# Properties of Polystyrene and Polymethyl Methacrylate Copolymers of Polyhedral Oligomeric Silsesquioxanes: A Molecular Dynamics Study

### Abstract

Molecular dynamics simulations were carried out on copolymers of both styrene and methyl methacrylate with polyhedral oligomeric silsesquioxane (POSS) derivatives to identify the origin of the property changes imparted upon the chemical incorporation of POSS. Simulations were carried out on these hybrid copolymers and the parent homopolymers to elucidate the effect of the T8, T10, and T12 POSS cages. These POSS comonomers were derivatized with a single polymerizable function and 7, 9, and 11 nonpolymerizable hydrocarbon moieties, respectively. Glass transition tem- peratures (Tg) were computed from speciﬁc volume versus temperature plots. The pack- ing of POSS units around the polymer backbone was analyzed via their radial distribu- tion functions. The effect of POSS on polymer motion was analyzed through the mean square displacement function. The improvements in the elastic moduli upon incorpora- tion of POSS were computed by employing the static deformation method. VVC 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 234–248, 2006

Keywords: molecular dynamics simulation; polyhedral oligomeric silsesquioxanes; polymethyl methacrylate; polystyrene

Author(s): REENA R. PATEL,1 RAJENDRAN MOHANRAJ,1 CHARLES U. PITTMAN JR.2

1. Engineering Research Center, Mississippi State University, Mississippi, 39762

2. Department of Chemistry, Mississippi State University, Mississippi, 39762

Corresponding Author: RAJENDRAN MOHANRAJ

Figure 1. Styrene/p-styryl-T8-POSS copolymers 2a–d with cyclopentyl (4 mol %), cyclohexyl (4 mol %), isobutyl (5 mol %), or phenyl (4 mol %) substituents on the POSS cage. (click on the image to enlarge).
Figure 2. Styrene/p-styrylnonaphenyl-T10-POSS-3 copolymer 5 and styrene/p-styryl- undecylphenyl-T12-POSS-4 copolymer 6. Each copolymer contains 4 mol % POSS. (click on the image to enlarge).
Figure 3. Methyl methacrylate copolymers with 1, 4, and 10 mol % of the heptaisobutyl-substituted POSS, 7a, or 4 mol % of the heptaphenyl-substituted POSS, 7b. (click on the image to enlarge).

### Methodology

Synthetic polymers have a limited temperature use range. Hybrid organic/inorganic polymeric materials have been developed to improve material properties.[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] Polymers like polystyrene and polymethyl methacrylate are typically used below their glass transition temperatures (Tg) in their glassy state, where they are hard and brittle. When a polyhedral oligomeric silsesquioxane (POSS) monomer is copo- lymerized into such plastics, their useable tempera- ture range can be increased due to an increase in the Tg or due to a rise in moduli in the rubbery region. Polymer nanocomposites containing POSS have been considered for applications in low dielec- tric constant materials,[15] synthesis of dendrimers,[16] coatings for lithography,[17] materials with increased resistance to oxidation,[7][18][19] processing viscosity modiﬁers,[20] and ﬂame retardants.[21][22]

POSS can be added pendant to the polymer backbone.[23] By retarding segmental motion, POSS units can raise the heat distortion temperature. However, the effect of POSS on material proper- ties is complicated to predict, because it is highly dependent on the nature of the polymer’s packing, the nature of the corner substituents on POSS, and on the aggregation of POSS moieties into small domains or clusters.[19][20][21][22][23][24] Molecular dynamic simulations may help understand the origin of the effects that POSS moieties impart to polymer properties.[25]

