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Fluid Phase Equilibria 261 (2007) 366–374

To the nanoscale, and beyond!Multiscale molecular modeling

of polymer-clay nanocomposites

Giulio Scocchi a,1, Paola Posocco a, Andrea Danani b,Sabrina Pricl a,∗, Maurizio Fermeglia a

a Molecular Simulation Engineering (MOSE) Laboratory, Department of Chemical,Environmental and Raw Materials Engineering, University of Trieste, Piazzale Europa 1, 34127 Trieste, Italy

b Institute of Computer Integrated Manufacturing for Sustainable Innovation (ICIMSI),University for Applied Sciences of Southern Switzerland (SUPSI), Centro Galleria 2, CH-6928 Manno, Switzerland

Received 10 May 2007; received in revised form 7 July 2007; accepted 8 July 2007Available online 20 July 2007

bstract

In this work, we present an innovative, multiscale computational approach to probe the behavior of polymer-clay nanocomposites (PCNs). Inetails, our modeling recipe is based on (i) quantum/force-field-based atomistic simulation to derive interaction energy values among all systemomponents; (ii) mapping of these values onto mesoscale simulation (MS) parameters; (iii) mesoscopic simulations to determine system density

istributions and morphologies (i.e., intercalated vs. exfoliated); (iv) simulations at finite-element levels to calculate the relative macroscopicroperties. Through these calculations, we could in principle modify the characteristic of the organic modifiers, polymers, and even substrate,nd thus isolate the factors that drive the polymers to permeate the clay galleries. This, with the ultimate goal of limiting the efforts of synthetichemists in following the inefficient Edisonian prescription of creating all possible mixtures in order to isolate the desired materials.

2007 Elsevier B.V. All rights reserved.

sipati

ihofArpp

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eywords: Polymer-clay nanocomposites; Multiscale molecular modeling; Dis

. Introduction

Polymer-clay nanocomposites (PCNs) are one of the mostxciting and promising classes of materials discovered in theast years [1–3]. A number of physical properties are success-ully enhanced when a polymer matrix is modified with a smallmount of layered silicate, on condition that the filler is dispersedo nanoscopic levels. Optical clarity, stiffness, flame retardancy,arrier properties, and thermal stability are mainly influenced,nd in some cases also the mechanical strength increases [4].

These composites are obtained by dispersion in a polymericatrix of small amounts (typically 1–5%) of a clay filler having

ne of the three dimensions of nanometric scale. Fabricat-

∗ Corresponding author. Tel.: +39 040 5583750; fax: +39 040 569823.E-mail address: sabrina.pricl@dicamp.units.it (S. Pricl).

1 Present address: Institute of Computer Integrated Manufacturing for Sus-ainable Innovation (ICIMSI), University for Applied Sciences of Southernwitzerland (SUPSI), Centro Galleria 2, CH-6928 Manno, Switzerland.

fonttsiw(

378-3812/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2007.07.046

ve particle dynamics; Finite-elements calculations; Materials design

ng these materials in an efficient and cost-effective manner,owever, poses significant synthetic challenges. PCNs can bebtained following three main different routes: intercalationrom solution, in situ polymerization, and melt intercalation.mongst these, the latter is the most appealing and, hence, the

oute most frequently adopted for both laboratory and industrialreparation of PCNs, as it can be realized by standard polymerroduction equipment.

The layered silicates most commonly used in PCN productionmontmorillonite being a prime example) belong to the structuralamily known as 2:1 phyllosilicates. Their crystal lattice consistsf 2-D layers where a central octahedral sheet of alumina or mag-esia is fused into two external silica tetrahedron by the tip, sohat the oxygen ions of the octahedral sheet do also belong tohe tetrahedral sheet. These layers organize themselves to form

tacks with a regular van der Waals gap in between them, callednterlayer or gallery. Isomorphic substitution of some elementsithin the layers – e.g., Al3+ with Mg2+ as in montmorillonite

MMT), or Mg2+ with K+ – generates negative charges that are

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G. Scocchi et al. / Fluid Phas

ounterbalanced by alkali or alkaline earth cations situated inhe interlayers. The lateral dimensions of these layers may varyrom approximately 200 A to several microns or more, depend-ng on the peculiar composition of the silicate, while the spacingetween the closely packed sheets is also of the order of 1 nm,hich is smaller than the radius of gyration of typical poly-ers. Consequently, there is a large entropic barrier that inhibits

he polymer from penetrating this gap and becoming intermixedith the clay.Thus, there are several issues that need to be addressed in

rder to optimize the production of these PCNs. Of foremostmportance is isolating conditions that promote the penetrationf the polymer into the narrow gallery. If, however, the sheetsltimately phase-separate from the polymeric matrix, the mix-ure will not exhibit the improved strength, heat resistance, orarrier properties. Therefore, it is also essential to determine theactors that control the macroscopic phase behavior of the mix-ure. To enhance polymer-clay interaction, the clay interlayerurfaces of the silicate are chemically treated to make the silicateess hydrophilic and therefore more wettable by the polymer. Aation exchange process accomplishes this, where hydrophilications such as Na+, K+ or Ca2+ are exchanged by a surfaceodifier, usually selected from a group of organic substances

aving one or more alkyl radicals containing at least six car-on atoms (usually quaternary ammonium salts, or quats). Theole of this organic component in organosilicates is to lower theurface energy of the inorganic host and improve the wettingharacteristic with the polymer. Intuitively, the nature and thetructure of these “surfactants” determines the hydrophobicityf silicate layers and, hence, the extent of their exfoliation.

