Effects of the nanotube length and network morphology on the deformation mechanisms and mechanical properties of cross-linked carbon nanotube films

A unique combination of exceptional thermal, electrical, and mechanical properties of carbon nanotubes (CNTs) stimulated intensive studies targeted at revealing the potential of CNT materials for advanced electronic and optoelectronic applications,¹ including on-chip thermal interface materials,^2–4 energy harvesting,⁵ flexible electronics and sensing,⁶ photoluminescence and imaging,⁷ and generation of terahertz radiation.⁸ The thin CNT films, in particular, are considered as promising candidates for low-cost flexible electronics.⁹ In these applications, the material performance is strongly affected by electronic, optical, and thermoelectric properties of individual nanotubes. The efficient use of the CNT materials in these applications also requires tunable mechanical properties. For instance, good resilience and high mechanical compliance are needed for thermal interface materials, while the ability to sustain cyclic large-strain, reversible bending and stretching deformations is important for flexible electronics.⁹ 

The light-weight CNT materials including CNT film, mats, sheets, and buckypaper,^10–20 forests, and vertically aligned arrays,^21–24 as well as sponges and aerogels,^25,26 demonstrate, however, poor elastic properties and strength. The experimentally measured Young's modulus and strength of CNT films vary in the range from $∼100MPa$ to $∼3GPa$ and from $∼2MPa$ to $∼30MPa$⁠, respectively.^{10–12,15,17,19,27–29} The light-weight CNT materials also demonstrate a pronounced viscoelastic behavior, e.g., stress relaxation under applied mechanical load.²⁹ In these materials, the individual CNTs form continuous networks of interconnected and entangled bundles of various thicknesses. The major mechanism of load transfer in the materials composed of pristine CNTs is friction at inter-tube sliding within thick bundles.¹⁶ The static and dynamic friction between pristine nanotubes, however, is low,^30–34 and the weak van der Waals interaction between CNTs introduces low barriers for inter-tube sliding. It represents a bottleneck for the load transfer in CNT materials with random networks of bundles of nanotubes.^35,36

The mechanical properties of light-weight CNT materials can be improved by increasing the degree of connectivity of the nanotube network. If the material density is fixed, then the CNT length and the degree of nanotube entanglement, as well as the characteristic bundle thickness and length, can be considered factors that affect the network connectivity and load transfer. The multiple experimental and theoretical studies of CNT-reinforced composites with polymer and metal matrix materials, e.g.,^37–40 reported an increasing reinforcing ability of nanotubes with increasing CNT length. The studies of the length-dependent structural and mechanical properties of “pure” CNT materials, e.g., buckypaper, are scarce, and the understanding of how the CNT length affects the network morphology and load transfer is still elusive. For instance, the coarse-grained mesoscopic simulations in Ref. 41 surprisingly predicted that Young's modulus of single-walled CNT buckypaper strongly decreases with increasing CNT length, while the porosity and pore size are found to increase with increasing CNT length. On the contrary, the authors of Ref. 42 found in coarse-grained molecular dynamics simulations that the CNT length only marginally affects the elastic properties of buckypaper in the frequency-independent regime. In the experimental study,⁴³ the authors fabricated buckypaper with controlled CNT lengths and found that the material composed of the millimeter-long CNTs has twice larger tensile strength compared to the material composed of micrometer-long CNTs, while the modulus does not change. The experimental study⁴⁴ revealed, contrary to Ref. 41, that the multi-walled CNT buckypaper with shorter nanotubes had larger pore sizes than that with longer nanotubes.

Another route to improving mechanical properties of CNT-based materials is offered by covalent cross-linking of atoms belonging to different nanotubes.^36–46 The cross-links (CLs) can be induced by irradiation of nanotubes with electron^36–50 and ion^51–56 beams or by chemical functionalization of nanotubes.^50,57 It was shown that the elastic modulus and strength of CNT bundles exhibit an order-of-magnitude increase after cross-linking by the electron beam irradiation.⁴⁹ The tensile modulus and strength of single-walled CNT sheets demonstrate up to threefold and fivefold increase after cross-linking by electron beams,⁴⁷ while the chemical functionalization combined with the electron beam irradiation improves the strength of CNT sheets by $∼50%$⁠.⁵⁰ The effect of the CNT length on the mechanical performance of cross-linked CNT materials, however, was not studied.

The load transfer in small CNT systems, including individual double and multi-walled CNTs, two nanotubes, as well as small bundles of nanotubes, is studied in molecular dynamics (MD) simulations based on empirical interatomic potentials^{47,51–56,58,59} and density functional theory,³⁶ as well as in combined MD-tight binding quantum mechanical simulations.⁴⁷ Although these results provide invaluable information on the theoretical limits of the load transfer improvement due to cross-linking associated with various types of defects, it is difficult to translate the obtained results into the properties of random CNT materials such as films, buckypaper, forests, and aerogels, where CNTs form continuous networks of bundles. The structure–property relationships for CNT network materials can be established with coarse-grained mesoscopic models that neglect atomic structures of individual nanotubes but retain geometrical information about the shape of nanotubes and their assembly into a network. Although various versions of such mesoscopic models for CNT materials were proposed, e.g., Refs. 41, 42, and 60–67, the effect of the material reinforcement through covalent cross-linking was studied only in Refs. 68–70. In these works, the CNT material is either represented by a system of straight and dispersed nanotubes,⁶⁸ a bundle with initial hexagonal packing of nanotubes,^69,70 or a film with a continuous network of CNTs.⁷⁰ None of the known mesoscopic simulations addressed the effect of the nanotube length on the mechanical properties of cross-linked CNT materials.

