ARTICLE pubs.acs.org/Macromolecules
Dynamics and Deformation Response of Rod-Containing Nanocomposites Gregory N. Toepperwein,† Robert A. Riggleman,‡ and Juan J. de Pablo*,† † ‡
Department of Chemical and Biological Engineering, University of Wisconsin, Madison, Wisconsin 53706-1691, United States Department of Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States
bS Supporting Information ABSTRACT: Theoretical and computational studies of polymer nanocomposites have largely focused on spherical inclusions in a polymer matrix. In order to address the influence of particle shape on nanocomposite behavior, extensive Monte Carlo and molecular dynamics simulations are used to examine the structure and deformation behavior of a model polymer upon addition of rods of varying aspect ratios. It is found that, at constant temperature, nanorod length does not meaningfully affect the elastic properties of composites but does affect postyield properties, such as the strain hardening modulus. In contrast, at constant T/Tg, several trends with additive length arise. Examination of the polymer bond autocorrelation function during deformation reveals that longer, dispersed rods induce a broadening of the relaxation spectrum. Nanocomposites show longer bond orientation relaxation times than the pure polymer during all stages of deformation. For the truly nanoscale additives used in this study, polymer mobility is found to be only a weak function of distance to the nearest nanorod so long as the additives did not aggregate.
’ INTRODUCTION It is generally accepted that the influence of nanoparticles on polymer behavior depends on a host of properties15 including concentration, particle geometry, particle size, interfacial interactions, and thermal history. Past experimental work, however, has led to diverse, and sometimes conflicting, rationalizations of nanocomposite behavior. It is only recently that molecular theory and simulation have been used to interpret experimental data and provide a molecular-level view of observed behaviors. The scarcity of molecular studies of polymer nanocomposites (PNC) can be largely attributed to the difficulty associated with generating large ensembles of equilibrium configurations for highly entangled polymeric materials, and the challenges involved in studying the structure and dynamics of such systems out of equilibrium (e.g., under tension or compression). A number of theories have been proposed to explain the influence of nanoparticles on nanocomposite behavior. The “interaction zone” hypothesis posits that a reduction in polymer mobility near the surface of a particle arises from a local reorganization of the polymer into a more structured or possibly glassy state.6,7 This ordered or glassy region imparts improved mechanical properties to the system, with smaller particles having a disproportionate influence for their concentration.6 A second hypothesis is that fillers may act as highly functional cross-links8 capable of forming bridges of polymer between them. A study of entangled PNC annealing below Tg9 suggested that aggregating particles formed more such bridges. Finally, it has been suggested that particles may directly10 or indirectly11 alter the entanglement state of the system. Recent simulations12 have shown that the entanglement lengths of attractive rod-containing r 2011 American Chemical Society
PNCs are significantly lower than that of the pure polymer for inclusions that are themselves immobile on the time scale of reptation. However, if the time scales of additive relaxation are much faster than reptation, the entanglement length of the PNC is comparable to that of the corresponding pure polymer. These results suggest that inclusions influence entanglement length by direct restriction of chain motion, and not through an alteration to the matrix configuration vis-a-vis that of the pure polymer. Simulation studies have examined several aspects of PNC reinforcement. Simulations of smooth spherical fillers, smaller than the polymer’s radii of gyration, showed better fracture resistance in the melt state than below Tg.13 Gersappe attributed this behavior to particle mobility, which allowed for better energy dissipation at higher temperature. A nanoparticle clustering study showed that the shear viscosity is higher for dispersed particle systems, in contrast to the behavior observed for macroscopic particles.14 That study further showed that deformation tends to disperse particles. Recent simulations have also examined the deformation behavior of PNCs and compared it to that of the corresponding pure glasses.15 It was found that nanoparticles stiffen the polymer glass (they increase the elastic modulus) and suppress creep response. Those simulations demonstrated that during deformation, the mobility of the polymer chains increases significantly in both the pure polymer and the nanocomposite systems. Received: July 26, 2011 Revised: October 12, 2011 Published: December 12, 2011 543
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Macromolecules Experimental studies of both carbon nanotube composites16 and polymers with clay inclusions17 suggest that particle size and shape are key factors in the mechanical response of polymer nanocomposites. However, as noted in a recent literature review,18 theoretical and computational studies of PNC with anisotropic inclusions have been limited. Buxton and Balazs19 used a network spring model whose spring constants were chosen to describe different shaped regions of matrix and filler. They used this model to simulate the behavior of spheres, rods, and disks in a polymer and demonstrated that rods and disks, when randomly aligned, provided superior mechanical properties in the elastic regime as compared to spheres. Knauert et al.20 examined the viscosity and isotropic tensile strength of isohedrals (i.e., many-sided polygons), rods, and sheets using a coursegrained beadspring model. Recognizing the difficulty of accurately measuring viscosity in the limit of zero strain rate, they opted to perform nonequilibrium