Indeed, the driving force for nanodispersion – that is the neg-tive change in free energy – depends on the delicate balanceetween the enthalpic term, due to the intermolecular interac-ions, and the entropic term, associated with the configurationalhanges in the constituents. Using a recently developed mean-eld model [5], it has been shown that the entropy loss associatedith the polymer confinement is approximately compensatedy an entropy gain associated with the increased conforma-ional freedom of the surfactant molecules as the gallery distancencreases due to the polymer intercalation. Therefore, enthalpyetermines whether or not polymer intercalation will takelace.

Finally, the properties of the hybrid commonly depend onhe structure of the material; thus, it is of particular interesto establish the morphology of the final composite. This cane defined exfoliated (only single platelets can be found inhe final bulk structure) or intercalated (where polymer pen-trates the galleries, but stacks remain clearly recognizable).owever, in the vast majority of cases, the final morphology

esults in a mixture of the two different situations describedbove.

Many papers have been published in recent years address-ng all possible issues of nanocomposites design, production,

nd characterization. After the pioneering work carried out athe R&D center of Toyota [6–9], experimental investigationsave focused especially on processing conditions and chemicalormulation of components, and their relative influence on mor-

Aowm

ilibria 261 (2007) 366–374 367

hology and final properties. Every possible aspect has beennalyzed, from matrix molecular weight to surfactant chem-stry [4]. Nonetheless, mechanisms that lead to clay intercalationr exfoliation within polymer matrix have not yet been fullynderstood, due to the amount of parameters involved and theifficulty of developing tools capable of monitoring intercala-ion processes in the melt. This is mainly due to the fact thatundamental variables defining final morphology and proper-ies of polymer nanocomposites need to be quantified at lengthcales that are difficult to probe experimentally. Accordingly,any efforts have been made also in the field of nanocompositesodeling and simulation [10]. In particular, multiscale simula-

ion techniques seem to be ideally suited to investigate mainicroscale and mesoscale characteristics of such materials, as

ong as the smaller parts of the systems must be solved with greatccuracy, but at the same time higher scales features needs to beodeled with lower level of detail in order to avoid excessively

igh computational times.Our recent efforts in polymer-clay nanocomposites simula-

ion were initially concerned with binding energy evaluations forell-characterized systems [10(c–e)], using atomistic molec-lar dynamics (MD) methods. At the same time, part ofur activities was focused on the development and applica-ion of mesoscale simulation (MS) tools to polymer blendsorphology investigations, and on the integration of these

ools with atomistic approaches [11–14]. In this work, weropose a global, innovative, multiscale molecular modelingM3) approach which allows us to calculate the macroscopicroperties of a given PCN material starting only from thehemical information on the nature of matrix and fillers. Theurther element of novelty of the M3 approach presentedere is that the concept of partial exfoliation and plateletrientation can be implemented in the corresponding modelorphology, and intercalated stack properties can be cal-

ulated from mesoscale (MS) simulations. Polyamide-basedanocomposite such as MMT/Nylon6 was selected as the idealeference system, by virtue of the numerous experimentaltudies available for this PCN in the scientific literature. Pre-isely, the following two systems were selected to test our

3 approach: MMT/Nylon6/trimethyl-dehydrogenated tallowuaternary ammonium chloride and MMT/Nylon6/dimethyl-ehydrogenated tallow quaternary ammonium chloride, respec-ively.

As far as simulation methods are concerned, we established aierarchical procedure [11] for bridging the gap between atom-stic and mesoscopic simulation for PCN design. The Dissipativearticle Dynamics (DPD) [15,16] is adopted as the mesoscopicimulation technique, and the interaction parameters of theesoscopic model are estimated by mapping the corresponding

inding energy values obtained from atomistic MD simulations.inally, we employ a finite-element (FE) software Palmyra [17]

n order to estimate some of the macroscopic properties foroth the intercalated stack and the exfoliated nanocomposite.

s some of the low-middle length scale computational recipesf M3 are reported in our previous work [10(c–e),11], here weill focus on the detailed description of the last part of ourultiscale procedure, i.e., from MS to FE calculations.

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68 G. Scocchi et al. / Fluid Phas

. Theory and calculations

Briefly, our modeling recipe is based on (i) quan-um/molecular mechanics-based atomistic simulation to deriventeraction energy values among all system components; (ii)

apping of these values onto mesoscale parameters and sub-equent simulations to determine system density distributions;iii) simulations at finite-element levels to calculate the relativeacroscopic properties.

.1. Quantum/molecular mechanics-based atomisticimulation

Atomistic simulations of the quat and polymer conforma-ions on MMT layers and binding energy calculations haveeen performed following a well-established route [10]. Accord-ngly, the procedure will only be briefly reviewed here. Asar as the MMT model is concerned, starting from relevantrystallographic coordinates [18] we built the unit cell of ayrophyllite crystal using the Crystal Builder module of theaterials Studio molecular modeling package (v. 4.1, Accelrys,

an Diego, CA, USA). The MMT model is characterized by aation exchange capacity (CEC) of approximately 92 meq/100 groughly corresponding to the CEC of the most commonlymployed commercial products), and the corresponding unitormula is (Al3.33Mg0.67)Si8O20(OH)4–Na0.67). The charges onhe individual atoms were placed following the charge schemeroposed by Cygan et al. [19], and this finally yielded the over-ll neutral MMT cell model that was used in all subsequentimulations.

