Hybrid Carbon Nanoparticles in Polymer Matrix for Efficient Connected

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Hybrid Carbon Nanoparticles in Polymer Matrix for Efficient Connected Networks: Self-Assembly and Continuous Pathways Ali Gooneie* and Rudolf Hufenus Laboratory of Advanced Fibers, Empa, Swiss Federal Laboratories for Materials Science and Technology, Lerchenfeldstrasse 5, CH-9014 St. Gallen, Switzerland S Supporting Information *

ABSTRACT: Functional polymer composites based on hybrid carbon nanoparticles (CNPs) offer synergistic properties and have recently received a lot of attention for various applications including photovoltaic cells. In this context, the size dispersity inherent in CNPs such as carbon nanotubes (CNTs) is still a controversial topic in light of new experimental findings when it comes to the formation of percolating networks. Here, we show how nanotube models with different aspect ratios (ARs) dispersed in polyamide 12 (PA12) matrix differ in their equilibrium nanostructures. To this end, large-scale dissipative particle dynamics simulations are carried out, and CNPs with different ARs representing fullerene-like isomers up to realistic CNTs are studied separately or in hybrids. The continuous pathways in the CNP nanostructures are further assessed in Monte Carlo calculations by random electrons transporting through the network quantifying its continuity for electrical conductivity. The results confirm that the morphology of the composite depends on the AR and, by increasing it, changes from a random dispersion to a selfassembled morphology and eventually to a bridging self-assembled network. The generic behavior predicted in the simulations is compared with the rheological and electrical conductivity measurements performed on PA12/CNT nanocomposites. Based on the results, the dispersion quality, the AR of CNPs, and the continuous pathways in the network are found to be interconnected in contrast to previous interpretations of hybrid nanocomposites.

1. INTRODUCTION Dispersions of carbon nanoparticles (CNPs), such as carbon nanotubes (CNTs) and fullerenes, in a solvent or a polymer matrix, can lead to the fabrication of synergistic functional materials with superior performance. Although the nanoparticles are expected to enhance some properties such as electrical conductivity, in many applications for such hybrid materials, use of low filler content is strongly desired in order to preserve e.g. optical and rheological properties of the host medium. Therefore, numerous works attempted to discover the correlation between nanoparticle concentration and the desired properties. Among the many instances in the literature, there are only few studies that have considered the role of the length and diameter dispersity of CNPs in their corresponding dispersions, albeit without directly discussing the patterns and mechanisms at the molecular level. The inherent size polydispersity of CNTs makes them a prime example of hybrid multifiller systems in the sense of their aspect ratios (ARs).1 Recent experiments have revealed the potential of assemblies of CNTs and fullerenes as hybrid nanocarbon systems for photovoltaic cells.2−4 These studies have shown how mixing of fullerenes and CNTs could lead to a more stable and finer dispersion of nanoparticles in the matrix. This has been shown to yield better electrical properties, where the donor electrons can transport much easier through the bulk within the network © XXXX American Chemical Society

of the acceptor nanoparticles. Other hybrid CNP systems, recently raising interest, combine CNTs with graphene nanoribbons, resulting in synergistic electrical and/or dielectric properties, e.g., applicable in controlled conductivity and/or charge storage applications.5,6 Such reports have provoked the current study where we consider similar hybrid systems as an opportunity for efficient materials. For CNT networks in particular, it is known that the high AR of the nanoparticles results in a reduced minimum interparticle distance necessary for the electrons to tunnel so that a tunneling connectivity is established throughout the entire bulk of the system.7−10 Although deviations from an ideally homogeneous dispersion of CNTs are common in polymer nanocomposites,11 its correlation with the length polydispersity of CNTs was only recently addressed.12 The CNT polydispersity has been studied before by applying the connectedness percolation theory to the dispersions of slender straight polydisperse rodlike particles. The predictions of this theory suggest that the critical filler loading to achieve the electrical percolation threshold is inversely proportional to the weighted average of rod lengths.13−15 Numerical simulations confirm this Received: March 19, 2018 Revised: April 17, 2018

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DOI: 10.1021/acs.macromol.8b00585 Macromolecules XXXX, XXX, XXX−XXX

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using dissipative particle dynamics (DPD).21 The coarsegraining degree in DPD can be adaptively adjusted so that an appropriate upper-scale nanoscopic representation of the underlying atomistic structures is developed. These coarser representations, in which several atoms are lumped into the socalled beads, pose a lower degree of freedom on the calculations.22,23 Moreover, the form of the DPD potentials between the beads is softer than traditional atomistic potentials, therefore allowing for larger time steps.22,24 These characteristics make DPD a promising method when it comes to simulate the nanoscopic equilibrium and nonequilibrium morphologies in complex soft materials. As a prime example, polymeric systems have been widely investigated using DPD.25−28 The time evolution of beads in DPD is controlled by the conservative force FCij , the dissipative force FDij , and the random force FRij , which are central pairwise forces. For any pair of i and j beads within the force cutoff radius rc, these forces are

observation and further extend it to homogeneous dispersions of straight rods with intermediate ARs.16,17 In a recent study, Meyer et al.16 utilized Monte Carlo simulations in combination with the connectedness percolation theory to study the connectivity percolation in suspensions of hard spherocylinders. They reported a cluster growth mechanism that differs between polydisperse systems with nonuniversal aspects. Analytical works by van der Schoot and co-workers13,18,19 investigated the impact of polydisperse fillers using both percolation and liquid state theories, reporting a very strong sensitivity of the percolation threshold on the degree of filler size dispersity. These studies so far have all predicted a quasi-universal dependence of the bulk conductivity on the weighted average length of nanorods/nanotubes, implying that a proper knowledge of the polydispersity is necessary to control the conductivity. However, the underlying mechanisms at the molecular level are widely overlooked in these hybrid multifiller systems. Here, we systematically test the clustering in mono-, bi-, and polydisperse systems to find their universal trends (if there is any) in self-assembly and network formation. Recently, we highlighted the role of length distribution of nanorods and nanotubes in the final properties of their developed polymer nanocomposites.20 The dispersion quality, the percolated networks, and the mechanical properties were discussed in particular. Furthermore, the electrical properties of the nanocomposites were linked to the formation of connected networks of such nanoparticles. In a recent experimental study, Majidian et al.12 reported their observations of the impact of length polydispersity of CNTs on their electrical properties in epoxy matrix. Their findings show how important it is to consider such effects when fabricating the nanocomposites. They further discussed the importance of length polydispersity of CNTs in correlation with their degree of clustering in the hosting matrix. According to their results, in a homogeneous dispersion the role of polydispersity is of critical significance and can enhance the electrical conductivity. On the other hand, clustering of CNTs in dispersions critically marginalizes the role of polydispersity. This correlation between polydispersity and dispersion quality, and their impact on the electrical response of the nanocomposite, demand detailed analyses in terms of fundamental nanoscopic phenomena. Unfortunately, a respective systematic study is generally hindered by experimental limitations in sample preparation and testing. In order to fill this gap and expand our knowledge of the role of each AR mode in a mixture, we perform extensive nanoscopic simulations of CNP dispersions with predefined monodisperse, bimodal, and multimode length distributions. A systematic variation in the CNPs length allowed for investigating a range of CNPs from fullerene-like nanoparticles up to realistic CNTs separately and in bimodal/hybrid mixtures. Monte Carlo (MC) calculations are employed to analyze the nanoparticles structure and continuous pathways in the polymer. At intermediate stages, we compare our simulations with rheological and electrical conductivity measurements of laboratory samples to highlight the similarities and contrast the differences. Such investigations on how the length dispersity contributes to a fine dispersion and network formation are extremely crucial, for instance, for the development of photovoltaic cells.