The goal of this work is to study how the CNT length, arrangements of nanotubes into a network of bundles, and CL density affect the connectivity of nanotubes and mechanical properties of free-standing cross-linked thin CNT films. For this purpose, we perform a series of large-scale mesoscopic simulation of in-plane film tension and compression using the mesoscopic model developed in Refs. 60, 63, and 70 for the film samples with a surface size of 2.5× 2.5 μm² consisting of up to ∼26 000 individual nanotubes with the constant CNT length varying from 200 nm to 1000 nm, as well as for samples with random CNT lengths, at small to moderate degrees of cross-linking. Our major findings are qualitatively different from the results obtained in Refs. 41 and 42 for the buckypapers composed of pristine CNTs. We found, in particular, that the tensile modulus and strength strongly increase with increasing CNT length. The simulations also reveal a significant effect of the network morphology on the transfer of mechanical load, when the nanotube dispersion without the formation of thick bundles can induce an order-of-magnitude increase in the tensile modulus. The calculated values of the tensile modulus and strength follow the power-type scaling laws with respect to the average number of CLs per nanotube with strongly different exponents. The variation of the mechanical properties of CNT films observed in mesoscopic simulations is explained by the effects of the CNT length, CL density, and network morphology on the network connectivity.

For in silico generation of thin CNT films and subsequent simulations of their mechanical loading, we use a coarse-grained mesoscopic model, which was initially proposed in Ref. 60 and recently extended to study mechanical properties of CNT materials with covalent CLs between nanotubes.⁷⁰ In this model, each nanotube of radius $RT$ is represented by a chain of stretchable cylindrical mesoscopic elements or segments of the same radius, where each segment constitutes a part of a CNT consisting of multiple carbon atoms. The position and length of each mesoscopic segment i of nanotube k is determined by a pair of the position vectors $(r(k)i,r(k)i+1)$ placed at the centers of the cylinder bases. The position of whole nanotube k in a material sample composed of $NT$ CNTs is then defined by nodes with the position vectors $r(k)i$ (⁠$i=1,..,Nk$⁠; $Nk$ is the number of nodes in CNT $k$⁠). The motion of individual nodes is described by the equations of motion of classical mechanics. Multiple nanotubes represented by means of tubular mesoscopic elements are depicted in Fig. 1, where the individual snapshots illustrate the structure of the film samples obtained in mesoscopic simulations further described in Secs. III and IV.

In the present work, the mesoscopic force field accounts for elastic stretching and bending of nanotubes,⁶⁰ bending buckling,⁶⁴ non-bonded van der Waals interaction between nanotubes,⁶³ and covalent CLs between nanotubes.⁷⁰ The total potential energy U of a simulated system during mechanical loading is equal to

where $Ustr$ and $Ubnd$ are the total energies of stretching and bending of individual CNTs, respectively, $UvdW$ is the potential energy of van der Waals interaction between nanotubes, and $UCL$ is the potential energy of CLs. In Eq. (1), each term is a function of positions of all mesoscopic nodes $X=(X1,…,XNT)$⁠, where $Xk=(r(k)1,…,r(k)Nk)$⁠.

All components of the force field given by Eq. (1) are directly parameterized for CNTs of various chirality based on results of atomistic simulations and continuum calculations of the inter-tube interaction potentials. The force constants in the harmonic potentials of stretching and bending of individual nanotubes are calculated for CNTs of various diameters based on semi-empirical relationships that fit the results of atomistic simulations performed with the AIREBO interatomic potential.⁶⁰ The van der Waals interaction between nanotubes is described based on the tubular potential method⁷¹ and mesoscopic tubular potentials⁶³ by means of forces distributed at surfaces of mesoscopic cylindrical elements of nanotubes. The tubular potential method ensures the absence of corrugation of the inter-tube potential for curved nanotubes, eliminates artificial inter-tube friction, and is capable of describing the self-assembly of nanotubes into continuous networks of entangled bundles.^63,64,70,72 Recently, the mesoscopic model adopted in the present paper and corresponding force field were implemented in the form of the USER-MESONT package of the LAMMPS simulator.⁷³ 

The CL energy in Eq. (1) is defined assuming that each CL in an in silico generated sample represents an individual CL in the real material. Then,

where $NCL$ is the total CL number in the material sample, $φ(r)$ is the potential energy of an individual mesoscopic CL of length r, and $rm(X)$ is the CL length, i.e., the distance between the end points of an effective bond that connects two mesoscopic elements and represents CL m in the mesoscopic model. Here, $φ(r)$ and $rm(X)$ are used in the form suggested in Ref. 70. The potential energy $φ(r)$ for an individual CL, in particular, is adopted in the form of the Morse potential with a smooth cutoff function. The calculation of $rm(X)$ can be based on various geometric bond models (GBMs) that define the variation of the CL bond length depending on the positions of mesoscopic nanotube segments that are “linked” by a CL. In Ref. 70, two GBMs were suggested, namely, the center-to-center GBM (C-C GBM), when an individual CL is assumed to “connect” points placed at the centerlines of corresponding CNTs, and the surface-to-surface GBM (S-S GBM), when an individual CL links points placed at the surfaces of CNTs. The exact formulation of these models for the case of a CL between two arbitrarily oriented cylindrical mesoscopic elements, as well as equations for forces exerted on individual mesoscopic nodes, are provided in Ref. 70. It has been shown that, with proper parametrization, both C-C and S-S GBMs predict the same mechanical properties in mesoscopic pullout simulations of a central CNT from a seven-tube bundle and at tension and compression of CNT films.

In the present paper, we use the S-S GBM with the parameters of the Morse potential and CL length function presented in Table 2 of Ref. 70. These parameters are obtained by fitting the results of mesoscopic simulations of the pullout of a central (26,0) nanotube at a linear CL density of $n0=1.2nm−1$ from a seven-tube bundle to the results of atomistic simulations performed in Ref. 56 for the same problem. In the bundles of nanotubes considered in Ref. 56, the CLs between CNTs are formed by the irradiation of bundles with energetic carbon atoms. The irradiation produces random structures of CLs, so that the bundle samples, obtained under fixed irradiation conditions, differ from each other by the total CL number and types of individual CLs, although the CLs formed by single interstitial atoms with one sp³–sp² and two sp²–sp² bonds represent the most abundant type of defects.⁵⁵ 

The samples of cross-linked CNT films with random continuous networks of bundles of nanotubes are generated based on a multi-step approach suggested and used in Refs. 70, 72, and 74. This approach is illustrated in Fig. 1 and outlined in this section. The further details about the sample preparation technique can be found in Ref. 70. All simulations of self-assembly of nanotubes into a network of bundles and subsequent mechanical loading of generated film samples are performed at a fixed mesoscopic temperature that is controlled using the Berendsen thermostat.⁷⁵ The mesoscopic temperature is defined as a measure of the average kinetic energy of a single mesoscopic node per one translational degree of freedom and does not characterize the vibrational motion of individual carbon atoms, as this motion is not explicitly accounted for in the mesoscopic model.

