Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

DOI: 10.1039/C9NR03317C
(Paper)
Nanoscale, 2019, Advance Article

Bharath Natarajan‡§
^{ab},
Itai Y. Stein§^{cd},
Noa Lachman^{ce},
Namiko Yamamoto^{cf},
Douglas S. Jacobs^{d},
Renu Sharma^{g},
J. Alexander Liddle*^{g} and
Brian L. Wardle*^{c}
^{a}Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
^{b}Corporate Strategic Research, ExxonMobil Research and Engineering, Annandale, NJ 08801, USA
^{c}Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. E-mail: wardle@mit.edu
^{d}Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
^{e}Department of Materials Science and Engineering, Tel Aviv University, Tel Aviv 6997801, Israel
^{f}Department of Aerospace Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
^{g}Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA. E-mail: james.liddle@nist.gov

Received
18th April 2019
, Accepted 16th June 2019

First published on 17th June 2019

Carbon nanostructure (CNS) based polymer nanocomposites (PNCs) are of interest due to the superior properties of the CNS themselves, scale effects, and the ability to transfer these properties anisotropically to the bulk material. However, measurements of physical properties of such materials are not in agreement with theoretical predictions. Recently, the ability to characterize the 3D morphology of such PNCs at the nanoscale has been significantly improved, with rich, quantitative data extracted from tomographic transmission electron microscopy (TEM). In this work, we use new, nanoscale quantitative 3D morphological information and stochastic modeling to re-interpret experimental measurements of continuous aligned carbon nanotube (A-CNT) PNC properties as a function of A-CNT packing/volume fraction. The 3D tortuosity calculated from tomographic reconstructions and its evolution with volume fraction is used to develop a novel definition of waviness that incorporates the stochastic nature of CNT growth. The importance of using randomly wavy CNTs to model these materials is validated by agreement between simulated and previously-measured PNC elastic moduli. Secondary morphological descriptors such as CNT–CNT junction density and inter-junction distances are measured for transport property predictions. The scaling of the junction density with CNT volume fraction is observed to be non-linear, and this non-linearity is identified as the primary reason behind the previously unexplained scaling of aligned-CNT PNC longitudinal thermal conductivity. By contrast, the measured electrical conductivity scales linearly with volume fraction as it is relatively insensitive to junction density beyond percolation. This result verifies prior hypotheses that electrical conduction in such fully percolated and continuous CNT systems is dominated by the bulk resistivity of the CNTs themselves. This combination of electron tomographic data and stochastic simulations is a powerful method for establishing a predictive capability for nanocomposite structure–property relations, making it an essential aid in understanding and tailoring the next-generation of advanced composites.

Fig. 1 Measured properties of epoxy-based nanocomposites with randomly dispersed or aligned carbon nanotubes (CNTs) and other prevalent carbon nanostructures (CNSs), i.e., graphene and graphene nano-platelets (GNPs) and morphology assumptions in property prediction tools for A-PNCs: (A) Elastic modulus (E) scaling with CNS volume fraction (V_{f}) for random and aligned CNTs,^{14–25} graphene,^{26} and GNPs.^{27} Fitting with a rule-of-mixtures equation shown in the legend demonstrates that high V_{f} aligned CNTs exhibit the highest effective CNS modulus, E_{o}. (B) Thermal conductivity (k) as a function of CNS V_{f} for random and aligned CNTs,^{28–32} graphene,^{33–37} and GNPs.^{27,31–33,38,39} Fitting with a rule-of-mixtures equation in the legend demonstrates that graphene and GNPs exhibit the highest effective CNS thermal conductivity, k_{o}, while the best overall thermal conductivity k is measured for the highest V_{f} CNT nanocomposite. (C) Electrical conductivity (σ) as a function of CNS V_{f} for random and aligned CNTs,^{30,40–42} graphene,^{27,43,44} and GNPs.^{38} Fitting with the power-law equation shown in the legend indicates that although CNT and graphene composites can reach similar σ, since their effective CNS electrical conductivity, σ_{o}, is of similar magnitude, their percolation exponent (t) can vary significantly. (A)–(C) demonstrate that although CNT-based polymer nanocomposites could exhibit the best mechanical and transport properties of all the CNSs, precise morphological descriptors are needed to avoid orders of magnitude over- or under-prediction of resultant physical properties. (D) Illustrations of the two commonly assumed morphologies of A-PNCs in comparison to the three-dimensional reconstruction and the morphology generated according to such reconstructions using the current simulation framework. |

