Simulation of interphase percolation and gradients in polymer nanocomposites

Composites Science and Technology 69 (2009) 491–499

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Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech

Simulation of interphase percolation and gradients in polymer nanocomposites Rui Qiao a, L. Catherine Brinson b,* a b

Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, United States Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, United States

a r t i c l e

i n f o

Article history: Received 4 August 2008 Received in revised form 25 October 2008 Accepted 20 November 2008 Available online 6 December 2008 Keywords: A. Polymer-matrix composites (PMCs) A. Nanocomposites B. Interphase C. Finite element analysis Interphase percolation

a b s t r a c t Experimental data suggests that well dispersed nanoparticles within a polymer matrix induce a significant interphase zone of altered polymer mobility surrounding each nanoparticle, which can lead to a percolating interphase network inside of the composite. To investigate this concept and the nature of the interphase, a two-dimensional finite element model is developed to study the impact of interphase zones on the overall properties of the composite. Thirty non-overlapping identical circular inclusions are randomly distributed in the matrix with layers of interphase surrounding the inclusions. The simulation results clearly show that the loss moduli of composites are either broadened or shifted corresponding to the absence or presence of a geometrically percolating interphase network. Our numerical study correlates well with experimental data showing broadening of loss peaks for unfunctionalized composites and a large shift of the loss modulus for functionalized nanotube polymer composites. Further, our results indicate the existence of a gradient in properties of the interphase layer and that incorporating this gradient into modeling is critical to reflect the behavior of polymer nanocomposites. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Polymer nanocomposites have attracted intense attention in the past decade, due to their unique properties and great potential as future materials. Recent experimental work demonstrated that by incorporating nanoscale inclusions, polymer nanocomposites can exhibit significant improvement in mechanical, electrical, thermal, and other physical properties in comparison to their parent polymer systems. Effects of nanoreinforcement include dramatically enhanced strength, stiffness, fracture toughness, increased thermal stability and heat distortion temperature, increased chemical resistance, and increased electrical conductivity while maintaining optical clarity [1–11]. Most important is that these superior properties are achieved at very low loading levels of inclusions, so the parent polymer does not sacrifice the advantages of low density and high processibility. These extraordinary behaviors make polymer nanocomposites a promising multifunctional material in many fields, including the aerospace, automotive, and medical device industries. A variety of nanoparticles morphologies have been considered, including spherical particles (e.g. silica), platelets (e.g. clay and graphite) and nanotubes. In addition to the nanoinclusions themselves, the interphase, a special region of polymer chains in the vicinity of the nanofillers, also plays an important role in the improvement of polymer composite properties. The existence of nanofiller surfaces in the poly* Corresponding author. Tel.: +1 847 467 2347; fax: +1 847 510 0540. E-mail address: [email protected] (L. Catherine Brinson). 0266-3538/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2008.11.022

mer alters the mobility of polymer chains surrounding them. Such perturbations in polymer molecular mobility extend several radii of gyration and create regions of polymer, the interphase, with properties and response different from that of the host bulk polymer. In conventional fiber or particulate composites, the interphase is only present an amount considerably smaller than the matrix and the reinforcements, thus the interphase is mainly considered for its contribution to the load transfer. However, due to the large surface to volume ratio of nanofillers (several orders of magnitude larger than conventional fillers), the amount of interphase polymer generated in nanocomposites can be substantial. Although direct measurements of interphase region in nanocomposites are lacking, recent work [12] has correlated thin film and nanocomposite data, providing quantitative evidence that local polymer properties are altered at substantial distances from nanoparticle surfaces and scale with the results of thin films confined by substrates of the same surface chemistry. Moreover, recent experimental work by Rittigstein et al. [13] observed Tg-confinement effects hundreds of nanometers away from confining surfaces in doubly supported thin films. These results on model nanocomposites suggest similar extents for interphase regions could exist in polymer nanocomposites. Thus, in nanocomposites with a large extent of interphase and large number of inclusions, the interphase region has the potential to penetrate through the nanocomposite to form a percolating altered polymer network that may fundamentally change the response of the bulk polymer. The remarkable change of viscoelastic properties is one example that cannot be well explained without considering the contribution