Then we modeled the quat molecules, which were choseno be dimethyl- and trimethyl-dehydrogenated tallow qua-ernary ammonium chloride, hereafter denoted as M2(C18)2nd M3C18, respectively. The quat conformational search wasarried out using the Compass force field (FF) [20,21] (asmplemented in the Discover module of Materials Studio),nd applying our well-validated combined molecular mechan-cs/molecular dynamics simulated annealing (MDSA) protocol10(c–e),22].

The procedure for polymer modeling consisted in build-ng and optimizing the polymer constitutive repeating unitCRU) using the Compass FF, which was then polymerizedo a conventional degree of polymerization (DP) equal to0. Then, 3 chains were used to build 10 different modelonformations, using the Amorphous Builder module of Mate-ials Studio. Explicit hydrogens were used in all modelystems. Each polymeric structure was then relaxed and sub-ected to the same MDSA protocol applied to model the quat

olecule. Each chain structure was assigned atomic charges23] using the charge scheme proposed by Li and Goddard24].

Subsequently, we employed MD simulations in the con-tant volume–constant temperature (NVT) ensemble in order to

etrieve information on the interaction and binding energy val-es between the different components of our PCN systems. Thisrocedure has been extensively described in previous papers10(c),11] and will not be reported here.

gi(l

ilibria 261 (2007) 366–374

.2. Mesoscale parameters mapping and simulation

In order to simulate the morphology of the organic speciesetween the montmorillonite layers at a mesoscopic level, wesed the DPD simulation tool as implemented in the Materi-ls Studio modeling package. In the framework of a multiscalepproach to PCN simulation, the interactions parameters neededs input for the mesoscale level DPD calculations have beenbtained by a mapping procedure of the binding energies val-es between different species obtained from simulations at thetomistic scale, as described in Section 2.1.

In the DPD method a set of particles moves according toewton’s equation of motion, and interacts dissipatively through

implified force laws; the force acting on the particles, which isairwise additive, can be decomposed into three elements: aonservative (FC

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

ij) force.ccordingly, the effective force fi acting on a particle i is giveny:

i =∑i�=j

(FCij + FD

ij + FRij

)(1)

here the sum extends over all particles within a given distancec from the ith particle. This distance practically constitutes thenly length scale in the entire system. Therefore, it is conveniento set the cutoff radius rc as a unit of length (i.e., rc = 1), so thatll lengths are measured relative to the particles radius. Theonservative force is a soft repulsion, given by:

Cij =

{aij(1 − rij)rij, (rij < 1)

0, (rij ≥ 1)(2)

here aij is the maximum repulsion between particles i and j,ij the magnitude of the particle–particle vector rij = ri − rj (i.e.,ij = |rij|), and rij = rij/rij is the unit vector joining particles ind j. The other two forces, FD

ij and FRij , are both responsible

or the conservation of the total momentum in the system, andncorporate the Brownian motion into the larger length scale.n the DPD model, individual atoms or molecules are not rep-esented directly by the particle, but they are coarse-grainednto beads. These beads represent local “fluid packages” able to

ove independently. Incorporation of chain molecules simplyequires the addition of a harmonic spring force between theeads. More detailed information about the DPD method can beound elsewhere [15,16,25].

Defining aij parameters and bead dimensions are probably theost questionable tasks in setting a DPD simulation. Accord-

ng to our procedure, repulsive DPD parameters are coupled tohe energy values resulting from the atomistic MD simulationshrough a simple combinatorial approach. The technique usedo define the beads and the method to calculate aij parametersave already been reported in Ref. [11]; accordingly, it will benly briefly summarized below.

The first step necessary to obtain the DPD input parameters

enerally consists in defining the DPD bead dimensions, thusmplicitly determining the characteristic length of the systemrc). As illustrated above, the interaction range rc sets the basicength-scale of the system; in other terms, rc can be defined as

G. Scocchi et al. / Fluid Phase Equilibria 261 (2007) 366–374 369

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tT

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E

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i

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E

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Table 1Self- and mixed DPD energies rescaled from MD simulations for the two PCNsystems considered

Eij P T H M

MMT/Nylon6/M2(C18)2

P −4.85 −0.24 −0.49 −0.021T −0.24 −0.026 −0.62 −0.87H −0.49 −0.62 333.1 −39.06M −0.021 −0.87 −39.06 0.00

MMT/Nylon6/M3C18

P −4.82 −0.30 −0.60 −0.02T −0.30 −0.57 2.91 −1.40

otacotaDtdetrcbtcinlaSMpobtained from the scaling procedure starting from atomistic MDsimulation energies are listed in Table 2. Finally, we made use ofthe option available in the DPD commercial software of includ-ing a smooth wall in the simulation box perpendicular to the

Table 2DPD interaction parameters for the two PCN systems considered

aij P T H M W

MMT/Nylon6/M2(C18)2

P 25 31.5 31.1 31.8 318T 31.5 31.8 30.9 30.6 306H 31.1 30.9 500 5 50M 31.8 30.6 5 15 5

MMT/Nylon6/M3C18

ig. 1. Schematic drawing of the surfactant mapping from atomistic represen-ation to DPD beads.