Numerous experimental studies have been performed on the synthesis and determination of the structural properties of polymers containing POSS.[6][7][8][9][10][11][12][13][14][26] However, few numerical simulations have appeared. Molecular dynamic simulations have been carried out on norbornene/norbornenyl-POSS copolymers.[25] Two recent numerical studies[26][27] have employed molecular dynamics simulations to investigate the properties of polymer nano- composites containing POSS moieties. Unlike the present study, which deals with copolymers containing chemically bound POSS moieties, these two studies focus on a polymer nanocomposite in which POSS moieties are blended, but not chemically bonded to the polymer matrix. Striolo et al.[27] studied POSS moieties, with either methyl groups or hydrogen atoms as the corner substituents, blended in a polydimethylsiloxane matrix. The other study[28] involved cyclopentyl-substituted POSS dispersed in a polyethylene matrix. Lattice Monte Carlo simulations have been used by Lamm et al.[29] to model polymer POSS nanocomposites by using rigid cubes and bead–spring chains to model the POSS moieties and the polymer backbone, respectively. The effect of introducing 10 mol % of mono-norbornenyl-substituted T8 POSS monomers with seven cyclohexyl or seven cyclopentyl func- tions on the remaining cage Si atoms was analyzed. Wei et al.[30] used molecular dynamics simula- tions to estimate the properties of a polymer nanocomposite containing carbon nanotubes and polyethylene. Molecular dynamics simulations of bulk atactic polystyrene in the vicinity of its Tg have been performed using the united atom model.[31] Mechanical properties of polyethylene were predicted using a molecular mechanics force ﬁeld for the interatomic potential and quasi harmonic lattice dynamics for the vibrational free energy.[32]Raaska et al.[33] conducted molecular mechanics and molecular dynamics simulations on amorphous polystyrenes with different tacticities and estimated their cohesive energies and solubility parameters.A molecular dynamics study of atactic polystyrene was performed to study the X-ray structure factor.[34] An aim of the present work is to predict the effect of small mole fractions of POSS comonomers, incorporated into either polystyrene or polymethyl methacrylate, on the glass transition temperatures, moduli, Poisson’s ratio, X-ray scattering intensity, and solubility parameters. Why does POSS impart these changes? The systems selected, herein, are based on the experimental data available in the literature whenever possible.[35][36]

### Material Model

Two polymer classes, polystyrene and polymethyl methacrylate, were studied. The systems were built as 100-mer units. Copolymers containing POSS moieties were built using 1, 4, 5, and 10 mol % of POSS. Seven polystyrene systems were selected (Figs. 1–3), including atactic poly- styrene and copolymers of styrene containing either 4 or 5 mol % of the mono-p-styryl-T8-POSS monomers 1a–1d. The seven corner substituents on the Si atoms in 1a–1d were cyclopentyl, cyclo- hexyl, isobutyl, or phenyl groups (see Fig. 1). Styrene copolymers, 5 and 6, of p-styryl-T10- POSS, 3, and p-styryl-T12-POSS, 4, respectively, (where the remaining nine or 11 Si atoms are bonded to phenyl groups) were also simulated with 4 mol % of the POSS monomer (see Fig. 2). Methyl methacrylate/POSS methacrylate copoly- mers 8a–8b were constructed where the seven R groups were isobutyl (8a) or phenyl (8b) (see Fig. 3). These copolymers contained 1, 4, and 10 mol % of 3-methacrylylpropylheptaisobutyl-T8-POSS, 7a, or 4 mol % 3-methacrylylpropylheptaphenyl-T8- POSS, 7b. Each T8-POSS methacrylate has a propyl-spacer connecting the inorganic cage to the polymer backbone, as is depicted in Figure 3.

### Input Data

Table 1. Simulated and Experimental Glass Transition Temperatures for Styrene/POSS Copolymers versus Atactic Polystyrene.[35][37][38] (click on the image to enlarge).

Table 2. Simulated and Experimental Glass Transition Temperatures of Methyl Methacrylate/POSS Copolymers versus Syndiotactic PMMA.[36][39][40][41][42] (click on the image to enlarge).

Molecular dynamics simulations were carried out using Cerius II software from Accelrys.[43] The COMPASS force ﬁeld was chosen based on the good agreement obtained in an earlier study [25] between predictions using this force ﬁeld and the corresponding experimental data. In particular,this force ﬁeld has been satisfactorily used to de- scribe interactions involving bonds such as Si--O and Si--C. 100-mer polymer units were packed in a cubic simulation box at a very low initial density (rv0.3 g/cm3). The system was minimized prior to performing dynamics runs. Each system was equi- librated for 1.5 ns before making production runs at 500 K and 0 atm using NPT (constant number of particles, pressure, and temperature). Once equilibrated, production runs were carried out for 300 ps. All systems were cooled in steps of 25 K. At each subsequent temperature, the system was equilibrated for 200 ps before making a 300 ps pro- duction run. In some cases, longer production runs were performed to ensure that every system was well equilibrated. For a sample case (polystyrene with 4 mol % phenyl T8-POSS, 2d, at 325 K), the value of density obtained using a 1.8 ns production run was within 0.5% of the corresponding value for a 300 ps production run.

