Excluded Volume Approach for Ultrathin Carbon Nanotube Network

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Excluded Volume Approach for Ultrathin Carbon Nanotube Network Stabilization: A Mesoscopic Distinct Element Method Study Yuezhou Wang, Grigorii Drozdov, Erik K. Hobbie, and Traian Dumitrica ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.7b01434 • Publication Date (Web): 27 Mar 2017 Downloaded from http://pubs.acs.org on April 1, 2017

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ACS Applied Materials & Interfaces

Excluded Volume Approach for Ultrathin Carbon Nanotube Network Stabilization: A Mesoscopic Distinct Element Method Study Yuezhou Wang,1 Grigorii Drozdov,2 Erik K. Hobbie,3 and Traian Dumitrica1, 2 ,4* 1

Department of Chemical Engineering and Materials Science, University of Minnesota, Twin Cities, MN 55455, 2

Scientific Computing Program, University of Minnesota, Twin Cities, MN 55455, 3

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Department of Physics, North Dakota State University, Fargo, ND 58108,

Department of Mechanical Engineering, University of Minnesota, Twin Cities, MN 55455 Corresponding Author *E-mail: [email protected].

ACS Paragon Plus Environment

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ABSTRACT.

Ultrathin carbon nanotube films have gathered attention for flexible electronics

applications. Unfortunately, their network structure changes significantly even under small applied strains. We perform mesoscopic distinct element method simulations and develop an atomic scale picture of the network stress relaxation. On this basis, we put forward the concept of mesoscale design by the addition of excluded-volume interactions. We integrate silicon nanoparticles into our model and show that the nanoparticle-filled networks present superior stability and mechanical response relative to pure films.

The approach opens new possibilities for tuning network

microstructure in a manner that is compatible with flexible electronics applications.

KEYWORDS: carbon nanotubes, ultrathin films, excluded volume, nanoparticles, mechanical load, flexible electronics


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INTRODUCTION Carbon nanotubes (CNTs)1 – graphene tubes roughly 1-10 nm in diameter2 and up to millimeters in length – are touted as game changing materials in electronics, aerospace and medicine.3-6 Depending on the chiral vector characterizing the symmetry of conceptually rolling a graphene sheet into a tube, single-walled (SW) CNTs can be either metallic or semiconducting. As-produced SWCNT materials typically contain a distribution of the two electronic types. However, recent advances in purification have ushered in an era of research focused on highly monodisperse SWCNTs with ideal colloidal dispersion.7-9 As thin flexible films, these purified materials feature a porous structure of SWCNT bundles that show tremendous potential for applications, including transparent conductors, field-effect transistors, and functional layers in photovoltaic devices.10-11 Ultrathin SWCNT networks (with coverages of up to a few SWCNT layers) are electrically conducting and one of the most feasible and promising applications of these films relates to flexible electronics.12 So far, only a handful of studies have focused on both the mechanics and electrical properties of these films.13-16 Flexural studies of length and type-purified SWCNT films deposited on polydimethylsiloxane substrates, for example, offer a glimpse of the remarkable TPa mechanics of individual SWCNTs,17 but with significant plasticity even at small strains.13-15 Because the network structure can change under strain, often irreversibly, the sheet resistance of flexible SWCNT films can be detrimentally impacted by flexure.18 Although this type electrical sensitivity can be useful for strain sensors19, most flexible electronics applications demand stability, and this type of strain response is clearly undesirable. Physically, there are two sources of electrical resistance in a percolated network.20-21 These are the intrinsic resistance of a SWCNT and the contact resistance between connected SWCNTs. Since the former is practically unchanged under the applied deformation, it follows that a dominant portion of the unwanted increase in


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resistance arises from a loss of SWCNT-SWCNT contacts. Thus, the mechanical and electrical response are tied together, and resolving the stability issue is vital to realizing the next generation of SWCNT-based materials for flexible electronics. Several strategies have been so far explored for enhancing the mechanical properties of CNT materials, including strengthening the CNT junctions by coating with graphene,22,23 and introducing covalent bonding via chemical functionalization24 and irradiation.25-28 Unfortunately, covalent bonding adversely affects the intrinsic CNT characteristics, and it is therefore a less suitable strategy for flexible electronics applications.24 Here we investigate a new approach that relies on the introduction of excluded volume - space within the mesoporous network that is made inaccessible to the SWCNTs. By filling the pores of the network with nanoparticles (NPs), we aim to effectively reduce the available volume that can accomodate SWCNT rearrangements. Because yielding requires such rearrangements, a stabilizing effect could be expected. In contrast to chemical crosslinking, the NP filler scheme relies only on the introduction of additional noncovalent interfaces and thereby preserves the exceptional properties of individual SWCNTs. Since all-atom simulations of SWCNT networks29 are computationally prohibitive, we explore this scheme via mesoscale-level simulations30-31 performed with the recently developed mesoscopic distinct element method32-33 (mDEM) for carbon nanotubes. In mDEM, the SWCNT network is represented by a collection of cylindrical elements interacting via bonded and nonbonded contacts, which represent the atomic-scale interactions. mDEM is a coarse-grained representation of the system in question as each element generally represents a segment of the detailed SWCNT containing many carbon atoms. The parallel-bond contacts32 are trained to capture the intra-tube covalent interactions responsible for the local linear elasticity of individual SWCNTs. In addition, we account for the microscopic van der Waals (vdW) interactions, which