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of interphase material. Recent experimental studies on the nanocomposites show that the transition zones of storage/loss modulus and tan d curves may be broadened or shifted in the time/temperature domain [8,14–16]. Since the properties of nanoinclusions are not time/temperature dependent within the experimental measurement scale, those enhancements cannot be simply attributed to the existence of nanoparticles. Moreover, some other observations indicate that the morphology of the interphase region should be taken into consideration to better explain the property changes in such composites [14,15,17,18]. For example, a remarkable shift in relaxation characteristics by 30 °C relative to the pristine polymer matrix is observed in the functionalized single-wall nanotube composite, while, in contrast, the unmodified nanotube composite retains the same relaxation characteristics as the bulk polymer [15]. Khaled et al. [14] also reported that the strong interfacial interactions between functionalized TiO2 nanofibers and a PMMA matrix leads to an increase of Tg. In these cases, a percolated network of interphase polymer surrounding the nanoparticles may be significantly impacting the Tg of the composites. Thus, a good understanding of the interphase effect on the thermomechanical properties is important for the accurate prediction of overall behavior. Micromechanical methods such as Halpin–Tsai, self-consistent scheme, and Mori-Tanaka have been applied to multiphase polymeric materials (polymer blends and composites) to understand and predict the thermomechanical properties. Although these models were developed for elasticity, it is straightforward, using correspondence principle, to invoke an analogy between elasticity solutions and viscoelasticity solutions. For example, Li and Weng studied the overall viscoelastic properties of a two-phase composite containing randomly oriented ellipsoidal inclusions using Mori-Tanaka method [19]. Colombini et al used a self-consistent scheme to analyze the mechanical properties of polymer blends [20]. An unavoidable difficulty for most of micromechanical models is that in order to obtain closed-form analytical solutions, complex shapes of inclusions are prohibited. Moreover, the assumption of a continuous matrix phase is usually fundamental in those models. For nanocomposites, both of these features of homogenization schemes are problematic because of the interphase: the interphase region surrounding the inclusion complicates the inclusion morphology and large amounts of interphase could form an irregular, connective network rendering the micromechanics approach inadmissible. In this work, we use a finite element approach to study the impact of interphase on the viscoelastic properties as well as thermal response of polymeric nanocomposites. We focus on analysis of the influence of a connective interphase network in the matrix to provide a qualitative explanation of Tg enhancement in the functionalized systems. The finite element method is employed in this study because it allows us to consider a very complex microstructure and is able to predict the macroscopic behaviors in a length/time scale close to the experimental tests. For example, the elastic properties of polymer nanocomposites have been predicted by finite element (FE) analysis and good agreement with experimental results have been reported [21,22]. The FE approach has been also adopted to study the viscoelastic properties in traditional polymer composites [23,24]. It is also noteworthy that, unlike some finite element simulations for polymer blends [25,26] in which each phase is discrete because the immiscibility of the component polymers leads to a heterogeneous multiphase system, a gradient of material properties is explicitly considered in this study. Our approach is designed to capture the evolution of the interphase in nanocomposites from discrete regions to a connective network and allows us to better understand its influence on the overall viscoelastic properties at different stages. This paper is organized as follows: The finite element model will be introduced in next section. Results and discussion will follow in Section 3. In Section 4, the conclusion will be presented.