In general, it is preferable to use a simulation time comparable to the longest relaxation time for the system. However, it is not clear what this relaxation time is for a copolymer containing bound POSS moieties with speciﬁc functional groups. Wei et al.[30] used 100 ps of equilibration time at each temperature to determine the den- sities of a nanocomposite containing carbon nano- tubes and polyethylene. In the present study, temperature and pressure were controlled by the methods of Andersen[44]and Berendsen and coworkers,[45] respectively. The velocity Verlet algorithm[46] was used to integrate Newton’s equation of motion. The cut-off distance was 10 A˚ for all systems. The interactions beyond the cut-off dis- tance were accounted by using the Van der Waal tail correction. Electrostatic interactions were taken into account by using the distance-depend- ent dielectric ‘‘constant’’ method41 with a dielec- tric value of 2.5.

## Glass Transition Temperature

Figure 4. Speciﬁc volume versus temperature curves for atactic polystyrene and the styrene copolymers 2d, 5, and 6 (4 mol % of POSS each) and 2c (5 mol % POSS).
Figure 5. Speciﬁc volume versus temperature curves for styrene copolymers 2a (4 mol % 1a) and 2b (4 mol % 1b).
Figure 6. Speciﬁc volume versus temperature curves for syndiotactic polymethyl methacrylate and methyl methacrylate copolymers 8a (containing 1, and 4 mol % of 3-methacrylylpropylheptaisobutyl-T8-POSS, 7a).
Figure 7. Speciﬁc volume versus temperature curves for methyl methacrylate copolymers 8a (10 mol % iso- butyl-substituted POSS 7a) and 8b (4 mol % of phenyl- substituted POSS 7b).
Table 3. Simulated Solubility Parameters (d) and Volumetric Coefﬁcients of Thermal Expansion (CTEV) for Styrene/POSS Copolymers.[47]

A sudden increase in segmental motion occurs in the glass transition temperature (Tg) region. Changes in both the heat capacity and speciﬁc volume versus temperature take place. The storage modulus drops. As the temperature in- creases above Tg, a pronounced increase in dV/dT (where V is the volume) occurs, caused by the introduction of free volume and driven by segmental motion enhancements. Simulations of the speciﬁc volume versus temperature are used to predict Tg (the temperature where this steep change occurs) for comparison to experimental Tg values. The speciﬁc volumes of pure atactic polystyrene, simulated, herein, at several temperatures using the all-atom model (0.96–1.04 cm3/g), were typically marginally lower than the corresponding simulated values obtained using united- atom model (0.98–1.05 cm3/g).[48]

Figure 4 shows the computed speciﬁc volume versus temperature curves for atactic polystyrene,the styrene copolymers 2d, 5 and 6 (containing 4 mol % of the p-styryl (phenyl) T8, T10, and T12- POSS monomers 1d, 3 and 4, respectively) and the styrene copolymer 2c containing 5 mol % of p- styrylheptaisobutyl(T8)POSS, 1c. The break in the slope for each system locates the predicted Tg. Tables 1 and 2 summarize these predicted Tg values for both the styrene and the methyl metha- crylate copolymers illustrated in Figures 4–7.

The differences in the simulated versus experi- mental Tg values obtained are relatively small. Importantly, the trends in the Tg changes have been captured using molecular dynamics simula- tions. The direction of the Tg change (i.e., raising or lowering of Tg) agrees with experimental results. The simulated Tg of 375 K for atactic poly- styrene agrees closely with experimental data.[37][38] The experimental value for DTg, (difference be- tween the experimentally measured Tg of the POSS-copolymer versus that of polystyrene) is given in the last column of Table 1. Note that these DTg are speciﬁed from a speciﬁc experimental study35 in which Tg ¼ 402 K was obtained for a polystyrene that had a speciﬁc tacticity. The simu- lated Tg for PMMA (415 K, see Table 2) is higher than experimental values. However, real syndio- tactic PMMA contains both stereochemical and regiochemical errors, while the simulated polymer’s structure was perfectly syndiotactic. The simula- tions concur with the experimental data for 8a and 8b, which show that POSS monomer (1 and 4 mol%) incorporation causes only small changes in Tg.

