Continuum percolation of carbon nanotubes in polymeric and colloidal media

Abstract

We apply continuum connectedness percolation theory to realistic carbon nanotube systems and predict how bending flexibility, length polydispersity, and attractive interactions between them influence the percolation threshold, demonstrating that it can be used as a predictive tool for designing nanotube-based composite materials. We argue that the host matrix in which the nanotubes are dispersed controls this threshold through the interactions it induces between them during processing and through the degree of connectedness that must be set by the tunneling distance of electrons, at least in the context of conductivity percolation. This provides routes to manipulate the percolation threshold and the level of conductivity in the final product. We find that the percolation threshold of carbon nanotubes is very sensitive to the degree of connectedness, to the presence of small quantities of longer rods, and to very weak attractive interactions between them. Bending flexibility or tortuosity, on the other hand, has only a fairly weak impact on the percolation threshold.

The physics of networks of electrically conducting carbon nanotubes (CNTs) in polymeric and in colloidal media has attracted much attention lately because of potential technological applications of CNT/polymer nanocomposites (1–5). The control over the formation of networks of such in some sense connected CNTs during the processing of the nanocomposite, which may include solvent-based as well as solvent-free but fluid (melt) stages, plays a key role in producing unique mechanical, thermal, and electrical properties of the final solid composite material (6). Of particular interest here is the so-called percolation threshold (PT), i.e., the minimal loading of CNTs required to form a network or cluster that spans the whole system (the “percolation network”) (7, 8). This critical loading as well as the physical properties of the CNT/polymer composite at concentrations beyond the PT depend on the polymeric matrix material and on the microscopic structure of the percolating network, the formation of which is influenced by the matrix material. Indeed, the polymer matrix has a strong impact on the interactions between the CNTs during processing but also on the conductive properties of the network after that because the CNTs turn out not to actually touch each other in the final product (1, 6). It stands to reason that a rational design of CNT/polymer nanocomposites should benefit from a deeper theoretical understanding of the combined influence of the polymer matrix and the detailed characteristics of the CNTs on the formation of a percolating network, one that arguably remains quite incomplete notwithstanding many decades of intensive research (1).

From a technological point of view, the central issue is to produce a nanocomposite with a controllable conductivity and as low as possible a loading of CNTs (6). Many factors, including the mean aspect ratio, the length polydispersity, the bending flexibility and tortuosity of the CNTs, as well as interactions between them and the polymer matrix, potentially affect the PT that has to be achieved in the liquid state before the structure freezes upon solidification of the polymeric host material. As is well known, the PT of conductive filler particles decreases in inverse proportion to their aspect ratio, a rule of thumb that is generally valid for elongated and flat particles alike (9, 10). Recent experiments on the percolation of single-walled carbon nanotubes (SWNTs) in aqueous dispersions (11, 12) have also shown that a remarkable lowering of the PT can be achieved by making use of weak attractive interactions between the SWNTs (11).

As already advertised, a comprehensive picture of the influence of the material parameters on the PT of CNT dispersions and composites remains elusive. In an attempt to come to a more realistic description, we focus attention on the effects of nonideality characteristic of real CNTs and apply the continuum percolation theory of physical clusters to elongated particles in an effective medium. The effective medium describes the presence of the host material, e.g., a polymer melt or an aqueous dispersion containing depletants such as surfactant micelles or polymer latex particles (11, 12).

By making use of a formal correspondence between continuum percolation theory and statistical theories of the structure of fluids, which for rod-like particles have reached a high level of sophistication (13–19), we are able to quite straightforwardly predict the effects of different kinds of nonideality on the PT of CNTs, including hard-core interactions, flexibility, length polydispersity, and (in principle) any medium-induced attractive interaction or “stickiness.” It is important to point out that the strength of the attractive interaction between the CNTs can be tuned, e.g., by the choice of host material, allowing one to manipulate the percolation of CNTs. The formalism that applies here provides a way to obtain analytical predictions in a field where guiding principles are hard to come by.

