Predictive modeling of conductivity for carbon black nanocomposites: influence of filler features, interfacial effects and network portion

Abstract

The conductivity of carbon black samples depends on more than one or two obvious variables; it arises from a complex interaction of structural and surface factors acting in concert. Considering these interdependencies in a model is essential for designing materials that achieve reliable performance. In this study, we developed a theoretical framework that integrates the filler geometry, interphase thickness, tunneling resistance, tunneling distance, tunneling diameter, percolation threshold, and intrinsic conductivity of the filler. Rather than considering these parameters in isolation, the model allows them to interact, and its predictions are validated using experimental datasets. When the carbon black radius drops to 10 nm and the networked fraction of nanoparticles is 1, the conductivity of the composite increases to 1 S/m. As the particle radius increases or the fraction of networks decreases, the conductivity of the system decreases sharply and, in some cases, returns to the insulating state. The results further show that the composite conductivity is strongly governed by the polymer tunneling resistivity (p) and the interphase conductivity (σ_int). The lowest σ_int (< 200 S/m) produces an insulated system, while the composite conductivity maximizes to 3 S/m at p = 10 Ω.m and σ_int = 8000 S/m. Accordingly, more interphase conduction and less polymer tunnel resistivity are desirable to improve the conductivity of samples.

Introduction

Carbon black (CB) composes of quasi-spherical particles, which rarely remain isolated; instead, they tend to aggregate and ultimately form conductive networks^1–5. Owing to its large surface area and the disordered character of its graphitic domains, CB functions effectively as a conductive filler within otherwise insulating polymers. The resulting composites can exhibit an enormous range of conductivities, sometimes differing by several orders of magnitude, depending not only on the amount of filler incorporated but also on subtle structural and interfacial characteristics that are often overlooked⁶. Although this study primarily focuses on spherical carbon black particles and their tunneling behavior, it is important to note that non-spherical particles, such as polygonal, oval, or irregularly shaped fillers, are commonly encountered in practical carbon black/polymer nanocomposite systems. Particle shape can influence both the effective tunneling distance and the percolation threshold by modifying the geometry of conductive networks and the spatial arrangement of neighboring fillers. From a physical perspective, non-spherical particles may introduce alternative tunneling paths and altered network connectivity compared to idealized spherical fillers. Such effects have been discussed in recent studies on percolation and transport in particulate composites⁷. A more explicit treatment of particle shape effects is beyond the scope of the present model and is left for future investigation. Particle size, for example, exerts a decisive influence on nanocomposite conductivity. Smaller or thinner particles present a larger surface area and collide more readily, which facilitates network formation at lower filler contents^8–11. It should be noted that, according to classical continuum percolation theory, the percolation threshold of an ideal mono-sized system is independent of the absolute particle size and is governed purely by geometrical connectivity. This conclusion, however, applies only to highly idealized systems that neglect interphase regions, tunneling transport, particle interactions, and size or shape polydispersity. In practical polymer carbon black nanocomposites, particle size influences the effective percolation threshold indirectly by modifying interparticle distances, excluded volume, interphase overlap, and tunneling probability. In such systems, the size effect arises through its coupling with additional characteristic length scales, such as interphase thickness and tunneling distance, rather than through a purely geometric scaling. Recent quantitative studies have demonstrated that the combined effects of size polydispersity and shape anisotropy can significantly shift the percolation threshold even at fixed filler volume fractions, highlighting the limitations of purely geometrical percolation models when applied to real nanocomposite systems. In practical terms, this reduces the percolating threshold, which is the point at which the material undergoes a sudden transition from an insulating to a conductive state¹². Previous studies have shown that particle morphology can significantly influence the percolation behavior of granular and particulate composites^8,13. In particular, Lin et al.^14,15 systematically investigated the effects of particle shape and size on percolation by comparing spherical, polyhedral, and ovoidal fillers, demonstrating that shape anisotropy and interfacial geometry can shift the percolation threshold and alter transport pathways in granular systems. These findings highlighted that particle shape may play an important role in percolation-controlled properties, although the present study focuses on spherical carbon black particles to enable a clear and tractable formulation.