simulations of unentangled PNC systems at high shear rates, in the shear-thinning regime. Of the three filler geometries considered in their work, they found that rods provide the largest increase in viscosity, but improved strain at fracture the least. They noted that particlepolymer interactions correlate well with the viscosity increase, but they were unable to provide an explanation for the observed changes in strain at fracture. Interest in rod-like additives is largely motivated by interest in carbon nanotubes, which have been considered extensively in the literature as polymer fillers.2124 However, for the comparatively short rods examined in this paper, the closest experimental analogue is provided by the work of Kalfus and Jancar,25 who examined spherical and platelet hydroxyapatite inclusions in poly(vinyl acetate) in order to examine the role of geometry. They found shape to have a negligible effect on storage modulus in the glassy state, but did note that the storage moduli of the spherical PNCs had stronger temperature dependence in the melt state. They concluded that filler geometry was a secondorder contribution to PNC reinforcement at constant temperature. As an alternative to inherently anisotropic fillers, Akcora et al. examined poly styrene grafted silica particles in a poly styrene matrix.26 These spherical additions formed various anisotropic aggregates depending upon the length and number of grafted chains. They found that particle sheets induced solid-like behavior during shear deformation, whereas dispersed spheres did not. In what follows, we provide a systematic study of the effect of particle aspect ratio on the mechanical behavior of nanocomposites. Computational models, which have been under-utilized in the study of anisotropic PNC systems, allow us to probe the effect of rod length without perturbing other variables, and they provide a reasonable description of the short length and time scales at the limits of experimental accessibility. Specifically, we examine the local dynamics of these systems in the melt state, and how they respond to constant strain-rate deformation in the glassy state. We find that particle geometry has little effect on elastic properties at constant temperature, but higher-aspect ratio additives improve the postyield strength of the composites. However, clear trends with rod length do occur in the elastic regime if materials are examined at the same temperature relative to their respective glass transitions. Nanocomposites show longer bond orientation relaxation times than the pure polymer during all stages of deformation. Additionally, for the small additives we consider, polymer mobility is not reduced meaningfully in the presence of the particle.
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Figure 1. Cooling curves for each PNC. Results above a temperature of 1.1 are erratic due to the quench from T = 1.75 to T = 1.2. Only data values shown in the figure were used for the determination of Tg.
’ MODEL AND CHARACTERIZATION This work utilizes a coarse-grained Lennard-Jones polymer model of fully flexible chains. Semiflexible nanorods of 1, 4, 6, 8, and 16 beads were inserted into these melts at concentrations of 10.0 wt %. Throughout this manuscript, nanocomposite systems are referred to by the length of their inclusions (i.e., 1-mer, 4-mer, etc.). The strength of polymerparticle interactions is three times stronger than that of polymerpolymer or particleparticle interactions, thereby promoting dispersion of the rods in the polymer matrix; i.e., εPN = 3.0εPP = 3.0εNN, where εPP represents the polymerpolymer LJ well depth, εNN is the nanorod nanorod well depth, and εPN is the cross interaction well depth. Both particles and polymer are constructed of beads with a diameter of σ. If we take σ to be on the order of 2 nm, a typical Kuhn length for a flexible polymer, then our nanorods have a diameter similar to that of single-walled carbon nanotubes. The bonds along the polymer backbone are modeled as harmonic springs with an equilibrium bond length of σ; the nanorod internal bonds have an equilibrium bond length of 2/3 σ, leading to nanorod aspect ratios of 1, 3, 4.33, 5.67, and 11. The filler has an additional bending potential that enforces a nearly linear configuration. Additional details on the model and its properties have been presented in previous work.12 In reduced units, temperature, pressure, and time are defined respectively as (kT)/(ε), (σ3P)/(ε), and ((εt2)/(mσ2))1/2. Here T, P, and t are temperature, pressure, and time, k is the Boltzmann constant, and m is the mass of a bead, taken to be unity in this study. Time in reduced units will be denoted as τ. Starting with the equilibrated melts at T = 1.75 produced previously,12 several independent configurations of each sample were used to prepare glasses. Systems were quenched to a temperature of 1.2, well above Tg, and then cooled to a reduced temperature of 0.3 over 900τ at a cooling rate of 103. From each cooling run, two configurations were extracted, corresponding to two different temperatures below the glass transition temperature. The first set were all extracted at the same absolute temperature of T = 0.3. The second set were extracted at the same temperature relative to their respective glass transitions, T/Tg = 0.9. For the T/Tg = 0.9 runs, Tg was obtained for each system from cooling runs by plotting density as a function of temperature. Results are shown in Figure 1; lines were fit to the high temperature and low temperature regimes, and their intersection 544
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Table 1. Number Density at Various Temperaturesa
a
density at
density at
density at
system
T = 1.75
T/Tg = 0.9
T = 0.3
Tg
pure polymer
0.667
0.984
1.024
0.519 ( 0.049
1-mer
0.772
0.999
1.036
0.620 ( 0.017
4-mer
0.797
1.026
1.063
0.610 ( 0.020
6-mer
0.796
1.027
1.065
0.610 ( 0.020
8-mer
0.792
1.028
1.067
0.607 ( 0.020
16-mer
0.770
1.036
1.073
0.557 ( 0.030