he side of a cube containing an average number of ρ beads.herefore:

c = (ρVb)1/3 (3)

here Vb is the volume of a DPD bead. Thus, even in a het-rogeneous system consisting of several different species, suchs PCNs, a basic DPD assumption is that all bead-types (eachepresenting a single species) are of the same volume Vb. Sincehe quat molecules basically consist of a strongly polar head andwo almost apolar tails, we assumed that they could be describedy two different kind of beads. Further, in the light of the aboveentioned assumption, we decided divide each single alkylam-onium molecule in five or three beads, one for the head and

wo for each tail, as illustrated in Fig. 1 for M3C18.In order to compute the aij parameters, we applied the fol-

owing combinatorial approach. Considering a system made upf single particles i and j, the total energy of the system isiven, in the hypothesis of neglecting the ternary contributionso interaction, by Eq. (4):

totsystem = niiEii + njjEjj + nijEij + njiEji (4)

here:

ii = nii(nii − 1)

2(5)

s the number of contacts between ni particle of type i, and

ij = ninj

2(6)

s the number of contacts between ni particles of type i and nj

articles of type j. Since the mixed energy terms Eij and Eji, andhe number of contacts nij and nji are the same, the expressionor the system total energy, i.e., Eq. (4), becomes:

totsys = niiEii + njjEjj + 2nijEij (7)

he values of self-interaction energies Eii and Ejj are eas-ly obtainable dividing the corresponding values calculatedhrough classical MD simulations by the appropriated numberf contacts, whilst the value of the system total energy Etot

sys iserived straightforward from the MD simulation. Accordingly,he remaining mixed energy term, Eij, is calculated by applyingq. (7). Table 1 reports all the values of the self- and mixed

escaled energies calculated in this work.

The bead–bead interaction parameter for polymer–polymer

nteraction was set equal to aPP = 25 for both systems, in agree-ent with the correct value for a density value ρ = 3 [25]. For

he nanocomposite characterized by M3C18 as quat, the value

H −0.60 2.91 279.3 −34.44M −0.02 −1.40 −34.44 0.00

f the head–head interaction (aHH = 500) was set with respecto polymer–polymer repulsion to describe both qualitativelynd quantitatively the effect of repulsion between two neatlyharged heads. This second value was set based on our previ-us experience [11], according to which such parameter provedo be effective to mimic a strong electrostatic repulsive inter-ction. Having fixed these two parameters, all the remainingPD interaction parameters aij could be easily derived from

he corresponding scaling law. This means that, once, we haveefined these two values, we can calculate the energy differ-nce matching a single unit difference in the aij parameters. Inhis way, we can determine a linear correspondence between theescaled energies and the corresponding DPD parameters. In thease of MMT/MMT interaction, since no energy values coulde extracted from the MD simulations, we set aMM = 15 in ordero mimic a rigid covalent structure. Finally, for the pair M/H theorresponding parameter aij was not calculated using the scal-ng law, but we set it to the value of 5, because of its strongegative value of the rescaled energy. Had we used the scalingaw described above, the parameter would have been negative,nd as such not utilizable as an input parameter in Materialstudio DPD simulation tool. For the nanocomposites based on

2(C18)2, we used the same parameters reported in our previousaper [11]. The final set of DPD parameters for both systems,

P 25 32.5 32 33 330T 32.5 32 37.8 30.7 307H 32 37.8 496.3 5 50M 33 30.7 5 15 0

370 G. Scocchi et al. / Fluid Phase Equilibria 261 (2007) 366–374

Table 3Palmyra solver values for stack properties for the two PCN systems considered

Lpa (nm) Hp

b (nm) Vpc (nm3) Hg

d (nm) #p/se Hsf (nm) Vs

g (nm3) ARsh #si

MMT/Nylon6/M2(C18)2

60 1 11,304 4 6 30 3,39,120 4 160 1 11,304 4 4 20 2,260,80 6 260 1 11,304 4 2 10 1,130,40 12 560 1 11,304 4 1 1 1,130,40 120 12

MMT/Nylon6/M3C18

60 1 11,304 4 4 20 2,260,80 6 260 1 11,304 4 2 10 113,040 12 860 1 11,304 4 1 1 113,040 120 12

a Lp: Platelet length.b Hp: Platelet thickness.c Vp: Platelet volume.d Hg: Gallery height.e #p/s: Number of platelets per stack.f Hs: Stack thickness.g

zbbsbaabsbitttc

2

gw(o[tnrYm

fPDict

edmtbdcstp

oscwhere resistors can be connected in series or in parallel. Thereare no clear rules available yet about which method should beused for multi-component mixtures. After several trials for dif-ferent morphologies and relative comparison to experimental

Vs: Stack volume.h ARs: Aspect ratio of stack.i #s: Number of stacks.

-axis at the origin. Accordingly, the wall interacts with eachead in the system with a potential of the same form as theead–bead conservative force. This force is soft and short-range,o the beads are not strictly forbidden from passing through thearrier. We set a very soft repulsion between the MMT beadnd the wall, in order to fill the space between the two surfaces,nd thus represent the MMT platelet. As far as the remainingeads/wall repulsive parameters are concerned, we decided toimply scale up the interaction parameters with the MMT beady a factor of 10. In this way, we preserved the proportion ofnteraction energies between clay and organic species and, athe same time, we prevented the beads from crossing the wall,aking an effective flat solid surface. The values for the interac-ion with the wall used in the simulations are reported the lastolumn of Table 2.