Our calculations show that the PT of CNTs is quite sensitive to the cluster (or connectedness) definition, and we settle the issue of how precisely polydispersity impacts upon it (10, 20): we find that long rods contribute more to the PT than short ones do, and that a length polydispersity therefore lowers the PT. This nonintuitive result is due to translation–rotation coupling of the rods. Rod flexibility, on the other hand, raises the PT, albeit by only a modest amount. By contrast, even very weak attractive interactions, caused, e.g., by the presence of the matrix material (21), significantly lower the PT, although we cannot rule out the existence of a regime outside the range of validity of our approach, where attraction might do the opposite (22). We confirm earlier speculation that there are two different regimes with a different local structure depending on the strength of the attractive interactions (11, 23), but in addition we make plausible the existence of two additional regimes at higher interaction strengths where the percolating network is either built up from nanotube bundles or is kinetically frozen and no longer in equilibrium.

Finally, we make connection between geometrical and electrical percolation by considering the tunneling of charge carriers (electrons) between the CNTs. We argue that any sensible cluster definition must be related to the tunneling distance, which depends on a combination of the nanotube and matrix material properties. A quasiclassical estimate of this length predicts it to increase with the dielectric constant of the host medium, implying that by using a host medium with a higher dielectric constant, one effectively decreases the potential barrier between adjacent nanotubes, producing an increase of the local conductivity of the composite and, through an increase in the effective connectedness of the CNTs, a decrease of the PT. It seems that the concept of connectedness percolation is not merely of theoretical interest but actually represents a useful toolbox that can assist in selecting routes on how to decrease the PT and/or increase conductivity levels by appropriate choice of polymeric matrix that regulates the strength of the attractive interactions as well as the charge transport between the nanotubes in them.

Continuum percolation was studied first for fluids of ideal (fully penetrable) particles and particles interacting by a harshly repulsive potential (Fig. 1b), and extensive literature on these models is available (10, 20, 24, 25). A formal analysis due to Hill (26), later extended by Coniglio (27), that considers the distribution of physical clusters in an equilibrium system of interacting particles is based on the pair connectedness analogue of the well known Ornstein–Zernike (OZ) equation of the liquid state theory of isotropic fluids. For monodisperse, rigid rods in the isotropic phase this equation reads (20, 27, 28)

where ρ denotes the mean number density of the nanotubes, i ≡ (r_i, u_i) their generalized coordinates with r_i and u_i the positional and angular degrees of freedom of test rod i = 1, 2, C⁺(1, 2) the so-called direct pair connectedness function of two test rods with coordinates 1 and 2, and P(1, 2) the pair connectedness function that describes the structure of the clusters. C⁺ (1, 2) measures short-range correlations (27), while P(1, 2) is defined such that ρ²P(1, 2)d1d2/(4π)² represents the probability of simultaneously finding a rod in a generalized volume d1 at 1 and another one in d2 at 2, provided they are part of the same cluster. Whether two particles belong to the same cluster depends on how a cluster is defined. Usually, two neighboring particles are considered “bound” if their mutual separation is less than a certain value called the “connectedness” or “overlap” criterion.

Fig. 1.

Configuration of two test rods and the connectivity criterion. (a) The configuration of two test rods of length L and diameter D skewed at an angle γ; r ≡ |r − r′| denotes the distance between the centerlines of the rods, and u and u′, their orientations. (b and c) The interaction potential u⁺ between the particles that belong to the same cluster vs. the shortest separation r between the centerlines of ideal (fully penetrable) particles (ε′ = 0) and hard-core repulsive particles (ε′ = ∞) (b), and attractive hard (fully impenetrable) particles (c); ε and σ denote the strength and range of the rod–rod attraction and Δ the connectedness criterion. In our calculations the connectedness criterion is chosen to coincide with the range of the potential, so Δ = D + σ.