The difference in Tg between the pure polymer and the nanocomposite is signiﬁcant for some, but not all cases. Thus, discussion of the estimate for errors associated with the measurement of the speciﬁc volume at the different temperatures and consequently in the prediction of Tg for all the cases in this study is pertinent. The uncer- tainty in the speciﬁc volume was mostly within the range of 0.01–0.02 cm3/g in terms of both the standard deviation in the speciﬁc volume during the production phase of the MD simulations, as well as the difference between the value at a spe- ciﬁc temperature and the corresponding linear ﬁt. The estimation of Tg, which is obtained as the point of intersection of two piecewise linear ﬁts, involves an error of rv3–7 K for the different cases. This estimate is based on the errors associ- ated with the two piecewise linear ﬁts of speciﬁc volume versus temperature. In some cases, the predicted change in Tg is in the vicinity of this error size, illustrating a limitation of this study.

## Solubility Parameters and Coefﬁcients of Thermal Expansion

The simulated solubility parameters (d) and volumetric coefﬁcients of thermal expansion (CTEV)for atactic polystyrene and the styrene/POSS copolymer systems are shown in Table 3. Solubil- ity parameters were determined from the square root of the simulated cohesive energy densities, taking into account the effect of the cut-off dis- tance on the nonbonded potentials. All the pre- dicted solubility parameters for styrene/POSS copolymers (2a–d, 5, and 6) were lower than those for polystyrene. Only the styrene/hepta- phenyl (T8) POSS copolymer 2d gives a d-value comparable to that of atactic polystyrene. This is due to the compatibility between the phenyl rings of styrene monomer units and those on the POSS corners (favorable p-stacking interactions).

Figure 8. Backbone MSD for atactic polystyrene at different temperatures.
Figure 9. Backbone MSD for styrene copolymer 2d(4 mol % of phenyl-substituted T8-POSS, 1d).
Figure 10. Backbone MSD for MMA copolymer 8a (10 mol % of isobutyl-substituted T8-POSS, 7a).
Figure 11. X-ray scattering intensity (arbitrary units) of pure atactic polystyrene as a function of the scatter- ing vector (A˚^-1).
Figure 12. X-ray scattering intensity (arbitrary units) as a function of the scattering vector (A˚ -1): (a) styr- ene/POSS copolymers 2b (4 mol % POSS 1b) and 2c (5 mol % POSS 1c), (b)styrene/POSS copolymers 2d (4 mol % POSS 1 d), 5(4 mol % POSS 3) and 6(4 mol % POSS 4).
Figure 13.The intermolecular RDFs at 325 K, based on the backbone carbon atoms, for atactic PS and copolymers 2a (4 mol % cyclopentyl-substituted T8- POSS), 2b (4 mol % cyclohexyl-substituted T8-POSS), and 2c (5 mol % isobutyl-substituted T8-POSS).
Figure 14. The intermolecular RDFs at 325 K, based on the backbone carbon atoms, for syndiotactic PMMA and the MMA copolymers 8a (1, 4, and 10 mol % isobutyl-substituted T8-POSS, 7a).
Figure 15. The intermolecular RDFs at 325 K based on all the atoms in the POSS moieties (Si, O, and R group) to the backbone carbon atoms in styrene/POSS copolymers 2a, 2b, and 2c.
Figure 16. The intermolecular RDFs at 325 K based on all the atoms in the POSS-moieties (Si, O, and R groups) to the backbone carbon atoms of MMA copolymers 8a (1, 4, and 10 mol % isobutyl-substituted T8- POSS, 7a).
Figure 17. The intermolecular RDFs based on the POSS cage Si and O atoms to other POSS, Si, and O atoms for copolymers 2a–c containing cyclopentyl-, cyclohexyl-, and isobutyl-substituted T8-POSS monomers (1a, 1b, and 1c), respectively.

The value of d is higher for cyclopentyl-substi- tuted POSS copolymer 2a, than that for copolymer 2b incorporating cyclohexyl-substituted POSS 1b. Cyclopentyl rings facilitate denser packing than cyclohexyl rings. The lowest predicted solubility parameter of the styrene/POSS copolymers was that of 2c, incorporating the iso- butyl-substituted POSS 1c. Isobutyl groups involve less dense packing due to their greater conformational freedom, which interferes with efﬁcient packing. The predicted value of d for the styrene/(4 mol %) phenyl-substituted-T8-POSS copolymer 2d is higher than those of the analogous phenyl-substituted T10- and T12-POSS co- polymers 5 and 6.