Those authors reported that strongly triaxial deformation usually leads to crazing, and that the exact transition between modes of failure is determined by a pressure-modified von Mises criterion. Addition of nanorods to the polymer imparts significant changes to the density of the resulting composite (Figure 1). Table 1 summarizes the density of the various PNCs in the equilibrium melt (T = 1.75), as well as in the two glassy states considered here (T = 0.3 and T/Tg = 0.9). Across all temperatures, the pure polymer has a lower density than the nanocomposites, but this densification effect is most pronounced in the high-temperature melt state. In general, the 4-mer, 6-mer, and 8-mer systems have similar densities, while the 1-mer is less dense. The density of the 16-mer exhibits a stronger temperature dependence relative to the other PNCs. The glass transition temperature decreases with increasing rod length for the nanocomposites, and the pure polymer has the lowest Tg. The nanocomposite systems with shorter rods (1-mer, 4-mer, 6-mer, and 8-mer) show excellent dispersion of the particles. It was previously shown12 that these nanorods exhibit a small degree of local alignment over a length scale approximately equal to the rod’s length, but that they adopt a random orientation globally (see Figure 2). The 8-mers show some regions of relatively higher and lower concentration, but no clear tendency to organize beyond such concentration fluctuations. For the 16mer nanocomposite, however, the rods self-assemble spontaneously to form large bundles or aggregates. These bundles consist of 8 to 20 parallel rods separated by a monolayer of polymer. Figure 2 shows a representative configuration of these aggregates. In all systems, the attractive polymerparticle interaction leads to a polymer layer surrounding the rods, but for the 16-mer aggregates it leads to trapped layers of polymer between the aligned rods.12 The bundles form transient end-to-side connections at several points during the simulation, but fail to organize on a global scale. Note that the locally organized regions of polymer around each rod are in contact with each other in such a way as to create a network that percolates the system. Because of the relatively high concentration of additives in our simulations, these polymerparticle connections span the entire system for all rod lengths considered here. The dynamics of the resulting composites are quantified in terms of the bond autocorrelation function, Cb(t), which previous studies have used to calculate segmental dynamics and relaxation times.15,30,31 This quantity is given by
All density values are (0.002.
Figure 2. Representative configurations of nanorods dispersed in a polymer matrix at T = 1.75. For clarity, only the nanorods are shown. The figures from left to right correspond to a 10 wt % filler concentration of 4-mers, 8-mers, and 16-mers. Only the 16-mers were found to selfassemble into bundles.
was identified as Tg. The values generated in this manner are given in Table 1. The Tg of the pure polymer is larger than reported in previous works,27,28 where a slower cooling rate was employed. The thermal history of the current samples is the same as that used in our previous examination of entanglement lengths during glassy deformation.12 Results obtained at the two different temperatures were similar. In the interest of brevity, the majority of this work focuses on the T/Tg = 0.9 systems. Discussion of the T = 0.3 results is limited to a subsection that highlights the main differences observed at the two temperatures. Complete results for the T = 0.3 systems are available in the Supporting Information. After cooling, systems were aged for 1000τ at constant pressure. The systems then underwent uniaxial compression or extension, performed by deforming one dimension at a constant true strain rate while the other dimensions were allowed to relax to maintain constant pressure. All systems maintained nearly constant volume during this process. Strain rates of 4.4 104 and 1.1 104 were applied for both modes of deformation until final true strains of 1.104 and 1.027 were attained, respectively. Note that all strain rates will be reported as positive numbers, although the true sign will be positive for tension and negative for compression. For the 16-mer nanocomposite systems, results are truncated at an absolute strain of 0.7, well before the smallest dimension of the simulation box begins to approach the length of the nanorods. Such deformations are described as simple uniaxial, in contrast to the triaxial deformations that induced cavitation in previous work.28 The systems considered here would undergo shear yielding if deformed further. A detailed exploration of loading conditions required to induce alternative modes of failure in PNC is beyond the scope of the current work; however, note that Roettler and Robbins29 have discussed the case of pure polymers.
Cb ðtÞ ¼ ÆP2 ½bðtÞ 3 bð0Þæ
ð1Þ
where P2 is the second Legendre polynomial, b(t) is a unit vector aligned along the bond of a polymer, and the angular brackets indicate an average over all bonds in the system. We quantify changes to the system dynamics by fitting the bond autocorrelation function to a KohlrauschWilliamsWatts (KWW) stretched exponential function of the form Cb ðtÞ ¼ Co eðt=τeff Þ
β
ð2Þ
where Co, τeff, and β are fitting parameters referred to, respectively, as the pre-exponential factor, unnormalized relaxation time, and the stretching exponent. Because various properties of our system are not constant during deformation (e.g., the stress), we find it useful to measure the dynamics within discrete “timewindows”, thereby facilitating comparison to changes in other properties. After performing a deformation, Cb is fit to simulation data spanning time-windows of 300, 600, and 900τ. The length of 545
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Figure 3. Results for deformation in compression (top) and extension (bottom) for 10 wt % filler at T/Tg = 0.9 under a strain rate of 4.4 104. Black lines represents pure polymer, other lines represent nanocomposites loaded with nanorods of indicated lengths.