Within the second-virial approximation, we have for the direct connectedness function C⁺(1, 2) = f⁺(1, 2) (20, 27–29), where f⁺(1, 2) = exp[−βu⁺(1, 2)] is the Mayer function of the particles that belong to the same cluster and interact through the potential u⁺ and β⁻¹ = k_BT is the thermal energy with k_B Boltzmann's constant and T the absolute temperature. We let u⁺ → ∞ for forbidden configurations and for configurations that do not meet the overlap criterion (27). The second-virial approximation is exact in the limit of infinite aspect ratios of the nanotubes, at least if they interact through a short-ranged, harshly repulsive potential (18), and is equivalent to a random phase approximation (30). Hence, rod–rod correlations are accounted for to all orders in the density albeit only linear graphs in a diagrammatic expansion are retained (30). It should be considered qualitative for L/D ≳ 20 and quantitative for L/D ≳ 100 for rods of length L and diameter D (18).

With this closure, Eq. 1 can be solved for P(1, 2). This is most conveniently done in Fourier space, albeit due to translation–rotation coupling this is still no trivial matter (13, 15, 17, 19). Fortunately, we can make use of methodology developed in the context of liquid-state theory because there is a direct correspondence between (i) the pair connectedness function, P, and the total correlation function, h, (ii) the direct pair connectedness function, C⁺, and the direct correlation function, C, and (iii) the mean cluster size, S, and the structure factor at zero wavevector k, lim_k→0S(k) (27). In short, connectedness percolation theory can be mapped onto liquid-state theory by putting P → h, C⁺ → C, and S → lim_k→0S(k).

The structure factor of rod-like particles has been calculated for various kinds of situation (13, 15–17, 19, 30), implying that we can directly obtain the corresponding cluster sizes for those cases. It is useful to note that explicitly dealing with the two-body OZ equation can be avoided by making use of the static form of the fluctuation–dissipation theorem that connects two-body correlators to the response of the single-particle density distribution to an external field (28). This produces an integral equation that for isotropic fluids can be solved exactly in Fourier space to give S(k) = 1 + ρ〈〈h̃(k, u, u′)〉〉′ = 1/[1 − ρ〈〈C̃(k, u, u′)〉〉′]. Here, u and u′ are the unit vectors in the direction of the main axes of the rods, the brackets 〈 … 〉 ≡ (4π)⁻¹ ∫ du (…) denote an orientational averaging with a similar prescription for the primed variables, and the tildes indicate a spatial Fourier transform, () ≡ ∫ dr exp[−ik · r] (…).

The mean cluster size is related to the pair connectedness function through S = lim_k→0 [1 + ρ〈〈P̃(k, u, u′)〉〉′], with P̃ the spatial Fourier transform of P(1, 2) (27). So, by invoking the correspondence between liquid-state and connectedness percolation theory, we immediately find S = lim_k→0 1/[1 − ρ〈〈f̃⁺(k, u, u′)〉〉′] within the second-virial approximation. At the PT the cluster size diverges, S → ∞, allowing us to calculate the volume fraction φ_p at which this happens,

where f̃⁺ is the Fourier transform of the connectedness Mayer function and V_rod is the volume of a rod. Although superficially simple, Eq. 2 is actually a nontrivial result because of the effect of translation–rotation coupling on the long-range correlations between the rods.

Keeping only the leading-order term in the aspect ratio L/D ≫ 1, $V_{rod}=\frac{\pi }{4}L{D}^2+O(D^3)$, where again L denotes the length and D the width of the rods. For any known potential u⁺, we are now able to predict φ_p (within the second-virial approximation). Fig. 1b presents a simple potential that interpolates between ideal (noninteracting) and hard rods, giving ${\lim }_{k\to 0}\langle \langle {\tilde{f}}^{+}\rangle \rangle =\frac{\pi }{2}{L}^2[\Delta -D(1-\exp [-{\epsilon }^{\prime }])]+\mathrm{\dots }$ to leading order in L/Δ ≫ 1, where Δ ≥ D represents the maximum separation of two connected rods and ε′ ≥ 0 represents a dimensionless repulsive potential for configurations where the cores of the rods overlap when their centerline-to-centerline separation is less than D. The connectedness criterion given by the overlap distance Δ is arbitrary and can be defined on the basis of geometrical or physical arguments. We return to this issue below.

