It is well known that carbon nanotubes (CNTs) possess extraordinary properties, such as regular electronic structure and high mechanical strength. However, since in their pristine state CNTs exist in a disordered agglomerated mass due to van der Waals (vdW) interactions (on the order of 0.5–1 eV/nm of nanotube–nanotube contact),1, 2 the theoretical potential of their extraordinary properties falls far short of the actual performance.3 The first challenge is, therefore, to obtain a homogeneous dispersion of CNTs for practical applications.
Different approaches are used to decrease nanotube aggregation, including ultrasonication or high shear mixing. These are usually combined with methods that alter the surface of the tubes either covalently (functionalization) or noncovalently (adsorption) to prevent reagglomeration of the CNTs. A notable drawback of covalent functionalization is the disruption of the extended π conjugation in CNTs, which may not significantly affect their mechanical and thermal properties but may profoundly affect their electrical properties.4
The potentially superb properties of CNTs are promising toward composite materials and, in particular, polymer composites, where they are expected to provide much higher reinforcement as well as electronic enhancement than conventional fillers.3, 4 The properties of the composite depends on the uniformity of their dispersion and alignment within the material, as well as the strength of interfacial bonding between the polymer and CNT. As with other surfaces,5, 6 the interfacial bonding of physically adsorbed polymers on CNTs may fail adhesively or cohesively, and one should therefore tailor the polymer as to address both of these issues. While oligomers and surfactants are expected to fail either adhesively (weak CNT-polymer interactions) or cohesively (strong CNT-polymer interactions), long polymer chains and, in particular, copolymer chains might be able to address both these failures.7
The curved surface of the CNT offers an additional advantage over, for example, a flat surface, since even weak polymer-surface interactions can lead to wrapping conformations,4 resulting essentially in irreversible adsorption.8 Moreover, the possibility of regular helical wrapping, displayed by a number of polymers9–13 may be used advantageously in a CNT/polymer composite, for example, through the helical pitch which directly controls the linear charge density of the complex and is predicted to be an important parameter controlling separation in ion-exchange chromatography.14, 15
Various factors are likely to contribute to the observed ordered helical conformations. These include vdW and hydrophobic interactions between polymer chains and the CNT, π stacking, electronic charge fluctuations, and CNT deformation. However, recent studies suggest that helical adsorption is driven less by specific monomer–CNT interactions,15, 16 and may be a result of the highly curved surface of the CNTs.8, 17, 18 A molecular dynamics study of DNA on CNTs15 revealed that in the limit of low ionic strength, the main contributors to the free energy of the system (and thus the predicted helical conformation) are adhesive interactions between the DNA and the CNT and electrostatic repulsion between charges on the DNA backbone, and not by π stacking interactions as previously suggested.19 The rather low energetic barrier for sampling various sites on the CNT surface by the bases on the DNA15 provides further evidence for the lesser importance of site-specific monomer–CNT interactions in determining the equilibrium adsorbed conformation.20 The competition between these two factors may result in an optimal helical wrapping geometry for small ionic strength.
Recently, using coarse-grained Monte Carlo (MC) simulations, in which only short-ranged weakly attractive nonbonded interactions were considered, we have shown that semiflexible chains will adsorb in nearly perfect helices on CNTs.8, 17, 18 When multiple chains are allowed to adsorb on CNTs, simulations predict the formation of multiple helices.8, 18 The multiple helix configuration was also conjectured by Smalley and coworkers21 as a result of the hydrophobic effect in case of an aqueous environment, so that surface coverage is maximized and the hydrophobic CNT is shielded from the aqueous solvent. Such conformational symmetry breaking is not surprising, however, in view of similar predictions for polymers under confined and/or constrained environments.22–26 Nonetheless, a notable difference between these studies and polymers adsorbed on CNTs is that the former cases are driven by the minimization of the free energy through maximizing energetically favorable contacts in the and thereby minimizing system entropy. In the polymer–CNT system, however, the helical conformations lead to an overall increase in entropy of the adsorbed polymer by allowing for greater mobility of fully adsorbed chains on the surface of the tube.8 As such, these ordered conformations have relatively high entropy as well as well as being energetically favored.
Block copolymers (BCPs) offer an additional parameter over homopolymers that can be used to control the distance between CNTs, and thus may be used to tailor the structure and performance of polymer/CNT composites.27 In a recent study,8 we showed that a sufficiently long hydrophobic block (H) will adsorb in a helical manner, whereas the polar block (P) extends to the bulk. Triblock (HPH) copolymers may provide additional advantage, in principle, by allowing for the formation of loops, anchored to the CNT surface by the wrapped hydrophobic blocks. The polar loops extending to the bulk may entangle with free polymer, leading to strong adhesion between the CNTs and bulk polymer.