2. Finite element model To examine the influence of the interphase on matrix dominated properties, a viscoelastic, 2D plain strain model was adopted. This configuration simplifies the computational requirements and, although it represents explicitly the case of aligned nanotubes, the results on time dependent properties are generalizable to other fiber arrangements. Since such composites can be idealized as transversely isotropic material, a representative volume element (RVE) with periodic structure is created to obtain the effective mechanical behavior of the composite. The results provide a qualitative understanding for the impact of interphase interactions on the viscoelastic response of nanocomposite. 2.1. Multiparticles unit cell generation A square unit cell containing a random dispersion of thirty nonoverlapping identical circles is generated to simulate the in situ configurations of aligned continuous nanotube embedded in polymer matrix. Each circle represents a nanotube in a transverse section of nanocomposite. To represent the continuum properties of the composite with many inclusions, periodic boundary conditions are applied to replicate an infinite but repetitive configuration so that the results are comparable to the experimental results. The periodic boundary conditions in two dimensions can be expressed as a function of the displacement u:

uðX 1 ; 0Þ þ U2 ¼ uðX 1 ; LÞ uð0; X 2 Þ þ U1 ¼ uðL; X 2 Þ

ð1Þ

where L is the length of square edge. U1 and U2 depend on the particular loading applied on the cell. For example, if the cell is loaded uniformly along the X1 direction with the deformation of d, the vectors U are written as U1 = (d, 0), U2 = (0, u2), where u2 is computed from the traction free condition:

Z

T 2 dC ¼ 0 on X 2 ¼ L

ð2Þ

C

where T2 stands for the normal traction acting on the boundary X2 = L. In order to properly enforce the periodic boundary conditions, the geometry of the unit cell must be periodic. This condition has particular impact on the inclusions close to or intersecting the edge of the unit cell. Moreover, the centers of inclusions must be generated randomly so that the particle arrangement is statistically isotropic. Finally, the microstructure should be suitable for finite element discretization. To fulfill all the requirements, the random sequential adsorption (RSA) algorithm with additional conditions is adopted to generate the coordinates of the inclusion centers [27,28]. According this method, the inclusions are generated randomly and sequentially inside of the cell such that each inclusion is accepted if it does not overlap any of the already existing inclusions. Given the radius of inclusions as r, the new particle i is accepted if the following conditions are satisfied:  The distance between the centroid of particle i and the centroids of all existing particles j = 1, 2, . . . , i1 have to exceed a minimum value s, where s > 2r. The reason to impose this condition is not only to avoid distorted elements between circles, but also to save sufficient space for interphase layers. By controlling the magnitude of s, we can generate distributions of particles that range from uniform to random dispersions (see below for details). Mathematically, we can express this condition as:

jxi  xj þ lj P s

ð3Þ

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where x is the centroid of each particle and l = (l1, l2). l1 and l2 are zero except for particles near the boundaries, in which cases l1 and/or l2 take on the values L or L to account for minimum distances to particles near the opposite edges (because of periodic boundary conditions).  The proximity of particle edges to the boundary of the unit cell is controlled to prevent the presence of distorted elements during meshing. Mathematically, this condition can be expressed as:

jxik  rj > t;

k ¼ 1; 2

jxik þ r  Lj > t;

k ¼ 1; 2

ð4Þ

where t is a small value depended on the quality of mesh. The modified RSA algorithm, as expressed by the conditions of Eqs. (3) and (4), was used to generate centers of particles in the unit cell. Herein, we assigned a consistent radius, one of unity, to all of inclusions since we want to isolate the effects of interphase percolation in the composite. In this study, the lateral dimension of the cell was chosen as 70 times the radius of the inclusions so that the volume fraction of nanoparticles is about 1.9% for all samples which is among the fractions frequently used in nanocomposites. Two models with different dispersion were generated as shown in Fig. 1. For the first sample (referred to as the ‘‘uniform