the time-window chosen had no statistically meaningful effect. Additional information about the time-window approach employed here is given elsewhere.30
’ GLASS DEFORMATION A simple and direct way of assessing mechanical properties from simulations is to perform deformations in the glassy state. Such nonequilibrium molecular simulations provide information about the dynamics of the system as well as a direct measurement of mechanical properties. For the remainder of this work “strain” will be used to refer to the absolute value of true strain, and “stress” will be used to refer to the absolute value of the true stress in units of εPP/σ3. For all PNCs considered here, the addition of particles drastically increases the stress required to deform the system. The stressstrain behavior of our PNC glasses is illustrated in Figure 3 at a strain rate of 4.4 104. Stressstrain behavior at a strain rate of 1.1 104 (not shown for brevity) is qualitatively similar; differences resulting from changes in strain rate will be covered in the following discussion. Strain softening, a decrease in stress following yield, occurs in all systems except the 16-mer. Strain softening in polymeric systems results from a structural reorganization and is associated with rejuvenation in glasses.32,33 For the spherical particles, the
Figure 4. Elastic modulus (top), yield strain (middle), and yield stress (bottom) as a function of rod length. Pure polymer results represented as horizontal lines. Black lines denote compression and red lines denote tension. Solid lines are used for systems with strain rates of 1.1 104 and dashed lines for systems with strain rates of 4.4 104. Note that the error reported for these systems is the error in the measurement at that given temperature; there is an additional error associated with identifying a specific glass transition temperature.
strain softening behavior is more pronounced than that observed in the pure polymer. This softening behavior decreases with increasing rod length, entirely disappearing in the 16-mer case. The stressstrain response to simple uniaxial deformation observed here stands in contrast to the response obtained during triaxial deformation observed in previous work.28 The stress peaks observed in triaxial deformation are 4 (PNC) to 6 times (pure polymer) larger than those reported here. This is due to the stress-peak representing fundamentally different behaviors. 546
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systems exhibit the lowest PNC yield stress. The 16-mer has a heterogeneous structure. That is, there are domains rich in nanorods and domains devoid of rods. Therefore, it is possible that domains devoid of nanorods require less stress to initiate flow. This reduction in yield stress upon aggregation runs counter to the observations of Akcora et al.26 They instead observed solid-like behavior not seen in their dispersed systems. However, the additive used by Akcora et al. was covalently bound to entangling polymer segments whereas polymer is bound to the 16-mer only by LennardJones interactions, albeit strong ones. Regardless of rod length, a higher stress is required to induce yield when the system is deformed more rapidly. The error in the pure polymer measurement of yield stress is (0.017. Local Stress Distribution. In addition to global stressstrain behavior, molecular simulations enable a detailed examination of the local stress state of each simulation bead. Here, local stress is calculated according to σ ¼
∑
1 fr 2V neighbors
ð3Þ
where σ is the stress tensor, V is local volume, f is the vector of force on a bead by a given neighbor, and r is the vector denoting the distance to the neighbor. For simplicity, local volume is treated as the average specific volume of the system. This results in a slight underestimation of the magnitude of the stress for additives as compared to the polymer, but has little qualitative effect on the observed behaviors. In order to better understand local stress behavior, it is useful to examine the spatial distribution of tensile stress in the direction of deformation (σxx) through the use of color maps, as shown in Figure 5. Here the local stress of polymers in thin slices is displayed with rod sites overlaid as black dots. The local stresses of the rods themselves (not included in production of the color maps) are strongly negative. Recall that the attractive interaction between rods and polymers produces an organized region of dense material around the additives. The rods pull polymer chains inward (negative stress) and the shell of polymer in the immediate vicinity of the rods is tightly packed, giving rise to positive stresses. This effect is plainly visible in the 16-mer aggregates, as denoted by red regions around the rods. The induced stress differential is less prominent for shorter rods, and difficult to discover through visualization alone. However, calculation of the average local stress of polymer as a function to the nearest rod (not shown) clearly reveals that the first solvation shell of polymer has significantly higher local stress in all PNCs. The pure polymer has some regions of higher and lower local stress, but these fluctuations are small as compared to the scale of stress gradients found around PNC rods. In PNCs, polymer far from the rods exhibits a distribution of stresses that is similar to those of the pure polymer. Two configurations of each system are shown in Figure 5, one prior to deformation and one at the yield point. Over this small strain, the positions of nanorods in each system do not change appreciably. In the pure polymer, regions of high and low local stress change position, suggesting that nonuniformity is the result of transient fluctuations and not the underlying structure. The same behavior is seen in PNCs in the polymer located far from the additive rods. The application of strain does not induce different stress responses across the system. Instead, all polymer and particles show, on average, a uniform increase in local stress that matches the global increase in stress for that system.
Figure 5. Cross section of local stress corresponding to 3σ thick slices. Images A, C, and E correspond to systems before deformation, and images B, D, and F correspond to systems at a strain of 0.055 (near yield point). Images A and B correspond to pure polymer, C and D show results for the 4-mer PNC, and E and F for the 16-mer PNC.
In a simple uniaxial deformation, the peaks around 0.05 strain correspond to the yielding transition between elastic and plastic behavior. In the triaxial deformation, the peaks denote the onset of cavitation, and thus failure. Failure under uniaxial deformation occurs at strains beyond those examined in the current work. Elastic Properties. The elastic properties of the PNCs as a function of particle length were extracted from deformation simulations, as summarized in Figure 4. On average, the elastic moduli (initial slope of the stressstrain curve) of the PNCs are a factor of 2 to 4 times higher than those of the pure polymer for short-rod systems. There is an overall trend of decreasing elastic modulus with increasing rod length. Systems subjected to strain rates of 4.4 104 had elevated values of elastic modulus as compared to those deformed at strain rates of 1.1 104. The yield point (i.e., the yield stress and yield strain) was determined by measuring where a line with an x-intercept of 0.02 and a slope equal to the elastic modulus intersects with the stressstrain curves. The yield strain increases monotonically with rod length, except for the 16-mer. There was a high degree of variability for pure polymer yield strain, reflected in errors in pure polymer yield strain of up to (0.0048. While a trend on increasing yield strain with increasing rod length is clear, the relation between PNC and pure polymer yield strain is not. The yield stress is about 0.7 higher for the PNCs than for the pure polymer, and changes only slightly with rod length. While the effect of rod length is minimal, it is statistically significant for rod lengths up to the 8-mer. In contrast, the aggregated 16-mer 547
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Figure 7. Polymer (top) and particle (bottom) orientation under extension for 10 wt % filler for T/Tg = 0.9.