We first generate a film sample that contains straight and dispersed nanotubes at a given material density with the periodic boundary conditions in the in-plane directions [Fig. 1(a)]. The CNTs are discretized into mesoscopic elements—cylindrical segments with an equilibrium length of 2 nm. Then, a dynamic simulation of the spontaneous self-assembly of CNTs into a quasi-equilibrium network composed of pristine nanotubes is performed for 1 ns at a mesoscopic temperature of 10 000 K. This temperature is chosen to increase the mobility of individual nanotubes and promote rapid formation of interconnected bundles that constitute a continuous network of nanotubes [Fig. 1(b)]. At this self-assembly stage, the volume of the film is constrained in the out-of-plane direction by two pistons that exert short-range repulsive forces on neighboring CNTs. The initial self-assembly is followed by an additional relaxation at a mesoscopic temperature of 300 K for 0.1 ns, when the overall bundle structure only marginally changes, but the arrangement of individual CNTs in thick bundles becomes close to the hexagonal one. After relaxation, the covalent CLs are homogeneously distributed inside the film sample at a chosen linear CL density $nCL$ (⁠$nCL$ is equal to the average number of CL ends per unit length of a CNT) using an acceptance-rejection approach [Fig. 1(c)]. For each CL, we generate a random point inside the sample, find a mesoscopic CNT segment closest to the generated point, and check whether it is possible to link a random point at the surface of that segment with another segment by a CL with the initial length equal to the equilibrium inter-tube gap between pristine nanotubes. If such a link is found, it is accepted as a CL, otherwise the point is rejected, and a new random point is generated. The cross-linked sample is relaxed during 0.1 ns at zero pressure and at a mesoscopic temperature of 300 K without constraints on the variation of the sample volume in the out-of-plane direction. During this relaxation, the bundle structure of the sample does not change, but some pre-straining of CLs is introduced, since the equilibrium length of the mesoscopic CL bond is different from the equilibrium gap between pristine CNTs.⁷⁰ The relaxation of the free-standing films changes the material density in less than 5%.

This approach is used to generate the CNT film samples composed of (26, 0) zigzag CNTs of radius $RT=1.0185nm$ and constant length $LT$ varying from 200 nm to 1000 nm at an initial material density $ρ$ of 0.1 g cm⁻³ with a surface size of 2.5× 2.5 μm² and film thicknesses of 40 nm. The chosen material density is within the typical range of density, from $∼0.05g/cm3$ to $∼0.6g/cm3$⁠, reported in multiple experimental studies of CNT films and buckypaper, e.g., Refs. 12, 16, and 47. The thickness of 40 nm is within the range of thickness from 10 nm to 100 nm for the ultrathin CNT films synthesized and studied, e.g., in Refs. 76 and 77.

The linear CL density $nCL$ varies from 0.05 nm⁻¹ to 2.5 nm⁻¹. The volume CL density, i.e., the average number of CLs per unit volume of the sample is equal to $nCLρ/(2ρT)$ (⁠$ρT=2πmRTnσ$ is the linear density of the nanotube material, m is the mass of a carbon atom, $nσ=4/(33lc2)=0.381Å−2$ is the surface density of atoms for a single-walled CNT, $lc=1.421Å$ is the lattice constant in graphene⁷⁸) and varies from $5.17×10−4nm−3$ to $2.59×10−2nm−3$⁠. All considered cross-linked samples are characterized by a small-to-moderate degree of cross-linking $fCL=nCL/(2πRTnσ)$⁠, which varies from $2.05×10−4$ to $1.03×10−2$⁠.

In addition, we also consider a film sample composed of (26,0) CNTs, where the individual nanotubes have random lengths $LT$ distributed according to the Weibull distribution with the probability density function (PDF),

The shape parameter $b=2.08$ and scale parameter $a=430nm$ are chosen to fit the experimental distribution of the nanotube length obtained in Ref. 37. We use Eq. (3) to generate random lengths of CNTs in the initial sample within the range of $100nm≤LT≤2000nm$⁠. For this cutoff distribution, the average nanotube length is equal to 620 nm. The CNTs shorter than 100 nm are excluded from consideration because such nanotubes cannot form a continuous network.⁶⁴ The nanotubes longer than 2000 nm are excluded since the chosen sample size in the direction of applied deformation (2.5 μm) must be larger than the length of an individual nanotube to avoid the effects related to the finite sample size and looping the nanotubes through periodic boundaries.

The major properties of the film samples considered in this work are listed in Table I. The samples with the continuous networks of nanotubes and constant CNT length are denoted as CN200–CN1000. The sample CNW with the continuous network of bundles and random CNT lengths is considered in Secs. IV D and IV F. This sample has only a few CNTs longer that 1500 nm. The total number of nanotubes in the samples varies from 5147 to 25 729. The integration time step $Δt$ during both self-assembly of nanotubes into a network and subsequent mechanical loading is equal to 10 fs for samples CN200–CNW. For samples DN200 and DN1000 with the dispersed network structure (the generation of these samples is described in Sec. IV E), a smaller value of the time step equal to 5 fs is utilized.