FijC = aijωC(rij)riĵ

(1)

FijD = −ξijωD(rij)rij[(vi − vj) ·riĵ ]riĵ

(2)

FijR = σijωR (rij)rijζij riĵ

(3)

In these equations, rij is the distance between the beads with the corresponding unit vector of r̂ij which points from bead j to bead i. vi and vj are the velocity vectors of beads i and j, respectively. The model parameters used in DPD are the maximum repulsion coefficient aij, the friction coefficient ξij, the noise amplitude σij, and a Gaussian random number ξij. ωC(rij), ωD(rij), and ωR(rij) are the conservative, dissipative, and random weight functions, respectively, which are defined as 2 ⎛ rij ⎞ ωD(rij) = [ωC(rij)]2 = [ωR (rij)]2 = ⎜1 − ⎟ rc ⎠ ⎝

(4)

A more detailed description of DPD and its parameters can be found in recent reviews on multiscale methods in polymers simulation.22,29 2.1. Polymer and CNP Models. To represent a flexible polymer chain, 25 DPD beads were connected using finitely extensible nonlinear elastic (FENE) springs with a bond potential of ⎛ ⎛ rij ⎞2 ⎞ UFENE(rij) = −0.5ksr0 2 ln⎜⎜1 − ⎜ ⎟ ⎟⎟ ⎝ r0 ⎠ ⎠ ⎝

(5)

in which ks is the FENE spring constant and was set to 50kBT/ rc2. r0 is the maximum extent of the bond and was set to 0.7rc in this work. These values ensure that unphysical bond crossings do not occur during the simulations.23,30 To preserve the flexibility of the chains, consecutive bonds were allowed to have any angle. CNPs were designed following Liba et al.,32 who developed a coarse-grained model of CNTs with an armchair (6,6) structure by lumping carbon atoms into DPD beads. In their model, these beads interact with each other through pair and triplet forces. This approach to simulate CNTs in DPD models has previously shown to be successful.28,33 As depicted in Figure 1, each CNT tube is composed of a series of triple-bead rings connected by harmonic bonds with the potential of

2. COMPUTATIONAL METHODOLOGY In order to investigate the evolution of CNP nanostructures in polymer, large-scale coarse-grained simulations were carried out B


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same as the typical temperature of melt processing of these nanocomposites. We also carry out our rheological experiments at this temperature for the sake of comparison. As a consequence of this set of units, the time unit τ is 18.6 × 10−11 s. In the simulations, the maximum repulsion coefficients between different pairs were carefully adjusted to represent a realistic dispersion of CNPs in the polymer matrix. The aii value was set to 25kBT/rc for all interactions between beads of the same type. The interactions between different types of beads were defined based on Groot and Warren’s approach.35 Accordingly, the excess repulsion parameter was related to the Flory−Huggins χij parameter by35 χij aij ≈ + aii (8) 0.306

Figure 1. Schematic representation of the coarse-grained CNP structure constructed by six triplet rings. The beads are illustrated in purple with their connecting bonds shown in green. The triplet rings are depicted in their surfaces shaded in gray for better visualization.

Uharmonic(rij) = k h(rij − r0)2

The solubility parameters δi and δj of the interacting beads were then incorporated to determine χij using

χij =

(6)

kBT

(9)

with Vref as a reference average bead volume taken equal to the average bead volume ∼2.58 × 104 Å3. The solubility parameter of PA12 was determined by the group contribution method to be 18.38 (J/cm3)1/2, which is in good agreement with the value derived from molecular dynamics simulations.36,37 For the CNPs, we used an average value of 19 (J/cm3)1/2 based on previous investigations that have been carried out on various CNT types and ARs.38,39 Consequently, the maximum repulsion coefficients between different pairs are calculated and given in Table 1. In this work, ξij was set to 4.5(mkBT/ rc2)1/2 between all bead types. Finally, the fluctuation− dissipation theory was used to define the noise amplitude as σij2 = 2ξijkBT.40 In all systems, the force cutoff radius was set to 1rc. The soft DPD potentials in comparison with classic molecular dynamics significantly disfavor kinetically trapped structures (though not in every case41). Here, we repeated some simulations randomly using different initial spatial configurations and found similar equilibrium states. On the basis of these attempts and previous equilibrium DPD simulations of similar dispersions,20,27,28,42,43 one can safely presume that the structures reported here are in fact at equilibrium.

where kh is the harmonic spring constant and was set to 2675.8kBT/rc2 after Liba et al.32 r0 is again the maximum extent of the bonds and was 0.6239rc for CNPs. Angular forces were also defined which allowed to correctly capture the stiffness of the CNTs against experimental results.32 The potential function of these forces is given by Uangle(θ ) = ka(cos θ − cos θ0)2

Vref (δi − δj)2

(7)

In this equation, ka is the angular potential constant and was 38.9kBT. Also, θ, and θ0 are the angle of triplets of neighboring beads and the equilibrium angle, respectively. The latter was set to 180°. In some simulations performed here, the flexibility was artificially increased by reducing ka to 19.45kBT so that the effects of this parameter can be elucidated. Three different CNPs with different ARs were developed in order to investigate their impact on the formed nanostructures: (i) low AR CNPs with an AR of 3.5, (ii) high AR CNPs with an AR of 33.8, and (iii) extra high AR CNPs with an AR of 107.4. To model the low AR CNPs, 15 beads were included in each tube constituting five triplet rings (L5 CNPs). For the high AR CNPs, 40 triplet rings were connected to each other containing a total of 120 beads (L40 CNPs). Finally, the extra high AR CNPs were modeled by 375 beads dispersed in 125 triplet rings along their length (L125 CNPs). The L5 CNPs can be considered as a fullerene isomer (for example, C360-D5h)34 considering their very low AR and having 360 carbon atoms in their full atomistic representation according to the coarsegrained CNT model of Liba et al.32 On the other hand, the L125 CNPs possess an AR close to that of some realistic CNTs. Therefore, valuable information can be provided on the nanoscale structures of a range of carbon nanoparticles and their hybrid systems by studying these CNPs. 2.2. DPD Units and Model Parameters. As the polymer matrix in the simulations, polyamide 12 (PA12) was selected with an average molecular weight of ∼400 kg/mol. Each chain was coarse-grained into 25 beads with an average volume of ∼2.58 × 104 Å3. This choice of coarse-graining degree yields a unit mass m of 1.35 × 10−23 kg for each bead and a length unit rc of 4.3 × 10−9 m. The energy unit is E = kBT at the temperature T = 513 K. The temperature is set so as to be the

Table 1. Maximum Repulsion Coefficients Aij [kBT/rc] for Different Pairs pair type

