Supporting Information

Mechanism of exfoliation and prediction of materials

properties of clay-polymer nanocomposites from

multiscale modeling

James L. Suter Derek GroenPeter V. Coveney∗

Centre for Computational Science, University College London20 Gordon Street, WC1H 0AJ, London, United Kingdom

∗E-mail: p.v.coveney@ucl.ac.uk

Contents

1 Introduction 1

2 Atomistic to coarse-grained particle mapping 2

3 Non-bonded potentials 43.1 Iterative Boltzmann Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 TMA in PEG polymer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 C2 in PEG polymer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Claybasal-TMA potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.5 Claybasal-C2 potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.6 Claycharge-NaCG potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.7 TMA-TMA, C2-C2 and TMA-C2 potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Bonded potentials 9

5 Validation 11

6 Clay structures used in this study 13

7 Density profiles 20

8 FabMD Software 20

1 Introduction

In this Supporting Information (SI), we report in detail the methodology we have employed in constructing ourmultiscale organoclay-polymer simulation system. Our aim is to retain chemical specificity while extending

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the time and spatial scales; it is therefore of great importance that the derivation of the interaction parametersis carefully considered. In our previous study, we described the procedure to generate coarse-grained (CG)interaction potentials for pristine clays and polymers [1]. Here we describe the techniques we have used togenerate the additional CG potentials required for the surfactant molecules interacting with polymer and claysurfaces. We list the simulations we have performed, both at the atomistic (all atom, AA) and CG level, tocreate all the parameters required to describe these complex systems. We then show how we validate ourmodels by considering their swelling characteristics. In Section 6, we report how we generated the initial stateof the models simulated in the main article and in Section 7 we describe our analysis of the interlayer structureduring the simulations. Finally, in Section 8, we describe our software toolkit used to manage the simulations.

We use a systematic (“bottom-up”) approach to derive effective pair potentials for a simplified CG systemwhere groups of atoms are represented by a single interaction site. To do so, we firstly defined a mappingscheme to represent the transformation from a small number of atoms (in this study, less than ten) to a singleinteraction site. In section 2, we describe the mapping implemented for the surfactants, polymer and claysystems in this study.

We assume that the total potential energy of the CG system can be separated into bonded and non-bondedinteraction energies; in section 4, we optimize the bonded parameters to match those of the atomistic targetsystem. For the non-bonded interactions (section 3), we have used a combination of techniques to constructthe CG effective pair potentials (which we refer to as interaction potentials), all of which are constructed toreproduce selected properties of fine-grained (atomistic) simulations. These include matching to potentials ofmean force and radial distribution functions.

2 Atomistic to coarse-grained particle mapping

In this section we describe the mapping operator that transforms selected groups of atoms into CG parti-cles. In the main article, we perform CG molecular dynamics of surfactant - treated clay interacting withpoly(ethylene) glycol (PEG) polymer molecules with a length of 100 monomer units. All CG potentials forpolymer-polymer and polymer-clay interactions are described in our previous study with pristine clays andhydrophilic polymers [1]. The mapping of atomic coordinates for PEG molecules and clay systems to their CGequivalents is shown in Figure 1a, reproduced from Suter et al. [1]. The blue circles enclose the atoms thatconstitute the terminal PEG polymer CG particles (PEGterminal), while the green ellipses enclose the PEGmonomer units of the polymer (PEGmonomer).

To compute the interaction potentials of surfactant molecules with polymers, we have used relatively short chainPEG polymers of 3 monomer and 2 terminal CG units, as shown in Figure 1. The short length of these polymermolecules and high temperature (500K) ensures that we can reach equilibrium within a few nanoseconds ofsimulation which can subsequently be used to generate CG interaction potentials. The temperature of 500Kis also chosen as it is at the higher-range of conditions used experimentally in melt processing of clay-polymercomposites. Through the remainder of the SI, all mention of PEG polymers used for CG parameterisationrefers to these short-chain polymer molecules shown in Figure 1. The longer polymers in the main article usethe parameterisation derived from these short-chain polymer molecules.

In this study, we considered two smectite aluminosilicate clays: the uncharged pyrophyllite Al2Si4O10(OH)2and the charged montmorillonite (where Al3+ ions are isomorphically substituted by Mg2+ ions, with Na+

counter ions). For simplicity, we have only considered montmorillonite clays with isomorphic substitution sites

2

(a) (b)

(c) (d)

Figure 1: The mapping scheme between the atomistic and coarse-grained representations for a) PEG molecules:the blue circles enclose the terminal units of the polymer (PEGterminal) while the green ellipses enclose themonomer units of the polymer (PEGmonomer), b) for the surfactant molecule, the tetra-methyl ammoniumgroup TMA is enclosed by the green circle, c) for the surfactant molecule, the C2 group of the alkyl chain isenclosed by the red circle (only the first unit in each chain is illustrated) and d) the various coarse-grainedclay particle types on a single clay sheet: claybasal (pink), claycharge (turquoise) and clayedge (yellow).

in the octahedral layer Al2−xMgxSi4O10(OH)2Nax. Each cation (Al3+ or Mg2+) in the octahedral layer of theclay is mapped to the centre of a CG particle (claybasal and claycharge respectively) For 5-coordinated Al3+

atoms on the edge of the clay, the position of the Al3+ ion is mapped to the CG unit clayedge. The clay CGtypes are shown in Figure 1.