In most CB nanocomposites, electron tunneling constitutes the dominant conduction mechanism^8,13. Electrons traverse the narrow polymer gaps separating adjacent particles, but the tunneling probability decays exponentially with distance. Even a fractional increase of a nanometer can reduce conductivity by orders of magnitude, and the polymer’s barrier height further compounds this effect^16,17. These features explain why minute differences in interparticle spacing or interphase chemistry can produce dramatic shifts in bulk conductivity. Wang et al.¹⁸ employed microstructural modeling and numerical simulation (DEM/FVM) to demonstrate that CB aggregates can be reconstructed in three dimensions and their conductivity estimated with reasonable accuracy. Their framework significantly reduced computational cost and even enabled a form of “virtual screening” of filler fractions to identify the levels required to achieve a target conductivity. One year later, Albright and Hobeck¹⁹ adopted a slightly different approach, developing a stochastic, data-driven model to predict electromechanical behavior. This method was anchored in microscopy observations and experimental data, which were used to construct realistic representative volume elements (RVEs). Remarkably, their predictions were consistent with both uniaxial tensile experiments and established analytical models, suggesting that their approach offers a practical route for forecasting microstructural evolution under load, although, as with any data-driven approach, its accuracy ultimately depends on the quality of the calibration dataset. On the theoretical side, Paredes-Madrid et al.²⁰ revisited the Simmons tunneling formalism, which remains the principal framework for describing interparticle tunneling resistance. This formalism expresses resistance as a function of barrier height, tunneling distance, and decay length and has since become the foundation of many conductivity models. Incorporating this tunneling picture into modern microstructural simulations reveals that parameters such as interphase thickness and tunneling distance do not merely adjust minor details but directly determine the macroscopic conductivity of CB/polymer systems. Moronkeji et al.¹⁷, using conductive atomic force microscopy on CB/PDMS composites, observed that the electrons initially flow through tunneling pathways, which gradually give way to more Ohmic-like conduction as strain is applied. Similarly, a three-dimensional conductive AFM study on CB/epoxy–amine composites²¹ revealed a combination of tunneling with local bottlenecks apparently dictating the overall network resistance. Simulations reported by Casanova²², based on Simmons’ tunneling formalism, suggested that an interparticle distance of approximately 3 nm is required for effective electron hopping, although real composites rarely maintain such uniform spacing. In a related study, experiments on CB-filled carbon gels identified a tunneling–percolation regime with thresholds between 8 and 12 wt%, reinforcing the conclusion that both network geometry and tunneling mechanisms jointly determine macroscopic conductivity.

Collectively, these findings indicated that subtle variations in interparticle spacing and interphase properties can shift the balance between tunneling-dominated transport and direct conduction in unpredictable ways. The primary contribution of the present study is the development of a unified conductivity model that integrates interphase transport, tunneling geometry, and surface-energy effects into a single predictive framework. Existing models typically address only one of these mechanisms, percolation, tunneling, or interphase phenomena, but rarely all three simultaneously. In the present work, all three effects were explicitly incorporated.

The current study introduces a unified, physics-informed model for the conductivity of polymer/carbon black composites that explicitly integrates filler geometry, interfacial transport, tunneling behavior, and percolation phenomena within a single predictive framework. Unlike existing models that treat percolation, tunneling, or interphase effects in isolation, the proposed approach couples all three mechanisms and preserves dimensional consistency while maintaining a clear physical interpretation. The framework is not merely curve-fitted but validated across four chemically and morphologically distinct polymer/CB systems, demonstrating its ability to generalize beyond a single dataset. By incorporating parameters such as tunneling distance, tunneling diameter, interphase thickness, and surface tensions, the model produces design maps that translate abstract equations into actionable levers for materials engineers. This combination of mechanistic breadth, cross-system validation, and practical guidance represents a significant step beyond descriptive models, moving toward a genuinely predictive tool for optimizing conductive polymer nanocomposites.