PA12

CNP

PA12 CNP

25 29.6

29.6 25

2.3. Simulation Method. All simulations were carried out in orthogonal boxes. The systems containing the L125 CNPs had the dimensions of 70rc × 70rc × 70rc representing a cube of real spatial dimensions of ∼300 nm on each side. Other systems without the extra high AR CNPs were simulated in a box with the dimensions of 50rc × 50rc × 50rc which represents a cube of real spatial dimensions of ∼213 nm on each side. In this way, we attempt to minimize the finite size effects in our simulations. Periodic boundary conditions were applied on all sides. A total of 375 × 103 beads of different types were dispersed in the smaller box producing a bead number density of 3, which is the C


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Figure 2. Equilibrium morphologies of various dispersions. L5 and L40 CNPs are shown in yellow and green, respectively.

typical value in most DPD studies after Groot and Warren’s.35 The same bead number density was achieved in the larger box by dispersing over 1 million beads in it. Such extensive largescale particle-based simulations on polymer/CNP systems can provide valuable insights into these materials. The systems contained various CNP concentrations of 0, 0.3, 0.5, 1, and 5 wt %. These amounts of nanoparticles were translated back to the number fraction of the beads out of the total number of beads based on a typical volume fraction approach frequently used in previous works. Subsequently, monodisperse systems containing nanoparticles with only one AR and bimodal dispersion containing a 50/50 vol/vol mixture of two ARs were constructed. For the bimodal dispersions, mixtures of L5 and L40 (named L(5 + 40)) and mixtures of L40 and L125 (named L(40 + 125)) were considered. These systems allow us to determine the behavior of each representative AR of CNPs in their monodisperse medium and later in combination with CNPs of higher ARs. Therefore, a multifiller hybrid system containing different ARs of CNPs can be pictured by the overlapping of their mechanisms in their individual and/or in combinatory forms. To further comment on such multimode dispersions, mixtures of L5, L40, and L125 CNPs (named L(5 + 40 + 125)) with equal proportions of each nanoparticle type

(one-third of the total loading) with different total CNP contents were simulated and analyzed. It should be noted here that the systems with higher flexibility are named similar to others except that their initials are “Lf” as opposed to “L”. For instance, the bimodal mixture of L5 and L40 CNPs with higher flexibility is named Lf(5 + 40). The simulations were performed in NVT ensemble which allows for a constant bead number density during the simulations and ensures the thermodynamic consistency of the coarse-grained model. To construct the systems, first a certain number of CNPs were randomly distributed in the box to correspond to the predefined contents. Afterward, PA12 chains were added to the system at random positions in order to satisfy the bead number density criterion. The resulting configuration was initially allowed to equilibrate for 2 × 106 steps. In this initial equilibration step, the aij values for all pairs were deliberately set to 25, except for the CNP−CNP interactions which was set to the large value of 500. In this way, equilibrated well-dispersed initial configurations of CNPs in polymer were prepared for the following equilibration runs with realistic repulsive forces (the data collection step). The data collection was then performed for 5 × 106 steps. The time step was set to 0.005τ in all simulations. Therefore, the initial D


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Figure 3. 2D structure factors of a representative self-assembly of L40 CNPs on a plane perpendicular to the bundle at different nanoparticle concentrations.

structures dominate at all concentrations with self-assemblies becoming larger with an increase in the nanoparticle content. To manifest the interparticle correlations in such selfassemblies, 2-dimensional structure factors were calculated on a normal plane to a representative self-assembly in the dispersion of high AR nanoparticles. The structure factors were calculated using the relation 1 N N S(k) = N ⟨[∑i = 1 cos(k·ri)]2 − [∑i = 1 sin(k·ri)]2 ⟩ in which N is the total number of beads in the bundle, with position vectors r. As shown in Figure 3, the honeycomb (hexagonal) structures gradually develop as more nanoparticles are added to the dispersions. Similar patterns are also reported for nanorods/ nanotubes with different ARs in an epoxy resin as well as at the interface of immiscible polymers.20,47 Considering that the thermodynamic ensemble and the force field are similar in all simulations, one can infer that the self-assembly process is governed mainly by the inherent anisotropy of the high AR nanoparticles in the absence of any explicit attractive forces in the DPD model.48 The random dispersion of low AR nanoparticles, on the other hand, is a direct consequence of their translational entropy. The self-assembly of high AR CNPs also depends on the interactions between polymer beads and nanoparticles. It has been shown before that by changing the maximum repulsion coefficient, systems with different sets of interactions can be simulated.20 A favorable dispersing interaction between the polymer and CNPs hinders selfassembly. This effect was also investigated in the simulations of the L40 CNPs in the 5% systems by altering aij of the polymer−CNP pairs to generic values of 25 and 10 [kBT/rc] (see Figure S1 of the Supporting Information). The former value represents a system with neutral interactions and the latter a system with favorable dispersing interactions. It is evident from the equilibrium snapshots that the self-assembled structures are still developed in the case of neutral interactions, whereas they do not appear in the system with favorable dispersing interactions. Finally, the self-assembly of CNPs from a well-dispersed initial state is also a function of the matrix viscosity. An increase in the polymer viscosity is generally expected to make it more difficult for the CNPs to translate in the matrix and thus hinders the formation of bundles. A further systematic study would be required to investigate the effect of matrix viscosity on the self-assembly. A continuous network of nanoparticles can be achieved provided that a certain degree of dispersion is reached in which the nanoparticles are within a certain distance from each other. These criteria are satisfied for the low AR nanoparticles only if their amount is sufficient, so that they locate closely to each other. On a separate note, knowledge of the persistence of such a network under mechanical deformation is indeed valuable, which will be addressed in a following paper. The high AR nanoparticles are generally considered to be more critical in network expansion.17 However, it is only the case where a good

equilibration and data collection runs correspond to the real times of 1.86 and 4.65 μs, respectively. The simulations were performed using LAMMPS (large-scale atomic/molecular massively parallel simulator)44 code package.

3. EXPERIMENTAL DETAILS 3.1. Materials and Sample Preparation. Polyamide 12 (PA12) was Grilamid L16 with a density of 1.01 g/cm3 obtained from EMSChemie, Switzerland. Carbon nanotubes (CNTs) were NC7000 supplied from Nanocyl S.A., Belgium. They were multiwalled and had an average AR of ∼158 according to the supplier. This average AR makes it possible to closely represent them by L125 CNPs in our simulations. PA12/CNT nanocomposites were melt-mixed in a twinscrew extruder with a temperature profile ranging from 230 to 245 °C. All samples were dried at 100 °C overnight in a vacuum oven prior to mixing. The nanocomposites contained different amounts of CNT, i.e., 0, 0.015, 0.15, 0.3, 0.75, and 5 wt %. The resulting nanocomposites were subsequently pressed into films with various thicknesses of 0.2 and 1 mm using a hot stage and then cooled down rapidly in cold press. 3.2. Rheological Experiments. The rheological measurements were performed using the rheometric mechanical spectrometer Anton Paar Physica MCR 301, Austria. All measurements were done in a nitrogen environment in a parallel plate fixture with a plate diameter of 25 mm and a constant gap of 1 mm. All samples were dried at 80 °C overnight in a vacuum oven prior to the measurements. Before starting a measurement, the samples were allowed to relax for 30 s in the rheometer at 240 °C to minimize the deformation history due to the sample loading. Strain amplitude sweeps were performed on all samples at a constant angular frequency of 10 rad/s at 240 °C to reveal the viscous/elastic melt behavior as well as the limit of the linear viscoelastic region. Afterward, angular frequency sweep experiments were carried out in the linear viscoelastic region at a constant strain of 0.5% and a temperature of 240 °C. 3.3. Electrical Conductivity Measurements. Electrical conductivities of the nanocomposites were measured at room temperature with a two-probe method. The accuracy of the two-probe method for the measurements is justified by considering the large differences in the electrical conductivities of the polymer nanocomposites and the metal probes.45,46 The measurements were carried out on two devices with different ranges of detection sensitivities. The nanocomposite with 0.15 wt % CNT content fell well within both these ranges and showed a reproducible electrical response in both setups, thus confirming the accuracy and consistency of this approach. The electrical resistivities measured by these setups are shown in the corresponding plot (Figure 10b) with different markers for clarity. The measurements were repeated on several samples from each nanocomposite.