The mapping of atomistic to CG surfactant molecules is shown in Figure 1. The CG surfactant molecules arecomposed of two CG units: 1.) the quaternary ammonium group (tetra-methyl ammonium, TMA), enclosedby the green circle in Figure 1b, and 2.) Two carbon atoms (and bound hydrogen atoms) in the alkyl backbonechain (which we have termed C2), enclosed by the red circle in Figure 1c (only the first two in each alkyl chainare shown).Therefore, in total, our CG simulations have 7 different CG types: 1. PEGterminal, 2. PEGmonomer, 3. claybasal,4. claycharge, 5. clayedge, 6. TMA and 7. C2.

In the schematic diagram shown in Figure 2, we illustrate how we construct the interaction potentials betweenall 7 CG units. Potentials between types 1 - 5, and associated non-bonded potentials, were identical to thoseused in our previous study and the reader is referred to Suter et al. [1] for these parameterisations. In this SI,

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Figure 2: A schematic diagram illustrating the workflow used to compute the large number of interactionpotentials required to simulate the coarse-grained montmorillonite platelet-polymer systems described in themain paper [1].

and in Figure 2, we describe the procedures for constructing the CG potentials for TMA and C2 interactingwith themselves and CG types 1-5. Note that these are short-range interactions and include all the van derWaals and electrostatic interactions “folded-in”; as such, there are no explicit charges on any CG species, whichare reflected in the species’s interaction potentials with other CG particles.

The simulations, both AA and CG, were performed at 500K. All atomistic simulations used the ClayFF [2]and CVFF [3] forcefields to describe the clay and the polymers respectively and were simulated at a pressureof 1 atmosphere (i.e. in the same thermodynamic state).

3 Non-bonded potentials

In the workflow shown in Figure 2, within each box we have indicated the potentials generated (and subse-quently used to produce the next set of potentials in the workflow) and the method employed. For many ofthe steps in the workflow, we used our software toolkit FabMD to handle the simulations. See Section 8 formore details.For the interaction potentials between TMA and PEG polymer molecules, and C2 and PEG polymer molecules(2nd column in Figure 2), we have used the Iterative Boltzmann Inversion (IBI) method 3.1. Subsequentinteraction potentials were generated either from the IBI method or by matching to potentials of mean force(PMF) calculated from constrained molecular dynamics simulations.To generate the interaction potentials for the surfactant CG units, we used a monomer approximation, i.e.,

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the potentials for TMA are calculated using a tetra-methyl ammonium molecule (CH3)3N+, and for C2 usingan ethane molecule C2H6 for potential of mean force (PMF) calculations and octane C8H18 for IBIs. Afterall CG interaction potentials have been constructed, an all-atom simulation of an organo-clay with completesurfactant molecules (as shown in Figure 1) - with the periodic boundaries on the clay sheets to simulate a clayinterlayer - is used to generate the bonded potentials of the surfactant molecules. This simulation is also usedas a check to ensure that the monomer approximation adequately reproduces the properties of the completesurfactant molecules when adsorbed on a clay surface. The details of this simulation are provided in section 4.All non-bonded interactions are constructed in tabular form, with an interval of 0.1 Å for potentials generatedusing IBI and an interval of 0.25 Å for all other potentials. Each potential has a cutoff of 19.75 Å.

3.1 Iterative Boltzmann Inversion

IBI is a structure-based coarse graining technique. We take an initial potential derived from atomistic sim-ulations, and apply it to a CG system [4]. We then run a CG simulation and extract the resulting radialdistribution functions (RDFs) from each type of interaction. We compare CG RDFs against target RDFs ob-tained from atomistic runs, and then update the CG potential U for each distance r according to the followingrule:

U(r, i+ 1) = U(r, i) + kBT ln

{g(r, i)

gAA(r)

}. (1)

Here U(r, i + 1) is the new CG potential, U(r, i) the potential used previously, kB the Boltzmann constant,T the temperature, g(r, i) the RDF obtained from the CG run and gAA(r) the atomistic target RDF. Our IBIimplementation does not use any ad hoc acceleration coefficient, as described for example in [5]. Once thepotential is updated, we run a new CG simulation, repeating this process until the RDFs have converged.To ensure rapid propagation of the method, we run each CG simulation long enough to ensure statisticalvalidity, but short enough to ensure rapid advancement of the algorithm. For the systems we study here, wechose to run the CG simulations for a minimum of 250,000 time steps.