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are represented at the mesoscale by the non-bonded contacts acting between elements in close spatial proximity.32 On the basis of extensive mDEM simulations, we develop a picture comprising the stress at the atomic level, stress relaxation, and microstructure evolution of SWCNT networks under uniaxial tension.34-35 We further include a NP phase, which is represented by a collection of spherical distinct elements interacting through non-bonded contacts, both with each other and with the SWCNT network. mDEM simulations indicate that the filling of the network pores with NPs is an effective approach for developing durable SWCNT films for flexible electronics applications. RESULTS AND DISCUSSION

Figure 1. (color) a) mDEM simulated SWCNT network, measuring 500 nm by 500 nm by 11 nm in size, after relaxation. Color reflects the magnitude of the bending moments stored by the parallel contact bonds. Callouts detail the entangled structure of SWCNT bundles and bent SWCNTs. b)


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Force chains (tension and compression) and pore size distribution in the relaxed SWCNT network. c) Energy and number of aligned pairs during the network relaxation process. Figure 1a-b displays the characteristics of a typical freestanding mDEM simulated network featuring a nanoporous structure of entangled SWCNT bundles and bent SWCNTs. This network was constructed from 350 (10, 10) SWCNTs each 475 nm in length, and has a density of only 0.2 g/cm3. It stores significant strain energy, 96 % of which is the bending visible also in the color code of Figure 1a. In addition to shear stress, the network also stores a small amount of tensile and compressive stress, which is isotropically distributed, Figure 1b. The pore size distribution, listed in the same figure, reveals the presence of many “medium” (100-to-200 nm2) pores and “large” (200-to-300 nm2) pores, in good agreement with the pore distribution measured over the same area in ultrathin networks prepared through the vacuum filtration of aqueous SWCNT solutions.18 To arrive at this mDEM representation of the network structure, we computer generated a random network of straight (strain free) SWCNTs, which were next allowed to evolve in time under the influence of the non-bonded interactions. As documented in Fig. 1c, mDEM relaxation leads to a significant total-energy lowering, in spite of the accumulation of strain in the SWCNTs. In the initial stages (not shown), the relaxation is dominated by the excluded-volume (i.e., repulsive nonbonded component) interactions, which act to quickly separate SWCNTs that are physically overlapped. On the ns time scale, the relaxation process is driven by attractive vdW interactions, which act to increase both the vdW cohesive and bending energy components. During the relaxation process leading to structure shown in Fig. 1a, there is a significant increase in the number of cylindrical distinct elements in vdW contact that are also aligned, Figure 1c. (Two elements were considered aligned when the crossing angle31 between them was less than 10 deg.) At the microstructure level, this behavior reflects network relaxation via zipping,37-38 a topological


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transformation in which crossed SWCNTs gradually bind. After 2 ns, we find that the increase in the number of aligned pairs slows down significantly, and that the network structure appears to stabilize at around 4 ns. In agreement with experiment,18, 37 we observe the formation of SWCNT bundles (see the inset of Fig. 1a), i.e. local aggregates of several SWCNTs. Note that the SWCNT bundling during the film relaxation is captured due to the realistic description of vdW interactions provided by the anisotropic mesoscopic vdW potentials.31-32 We have further probed the mechanical stability of the mDEM-represented networks to uniaxial tensile loadings. As in our previous simulations,32 two thin layers of distinct elements at the left and right edges of a film were designated as grips. Displacement-controlled loading was enabled by prescribing both grips to accelerate from 0 to the given velocity during 0.6 ns. This acceleration period is used to reduce the undesired dynamic response, which is significant in the case of instantaneous acceleration of the grips. In the series of simulations discussed, the network samples were elongated at the constant strain rate of 108 𝑠 −1 . Our simulations considered the response of the networks over the 0%-to-20% strain range, and revealed significant microstructural changes. As the network elongates, it decreases its lateral dimensions significantly. Although the tensile stress in individual SWCNTs is modest even at large strains (for the 20 % elongated sample see the color code in Figure. 2a), a neck develops. As can be seen in the same figure showing the 200 % elongated network, the neck eventually leads to failure via massive disentanglement and SWCNTs bundling along the applied strain direction.