4. RESULTS AND DISCUSSION 4.1. Equilibrium Morphology: Self-Assembly and Connected Networks. The equilibrium morphology of CNPs with different length distributions were simulated and are shown in Figure 2. The low AR CNPs disperse randomly in polymer and do not self-assemble into bundles while the high AR CNPs form honeycomb-like self-assemblies. These microE


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± ± ± ± 0.0001 0.0002 0.0003 0.0003 0.0458 0.0743 0.1425 0.5366

n.s. = no converging simulation was achieved. a

Lf(40 + 125)

0.0246 0.0384 0.0512 0.1340 0.0001 0.0002 0.0002 0.0003 ± ± ± ± 0.0221 0.0332 0.0426 0.1230

Lf125 L(5 + 40 + 125)

0.0332 ± 0.0002 0.0508 ± 0.0002 0.0826 ± 0.0002 n.s.a 0.0001 0.0002 0.0001 0.0003 ± ± ± ±

L(40 + 125)

0.0271 0.0372 0.0495 0.1380 0.0001 0.0002 0.0003 0.0004 ± ± ± ±

L(5 + 40)

0.0345 0.0548 0.1033 0.3749 0.0001 0.0001 0.0002 0.0004 ± ± ± ± ± ± ± ± ± ± ± ± 0.3 0.5 1 5

L125

0.0200 0.0309 0.0507 0.1330

L40 L5 nanoparticle content (%)

0.0002 0.0002 0.0002 0.0003

type of system

Table 2. MC Evaluations of the Dispersion Quality Index F

0.0264 0.0266 0.0431 0.1318

dispersion state is achieved.12 Because of the self-assembly of high AR CNPs, often higher nanoparticle concentrations are required in order to form a connected network of such bundles throughout the matrix. Therefore, for an efficient network to be formed, one needs to combine certain elements from both ARs. For an electrical percolated network, while the low AR nanoparticles provide a more evenly dispersed mixture in which almost all regions of the matrix are influenced by their presence, the high AR nanoparticles form efficient pathways spanning longer distances through the matrix for electron transport. For CNTs in particular, it is known that the high AR of the nanoparticles results in a reduced minimum interparticle distance necessary for the electrons to tunnel, so that a tunneling connectivity is established throughout the entire bulk of the system.7−10 A similar analogy can be imagined for a rheological percolated network in which the high AR CNPs can form a larger number of topological entanglements with the polymer chains, hindering their dynamics in comparison with the low AR CNPs.49 We simulated 50/50 bimodal mixtures of low and high AR nanoparticles to pursue their benefits and merits (see Figure 2). According to the equilibrium morphologies, the mixtures show a relatively lower degree of self-assembly in comparison with their monodisperse high AR counterparts. Recent theoretical investigations on the monodisperse rods using Bethe lattice models have shown that local variations in the effective volume fraction and the dispersion quality alter the macroscopically averaged volume fraction at the percolation threshold.50,51 Here, stochastic MC calculations were employed to quantify the dispersion quality of all simulated systems. In these calculations, 105 beads were randomly placed in the simulation box one by one. If any of these beads fell within the force cutoff distance of a CNP bead, it was counted as a hit. Otherwise, it was counted as a miss. The ratio of the total number of hits to the total number of random points was defined as the dispersion quality index. This stochastic calculation was repeated 10 times for each simulated system, and the average of all calculations is reported here. This index with a possible maximum of 1 quantitatively represents the dispersion quality of the systems. Moreover, it provides a measure to estimate the effective volume of the nanoparticles in the mixtures by evaluating the probability of encountering a nanoparticle for a random bead in the total volume. The results of such calculations are given in Table 2. At a constant nanoparticle AR, the dispersion index becomes larger by increasing the nanoparticle content. However, this increase is more pronounced for L5 nanoparticles as opposed to L40 nanoparticles. Furthermore, the calculations show that the dispersion index increases in the L(5 + 40) mixtures with a bimodal length distribution compared to their monodisperse L40 counterparts. These trends in the quantified dispersion qualities are in agreement with the observations from trajectories in Figure 2. In order to provide information on a wider spectrum of ARs, monodisperse L125 and 50/50 bimodal dispersions of high AR (L40) and extra high AR (L125) nanoparticles were simulated as well. The L125 nanoparticles resemble the dimensions of realistic CNTs and allow for a proper comparison with experimental samples. The corresponding equilibrium morphologies of these CNPs in monodisperse systems and bimodal mixtures are shown in Figure 4. The L125 CNPs show more bent states due to their higher AR. These extra high AR nanoparticles form a network of physically entangled CNPs in which some single (or self-assembled) nanoparticles bridge

0.0001 0.0001 0.0002 0.0003

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Figure 4. Equilibrium morphology snapshots of extra high AR CNPs and their bimodal mixtures at different concentrations and flexibilities. L40 and L125 CNPs are shown in green and red, respectively.

with fullerene) and is more suitable for photovoltaic cells.3 A similar behavior is observed in multimode L(5 + 40 + 125) mixtures in which the dispersion index is larger than the monodisperse L40 and L125 systems, thus confirming a higher effective volume of the nanoparticles in the matrix.20,43 The effect of the initial configuration of CNPs on their selfassembly was examined by simulating similar systems to those in Figure 4 but with a moderate starting dispersion quality. This moderate dispersion quality was achieved by forcing the centers of mass of the CNPs to be within a centered smaller box with each side being half of the simulation box. The equilibrium morphologies (Figure S2) and the MC dispersion indices (Table S1) confirm that a finer initial dispersion quality is in favor of finer equilibrium dispersion in general. However, the overall trends do not change significantly. Real CNTs with very high ARs show significant semiflexibility and form spaghetti-like bundles in their dispersions with local ordered self-assemblies similar to our simulations.53 Therefore, it would be interesting to look into the role of flexibility on the final nanostructures and networks. To do this, the angle force constant was reduced to half its value to investigate the effect of the flexibility of individual CNPs on the network quality. One can see almost no significant difference in the dispersion qualities of these samples with their previous counterparts. This is probably due to two main reasons: (i) the highest AR in the current simulations is still not large enough to reflect significant differences in the flexibility behavior, and (ii) the altered flexibility is not imposing a critical change on the initial flexibility of CNPs. These arguments are supported by the incorporated CNT model in DPD simulations. The flexibility of CNTs is usually