The structures of continuous networks generated in samples with a constant CNT length of 200 nm, 400 nm, and 1000 nm are shown in Fig. 2. The networks consist of interconnected bundles that include from a few nanotubes to a few tens of nanotubes. Visually, the obtained network structures resemble the structures of thin CNT films and buckypaper revealed in the experimental SEM and TEM images.^11–13,16 At the same time, films structures in Figs. 1 and 2 (see also Fig. 7) are significantly different from the samples of buckypaper in silico generated in Refs. 41 and 42. The ability to simulate the dynamic process of a spontaneous self-assembly of nanotubes into thick interconnected bundles is a distinct feature of the mesoscopic force field used in the present study, which completely eliminates artificial barriers for inter-tube sliding. The bead-and-spring coarse-grained model initially suggested for simulations of CNT material in Ref. 62 and then used in Refs. 41 and 42 introduces barriers for inter-tube sliding at a finite size of spherical beads representing mesoscopic elements of CNTs. It can preclude dynamic formation of thick bundles.

The relaxed structures of the in silico generated CNT films are somewhat different depending on the CNT length. The average bundle size is one of the major structural properties of the CNT network, as the bundle thickness affects the number of interconnects or junctions between bundles and determines the degree of the network connectivity. It was experimentally shown that the variation of the bundle size correlates with the variation of mechanical properties of films composed of single-walled CNTs.¹⁹ To characterize the bundles sizes in the in silico generated CNT films, we determine the number of nanotubes $NB$ in the bundle cross sections corresponding to each mesoscopic element, find PDFs of $NB$⁠, and then use these PDFs to calculate statistical properties of the bundle thickness distributions. The distributions of $NB$ in samples CN200 and CN1000 are shown in Fig. S1 of the supplementary material. The results of these calculations (Table I) indicate that, with an increasing CNT length, the thickness of nanotube bundles $NB$⁠, in average, slowly increases, from 9.48 at $LT=200nm$ to 11.8 at $LT=1000nm$⁠.

For given $NB$⁠, an equivalent diameter of a circular bundle can be calculated as $DB=(2RT+δh0)[6NB/(π3)]1/2$⁠.⁷⁹ The average $⟨D⟩$⁠, most probable $D0$⁠, and maximum $Dmax$ diameters of equivalent circular bundles are listed in Table I. All these bundle diameters only marginally increase with increasing length of individual nanotubes in samples CN200–CN1000. For instance, $⟨D⟩$ increases less than 10% when the CNTs become five times longer. Thus, the adopted approach for the generation of CNT films with nanotubes of different lengths allows us to obtain films with nearly the same bundle size distributions.

For our in silico generated samples, the typical bundle diameters are somewhat smaller than the experimentally measured values reported in the literature. For instance, the values of $D0=15nm$⁠, $⟨D⟩=13.5nm$⁠, and $Dmax=34nm$ can be deduced from the experimentally measured bundle size distribution in a low-density single-walled CNT network in aqueous solution.¹⁸ The average bundle diameters from 9.8 nm to 21 nm are experimentally measured for various single-walled CNT films in Refs. 47 and 19. The boundaries of the experimental range, however, are assumed to be somewhat overestimated in Ref. 19, as the experimental resolution did not allow to account for bundles with diameters less than $∼4nm$⁠. The average bundle diameter $⟨D⟩$ in our CNT films vary from 7.6 nm to 8.48 nm. A decreased average bundle size calculated for the in silico generated films can be explained by a few-fold larger density of the CNT films used in experiments of Refs. 19 and 47 compared to the density of films considered in the present work, as the average bundle size increases with increasing material density.⁷⁹ 

The in silico generated CNT films are subjected to the in-plane tensile or compressive deformation under conditions of a constant mesoscopic temperature equal to 300 K. To deform samples quasi-statically, we first tried to apply the approach developed in Ref. 70. In this approach, the periodic boundary conditions in the direction of applied deformation $Ox$ (the horizontal direction in Fig. 2) are released by “cutting” the nanotubes that cross the sample boundaries perpendicular to that direction. Then, we clip on the CNT parts inside two layers of 50 nm thickness along the sample boundaries perpendicular to the deformation direction and move these parts in opposite directions along $Ox$ with constant velocity $VD/2$ so that the total deformation velocity is equal to $VD$⁠.

The preliminary simulations showed that, for relatively large computational samples considered in the present work, this approach provides the results independent of the strain rate and sample size only if the deformation velocity is smaller than 1 ms⁻¹. This excessively small value of $VD$ precludes us from obtaining the tensile mechanical properties based on the existing deformation approach under conditions of quasi-static deformation. To overcome this difficulty, we use a modified approach for sample deformation, when, at each time step of the computational algorithm and before solving the equations of motion, the positions of all mesoscopic nodes in the deformable part of the film, as well as position of the clip layers, are scaled with the scaling coefficient $(VDΔt+LD0)/LD0$⁠, where $LD0=2400nm$ is the initial size of the deformable part of the sample between the clip layers. In the preliminary simulations, we found that this modified approach allows us to obtain the tensile mechanical properties independent of the strain rate $ε˙=VD/LD0$ and sample size if $ε˙≤0.025ns−1$⁠. We also found that the original and modified approaches result in the same initial slope of the stress–strain curves if $VD$ is smaller than 1 ms⁻¹. All stretching simulations described below in Secs. IV B–IV E are performed with the modified approach at $VD=25ms−1$⁠. All simulations of in-plane compression (Sec. IV F) are performed with the original approach at $VD=10ms−1$⁠.

To quantify the mechanical response of samples on the applied engineering strain $ε=VDt/LD$⁠, we calculate the engineering stress $S=(|F1|+|F2|)/(2Ax)$⁠, where $F1$ and $F2$ are the total forces exerted on the left and right clip layers from the deformable part of the sample along the deformation direction and $Ax$ is the area of a sample cross section before the onset of deformation. To distinguish between the tensile and compressive deformations, it is assumed, whenever necessary, that the tensile and compressive deformations correspond to positive and negative $ε$⁠, respectively.

The variation of the engineering stress S with engineering strain $ε$ in samples with various lengths of nanotubes and various CL densities is illustrated in Fig. 3. The noisy variation of stress in this figure is explained by the finite size of the film samples. The magnitude of the noise depends on both sample size and CL density, with the smallest magnitude observed in samples with the largest CL density [Fig. 3(d)]. The calculated stress–strain curves with gradual decrease in stress at strains that are larger than the strain $εmax$ at the maximum stress $Smax$ are typical for viscoelastic materials.