Increasing the size of the POSS cage going from T8 to T10 to T12 cages increases both the weight fraction and volume fraction of POSS in the copolymers at the constant POSS mole fraction loading (4 mol %). Thus, the T10 and T12 POSS cages might be expected to retard segmental motion more effectively than T8-POSS. In Table 1, we see that the Tg values of 5 and 6 are predicted to be 50 K higher than those of poly- styrene. The reasons for the predicted decrease in d values for 5 and 6 versus 2d are not obvious. While the large T10 and T12 cages retard segmental motion, they may introduce more free volume. These counteracting features affect Tg and solubility behavior differently. These predictions of Tg and d values for 2d, 5, and 6 are tempting targets for future experimental veriﬁcation. Atactic poly- styrene’s cohesive energy density[49] (8.1–9.9 (cal/ cm3)1/2) is rv25% higher than our simulated experimental value of 6.15 (cal/cm3)1/2.

The solubility parameters and volumetric ther- mal expansion coefﬁcients for the methyl metha- crylate copolymers are shown in Table 4. The predicted solubility parameter, d, was highest for syn- diotactic PMMA and progressively decreased as the isobutyl-substituted-POSS 7a content increased from 1 to 4 to 10 mol % in the 8a copolymer series.The d-value for MMA/phenyl-substituted POSS 7b (4 mol %) copolymer, 8b, is also lower than that for syndiotactic PMMA, but higher than the d values of all the 8a copolymers, where seven iso- butyl groups adorn the T8 POSS cage.

## Mean Square Displacement

The mean square displacement (MSD) of each backbone carbon atom was summed over all the backbone carbon atoms and then averaged. Polymer backbone motion can be compared both with and without the POSS moieties present, to help understand the restrictions imposed by the massive POSS cage on segmental motion. The high mass (rv1000 amu) and large volume of pendant POSS moieties greatly restrict thermal activation of chain motion. Thus, POSS cages act as ‘‘anchors.’’ The MSD of the polymer backbone, plotted as a function of time at different temperatures, illustrates the effect of temperature on backbone motion. MSD is computed using the following equation

$\boldsymbol{MSD}=< \begin{vmatrix}{r_i}(t)-{r_i}\end{vmatrix}^2 >$

where ${r_i}$ denotes the coordinates of the i th atom.

Atactic polystyrene backbone MSD plots are shown in Figure 8. Backbone motion increases with temperature. A sudden increase in the back- bone motion occurs at the glass transition temperature. Figure 9 displays the MSD of the backbone polystyrene carbon atoms for the styrene copolymer 2d (4 mol % of phenyl-substituted T8- POSS monomer 1d). This case is characterized by lower MSD at 450 K (compare Figs. 8 and 9) and has a higher Tg than that for atactic polystyrene (45 K higher, see Table 1). This suggests that the presence of POSS moieties could retard the motion of the polymer backbone. Figure 10 shows the backbone MSD at different temperatures for MMA copolymer 8a (10 mol % of isobutyl-substituted T8-POSS, 7a). Simulations of longer durations are required to observe diffusive behavior at high temperatures.

## X-Ray Scattering Intensity

The scattering intensity (for example, the work of Guinier[40]) is given by the following equation

$\boldsymbol{I(Q)}=\sum_{j}\sum_{k}\frac{f_jf_k(Sin\boldsymbol{Q}r_{jk})}{\boldsymbol{Q}r_{jk}}$

where Q, the magnitude of the scattering angle, is given by

$\boldsymbol{Q}=\frac{4\pi Sin\theta}{\lambda}$

where $2 \theta$ is the scattering angle and $\lambda$ is the X-ray wavelength. The indices j and k range over all the atoms in the molecule.

Figure 11 displays a plot of the simulated X-ray scattering intensity versus scattering vec- tor for atactic polystyrene. The X-ray scattering intensity for atactic polystyrene exhibits an ‘‘amorphous’’ peak at 1.4 $\dot A^{-1}$ and a small polymerization peak at 0.75 $\dot A^{-1}$ in both the molecular dynamics simulation of Ayyagiri et al.[34] and in experimental measurements.[50] We observed the amorphous peak at 1.24 $\dot A^{-1}$ (Fig. 11), but the smaller ‘‘polymerization’’ peak is not captured. It might be possible to improve the estimate of the scattering intensity by using averages based on multiple trajectories and perhaps an improved force ﬁeld developed speciﬁcally for polymers containing POSS.