.3. Finite-elements calculation of macroscopic properties

In order to calculate macroscopic properties of the investi-ated nanocomposite systems we performed FEM simulationsith fixed and variable grid using the software Palmyra software

v. 2.5, MatSim, Zurich, CH). This software has been validatedn different composite material morphologies by several authors12,26,27], yielding reliable results. In this work, we appliedhe software solver in order to analyze both platelet stack andanocomposite overall properties, using fixed and variable grid,espectively. In particular, we decided to focus our attention onoung modulus and permeability, since in these properties liesost of the industrial interest towards these new materials.As far as platelet-stack modeling is concerned, we success-

ully imported the density fields obtained with DPD MS intoalmyra after an appropriate formatting of Materials Studio

PD output files. Once imported, the density fields are divided

nto mesh elements. The mesh is represented as a fixed regularubic lattice, but the solver works on the basis of a space-fillingetrahedral mesh, i.e., without voids. The mapping of cubic grid

Fn

lements to a tetrahedral mesh results in six isochoric tetrahe-rons for each cubic grid element [17]. For arbitrary compositeorphologies, Palmyra uses a numerical method to determine

he overall properties from the properties of the componentsased on small homogeneous grid elements. The morphology isefined by a number of phases in a periodically continued baseell of cubic or orthorhombic shape where the phases may con-ist of any material. For each of the grid elements it is possibleo specify the fraction of each phase contained in that particularosition.

Two methods can be chosen for calculating the propertiesf each grid element that is a combination of different phases:erial (arithmetic) averaging or parallel averaging. These namesome from the meaning of the method in electronic circuits,

ig. 2. Finite-element meshed volume from DPD simulation of M3C18-basedanocomposite.

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G. Scocchi et al. / Fluid Phas

ata, we decided to use the serial averaging system in ordero calculate mechanical and permeation properties of our stack

odel. The meshed volume from DPD simulation of M3C18-ased nanocomposite can be seen in Fig. 2.

The properties of the quat molecules where chosen to matchhose of MMT and Nylon6 for the cation heads and tails, respec-ively. Mechanical and permeation properties for the silicate andolymer were taken from the available literature [28–30]. OurPD box resulted in a 100 × 100 × 30 Palmyra volume ele-ent, containing 300,000 nodes and 1,800,000 tetrahedrons (seeig. 2). After properties are assigned to each grid element, stackroperties are calculated using the method described below forhe variable grid mesh. The corresponding solver results for stackroperties are reported in Table 3.

Subsequently, we retrieve overall nanocomposite propertiesy defining a morphology similar to the corresponding TEMmages [28,29], applying a variable grid mesh to our model,nd finally run Palmyra solver. The solver method is the samemployed in fixed grid calculations. For the mechanical prop-rties, it applies six different infinitesimally small deformationso the composite and minimizes the total strain energy for eachf these deformations in order to calculate the elastic compositeroperties. Since composite materials are usually anisotropic,he symmetry class of the tensor is triclinic; if, however, the

aterial has some symmetry, the corresponding tensor specifi-ations may be simplified. The total elastic energy FE, which is auadratic function of the displacements of the nodes, is defineds the sum over the element contributions:

=nTetr∑t=1

ft = 1

2

nTetr∑t=1

VtCikeiek (8)

here ft is the elastic energy of tetrahedron t, V its volume, Cikhe elastic constants, and ei is the local strains. In general, thei are made up of mechanical and thermal parts. The units ofhe energy F are [L3 GPa], where L is the length unit of theimulation cell.

The local stresses are calculated from the nodal displace-ents at the end of each minimization run, assuming linear

lastic behavior. The local hydrostatic pressures P are calculatedor each tetrahedron as the sum of the principal components ofhe stress tensor of that volume element:

= 13 (σXX + σYY + σZZ) (9)

he deviatoric von Mises stress for a mesh element is defineds:

=√

[(σxx − σyy)2 + (σyy − σzz)2 + (σxx − σyy)2]

2+ 6(σ2

xx +

his value is listed in the output file for every applied defor-ation; the units are the same – [GPa] – as for the elastic

onstants. On the other hand, to calculate permeability prop-rties, a Laplace solver is used, that applies a field in the three

ain directions to the finite-element mesh and minimizes the

nergy of the composite [26].In order to build our nanocomposite model for FEM simu-

ation, we started defining an appropriate number of platelets

Tt20

ilibria 261 (2007) 366–374 371

+ σ2zz)

2 (10)

o be contained in our simulation boxes. Relying on previoustudies on determination of relevant volume element for similarystems [17,26] for both nanocomposites models we decidedo build boxes containing 36 MMT platelets, characterized by ahickness of 1 nm and a diameter of 120 nm, respectively. Twelvef these were considered to be exfoliated, while the other 24ere considered to be part of stacks of different sizes, accord-

ng to the data reported in Table 3. The simulation cubic cellas characterized by a length of 0.3135 �m (and a volume of.0308 �m3) for the M2(C18)2-based system, and by a length of.3052 �m (and a volume of 0.02842 �m3) for the M3C18-basedCN. These values stem from the choice of trying to reproduce

he same nanocomposites described in Refs. [28,29] in order toest the simulation results against experimental data. The vol-me fractions for the platelets were calculated considering theensity values reported in Ref. [29] and, for stacks, an inter-amellar gallery height of 4 nm. These considerations yielded a