The values of the tensile modulus E, defining the slope of the stress–stain curves at zero strain, strength $Smax$⁠, which is equal to the maximum engineering stress observed in a simulation, and strain $εmax$ at $S=Smax$ are shown in Fig. 4. In order to define E, $Smax$⁠, and $εmax$ under conditions when the stress–strain curves are noisy, we fit these curves with polynomials as described in Ref. 70. The values of E and $Smax$ monotonically increase with increasing nanotube length and CL density. At a fixed CL density, E and $Smax$ increase roughly proportionally to the nanotube length. The rate of this increase is not constant but strongly changes depending on $nCL$⁠. For example, at $nCL=2.5nm−1$⁠, $Smax$ demonstrates fivefold increase when $LT$ increases from 200 nm to 1000 nm, while at a smaller $nCL$⁠, the slopes of curves in Fig. 4(b) are smaller. The strain $εmax$ is a more conservative parameter. The simulations reveal only moderate changes in $εmax$ around a value of 0.02 depending on the nanotube length and CL density when $nCL≤0.5nm−1$ [Fig. 4(c)]. Further increase in the CL density to 2.5 nm⁻¹, however, induces from two to three times increase in $εmax$⁠.

The predicted values of the modulus and strength of cross-linked CNT films agree on the order of magnitude with the values reported in the experimental studies on cross-linked CNT films and buckypaper. In Ref. 47, the measured tensile modulus, strength, and strain at failure of single-walled CNT films at a density of 0.6 g cm⁻³ with the electron irradiation-induced CLs vary from $∼1.5GPa$ to $∼3.5GPa$⁠, from $∼15MPa$ to $∼75MPa$⁠, and from $0.015$ to $∼0.037$⁠, respectively, depending on the irradiation dosage. The small values of $Smax$ deduced from the simulations for the films at a small CL density of 0.05 nm⁻¹ agree on the order of magnitude with the experimental measurements for single-walled CNT films. The strength at break of various CNT films was found to decrease exponentially with the increase in the material porosity determined as $P=1−ρ/ρCNT$ (here $ρCNT=1.5gcm−3$⁠), and is equal to $∼2MPa$ at a porosity P of 0.72.¹⁹ In our films, $P=0.93$⁠, and the value of $Smax$ for our pristine films, thus, is expected to be even smaller than 2 MPa.

The results presented in Figs. 4 and 5 indicate that the effect of the nanotube length on the major mechanical properties of cross-linked CNT films is altered by the CL density. The analysis of the computational results shows that the average number of CLs per nanotube $N=LTnCL$ is the major factor that determines the modulus and strength. The values of E and $Smax$ found in the mesoscopic simulations at various $LT$ and $nCL$ tend to fall on a universal curve for each parameter as a function of N (Fig. 5). These universal dependences can be approximated by the power-type scaling laws

The least-square fitting of Eqs. (4) and (5) to the data points shown in Figs. 5(a) and 5(b), respectively, results in $A=0.23GPa$⁠, $N∗=8$⁠, $α=0.49$⁠, $B=0.34MPa$⁠, and $β=0.86$⁠. Equations (4) and (5) with these best-fit parameters are shown in Fig. 5 by dashed curves. Thus, the scaling of E and $Smax$ is described by the power laws with quite different exponents, $α∼1/2$ for E and $β∼1$ for $Smax$⁠. The parameter $N∗$ in Eq. (4) can be considered as a such critical CL number per nanotube that the CNT films cannot sustain elastic deformation at $N<N∗$⁠. The prefactors A and B in Eqs. (4) and (5) are not universal constants but can depend on the structural parameters of the CNT networks, e.g., average bundle size. The data in Fig. 5 are moderately well approximated by Eqs. (4) and (5) with unique values of A and B, because all data points are obtained for CNT film samples CN200, CN400, and CNT1000 with similar structural parameters.

As shown in Sec. S2 of the supplementary material, the moderate accuracy of approximation of the calculated data points with Eqs. (4) and (5) can be explained by the fast and strongly non-linear variation of E and $Smax$ with the CL when density $nCL$ is less than $0.5nm−1$⁠. To reduce the discrepancies between the fitting equations for E and $Smax$ and calculated data points, one can use the scaling variables in the form of $N/nCLω$⁠, where $nCL$ is in nm⁻¹ and the exponent $ω$ is different for E and $Smax$⁠. Then, the approximate equations for E and $Smax$ can be used in the form

The least-square fitting of Eqs. (6) and (7) to the data point shown in Figs. 4(a) and 4(b), respectively, results in $C=−2.26GPa$⁠, $A=0.6GPanm−γ$⁠, $γ=0.25$⁠, $α=0.4$⁠, $D=3.17MPa$⁠, $B=0.2MPanm−δ$⁠, $δ=0.07$⁠, and $β=0.94$⁠. The values of E and $Smax$ calculated based on Eqs. (6) and (7), respectively, are shown in Fig. 4 by empty symbols. Although the best-fit values of the exponents $α$ and $β$ for the fitting equations in the form of Eqs. (4)–(5) and (6)–(7) are somewhat different, the ratio $β/α$ is large in both cases and varies between 1.76 and 2.35. It indicates a strong difference in scaling between the modulus and strength as functions of the CNT length and CL density.

The inelastic deformation of CNT networks is accompanied by two major types of irreversible structural changes, namely, breaking of CLs and bundling/de-bunding of nanotubes. The fracture of nanotubes at tension is accounted for in our mesoscopic model but never happens under conditions considered in the present work.