Theory

Dang and Zhang²³ proposed a model linking percolation behavior to electron transport in polymer-based nanocomposites. The general structure of Eq. (1) follows the percolation-based formulation reported in²³; however, the present form is adapted and extended to ensure compatibility with the subsequent incorporation of interphase and tunneling contributions for CB-filled samples. Their approach successfully captured the sharp increase in conductivity near the percolation threshold. The general form of their equation can be expressed as:

1

σ _c denotes the effective electrical conductivity of the nanocomposite, σ_matrix represents the intrinsic conductivity of the polymer matrix, σ_eff is the effective conductivity of the filler network, ϕ_eff is effective fraction of nanoparticles and ξ is the fraction of percolated particles. The parameter η accounts for the waviness of the conductive pathways, while H (η, p) represents the geometric influence of filler morphology. All parameters are defined consistently throughout the model to preserve dimensional correctness and physical interpretation. The baseline interphase model estimates the longitudinal conductivity through a representative matrix–interphase stack as described by²⁴:

2

Here, d denotes the diameter of the carbon black particles, and is defined as d = 2R (R is filler radius). The parameter σ_z represents the baseline longitudinal conductivity of the matrix–interphase pathway prior to the incorporation of tunneling contributions. The interphase thickness is denoted by t, σ_f corresponds to the intrinsic conductivity of the filler network, and σ_int represents the interphase conductivity. While this formulation accounts for the geometric weighting of matrix and interphase conduction pathways, it does not explicitly include electron tunneling effects. In particular, the tunneling distance (δ_a) and the tunneling diameter (D_t) were not considered at this stage. The tunneling diameter D_t represents the characteristic cross-sectional area available for electron tunneling between adjacent aggregates and should be distinguished from the geometric contact diameter, which corresponds to direct physical contact. Consequently, the baseline σ_z tends to under-predict conductivity at low to moderate loadings and cannot reproduce the sharp increases in conductivity associated with barrier narrowing. To address this limitation, the pathway is augmented with an explicit tunneling branch characterized by a tunneling conductance across a gap of _a and D_t. The tunneling conductivity can be written as:

3

where p is polymer tunneling resistivity. Geometrically, the tunneling contribution is weighted by the cross-sectional area of the junction, while the series pathway length increases with the additional gap. By integrating the tunneling distance (δ_a) and the effective tunneling diameter (D_t) into the transport model, Eq. (2) is generalized to a more comprehensive form as:

4

Here, the first term represents direct filler network transport scaled by , the second term accounts for ohmic transport through the interphase, and the third term captures barrier-limited tunneling with explicit dependence on D_t. The enlarged denominator reflects the longer effective path length introduced by the tunneling gap. This modification preserves the original geometric logic while restoring sensitivity to tunneling geometry, thereby enabling the model to reproduce both the early percolation-like rise and the pronounced dependence of conductivity on nanoscale gap closure^25,26. An illustrative schematic is provided in Fig. 1, highlighting the roles of tunneling distance and tunneling diameter in comparison with the classical model.

Fig. 1.

Conductive network formation via interphase and tunneling in CB-filled composites.

Filler loading must be expressed as a volume fraction when used in the conductivity model. If the experimental input is given as a weight fraction (or wt%), it should be converted to the volume fraction using relation (5):

5

Here, w denotes the weight fraction of carbon black in the composite. p_f and p_p are the densities of the filler and polymer, in that order. Surface tension exerts a strong influence on the conductivity of samples by governing wetting, dispersion, and interfacial adhesion.