For ideal rods ε′ = 0, so φ_p = ½Δ/L in the long-rod limit L/Δ ≫ 1, whereas for hard rods ε′ → ∞, giving in the same limit φ_p = ½(D/L)(1/($\frac{\Delta }{D}$ − 1)). For ideal (penetrable) rods it is customary to put Δ ≡ D, producing the familiar expression φ_p = ½ D/L. These predictions agree with those obtained from excluded-volume considerations (10, 20, 31). The scaling φ_p ∝ D/L for L/D ≫ 1 is in accord with computer simulations on both penetrable and hard rods (32, 33), and with experiments on multiwalled carbon nanotubes (MWNTs) (34), whereas the scaling φ_p ∝ ($\frac{\Delta }{D}$) − 1)⁻¹ seems to be consistent with very recent Monte Carlo simulations (T. Schilling and M. A. Miller, personal communication). The aspect ratio of SWNTs is typically L/D ≈ 10³, and the very low theoretical PT of order 10⁻³ is indeed seen experimentally (1, 12, 35). It is worth mentioning that some computer simulations of the percolation of ideal and also hard rods give numerical prefactors slightly larger than the classical value of 1/2, namely 0.5764 (36) and 0.6 (37). The reason for this discrepancy is not quite clear, but may perhaps be attributed to the finite size of the simulation box (36, 38). The connectedness analogue of the reference interaction site model (RISM) (25), which preaverages angular correlations (25, 28, 32), overestimates this numerical prefactor by ≈20% in the slender-rod limit (38).

In reality, SWNTs are not monodisperse, nor are they infinitely rigid or even straight. By again exploiting the relation between connectedness percolation theory and the statistical theory of rod fluids (13, 15, 19), we can directly establish how length polydispersity and bending flexibility modify the results obtained for hard rods. The structure factor of polydisperse hard rods was calculated by one of us with the help of the fluctuation–dissipation theorem (15), and from that we obtain in a similar manner as sketched above for the monodisperse case a PT that obeys

where 〈L〉_w ≡ ∫ dLL²P(L)/∫ dLLP(L) is the weight average of the nanotube lengths given their distribution function P(L), again a result of translation–rotation coupling. Eq. 3 settles an earlier discussion on the type of average to be taken, i.e., number versus weight average (10, 15, 20). This remarkably simple result can in principle also be obtained by solving the multicomponent OZ equations in the zero-wavevector limit, but even for the simplified case of a bidisperse system consisting of rods of two different lengths this proves to be not so straightforward (15).

We conclude that a polydispersity in the nanotube lengths lowers the PT at equal average length because longer rods are weighed more heavily than shorter ones. Indeed, if P(L) were, say, a Gaussian with mean L̄ and variance σ², then 〈L〉_w ∼ L̄ + (σ²/L̄) > L̄ for all 0 < σ² ≪ L̄². The effect is actually quite strong, as we show in Fig. 2 for the simplified case of a bidisperse system consisting of rods of equal diameter but different length. The figure shows that adding quite small quantities of longer CNTs can dramatically lower the PT.

Fig. 2.

Impact of the relative amounts of long and short rods on the percolation threshold of a bidisperse system of rods of equal diameter but different length L_short and L_long. Plotted is the PT φ_p(x) as a function of the mole fraction x of long rods for various length ratios L_long/L_short. The larger the length difference the larger the reduction of the PT even at very low loadings of the longer species. From top to bottom: L_long/L_short = 2, 4, 8, and 16.