Figure 6. Polymer (top) and particle (bottom) orientation under compression for 10 wt % filler at T/Tg = 0.9.
This suggests that stress propagates rapidly in the PNC as compared to the rate of deformation. The results in the figure are for compression at constant T/Tg, but the same behavior occurs at constant T and during tensile deformation. Additionally, the slices shown in the color map are perpendicular to the axis of deformation, but slices through other planes reveal the same behavior. At higher strains than those presented in the color maps, we find the same trends, albeit rearrangement of particles and polymer chains make side-by-side comparison less clear. The current simulations do not continue to the point where local fluctuations can organize to produce failure. A discussion of failure in shear-yielding is beyond the scope of this work. Rod and Polymer Orientation. Rods and polymer molecules exhibit a tendency to align either perpendicular (in compression) or parallel (in extension) to the direction of deformation. We quantify alignment during deformation by measuring correlations between the end-to-end vector of the polymer or particle with the direction of deformation. Using a second Legendre polynomial of their product we have 1 ð3ðbi 3 ux Þ 1Þ ð4Þ Px ¼ 2
alignment, to zero for random orientation, to 1.0 for perfect parallel alignment. We examine both the end-to-end alignment of polymers and the end-to-end alignment of the nanorods. The changes in alignment of the polymer chains during compression are shown in Figure 6a; the average alignment with the deformation axis goes smoothly from ∼0.0 (random) to ∼-0.42 (highly perpendicular) for all systems. As we are deforming a finite number of chains, the initial extent of alignment differs slightly from zero for some systems. There are only small differences among the various nanocomposite systems at small deformations, and there is essentially no difference for large deformations. Figure 6b shows the orientation of the nanorods. At low strains, shorter rods reorient themselves more than longer rods. However, at high strains, the alignment of the 4-mers reaches an asymptotic Px value of approximately 0.32. At these strains, the higher rotational mobility of shorter rods is balanced by the aligning force exerted on the system. The onset of similar behavior is visible for the 6-mers. The clustering of 16-mers into bundles results in large numbers of rods having the same orientation, and we therefore have less independent orientational vectors for the 16-mer systems. Having relatively few independent vectors, the initial average orientation is not random, and it is difficult to draw meaningful conclusions from its orientational changes. In extension, the polymer chains align parallel to the axis of deformation, as shown in Figure 7a. As was the case under compressive deformations, there is no clear difference between the pure polymer and the polymer in the nanocomposites.
where Px measures the correlation, bi is the unit vector along the end-to-end vector of the polymer or particle i, ux is a unit vector aligned in the x-direction (the axis of deformation), and the angular brackets indicate an average over all particles (or polymer chains). Values of Px scale from 0.5 for perfect perpendicular 548
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Figure 9. Hardening modulus at a strain rate of 4.4 104 as a function of sites per nanorod. Pure polymer results represented as horizontal lines. Black lines denote compression and red lines denote tension. Solid lines are used for systems with strain rates of 1.1 104 and dashed lines for systems with strain rates of 4.4 104.
By plotting the stress response of Figure 3 against λ2 λ1, a linear relationship is obtained in the postyield regime (see Figure 8); the hardening modulus is the slope of this line. Significantly more advanced rubber-elasticity models that include nongaussian response (e.g., finite extensibility) and couple chain relaxation to account for nonaffine behavior have also been developed; Boyce and Arruda have compiled a review36 of these models. However, a simple Gaussian model is sufficient to describe the phenomenology of interest in the present work. The applicability of such a simple model to a polymer undergoing large deformations is consistent with the experimental work of Tervoot and Govaert.37 We examine the hardening modulus at a deformation rate of 4.4 104 in Figure 9. The PNCs under compression show an increase of hardening modulus with rod length, with the 16-mers exhibiting a hardening modulus that is twice as high as that observed with spheres. In tension, all PNC exhibit the same hardening modulus within the statistical uncertainty, except for the 1-mer. However, we see that all PNCs have higher toughening moduli than the corresponding pure polymer. A number of shortcomings of rubber-elasticity models for hardening modulus have been noted in the last several years.3842 These include the observation of hardening in chains of onequarter the entanglement length, a temperature dependence that is in conflict with experiments, and inconsistencies in the order of magnitude of the predicted response. A polymeric system is more complicated than the simplistic description provided by eq 5. Likewise, the addition of rods, which distort the chain distributions on short length scales,12 violate the Gaussian assumption behind this neo-Hookean model. Despite these disagreements regarding the validity of the underlying physical model, however, polymer and PNC systems can be described qualitatively by the relation prescribed in eq 5. In order to test whether the observed hardening modulus trends are consistent with the assumptions of the rubberelasticity model, we compare our current result with our previous examination of Ne in the melt state at T = 1.75.12 As explained in the literature, evaluation of entanglements in the presence of filler materials was addressed in the limits of high and low mobility on the time scales of reptation. Note that the exact value obtained for Ne depends on the method of entanglement
Figure 8. Rescaled results for deformation in compression (top) and extension (bottom) for 10 wt % filler at T/Tg = 0.9. Black lines represents pure polymer, other lines represent nanocomposites loaded with nanorods of indicated lengths.