MT volume fraction of 0.01863.Exfoliated platelets were taken to exhibit the mechanical

roperties reported in Ref. [29], while stacks were assigned val-es according the procedure described above, i.e., starting fromhe corresponding DPD density profiles. Moreover, we decidedo define an orientation for stack and single particles. Each ori-ntation can be described by the three Cartesian componentsp1, p2, p3} of a unit vector p pointing along the particle axis.he components of the orientation vector p can be expressed by

he angles ϕ and θ in the following way:

p1 = sinθ cosϕ

p2 = sinθ sinϕ

p3 = cosθ

(11)

nly two vector components are independent since the lengthf p is normalized to 1. Orientation tensors can be defined byorming dyadic products for each vector p that characterizes therientation of an object and subsequently integrating the productf the resulting tensors with the distribution function over allossible directions of p. The second order orientation aij tensoran be calculated as following:

ij =∮ρjρjψ(ρ) dρ (12)

or any orientation state we can define a coordinate systemhere all off-diagonal components of the orientation tensorecome zero. In this case, the diagonal components correspond

o the eigenvalues and their sum equals 1 because we deal withormalized orientation vectors p. In Palmyra, the adjustment ofhe orientation tensor is done by predefining the targeted secondrder orientation tensor, entering two eigenvalues of the tensor.

he third eigenvalue is determined by the fact that the sum of

he eigenvalues must be 1. In our study, we set eigenvalues 1 and(corresponding to the x and y direction, respectively) equal to.06. Accordingly, eigenvalue 3 (relative to z direction) was 0.88.

372 G. Scocchi et al. / Fluid Phase Equilibria 261 (2007) 366–374

Fig. 3. Frame extracted from the equilibrated MD trajectory of the PCN system made up by a MMT platelet (polyhedron style), Nylon6 (stick style), and M3C18

quat molecules (CPK style).

Fig. 4. Morphology of the PCN system made up by a MMT platelet, Nylon6, and M3C18 quat molecules obtained via DPD simulation. (Top) Ternary system;(bottom) detail of MMT and quat species. Color code: orange, MMT; blue, surfactant heads; light green, surfactant tails; dark green, polymer. (For interpretation ofthe references to color in this figure legend, the reader is referred to the web version of the article.)

G. Scocchi et al. / Fluid Phase Equilibria 261 (2007) 366–374 373

Table 4Values of the Young modulus (E) and O2 permeability (P) in the x, y, and z directions for the DPD models and the 36 particles boxes of both PCNs considered

Exx (GPa) Eyy (GPa) Ezz (GPa) Pxx (barrer) Pyy (barrer) Pzz (barrer)

DPD model for PCN with M2(C18)2

1.30 × 102 1.30 × 102 2.18 × 101 3.59×10−2 3.59 × 10−2 0.00

DPD model for PCN with M3C18

1.26 × 102 1.26 × 102 2.00 × 101 3.47 × 10−2 3.47 × 10−2 0.00

36 particle box for PCN with M2(C18)2

3.85 × 102 3.76 × 102 3.20 × 100 1.52 × 10−2 1.52 × 10−2 1.37 × 10−2

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6 particle box for PCN with M3C18

4.04 × 102 4.03 × 102 3.23 × 100

n this way, we created a morphology in which the platelets areriented parallel to the x–y plane, as if the nanocomposite werextruded perpendicular to the z direction. Finally, after havingreated the appropriate volume mesh, we run the Palmyra solver.

. Results and discussion

The M3 strategy to obtain some macroscopic, industriallyelevant properties of polymer-clay nanocomposites rely on thebtainment of interaction energy values among the system com-onents (i.e., polymer, clay and organic surface modifier oruat) from atomistic MD simulation. As an example, Fig. 3hows a frame of the equilibrated MD trajectory of the PCNade up of montmorillonite, Nylon6 and M3C18. As the sim-

lation proceeds in time, the surfactant chains flatten onto thelay surface. However, contrarily to what happens in the casef the “double-tailed” surfactant [11], in the case of the organicearing a singly hydrocarbonic chain some parts of the MMTurface remain available to the polymer. This constitutes theest balance between attractive and repulsive forces in the over-ll system, leading to a complete exfoliation of this PCN, aserified experimentally by Fornes and Paul [28].

The interaction energy values calculated for these systems,nd aptly scaled via the procedure described in the previousections (see Table 1), lead to the definition of the Dissipativearticle Dynamics input parameters listed in Table 2. According

o the M3 recipe, then, the corresponding DPD simulations areerformed, and the relevant MS morphologies of the systemsre obtained. Fig. 4 shows the DPD simulation results for theystem with M3C18 as quat.

As can be seen comparing Figs. 3 and 4, a good agreementetween the atomistic and the mesoscale predictions is found.fixed wall of nanoclay is obtained, by which the polar heads

f the surfactant molecules are attracted to, and onto which theong, apolar hydrocarbonic surfactant tail is flattened. As cane appreciated from Fig. 4, the charged parts of the surfactantemain close to the MMT layers, leaving the tails of the organicolecules to interact with the Nylon6 chains. However, areas

re available on the clay surface for entering in contact with

he polymer, thus rendering the effective binding between theomponents of the nanosystem almost ideal.