between some other single (or self-assembled) nanoparticles. The equilibrium morphologies are evidently different from previous DPD simulations which did not consider the role of AR.27 The extra high AR nanoparticles follow a multiscale organization proposed before based on transmission optical microscopy and small-angle X-ray scattering investigations for CNTs.52 In this multiscale organization scheme, individual CNTs organize in nanosized self-assembled bundles. These bundles are then connected to form networks which determine the macroscopic properties such as electrical conductivity. Here, L125 CNPs self-assemble into hexagonal structures in their monodisperse and bimodal dispersions. However, a clear distinction between their L(40 + 125) mixtures and the L(5 + 40) bimodal dispersions is that the L40 nanoparticles also contribute to the self-assemblies of the L125 CNPs. The dispersion index for these systems is also given in Table 2. The differences between the dispersion qualities of the monodisperse L40 and L125 and bimodal L(40 + 125) mixtures at different concentrations do not seem to follow a meaningful trend. Considering all of the simulated systems, an ARdependent mechanism is observed in the network formation of hybrid CNP systems starting from a well-dispersed state. While the L(5 + 40) mixtures possess a higher dispersion index than to their L40 monodisperse counterparts due to the random dispersions of L5 CNPs, the L(40 + 125) mixtures show no significant difference in their quality of dispersion. This further supports the experimental observations that a mixture of fullerenes and CNTs results in more homogeneous dispersions than only CNT-containing dispersions (with their inherent length dispersity, but on a relatively larger AR range compared G


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Figure 5. Rheological measurements of PA12/CNT nanocomposites with different CNT loadings: (a) the strain amplitude sweep experiments and (b) the phase angle δ measured in frequency sweep experiments in the linear region.

Figure 6. (left) Radial distribution function g(r) of the system containing 1 wt % L40 CNPs at the start (red) and end (blue) of the simulations. The intraparticle subtracted radial distribution function g(r,t) − g(r,t0) at the end of the simulations in also shown in black. (right) g(r,t) − g(r,t0) profiles calculated at several time steps during the equilibrium runs for the system containing 1 wt % L40 CNPs. The gray zone shows the average location of the first correlation peak ∼1.8rc.

described in terms of their persistence length.54 In the present simulations, the angle force constant ka of coarse-grained CNT was set to 38.9kBT (2.75 × 10−19 J in real units). This value is ∼15 times smaller than the value derived by Liba et al.32 (41.4 × 10−19 J) based on the deflection measurements on singlewalled CNTs. It has been shown that the persistence length of single-walled CNTs typically exceeds 30 μm.54−56 Assuming a direct linear relationship between the persistence length and the angle force constant,55 the minimum persistence length of our CNTs is ∼2 μm. This length is more than 7 times larger than the longest CNT simulated here (L125). Consequently, the flexibility, even though overestimated in our models, cannot be easily detected. For the ka value of 19.45kB, while the difference is smaller, the CNTs lengths are still too short for a dramatic change in the flexibility behavior. The increase in the dispersion index with an increase in the CNP content suggests the gradual formation of the percolated network of nanoparticles within the PA12 matrix. For CNTs, it has been shown that a better dispersion quality results in a more efficient network at lower CNT concentrations.5 This network forms a solidlike elastic structure in the polymer matrix

by significantly hindering chain movements. Since the polymer chains in our models are much smaller than the real polymer chains (due to computational limitations23), we carried out rheological test on laboratory samples to show the formation of the double percolated network in such nanocomposites. Oscillatory amplitude and linear frequency sweep tests were performed on PA12/CNT nanocomposites with different CNT contents (see Figure 5). In the amplitude sweep experiments, the loss modulus is larger than the storage modulus for samples containing less than 0.3% CNT. This trend is then reversed at higher CNT contents, suggesting a shift in the behavior of the nanocomposites from a viscous material to an elastic material as the CNT content increases. Furthermore, the limiting strain of the linear viscoelastic region becomes smaller with CNT content, indicating that a more complex microstructure (which is more sensitive to the applied strain) is being developed in the nanocomposites. The frequency sweep measurements also show substantial decrease in the phase angle at all frequencies as the CNT amount is increased. A decrease in the phase angle further supports the gradual formation of solidlike elastic microstructures within the nanocomposites. In particular, the H


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Macromolecules 3D percolated network of CNTs in PA12 can be observed at lower frequencies where it significantly influences the longer relaxation times. At CNT contents between 0.15 and 0.3%, the developing network percolates in the matrix and dominates the long-range microstructural characteristics of the nanocomposites. This can be witnessed in the frequency-independent phase angle at low frequencies region. Therefore, a critical rheological mass concentration of approximately 0.3% is needed to achieve the double percolated network of CNTs and polymer chains. 4.2. Nanoscopic Morphology Development. Considering the differences between the dispersion qualities of bimodal and multimode mixtures versus their monodisperse counterparts, it is important to address the underlying mechanisms of morphology development at the nanoscale. The development of the CNPs morphology in the dispersions can be envisioned in the evolutions of the radial distribution function, g(r). This function was calculated for all simulations with time, g(r,t), in order to capture the kinetics of CNP−CNP correlation developments. One should notice that in the course of morphology evolutions in simulations we are interested to follow the interparticle structural evolutions rather than the intraparticle correlations. Consequently, the radial distribution function at the start of the simulation, g(r,t0), was subtracted from its time-dependent evolutionary profile to remove the intraparticle correlations. In this way, all the peaks in the g(r,t) profile originating from the bead−bead counts within the same CNP molecule are removed from the overall distribution profiles. The positions of these peaks were checked by calculating g(r) of an individual CNP at different ARs. It was verified that all of the removed peaks belong to such intraparticle structures rather than to the interparticle structures. The single-particle g(r) functions are given in Figure S3. Such a calculation is shown in Figure 6a for a 1% content of L40 nanoparticles at the end of the simulation run. Similar calculations were performed for all systems at several time steps in the equilibrium simulation runs (see Figure S4 for similar calculations of the L40 nanoparticle at various concentrations). As a result of such evaluations, the interparticle structural developments can be followed with time. These changes attribute to the spatial correlations and packing of CNPs at different time steps, thus revealing the selfassembly kinetics. The evolution of the g(r,t) − g(r,t0) with time is shown in Figure 6b. The first correlation peak at ∼1.8rc (the gray zone of the plot) is indicative of the packing of CNPs within the bundles with an almost constant average interparticle distance at all times and concentrations (see the Supporting Information). This separation distance corresponds to ∼7.68 nm in real units, which is in good agreement with the measured values using synchrotron small-angle X-ray scattering by Fei et al.,53 who reported it to be 7.26 nm for CNT bundles in polystyrene matrix. Moreover, it shows that any increase in the CNP concentration only alters the overall size of the selfassemblies and does not influence the intraassembly correlation distances. Indeed, the overall increase of the profile at all separation distances suggests the addition of new nanoparticles to the assemblies, which leads to their further enlargement. These trends are in agreement with 2-dimensional structure factors shown before. By integrating the subtracted radial distribution function at all separation distances, the total probability of finding a bead of nanoparticle i close to a bead of nanoparticle j is calculated and weighted with their corresponding pair separation distance.

Figure 7. Time-dependent interparticle correlations for different systems with varying CNP loadings.