The fraction of broken CLs $fb$ for samples CN200–CN100 is shown in Fig. 6(a). Although the number of broken CLs is relatively small for the range of strain considered in this figure, it has a crucial effect on the load transfer through the network, as most of the broken CLs connect the nanotubes that serve as interconnects between thick bundles. The rate of CL breaking increases with increasing length of nanotubes. The fraction of broken CLs remains negligible until $ε=0.005$ and after that it starts to grow fast. A relatively early onset of CL breaking at tension pre-determines the low elastic limit $εe=0.005$⁠. At $ε>εe$⁠, the mechanical response of CNT films on stretching is not purely elastic. We confirmed it in additional simulations, where we performed relaxation of films at constant elongations above the elastic limit, i.e., at $ε>0.005$⁠, and found that the stress relaxes with time, as it is expected for viscoelastic materials.²⁹ At $ε>εmax$⁠, the rate of CL breaking decreases, and the fraction of broken CLs tends to saturate.

To characterize the degree of non-reversible changes in the film network structure at stretching due to bundling and de-bundling, we calculate the average bundle size and plot it as a function of strain in Fig. 6(b). The obtained results suggest that the net result of the structural changes at tension corresponds to de-bundling, which provides, however, a relatively small contribution to the overall non-reversible changes in the network structure. The effect of de-bundling is more pronounced for the film composed of 1000 nm long CNTs, while only a marginal de-bundling occurs in the film composed of 200 nm long CNTs.

To characterize the involvement of individual nanotubes into the transfer of mechanical load, we plot the stretched sample CN1000 in Fig. 7, where the individual nanotubes are colored according to their local strain $ϵ=Δl/l0$ (⁠$l0=2nm$ is the equilibrium length of a mesoscopic CNT segment and $Δl$ is its elongation). In the relaxed non-deformed samples, the absolute majority of nanotubes are only marginally stretched, when $ϵ$ is well below $0.005$⁠. It can be concluded, e.g., based on the analysis of the PDFs of $ϵ$ at various degrees of deformation provided in Sec. S3 of the supplementary material. In the deformed sample shown in Fig. 7, the red and blue nanotube elements have the absolute value of the local strain exceeding 0.005. The load transfer between two clip layers in the deformed sample occurs mostly through the nanotube parts that are stretched and compressed above the level of strain in the non-deformed sample. For a sample shown in Fig. 7, a large fraction of the film material is only marginally stretched and, thus, produces small contribution to the overall load transfer. The nanotube segments that provide significant contribution to the overall load transfer are connected through CLs with the clip layers and, thus, form a percolating network in the direction of applied deformation.⁷⁰ 

To describe the variation of the degree of the network connectivity at evolving deformation, we develop an approach which allows us to identify the part of the whole film that forms a load transfer network percolating the sample in the direction of applied deformation. Here, we only briefly describe the approach to define that load transfer network as it will be described in detail and analyzed elsewhere. To define the percolating load transfer network, we first create a list of all mesoscopic elements that are stretched or compressed above the threshold strain $ϵth$⁠, i.e., all mesoscopic elements with $|ϵ|>ϵth$⁠. Then, we retain only such elements in that list, which are “connected” through the closest neighbor elements of the same CNT or by CLs with other elements from the list to the CNT segments crossing the clip layer boundaries. Quantitatively, the number of mesoscopic segments and mass fraction $fP$ of the material included into such percolating networks essentially depend on the threshold strain $ϵth$⁠. The choice of $ϵth$ for comparison of connectivity in samples with strongly different network structures, e.g., films with continuous networks and films with dispersed CNTs (the latter are considered in Sec. IV F) is a non-trivial problem, as the different magnitudes of strain of individual nanotubes are expected in samples with strongly different structures. At the same time, for samples with roughly the same structure of bundles, like samples CN200–CN1000 in Table I, the particular value $ϵth$ used for the analysis of the percolating network does not affect the conclusions. Our calculations of the percolating networks in sample CN200–CN1000 at various values of $ϵth$ show that even a relatively small values of $ϵth$ can be used for the analysis of the connectivity variation if these values correspond to no percolating network in the non-deformed samples. The threshold value $ϵth=0.001$ satisfies this requirement and is used for further analysis. The percolating load transfer network in Fig. 8 is obtained with this $ϵth$ for the stretched film shown in Fig. 7. Although the sample contains a substantial fraction of segments stretched and compressed above this level of strain, many of these segments do not form a percolating network.

The fraction $fP$ of material included into the percolating network can be used as a measure of the evolving connectivity in the network of cross-linked nanotubes, which predetermines the resistance of the network to the applied deformation. The effect of the CNT length on the formation of the percolating load transfer network is illustrated in Fig. 9. An increase in $LT$ results in much broader percolating networks involving larger fraction of the whole network. The stress–strain curves correlate with the variation of $fP$ such that an increase in $fP$ always corresponds to the increasing stress in stretched films with increasing CNT length or CL density. For instance, when stretched samples CN200–CN1000 are compared at various deformation states, the films with larger tensile stress always have larger fraction $fP$ of materials included into the percolating network. The data shown in Figs. 4(a) and 10(a) exemplify such a relationship, since the elastic modulus E and $fP$ are increasing functions of the nanotube length.

The fraction of material involved in the percolating network $fP$ is shown in Fig. 10(b) as a function of strain $ε$ for the film samples with the nanotube length equal to 200 nm and 1000 nm. In both cases, the maximum of $fP$ occurs at $ε=εmax$ and, thus, correlates with the maximum of the tensile stress. An initial increase of $fP$ under conditions of CL breaking is explained by the changes in the load transfer network, which does not remain constant and evolves during the deformation process. The breaking of CLs deactivates existing paths of load transfer through a network and simultaneously activates new ones. The fraction of CLs that connect CNT parts included into the percolating network (not shown in Fig. 10) and, thus, participating in the load transfer, varies like the fraction of CNT materials included into the percolating network. The existence of the percolating load transfer network during the whole deformation process explains the gradual decrease in stress at $ε>εmax$ instead of distinct fracture behavior. It is characteristic for all stress–strain curves obtained in our simulations.