Recent work on three-dimensional morphological quantification^{45,46} and modeling of CNS reinforced polymers^{47,48} has shown that a significant amount of stochastically-varying local curvature is exhibited by the CNSs present in the polymer matrix.^{45} These works also show that such structural randomness leads to the formation of a large number of van der Waals dominated junctions that may contribute to the mechanical and transport properties of such architectures. However, the formation of such junctions cannot be natively described by the assumptions that dominate previous theoretical frameworks, which primarily include simple functional forms, whereas stochastic descriptions are more representative of the kinetic and diffusive processes by which CNSs are synthesized,^{5,49} e.g., CNTs via chemical vapor deposition.^{50} See Fig. 1D for an illustration of the simple morphological assumptions previously used to model an exemplary aligned CNT system, in addition to the visualized three-dimensional CNT topology and the stochastic descriptions used herein. Further, while the number of junctions formed in CNS architectures is strongly dependent on the CNS volume fraction, very little is currently known about how the junction density evolves with CNS packing proximity (i.e., V_{f}).

3D descriptions of CNS morphologies in nanocomposites have recently been provided using quantitative electron tomography^{45,46} and small angle X-ray scattering.^{50} Specifically, we have developed an electron tomographic approach that permits the extraction and quantification of the 3D arrangement of aligned CNT forests embedded in a polymeric matrix (Fig. S1†).^{45} Morphological metrics describing the network structure such as alignment, proximity, junction density, and waviness may be obtained from the resultant digitized reconstructions with nanometer resolution. Since aligned CNTs exhibit the best combination of mechanical, thermal, and electrical properties among the previously studied CNS reinforced polymer systems (see Fig. 1A),^{17,29,42} and were also the subject of our recent experimental and computational work on their three-dimensional morphology, this system was selected to be the focus of the current study.

Here, we use newly available morphogenesis (evolution of morphology with increasing V_{f}) data from electron tomography and stochastic simulations, to develop more representative mechanical, thermal, and electrical property prediction tools for aligned CNT polymer nanocomposites (A-PNCs) in directions that are both parallel and transverse to the CNT primary axis. Firstly, using the tortuosity measured by 3D TEM, we introduce a new 3D stochastic definition of waviness, which is found to provide the best match to previously-measured modulus data. We then establish the scaling of the CNT–CNT junction density with CNT V_{f}. This scaling is used to shed light on the dependence of transport properties on CNT network structure and its evolution with V_{f}. We believe that the structure–property correlations established here will provide valuable insights into many other past and future results from studies of similar nanocomposites and aligned CNT structures. More broadly, the findings reported herein are expected to be generalizable to other CNS-PNC systems with well characterized 3D morphology.

Our imaging method enables, for the first time, the accurate measurement of A-CNT V_{f} in the nanocomposites (nominally the in situ V_{f}), among other quantitative features.^{45} Comparing the ex situ and in situ V_{f}s we observe that polymer impregnation into the porous A-CNT forest causes it to expand laterally (normal to the alignment axis), resulting in a lower V_{f} than that estimated post densification (in situ V_{f}s of (0.44 ± 0.01) %, (2.58 ± 0.25) %, (4.04 ± 0.19) %, (6.89 ± 0.43) %, ex situ V_{f}s of 1.00%, 5.00%, 6.00% and 11.70%, respectively). From the plot of in situ composite V_{f} versus the ex situ V_{f} of the densified starting material (Fig. S1f†), we find that a corrective factor of 0.59 can be applied to all the V_{f} values reported earlier,^{17,29,42} which did not account for the polymer-induced expansion. The inter-CNT spacings or CNT proximity (Γ) is the second morphological factor and can be readily calculated from the tomography data (Fig. S2†). With increased V_{f} we observe a non-linear reduction in Γ, which is in excellent agreement with the trends observed in the data obtained from SEM images of as-grown densified CNT arrays (Fig. S2†).^{56} As suggested by Stein et al., the mean spacings take the form:

(1) |

3D tortuosity of the CNTs, the third morphological factor, is key to stochastic simulations of morphology and can also be quantified from our tomography data. From the skeletonized 3D volumes of the 4 different V_{f} A-PNCs studied (ex situ V_{f}s of 1.00%, 5.00%, 6.00% and 11.70%), we extract tube arc length (l) to Euclidean distance (d) ratios i.e., tortuosity (l/d), for tube segments in the tomographic volume whose lengths are at least 10 times larger than the diameter of the CNTs (7.65 nm) (Fig. S3a†). The segments have to be at least of this length in order for us to extract a meaningful, representative waviness. We observe the 3D tortuosity to decrease with increasing V_{f}, suggesting a straightening of CNTs due to increased confinement (Fig. S3b†). However, the tortuosity measured here does not explicitly capture the nature or functional form of waviness, which is also an essential morphological descriptor. The waviness of CNTs has been previously described using simple sinusoidal and helical functions that neglect the stochastic nature of CNT growth.^{57,58} Here, we employ our novel definition of waviness, which provides a more realistic representation CNT morphology in the form of a “random” helical function. The usual waviness ratio (w) is a scalar measure of waviness that is typically defined as the amplitude over the wavelength of the chosen waviness function. For the random helical definition this ratio is modified by a pre-factor as described in other work.^{47,59} Depending on the assumed geometry (i.e. sinusoidal/helical/randomly helical), the 3D tortuosity for a given w will vary drastically. The tortuosity versus w plot in Fig. 2B shows how the tortuosity may be converted to an analytical/deterministic waviness (i.e. w) and vice versa.^{48} Using the 3D morphology data, we first consider the elastic modulus of the A-PNCs.

Fig. 2 Deformation modes, morphological descriptors, and modulus modeling: (A) Illustration of the dominant physical mechanisms for CNT mechanical response. (B) Plot of tortuosity versus the analytical waviness ratio (w) for sinusoidal, helical and random helical formulations of waviness.^{48} The shaded region represents the range of tortuosity (τ) values calculated from tomography data. (C) Elastic modulus (E) of A-PNCs in the longitudinal (||, parallel to CNT alignment) and transverse (⊥, perpendicular to CNT alignment) directions, and the respective model predictions, as a function of the in situ CNT volume fraction V_{f} and accounting for morphology evolution. E scaling for A-PNCs shows that higher V_{f} mediated waviness reduction leads to significant enhancement in the E previously measured^{17} in the parallel (||) direction, whereas the perpendicular (⊥) direction sees little benefit with higher V_{f}. |

Most prior studies have employed a constant waviness ratio to fit moduli measured over a range of V_{f}s, which leads to a strong under- or over-estimation of the elastic modulus depending on the V_{f} at which the waviness was measured. Since waviness changes with the V_{f}^{45,60} – decreasing at higher V_{f}s due to increased crowding – a careful study of the evolution of waviness with CNT packing is essential to interpret the complex dependence of the modulus on V_{f}. Using the measured trend in waviness with V_{f} (Fig. S3†), we generate sinusoidal, helical and randomly helical CNT morphologies at the V_{f}s studied by Handlin et al.^{17} The reinforcement modulus of these CNTs (E_{f}) at various V_{f}s is measured using our simulation technique^{47} and inserted into the rule-of-mixtures formula (E_{A-PNC} = V_{f} × E_{f} + (1 − V_{f}) × E_{Matrix}) to estimate the composite elastic modulus. From Fig. S5,† we note that, despite accounting for the change in waviness with V_{f}, the sinusoidal and helical assumptions strongly underestimate the A-PNC properties. This indicates that such simplistic definitions of waviness are inadequate in describing the mechanical performance of these systems (see discussion in Fig. S5†). Remarkably, the random helical definition, which is the first to account for the stochastic CNT morphology, provides the best match to the measured nanocomposite modulus (Fig. 2B and Fig. S5†). This result strongly supports the need to explicitly consider the stochastic nature of CNT morphology, and its evolution with increased crowding.