γ _p represents the polymer surface tension, γ_f ​ is the CB surface tension, and γ_pf is the polymer-filler interfacial tension. A common approximation for the interfacial tension is given by²⁷:

6

The percolation volume ​) denotes the critical filler volume fraction at which a continuous conductive network is formed, which can be expressed as:

7

This expression was constructed to reflect the primary factors that govern the percolation threshold, namely particle size, surface energies, and interphase thickness. Although its form appears deceptively simple, it arises from a combination of scaling arguments and empirical observations reported in the literature. In general, the percolation threshold increases as the particle radius grows or as the polymer surface energy rises, whereas it decreases when the interphase becomes thicker or when the polymer and filler are more compatible (captured here through ​)²⁸. The small numerical pre factor (0.001) and the algebraic structure were not derived from first principles; instead, they were tuned to align the predictions with percolation values measured for spherical carbon black systems, which will be evaluated in the next section. Thus, although the equation resembles a physical law, it should be regarded as a semi-empirical model whose accuracy depends on how closely the system under study matches the experimented cases.

The effective filler volume fraction accounting for the interphase is given⁸ by:

8

Furthermore, the percentage of percolated nanoparticles, denoted as ξ was determined using the following equation²⁸:

9

When the effects of tunneling distance, tunneling diameter, filler concentration, and networked nanoparticle fraction are incorporated, the proposed model for the electrical conductivity of CB–polymer nanocomposites can be expressed as Eq. (10):

10

It should be noted that, in the present formulation, carbon black particles are assumed to be spherical. Accordingly, Eq. (10) is derived under the spherical CB assumption, and particle shape is not treated as an independent morphological variable. Explicit shape anisotropy effects associated with non-spherical fillers are beyond the scope of the current model and would require modified geometric descriptors. At very high filler loadings, additional terms such as particle agglomeration, contact saturation, and excluded-volume effects may influence conductivity and are not explicitly considered in the present model. The tunneling distance and tunneling diameter were introduced as effective parameters and defined within the framework of the proposed model. The interphase conductivity was also treated as an effective parameter that may depend on the processing conditions.

Results and discussion

Assessing the equations by real observations

The validity of the proposed model (Eq. 10) is examined using experimental data from four different nanocomposite systems: poly(vinylidene fluoride)/CB (PVDF/CB)²⁹, thermoplastic elastomers)/CB (TPE/CB)³⁰, ultra-high-molecular-weight polyethylene/CB (UHMWPE/CB)³¹, and soy protein isolate/CB (SPI/CB)³². In each case, the measured conductivity–filler trends are compared with the model predictions to assess the degree of alignment and, equally importantly, to evaluate the role of interphase and tunneling parameters. Table 1 lists the parameters used in the calculations. Figure 2(a–d) shows the comparisons among the predicted and experimented conductivities of samples. All predictions well follow the experimental data verifying the developed model.

Table 1.

Model inputs and percolation threshold for carbon black–filled polymer nanocomposites.

Samples	R (nm)	t (nm)	(mN/m)	Exp.	Equation 7	a (nm)	D t (nm)	p (Ω.m)	(S/m)
PVDF/CB29	10	10.1	34	0.0134	0.0134	2	40	100	8000
TPE/CB30	25	4.9	33	0.0600	0.0600	20	5	70	750
UHMWPE/CB31	10	13	28	0.0048	0.0048	0.5	40	20	4000
SPI/CB32	7.5	7.9	30	0.0076	0.0076	22	24	50	800

Fig. 2.

Experimental conductivity data and model predictions for: (a) PVDF²⁹, (b) TPE³⁰, (c) UHMWPE³¹, and (d) SPI³² CB systems.

In the PVDF/CB system, the measured percolation threshold (0.0134) is reproduced exactly by Eq. (7), a level of agreement that is not always guaranteed in such models. The interphase thickness of 10.1 nm falls comfortably within nanoscale values reported elsewhere, which lends credibility to the parameter selection. Once the threshold was crossed, the conductivity increases sharply. The model captures this rise well, suggesting that the combination of a modest interphase thickness and a short tunneling distance is sufficient to support efficient electron transport once a percolated fraction formed. The polymer tunnel resistivity (100 Ω.m), interphase conductivity (8000 S/m), and tunneling diameter (40 nm) also appear reasonable, reinforcing the developed model. TPE/CB system displays a very similar behavior. Its percolation threshold of approximately 0.06 shows good agreement with the model predictions. The interphase thickness (4.9 nm) and tunneling dimensions are also at nano-scale.