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model” in the remainder of this paper), the value of s in Eq. (3) was set to be 10r, causing a large excluding volume of each particle, so the dispersion is very uniform. Meanwhile, the value of s was set to be equal to 2r for the second sample (called ‘‘random” in the following) which produces randomly dispersed particles. The interphase region in the polymeric composites was modeled as an annulus surrounding the particle with the thickness varying from half to several times diameter of the nanotube. Subsequently, several cells with different volume fractions of interphase were created. In this study, perfect bonding between matrix and inclusions was assumed, thus no interface contact models need be considered in the simulations. Fig. 2 shows two cells with uniformly dispersed particles surrounded by interphase layers: in the left plot, the thickness of interphase layers is set to be 1.5 times the nanotube diameter which leads to a volume fraction of interphase (Vt) as 28.8%, while, in the right plot, the thickness is 3 times the nanotube diameter and Vt is 81.4%. Herein, all volume fractions are calculated based on the mesh so there is no error induced by finite element discretization. In the model construction, the overlapped annuli were merged to form a union and the overlapped interphase area was only counted once. From Fig. 2, we can clearly see a percolated network of the interphase region appears in the matrix as the Vt exceeds the percolation threshold, which

Fig. 1. Schematic of the unit cell for (a) uniformly (b) randomly dispersed nanocomposite where nanotubes are represented by black disks. The volume fractions of nanotubes in both configurations are 1.9%. See text for details.

Fig. 2. Schematics of composite configurations with uniformly distributed inclusions (from Fig. 1a) and different volume fractions of interphase layers. The black interior inclusions represent nanotubes, while the blue annuli surrounding the interior nanotubes represent the interphase layers and rest of area is the bulk matrix. (a) Before interphase percolation, Vt = 28.8%. (b) After interphase percolation, Vt = 81.4%.

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is 70.6% for this uniform dispersion (as shown in Fig. 1a). It should be noted that the percolation threshold in this paper refers to the minimum volume fraction of the interphase to form a connective network throughout the matrix under given locations of each particle. Particularly, this threshold is calculated numerically by detecting geometrical percolation in the composite given the dispersion showing Fig. 1a. Therefore, the percolation threshold defined here strongly depends on the geometry of the RVE and varies as the distribution of inclusions changes, differing from classic geometrical percolation theory. The percolation threshold for the random case shown in Fig. 1b is also calculated and is much lower, only 43.0%. 2.2. Material properties The frequency domain response of polycarbonate (PC) measured by dynamic mechanical analysis [29] was used as the material property of matrix in this model. The master curves of bulk PC at a reference temperature of 150 °C was subsequently trimmed and approximated with a 28-term Prony Series using the DYNAFIT program previously developed in this lab [30]. The comparison between the experimental results and the Prony Series approximation is shown in Fig. 3a. In addition, the Williams–Landel–Ferry (WLF) equation was utilized to interpolate shift factors for temperatures greater than Tg of PC in order to smooth the simulation results in the temperature domain (Fig. 3b, see text in section on ‘‘Impact on glass transition temperature” for details). To meet the requirement of viscoelastic material input of ABAQUS, the complex Young’s modulus of PC must be converted to complex shear and bulk moduli and the values of the instantaneous properties (the glassy equilibrium values at zero time) must be provided. The instantaneous Poisson’s ratio of PC is 0.4 [31] and the instantaneous bulk modulus K* was calculated from this value and the instantaneous Young’s modulus using elasticity conversion expressions. Since the variation of bulk modulus is relatively small in the frequency domain, while the changes in the Poisson ratio are known to be important, K* was assumed to be constant. Therefore, the complex shear modulus can be calculated at each frequency point via the correspondence principle [31] using the data for Young’s modulus, E*, and the constant K*:  