This quantity describes the overall self-assembly state in the system at any time relative to its initially well-dispersed microstructure. It is therefore possible to track the overall selfassembly of a system with simulation time. The results of such calculation are shown in Figure 7 for dispersions of L5, L40, and the L(5 + 40) mixture at all concentrations. According to the results, the L5 nanoparticles do not form aggregated microstructures in the PA12 during the entire equilibrium runs. This statement holds for all of the simulated concentrations depicting the Brownian motions typical of low AR nanoparticles driven by entropy considerations.48 The L40 nanoparticles, on the other hand, start to self-assemble at a certain onset time, which decreases to some extent with CNP loading. The bimodal mixtures, however, behave in between these two extremes. The self-assembly process in the bimodal mixtures is closer to the L5 nanoparticles and slowly transforms toward an average of the monodisperse counterparts as the CNP concentration increases. It should be noted that the selfassembly process has not completed yet, even though the total potential energy of the simulated systems had already reached its steady state. One would expect to eventually reach a state in which all L40 CNPs form one large self-assembly altogether. I


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hindered or not influenced in their bimodal mixtures depending on the nanoparticle loading. At low concentrations, the selfassembly in monodisperse system takes place faster than its corresponding bimodal mixture similar to previous lower AR dispersions and their mixtures. As the concentration increases, however, this difference becomes less prominent until it almost disappears for contents above 0.5 wt %. The MC calculations of the dispersion index showed that the CNP nanostructures become more dominant with the addition of more CNPs to the simulation box. Furthermore, according to the rheological tests, the percolated network dominates the overall dynamics of the chains as well as the CNPs as the concentration is increased from 0.3 to 1%. Therefore, it is expected that at higher CNP loadings the movements of the nanoparticles are dramatically hindered by such percolated networks. Consequently, CNPs are increasingly more tangled to each other as well as to polymer chains. As a result, the self-assembly kinetic does not rely on the length distribution after a certain AR, as also seen in L(40 + 125) bimodal mixtures. Furthermore, it seems that the increased flexibility of CNPs does not alter the equilibrium morphology formation, since the CNPs do not show faster or slower disentanglement (e.g., from the network) and translation (e.g., into the self-assemblies) kinetics. A comprehensive picture of the microstructure and networking of the CNPs in polymers, whether in monodisperse systems, bimodal, or multimode mixtures, can be drawn by considering the insights provided so far. At a constant nanoparticle concentration, the low AR fullerene-like nanoparticles disperse randomly in the matrix influencing its entire volume almost uniformly. The high AR CNPs (medium length nanotubes), on the other hand, tend to self-assemble and form bundles of honeycomb structures. The bimodal mixtures of these nanoparticles show a combined microstructure in which the fullerene-like nanoparticles are well-dispersed in the matrix, and the high AR CNPs undergo self-assembly. However, this self-assembly is somewhat hindered by the interferences of fullerenes. This notion is supported by the equilibrium microstructures of the L40 CNPs in the bimodal mixture at 1% and the monodisperse L40 CNPs at 0.5% shown in Figure 2 and their self-assembly kinetics shown in Figure S5. These systems both contain the same amount of L40 CNPs, although in the former the self-assembly is also influenced by the presence of fullerene-like nanoparticles. Clearly, the presence of the low AR nanoparticles hinders the self-assembly of the same number of high AR nanoparticles. Recently, we also showed how the interferences of short and long nanorods on the movements of each other slows down the self-assembly of long nanorods in an epoxy matrix.20 Hence, one can conclude that the microstructure of these bimodal mixtures is not merely a linear combination of phase-separated short and long nanoparticles but that there are mutual interactions between these modes. For L125 CNPs, the emergence of an entangled network is observed besides the aggregation with self-assembly patterns similar to L40 CNPs. The bimodal mixtures of L125 CNPs with the L40 CNPs show a slowed down or unaffected overall self-assembly process depending on their total concentration in connection with their ability to translate freely in the matrix. In these mixtures, the coassembly of L40 and L125 CNPs is observed in their equilibrium morphologies shown in Figure 4. On the basis of these simulations and our previous work in this field, it is anticipated that the bimodal mixtures of fullerene-like nanoparticles with L125 CNPs possess identical morphological elements for each AR, i.e., a

Although this scenario gives the ultimate equilibrium morphology, it does not critically change the total potential energy of the systems. This is simply due to the fact that the joining of two medium-sized bundles into a large bundle does not considerably alter the overall interactions of nanoparticle− nanoparticle, nanoparticle−polymer, and polymer−polymer pairs. Large-scale simulations of CNTs in polymer based on the hybrid particle-field molecular dynamics technique have also shown that the self-assembled bundles do not undergo any significant changes on longer equilibration times (up to the order of a few milliseconds) after the potential energy of the system has reached a steady state.42 In addition, real nanocomposites may hardly reach such an ultimate state since the nanoparticles need considerable spatial freedom and could take unrealistically long times (if ever) to form a single large bundle. Therefore, one can safely take these morphologies as the equilibrium morphologies of such systems and compare them with experimental materials. For extra high AR L125 CNPs, a different trend was observed as opposed to their 50/50 mixtures with L40 nanoparticles. As shown in Figure 8, the self-assembly is either

Figure 8. Time-dependent interparticle correlations for different systems with varying CNP loadings, containing extra high AR CNPs. J


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Figure 9. Equilibrium morphologies of multimode systems with total CNP contents of (a) 0.3, (b) 0.5, and (c) 1 wt %. L5, L40, and L125 CNPs are shown in yellow, green, and red, respectively. In (d) the evolution of the morphology is plotted with time for the monodisperse systems as well as the multimode systems at similar concentrations.

4.3. Continuous Pathways in the Network for Electron Transport. So far, it was shown that the incorporation of bimodal and/or multimode CNPs can result in different morphological patterns on the nanoscopic scale. The efficiency of a network of CNPs within a polymer matrix is in fact concerned with its connectedness. To examine the connectivity of the networks in our simulations, we used MC calculations to estimate the probability of electron transport in graph representations of CNP networks. The continuity of the pathways in these networks can be revealed by the transport of a hypothetical electron through the simulation box. If a random electron starts to transport from one side of the simulation box (e.g., x = 0) and reaches the other side (e.g., x = L with L being the size of the box) by passing through the conductive CNPs network, a continuous pathway exists between the two corresponding sides (e.g., in the x direction). Extensive MC calculations were performed on the simulated equilibrium nanostructures following a similar approach to investigate the continuity of the CNP pathways. These calculations were carried out for 105 random electrons passing through the box in either x, y, or z directions (105 random electrons for each direction). If there is any connected pathway for such a transport, it is counted as a continuous path and as an isolated path if otherwise. In this way, the probability of the electron transport was calculated in all directions for all simulated systems. Each system was evaluated in this manner for 10 times, and the averages of all calculations are reported here. Two mechanisms were considered for an electron to transport from bead to bead. First, it was assumed that electron transports between any two beads if there is a bond between them. Second, the nonbonded beads could still contribute to the electron transport via the tunneling