We note that the model CNT ensembles generated using the 3 primary structural descriptors (V_{f}, proximity and waviness/tortuosity) obtained by tomography, also show agreement with secondary or derivative structural features such as CNT–CNT junction density (N_{J}) and the average distance between nearest contact points (ξ) (Fig. 3). This agreement further validates the random helical formulation developed herein. We obtain N_{J} (m^{−3}) by skeletonizing the reconstructed 3D volumes (Fig. S1 and S6†) and tallying the number of CNT–CNT contacts i.e., branches in the skeletonized structures (Fig. 3A).^{61} We find that, as the CNTs are brought closer, the short range van der Waals attractions cause them to cluster,^{45} causing a non-linear increase in N_{J}, best fit by a power law equation, N_{J} (m^{−3}) = 3 × 10^{21} V_{f}^{1.5}. N_{J} increases by nearly two orders of magnitude ((7.24 ± 1.13) × 10^{20} m^{−3} to ((5.38 ± 0.63) × 10^{22} m^{−3}) for an order of magnitude increase in in situ V_{f} ((0.44 ± 0.01) % to (6.89 ± 0.43) %) (Fig. 3B). We note that these experimentally measured N_{J} values are in good agreement (in the order of magnitude, ∼10^{22} m^{−3}) with earlier computational calculations of N_{J} in CNT films.^{62} However, these calculations predict a quadratic dependence of N_{J} on V_{f} whereas our data shows N_{J} ∼ V_{f}^{1.5}. That said, the earlier work simplistically assumed that CNTs are rigid rod-like objects, whereas they are only stiff along the local plane of the CNT wall, and rather overall compliant due to shear and bending modes (see Fig. 2A).^{62,63} On the other hand, here we have real estimates from high quality tomography data. The average distance between nearest contact points (ξ) may also be readily calculated from N_{J} using:^{62}

(2) |

We now turn our attention to the transport properties (thermal and electrical conductivity) of the A-PNCs utilizing the junction density and distance quantification discussed above.

We plot the thermal conductivity values (k) reported in Marconnet et al. against the corresponding corrected, or in situ V_{f}s, (Fig. 4B).^{29} We find that the best approximation to the trend in the corrected data is given by (k − k_{m}) = (115 ± 17) (W m^{−1} K^{−1}) V_{f}^{1.56}, as opposed to (k − k_{m}) = 73 (W m^{−1} K^{−1}) V_{f}^{1.72}, suggested in the original paper (Fig. S7a†).^{29} The increase in the prefactor in this relationship suggests a stronger contribution of the CNTs to conduction than originally calculated.^{29} While the power-law dependence might appear similar to percolation behavior, we note that such scaling at the high CNT loadings investigated here has been found to be due to network characteristics dominated by CNT–CNT contact resistance.^{63} Marconnet et al. speculated, in accordance with previous findings, that the upward concave trend in the data was indicative that the number density of CNT–CNT contacts (N_{J}) was increasing (although the nature of this increase was then unclear). Now that CNT–CNT contact density can be quantified, it appears that the scaling of N_{J} with V_{f} (N_{J} ∼ V_{f}^{1.50}) in our tomography data (Fig. 4C) is identical to the thermal conductivity power law behavior (k ∼ V_{f}^{1.50}) in the corrected property data, which validates the original interpretation of the data. Additionally, in the plot of the thermal conductivity enhancement versus the estimated N_{J} for the corrected V_{f}s in Marconnet et al., we observe a close-to-linear dependence (discounting the clear outlier indicated by red data point in Fig. S7b†). Indeed, the CNT networks, growing and interconnecting with the increased V_{f}, act as “heat pipes” throughout the composite, thus increasing thermal conductivity as their cross-sectional area contribution to the nanocomposite increases with V_{f}.

From our tomography data we are able to provide the first quantitative support to hypotheses linking morphological changes and the non-linear increase in thermal conductivity in aligned CNT composites. While it is possible for other sources of non-linearity to exist (see details in ESI, Fig. S8 and S9†), the 3D data reveals that the primary source of non-linearity is the non-linearly increasing density of CNT–CNT contacts with V_{f}. A natural conclusion from this newly exposed data is that, contrary to the original interpretation, in order to increase the thermal conductivity in a realistic system, with imperfect alignment and continuity, the number of CNT–CNT contacts needs to be maximized, as it provides more axial pathways for phonon transport. These findings further emphasize the importance of network visualization, and allow for better understanding of the origin and limitations on A-PNC thermal conductivity.

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## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9nr03317c |

‡ This work was primarily done while at the Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA. |

§ These authors contributed equally to this work. |

This journal is © The Royal Society of Chemistry 2019 |