The UHMWPE/CB composite shows the percolation threshold of 0.0048 again matches the model prediction. The large interphase thickness (13 nm), a tunneling distance of 0.5 nm, p = 20 Ω.m and a tunneling diameter of 40 nm provide the measured conductivity of about 10 S/m at ϕ_f = 0.1, which is acceptable. SPI/CB sample also has the acceptable data for interphase aspects, percolation onset and tunneling features which further validate the current equations. Even so, the agreement between measured and predicted values reinforces the ability of the developed model to handle systems with markedly different interfacial and morphological characteristics. Taken together, these comparisons indicate that the model is not only consistent with experimental data but also sensitive to structural details often overlooked, including interphase thickness, tunneling distance and tunneling diameter. The interphase thickness values fall neatly within the expected nanoscale range, which lends further confidence to the parameterization.

Effect of factors on the conductivity

Equation (10) is employed to evaluate the effects of various factors on the nanocomposite conductivity. The following average are: t = 9 nm, R = 20 nm, δ_a = 5 nm, D_t = 22 nm, g/cm³³³, ϕ_f = 0.04, σ_f = 10⁴ S/m^18,33, σ_int = 3000 S/m, γ_p = 34 mN/m, p = 400 Ω.m and γ_f = 55 mN/m³⁴.

The interplay between nanoparticle radius (R) and the percolated particle fraction (ξ) in determining nanocomposite conductivity is shown in Fig. 3(a, b). In the three-dimensional surface plot, conductivity increases sharply when particles are small and ξ is high. At R = 10 nm and ξ = 1, the conductivity can reach 1 S/m. Smaller nanoparticles and higher network portion produce a higher conductivity in the samples. It should be noted that, while smaller carbon black aggregates increase the specific surface area and facilitate network formation, excessively small particles may also promote the agglomeration. The present model captures the favorable geometric and tunneling-related effects of particle size but does not explicitly account for agglomeration phenomena.

Fig. 3.

Conductivity influenced by ξ and R: (a) 3D view and (b) contour plot.

The radius of CB aggregates plays a central role in determining the conductivity of nanocomposites, although its influence is sometimes underestimated. Small aggregates bring neighboring particles closer together, increasing the likelihood of electrons tunneling or hopping across polymer gaps. Because tunneling probability decays exponentially with distance, even a minute reduction in spacing can produce dramatic gains. Smaller aggregates also expose a larger surface area, naturally increasing the number of potential contact junctions. These two effects shorten the gaps and enhance the contact points simultaneously promoting the formation of conductive pathways. As the aggregate radius increases, the situation reverses. Fewer interparticle junctions become available, typical separations increase, and the composite struggles to establish a continuous network. At very large radii, even relatively high filler contents may fail to produce a conducting path because the particles are simply too far apart for tunneling to bridge the gaps effectively. In this sense, large radii impose an intrinsic limit on conductivity that other factors may not be able to overcome. In simple terms, small aggregates enhance conductivity by providing denser connectivity and shorter tunneling paths, while large aggregates suppress it by widening gaps and reducing contact opportunities. This makes CB size one of the most practical levers for improving the conductive performance of nanocomposites.

The percolated fraction of nanoparticles represents the proportion of filler particles actively connected within a system-spanning conductive network. Its effect on conductivity is central to the percolation theory of composite materials. When ξ is very low, conductive particles are too sparsely distributed, and their clusters remain isolated. In this regime, even when particles are small and favorable for tunneling, the absence of connectivity prevents long-range electron transport, and the material behaves as an insulator. As ξ increases, the probability of forming connected clusters rises. At a critical value, known as the percolation threshold, a dramatic change occurs: small, isolated clusters suddenly merge into a continuous, system-spanning network that allows electrons to travel throughout the composite. This threshold marks the transition from insulating to conductive behavior. In practice, conductivity can increase by several orders of magnitude with only a small rise in ξ once the system crosses this critical point. Beyond the percolation threshold, further increases in ξ continue to improve conductivity, but the effect becomes less pronounced. Once the majority of particles are part of the conductive backbone, adding more connections provides redundancy rather than creating entirely new pathways. At very high ξ values, conductivity approaches a saturation level where almost all possible paths are already established^35–37. Overall, the analysis of Fig. 3 is consistent with Eq. (10) and with experimental evidence, the largest increases in conductivity occur when particle size and network connectivity are optimized simultaneously.