The final step in the M3 ansatz consists in setting up and per-orming the FE simulations based on the system morphologies

bTiu

1.52 × 10−2 1.54 × 10−2 1.34 × 10−2

btained at the previous, MS step. The results of FE calculationsf the mechanical and permeability properties for the DPD twolatelet-stack structure, and the 36 particles box for both PCNystems considered in this work are reported in Table 4.

The calculations of the Young modulus yielded results whichre comparable to experimental data if we consider the val-es along the x and y directions (i.e., parallel to the simulatedxtrusion direction). Eventual discrepancies between virtual andxperimental data could be due, for instance, to a different degreef crystallinity or different crystalline forms of Nylon6 in pres-nce of MMT nanofillers, as claimed by some authors [31,32].ll estimated values for the nanocomposite with M3C18 as quat

re higher than the corresponding ones relative to the PCN with2(C18)2, in harmony with the different degree of exfoliation

hat has been modeled.DPD models analyzed with Palmyra present very similar

esults in both cases. It is interesting to note that, within thetack, O2 permeability appears to be enhanced instead of reducedlong the x and y directions. A possible cause for this behavioran be traced to the polymer matrix structure in the proxim-ty of the MMT platelet. Indeed, previous studies [10(f, g),11]ave already pointed out the presence of alternating high andow polymer density planes next to platelet surfaces. Accord-ngly, in the lower density planes, the oxygen molecules cannd a preferential diffusion path. Finally, as far as permeability

n the 36 particle boxes is concerned, it is possible to note that,s expected, O2 diffusion is hindered only in the z direction,.e., perpendicular to platelet x–y plane. Permeability is anywayeduced only to a small extent (approximately 10–15%), becausef the low nanofillers content (less than 2% in volume).

. Conclusions

There are many levels at which molecular modeling andimulation can be useful, ranging from the highly detailed ab ini-io quantum mechanics, through classical, atomistic molecular

odeling, to process engineering modeling. These computationsan significantly contribute to reduce wasted experiments, allowroducts and processed to be optimized, and permit large num-

ers of candidate materials to be screened prior to production.hese techniques are currently used to obtain thermodynamic

nformation about a pure or mixed system. Properties obtainedsing these microscopic properties assume the system to be

3 e Equ

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74 G. Scocchi et al. / Fluid Phas

omogeneous in composition, structure and density, which islearly a limitation. When a system is complex, comprising sev-ral components, eventually sparingly miscible, PCNs beingrime examples, peculiar phases with remarkable propertiesan be observed. These so-called mesophases comprise far tooany atoms for atomistic modeling description. Hence, coarse-

rained methods are better suited to simulate such structures.ne of the primary techniques for MS modeling is Dissi-ative Particle Dynamics, a particle-based method that usesoft-spheres to represent groups of atoms, and incorporatesydrodynamic behavior via a random noise, which is coupledo a pairwise dissipation. However, retrieving information on

esophase structures is not enough for predicting macroscopiceatures of such materials. This is possible if mesophase mod-ling is coupled with appropriate FE tools like Palmyra thatprovided properties of pure components are given or can be

n turn obtained by simulation – allows obtaining a realisticstimation of many nanocomposites features, if integrated withxperimental morphological data.

In this work we presented a hierarchical procedure for obtain-ng the input parameter necessary to perform polymer-clayanocomposite DPD simulations from molecular dynamics sim-lations, thus bridging the gap between mesoscopic (i.e., higherength scale) and atomistic (lower length scale) modeling. Sub-equently, we use a FE approach to calculate mechanical andermeability properties of both mesoscopic structures (stacks)nd overall nanocomposite. Certainly, the presented approachan be improved in many ways: first of all, in order to realize arue multiscale modeling approach, platelet spatial distributionhould be calculated and not defined starting from experimen-al data. Moreover, different morphologies should be analyzed

ore precisely, and size distribution for platelets should be takennto account, as reported by other authors [29]. Finally, matrix

orphology next to the single exfoliated platelet surface shoulde considered, and exfoliation rate should be investigated moreeeply. All of these issues will be addressed in future works byur group.

eferences

[1] P.C. LeBaron, Z. Wang, T.J. Pinnavaia, J. Appl. Clay Sci. 115 (1999) 11–29.[2] M. Alexandre, P. Dubois, Mater. Sci. Eng. 28 (2000) 1–63.[3] D. Schmidt, D. Shah, E.P. Giannelis, Curr. Opin. Solid State Mater. Sci. 6

(2000) 205–212.[4] T.J. Pinnavaia, G.W. Beall (Eds.), Polymer-Clay Nanocomposites, John

Wiley & Sons, New York, 2000.[5] E.P. Giannelis, Appl. Organomet. Chem. 12 (1998) 675–680.[6] Y. Fukushima, A. Okada, M. Kawasumi, T. Kurauchi, O. Kamigaito, Clay

Miner. 23 (1988) 27–34.[7] A. Usuki, M. Kawasumi, Y. Kojima, A. Okada, T. Kurauchi, O. Kamigaito,

J. Mater. Res. 8 (1988) 1174–1178.[8] Y. Kojima, A. Usuki, M. Kawasumi, A. Okada, T. Kurauchi, O. Kamigaito,

J. Polym. Sci. Part A: Polym. Chem. 31 (1993) 1755–1758.[9] Y. Kojima, A. Usuki, M. Kawasumi, A. Okada, Y. Fukushima, T. Kurauchi,