3.2 TMA in PEG polymer

We calculated the interaction potentials between PEG polymer species and TMA ions by using the IBI tech-nique to match g(r) between these species. To construct the atomistic system, we inserted a TMA and achloride ion into an equilibrated bulk polymer system (as described in the SI of Suter et al. [1]). The chlorideion was initially placed at least 30 Å from the TMA ion and the forces it experienced during the simulationwere not integrated (leading to the ion remaining stationary). A harmonic potential of 4.7 kcal−1 was used totether the centre of mass of the TMA to the origin of the simulation cell, thereby allowing us to simulate theinteraction of TMA with the PEG polymer, without the counter-ion forming an ion pair.

Atomistic details: The bulk PEG system consists of 33728 atoms (1088 molecules of 31 atoms, each moleculecorresponding to 2 terminal CG units and 3 monomer CG units). One TMA and one chloride ion were added(total number atoms: 33746). The simulation cell was cubic and the system was kept at a constant pressure of1.0 atmosphere and a temperature of 500K via a Nosé-Hoover thermo/barostat. The total simulation time was10ns, corresponding to 10,000,000 timesteps with a timestep 1.0fs. The average x, y, z simulation supercelllattice parameters was 77.45 Å.The CG simulation consisted of 5441 CG units and each iteration in the IBI was simulated for 1ns, using a5fs timestep in the NV T ensemble.The radial distribution functions at the atomistic and CG levels are shown in Figure 3.

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(a) (b)

Figure 3: The radial distribution functions at the atomistic (black lines) and coarse-grained (red lines) levelsfor PEG - TMA: (a) PEGterminal and TMA; (b) PEGmonomer and TMA.

(a) (b)

Figure 4: The radial distribution functions at the atomistic (black lines) and coarse-grained levels (red lines)for PEG - TMA, for (a) PEGterminal and C2, and (b) PEG monomer and TMA.

3.3 C2 in PEG polymer

In the same manner as the TMA - PEG interaction potentials, we calculated the interaction potentials betweenPEG polymer species and C2 molecules by using the IBI technique to match g(r) between these species. Toconstruct the atomistic system, we inserted an octane molecule in an equilibrated bulk PEG polymer system.Atomistic details: The bulk PEG system consists of 33754 atoms (1088 molecules of PEG and one octanemolecule C8H18). The system was kept at a constant pressure of 1.0 atmosphere and a temperature of 500K viaa Nosé-Hoover thermo/barostat. The total simulation time was 7.5ns, corresponding to 7,500,000 timestepswith a timestep 1.0fs. The average lattice parameter of the cubic supercell was 77.54 Å.The CG simulation consisted of 5444 CG units and each iteration in the IBI was simulated for 1ns, using a5fs timestep in the NV T ensemble.The radial distribution functions at the atomistic and CG levels are shown in Figure 4.

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Figure 5: An illustration of the potential of mean force calculation of a TMA ion adsorbing onto a pyrophylliteclay sheet, used to compute the claybasal-TMA interaction potential. The black arrow is the reaction coordinatefor the PMF, which corresponds to the difference in z coordinate between the centre of mass of the clay sheetand the TMA ion.

3.4 Claybasal-TMA potentials

To calculate the claybasal-TMA interaction potential, we matched the atomistic and coarse-grained PMF for aTMA ion adsorbing on an uncharged (pyrophyllite) clay surface. In the atomistic system, a charge balancingchloride ion is also present. This chloride ion was initially placed at least 30 Å from the TMA ion and theforces it experienced during the simulation were not integrated (leading to the ion remaining stationary, awayfrom the clay sheets). In this case, the reaction coordinate for the PMF calculation is the difference in the zcoordinates from the clay sheet (calculated as the average z coordinate of the octahedral aluminum atoms)to the TMA ion. An illustration of this process is shown in Figure 5. The PMF calculation is achieved usingumbrella sampling, which requires an ensemble of simulations, each constrained to a value on the reactioncoordinate using a harmonic potential. The PMF can then be reconstructed using the weighted histogramanalysis method (WHAM) [6].Atomistic details: Initially, the TMA ion was inserted into the final snapshot of the atomistic simulation ofthe PEG polymer adsorbed onto the pyrophyllite surface (as described in the SI of Suter et al. [1]), with thechloride ion inserted as described as above. Each system contained 6754 atoms, with lattice dimensions 41.78Å × 36.09 × 57.67 Å. A harmonic constraint of 4.7 kcal mol−1 Å−2 was used to restrain the centre of mass ofthe TMA ion at each constraint value. As the restraint only acted on the difference in z coordinates, the TMAion was free to move in the xy plane. The TMA ion was manually moved to the exact constraint value at thebeginning of each ensemble run. The interval between constraint values was 0.25 Å, ranging from 20 Å to 2.5Å (70 constraint values). The lowest constraint values are within the van der Waals radius of the clay surfaceatoms. Each constraint run lasted for 3,000,000 timesteps with a timestep of 1.0fs (3ns) after initial energyminimization. The coarse-grained ensemble runs were performed for 2,500,000 timesteps at a 5fs timestep.The atomistic and coarse-grained PMFs for a TMA ion adsorbing on a pyrophyllite surface in PEG is shownin Figure 6.