We compare the results of a stretching simulation performed for the sample with the random distribution of the nanotube length (sample CNW) with the results obtained for the sample with the constant CNT length (sample CN620), where $LT=620nm$ is equal to the average CNT length in the sample CNW. Compared to the sample CN620, the network in the non-deformed sample CNW includes significantly larger number of thin bundles with $NB∼5$ and smaller number of thicker bundles with $NB∼20$ [Fig. 11(a)]. The overall structure of the sample CNW is close to the structure of the sample CN200 composed of 200 nm long nanotubes with similar values of the average and most probable equivalent bundle diameters (Table I).

The simulations of cross-linked samples CNW and CN620 at a CL density of 0.5 nm⁻¹ predict practically the same tensile modulus E [Fig. 11(b)]. Moreover, the stress–strain curves for both samples are only marginally different beyond the elastic limit up to a strain of $∼0.008$⁠. For the film with random CNT lengths, the strength and strain $εmax$ are moderately larger compared to the film with the constant length of nanotubes. The increased values of $Smax$ and $εmax$ in the sample CNW are attributed to the effect of relatively small number of nanotubes, which are much longer than the average CNT length. The obtained results suggest, however, that the overall effects of the dispersion of the nanotube length on the structural and mechanical properties are only moderate.

To study the effect of the network structure on the mechanical properties of CNT films, we perform simulations with the samples, where the spontaneous self-assembly of nanotubes into bundles is impeded by CLs and the nanotubes remain disperse before the onset of deformation. For this purpose, we add CLs to an initial sample composed of straight and dispersed nanotubes. Then, the sample is relaxed in a thermostat at a mesoscopic temperature of 300 K during 0.2 ns with pistons constraining the film volume. Finally, the pistons are removed, and the free-standing film is relaxed additionally during 0.02 ns at the same temperature of 300 K and at zero pressure. At a CL density $nCL=0.05nm−1$⁠, this approach leads to films with a steady-state structure, where almost all nanotubes are straight and dispersed, and the average bundle size is less than two (Table I). The structure of such samples with the nanotube lengths of 200 nm and 1000 nm before the onset of deformation is presented in Sec. S4 of the supplementary material. During relaxation of the sample DN200, only 27 CLs out of 128 619 total CLs are broken. No CL breaking is observed during relaxation of the sample DN1000.

Contrary to films with continuous networks of bundles of nanotubes, the deformation of films with dispersed CNTs results in a strong reformation of the network structure, as it is illustrated in Fig. 12 for a strain of 0.032. On average, the rate of CL breaking is much higher in films with dispersed nanotubes [Fig. 13(a)]. The massive breaking of CLs during stretching releases constraints for nanotube self-assembly so that the tension of the films is accompanied with a rapid bundling of individual nanotubes [Fig. 13(b)]. The process of bundle formation results in the formation of large pores.

The films with dispersed nanotubes exhibit an order-of-magnitude increase in the initial slopes of the stress–strain curves, as well as from threefold to sixfold increase in the strength compared to films with the continuous networks of bundles of nanotubes at the same CNT length and CL density (Fig. 14 and Table II). We associate this increase in the mechanical properties with better network connectivity provided by CLs in the samples of dispersed nanotubes. The difference in the tensile modulus and strength between samples with continuous networks and dispersed nanotubes at the same CNT length and CL density indicates that at least the prefactors A and B in the scaling laws in the form of Eqs. (4)–(7) are not the universal constants but depend on the structural parameters of the CNT films. The strain $εmax$ only weakly depends on the sample structure. At the same time, even initial phase of deformation of samples with dispersed nanotubes is inelastic, since the process of CL breaking starts immediately after the onset of deformation [Fig. 13(a)] and is accompanied with fast increase in the average bundle size [Fig. 13(b)]. The bundle formation that induces the nucleation of large pores creates a “weak cross section” in the sample and results in a fast drop of the stress with increasing strain at $ε>εmax$ so that at $ε∼0.04$ the stress in films of dispersed nanotubes reduces to the stress in the corresponding films with continuous networks of bundles of nanotubes.

In this section, we describe the selected results obtained for in-plane compression of CNT films. For all parameters considered in the simulations, we observed relatively small material resistance to compression, when the maximum stress is at least a few times smaller than the strength of the same film at tension as was already shown in Ref. 70. At a fixed CL density, films with various CNT lengths demonstrate similar response to the applied compressive deformation in terms of the magnitude of stress, but the deformation can proceed through the formation of different surface landscapes. The discussion of computational results is focused in this section, therefore, on the effect of the CNT length and network structure on the qualitative picture of compressive deformation and associated irreversible changes in the network structure.

At compression, the CLs can prohibit relative sliding of nanotubes without inducing significant strains inside CNTs, and, thus, the presence of CLs can qualitatively affect the shape of the compressed film. Under all conditions considered in simulations, the films deform at compression through the collective bending of nearby bundles which leads to collective folding of the whole films as shown in Figs. 15 and 16. The simulations show that even small amount of CLs can induce collective bending of bundles and folding of the films, as the degree of cross-linking $fCL$ for the samples considered in Figs. 15 and 16 is equal to $2.05×10−4$⁠. These results suggest that the collective folding of the CNT films composed of sufficiently long nanotubes at compression can be observed also without CLs, when a small fraction of nanotubes with defective atomic structures or bending buckling kinks,⁶⁴ as well as impurities, e.g., residual metal nanoparticle, can serve as obstacles for the inter-tube sliding. Our preliminary simulations, however, show that the mode of deformation can change qualitatively depending on the material density, bending rigidity of nanotubes, their length, and film thickness, and the complete picture of the compressive deformation of thin CNT films is complicated.

The folding explains the reduced resistance of thin films to compression compared to tension. At the smallest CL density, $nCL=0.05nm−1$⁠, the film wrinkles form a complex two-dimensional landscape in a case of a film composed of short, 200 nm long nanotubes [Figs. 15(a) and 16(a)], when positions of individual bumps and depressions are not correlated along the direction perpendicular to the direction of applied deformation (⁠$y$ direction in Fig. 15). For films composed of longer nanotubes, the compression results in the formation of a nearly perfect one-dimensional wavy pattern of bumps and depressions [Figs. 15(b) and Figs. 16(b)–16(c)]. The maximum height of the landscape h (the difference between the height of top film surface at the tallest bump and deepest depression) is practically independent of the CNT length. For all films shown in Fig. 16, h varies in the range of 210–230 nm. The maximum height of the landscape h increases roughly proportionally to strain and reaches 390 nm at $ε=0.125$ for the sample CN200. A similar increase in the relief height without breaking the network structure is observed for other samples composed of longer nanotubes.

