Multiscale Shear-Lag Analysis of Stiffness Enhancement in Polymer

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A Multiscale Shear-Lag Analysis of Stiffness Enhancement in Polymer-Graphene Nanocomposites Asanka Weerasinghe, Chang-Tsan Lu, Dimitrios Maroudas, and Ashwin Ramasubramaniam ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.7b03159 • Publication Date (Web): 16 Jun 2017 Downloaded from http://pubs.acs.org on June 19, 2017

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ACS Applied Materials & Interfaces

A Multiscale Shear-Lag Analysis of Stiffness Enhancement in PolymerGraphene Nanocomposites

Asanka

Weerasinghe,1 Chang-Tsan

Lu,2,†

Dimitrios

Maroudas,2,*

and

Ashwin

Ramasubramaniam3,* 1

Department of Physics, University of Massachusetts, Amherst, MA 01003, U.S.A.

2

Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003,

U.S.A. 3

Department of Mechanical and Industrial Engineering, University of Massachusetts, Amherst,

MA 01003, U.S.A. † Present Address: Third Wave Systems, Eden Prairie, MN 55344, U. S. A.

ABSTRACT Graphene and other 2D materials are of emerging interest as functional fillers in polymermatrix composites. In this work, we present a multiscale atomistic-to-continuum approach for modeling interfacial stress transfer in graphene–HDPE nanocomposites. Via detailed characterization of atomic-level stress profiles in sub-micron graphene fillers, we develop a modified shear-lag model for short fillers. A key feature of our approach lies in the correct accounting of stress concentration at the ends of fillers that exhibits a power-law dependence on filler (“flaw”) size, determined explicitly from atomistic simulations, without any ad hoc modeling assumptions. In addition to two parameters that quantify the end-stress concentration, only one additional shear-lag parameter is required to quantify the atomic-level stress profiles in graphene fillers. This three-parameter model is found to be reliable for fillers with dimensions as 1 ACS Paragon Plus Environment


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small as ~10 nm. Our model predicts accurately the elastic response of aligned graphene–HDPE composites and provides appropriate upper bounds for the elastic moduli of nanocomposites with more realistic randomly distributed and oriented fillers. This work provides a systematic approach for developing hierarchical multiscale models of 2D material-based nanocomposites and is of particular relevance for short fillers, which are currently typical of solution-processed 2D materials.

Keywords: polymer nanocomposites, graphene, molecular dynamics, shear-lag models, mechanical properties

Corresponding Authors *

[email protected]

*

[email protected]

2 ACS Paragon Plus Environment

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1. Introduction Graphene—an atomically-thin sheet of sp2-bonded carbon—has been widely investigated for its exceptional mechanical and electronic properties.1-3 Polymer-matrix composites are poised to be among the early technological applications for graphene as suggested by several reports of improved stiffness and strength, enhanced electrical and thermal conductivity, elevated glass transition temperatures, and reduced gas permeability at relatively low graphene loading.4–8 While graphene is not unique as a filler in these aspects—silicate or carbon-nanotube fillers also being competitive—the potential for achieving a multiplicity of enhanced material properties with a single filler make graphene an attractive candidate for enabling new multifunctional polymer nanocomposites.4, 9 For mechanical reinforcement, in particular, graphene is an attractive filler due to its high elastic modulus (~1 TPa) and intrinsic strength (> 100 GPa).10 Several studies of model polymer–graphene systems have sought to understand how filler size, 11,12,13 orientation, 14 and surface functionalization 15 control the mechanical response. In general, these studies indicate that large, well-aligned, few-layer graphene nanoflakes with strong interfacial adhesion to the matrix provide the best composite properties, 16 which is consistent with decades of experience in fiber-reinforced composites. Indeed, continuum mechanics-based shear-lag models17 have been used successfully to model strain profiles along graphene fillers from Raman measurements over micron scales in PMMA-graphene composites11 with suitable modifications to account for interfacial sliding and delamination.18 However, a recent experimental study suggests that shear-lag models could break down for polymer– graphene nanocomposites with filler dimensions at the sub-micron level due to residual stresses, chemical doping, or edge effects.19 Furthermore, both theory20,21,22 and experiment23 indicate that the detailed structural characteristics of the polymer-matrix interface (the “interphase”) become 3 ACS Paragon Plus Environment