The simulated X-ray scattering intensities versus scattering vector plots of styrene/POSS copolymers 2b and 2c containing 4 mol % of cyclohexyl- and 5 mol % of isobutyl-substituted POSS monomers 1b and 1c, respectively, are shown in Figure 12(a). These POSS moieties generate a second scattering peak at 0.5 $\dot A^{-1}$. The amplitude of this 0.5 $\dot A^{-1}$ peak is higher for copolymer 2c with 5 mol % isobutyl-substituted POSS 1c than for copolymer 2b with 4 mol % cyclohexyl-substituted POSS 1b. Figure 12(b) shows the corresponding plots for styrene/POSS copolymers 2d, 5, and 6, which contain the phenyl functional group, and contrasts the cases involving T8, T10, and T12 POSS cages. The amplitude of the second peak for the T12 cage is noticeably higher compared to the corresponding values for cases involving T8 and T10.

The radial distribution function (RDF), also referred to as pair correlation function, is denoted by g(r). Analysis of g(r) helps to understand the distribution of atoms in the polymer and the packing details. The RDF is deﬁned as the ratio of the probability of ﬁnding a pair of atoms that are separated by a distance r to the corresponding probability that is expected for a completely random distribution at the same density.54 Knowing the RDF helps analyze the local packing of polymers. In the examples shown here, the intermolecular radial distribution functions are plotted (Figs. 13–17), and they have been computed based on a speciﬁc subset of atoms in the system. At large distances, this has a value less than unity, but the total RDF attains the value of unity, as expected.

The RDFs are plotted in Figure 13 based on all the backbone carbon atoms for atactic PS and styrene/POSS copolymers 2a (4 mol % of cyclo-pentyl-substituted T8-POSS), 2b (4 mol % cyclo-hexyl-substituted T8-POSS), and 2c (5 mol % iso-butyl-substituted T8-POSS). Each copolymer exhibits a diffuse peak rv10 A˚ , indicating an average interchain spacing in the vicinity of the 10 A˚ spacing found for polystyrene.[51] The higher g(r) values for atactic polystyrene versus copolymer 2b, over the region graphed, implies that there are a smaller number of short interchain distances per unit vol- ume when cyclohexyl-substituted POSS 1b moieties are present. However, differences in the average interchain spacings are small in this series of polymers. Styrene/cyclopentyl-substituted POSS copolymer, 2a, has a higher g(r) value than styrene/cyclohexyl-substituted POSS copolymer, 2b. Cyclopentyl substituents are less ﬂexible, and so, they pack more efﬁciently than cyclohexyl substituents. Hence, the number of intermolecular contacts is higher per unit volume for 2a.

The RDFs based on the backbone carbon atoms are displayed in Figure 14 for syndiotactic PMMA and the three MMA/POSS copolymers 8a (1, 4, and 10 mol % of the isobutyl-substituted T8-POSS monomer 7a). Pure PMMA has the highest RDF for radial distances of 7.8–11 A˚ . A diffuse peak at 8.5–9 A˚ corresponds to its average interchain spacing. The magnitudes of these curves decrease with an increase in the POSS 7a content. Thus, introducing POSS 7a into the polymers reduces the number of interchain contacts relative to syndiotactic PMMA.

The intermolecular RDFs at 325 K of all the POSS-functions’ atoms, including the R-groups (substituted with cyclopentyl (1a), cyclohexyl (1b), and isobutyl (1c) groups) of styrene copoly- mers 2a–c, to the backbone carbon atoms are plotted in Figure 15. Lower values for the RDF were obtained for the styrene/cyclopentyl POSS 1a copolymer, 2a than for cyclohexylPOSS 1b- containing copolymer 2b, showing that the larger cyclohexyl substituents cause greater average backbone-to-POSS distances. The curve for styrene copolymer 2c, with 5 mol % isobutyl-substituted POSS, is relatively linear and does not show any special features. Bharadwaj et al.[25] obtained a higher simulated value of the pair correlation function for a norbornene/POSS copolymer with a cyclopentyl-substituted POSS comonomer versus that with a cyclohexyl-substituted POSS in a molecular dynamics study.

Figure 16 shows the intermolecular RDFs at 325 K of all the POSS atoms, including the R groups, in the MMA copolymers 8a (containing 1, 4, and 10 mol % isobutyl-substituted POSS 7a) relative to the backbone carbon atoms. The magnitude is higher for the 1 mol % isobutyl-substituted POSS copolymer and decreases upon increasing the POSS content to 4 and 10 mol %. Increasing POSS, 7a content decreases packing efﬁciency.