The role of polymer and filler surface tensions ( and ) in the nanocomposite conductivity is illustrated in Fig. 4(a, b). The conductivity rises gradually as γ_p decreases and γ_f increases, reaching about 2.3 S/m when γ_p is low (5 mN/m) and γ_f is high (80 mN/m). At the opposite corner of the plot, where γ_p is high (40 mN/m) and γ_f is low (50 mN/m), the conductivity falls to 2.13 S/m. This plot indicates that conductivity is less sensitive to changes in the polymer and filler surface tensions.

Fig. 4.

Conductivity predictions as a function of γ_p and γ_f: (a) 3D view and (b) contour plot.

Reducing γ_p consistently leads to higher conductivity, far more effectively than increasing the filler’s surface tension alone. This suggests that polymer–filler compatibility is dictated primarily by the polymer’s ability to wet the filler surface. Although γ_p dominates, conductivity does not reach its maximum unless both conditions are satisfied: low γ_p combined with high γ_f. This synergy underscores that the two parameters act in concert. These trends can be explained mechanistically by considering wetting and particle distribution. A low-energy polymer surface does not conform to the high-energy filler particles, producing poor interfacial contacts, which weaken the filler dispersion and increase the probability of electron tunneling or direct contacts. Conversely, improved wetting by high γ_p and low γ_f grows the particle dispersion throughout the matrix. Better dispersion increases the inter-particle distances, thereby undermining the networking and conductivity. This explanation shows the correct effect of both γ_p and γ_f on the nanocomposite conductivity.

The composite conductivity is governed by the combined effects of polymer tunneling resistivity (p) and interphase intrinsic conductivity (), as presented in Fig. 5(a, b). The plots show that the conductivity consistently decreases as p increases. As p increases and σ_int diminishes, σ_c approaches values close to zero. This highlights the dominant effect of tunneling barriers and interphase conductivity in the charge transport.

Fig. 5.

Effect of tunneling resistivity and interphase conductivity on the conductivity: (a) 3D view and (b) contour plot.

Tunneling resistivity originates from nanoscale gaps or thin insulating layers between adjacent conductive particles^12,38–40. These barriers require electrons to cross the junctions via quantum tunneling, which is inherently sensitive to distance and barrier height. Consequently, even small increases in p significantly reduce the likelihood of charge transport across filler–filler contacts, thereby lowering σ_c. Numerical analysis showed that at high p values, σ_c became almost independent of σ_int. This indicates that, regardless of whether the filler has metallic or graphitic conductivity, the composite cannot achieve significant charge percolation if tunneling barriers dominate.

The interphase conductivity manages the electron transfer through the interphase. A higher interphase conductivity provides more electron transferring, while a low interphase conductivity limits the charge transport. Also, higher interphase conductivity produces a network with higher conduction, while a low conductivity of interphase restricts the conduction of networks. Since the conductivity of interphase and networks governs the conductivity of whole system, the interphase conductivity directly handles the nanocomposite conductivity.

The influence of the parameters and on the conductivity of samples is depicted in Fig. 6(a, b). Notably, at ϕ_p = 0.005 and ϕ_f = 0.06, the highest conductivity is 1 S/m. As the concentration of carbon black increases and percolation onset reduces, the particles move closer together and the probability of forming continuous paths rises sharply. These paths act as electron-transport channels, transforming the composite from an insulating material to one capable of supporting charge transport.

Fig. 6.