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important when considering reinforcement mechanisms in nanocomposites; such details are typically missing in models using the sharp-interface approximation. Thus, there is need for understanding atomic-scale mechanisms of stress transfer at graphene–polymer interfaces— including the role of the interphase in this process—and identifying regimes of applicability or breakdown of continuum models. In this article, we conduct a detailed analysis of the interfacial stress transfer in polymer– graphene composites within the small-strain regime (elastic response with no interfacial sliding). Using molecular-statics and molecular-dynamics (MD) simulations, we study the mechanical response of a model polymer nanocomposite, namely, graphene-reinforced high-density polyethylene (HDPE). By systematically controlling filler size and performing extensive statistical sampling, we obtain atomic-level stress profiles along graphene fillers with sub-micron dimensions and develop a connection with shear-lag models for short fillers.24 In obtaining this atomistic–continuum connection, we find that it is important to account for both the filler end stresses as well as the presence of a denser polymer interphase that mediates stress transfer to the filler. A key finding of our work is that, within the elastic regime, the behavior of monolayer graphene fillers is similar to that of flaws in bulk materials, and we present an approach for explicitly calculating the flaw-size dependence of the end-stress concentration fully from atomistic simulations. We find that, in addition to two parameters that quantify the end-stress concentration, only one additional shear-lag parameter is required to quantify the atomic-level stress profiles in graphene fillers, and this three-parameter model to be reliable down to graphene filler dimensions of ~10 nm. We demonstrate the accuracy of our atomistically-informed, modified shear-lag model in predicting the elastic response of aligned graphene-HDPE nanocomposites and in providing appropriate upper bounds for the elastic moduli of random 4 ACS Paragon Plus Environment

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graphene–HDPE nanocomposites. Our work provides a systematic approach for developing hierarchical multiscale models of 2D material-based nanocomposites and is of particular relevance for short fillers, which are the typical outcome of solution processing of 2D materials.8,25

2. Results and Discussion We consider a model graphene-HDPE nanocomposite (Figure 1) that consists of a regular array of graphene nanoribbons (GNRs) represented by a narrow graphene strip of width 2l (ranging from 1 nm to 50 nm) embedded in a larger polymer matrix supercell whose dimensions in the y- and z-directions are chosen to be sufficiently large to avoid filler–filler interactions. As

Figure 1: (a) 3D view of the atomic configuration of a model graphene-HDPE nanocomposite with regular nanoribbon arrays with a 20-nm-wide graphene nanoribbon extending infinitely along the x-direction equilibrated at 150 K. (b) Side view of the model graphene-HDPE nanocomposite and a magnified view of the interface, which highlights the polymer densification in the vicinity of the graphene filler. (c) Schematic representation of the graphene filler with the dimensions used in the continuum shear-lag model. (d) Normalized atomic density profile of the polymer, ρ(z/Lz)/ρ0 where ρ0 is the normal density of the polymer matrix at 150 K, in the direction normal to the plane of the graphene filler. The dashed red line represents an optimal Gaussian envelope to the density profile. 5 ACS Paragon Plus Environment


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the GNR is periodic along its length, the ribbon width 2l is the only geometric parameter that controls the filler size. This atomic-scale model ensures that, upon straining along the ydirection, the stress profile in the filler is one-dimensional (to within statistical variations along the ribbon length), which is readily amenable to a shear-lag analysis. We considered several graphene nanoribbons with widths ranging from 1 nm to 50 nm (Table S1), embedded in an HDPE matrix and subjected the composites to quasi-static (0 K) strain increments in the ydirection of 0.2% up to a peak value of 0.8% considering 10 different equilibrated initial configurations in each case to account for statistical variations. The ribbons were chosen to have zigzag edges without any loss of generality. Figure 2(a) shows the results for normal stress profiles along the ribbon width, σyy(y) for a 30-nm-wide GNR filler at a few different strain levels; each data point in these 1D profiles corresponds to the virial stress averaged over a zigzag row. It is immediately apparent that the normal stress distribution is qualitatively similar to that expected from a shear-lag analysis,26 with the superposition of a non-zero end stress whose magnitude grows with applied strain.24 Thus, both interfacial shear stresses and normal end stresses must be considered when analyzing the stress-transfer process. There is also clear evidence of a spike in the normal stress confined to the very edges of the fillers in our model nanocomposites that arises from the intrinsic edge stress of the graphene nanoribbons,27 which we also observe in our calculations of isolated GNRs, and is unimportant for our present purposes. It should be mentioned that the edge stresses are sensitive to passivating species used for edge termination and that these stress spikes may be quenched28,29 or enhanced30 depending on the passivating species. Our calculations show that for sufficiently wide ribbons there is no clear correlation between edge stress and ribbon width, as expected for a localized edge effect.