Figure 7b shows the alignment of the particles; the smallest rods again show greater changes in Px than long rods at low strain. However, the alignment of the 4-mer begins to saturate at higher values of strain then it did under compression, with the final value lying slightly above Px = 0.6. Postyield Behavior. In the postyield regime, the hardening modulus provides a measure of how much additional stress is required to deform a material as strain increases (in the plastic regime). Historically, the increased resistance to deformation in polymer materials with increasing strain has been examined under the framework of entropic network models.34,35 Under such models, a polymer melt is treated as a cross-linked network of polymer chains, where the length of segments between crosslinks is the polymer entanglement length (Ne). The stress (σ) required for deformation arises from the entropic resistance to straightening the polymer segments. The simplest possible such model assumes ideal Gaussian behavior for the chain segments as they undergo affine deformation. The Gaussian or neo-hookean model will exhibit a stressstrain response of the form:34,35 σ ¼ σ0 þ GR ðλ2 λ1 Þ
ð5Þ
where σ0 is flow stress to account for preyield behavior, λ is the stretch tensor defined as the ratio of the current box length to the original box length, GR is the hardening modulus, and the term in parentheses results from the entropy of a Gaussian chain. 549
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Figure 10. Unnormalized relaxation time (top) and stretching exponent (bottom) at T/Tg = 0.9 during compressive (left) and tensile (right) deformation.
identification;43 values reported here correspond to a contact definition of entanglements. At T = 1.75, the pure polymer has an Ne of 56.0 ( 3.5. In the limit of fast particle relaxation, the PNC Ne’s were statistically indistinguishable from that of the pure polymer. As such, no difference in hardening modulus between the different systems would be predicted. This is consistent with the lack of trend with rod length observed for hardening modulus during tension, but cannot explain the compressive behavior. In the limit of slow particle relaxation, Ne monotonically increases with rod length from 19.65 ( 0.11 to 26.23 ( 0.42, suggestive of a much more entangled system than the pure polymer in all cases. In this case, the hardening modulus would be predicted to decrease with rod length; instead, an increase in hardening modulus with rod length is observed in our simulations. Regardless of which particle relaxation assumption is used, previously calculated Ne values cannot be used to explain the trends in hardening modulus observed here. This finding is in contradiction with rubber-elasticity models, but is not unprecedented.3842 Bond Autocorrelation. The dynamics of the nanocomposites were further examined in the glassy state during tensile deformation by calculating the bond autocorrelation function, Cb(t). Previous studies have shown that Cb(t) provides a useful measure of the segmental dynamics in polymer glasses, similar to the incoherent dynamic structure factor, Fs(q,t).31 Here we apply the time-window approach described earlier. Similar approaches have been employed in previous simulations30 and have been shown to be consistent with optical measurements of the segmental dynamics of PMMA during creep deformations.44,45 As described earlier, we extract fitting parameters for τeff, β, and Co from plots of Cb(t) in specified windows. We focus on
Figure 11. Average particle diffusion from sample runs for 10 wt % filler.
four such windows corresponding approximately to the elastic regime (strain 0.0020.035), near yield (strain 0.0370.069), early postyield including strain softening if present (strain 0.0720.104), and late postyield (strain 0.1410.171). Several ranges were tested for the late postyield, and all gave similar results. Figure 10 summarizes the results for the unnormalized effective relaxation times, τeff, and the stretching exponent, β, during extension for the PNC systems at a deformation rate of 1.1 104. The value of the strain associated with τeff is chosen as the strain in the middle of the window used to determine τeff. Under both tension and compression, there is a clear trend of increasing relaxation time with rod length, and all PNCs have slower polymer relaxation than the pure polymer. The 16-mers 550
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Figure 12. Dynamic structure functions for particle components of the 1-mer (top left), 4-mer (top right), 8-mer (bottom left) and 16-mer (bottom right) evaluated at different wave vectors. Black, red, green, and blue curves represent q values of 7.0, 1.6, 0.65, and 0.5 respectively.