G ¼

3K E 9K   E

ð5Þ

The axial modulus of nanotubes is generally two orders of magnitude higher than the transverse modulus [32] and the absolute values of the moduli depend significantly on the type of nanotube and its processing method. However, our simulations (results not shown here) illustrate that neither the anisotropy nor the magni-

tude of the nanoparticle modulus has measurable impact on the viscoelastic properties of the nanocomposites. Thus, since our results exclusively focus on the viscoelastic properties of the nanocomposites, for simplification, the material properties of nanotubes were assigned to be linear, elastic, and isotropic with Young’s modulus of 1 TPa and Poisson’s ratio of 0.3. Although the properties of nanoparticles and bulk polymer are readily available, it is more of challenge to determine the interphase properties. First, it is not yet possible to measure those properties using current experimental techniques without biasing the results with significant assumptions. On the other hand, the numerical calculation of molecular dynamics (MD) can capture some structural and dynamical details of interphase only at the molecular time and length scale, while the continuum modeling requires material properties in many orders of magnitudes longer. Nevertheless, both numerical and experimental work provides guidance for the trend in continuum properties of interphase compared to the host bulk polymer. It has been noted that the attractive interactions between nanoinclusions and matrix can restrict mobility of polymer chains [33–38]. For example, Wei et al. [36] performed MD study that demonstrated an increase of 20 °C to 60 °C in Tg for a polyethylene cell containing nanotubes. Other MD simulations also showed Tg to increase near the nanoparticles for attractive interaction [33–35]. Similarly, at the other end of the spectrum, experimental observations show that the Tg of a nanocomposite can be on the order of ten degrees higher than the pure matrix materials [15,39]. Therefore, based on such simulation and experimental results, we hypothesize that the interphase properties are related to those of the bulk polymer matrix by a shift in the relaxation times. This approximation captures one of the most important characteristics of interphase, that of altered polymer mobility, by introducing the simplest possible assumption on the properties. In this paper, the interphase properties were determined by shifting the PC master curve two decades lower in the frequency domain, representative of strong positive interactions between the nanoparticle and the host matrix material. 3. Results and discussion Finite element models were created with different interphase volume fractions and the calculation was performed using the commercial FEA package, ABAQUS. The impact of interphase volume and structure were analyzed separately in this study. 3.1. Frequency response Based on the particle distribution from the uniform case (Fig. 1a), seven models were created with different thicknesses

Fig. 3. (a) Experimentally measured complex modulus of PC and its Prony Series approximation. (b) The shift factor versus temperature where the red line is the data fit by WLF equation (original data is from reference [23]).

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of interphase layers resulting in volume fractions of Vt = 5.7%, 28.8%, 46.1%, 57.7%, 65.2%, 81.4%, and 87.4% (the corresponding interphase thicknesses are 0.5d, 1.5d, 2d, 2.25d, 2.5d, 3d, and 3.25d, where d is the diameter of inclusion). The comparisons of the transverse storage and loss moduli for those models are shown in Fig. 4. The results clearly show that the transition zones of nanocomposites shift towards lower frequency as compared with pure PC. The loss modulus peak, shown in Fig. 4b, is broadened compared to pure matrix, and the broadening becomes more apparent as the volume fraction of interphase increases. In addition, the loss peaks are shifted to lower frequency corresponding to the transition zone of interphase. Note that the relaxation characteristics of the nanocomposite transitions from being dominated by the bulk PC to the interphase when the volume fraction of interphase exceeds the percolation threshold (70.6% in this configuration). This result is more evident by comparing the tan d curves of samples with different interphase volume fraction (Fig. 4c). For comparison, we can also consider the configuration of the random sample (Fig. 1b). In this case, we can expect a lower percolation threshold of interphase compared to the uniform dispersion. Fig. 5 shows the tan d results for these simulations with several different volume fractions of interphase. Consistent with the geometric percolation threshold of 43.0% in this configuration, we also observe that the simulation results demonstrate a transition from PC to interphase dominated behavior once this percolation threshold is reached. Taken together, these results indicate that not only is the amount of interphase important, but critically the connectivity of the interphase (its percolation) has significant impact on the overall viscoelastic properties of nanocomposites. In order to further verify the impact of the interphase percolation, a comparison between the finite element simulation and the simple rule of mixture (ROM) results has been conducted. For the ROM, the transverse formulation is used as appropriate for the transverse modulus of unidirectional composites, which is:

1 mf mi mm ¼ þ þ Ec Ef Ei Em

ð6Þ

where subscripts, c, f, i, and m represent the composite, nanotube, interphase, and matrix, respectively. Since both moduli of interphase and polymer matrix are complex values, the modulus of composite calculated by Eq. (6) is also a complex number. Fig. 6 shows the comparison of tan d in frequency space that is computed by the FE models and ROM method respectively. Note that the FE models used here are exactly same as those shown before, thus the percolation thresholds are 70.6% and 43.0% with respect to uniform case and random case. Fig. 6a shows that in the case with low interphase volume fraction no percolation occurs in either model, from FE or ROM predictions. Fig. 6b displays the high volume case in which the interphase is percolated in both of the models. Since the transverse ROM treats the composite in a ‘‘sandwich” mode, the overall properties are dominated by the weakest phase. Thus, because interphase modulus exceeds that of pure PC, the ROM predictions for tan d of the composite are still dominated by the characteristics of the pure PC and do not capture the shift of tan d peak. This comparison to the ROM provides additional evidence that the percolation of the interphase significantly affects the properties of polymeric nanocomposites, in such a way that a simple ROM prediction can no longer capture the essence. 3.2. Impact on glass transition temperature (Tg) As the vast majority of thermomechanical experimental data on nanocomposites is in the temperature domain, we also use the FEM approach to predict the temperature dependent response of the nanocomposites with different interphase fractions. The finite

Fig. 4. Predicted moduli of nanocomposites of uniformly distributed inclusions (Fig. 1a) with different volume fraction of interphase (Vt). Properties for bulk PC and the interphase are also shown. (a) Comparison of storage moduli. (b) Comparison of loss moduli. (c) Comparison of tan d.

element model and material properties used are identical to the uniform dispersion case described earlier. The material behavior in temperature domain was obtained by calculating the complex modulus of nanocomposites at a fixed frequency, in this case 1 Hz dynamic loading, at different temperatures. The material properties for each phase at each temperature were obtained by shifting the master curves at the reference temperature by the appropriate shift factors. The shift factors were obtained in the

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Fig. 5. Tan d in frequency domain of nanocomposites with randomly distributed inclusions (Fig. 1b).

time–temperature superposition (TTSP) process for frequency domain response master curves of the pure PC material (Fig. 3) and the WLF equation was used to fit the experimental curve of shift factors for T > Tg of PC. The temperature domain responses for two nanocomposites with different volume fraction of interphase, Vt = 15% and 81.4%, were plotted in Fig. 7. The glass transition temperature (Tg) of the nanocomposite with larger interphase, indicated by the tan d peak, clearly shifts to higher temperature compared to the pure PC and is dominated by the interphase properties. This result is again consistent with the concept of a percolated interphase dominating the effective viscoelastic properties of the composite. Experimental data for composites with surface modified nanotubes indicates that the larger region of reducedmobility polymer (i.e. interphase) in those systems leads to a higher Tg than unfunctionalized nanocomposites [15,39]. The broadening of the transition zone is also observed for the composite with 15% interphase, but is less pronounced, which could be attributed to the low volume fraction of interphase in that model. Note that a broadened tan d peak is not always observed [15] and further work will focus on the origin of the broadening in the temperature domain.

Fig. 7. Prediction of glass transition temperature of nanocomposites with uniform nanotube distribution (Fig. 1a). The data at each temperature point is calculated at frequency of 1 Hz and the tan d curves have been normalized.