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Figure 2: (a) Axial stress profiles, σyy(y/l), along a 30-nm-wide graphene nanoribbon (2l = 30 nm) computed from quasi-static straining simulations at various indicated composite strain levels for a model graphene-HDPE nanocomposite with regular nanoribbon arrays. (b) Dimensionless axial stress profiles, σyy(y/l)/(Eg εm), across graphene nanoribbons with widths (2l) of 8, 12, 20, and 30 nm computed as in (a) in model graphene-HDPE nanocomposites. The stress profiles at various composite strain levels, εm (up to 0.6% in each case) nearly collapse on to a single curve for each nanoribbon. The solid lines represent optimal fits to the simulation results according to the modified shear-lag model (Equation 2), excluding the results in the immediate vicinity of the sheet edge (y ≈ l). The stress profiles are symmetric over the domain -l ≤ y ≤ l and plotted only over 0 ≤ y ≤ l. (c) Shear-lag parameter, β, and dimensionless end stress, σ0/(Eg εm), as a function of filler aspect ratio, s = l/t, from non-linear fitting of the stress profiles in (b) according to Equation 2. The red dashed line represents the power-law fit, σ0(s)/(Egεm) = αsδ ~ sδ, from which we obtain α = (2.89 ± 0.09)×10-3 and δ = 0.80 ± 0.01 (first two points excluded). (d) Composite elastic moduli computed from quasi-static straining simulations and comparison with their estimates as a function of s from short-filler (SFT; Equation 3) and long-filler (LFT; Equation 4) theories.



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Figure 2(b) shows analogous stress profiles (normalized as per Equation 2 below) for a few different ribbon widths, demonstrating the similarity to short-fiber stress profiles with non-zero end stresses. It is also evident that the stress profiles for the narrower nanoribbons are shallower than those for the wider ones, which is fully consistent with expectations from a classical shearlag analysis. Based on our observations above, it is plausible that a shear-lag model could indeed describe stress transfer in graphene–HDPE nanocomposites provided that the end stresses are correctly accounted for. To this end, we note first that the normal stress distribution for fillers that are fully embedded in the matrix is given by11, 26 (SI)

⎛ β y⎞ ⎛ β y⎞ σ yy ( y) = Eg ε m + C sinh ⎜ ⎟ + D cosh ⎜ ⎟ , 0 ≤ y ≤ l , ⎝ t ⎠ ⎝ t ⎠

(1)

where Eg (970 GPa) is our calculated value for the elastic modulus of graphene, which is consistent with previous reports;10,31 t (0.335 nm) is the thickness of the nanoribbons, chosen to be the typical interlayer spacing in graphite;32,33 εm is the matrix strain; and β = 4

Geff

t Eg (Lz − t)

is a dimensionless parameter, the so-called shear-lag parameter, with Geff being the effective shear modulus of the matrix including the interphase (Equation S3), and Lz being the height of the simulation supercell in the direction perpendicular to the plane of the filler. The integration constants C and D in Equation 1 are to be determined by satisfying dσyy/dy = 0 (σyy is symmetric over the width of the ribbon and attains a maximum at y=0) at the center of the ribbon and by a suitable boundary condition for σyy at the edge of the filler (y = l). While various approximations have been suggested to model the end stress,24,34,35 an appropriate representation of the stress concentration at the filler ends and a proper accounting of the interphase introduce a significant degree of uncertainty. However, the end stress is accessible unambiguously in our atomistic 8 ACS Paragon Plus Environment

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simulations and so we simply set the boundary condition at y = l as σ yy

y=l

≡ σ 0 (l) and extract

σ0(l) from our simulation data; the dependence of σ0 on filler width (l) introduced here anticipates a scaling law that will be discussed later and can be understood intuitively as the influence of “flaw” size on the stress concentration. The resulting (normalized) normal stress distribution in the filler may, hence, be written as (SI)