random dispersion of L5 nanoparticles coexisting with partially entangled partially aggregated L125 CNPs. As the nanoparticle concentration is increased, the low AR nanoparticles impact a larger portion of the matrix, and the self-assembled bundles of the high AR nanoparticles grow in size. Consequently, the formation of a continuous network of nanoparticles becomes easier with concentration, and it begins to dominate the relaxation behavior of the resulting nanocomposites. Such a network formation is also observed in the rheological measurements of laboratory samples with an intrinsic length polydispersity in the CNTs. Multimode L(5 + 40 + 125) systems containing CNPs with all ARs were also simulated (see Figure 9). The results clearly show slowed down self-assembly kinetics at all concentrations and a finer equilibrium dispersion compared with monodisperse L125 systems. Hence, one can draw the conclusion that the length polydispersity can be harvested in order to make efficient networks and microstructures in nanocomposites. In such synergistic materials, the shorter nanoparticles influence a large portion of the bulk in a uniform pattern while, at the same time, they hinder the aggregation of longer nanotubes. These uniform dispersions of short nanoparticles could participate in the overall network efficiency through various complementary mechanisms such as increasing the effective volume of CNP networks or an easier electron transfer from donor to acceptor in photovoltaic cells. The longer nanotubes, on the other hand, provide short continuous pathways, for instance for electron transport and/or form physical entanglements, for instance, to hinder the polymer chain dynamics. It should be noted that the mentioned mechanisms are not only limited to a certain AR of CNPs, as all ARs participate in them, but some ARs might play a bigger role than others in each case. K


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Table 3. MC Evaluations of the Electron Transport Probabilities in Different Directions through the Simulation Boxa type of system nanoparticle content (%) 0.3

L5

L40

L125

L(5 + 40)

L(40 + 125)

L(5 + 40 + 125)

Lf125

Lf(40 + 125)

x-dir y-dir z-dir

0.0196 0.0343 0.0432

NT NT NT

NT NT 0.0103

NT NT NT

0.0705 NT NT

0.0853 0.0203 0.0932

0.0217 0.0582 NT

NT NT NT

x-dir y-dir z-dir

0.0324 0.0086 0.0551 NTb

0 NT NT NT

0.0034 NT NT 0.0086

0 0.0153 NT 0.0393

0.0235 0.0257 0.0504 0.0597

0.0663 0.0597 0.0173 0.0619

0.0399 NT NT 0.0219

0 NT 0.0447 0.0352

x-dir y-dir z-dir

0.0212 NT 0.2187 0.0494

0 NT NT NT

0.0029 0.0690 NT 0.0753

0.0273 0.0886 0.0519 0.0400

0.0453 NT NT 0.0287

0.0463 0.2256 0.2488 0.1184

0.0073 0.0467 NT 0.1228

0.0266 0.0121 NT 0.0226

x-dir y-dir z-dir

0.0894 0.8803 0.8305 0.8212

0 0.1792 0.1323 0.1980

0.0481 NT 0.0877 0.2332

0.0602 0.5915 0.6172 0.6564

0.0096 0.1332 0.0641 0.1940

0.1976 n.s.c

0.0848 0.2772 0.2643 0.1898

0.0174 0.0917 0.0291 0.1367

0.8440

0.1698

0.1070

0.6217

0.1304

0.2438

0.0858

ave 0.5

ave 1

ave 5

ave a

b

c

In these calculations the Ftunneling was 5.16. NT = no transport. n.s. = no converging simulation was achieved.

approach are given in Table 3 (and for the former approach in Table S2) where the probability of electron transport in the pathway is given for all directions. The average transport probability is also evaluated for each system and given in the table. At this point, it should be noted that the electron tunneling in reality indeed includes more complex phenomena rather than being a simple hopping between CNP beads within a certain cutoff distance. A critical difference between our calculations and the reality lies in the fact that a certain energy barrier must be overcome for an electron to tunnel over the thickness of the polymer film between nanoparticles. However, the current simulations and MC calculations do not have the required resolution to account for such phenomena. Consequently, the evaluated transport probabilities are generic representations of the overall connectedness of the networks. In general, the results show that the L5 nanoparticles display a higher probability of electron transport in comparison with the L40 and L125 nanoparticles. The monodisperse L40 and L125 CNPs with comparable dispersion qualities at all concentrations (Table 2) are of particular interest. The simulations suggest that L125 CNPs with higher AR than L40 CNPs show a higher electron transport probability at the same nanoparticle loadings. L125 CNPs show a sudden rise in the transport probability at a concentration between 0.5 and 1 wt % while such an increase only appears at higher L40 loadings between 1 and 5 wt %. Therefore, the higher AR nanoparticles show a higher electron transport probability provided that the dispersion quality is almost identical in all samples. These insights from the simulations are supported by recent experiments on epoxy/CNT nanocomposites.12 As opposed to the traditional belief that the higher AR CNTs could provide a better electrical conductivity, one should always consider the importance of the dispersion quality. Otherwise, it should have been expected to have the lowest electron transport probability in the monodisperse L5 systems while the simulations show the opposite. Moreover, the bimodal mixtures L(5 + 40) and L(40 + 125) show a higher probability of electron transport as

mechanism. In order to include the tunneling mechanism in the MC calculations, one needs to define the maximum tunneling distance of electrons in the coarse-grained systems. The maximum tunneling distance of electrons in reality in polymer nanocomposites is ∼1.4 nm.57 This tunneling distance is almost 10 times the length of a carbon−carbon bond in a CNT molecule. Therefore, one can take a maximum tunneling factor f tunneling of 10 times the bead−bead bond length in the coarsegrained simulations to represent the maximum tunneling distance in the MC calculations. However, one might argue that in the coarse-graining procedure used in our simulations each CNT bead represents 24 carbons.32 Consequently, one should initially divide the bead−bead bond length to the number of constituting bonds and then use the f tunneling of 10 with this corrected bond length. In this way, the maximum tunneling length is calculated to be ∼0.3rc which is meaningless since there are no pair of CNT beads closer than ∼0.6rc during our simulation runs (as can be seen in g(r) profiles). This observation is simply a consequence of the repulsive forces which are active for any pairs closer than 1.0rc. The MC calculations performed using this tunneling factor led to almost full isolation against any electron transport in all systems. A third possible strategy to scale the maximum tunneling distance of electrons in the DPD units is to scale it the same as the repulsive forces are scaled in comparison with the carbon− carbon Lennard-Jones forces. In the consistent valence force field (CVFF), the force cutoff distance for carbon−carbon pairs is 3.87 Å. In our coarse-grained models with a force cutoff distance of 1.0rc, the maximum tunneling distance should be set to ∼3.61rc to keep the scaling proportional to that of the conservative forces. This value yields a f tunneling of 5.16 for the MC calculations. Here, we performed the MC calculations using all three approaches and report the results from the first and third methods since the second scaling did not yield any conductivity. In this way, the maximum tunneling distance is scaled based on the characteristic carbon−carbon bond length and/or force cutoff distance. The results based on the latter L


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Figure 10. (a) Average electron transport probabilities calculated by MC method using f tunneling values of 5.16 and 10. The lines in the figure are guides to the eye. (b) Electrical resistivity measurements of PA12/CNT samples with different CNT concentrations. The markers show different measuring setups with different ranges of detection sensitivities as discussed in Experimental Details section. Note that the error bars are in some cases too small to be visible.