To investigate the influence of a finite bending flexibility on the PT, we make use of the statistical theory for fluids of slightly flexible hard rods (30), worked out in detail for rods in the nematic liquid-crystalline phase (19). Within the worm-like chain model near the rod limit, corrections because of a finite bending rigidity enter the description by renormalizing the direct correlation function C. We find that to first order in L/p ≪ 1, where p is the persistence length of the rods, only that part of the direct correlation function that describes the hard-core interactions is affected by bending fluctuations, at least in the relevant limit k → 0. The Mayer function Boltzmann averaged over weak bending fluctuations becomes slightly smaller than that of rigid rods by a relative amount of order L/p ≪ 1. Applying the same recipe as before then gives φ_p = ½ D/L($\frac{\Delta }{D}$ − 1)⁻¹ (1 + O(L/p)), i.e., the PT goes up for slightly flexible impenetrable rods. This finding one would expect because bent rods have a smaller effective aspect ratio than straight ones, and it seems to be in agreement with recent predictions of computer simulations done by Dalmas et al. (33), although in those simulations the bent (or “tortuous”) configurations were frozen (quenched) and not subject to annealing (equilibration). Formally, one has to distinguish between annealed and quenched tortuosity.

The question arises what happens in the other limit, when L ≫ p and the cylinders become so flexible that they fold back on themselves. If p/D ≫ 1, they are locally rigid and may be considered “semiflexible.” Once more making use of the formal correspondence of the statistical theory of fluids and percolation theory (20, 27), we deduce that for semiflexible chains, i.e., in the limits L/p ≫ 1 and p/Δ ≫ 1, the PT must obey φ_p ≈ ½ D/L(1/($\frac{\Delta }{D}$ − 1)) and be comparable to the rigid-rod result. The reason is that within the second-virial approximation the excess free energy of self-avoiding rods is approximately the same as that of semiflexible chains (30, 39). It appears that the approximate identity φ_p ≈ ½ D/L($\frac{\Delta }{D}$ − 1)⁻¹ should be valid for all values of L/p, provided L/Δ ≫ 1 and p/Δ ≫ 1. RISM calculations seem to (approximately) bear this out (40). This conclusion is also corroborated by computer simulations of tortuous rods (33, 41, 42).

We now turn to the case where perfectly rigid cylinders not only interact sterically but in addition attract each other mutually. This attraction or stickiness may be due to, e.g., van der Waals and structural interactions (in the melt) and/or depletion interactions (in solution) (21). The functional form of the attractive interaction is likely to be algebraic (for van der Waals and depletion interactions) or exponential (for hydrophobic interactions), but more complex medium-induced potentials may also arise (21). Fortunately, the spatial integrals of the Mayer function we consider here turn out quite insensitive to the precise form of the interaction potential (13), so a simplified potential will do. Here, we focus on the square well-type of “sticky” potential analyzed by one of us in a different context (13). For sticky hard rods, we presume the connectedness potential u⁺ between two rods to consist of the hard-core repulsive (u⁺ → ∞) and a short-range attractive part (u⁺ < 0). We suppose that two neighboring rods are part of the same cluster if their shortest centerline-to-centerline distance does not exceed Δ = D + σ, where σ is the range of the attractive potential, to reduce the number of free parameters in the theory to an absolute minimum. As in the case of hard rods u⁺ → ∞ if the rods overlap or if they are not connected. See also Fig. 1c.

For separations in the range D ≤ r ≤ Δ, we choose for the sticky potential one proportional to the overlap area. This gives βu⁺ = −((Δ/D) − 1) ε|sin γ|⁻¹ for skewed configurations with γ > (D/L)$\sqrt{(\Delta /D)-1}$, where γ is the angle between the rods and ε is the dimensionless strength of the attraction, and βu⁺ = −((Δ/D) − 1)ε(L/D)(1 − |z/L|) for parallel rod configurations with γ < (D/L) $\sqrt{(\Delta /D)-1}$, where z is the distance between the centers of mass along the centerlines of the rods (13, 14). In these expressions, the factor $\sqrt{(\Delta /D)-1}$ corrects for curvature of the rods. For a detailed discussion of this and another, more realistic, potential, see refs. 13 and 14.