σ yy (η ) Eg ε m

⎛ σ (s) ⎞ cosh( β sη ) = 1+ ⎜ 0 − 1⎟ , 0 ≤ η ≤ 1, ⎝ Eg ε m ⎠ cosh( β s)

(2)

where we have introduced the dimensionless coordinate η = y/l, and s = l/t is the filler aspect ratio. When plotted in this dimensionless form, the normal stress profiles for a particular graphene nanoribbon (fixed l) practically collapse onto a single master curve (Figure 2b); a ~5% deviation due to systematic error with respect to the strain level is attributed to small variations in the GNR morphology, such as the warping or rippling of the GNR observed for larger ribbons. Shear-lag model generalizations to capture filler deviations from a planar morphology are beyond the scope of this study, but constitute an interesting topic to be addressed within perturbation theory in future theoretical investigations. The dimensionless parameters σ0(s)/(Egεm) and βs can now be determined via nonlinear regression. The results from fitting of the atomic stress profiles including those of Figure 2b according to Equation 2 are listed in Table S3 and plotted in Figure 2c for the various filler sizes studied here with sampling over ~30 stress profiles per filler. Examples of data fits in Figure 2b show that the modified shear-lag model captures accurately the axial stress distribution in the fillers with the exception of the intrinsic edge stress that is beyond the purview of the model. A closer inspection of Table S3 and Figure 2c reveals that the shear-lag parameter β decreases rapidly with filler width and attains a near-constant value beyond 2l ~ 10 nm. As β is, by definition, independent of 9 ACS Paragon Plus Environment


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l, we interpret 10 nm to be the lower bound for applicability of the shear-lag model in our model nanocomposite. Furthermore, Table S3 and Figure 2c also indicate that σ0(s) increases monotonically with aspect ratio s (equivalently, l, as t remains constant) closely following a power-law relation for larger fillers. Hence, we make the scaling ansatz σ0(s)/(Egεm) = αsδ ~ sδ and determine α and δ from the calculated values of σ0(s). Here, α depends upon the material properties of the filler and the interphase, whereas δ controls the flaw-size effect and remains constant (δ ≈ 4/5) for this particular “flaw”, i.e., monolayer graphene filler. Finally, the elastic modulus for our model composite system with aligned short fillers of volume fraction ϕ is given by (SI)

⎡ tanh( β s) ⎤ EcSFT = (1− φ )Em + φ Eg ⎢1+ (α sδ − 1) . β s ⎥⎦ ⎣

(3)

As seen from Figure 2d, the predictions of Equation 3 (short filler theory, SFT) are in excellent agreement with the simulation results. We also note that a simplistic estimate according to the rule of mixtures, Ec = (1− φ )Em + φ Eg , severely overestimates the elastic modulus of the nanocomposite [Figure S3(a)], which underscores the importance of accounting properly for the flaw-like behavior of the fillers.36 While our atomistic simulations are restricted to narrow graphene nanoribbons fillers—the largest filler of 50 nm already requiring a heavy computational cost—we outline briefly how our results can be extrapolated to larger fillers. Firstly, we recall that the shear-lag parameter, β, does not depend upon filler width beyond a lower bound of ~10 nm (Figure 2c; Table S3). Thus, if the filler separation Lz is held constant the shear-lag parameter β remains constant, provided that a homogeneous state of matrix strain, εm, is achieved between fillers (well-separated fillers). To allow for greater flexibility in the model, β could further be calculated as a function of Lz. In 10 ACS Paragon Plus Environment

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Figure S4, we show the results of numerical tests for three different values of Lz for a fixed ribbon width of 2l=12 nm. It is evident from these results that the weak scaling of β with filler separation Lz ( β ∼ 1/ Lz − t ) is barely distinguishable, giving essentially a constant value of β to within statistical error; the same is true of the end stress, σ0. These observations suggest that the mechanical response of the composite discussed above can be taken to be dependent primarily on filler size.