The RDF based on POSS cage to POSS cage (Si and O atoms) distances for copolymers 2a–c are illustrated in Figure 17. The RDF of copolymer 2a (4 mol % cyclopentyl-substituted POSS 1a) is lower than that of its cyclohexyl- and isobutyl-substituted POSS analogs, 2b and 2c, respectively, at radial distances shorter than 11 A˚ . How- ever, at radial distances greater than rv11.5 A˚ , copolymer 2a has the highest pair correlation function followed by the styrene copolymers 2b and 2c.

The aggregation of POSS groups within thermoplastic and thermoset polymers is well known. For example, Coughlin and coworkers[19][52] prepared ethylene copolymers of mono-norbornenylethyl T8-POSS. When solidiﬁed from the melt or precipitated from solutions, varying degrees of POSS aggregation was observed, and the aggregates had varying degrees of crystalline order. It is beyond the scope of the present investigation to observe any tendency for aggregation of POSS units because of the limited simulation time durations employed in this study.

The large number of peaks in the intermolecular RDF for cyclohexyl POSS (see Fig. 17) suggests several preferred arrangements exist for the relative locations of the cyclohexyl-substi- tuted POSS moieties. The occurrence of such peaks is less prominent for cases with cyclopentyl- and isobutyl-substituted POSS moieties com- pared to the cyclohexyl case. Capaldi et al.[28] observed that at 300 K there were several peaks in the RDF computed based on the centers of mass of POSS moieties and silicon atoms belonging to the other POSS moieties (for a system com- posed of cyclopentyl-substituted POSS in a poly- ethylene matrix). This indicates the tendency for a particular organization of the neighboring POSS moieties.

In the study by Striolo et al.,[27] which involves POSS substituents (hydrogen atoms or methyl groups) that are small in size and do not have signiﬁcant conformational variation, the RDF was not characterized by a large number of peaks. In a simulation window of 2 ns, there were a few occasions during which the displacement of the POSS moiety was characterized by sharp changes (referred to as ‘‘hopping’’ events). This was explained by POSS conﬁnement within a void in the amorphous polymer matrix for a signiﬁcant duration (corresponding to small change in dis- placement with time). Occasionally the POSS moiety had the opportunity to move to an adjacent void, thereby resulting in a sharp change in displacement. In the present study, production runs were typically performed for a shorter dura- tion of 0.3 ns.

The occurrence of several peaks in the RDF (Capaldi et al.[28], present study) and the observed hopping motions of POSS moieties[27] are all probably closely related to the dynamics of the POSS moieties in the presence of polymer chains. Such phenomena can be expected to depend upon fac- tors such as (i) the conformational ﬂexibility of the POSS moieties (including those of the substitu- ents) in the environment consisting of the polymer backbone (or matrix for blended nanocomposites),(ii) the temperature, and (iii) the choice of either copolymerization (the present study) or blending of POSS moieties in a polymer matrix[27][28] when the polymer nanocomposite is formed.

## Mechanical Properties

Molecular dynamics studies of amorphous polymers can examine interactions occurring at the atomic level. These simulations can be used to examine the elastic coefﬁcients, which are com- puted from the numerical estimates of the following equation

$\boldsymbol{C_{ij}}=\frac{d^2 U}{d \epsilon _i \epsilon _j}= \frac {d \sigma _i}{d \epsilon _j}$

The static deformation method uses the second derivative of the deformation energy (U) with respect to strain to estimate the elastic stiffness coefﬁcients. The mechanical properties were determined by choosing the minimum energy frame of the trajectory, minimizing it, and run- ning static deformation simulations on the mini- mized frame. Calculations of elastic constants were performed at 500, 400, and 325 K. The low- est common temperature at which computations were performed for all the systems in the simu- lated annealing procedure was 325 K, which was sufﬁciently below the Tg value for all the cases studied. Hence, mechanical properties were com- puted at 325 K, instead of 300 K.

Table 5 lists the calculated elastic constants of atactic polystyrene at 500, 400, and 325 K. The tensile modulus decreases with an increase in temperature. The value of the tensile modulus, computed at 325 K, is rv4.3 GPa, and the experimental value[49] is in the range of 3.2 GPa (unoriented case[53]) to 4.2 GPa (oriented monoﬁlament[54]). The use of a dynamic method for determining mechanical properties would give more accurate results, but it involves signiﬁcant additional computational costs, since separate molec- ular dynamics simulations must be performed under certain constraints. Hence, it was not used in the present study. Estimation of mechanical properties through molecular dynamics simula- tions is less accurate for amorphous systems, which might contain large unoccupied regions in the computational cell (e.g., systems in this study), compared to crystalline systems.