3.3. Gradient of Interphase properties One concern that arises from the previous results compared to experimental data is the dual-peak appearing on the tan d curves for nanocomposites with moderate volume fraction of interphase (e.g. 50% interphase for the uniform distribution case in Fig. 4c). One possible reason for such a sharp distribution is that only two relaxation modes were used in our models, one for bulk PC and the other for an interphase with two decades less mobility. In the geometric configuration of the numerical model, the interphase properties were assigned to a single discrete domain around each nanoparticle with a sudden discontinuous change to the bulk matrix properties at its boundary. Considering experimental data on ultrathin polymer films on an attractive surfaces [37], the polymer mobility changes in a gradient fashion away from the polymer–surface interface. Thus, to determine if the gradient in properties could lead to a change in response which would diminish the artificial dual-peak at certain volume fractions, we considered the simplest possible gradient: two concentric layers of interphase surround the nanotubes with modulated properties (Fig. 8). Maintaining the basic geometry of previous models, the interphase annulus was divided

Fig. 6. Comparison between the FE simulations and rule of mixtures (ROM) results. (a) Shows the case for a low interphase volume fraction which is below the percolation thresholds for both uniform and random distributions. (b) Shows the high volume fraction case in which the interphase is percolated in both configurations.

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Fig. 8. Schematic of unit cell for a uniform nanotube distribution (Fig. 1a) with double-layer of interphase surrounding the nanotubes. The ratios of thickness (inner:outer) are 1:1 for all models. The total interphase volume fractions (Vt) are 46.1%, 65.2%, and 81.6%, respectively.

into two concentric annuli of the same thickness, such that the sum of volume fractions of the two interphase layers remains the same as the interphase volume fraction of the corresponding single-layer interphase models showed earlier. The properties of the first interphase, that closest to the nanotube and denoted as ‘‘Interphase-1”, remained the same as before (shifted 2 decades toward lower frequency). The properties of the second interphase, closest to the PC and denoted as ‘‘Interphase-2”, were also derived from bulk PC but shifted only one decade in frequency, thereby simulating the decay of the properties toward that of the matrix material. Three examples are studied here for the uniformly distributed inclusion configuration (Fig. 1a). In the first case (Fig. 8a), the inner layer and outer layer thicknesses are identical to the diameter of the nanotube, and the total volume fraction of two layers is 46.1%, the same as the Vt in one sample studied earlier. In the second sample (Fig. 8b), the thicknesses of inner layer and outer layer are 1.25 times the nanotube diameter, resulting in a total volume fraction for two interphase layers of 65.2%. In the third model, both thicknesses of the two interphase layers are equal to 1.5 times the nanotube diameter for a total volume fraction of 81.6%. The dynamic responses of new models were calculated and tan d curves of these new models compared to those of the previous singlelayer models are plotted in Fig. 9. The results clearly show that, for the double-layer cases, the tan d curves are much smoother and double peaks are substituted by a single peak which moves towards to the bulk PC region. These results are important, as they agree fundamentally with experimental data on nanocomposites in which gradual shifting of the tan d peaks are observed, but never double peaks. The implication then is that the gradient interphase ultimately leads to a unique relaxation characteristic for the nanocomposites. Comparing the double-layer model tan d curves with those of previous single-layer models, we see that the peaks for the gradient interphase shift to higher frequency. This phenomenon is more pronounced for the samples where the interphase dominates the overall viscoelastic properties (i.e. Fig. 9c). This result occurs because the outer layer interphase, which plays important role in the double-layer model, has a transition zone occurring at higher frequency than the inner layer interphase. The inner layer interphase, with a lower frequency loss peak, is used only for half of the total interphase. In contrast, the single-layer model uses the inner layer interphase properties for the full extent of the interphase zone. It is also observed that the magnitudes of the tan d peaks differ for the single-layer and double-layer interphase models. Since the properties of interphase were obtained by shifting the master