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Figure 2. (color) a) SWCNT network (same as in Fig.1) under 20 % strain showing 37% transverse shrinking in the middle (compared to the initial state). The 200 % stretched network shows that failure occurs as this shrinkage continues. Color shows the magnitude of the tensile force (above) and bending moment (below) in the parallel contacts. b) Number of aligned pairs during stretching. c) SWCNT network with 4,000 NPs (NPs are not shown) under 20 % strain. d) Pore size distribution in the relaxed and stretched networks. The number of “very large” pores is shown on the secondary scale. In a and c, the double arrow indicates the applied strain direction.


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The observed microstructural changes originate in the attractive component of the vdW interaction: Because the load transfer between the gripped SWCNTs and the center of the network is poor, these gripped SWCNTs along with some entangled SWCNTs, are pulled out of the network. This triggers a massive zipping relaxation, which is visible by inspection in the thickening of the bundles. The evolution of the number of aligned pairs, Figure 2b, indicates that zipping-relaxation starts at 3 % strain and becomes substantial at larger strains. For example, at 20 % strain, the microstructure evolves to almost double its number of aligned contacts, and thus lowers the vdW cohesive energy. This zipping relaxation process is associated with a significant increase in bending. For example, we find that the 20 % stretched network doubled its amount of bending energy. At the same time, the network acquired a negligible amount of tensile stress Having identified the key mechanism of stress relaxation under tension, we now investigate the possibility of designing interactions at the mesoscale in order to enhance the network stability and mechanical response. Clearly, an increase in the attractive vdW interactions is not effective as it promotes bundling, and thus lowers of the network’s plasticity threshold. On the other hand, a decrease in vdW attraction will lower the load transfer capabilities of the SWCNT junctions. We seek here to arrest the initial affine deformation of the network using a simple excluded-volume scheme with the addition of a low concentration NP phase that will also maintain the lightweight attribute of the film. Specifically, we have considered silicon (Si) nanospheres measuring 5 nm in diameter. With this choice in dimension, we can fill a large number of “medium” and “large” pores without significantly perturbing the relaxed pristine SWCNT network. As an engineering material, Si is nontoxic, abundant, and it currently dominates the microelectronics industry. At the nanoscale under quantum confinement, silicon also acquires attractive photonic traits, such as a bandgap photoluminescence. In terms of functionality, mixing Si NPs with SWCNTs thus has


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potential relevance to a variety of devices, such as nanoscale heterojunctions and light-emitting diodes. Figure 3. a) Atomistic and mDEM representations of NP-SWCNT at the equilibrium position. The magnitude and direction of the vdW forces between NP and each mesoscopic element of the SWCNT is shown with colored sticks. b) Plot of the NP-SWCNT vdW potential as obtained with the developed mDEM contact. The MD data (dots) is shown for a comparison. The depth of the

potential well is 4.7 eV (108.4 kcal/mol). c) Relaxed SWCNT network filled with 2,000 NPs (volume fraction 4.7 %), as obtained with the developed model. The filled network density is 0.33 g/cm3. With mDEM, each Si NP was coarse-grained as a spherical distinct element of the same mass and size. To this end, we have developed additional vdW contact models for the NP-NP and NPSWCNT vdW interactions. Note that the NP-SWCNT contact potential necessitates an anisotropic closed form U (R, ), where R is the distance between the centers of the elements and  the