Secondly, we note that the end stress σ0(l) cannot grow unbounded

with increasing filler size, l, and failure must eventually occur either by debonding between the filler and the interphase or by failure (yielding, void growth, etc.) within the interphase/matrix. Thus, for long fillers, it is reasonable to invoke the usual approximation of zero end stress, σ0 = 0; with these assumptions, we recover the classical result for the composite modulus17

⎡ tanh( β s) ⎤ EcLFT = (1− φ )Em + φ Eg ⎢1− . β s ⎥⎦ ⎣

(4)

As long as end stresses are transmitted from the polymer to the filler, it is evident from Equations 3 and 4 that EcSFT > EcLFT (as α, δ, and s > 0) and the reinforcement effect is substantially higher than would be expected from a classical shear-lag analysis (Figure 2d). We emphasize that this result indicates the correct scaling of SFT, as compared to LFT, for very short fillers and does not imply that short graphene fillers offer better reinforcement than large fillers. The details of the crossover from the short-filler to the long-filler regime are specific to the material system and, given a reliable microscopic model, could be estimated from more detailed MD simulations that pinpoint the onset of failure at the filler ends. Finally, we have also performed finite-temperature MD simulations of graphene–HDPE nanocomposites consisting of randomly distributed circular graphene nanoflakes in HDPE—a more realistic nanocomposite structure than the model system considered thus far—and 11 ACS Paragon Plus Environment


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(a)

(b)

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(e) 1.6 1.5

Simulation SFT LFT

1.4 1.3 1.2 1.1 1

(c)

(d)

0.9 0

5

10

15

20

Figure 3: (a-d) Representative atomic configurations of graphene-polymer nanocomposites with randomly distributed circular graphene flakes (black disks) in an HDPE matrix equilibrated at 150 K. The filler radii are (a) 0.86 nm, (b) 4.39 nm, (c) 8.69 nm, and (d) 20.16 nm. (e) Composite elastic moduli computed from MD simulations of tensile straining tests as a function of filler radius, r, along with the upper bounds calculated according to short-filler theory (SFT; Equation 3) and estimates from long-filler theory (LFT; Equation 4). Dashed lines are a guide to the eye; for clarity, only one error bar is shown in each case for the SFT and LFT data, the other error bars being of comparable size. calculated the elastic moduli of the nanocomposites as a function of filler size. Figures 3a-d display representative simulation supercells (details in Table S2) for these cases while Figure 3e shows the scaling of the composite modulus with the graphene filler radius. The monotonic increase of the composite’s elastic modulus with graphene nanoflake radius, r, in Figure 3e is analogous to the composite modulus scaling with nanoribbon width, l, of Figure 2d and consistent with the modified shear-lag picture for stress transfer. This observation was also reported in our previous work20 and is now confirmed for more dilute fillers (2 wt. %) and 12 ACS Paragon Plus Environment

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significantly larger filler sizes. Similar to the model composite, we find the presence of a denser interphase in the vicinity of the nanoflake as well as the presence of end stresses at the nanoflake edges (Figure S2) with no evidence of debonding over the range of strains (< 0.8%) applied in the mechanical testing simulations. The analytical models discussed above (Equations 3 and 4) are directly applicable only to fillers aligned with the load axis; nevertheless, these models furnish upper bounds for a composite with randomly oriented fillers. As seen from Figure 3e, the short-filler theory (SFT; non-zero end stresses) indeed provides an upper bound for our simulation results whereas the long-filler theory (LFT; zero end stresses) severely underestimates the composite modulus. Once again, the simple rule-of-mixtures estimate is seen to perform poorly as compared to the shear-lag-based model [Figure S3(b)] underscoring the flaw-like nature of these fillers. More rigorous models 37 , 38 for the mechanical response of these nanocomposites that take into account both the random orientation as well as the geometrical details of flakes are beyond the scope of this work and will be considered elsewhere.

3. Conclusions We have simulated and analyzed the elastic response of graphene-HDPE nanocomposites using molecular-statics and molecular-dynamics simulations. Using model graphene nanoribbons as fillers, we have developed a hierarchical multiscale shear-lag model that captures accurately stress profiles in these short fillers as well as the macroscopic elastic response of the aligned nanoribbon-HDPE composite. A key feature of our model is the correct description of the endstress concentration from fully atomistic modeling without any ad hoc approximations. The endstress concentration σ0 scales with filler size l as σ0 ~ l4/5, indicating the flaw-like character of monolayer graphene fillers. The inclusion of this end stress in the modified shear-lag model 13 ACS Paragon Plus Environment