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(a) (b)

Figure 6: The potential of mean force for adsorption of (a) TMA ion and (b) C2 onto a pyrophyllite claysurface at the atomistic level (solid line) and the coarse-grained level (red line).

3.5 Claybasal-C2 potentials

In a manner analagous to the simulations to parameterise the claybasal-TMA interaction potential, an ethanemolecule was inserted in the final snapshot of the atomistic simulation of the PEG polymer adsorbed onto thepyrophyllite surface. Each system contained 6744 atoms, with lattice dimensions 41.78 Å × 36.09 × 57.67 Å.A harmonic constraint of 4.7 kcal mol−1 Å−2 was used to restrain the centre of mass of the ethane moleculeat each constraint value. Each constraint run lasted for 3,000,000 timesteps with a timestep of 1.0fs (3ns)after initial energy minimization. The coarse-grained ensemble runs were performed for 2,500,000 timestepsat a 5fs timestep. The atomistic and coarse-grained PMFs for an ethane molecule adsorbing on a pyrophyllitesurface in PEG is shown in Figure 6. As ethane is a non-polar molecule, we have used the same potential forC2 interacting with a charge site (claycharge-C2), and for a clay edge (clayedge-C2).

3.6 Claycharge-NaCG potentials

To calculate the interaction parameters for the TMA ion interacting with a charge site in the clay sheet, wecomputed the PMF of a TMA ion adsorbing directly onto a charge site in the clay framework (a Mg2+ ionisomorphic substitution in the octahedral layer, i.e. a montmorillonite clay). The reaction coordinate is thedistance from the charge site (i.e Mg2+ ion / coarse-grained claycharge ion) and the centre of mass of the TMAion. The interval between constraint values was 0.25 Å, ranging from 20 Å to 2.5 Å (70 constraint values)with a harmonic constraint of 4.7 kcal mol−1 Å−2 used to constrain the position of TMA ion. IBI was usedto update the claycharge-TMA potential at each update. The atomistic and coarse-grained PMFs for TMAadsorbing on a montmorillonite surface in PEG are shown in Figure 7.

3.7 TMA-TMA, C2-C2 and TMA-C2 potentials

To calculate the interaction potentials between the surfactant CG atom types in the PEG polymer, we haveperformed PMF calculations where the reaction coordinate is the distance between two species. For the TMA-TMA interaction potential, we manually inserted two TMA ions and two chloride ions into the final snapshotof the bulk PEG system described above. The chloride ions were placed at least 25 Å away from the TMAions and their forces were not integrated during the molecular dynamics simulation to render them stationary.

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Figure 7: The potential of mean force for adsorption of a TMA ion onto a montmorillonite clay surface at theatomistic level (black line) and the coarse-grained level (red line).

For the TMA-C2 interaction, one TMA and one ethane molecule were inserted, along with a charge balancingchloride ion. In both cases, one of the TMA ions was tethered using a harmonic potential of 4.7 kcal mol−1

Å−2 to ensure the TMA ion did not drift towards the chloride ion. In each case, we performed an ensembleof 70 simulations, each with a harmonic potential of 4.7 kcal mol−1 Å−2 constraining the distance betweenthe TMA ions to a value ranging from 20 Å to 2.5 Å. Each ensemble simulation lasted 0.5ns. The resultingpotentials of mean force at the atomistic and coarse-grained levels are shown in Figure 8.To compute the C2-C2 interaction potential, we inserted ten octane molecules into an equilibrated systemof a small single pyrophyllite clay sheet immersed in a PEG polymer melt. The platelet had a diameter ofapproximately 40 Å. It was placed in a box of initial dimensions 60 × 60 × 60 Å3, lying on the xy plane. Theclay consisted of 1280 atoms. Short-chain PEG polymers (shown in Figure 1) were inserted into the simulationbox using the Monte Carlo growth method described in Suter et al. [1], up to a density of approximately 0.6 gml−1. In total, 327 PEG molecules (each consisting of 31 atoms) were inserted into their respective systems.After energy minimization, the atomistic simulations were performed for 2ns at 500K and 1 atmosphere.The final lattice dimensions are: 52.1 × 52.1 × 52.1 Å3. Ten octane molecules were added to this systemand equilibrated for 15ns at 500K and 1 atmosphere. IBI was used to match the C2-C2 radial distributionfunction: it should be noted that during the IBI procedure, the C2-C2 RDF was very sensitive to updates,which sometimes led to the octane molecules forming clusters. The resulting radial distribution functions areshown in Figure 9, along with a snapshot from the AA molecular dynamics simulation.