Table 6 lists the calculated elastic constants for the atactic polystyrene, syndiotatic methyl methacrylate, and the different POSS copolymers at 325 K from static deformation simulations. The tensile modulus of polystyrene increases when cyclohexyl-substituted POSS monomer 1b is incorporated. In contrast, decreases in the tensile moduli are predicted for cyclopentyl- and isobutyl-substituted T8 POSS (4 mol %, 1a, and 5 mol%, 1c, respectively) incorporation. The copoly- mers, with 4 mol % of phenyl-substituted T10 and T12 POSS cages (see values for 5 and 6), exhibit large values for the predicted elastic moduli.

The estimated value of 4.39 GPa for the tensile modulus of syndiotactic PMMA is higher than the experimental value of 3.3 GPa.[49] An increase in tensile modulus is predicted when 1 or 4 mol % isobutyl-substituted POSS 7a is incorporated into PMMA. However, incorporating 10 mol % of iso- butyl POSS-7a or 4 mol % of phenyl-substituted POSS-7b lowers the tensile modulus. Experimentally, we have observed higher moduli in many cases where small mole percents of a POSS comonomer was incorporated into organic polymers followed by lower moduli when the amount of POSS incorporation increased.[24][26][36][55][56] Although the absolute values of the simulated tensile mod- uli for the homopolymers polystyrene and PMMA are not very close to the experimental data, it is of interest to note the relative trends in elastic constants for the POSS copolymers compared to the corresponding homopolymers. Future studies will compare predictions using the dynamic method and the ﬂuctuation method to compute the elastic constants.

## Conclusions

Molecular dynamics simulations were used to study the changes in properties of polystyrene and poly(methyl methacrylate), PMMA, achieved by the incorporation of T8-POSS monomers with various organic substituents on the outer surface. Polymer segmental motion was analyzed through MSD. A signiﬁcant increase in the polymer motion was witnessed above the glass transition temperature.

Glass transition temperatures were predicted from speciﬁc volume versus temperature plots. Incorporation of small mole fractions of T8-POSS compounds raised the predicted change in Tg for styrene copolymers, but not for MMA copolymers. Phenyl-substituted T8, T10 and T12-POSS monomers were predicted to substantially raise the Tg of their atactic polystyrene copolymers. The trends in Tg predicted upon incorporating various T8-POSS monomers into styrene copolymers agreed with experimental results. When the POSS substituents were phenyl groups, the Tg increases in styrene copolymers were much larger than when the substituents were alkyl groups. Such R- group compatibility allows the massive pendant POSS moieties to exert larger restrictions on seg- mental motion. When the R groups are alkyl functions, they are less compatible with the styrene polymer’s phenyl rings and pack less efﬁciently. This effect operates to lower Tg, counter- acting the effect of the large pendant masses and volumes of the pendant POSS groups in restricting segmental motion. Therefore, the Tg increases are smaller. PMMA is a polar polymer, and all of the R substituents used (e.g., cyclopentyl, cyclo- hexyl, isobutyl, and phenyl) are nonpolar. Thus, R-group compatibility with the PMMA medium is less than that with polystyrene. Consequently, the Tg values exhibit small decreases or no change upon incorporating 4 or 5 mol % of 1a, 1b, 1c, or 1d (POSS with cyclopentyl, cyclohexyl, iso- butyl, and phenyl groups, respectively) into PMMA.

The approach used in the present study may not predict highly accurate material properties, but it does provide semiquantitative insight on microstructure and local dynamics for these polymer nanocomposites. Averages obtained using multiple trajectories with uncorrelated initial conditions, longer simulation times, will aid in improving the accuracy of the predictions.

The occurrence of several peaks in some of the radial distribution function plots suggests the existence of preferred arrangements of POSS moieties. X-ray scattering curves were also obtained, and a new peak due to the POSS moieties was predicted. These curves can be compared with future experiments to determine the accuracy of the force ﬁeld used in this simulation. Mechanical properties, computed using static deformation method, predict that incorporating POSS monomers with T10 and T12 cages containing phenyl groups into polystyrene will result in higher values for the moduli. These results should be compared to both experimental data and simulations based on dynamic method in the future. Trends in the solubility parameters, d, were also predicted and are available for comparison to future experimental studies.

### Acknowledgments

The authors acknowledge the National Science Foundation grant number EPS 0132618 for ﬁnancial support; the National Center for Supercomputing Applications at the University of Illinois, Urbana Champaign is thanked for providing computer time for a portion of the simulations performed in this study.

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