are an exception to the order, having relaxation times closest to the pure polymer. Given that these relaxation times were calculated from relatively short windows, the fastest polymers to relax in the system are going to strongly influence the measured time, and the 16-mer systems (unlike the shorter rods) have large domains without inclusions. For both compression and tension, there is a strong trend of decreasing β with increasing rod length, with values for the pure polymer falling near or slightly above those for the 4-mer. As a decrease in β corresponds to a broadening of the relaxation spectrum, these results show that longer rods either introduce additional time scales for relaxation or alter ones that are normally less important in the pure polymer. Only the 1-mer system has a higher β than the pure polymer. The 16-mer system follows the same trend as the other rods, despite its aggregated nature. Brief Comment on Constant Temperature Trends. As noted before, results obtained for systems at T = 0.3 are broadly in line with those discussed for systems at T/Tg = 0.9, and the complete details of the T = 0.3 results are given in the Supporting Information. However, a few distinct differences do exist, necessitating a brief comparison of results here. Recall that T = 0.3 is a lower temperature than T/Tg = 0.9 for all systems examined here (as per Table 1). It is also important to note that the glass transition temperature changes with rod length in a nonlinear away. The overall stressstrain curves are in good qualitative agreement between the two temperatures, although the stresses obtained during deformation at T = 0.3 are higher in order to overcome the reduced mobility of these systems. Similar to the deformations at T/Tg = 0.9, strain softening is more pronounced in compression than in extension. The rod length dependence of elastic properties does not exhibit clear trends at T = 0.3. The general insensitivity of elastic properties to geometry at constant
temperature is consistent with the experimental work of Kalfus and Jancar.25 However, values obtained for elastic modulus are significantly different for different sets of rod length. The apparent lack of trends with rod length could be the result of conflicting influences between rod length and degree of cooling below Tg. It is also possible that the role of the additive in the elastic regime is generally more pronouced near Tg than at lower temperatures. Rod and polymer orientation differ trivially between T = 0.3 and T/Tg = 0.9. Examination of bond autocorrelation shows the most profound differences between T = 0.3 and T/Tg = 0.9. The qualitative shape of the strain versus relaxation time curve is not temperature dependent, however the T = 0.3 systems exhibit longer relaxation times, as expected.
’ MELT DYNAMICS The dynamic behavior of the PNCs above Tg was characterized in two ways: analysis of particle mean-squared displacement (msd) and calculation of dynamic structure factors for both particles and polymers. These calculations were performed on the equilibrated melts at T = 1.75. Figure 11 shows average particle displacements as a function of time. As expected, short rods diffuse more quickly than long rods, with spheres achieving 100σ2 msd in only 285τ. For the 16-mer system, as mentioned previously, significant aggregation of the particles occurs. The resultant aggregates have drastically reduced mobility. At long times, all inclusions other than the 16-mer are seen to achieve a slope of unity, consistent with diffusive motion. The 16-mer has a final slope of 0.47 ( 0.02, which corresponds to subdiffusive motion. Given the intertwined nature of the polymerparticle bundles, this subdiffusive behavior is not surprising. 551
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Figure 13. Dynamic structure functions for polymer components of 4-mer PNC (top) and 16-mer PNC (bottom) evaluated at different wave vectors. As before, black, red, green, and blue curves represent q values of 7.0, 1.6, 0.65, and 0.5 respectively.
Figure 14. Dynamic structure functions for q values of 7 and 1.6 for polymer components of the 4-mer PNC (top) and 16-mer PNC (bottom). Each line represents a different layer of polymer beads about the nearest particle.
To further evaluate the mobility of polymer and particles, we calculated dynamic structure factors according to 1 Fs ðq, tÞ ¼ N
∑j eiq 3 ðr ðtÞ r ð0ÞÞ j
j
polymer, the 1-mer, and the 8-mer systems. For the 16-mer system, however, the curves at q = 7 and q = 1.6 show the presence of an additional mode of decay, as indicated in Figure 13b by the slow approach of Fs to zero after an initial, rapid decay. To investigate the origin of this decay mode, we examined the relaxation of monomers as a function of their distance from the nearest particle in the initial configuration. The dynamic structure factors for q = 7 and q = 1.6 are shown in Figure 14 for each of the first four solvation shells of polymer about the particles, for 4-mer and 16-mer PNCs. A bead is tagged with regard to which shell it is in when the simulation begins, regardless of its eventual position. Polymer relaxation in the presence of smaller particles shows some spatial-temporal correlation for q = 7 and 1.6, with closer shells having slower dynamics. For the 16-mer nanocomposite, though, the beads in direct contact with the particles exhibit substantially hindered dynamics. The effect is much stronger than in the 4-mer case, and relevant across several values of q. If one thinks of monomers as being weakly bound to the surface of particles, it follows that the dynamics of polymer material in the vicinity of the particle would be similar to that of the particle itself. For the dispersed nanocomposite systems, the particles are highly mobile and hence do not reduce polymer mobility substantially. For the 16-mer nanocomposite, the polymer trapped in the aggregated bundle is dynamically hindered.
ð6Þ
where Fs is the dynamic structure factor, q is the wave vector, t is time, rj is the position of the jth bead, and i is the root of 1. The dynamic structure factors were calculated at several q, corresponding to length scales between 0.9σ and 12.6σ, and the real component plotted in Figure 12. Fs probes mobility on length scales of 2π/q, and can be measured experimentally by incoherent inelastic neutron scattering. Examination across all length scales considered here show a trend of decreasing additive particle mobility with increasing particle length. The times required at q = 7 to decay below 0.001 for the 1-mer, 4-mer, and 8-mer nanocomposite systems are 2.5τ, 6.3τ, and 14τ, respectively. Larger particles are observed to move more slowly; the 16-mer bundles have drastically reduced mobility and show a dynamic structure function value of 0.018 at 100τ for q = 7. The motion of the polymers was also characterized by dynamic structure factors. The color scale remains the same. The pure polymer (not shown) and the polymer component of the PNCs had similar behavior in general. Figure 13a shows the dynamic structure factor for the polymer component of the 4-mer PNC, which is generally representative of the pure 552
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As such, polymers in the vicinity of the 16-mer particle relax more slowly than in the bulk, leading to an additional mode of decay in the polymer dynamic structure function. A similar analysis of polymer mobility near the surface of comparatively large icosahedron-shaped particles was performed by Starr et al.46 They found polymer relaxation in their systems to exhibit the same slowing of polymer dynamics in the vicinity of the particles that we found for the 16-mer PNC, despite their polymers not being strictly “trapped”. As in our case, this led to an additional mode of decay in the averaged dynamic structure factors at short length scales. This similarity implies that polymers on the surface of a large particle are just as restrained as the polymers in our aggregates. It should also be noted that the lack of difference in polymer mobility with respect to nanorod position for our other PNCs suggests that our smallest particles may affect the polymer matrix by different mechanisms than those of other PNC studies.