4 Bonded potentials

In the previous section, we have shown the parameterisation of all the non-bonded interactions of the surfactantmolecules. To calculate the bonded interactions (bond, angles and dihedrals), we performed an AA molecularsimulation of a montmorillonite clay interlayer with a separation of approximately 59 Å and sodium ionsreplaced by the complete surfactant molecules, dimethyldioctadecylammonium cations.Atomistic details The clay system consists of 2 interlayers, with 14191 atoms in total. Of the 256 octahedralaluminum atoms in the clay framework, 34 are substituted for Mg2+ ions (a ratio of one in 7.5). There aretherefore 34 surfactant molecules in the simulation (17 in each interlayer). 237 PEG molecules are present inthe system, divided between the two interlayer spacings. The average lattice dimensions are: 41.63 Å × 35.96Å × 117.42 Å. The average d-spacing is therefore 58.71 Å. The simulation was run for 22,000,000 timesteps

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(a) (b)

Figure 8: The potential of mean force as a function of the distance (a) between two TMA ions in PEG and(b) between TMA and C2 in PEG. The atomistic PMFs are black lines and the coarse-grained PMFs are redlines.

(a) (b)

Figure 9: (a) A snapshot from AA molecular dynamics simulation of an isolated pyrophyllite platelet, immersedin PEG polymer, with 10 octane molecules inserted. The octane molecule’s colors are: turquoise = C atoms,white = H atoms. For the clay sheet, yellow = Si atoms and pink = aluminum atoms. (b) The RDFs for theC2-C2 interaction at the AA level (black) and the CG level (red).

at 1fs per timestep (22ns), under NpT conditions. A snapshot from the simulation is shown in Figure 10.From the data generated we used Iterative Boltzmann Inversion to optimize the bonds, angles and dihedrals ofthe surfactant molecules. The bonded coarse-grained potentials for bonds and angular terms were calculatedas tabulated functions, with an interval of 0.1 Å. Spline functions are fitted to these potentials (and all thetabulated potentials described here) within the LAMMPS code. The dihedral potentials for the PEG systemwere fitted to the following function:

U(θ) =∑n=1,5

An cosn−1(θ) (2)

where An are five energy coefficients.

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(a) (b)

Figure 10: (a) a snapshot from AA molecular dynamics simulation of a montmorillonite clay with a separationbetween clay layers of approximately 59 Å. The clay contains 34 surfactant molecules and 237 PEG molecules(shown as transparent). The colors are: turquoise = C atoms, white = H atoms, red = oxygen atoms, blue =nitrogen atoms. For the clay sheet, yellow = Si atoms and pink = aluminum atoms. (b) A snapshot from theequivalent CG simulation, with the C2 CG types in red, TMA = blue, PEGterminal = white PEG monomer =turquoise, claybasal = pink and claycharge = cyan.

The agreement between the probability distributions of the bonded degrees of freedom is shown in Figure 11.

5 Validation

The atomistic simulation we describe in section 4 to parameterise the bonded degrees of freedom also allowsus to compare the properties at the AA and CG level of a realistic clay interlayer to determine whether theCG surfactant potentials described previously adequately represent the AA system.Firstly, we can compare the atomic density perpendicular to the clay surface. This illustrates the ability ofthe CG potentials to capture the adsorption properties of the clay surface. Note, the CG potentials have notbeen optimized to match the AA density profiles. A comparison between the CG and AA density profiles isshown in Figure 12. We find good agreement in the position of the surfactant molecules and we capture thePEG polymer molecules also adsorbing on the surface.

We can also compare radial distribution functions for the surfactant and PEG polymer with the clay surface,

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(a)

(b)

Figure 11: The bonded degrees of freedom for the surfactant molecules at the atomistic (solid lines) and coarse-grained level (dashed lines). (a) The bonds in the system: the black lines are polymerterminal-polymermonomer

bonds and the red lines are polymermonomer-polymermonomer. (b) The angles in the system: the black lines areC2 - C2 - C2 angles, the red lines are C2 - C2 - TMA angles and the blue lines are C2 - TMA - C2 angles. (c)the dihedrals in the system: black = C2 - C2 - C2 - C2, red = C2 - C2 - C2 - TMA, green = C2 - C2 - TMA- C2, blue = PEG - PEG - PEG - PEG.

shown in 13. We see very good agreement for the C2 and PEG CG units; while the TMA - clay RDFs have thecorrect first and second peak positions, there is some over structuring in g(r), with higher maxima and minimavalues in the CG g(r). As the density profiles perpendicular to the surface are in good agreement for TMA,we can assume this is due to greater mobility on the surface of AA TMA, which is lost in the coarse-graining.Another test of our CG interaction potentials is how well they reproduce the AA clay swelling behavior; thatis, the changes in spacing between the layers as the amount of intercalated polymer increases. In Figure 14, weshow the computed d spacing (i.e. the separation between the clay sheets) as the percentage PEG polymer by