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misalignment angle (Figure 3a), in order to render a corrugation-free vdW energy potential, Figure 3b. The mesoscopic potential, detailed in the SI, was trained to reproduce the underlying molecular dynamics (MD) vdW landscape obtained with the UFF description the atomistic interactions.38 Relying on the enhanced mDEM model, we have filled the relaxed networks with NPs and conducted further relaxation simulations. During this process, the NPs fill the natural pores of the network, Figure 3c, and find their minimum energy locations near junctions.39 We have next queried the stability to uniaxial strains of the relaxed filled SWCNT networks. The results presented in Figure 2 can be directly compared with the corresponding behavior of the same network without NPs in order to evaluate the effectiveness of the added contacts. As can be seen in Figure 2c, necking is totally absent in the 20 % elongated hybrid network. By inspecting the color code, one can observe that the individual SWCNTs carry more tensile load. Compared with the pristine case, there is significantly less non-affine bending deformation. The evolution of the number of aligned pairs, Figure 2b, indicates that the zipping relaxation process is significantly inhibited. For example, under 20 % strain, the increase in the number of aligned pairs is about 50 % less than in the pristine case. In the strained pristine network, the zipping relaxation leads to an increase in the number of “medium” and “large” pores, and to the creation of sixteen “very large” (300-to-1600 nm2) pores, Figure 2d. Some of these “very large” pores are emerging from the zipping-mediated union of “medium” pores, Figure S1. A comparison of the number of pore sizes of the pristine and hybrid networks demonstrates that the addition of NPs makes the network more stable as it is very effective in limiting the increase in the number of “large” pores. The secondary plot of Figure 2d further reveals the emergence in the hybrid network of only seven “very large” pores, with sizes distributed toward the smaller range of this category. Since the number of “medium” pores is


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larger in the strained filled network than in the pristine one, we conjuncture that the NPs are arresting the zipping-mediated union of “medium” pores into “very large” ones. Figure 4. (color) mDEM simulated SWCNT network, measuring 1,000 nm by 1,000 nm by 11 nm in size, under small tensile strains: a) and b) are pristine, while c) and d) are filled with 10,000 NPs (volume fraction 6 %). For clarity, the NPs are not shown. The double arrows indicate the applied strain directions. In a) and c), the load transfer in the network is reflected by the magnitude and distribution of the tensile forces stored by the parallel-contacts. (See the color scale bar). The force chains presented in b) and d) show that the pristine and filled networks are accommodating the imposed 2 % elongation through tension and compression filaments oriented along and

perpendicular to the strain direction.


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Figure 5. Engineering stress-strain curves in pristine and filled SWCNTs measuring a) 1,000 nm by 1,000 nm by 11 nm, and b) 500 nm by 500 nm by 11 nm. The added attractive vdW through the NP interactions are effective in enhancing the tensile stress in the network. As an illustration, Figure 4 presents a detailed comparison of the tensile stress development and array of force chains in a larger network constructed from 500 SWCNTs, each 950 nm in length. The network has a density of 0.14 g/cm3 and a porosity of 83.5 %. In the 0 % to 2 % strain range considered, there are little network topology changes. Comparison of panels a and c reveals that in the filled film, the SWCNTs at the left/right edges develop significant tensile stress when the “grip” elements are being gradually displaced. By contrast, the same SWCNTs are pulled out from the pristine network while carrying very little tension. Further, panels b and c reveal a higher density of tensile force chains throughout the filled SWCNT sample. The significantly lower density of compression chains in the transverse direction is likely a signature of the excluded-volume interactions.


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The differences in microstructural evolution between the pristine and filled networks are reflected in the stress-strain curves presented in Figure 5. The plotted engineering stress, which was derived from the unbalanced force (sum of the contact force vectors) on the grip distinct elements, doesn’t account for the transverse shrinking effect in the cross-sectional area. As expected from the analysis of Figure 4, the filled networks are stiffer than the pristine ones. In Figure 5a, we measured over the 1 % to 2 % deformation range a Young’s modulus of 8.6 GPa in the pristine large network, and 12.4 GPa and 13.1 GPa for the SWCNT network with 5,000 NPs and 10,000 NPs respectively. These elastic constant predictions compare well with the 8 GPa value measured in buckypaper34 and are smaller than the ~24.0 GPa value reported40 for sheets with similar densities and porosities, but comprising longer and larger diameter CNTs. Due to the excluded-volume interactions, the filled networks are able to resist higher tensile loads until they yield. In Figure 5b, the yield stress of 0.4 GPa is remarkably about 50 % larger than in the pristine case. After yield, the massive relaxation of the pristine network is reflected in a plateau of the stress strain curve. In the filled networks, the partial hindering of the vdW strain relaxation is reflected in stress strain curves above the pristine case curve.

CONCLUSION In conclusion, our mDEM simulations indicate that the nanoparticle-filled networks present superior stability and mechanical response relative to pure films.

From an experimental

perspective, incorporating the nanoparticles into CNT materials can be achieved using several possible approaches, including in situ synthesis41,42 and two-step processes such as an initial assembly of the nanotube network followed by the subsequent deposition of the NP phase on top


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of the SWCNTs. Some potential deposition routes include vacuum filtration,43 direct deposition of the nanoparticles from a gas phase immediately following their synthesis,44 or Langmuir-Blodgett type approaches that exploit the formation of a nanoparticle monolayer on the surface of an immiscible fluid, such as water.45 The precise choice of deposition scheme will be dictated by the nature of the NPs, such as solubility and surface treatment. Each approach will likely yield variations in the precise morphology of the hybrid. In general, optimizing the properties of interest of these hybrids implies exploring a large parameter space.