In this work, we further study the physical association of polymers with CNTs, using coarse-grained MC simulation. We study how the helical pitch changes with polymer length and stiffness and develop a simple model that describes this behavior. In our study of BCPs, we focus on the effect of different chain architectures on the adsorbed polymer conformation.
We utilize a minimalist MC model to concentrate on the physical effects that drive long polymer chains to associate with CNTs and adopt ordered adsorbed conformations. Below, we provide a brief description of the model and assumptions. Details of the simulation are described elsewhere.18
The monomers are modeled as soft spheres, with Lennard-Jones (LJ) interaction potential between nonbonded monomers,
where k is the Boltzmann constant, T is the temperature, εij is the average 2-body LJ interaction parameter between monomers i and j (εij = (εii εjj)1/2), σij is the average effective size of monomers i and j (σij = (σi + σj)/2), and rij is the distance between the two interacting nonbonded monomers. The well potentials used for the hydrophobic and polar interactions are εH = 1 and εP = 0.01, respectively. A cut-off radius of 21/6σ for the monomer–monomer interactions is used to allow for the effect of short-range attractive forces at a low computational cost. We note that for the monomer–monomer LJ interactions, the potential cut-off has a minor effect on chain conformational behavior when compared with the bending potential (eq 2, below) at the temperature, chain length, and concentration studied, especially for the stiffer chains. Thus, the attractive well of the potential effectively comes into play when the chains are adsorbed on the cylinder.
Chain stiffness is modeled by the following harmonic expression for the energy required to bend the angle θ formed between consecutive bond vectors
where κ characterizes the resistance of the bond angle θj about monomer j to bending. θj is defined by cosθj = bj−1 · bj/(|bj−1‖bj|), where bj = rj+1 − rj, and rj is the absolute position of monomer j. The bending stiffness parameter κ appearing in eq 2 can be easily related to the persistence length, lp, through a bond angle correlation function which decays exponentially with lp, thereby obtaining a power law dependence of lp on κ.22, 18, 28, 29 Note that using cosθ instead of θ in eq 2 simply provides a different correlation between κ and the persistence length of the polymer, but simplifies the computational calculation by avoiding the use of the inverse cosine function. In Figure 1, we show a plot of the radius of gyration as a function of chain length for various values of κ. It can be seen that our simulation conditions correspond to a good solvent.
Since CNTs exhibit an even charge distribution,30 there is hardly any electrostatic interaction between nanotubes and polymers, and therefore in the absence of chemical functionalization, the polymer–CNT interaction is governed by vdW forces.31, 32 Therefore, we model the CNT as a smooth infinitely long and impenetrable cylindrical shell with a continuous LJ potential field averaged over the tube length. The polymer–tube interactions are obtained by integrating the dimensionless LJ potential over the length of the tube,18, 33 to obtain a potential that depends on tube dimensions (inner radius ρi and outer radius ρo) and the perpendicular distance of a monomer from the surface of the tube, D. Integration of eq 1 over the vertical z-axis of the tube can be carried out analytically, obtaining:
where εw is the tube-monomer LJ interaction parameter and x = (D + R)2 + ρ2 − 2ρ(D + R)cosφ. Integration over φ and ρ is performed numerically once at the beginning of the simulation for each of the outer radii considered. A tube thickness of 1σ was considered in all our calculations and the potential cut-off was fixed at 3σ. εw was adjusted such that the potential minimum of tubes of different radii correspond to the potential minimum of a tube with ρo = 2σ (Ucyl,min ≅ −1.46) to concentrate on the effect of surface curvature. Although we studied nanotubes of various radii, we concentrated on ρo = 2σ, which corresponds to the dimension of single-walled CNTs.18 Moreover, in a previous communication, the value of ρo = 2σ was shown to be the minimum tube radius beyond which adsorption occurs for semiflexible chains.18 This value of Ucyl,min is slightly lower than the potential minimum, ULJ,min = −1, taken for monomer–monomer interactions, and is a direct result of integration of eq 3 with εw = 1.