Influence of percolation threshold and CB volume fraction on the conductivity: (a) 3D view and (b) contour plot.

A lower percolation threshold forms the conductive networks by a smaller volume fraction of CB nanoparticles. So, a lower percolation onset provides the bigger networks and higher conductivity. Equation (9) demonstrates that a lower percolation onset produces the higher net portion extending the networks. Conversely, a high percolation threshold negatively impacts the conductivity, because it reduces the extent of networks and restricts the conductivity. Therefore, the percolation onset adversely governs the conductivity confirming this model.

A higher CB concentration provides a higher extent of filler–filler contacts and electron tunneling, allowing the formation of bigger conductive networks. In fact, a high CB concentration exceeds the percolation threshold and builds the big networks improving the conductivity. However, a low CB concentration weakens the filler–filler contacts and extends the tunnels, restricting the formation of conductive networks. A low CB amount cannot yield the connected networks for electron flow reducing the conductivity. Accordingly, the developed model properly links the conductivity to CB amount. This trend was reported for the experimented conductivities of many CB-filled examples, as stated in Fig. 2.

Figure 7(a, b) shows the composite conductivity by tunneling diameter (D_t) and tunneling distance (δ_a). The highest conductivity (4 S/m) occurs at the corner where δ_a is at its minimum (0.001 nm) and D_t reaches its maximum value (close to 100 nm). This regime corresponds to an extremely thin tunneling barrier combined with a broad effective tunneling junction area, allowing electrons to transfer efficiently between carbon black particles and form continuous conductive channels across the polymer matrix. By contrast, the lowest conductivity appears as 0, where δ_a is largest (> 12 nm) and D_t is smallest (< 25 nm). Under these unfavorable conditions, the predicted conductivity is reduced to 0. Big tunneling barriers and narrow contact areas suppress electron tunneling, leaving the composite in a near-insulating state.

Fig. 7.

Conductivity variation by tunneling distance and tunneling diameter: (a) 3D view and (b) 2D plot.

The importance of tunneling distance lies in its exponential effect on charge transport. Even a nanometer-scale increase in δ_a drastically reduces the probability of electron transfer between adjacent carbon black particles. Reducing tunneling distance produces the thinner polymer layers separating the fillers lowering the percolation threshold and triggering conductivity in the composite. Additionally, the tunneling diameter (D_t) controls the effective cross-sectional area available for tunneling. A larger D_t provides broader junctions through which many electrons can transfer in parallel, thereby strengthening each conductive link and stabilizing the percolated network once tunneling is possible. While its influence is less dramatic than that of δ_a, the tunneling diameter acts as a key amplifier: when δ_a is already small enough to permit tunneling, increasing D_t significantly enhances the efficiency and robustness of conductivity. Taken together, the two plots and their numerical extremes highlight the complementary roles of these tunneling parameters. The tunneling distance sets the fundamental limit on whether electrons can cross the insulating matrix, while the tunneling diameter defines how effective and stable those crossings become once tunneling is allowed^32,41,42. For carbon black nanocomposites, achieving high conductivity therefore requires simultaneously minimizing δ_a and maximizing D_t to ensure both the initiation and reinforcement of conductive pathways.

The effects of filler conductivity () and interphase thickness () on the overall nanocomposite conductivity () are demonstrated in Fig. 8(a, b). The plots highlight that σ_c remains negligible when σ_f or interphase depth is low. Only when these parameters approach higher values, the conductivity begin to rise significantly. When t is below 5 nm, σ_c is almost zero regardless of σ_f, indicating insufficient network connectivity. As t increases beyond 15–20 nm, conductivity rises sharply, particularly for highly conductive CBs.

Fig. 8.

Conductivity affected by σ_f and interphase thickness: (a) 3D view and (b) contour plot.