Since an exhaustive experimental

study of how nanoparticle size, composition (metallic vs. semiconducting) and concentration act in concert to impact targeted properties (modulus, conductivity, durability) is prohibitive, mDEM opens the possibility for a computational-guided approach for the processing of stable SWCNT systems, including aerogels.22,46 NP-SWCNT vdW contact models can be developed in the same manner as here. Additionally, we anticipate that mDEM can be coupled with a graph theory approaches47 in order to identify and characterize probable conduction pathways in the SWCNT network, and thus to predict the electric transport of the network.

COMPUTATIONAL METHODS In the presented mDEM simulations, each SWCNT is represented by a linear sequence of rigid cylindrical elements. This constitutes a coarse-grained model of the SWCNT in question, as each element represents a segment of the SWCNT.33 Specifically, we have discretized a (10,10) SWCNT such that each cylindrical element with length 1.36 nm and radius 0.68 nm, and thickness 0.335 nm represents 220 carbon atoms. These mesoscopic elements are tracked in the Lagrangian frame. In the PFC3D implementation,48 they are evolved deterministically in time as rigid bodies


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with a damped scheme based on a velocity Verlet algorithm for translations and a fourth-order Runge–Kutta algorithm for rotations. The time step is 10 fs. The cylindrical elements are coupled by parallel-bond contact, which introduce normal and shear forces (Fn and Fs) and moments (Mn and Ms) to resist relative displacements and rotations. The parallel-bonds were calibrated with MD data31 and the chain of parallel-bonded elements describes the individual SWCNT as a Euler– Bernoulli beam. Mesoscopic elements located on two different SWCNTs may interact via vdW and viscous contacts. The vdW contact has been developed by us via a procedure that involves the integration of the atomistic vdW interactions between two CNTs.32 Our vdW contacts capture the general case when the SWCNTs are simultaneously shifted and crossed. Unlike isotropic potentials,37 they reproduce the smooth sliding of two parallel SWCNTs vs. offset. References32 and 33 give the explicit form for these three contacts. The new NP-SWCNTs and NP-NP developed here are presented in the SI. In Figs. 1c and 2d, the pore size measurement was performed by image processing the pixels of vacancies on a 2D-projected black and white network image. The force chains shown in Fig. 1b and Fig. 4 b and d display the network structure formed by parallel contacts in tension and compression. Because computational strain rates comparable to those applied in experiment are prohibitive, we probed the SWCNT networks at a strain rate of 108 s−1 . We expect that with this choice, mDEM simulations capture well the elastic regime and the initial stages of the network failure. A high strain rate is less desirable when massive network relaxation occurs. Nevertheless, the failure mechanism obtained in the pristine network, Fig. 2a, compares well with the experimental observations.35


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ASSOCIATED CONTENT Supporting Information. The following files are available free of charge: A Supporting Information file gives the technical details for the construction of the NP-SWCNT and NP-NP mDEM contacts, and Figure S1.

Movie 1: Pristine SWCNT network (500 nm by 500 nm by 11 nm in size) stretched to 20 % strain. Tensile load conveyed by individual SWCNTs are colored. The last frame is shown in Fig. 2a.

Movie 2: Filled SWCNT network (500 nm by 500 nm by 11 nm in size, filled with 4,000 NPs) stretched to 20 % strain. The last frame is shown in Fig. 2b.

Movie 3: Pristine SWCNT network stretched to 10 %. Zoom-in around the network center showing structure collapse and massive bundling. The first and last frames are shown in TOC.

Movie 4: Filled SWCNT network stretched to 10 %. Zoom-in around the network center showing stabilized network and lack of bundling. The first and last frames are shown in TOC. ACKNOWLEDGMENTS This work was supported by an Early Stage Innovations grant from NASA’s Space Technology Research Grants Program. Y.W. gratefully acknowledges support from the University of Minnesota Informatics Institute. G.D gratefully acknowledges partial financial support from Russian Foundation for Basic Research under grants RFBR 16-31-00429 and RFBR 16-31-60100. We thank the Itasca Consulting Group for the PFC3D software support.


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Our test mDEM simulations on two 300 crossed SWCNTs indicated that the zipping relaxation time is sensitive to in-plane nm deviations of the NP from its minimum energy position. It changed from 4 ns (equilibrium) to 1.5 ns when the NP was shifted away by 1


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Excluded Volume Approach for Ultrathin Carbon Nanotube Network

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