Simulations were carried out on chains consisting of Nm monomer units consisting of b blocks, where each block is made up of NH hydrophobic (H) segments or NP polar (P) segments. Bond length was taken to be 1σ in all our simulations. All segments were taken to be of equal size σ. A cubic simulation box with periodic boundary conditions was considered with the nanotube placed in the center of the box, aligned along the z-axis. The simulation is initialized by placing Nc identical chains randomly within the simulation box. We use Nc = 3 for the homopolymer and Nc = 1 for the BCP (unless otherwise stated). Equilibration is achieved using a combination of reptation and kink-jump moves,34 tried with ∼0.1 and 0.9 probability, respectively, during the production stage of the simulation. The move probability is adjusted at the course of equilibration to achieve ∼50% Metropolis acceptance rate. To ensure equilibration, 106 MC steps are performed prior to calculating average properties. Averages are calculated over additional 106–107 MC steps, sampled every 102 MC steps.
RESULTS AND DISCUSSION
Recently, we have shown that there exists a range of parameters for which polymers will adsorb on CNTs in an ordered helical manner.8, 17, 18 In addition, we presented evidence that points to an adsorption transition followed by a sharp order transition to nearly perfect helices as chain stiffness is increased.8, 18 This symmetry breaking of the configurational space of the polymer to helical adsorption is thermodynamically driven. While adsorption leads to maximization of polymer–CNT interactions, reorganization into a helical conformation leads to an entropic gain by allowing for greater mobility of the chains on the surface of the tube.8 That the system has reached thermodynamic equilibrium can be inferred from Figures 2 and 3. The energy reaches its equilibrium value quite quickly (Fig. 2) and adsorption is reversible with temperature (Fig. 3), indicating that the observed helical conformations are a thermodynamically stable state, also at finite temperatures. Desorption of the polymers with increasing temperature has also been reported experimentally.10 The snapshots in Figure 2 show polymers at different stages of adsorption. In a previous communication,8 we showed that the wrapping conformations at finite temperatures allow the polymers to retain relatively high diffusivity (calculated from the mean squared displacement) in the direction parallel to the tube axis, when compared with the radial direction. In Figure 4, we follow the vertical and azimuthal position (z vs. φ) of the central monomer in an adsorbed homopolymer chain during the course of 105 MC iterations, after equilibrium has been achieved. The monomer follows the helical path of the polymer as it spirals up and down the surface of the tube. This behavior suggests that the polymers adopt an adsorbed conformation with an optimal pitch.
The helical pitch is predicted to be an important parameter controlling, for example, separation in ion-exchange chromatography5 since it directly controls the linear charge density of the polymer–CNT surface, and thus strongly influences the electrostatic field near the surface.30, 35 Though π stacking has been considered a dominant factor in determining polymer helicity,19, 31, 36–38 several studies have argued that π conjugation may not contribute significantly to polymer wrapping of CNTs.15, 16, 32, 39 Thus, nonspecific association that is responsible for helical wrapping may also result in an optimal pitch for a particular polymer–CNT configuration. That an optimal helical pitch may be determined by nonspecific structure and interactions is important since all known preparations of CNTs give mixtures of nanotube chiralities.4 The following analysis suggests that the helical pitch is indeed optimized for polymer–CNT systems displaying helical wrapping.
The average pitch angle α of the adsorbed polymers, schematically illustrated in Figure 5, is found to be nearly independent of chain length (Fig. 6). α can be approximated from the Kratky-Porod40 (KP) worm-like chain assumption that the angle θ formed by bond vectors separated by x segments decays exponentially,
where lp is the persistence length of the chain. We assume that the optimal conformation occurs when the average bond angle between two consecutive segments (x = 1) corresponds to the average bond angle θ of the adsorbed polymer. The relation between the pitch angle α and bond angle θ is obtained from geometric considerations,
According to eq 5, the helical pitch should not depend on chain length. The dependence of the adsorbed fraction and of α on chain length obtained from the simulation is shown in Figure 6. The dashed line represents the adsorption transition marking the approximate location of the peak in the fluctuations Cα of the adsorbed fraction, that is, Cα = 〈f〉 − 〈fads〉2. Figure 6(a) suggests that short chains do not experience sufficient energy gain to overcome the entropic penalty upon adsorption, and thus remain largely in the bulk, while longer chains adsorb readily. Chain length then plays an important factor in determining the adhesion between the polymers and CNT, with the transition between oligomeric and polymer behavior in this study occurring at Nm ≈ 15 (dashed line). Polymer chains are therefore expected to be much more efficient at solubilizing CNTs then their oligomeric counterparts.11 We also learn from Figure 6(b) that oligomeric chains of all flexibilities as well as long flexible polymers adsorb in random orientations (α = π/2), whereas the longer stiff polymers adsorb in a conformation characterized by α < π/2. The random adsorbed orientation of the short stiff oligomers results from their relatively low entropic state (cf. a long polymer chain), allowing oligomers to readily fluctuate between adsorbed and desorbed states.