curve of bulk PC horizontally to lower frequency, the mechanical damping capabilities are identical for all those materials. Therefore, we should expect the same areas under the tan d curves for both single-layer and double-layer models, as shown in Fig. 9. For the cases where two maxima appeared on the tan d curve of the single-layer model (a and b in Fig. 9), the main peak is necessarily lower than the peak of the corresponding double-layer model to maintain the same damping effect. In the case of Fig. 9c, both single- and double-layer models are fully percolated with only one peak, suppressing the relaxation signature of bulk PC. Thus in this case, both models have the same intensity. Comparing the movement of the tan d for each of the double-layer cases, we observe that the tan d peaks gradually shift to lower frequency as the volume fraction of interphase increases. In particular, when a percolated interphase network forms inside of the composite (Fig. 8c), the predicted tan d peak moves to a position between two interphase peaks (Fig. 9c), indicating that the interphase dominates the overall relaxation properties of the composite. 4. Conclusion The influence of the interphase and its structure on the viscoelastic response of polymer nanocomposites was studied via a two-dimensional finite element analysis. A unit cell with thirty dispersed particles was created to represent the transverse section of unidirectional nanotube reinforced polymer composites.1 Each particle was coated by a layer of interphase, whose properties were obtained by shifting the bulk matrix properties in the frequency domain, corresponding to altered mobility of the polymer in the vicinity of nanotubes. Given the experimentally measured frequency domain response of bulk polymer, the viscoelastic properties of the nanocomposites in both frequency and temperature domains were calculated. Furthermore, varying the thickness of interphase layers and modifying the dispersion of nanotubes caused changes in the interphase connectivity in the polymer. The simulation results demonstrated that relaxation characteristics of nanocomposites are greatly dependent upon the volume of interphase and formation of an interphase network. Quantitative comparison between the experimental data and the model prediction is not possible at current stage, due to many uncertainties, including nanotube and interphase properties, in situ nanotube morphology, and the 2D nature of the simulation. Nevertheless, 1 We note that unit cells with different numbers of particles were also studied, all resulted in the same conclusion.

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 For the case of low volume fraction of interphase such that the interphase domains remain discrete, the predicted transition zones of the nanocomposites are broadened but no significant shift occurs in frequency or temperature domains compared to the properties of the bulk polymer. A discrete interphase situation is consistent with relatively thin interphase layers, very low volume fractions of nanotubes, or the nanotubes aggregated to clusters. Results correspond to data on unmodified surface of MWNT nanocomposites and unfunctionalized SWNT nanocomposites.  For the case of high volume fraction of interphase such that the interphase is percolated through the composite, a clear shift of the predicted relaxation peaks of the nanocomposites occur compared to the peak of the pure polymer. A percolated microstructure corresponds to the results of functionalized SWNT nanocomposites and MWNT nanocomposites with surface modification which have better dispersion of nanotubes and a thicker region of reduced-mobility polymer around the nanotubes.  It is important to use a gradient in the properties of the interphase to avoid an artificial dual-peak phenomenon. While an artificial dual-peak was observed at some intermediate volume fractions for the single-layer interphase cases, implementation of a simple double-layer interphase with modulated properties completely removed this anomaly. The results from this work provide powerful insight into the importance of percolating interphase in nanocomposites for viscoelastic properties. However, a significant limitation of the work is the 2D nature of the models. The two-dimensionality does not permit us to study the influence of aspect ratio or shape of the nanoparticles, which is known to be important. The percolation thresholds for 2D models are also significantly higher than for 3D network structures and the resulting connectivity of the interphase domains is more limited. Thus, a 3D model is necessary to better study the properties of nanocomposite with different inclusions. Moreover, a quantitative comparison between the numerical and experimental results would be enhanced by the extension to a 3D system. Acknowledgment The authors wish to thank the support of the National Science Foundation NIRT program under Grant No. 0404291. References

Fig. 9. Comparison of tan d in frequency domain for single-layer model and doublelayer model. The interphase volume fractions (Vt) of these models are: (a) 46.1%, (b) 65.2%, and (c) 81.4%, respectively, refer to text for details.

the predicted pattern of the impact of the interphase on the overall performance of the nanocomposites is consistent with experimental observations [15,29]. Significant observations are:

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