The compression of the film with random distribution of CNT lengths also results in the correlated bending of the whole film [Fig. 16(d)]. It indicates that the response of the CNT films to compression depends on the length of nanotubes that form the scaffold of large bundles, while the presence of large number of relatively short nanotubes does not affect the regime of compressive deformation. In the sample CNW shown in Fig. 16(d), the scaffold of the network is formed by relatively long nanotubes (the average CNT length is equal to 620 nm) so that the pattern of the compressed films is similar to the patterns of films composed of constant-length nanotubes with $LT≥400nm$⁠.

The rates of CL breaking and bundling strongly decrease after a short initial stage of compression, when the absolute strain |$ε$| becomes larger than 0.02 as shown in Fig. 17. Contrary to tension, the initial rate of the CL breaking at compression does not depend on the CNT length and compression induces bundling of nanotubes. At $LT≥400nm$⁠, the bundling process, however, stops when the absolute strain exceeds 0.02. At larger strains, the compression of films composed of relatively long nanotubes is a practically reversible process, since it is accompanied with small rate of CL breaking and almost zero rate of bundling of nanotubes. At an absolute strain of 0.1, the number of broken CLs in the folded films is in the order of magnitude smaller than in stretched films.

The compression simulations of the films with a network of dispersed nanotubes always result in the one-dimensional wavy pattern for all CNT lengths considered. The surface landscapes of the folded films composed of dispersed nanotubes are quantitively similar to the films with the network of bundles composed of long nanotubes and shown in Figs. 15(b) and 16(c). These results indicate that the degree of connectivity of the network is the key factor that affects the regime of compressive deformation of thin cross-linked CNT films.

The picture of compressive deformation of the free-standing films obtained in mesoscopic simulations resembles the experimentally observed wrinkling or folding of thin CNT films on a polymer substrate.⁷⁶ The experiments reveal hierarchical folding of films, when the surface landscape is a superposition of multiple harmonics with progressively decreasing length scales. At $ε=−0.05$⁠, the maximum experimental relief height is equal to $∼200nm$⁠, and the wavelength of the dominant wrinkling mode is larger than $∼2μm$⁠.⁸⁰ Our simulations predict the single-mode deformation with the practically same maximum relief height equal to 210–230 nm and twice smaller wavelength equal to $∼1μm$ at $ε=−0.055$⁠. The simulations results, however, are not expected to ideally coincide with the experimental observation and measurements of Ref. 76 as the simulations are performed for free-standing films with somewhat different physical properties. Since the wavelength of the wrinkling mode in our simulation is about a half of the sample size, we believe that the simulations with samples of larger size are necessary to capture the wrinkling modes with the largest wavelength.

The mechanical properties of free-standing thin films composed of single-walled carbon nanotubes with covalent cross-links are studied in large-scale mesoscopic simulations. Two types of the films with different morphology of the underlying network of nanotubes are considered, namely, the films with continuous networks of bundles of nanotubes, which are obtained in preliminary dynamic simulations of spontaneous self-assembly of nanotubes driven by the van der Waals attraction, and the films consisting of intentionally dispersed nanotubes, where the cross-links preclude formation of bundles.

The simulations reveal the strong effects of the nanotube length, morphology of the network of nanotubes, and cross-link density on the load transfer and mechanical properties. In particular, the tensile modulus and strength increases with increasing CNT length and cross-link density. It is shown that the average number of cross-links per nanotube is the primary parameter that controls the elastic and inelastic properties of CNT films at tension. The values of the tensile modulus and strength obtained in the mesoscopic simulations tend to follow the power-type scaling laws depending on the average number of cross-links per CNT. The exponents in the scaling laws for the modulus and strength are strongly different form each other. The dispersion of nanotubes in the films without formation of thick bundles can increase the modulus and strength from a few times to an order of magnitude. The structural and mechanical properties of a CNT film with a random distribution of the nanotube length are found to be only moderately different from the corresponding properties for a film, where all CNT have a constant length equal to the mean value of the random nanotube lengths. The tension of cross-linked CNT films is accompanied by strong irreversible structural changes associated with breaking of cross-links and, to a small extent, with de-bundling. As a result, the elastic limit of the in silico generated films is found to be small and corresponds to a strain of ∼0.005. These findings are explained by the effect of CNT length, cross-link density, and dispersion of nanotubes on the degree of connectivity of the load transfer network that appears in cross-linked CNT films in response to applied deformation.

During compression, the simulations reveal the collective bending of bundles, which leads to folding of the whole films. The surface landscape of the compressed films depends on the CNT length. For films composed of short nanotubes and at moderate compressive strains, the landscape represents a complex two-dimensional surface pattern of bumps and depressions. The films composed of long nanotubes are folded into a one-dimensional structure, where bumps and depressions span the whole sample in the direction perpendicular to the deformation direction. The compression of the films induces breaking of relatively small fraction of cross-links. The process of bundling of nanotubes occurs only during an initial stage of the deformation when the absolute strain does not exceed 0.02. At larger strains, the folding of the films composed of relatively long nanotubes is found to be a practically reversible process.

See the supplementary material for the auxiliary plots supporting conclusions made in the present study and analysis of the scaling of the tensile modulus and strength as functions of the CNT length and CL density.

This work was supported by the National Aeronautics and Space Administration (NASA) through an Early Stage Innovations grant from NASA's Space Technology Research Grants Program (No. NNX16AD99G) and by the National Science Foundation (NSF) through Award No. CMMI-1554589. The computational support was provided by NASA's Advanced Supercomputing Division and Alabama Supercomputer Center.

The data that support the findings of this study are available within the article and its supplementary material. The raw results of computations are available from the corresponding author upon reasonable request.