’ CONCLUSION The polymer nanocomposite literature has focused on spherical inclusions, providing comparatively less information on anisotropic fillers such as rods. To address the influence of inclusion shape on composite behavior, we characterized polymer melts containing short, truly nanosized rods. We found that a combination of rebridging Monte Carlo and molecular dynamics moves used in tandem provide the best means of sample equilibration. In this paper, nanorods ranging from 1-mer to 16-mer were examined. In all PNCs (except the 16-mer), the additives were dispersed throughout the system and maintained a random orientation in the melt state. The 16-mer additives, however, aggregated into bundles having a high degree of local order. The mechanical strength of PNC glasses was found to be influenced by rod length. We performed deformation simulations at T = 0.3 and at T/Tg = 0.9. In general, the elastic properties at constant temperature showed only weak trends with rod length, although all PNCs were considerably stronger than the corresponding pure polymers. This is in agreement with the experimental work of Kalfus and Jancar.25 Past the yield point, however, differences with rod length did emerge. First, the spherical inclusions showed significant strain softening. This phenomenon became less pronounced with increasing rod length. Second, longer rods showed nearly a 2-fold increase in the hardening modulus as compared to the shortest rods. In contrast, at constant T/Tg clear trends with rod length did occur in the elastic regime. Specifically, elastic modulus decreased with increasing rod length, and yield strain increased with increasing rod length. In order to examine postyield behavior, the hardening modulus was calculated. The origin of this measure is rooted in entropic network models, necessitating adoption of their formalism to define hardening moduli despite conflicts associated with the underlying physical mechanisms. It was found that the hardening moduli were higher for PNCs than for pure polymers, and that hardening modulus increased with increasing rod length for compressive deformation. These trends could not be reconciled with our previous examination of Ne12 under rubber-elasticity models. Previous literature has noted several short-comings in using rubber-elasticity models to describe hardening behavior.3842 The disconnect between entanglements and hardening moduli shown here is another such discrepancy.
Bond relaxation dynamics during deformation reveal several results; the following results apply to T/Tg systems. Additives induce slower polymer orientational relaxation than found in the pure polymer, and the increase in unnormalized relaxation time is highest in the elastic regime. Increasing rod length strengthens this effect, with the exception of the aggregated 16-mers, most likely due to the relatively higher mobility of polymer-rich regions away from the rod bundles. There is also a strong trend of decreasing stretching exponent with increasing rods length, indicative of a broadening of the relaxation spectrum. Only the 1-mer system had a higher stretching exponent than the pure polymer. Despite the additional statistical uncertainty associated with the estimation of the glass transition temperature, analysis of our results at T/Tg = 0.9 provides better insights into the immediate role of additive length. The elastic properties of the polymer, while drastically affected by additives, are generally insensitive to rod length if examined at constant temperature. However, evaluation at constant T/Tg reveals clear trends, indicating that rod length affects the system beyond the apparent change in Tg. Lastly, we examined the dynamics of the PNC in the melt. We found that particle mobility is reduced with increasing rod length, especially the aggregated 16-mer system. Polymer mobility in the PNCs remained largely independent of particle length, except in the 16-mer case. In that system, there was an appreciable slowing of polymers near the surface of 16-mer aggregates. Only weak position-mobility correlation was found for other nanocomposites, likely due to rod mobility approximately matching polymer mobility. The ability of particles to provide reinforcement without the reduction of local dynamics is in contradiction to zonetheory type models. It has been shown in other studies that larger additives reduce polymer mobility and this correlates well with improved mechanical properties. However, our results indicate for smaller additives that reinforcement mechanisms not resulting in an appreciable reduction in mobility are applicable.
’ ASSOCIATED CONTENT
bS
Supporting Information. Results for systems examined at a constant temperature of T = 0.3. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We would like to thank the Grid Laboratory of Wisconsin (GLOW) for use of their computation resources and the Semiconductor Research Corporation (SRC) for their generous funding. The authors are also grateful to the National Science Foundation for support through the Nanoscale Science and Engineering Center at the University of Wisconsin. ’ REFERENCES (1) Komarneni, S. J. Mater. Chem. 1992, 2, 1219–1230. (2) Kamel, S. eXPRESS Polym. Lett 2007, 1, 546–575. (3) Huang, Z. M.; Zhang, Y. Z; Kotaki, M.; Ramakrishna, S. Composites Sci. Technol. 2003, 63, 2223–2253. (4) Gangopadhyay, R.; De, A. Chem. Mater. 2000, 12, 608–622. (5) Crosby, A. J.; Lee, J. Y. Polym. Rev. 2007, 47, 217–229. 553
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