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leads to excellent agreement with composite elastic moduli computed from atomistic simulations, in distinct variance with predictions from conventional shear-lag models or simple averaging through rule of mixtures. Moreover, the predicted composite moduli from the modified shear-lag theory also furnish reasonable upper bounds for elastic moduli computed from MD simulations of straining of nanocomposites with random-oriented graphene flakes in HDPE unlike standard shear-lag models or rule of mixtures. The hierarchical multiscale modeling approach developed here for graphene-HDPE nanocomposites, including proper atomistically-based parameterization of coarse-grained models and multiscale linking through an atomistically-supplied boundary condition for well-posedness of a continuum mechanics boundary-value problem, is readily applicable to other graphenebased nanocomposites as well as to 2D fillers beyond graphene. It is also of particular relevance in the short-filler limit, which is the typical situation with solution-processed 2D materials.8,25 Furthermore, accurate, atomistically-informed continuum models of stress transfer at the elementary filler level can now enable more sophisticated continuum analyses37,38 with randomly-oriented fillers and/or various filler shapes, which will be considered in the future.

COMPUTATIONAL METHODS Graphene–HPDE nanocomposites were modeled using the LAMMPS simulation software.39 The polymer matrix was described using the united-atom model as parameterized by Buell et al.40 while the graphene fillers were described using the Dreiding potential of Cornell et al.41 Filler– matrix interactions were described as non-bonded interactions using Lorentz-Berthelot mixing rules.42 We estimate the filler–matrix gap in all of our equilibrated structures to be in the range of 0.3-0.4 nm, which is consistent with estimates (0.2–0.4 nm) from prior work.43,44 The mixing 14 ACS Paragon Plus Environment

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rules adopted in this study are merely a matter of convenience for a generic representation of filler–matrix interactions and are not intended to model a specific polymer–matrix composite. Different choices of filler–matrix interactions will ultimately affect the precise values of the shear-lag parameter β and the end-stress σ0 within the continuum model but do not affect qualitatively the short-filler theory developed here. All relevant functional forms and parameters for the potentials can be found in our previous work.20 The HDPE matrix was modeled using unbranched CH3-(CH2)n-CH3 (n = 98; “100-mer”) chains. The graphene fillers are modeled as pure carbon structures without any edge passivation. We note that while edge passivation changes the local geometry of the atomic configuration, the corresponding dominant mechanical effect is to alter the edge stress;28-30 in the absence of explicit covalent bonding between the filler and the matrix, the atomic details of the edge structure only perturb the end stress without affecting the overall stress transfer in short fillers. The computational protocols for preparing and equilibrating the composite supercells followed closely those in Ref. 20 and are provided in further detail in the SI. The simulation supercells were equilibrated at 150 K to ensure that all of our samples are in the glassy regime20 and subjected subsequently to uniaxial tensile straining tests, detailed protocols for which are provided in the SI. Stress distributions in the graphene fillers were calculated from atomic-level virial stresses42,45 assuming an atomic volume of Ω=0.0084 nm3, which is equal to the atomic volume in graphite according to the interatomic potential chosen for this study in conjunction with an assumed sheet thickness of 0.335 nm. Density profiles normal to the fillers were calculated as described in the SI.



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SUPPORTING INFORMATION The Supporting Information is available free of charge on the ACS Publications website at DOI: [to be inserted by publisher] Detailed description of methods for preparation and equilibration of nanocomposite samples; derivations of analytical models; data tables for fitted shear-lag parameters; atomic models of graphene-HDPE nanocomposites used in MD simulations, density profiles, and stress profiles; and comparison of rule of mixtures with modified shear-lag models for prediction of elastic moduli of nanocomposites

ACKNOWLEDGMENTS This work was supported by the Army Research Laboratory (Grant Nos. W911NF-11-2-0014 and 911NF-15-2-0026). Access to computational resources through the Department of Defense High-Performance Computing Modernization Program (HPCMP) Open Research Systems as well as the Massachusetts Green High-Performance Computing Center (MGHPCC) is gratefully acknowledged.

REFERENCES

1. Geim, A. K.; Novoselov, K. S. The Rise of Graphene. Nat. Mater. 2007, 6, 183-191. 2. Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. The Electronic Properties of Graphene. Rev. Mod. Phys. 2009, 81, 109-162. 3. Dreyer, D. R.; Ruoff, R. S.; Bielawski, C. W. From Concept to Realization: An Historical Account of Graphene and Some Perspectives for Its Future. Angew. Chem. Int. Ed. 2010, 49, 9936-9344.


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Table of Contents Figure


Multiscale Shear-Lag Analysis of Stiffness Enhancement in Polymer

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