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(a)

Figure 12: Density profile perpendicular to the clay surface for the AA and CG simulation of a expandedclay interlayer simulation, shown in Figure 10. AA density profiles are solid lines and CG density profiles aredashed lines. The colors are: black = C2, red = TMA, blue = PEGterminal, green = PEGmonomer.

weight of the system increases, up to 50%. The various polymer weight fractions are computed by removingthe required number of PEG molecules from the montmorillonite system described previously at both the AAand CG levels. The systems were simulated at 500K and 1 atmosphere pressure. We find good agreementwith a slight systematic under estimation of the clay expansion at the CG level (< 10 %) at low polymerloadings. Our coarse-grained model has not been fitted to the swelling behavior; therefore the results ofFigure 14 demonstrate that the approach described here of constructing coarse-grained interaction potentialsby reproducing the properties of individual interactions in turn captures the behavior of surfactant treatedclay-polymer nanocomposites.

6 Clay structures used in this study

The clays commonly used in nanocomposites are 2:1 silicates: they possess a two-dimensional layered structurewith a thickness of approximately 0.95 nm comprising two silica tetrahedral sheets either side of an octahedrallycoordinated sheet of aluminum oxide. The atomic structure of 2:1 silicate clays is shown in main article Figure1. Individual clay layers have a very high aspect ratio; typically they have a diameter of the order of 0.01 - 1.0micrometres, giving an aspect ratio between 10 - 1000 [7]. As a result of the extensive interfacial area, dispersedclay layers can provide significant reinforcement to polymer matrices through the addition of small volumefractions (typically less than 5%), which may lead to very strong, yet lightweight, materials [8]. However, clayplatelets are typically found in large stacks, called tactoids, comprising several hundred clay layers. In thisaggregated state, the majority of the surface area is not available to the polymer matrix and the potential forreinforcement is diminished. It is therefore of great importance to understand how we can disperse these claylayers from their aggregated, immiscible state into a fully exfoliated, dispersed state (see main article Figure2.) One method to address this is through chemical modification of the clay surface; isomorphic substitutionsin the clay framework give the clay a negative charge, which is balanced by interlayer counterions such as Na+

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(a) (b)

(c)

Figure 13: Radial distribution functions for the AA and CG simulation of an expanded clay interlayer simula-tion, shown in Figure 10. AA RDFs are solid lines and CG RDFs are dashed lines. (a) RDF for C2 - claybasal

(black) and for TMA - claybasal (red) (b) RDF for C2 - claycharge (black) and for TMA - claycharge (red) (c)RDF for PEGmonomer - claybasal (black).

or Ca2+. Replacement of these counterions by organocations, such as quaternary ammonium ions, creates anexpanded clay tactoid with much greater affinity for the polymer [9, 10, 11, 12, 13]. Such organophilic clays, incombination with polymer under various processing conditions such as hydrostatic pressure or shear, can leadto exfoliated clay systems. However, in many cases, the layers still remain stubbornly aggregated; in general,most nanocomposites are experimentally observed to comprise intercalated stacks and some single layers [14].In our models, the Na+ cations of the pristine clay are replaced with alkyl ammonium chains (C16)2-N+-(CH3)(CH3) to create an organoclay, parameterised as described in Sections 3 and 4. We consider only chargesites in the octahedral layer of the clay, where Al3+ ions are replaced by Mg2+ ions, which are the majority ofcharge sites for montmorillonite clays, the archetypical swelling clay. These charge sites are counterbalancedby the organic modifiers, resident in the inter-layer spacing. As described in Section 2, in our model thereare 5 CG clay types: neutral clay (corresponding to a Al3+ octahedral ion), charged clay (Mg2+), edge clay(five coordinate Al3+), quaternary ammonium groups ((Me)4N+) and ethyl groups (C-C). As described inthe main article, we simulated three models using CG molecular dynamics , with each having different levelsof substitution of the six coordinate Al3+ ions by Mg2+ ions: 5%, 11% and 17%. As there is an organicmodifier for each charge site in the clay framework, the number of charge sites determines the grafting density

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Figure 14: The changes in d-spacing with increased fraction of polymer intercalated between the clay layersat the atomistic level (black solid line) and the coarse-grained level (red line).