O. Kamigaito, J. Mater. Res. 6 (1993) 1185–1189.

[

[[

ilibria 261 (2007) 366–374

10] See for instance;(a) A.C. Balazs, Curr. Opin. Solid State Mater. Sci. 7 (2003) 27–33;(b) G. Tanaka, L.A. Goettler, Polymer 43 (2002) 541–553;(c) M. Fermeglia, M. Ferrone, S. Pricl, Fluid Phase Equilib. 212 (2003)315–329;(d) M. Fermeglia, M. Ferrone, S. Pricl, Mol. Simul. 30 (2004) 289–300;(e) R. Toth, A. Coslanich, M. Ferrone, M. Fermeglia, S. Pricl, S. Miertus,E. Chiellini, Polymer 45 (2004) 8075–8083;(f) F. Gardebien, A. Gaudel-Siri, J. Bredas, R. Lazzaroni, J. Phys. Chem.B 108 (2004) 10678–10686;(g) F. Gardebien, J. Bredas, R. Lazzaroni, J. Phys. Chem. B 109 (2005)12287–12296;(h) D. Sikdar, D.R. Katti, K.S. Katti, Langmuir 22 (2006) 7738–7747;(i) D. Sikdar, D.R. Katti, K.S. Katti, R. Bhowmick, Polymer 47 (2006)5196–5205.

11] G. Scocchi, P. Posocco, M. Fermeglia, S. Pricl, J. Phys. Chem. B 111 (2007)2143–2151.

12] P. Cosoli, G. Scocchi, S. Pricl, M. Fermeglia, Microporous MesoporousMater., 2007, doi:10.1016/j.micromeso.2007.03.017.

13] M. Fermeglia, S. Pricl, Prog. Org. Coat. 5 (2007) 187–199.14] M. Fermeglia, P. Cosoli, M. Ferrone, S. Piccarolo, G. Mensitieri, S. Pricl,

Polymer 47 (2006) 5979–5989.15] P.J. Hoogerbrugge, J.M.V.A. Koelman, Europhys. Lett. 18 (1992) 155–160.16] J.M.V.A. Koelman, P.J. Hoogerbrugge, Europhys. Lett. 21 (1993) 363–368.17] A.A. Gusev, Macromolecules 34 (2001) 3081–3093.18] S.I. Tsipursky, V.A. Drite, Clay Miner. 19 (1984) 177–193.19] R.T. Cygan, J.J. Liang, A.G. Kalinichev, J. Phys. Chem. B 108 (2004) 1255.20] H. Sun, J. Phys. Chem. B 102 (1998) 7338–7364.21] S.W. Sun, H. Sun, J. Phys. Chem. B 104 (2000) 2477–2489.22] See for instance;

(a) M. Fermeglia, M. Ferrone, S. Pricl, Bioorg. Med. Chem. 10 (2002)2471–2478;(b) F. Felluga, G. Pitacco, E. Valentin, A. Coslanich, M. Fermeglia, M.Ferrone, S. Pricl, Tetrahedron: Asymmetry 14 (2003) 3385–3399;(c) S. Pricl, S. Vertuani, L. Buzzoni, A. Coslanich, M. Fermeglia, M. Fer-rone, M.S. Paneni, P. La Colla, G. Sanna, A. Cadeddu, M. Mura, R. Loddo,in: R.F. Schinazi, E. Schiff (Eds.), Framing the Knowledge of Viral Hep-atitis, IHL Press, Arlington, MA, USA, 2005, pp. 279–318;(d) M.G. Mamolo, D. Zampieri, L. Vio, M. Fermeglia, M. Ferrone, S. Pricl,G. Scialino, E. Banfi, Bioorg. Med. Chem. 13 (2005) 3797–3809;(e) A. Carta, G. Loriga, S. Piras, G. Paglietti, P. La Colla, G. Collu, M.A.Piano, R. Loddo, M. Ferrone, M. Fermeglia, S. Pricl, Med. Chem. 2 (2006)577–589;(f) G. Scialino, E. Banfi, D. Zampieri, M.G. Mamolo, L. Vio, M. Ferrone,M. Fermeglia, M.S. Paneni, S. Pricl, J. Antimicrob. Chemother. 58 (2006)76–84;(g) P. Posocco, M. Ferrone, M. Fermeglia, S. Pricl, Macromolecules 40(2007) 2257–2266 (and references therein).

23] B.J. Delley, J. Chem. Phys. 92 (1990) 508–517.24] Y. Li, W.A. Goddard III, Macromolecules 35 (2002) 8440–8455.25] R.D. Groot, P.B. Warren, J. Chem. Phys. 107 (1997) 4423–4435.26] M.A. Osman, V. Mittal, H.R. Lusti, Macromol. Rapid Commun. 25 (2004)

1145–1149.27] M. Heggli, T. Etter, P. Wyss, P.J. Uggowitzer, A.A. Gusev, Adv. Eng. Mater.

7 (2005) 225–229.28] T.D. Fornes, D.L. Hunter, D.R. Paul, Macromolecules 37 (2004)

1793–1798.29] T.D. Fornes, D.R. Paul, Polymer 44 (2003) 4993–5013.

30] G.W. Ehrenstein, Polymeric Materials: Structure–Properties–Applications,

Hanser Gardner, 2001.31] T.D. Fornes, D.R. Paul, Polymer 44 (2003) 3945–3961.32] A. Ranade, N.A. D’Souza, B. Gnade, A. Dharia, J. Plast. Film. Sheeting

19 (2003) 271–285.