Inserting this potential in Eq. 2 gives for the PT of the sticky nanotubes an expression of the following form

in the limit of long aspect ratios L/D ⩾ L/Δ ≫ 1, with

a term stemming from parallel-rod configurations and

from skewed ones. Eqs. 4–6 generalize an earlier result of Poulin and coworkers (11) deduced from excluded-volume considerations for the specific case Δ = 2D, apart from an erroneous numerical prefactor that we have now corrected. Note that we presume the second-virial approximation to hold, an presumption that for large enough values of the stickiness parameter ε breaks down (13). (See also the discussion below.)

The dependence of the PT of nanotubes on their stickiness is shown in Fig. 3a for the realistic case of L/D = 1,000 (11), and for different connectedness criteria. From Fig. 3a we see that the critical volume fraction decreases with increasing values of Δ, as one would expect. The dependence on Δ is strong, in fact more so for sticky rods than for hard ones, a conclusion that has far-reaching consequences because it is a parameter that can in principle be tuned experimentally (see below). This finding also implies that the value of the PT must depend on the way it is measured, because different physical quantities probed to establish the PT are sensitive to particle separations. Finally, we see that attractive interactions between nanotubes can lower the PT quite substantially. With an increase in stickiness of the nanotubes, the PT first decreases linearly with ε to suddenly drop exponentially for stronger interactions. Hence, by optimizing the stickiness of the rods by an appropriate choice of polymer matrix, a significant reduction of the PT may be achieved.

Fig. 3.

Predictions for the PT of sticky rods. (a) The critical volume fraction (PT) of the nanotubes, φ_p, as a function of the stickiness energy of the nanotubes, ε, for various connectedness criteria, Δ, measured in particle widths D. The dot-dashed lines indicate the onset of the exponential regime given by Eq. 5 and the breakdown of the second-virial approximation for each case. The aspect ratio L/D of the rods is fixed at 10³. (b) Three fits to the experimental data points (indicated by circles) of Poulin and coworkers (11) according to Eq. 4 with two fitting parameters, L/D and Δ/D, presuming the attraction to be due to depletion interactions caused by the presence of surfactant micelles. From top to bottom: L/D = 1,200, Δ/D = 1.19; L/D = 1,000, Δ/D = 1.25; and L/D = 800, Δ/D = 1.34.

The existence of two percolation regimes evident from Fig. 3 is a result of the predominance of entropically favored skewed rod configurations for weak interactions, i.e., when ε$\sqrt{\Delta -DL}/\sqrt{D^3}$ ≲1 and I_∥ ≪ I_⊥, and energetically favored parallel configurations for stronger ones, when ε$\sqrt{\Delta -DL}/\sqrt{D^3}$ ≫ 1 and I_∥ ≫ I_⊥. Because the direct correlation function C̃⁺ = f̃⁺ measures local correlations, it follows that the underlying percolation networks must for this reason have different structures depending on the interaction strength: weakly interacting nanotubes form locally isotropic networks (Fig. 4a), whereas more strongly interacting ones should produce locally anisotropic but globally isotropic networks (Fig. 4b), confirming earlier speculation (11, 23). Note that although angular correlations do become longer ranged with increasing stickiness, they do not actually diverge except for densities very much higher than the PT (13).

Fig. 4.

Percolation regimes and underlying percolation networks. (a) Equilibrium locally isotropic network of individual nanotubes. (b) Equilibrium locally anisotropic but globally isotropic network of individual nanotubes. (c) Equilibrium isotropic network of percolating bundles of nanotubes. (d) Nonequilibrium fractal network.