of surfactants in the interlayer region, that is the number of surfactant chains per unit area of clay layer.Each model is composed of 8 tactoids, with each tactoid composed of 4 layers (32 layers in total). As thereare 4 clay layers in each tactoid, the tactoid contains two types of interlayer gallery: an outer interlayer andan inner interlayer. Each clay layer is of hexagonal shape with a diameter of approximately 100 Å; in totalthey comprise approximately 2 % of the volume of the simulation cell. The models are listed in main articleTable 1 and the size of the simulation cells are approximately 400 - 600 Å3.To create the initial conditions for the CG models, we used a Monte Carlo procedure to produce a low energystate, drawing on the methods of Theodorou and Suter [15] and the look-ahead procedure developed byMeirovitch [16]. The details of this method are in our previous publication [1]. This method generates anamorphous polymer system in a relatively low energy state and avoids high energy overlaps.We firstly generated the 4 layer tactoid with surfactants at the atomistic level for each of the three models. Thetactoid was placed in the centre of a large empty simulation box and subsequently simulated using LAMMPSfor approximately 5ns, allowing the system to equilibrate. This tactoid was subsequently coarse-grained usingthe mapping operator defined in Section 2, and 100 monomer PEG polymers were built around the tactoidusing the Monte Carlo procedure described above [1]. These models were replicated in a 2 ×2 × 2 array to formthe eight tactoid (32 layers) initial starting models. This structure was subsequently energy minimized using asteepest decent algorithm before being used in the CG molecular dynamics simulations. All simulations wereperformed at 500K and at relatively high pressure (300 atms) to replicate the conditions of melt processing.The lattice parameters were allowed to vary under constant pressure and temperature conditions (NPT ) for thefirst 100ps, after which the simulation was run with constant volume (NV T ). The simulations were performedusing the LAMMPS molecular dynamics code [17, 18], employing a timestep of 1.75 - 2fs. Although longertimesteps were used during parameterisation, these relatively short timesteps were used during production toensure the stability of the simulations. For analysis, thermodynamic data was collected every 100 timesteps andpositional data every 20,000 timesteps. Each simulation was run for a minimum of 400ns, which correspondsto greater than 2 microseconds when the CG timesteps are introduced (due to the reduced friction of the CGinteraction potentials).Materials properties. To create the initial conditions for the uniaxial simulations, the final positions fromour models at 500 K were cooled down to 100 K over 4 ns and subsequently simulated at 100 K for 2 ns, with

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no drift in the lattice parameters observed at the end of the simulation.To estimate the glass transition temperature of our clay-polymer system, we plotted the volume of the sim-ulation cell of model III as a function of temperature. We estimated the glass transition temperature to beapproximately 220K, corresponding to the intersection of two linear trend lines fitted to the ends of the data(Figure 15). In our stress-strain simulations, we wish to probe the clay-polymer system in a glassy state. Wetherefore performed simulations at 100K, significantly below the glass transition temperature.

Figure 15: Plot of simulation cell volume as a function of temperature for model III, used for determiningthe glass transition temperature. The red line is the volume dependence of model III as we cool the systemto 100K. The two black lines are linear regressions fitted to the low and high temperature behaviour. Theirintersection is at approximately 220K, indicating the Tg of our clay-polymer systems.

The CG nanocomposite models were deformed in the x directions under a uniaxial tensile strain at a constantstrain rate (1×108 s−1) with a zero pressure condition imposed on the lateral faces. The stress componentswere determined from the pressure tensor, calculated via the virial stress, to give the stress-strain behaviorreported in Figure 6 in the main paper. This is the same method as that of Hossain et al. in their study ofamorphous polyethylene [19].In Figure 6 in the main paper, we plot the instantaneous stress while the strain increases. To check the validityof this approach, we also calculated the stress-strain behaviorof model III by averaging the stress over 1 ns ofsimulation at varying strains (with each increasing strain created by scaling the final snapshot of the previousstrain simulation), with the stress outputted every 200fs. The stress-strain curve calculated in this manner isshown in Figure 16. Using a linear fit to between strains of -0.005 and 0.005, we find a Young’s modulus of0.224 ± 0.01, in good agreement with the Young’s modulus calculated using the continuously increasing strainmethod (0.266 GPa).Note that the calculations reported for the studies at 100K are not expected to be quantitatively correct,as they employ potential parameterizations obtained at higher temperatures (both those from the all-atomClayFF and the CG parameterization).For comparison with micromechanical models, we have also computed the longitudinal stiffness, E11 for com-posites filled with unidirectional disk-like particles using the Halpin-Tsai equation. We have used the expres-

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Figure 16: Stress-strain plot for uniaxial compression and extension of organo-clay model III. The stress valuesare calculated as an average of a 1ns simulation at each fixed strain point. We calculate elastic properties (theYoung’s modulus) from the gradient of the stress-strain curve between -0.005 and 0.005 strain values.

sions given in Sheng et al. [20],

E11

Em=

1 + 2(L/t)fcη

1− fcη(3)