The prediction of the helical pitch obtained by solving eq 5 as a function of chain stiffness is shown by the dashed curve of Figure 7. The symbols show the calculated pitch angle from simulations, using the relation lp ≈ κ0.6.18 A better agreement is obtained when θ is corrected by 2° (solid curve). That is, the adsorbed chains have a somewhat smaller bending angle (they are more flexible) then predicted by the unconfined KP model. The probable cause is the interaction with the CNT surface, which competes with the bending interactions and thereby reduces the effective size of the chain. The transition to extended conformations (α < π/2) in Figure 7 occurs when chain stiffness, determined from the average bending angle, θ = cos−1(−e), approximately equals the minimum radius of curvature of the cylinder, θc = 2cos−1(l/2ρ). Polymers with lp beyond this critical value will adsorb in extended helical conformation to reduce the effect of surface curvature.
Our previous simulations indicate that semiflexible homopolymers under the conditions studied adsorb in a monolayer on the surface of the CNT, for the range of concentrations studied.8 Such conformation may not be ideal for polymer reinforcement, where strong adhesion between the polymer matrix and CNT is required. For such applications, BCPs consisting of monomers with different solvent selectivity may provide a stronger interface between the CNT and polymer.41 In such a scenario, one block is anchored to the CNT surface through strong physical association, while the other block extends to solution. This configuration not only provides steric repulsion between CNTs but also increases adhesion between the CNT–polymer interface27 by allowing for entanglement between the adsorbed polymers and matrix polymers.
We studied the adsorption and wrapping mechanism of diblock and triBCPs on CNTs consisting of hydrophobic (H) blocks that favorably interact with the CNT (εH = 1) and ‘polar’ (P) blocks that interact with the CNT through weakly attractive interactions (εP = 0.01). Such a choice of interactions represents adsorption in a hydrophobic solvent, and results in strong microphase separation of the hydrophobic and polar segments of the polymer in absence of the CNT.42 In general, adsorbed BCPs show significant loss of order when compared with homopolymers under the same conditions, though helical conformations of the H blocks may also be observed (Fig. 8). However, at the relatively short simulation times considered, helical conformations appear to occur fortuitously if H segments near the interface between the two blocks adsorb first, allowing for sequential wrapping of the other H segments. Thus, a mixture of helical and ordered configurations might be expected, as has also been reported experimentally for alternating copolymers.43 Similar to the homopolymer, the randomly adsorbed state of the copolymer is likely to be a metastable state. However, for the copolymer, rearrangement to the helical state requires significantly longer (simulation) times due to the competing interactions of the polar block.
Whether adsorbed in a random or ordered conformation, BCPs are much less mobile than the homopolymers since a wrapping spiral movement on the surface of the CNT, observed with the hompolymers (Fig. 4) is not favored energetically. Such movement would necessitate an energetically less favored contact between the CNT and P segments. Nonetheless, the possibility of formation of loops with the triBCPs, shown in Figure 8(b), might offer particular advantage toward interface reinforcement, by addressing both adhesive and cohesive failure in polymer/CNT composites.
For successful dispersion of CNTs, one clearly needs strong binding between the polymer and the CNT that can overcome strong vdW interaction between the tubes themselves. The sequential wrapping mechanism revealed by our simulations suggest that the separation of CNT bundles may occur through a “zipping” mechanism,44 where penetration of polymer segments into the spaces between CNTs as they wrap around the CNT surface leads to debundling of CNT aggregates. In addition, the spiral path characterizing the movement of the helically adsorbed polymer on the surface of the CNT might also contribute to such a mechanism.
The equilibrium adsorbed conformation of the homopolymer has an optimal helical pitch, determined by chain stiffness, CNT radius, and probably polymer–CNT interactions. BCPs, on the other hand, adsorb in more random wrapping conformations, with the solvent-selective block extending to solution. The competing interactions of the different blocks result in lower mobility of BCPs on the nanotube surface and a lower ability to rearrange into ordered conformations. Nonetheless, such configurations exhibiting tails and loops may allow for entanglement and improved interfacial adhesion in polymer–CNT composites.