We are now in a position to compare our results with the experimental data of Poulin and coworkers (11). In their experiments, the nanotubes attract each other by means of depletion-induced attractive interactions, where spherical micelles of sodium dodecyl sulfate (SDS) act as depletants. The relation between the stickiness of the nanotubes, ε, and the depletant volume fraction, φ_m, can be established by using the dependence of the depletion potential u⁺ = −ΠV_overlap on the osmotic pressure, Π, and the overlap of the excluded volumes between rods and micelles, V_overlap, at close separations between the rods (21). To lowest order in the volume fraction of micelles, Π = β⁻¹φ_m/$\frac{\pi }{6}$ a³, with a their diameter. At contact, so for r = D, the depletion potential obeys ε ≈ $\frac{6}{\pi }$ φ_m ≈ 2φ_m with φ_m their volume fraction. Our model potential, u⁺, extends the contact value of the depletion potential from r = D to r = Δ > D, where we now use Δ as a fitting parameter, as in fact the aspect ratio L/D. In Fig. 3b, we present our comparison to the experimental results for various values of the fitting parameters L/D and Δ/D. We obtain good agreement, although three experimental data points are obviously not sufficient to accurately estimate the nanotube characteristics. More data points are sorely needed, especially in the limit of vanishing depletion interaction for φ_m → 0, because then we can fix the effective aspect ratio of the rods. The calculated fitting parameter Δ/D ≃ 1.2–1.4 is realistic but somewhat smaller than the parameter Δ/D = 1 + (a/D) = 2 implicitly taken by Poulin and coworkers (11). The reason is that for curved particles the depletion potential is not actually a square well, and a somewhat smaller value of Δ corrects for this.

A moot point that needs discussion is the validity of the second-virial approximation for sticky rods. At the level of the third-virial approximation, we have , where g⁺ ≡ f⁺ * f⁺ is a (spatial) convolution of two connectedness Mayer functions (20). The second-virial approximation breaks down when the dimensionless quantity equals unity, where we insert the second-virial prediction we have , showing that in that regime the second-virial approximation is indeed not valid unless Δ → D. Where precisely the second-virial approximation breaks down depends on ε, L, D, and Δ, and is indicated in Fig. 3a.

When τ ≳ 1 we expect the percolation network to be built up not from individual nanotubes but from nanotube bundles, which are known to form as a result of the strongly anisotropic nature of the attractive interactions between them (13, 43). These bundles (shown in Fig. 4c) have a smaller effective aspect ratio than the individual rods, so one would surmise that the PT then increases again with increasing (but still very weak) attraction. This would explain the simulation results of Grujicic et al. (22). With further increase of the stickiness, nonequilibrium aggregation and the formation of kinetically frozen structures is likely to occur (illustrated in Fig. 4d), not dissimilar to that found in dispersions of spherical particles such as carbon black (44). The PT of fractal aggregates is very much lower than that of the individual particles, suggesting that a significant lowering of the PT of CNTs should be possible under conditions of strong attractive interactions, i.e., out of equilibrium. It is not clear, however, whether these two regimes are experimentally accessible because strongly interacting CNTs are not easily dispersed in a liquid medium.

We end our analysis with a discussion of the connectedness criterion associated with the distance Δ, a phenomenological parameter in the theory. We put forward that when applying the theory to experimental situations, the choice of this parameter is dictated by other physical phenomena in the CNT/polymer composite, e.g., the way the PT is measured. The usual requirement for CNT/polymer composites is good electrical properties, although in general connectedness percolation does not necessary mean electrical percolation. To have good conductive properties the underlying percolation network has to be conductive but, as already alluded to, nanotubes in a composite do not appear touch each other (1, 45). Hence, the conductivity of the percolation network in the final solid product must be determined by the tunneling of the charge carriers (electrons) from one nanotube to another, and therefore by the typical some average distance of closest separation between the nanotubes in the composite (11). (See Fig. 5.) The corresponding local conductivity because of tunneling is σ = σ₀exp [−r/ξ], where ξ is the characteristic tunneling distance and σ₀ some constant (31, 46). One way to relate the connectedness percolation theory to electrical percolation is to choose the connectedness criterion Δ − D ≈ ξ, which would guarantee the underlying percolation network in the polymer composite to be conductive (42, 47).

