading and the aspect ratio of the nanoparticles, while it should be independentof the nature of the surface modifier. Satisfactorily, a k value of the same order ofmagnitude with respect to the PP/C20A system was found.

Applying the multiscale simulation approach also allows to predict to the effectof the degree of clay exfoliation as well as of clay loading in a given polymericmatrix on the resulting macroscopic properties. As an example, Fig. 15 presentsthe scheme for the construction of models with different degrees of exfoliation (or,in other words, the ratio between single clay platelets and stacks) for the PP/C20Asystem. Six different models were generated to represent the different degrees ofexfoliation. To study the loading effect, systems with Φw = 2.0 and 4.6% w/w offiller in matrix were also simulated.

Figure 16 shows the dependencies of the different macroscopic composite prop-erties on the degree of exfoliation. In all cases analyzed in this work, the calculated

Fig. 15. Model schema adopted to study the effect of the degree of exfoliation for MMT-basednanocomposites systems.

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(a) (b)

(c)

Fig. 16. Enhancement factor Ef of the Young modulus (a), thermal conductivity (b), and oxygenpermeability (c) as function of the degree of exfoliation for the system PP/C20A.

Fig. 17. Enhancement factor Ef of the Young modulus as function of the nanofiller loading forthe PP/C20A PCN systems at different degree of exfoliation.

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data confirm the trend observed in the corresponding experiments. In particular,the Young modulus increases with increasing level of exfoliation, while the oppo-site behavior is seen in the case of thermal conductivity and permeability. In thesame figures, the effect of the nanoparticle loading is also depicted, from which itis evident that higher loading result in an increase of Young modulus and thermalconductivity, while the permeability is reduced by the inclusion of higher amountof the filler. Interestingly, the presence of higher amount of nanoparticle does notchange the trend induced by the degree of exfoliation.

Figure 17 shows the enhancement factor of the Young modulus as a function ofnanofiller loading for the PP/C20A PCN systems at different degree of exfoliation.The results show a certain degree of nonlinearity of the mechanical property as afunction of loading, particularly at high degree of exfoliation.

4. Conclusions

In this work we presented the derivation and application of a multiscale molec-ular modeling procedure to characterize polymer/clay nanocomposite materialsobtained with full or partial dispersion of nanofillers in matrix. This approach relieson a step-by step message-passing technique from atomistic to mesoscale to finiteelement level; thus, computer simulations at all scales are completely integrated andthe calculated results are compared to the experimental evidences obtained fromour partners or the literature. In particular, given polymer nanocomposite systemshave been studied by different molecular modeling methods, such as “atomistic”Molecular Mechanics and Molecular Dynamics, the mesoscale Dissipative ParticlesDynamics and the “macroscale” Finite Element Method. The entire computationalprocedure has been applied to a number of diverse polymer nanocomposite systemsbased on montmorillonite as clay and different clay surface modifiers, and theirmechanical, thermal and barrier properties were predicted in agreement with theavailable experimental data.

Acknowledgments

This work is part of the 6th Framework EC Program through the MultiHybridsIP 026685-2IP project, of which the financial support is gratefully acknowledge bySPP, RT, PP, SP, and MF. DRN wishes to acknowledge the financial support fromCONACYT Mexico (Grant No. 146624).

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