A higher conductivity of the filler ensures that electrons encounter minimal internal resistance as they enter the carbon black domains, allowing for more efficient charge transport through the filler backbone. When σ_f is low, the potential for forming a conductive network is limited and the composite exhibits almost insulating behavior. In contrast, when σ_f is high, the filler can support stable and efficient electron flow, allowing the macroscopic conductivity of the composite to be significantly enhanced, especially when combined with favorable tunneling distances and interfacial conditions^43–45. Therefore, the thickness of the interphase and the conductivity of the filler are both crucial in determining the final conductivity of the nanocomposite.

The thickness of the interphase strongly affects the stability and reliability of the conductive paths in carbon black nanocomposites^46–48. A thicker interphase provides a larger area where electron waves from one particle can overlap with those of another, which enhances the probability of tunneling. This not only increases conductivity but also makes the conductive network less sensitive to fluctuations in filler distribution. In contrast, when the interphase is too thin, the conductive paths become brittle and discontinuous, meaning that small changes in particle spacing or orientation can easily break the network and suppress charge transport. The role of the interphase thickness should therefore be considered as both a structural and functional parameter.

Conclusions

In this study, a comprehensive model was developed and validated for predicting the conductivity of carbon black composites. Neither factor alone guarantees the formation of a conductive network; rather, their synergy determines whether insulating or conductive behavior emerges. Numerical investigations revealed several key trends. The highest conductivity is observed as 1 S/m when the filler volume fraction (ϕ_f) is 0.06 and the percolation threshold (ϕ_p) is 0.005. The maximum conductivity of 4 S/m occurs at a minimal tunneling distance (δ_a) of 0.001 nm and a maximum tunneling diameter (D_t) of 100 nm. However, the lowest conductivity as 0 is observed as δ_a is largest (> 12 nm) and D_t is smallest (< 25 nm). Also, the conductivity remains nearly zero when filler conductivity is too low or the interphase is very thin. However, a higher filler conductivity and thicker interphase yield a higher conductivity. Finally, it should be noted that the present model is formulated for composites containing spherical carbon black particles and is applicable within moderate filler volume fractions, below the regime where particle agglomeration and contact saturation dominate. High filler contents beyond this range, as well as non-spherical particle morphologies, may introduce additional mechanisms not captured in the current framework. Moreover, environmental effects such as temperature and humidity, as well as conductivity evolution under dynamic or cyclic loading, are not considered in the present study. These aspects define the current applicability limits of the model and represent important directions for future extensions.

Abbreviations

σ_c

The effective conductivity of the nanocomposite

σ_matrix

Intrinsic electrical conductivity of the polymer matrix

σ_eff

Effective conductivity of the filler network

ϕ_eff

Effective filler volume fraction

ξ

Fraction of percolated

η

Waviness of the conductive pathways

H

Geometric influence of filler morphology

R

Radius of carbon black

d

Particle diameter

σ_z

Baseline longitudinal conductivity of the matrix–interphase pathway

t

Interphase thickness

σ_f

Intrinsic conductivity of the filler

σ_int

Interphase conductivity

δ_a

Tunneling distance

D_t

Tunneling diameter

p

Polymer tunneling resistivity

w

Weight fraction of carbon black in the nanocomposite

ρ_f

Filler density

ρ_p

Polymer density

γ_p

Polymer surface tension

γ_f

Filler surface tension

γ_pf

Polymer–filler interfacial tension

ϕ_p

Percolation threshold

Author contributions

**Mohammad Boomhendi: ** Formal analysis; Investigation; Methodology; Roles/Writing - original draft.**Mostafa Vatani: ** Formal analysis; Methodology; Visualization; Roles/Writing - original draft.**Yasser Zare** : Conceptualization; Formal analysis; Investigation; Methodology; Visualization; Supervision; Roles/Writing - original draft.**Muhammad Tajammal Munir: ** Software; Validation; Writing - review & editing.**Jin-Hwan Choi** : Supervision; Visualization; Validation; Writing - review & editing.**Kyong Yop Rhee: ** Project administration; Supervision; Validation; Writing - review & editing.

Funding

No funding was received.

Data availability

The data that support the findings of this study are available on request.

Declarations

Competing interests

The authors declare no competing interests.