CNPs in the current DPD models has hardly any systematic influence on the overall dispersion state and network formation. Even though the simulations show that the probability of electron transport dramatically increases at a certain nanoparticle loading for each system, it is not possible to comment on the electrical percolation threshold due to the limitations of the simulations. Therefore, to provide an experimental insight into the critical concentration for electrical percolation, the electrical resistivities of hot pressed PA12/CNT films were measured, and the results are shown in Figure 10b. The resistivity of the samples decreases as the connected network of CNTs develops with increasing CNT concentration. A change in the slope of the samples resistivity versus CNT content is observed at a CNT loading of ∼0.43 wt %. This concentration corresponds to the onset of the fully developed network of nanoparticles within the matrix and is in good agreement with other well-dispersed melt-mixed polymer nanocomposites (see ref 1 and the references within). The differences between this experimental concentration and MC calculations can be due to several factors including the scaling of f tunneling, the different length polydispersity profiles used in the simulations versus the experiments, etc. However, the overall behavior of the simulated systems and the measured samples is in good generic agreement. Existing theoretical studies predict that the tunneling conductivity in polydisperse rod suspensions depends on the length distribution of rods through their weighted length average Lw = ⟨L2⟩/⟨L⟩.10,59 The critical tunneling distance dc in such systems is approximated by dc = ⟨D2⟩/⟨2LwΦ⟩ where D is the diameter of particles and Φ is their volume content in the systems.8,10 However, recent experiments showed that this inverse relation with Lw is only dominant in homogeneous dispersions and does not hold for clustering CNTs.12 A refined model was then developed based on the mean coordination number of the nanotubes at contact Zcl to account for the degree of clustering. Based on this approach, critical tunneling distance was redefined to be

opposed to their monodisperse counterparts containing L40 and/or L125 nanoparticles at the same concentration. More interestingly, the probability of electron transport in the hybrid L(5 + 40 + 125) system is at all concentrations higher than the monodisperse L40 and L125 and comparable to the L5. Consequently, more efficient dispersions in terms of connectedness with an improved pathway networking are generally achieved when hybrid nanocomposites are fabricated from multimode CNPs. It should be noted that the directional calculation of electron transport allows considering the anisotropic geometry of the nanoparticles, which becomes important particularly at higher ARs. The average of these directional possibilities should consequently give a good representation of the 3D network continuity in terms of electron conductivity. These averages are also given in Table 3 (and Table S2) and are plotted in Figure 10a for various systems and tunneling factors. In the figure, the development of nanoparticle networks for L5 and L(5 + 40) systems becomes increasingly pronounced at low concentrations (∼0.5 and 1 wt % for f tunneling of 10 and 5.16, respectively) as the transport probability increases abruptly. However, the L40 system hardly starts to develop an electrically percolated network up to the highest nanoparticle concentration simulated. For the systems with increased flexibility of CNPs, no systematic trend is found when considering both f tunneling values and all simulated systems. We repeated these simulations to check the reproducibility and found no significant changes in the results. Therefore, the change in the flexibility of the CNPs does not systematically affect the formation of their dispersions and the continuous pathways. It was pointed out before that this observation could be due to either the limited ARs of CNPs or the excess by which the flexibility is changed. Although these factors could be of relevance for CNPs with higher ARs, for ARs below 100, recent theoretical investigations have shown that deviations of a deformed nanoparticle from a perfect straight rodlike shape has very little effect on the percolation threshold.58 These findings report that for small or moderate nanoparticle deformations the universal scaling of the percolation threshold is hardly influenced by the precise particle shape. Hence, it is in agreement with predictions of the connectedness percolation theory that the higher flexibility of

dc =

1 − Zcl ⟨D2⟩ 2Φ (1 − Zcl)Lw + ZclLn

(10)

in which Ln = ⟨L⟩ is the number-average length. For an ideally homogeneous system Zcl = 0, and the previous correlation is M


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(providing finer dispersions) and the continuous pathways are provided by the higher ARs (providing entangled networks spanning throughout the matrix). On the basis of the results, it was argued that the length polydispersity and dispersion homogeneity mutually influence each other and are thus interconnected. This conclusion provides a general framework for a comprehensive understanding of various hybrid nanoparticle systems reported before.

recovered. For CNT dispersions, Zcl is typically between 0.6 and 0.9, thus rendering the polydispersity ineffective to the overall electrical conductivity.12 Our simulations show that if the polydispersity does not lead to a better dispersion quality, then the polydispersity does not have any major impact on the probability of electron transport through the systems. Nevertheless, for bimodal and multimode hybrid dispersions in which the dispersion quality is improved (for instance, mixtures of short and high AR CNPs), the polydispersity plays a key role in the electron transport and hence on the overall connectedness of the electrical network. This is a clear distinction from previous studies where the roles of polydispersity and dispersion homogeneity were considered independently (or not considered at all; see ref 12 and the references within) whereas they are interconnected according to the present simulations. In other words, the length distribution modifies the dispersion homogeneity and the dispersion homogeneity determines how important the role of polydispersity is in the network formation. This realization based on DPD simulations enables a general framework for the understanding of real hybrid systems in which the dispersion state is shown to be controlled by polydispersity3 as well as other real hybrids in which the effectiveness of polydispersity on network formation is shown to be controlled by homogeneity.12

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00585. Equilibrium morphologies, mechanism of morphology development, assessment of continuous pathways (PDF)

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AUTHOR INFORMATION

Corresponding Author

*(A.G.) E-mail [email protected]. ORCID

Ali Gooneie: 0000-0003-0703-9437 Notes

5. CONCLUDING REMARKS The incorporation of hybrid carbon nanoparticles in a hosting polymer matrix recently has raised a lot of interest due to the synergistic properties that can be exploited in such systems. Use of CNTs along with other nanoparticles such as fullerenes is indeed a viable route toward functional materials, for instance, for photovoltaic applications. CNP hybrids having different ARs are simulated in this paper to reveal their individual as well as combinatory characteristics while well-mixed in a polymer matrix. To this end, the self-assembly and network formation were studied in monodisperse, bimodal, and multimode dispersions of CNPs. The results clearly indicate an ARdependent morphology developing in the polymer nanocomposites under equilibrium. While the low AR CNPs (fullerene-like nanoparticles) randomly disperse in the matrix, the medium and high AR CNPs (short and long CNTs) selfassemble and form bundles. If the AR is large enough (e.g., in real CNTs), such bundles bridge each other and contribute to an overall network, thus controlling the macroscopic behavior such as electrical conductivity. The low AR CNPs contribute to a well-dispersed morphology in which properties such as electron tunneling and electron hopping from donor to acceptor are evidently facilitated, even though their network can be unstable and sensitive to mechanical deformations. The high AR CNPs, on the other hand, form tightly entangled networks with each other as well as with polymer chains, as was also witnessed in rheological measurements on PA12/CNT nanocomposites. It was discussed that bimodal and multimode mixtures of different ARs can yield efficient networks in which a combination of advantageous properties of these ARs can be harvested. In particular, the self-assembly of high AR CNPs can to some extent be controlled by the interferences of the low AR CNPs, thus lowering the critical concentration for the percolation threshold. The continuity of these networks was further assessed by MC calculations of random electrons transporting through such nanostructures. The results confirmed an improved electron transport in the hybrid systems where the electron tunneling is facilitated by the lower ARs

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

This research was cofunded through a grant by the Swiss Commission for Technology and innovation CTI (Grant 18816.1 PFIW-IW). We thank Dr. Amin Sadeghpour for valuable discussions and Pietro Simonetti for assistance in the experiments.

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