η =(Ec/Em)− 1

Ec/Em) + 2(L/t)(4)

where fc is the clay volume fraction, the clay aspect ratio is L/t, Ec and Em are the Young’s moduli for theclay particle and polymer matrix respectively. To calculate the volume fraction, we estimated that the surfacearea of each individual clay layer to be approximately 7400 Å2. For model III, the initial separation betweenclay layers is 28 Å. The volume, therefore, of each 4 clay layer platelet is 830,000 Å3. The volume of modelIII at 100K is 464.4 Å × 470.8 Å × 580.5 Å = 1.26 × 108 Å3. The clay volume fraction is therefore 8 ×830,000 / 1.26 × 108 Å3 = 0.0524. If we consider the volume the thickness of a singe clay layer to be 10 Å bynot considering the interlayer spacing (rather than 28 Å), the volume fraction decreases to 0.019.We previously reported Em as 0.127 GPa [1]. In Figure 17, we show the stress-strain relationship for in-planeextension of a single clay sheet (xy dimensions 41.36 Å × 35.82 Å) under a uniaxial tensile strain at a constantstrain rate (1×108 s−1) with a zero pressure condition imposed on the lateral faces. Using a linear regressionto the stress-strain curve between strains of -0.01 and 0.01, we find an in-plane Young’s modulus (Ec) of 4.54± 0.02 GPa. Using equation 3, and a clay volume fraction of 0.0524, we determine E11/Em to be 1.74.

We do not expect crystallization to occur in our models when cooled down to 100K. The timescales involvedin the crystallization of relatively long polymers in terms of molecular simulation of the kind studied here canbe on the order of seconds, minutes or even hours. Our simulations are cooled down from 500K to 100K overapproximately 4 nanoseconds, effectively quenching the amorphous polymer melt. To illustrate that there isno crystallization in our simulations, we have plotted the end-to-end lengths of the polymers after simulation

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Figure 17: Stress-strain behavior for uniaxial compression and extension of our clay models. Using a linearregression to the stress-strain curve between strains of -0.01 and 0.01, we find an in-plane Young’s modulus of4.54 ± 0.02 GPa.

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at 500K and 100K for model III (Figures 18a and b), and observe a wide range of distributions for bothtemperatures. The crystal structure is known to possess long helical fibers [21], so if any crystallization isoccurring, we would expect the end to end distances to increase substantially as the polymers align alongtheir fiber (c) axis. We have also converted the end-to-end vectors into spherical coordinates; again there isno ordering in the θ and φ angles we observe (Figure 19).

(a) (b)

Figure 18: End-to-end distance distribution for molecules in model III (a) at 500k and (b) when cooled to100K.

(a) (b)

Figure 19: Volume normalised distribution functions of the θ and φ spherical coordinates of the end-to-end vector of the PEG molecules in model III after cooling to 100K. (a) θ distribution function (Pθ(θ) =

H(θ)/sin(θ)), where H(θ) is the histogram of θ angles and (b) φ angles, Pφ(φ) = H(φ), where H(φ) is thehistogram of φ angles.

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7 Density profiles

The density profiles perpendicular to the clay surface within each tactoid were calculated by fitting a plane toeach clay sheet; for each sheet, the CG coordinates from the simulation were multiplied by the rotation matrixthat transforms the fitted plane onto the xy plane. The density profiles were then computed as the numberof atoms of a given CG species between planes at z and z + dz, of CG atoms with x and y coordinates thatwere within the footprint of the clay sheet. The density profiles for models II and III when in an intercalatedstate are shown in Figure 20.

8 FabMD Software

We use the FabMD software toolkit to manage molecular dynamics simulations across multiple resources.FabMD is a domain-specific version of FabSim [22], which in turn relies on basic Python libraries (numpy,scipy) and the Fabric library (see http://www.fabfile.org). FabMD provides automated administration of inputand output directories, and simulation-based workflows that use a range of resources (e.g., local workstationsand remote supercomputers). It allows users to run sequences of jobs and script execution activities, orcombinations using the two, by invoking a one-line command. We have run FabMD-based jobs (which primarilyconsisted of running LAMMPS) for numerous tasks in this work. For example, we used FabMD to managethe iterative Boltzmann inversion (IBI) required for parameterizing the potentials, allowing us to send IBIruns to available remote resources. The runs that we performed for each IBI iteration typically used 32 coresand lasted between 5 to 25 minutes each. FabMD is available under an open-source LGPL license, as part ofFabSim [22].

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(a)

(b) (c)

Figure 20: Density profiles perpendicular to the clay layers when in an intercalated state for the CG simulationsdescribed in the main article. In a) we show a snapshot from the CG simulation at approximately 10ns of asingle intercalated interlayer in model II (right) used to calculate the density profile: the density is calculatedperpendicular to a plane fitted to the clay layer for atoms within the footprint of the clay layer. The atomcolors are the same as main article Figure 1, with the surfactant molecules transparent. The correspondingdensity profile is on the left. It is clear there exist several PEG molecules which bridge across the interlayer. b)The density profile for model II and c) model III when intercalated. The density profile colors are as follows:green = clay CG atoms, red = ammonium surfactant groups, blue = surfactant alkyl chain atoms, black =PEG molecule atoms. In c), we see that the PEG molecules are predominantly resident on the clay surface,with much less density in the middle of the clay layer in model III than in model II.

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