Fig. 5.

The conductive percolation network in a nanocomposite and the picture of electron transport via quantum tunneling. The connectedness criterion has to be of the order of the tunneling length, so Δ − D ≈ ξ.

There is no universally valid theory of electron transport in CNTs/polymer composites (45). Indeed, it is a formidable quantum-mechanical problem that, in principle, should be solved from first-principles calculations. The tunneling distance will be determined by factors such as the structure of the energy levels of the electronic states in two adjacent nanotubes immersed in the polymer matrix, the contact potential barrier, and the electrostatic charging energies of the nanotube and the polymer (48). To get an order of magnitude estimate for ξ and its functional dependence on a single material parameter of the polymer matrix, the dielectric constant, we ignore all this and consider a scaling law based on the time-independent Schrödinger equation, telling us that ξ ≈ /$\sqrt{2m_e(E_e-E_f)}$ (48). Here, h̄ is the reduced Planck constant, m_e is the electron mass, and E_f and E_e are the energies of a conduction electron on a tube and in the matrix. E_e − E_f represents our estimate for the energy barrier between two nanotubes.

We insert for E_f the estimated Fermi energy of an electron on a bare SWNT, E_f ≈ −3 eV (48). For E_e, we presume the Born energy of an electron in a dielectric matrix to be the relevant energy scale, implying E_e ≈ k_BTl_B/2Λ with l_B the Bjerrum length and Λ the thermal wavelength of the spatially delocalized electron in the matrix (21). The Bjerrum length measures the distance at which the Coulomb energy of two unit charges equals the thermal energy and scales with the inverse of the dielectric constant of the polymer matrix. We deduce from this functional dependence that in a more polarizable medium, i.e., with a higher dielectric constant, the tunneling distance should be larger and therefore the percolation threshold lower. Of course, the dependence of ξ on the dielectric constant cannot be very strong because |E_f| ≫ |E_e|, but because the PT of sticky rods is very sensitive to the value of Δ, the impact of the polarity of the medium could still be appreciable. For a typical polymer matrix the relative dielectric constant is about 3, giving ξ ≈ 0.1 nm, consistent with estimates suggesting that the typical distance between two nanotubes should to be less than a nanometer or so to get reasonable conductance (31, 35, 49). This distance suggests that for SWNTs in a polymer matrix Δ/D ≈ 1.1.

In conclusion, we put forward that connectedness percolation theory can be a useful predictive tool in the rational design of conductive, nanotube-based composite materials. Indeed, according to our calculations, the choice of host material influences the interaction range and strength as well as the degree of connectedness of the nanotubes during the fluid stages of the processing, opening up ways to manipulate the percolation threshold of the composite. We have been able to calculate the functional dependence of the PT in realistic systems on nonidealities such as flexibility, length polydispersity, and medium-induced attractive interactions. Increasing the attraction between the nanotubes or the length polydispersity can significantly lower the percolation threshold, and a finite bending flexibility or tortuosity should increase it, albeit only weakly so. This finding implies that the nanotubes need not be perfectly straight or monodisperse to be useful as a conductive filler. It appears that in a host medium with higher dielectric constant the tunneling distance should be larger, resulting in a lower PT and a higher local conductivity of the composite, suggesting that one has some control over the level of conductivity in the final CNT/polymer composite. Finally, our predictions are based on the tacit assumption that equilibrium is reached during processing, which need not be the case. At present, very little seems to be known about the time scales required to reach equilibrium, an interesting and important topic in its own right.