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 (σint). The lowest σint (< 200 S/m) produces an insulated system, while the composite conductivity maximizes to 3 S/m at p = 10 Ω.m and σint = 8000 S/m. Accordingly, more interphase conduction and less polymer tunnel resistivity are desirable to improve the conductivity of samples.
Data availability
The data that support the findings of this study are available on request.
Abbreviations
- σc :
-
The effective conductivity of the nanocomposite
- σmatrix :
-
Intrinsic electrical conductivity of the polymer matrix
- σeff :
-
Effective conductivity of the filler network
- ϕeff :
-
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
- σz :
-
Baseline longitudinal conductivity of the matrix–interphase pathway
- t:
-
Interphase thickness
- σf :
-
Intrinsic conductivity of the filler
- σint :
-
Interphase conductivity
- δa :
-
Tunneling distance
- Dt :
-
Tunneling diameter
- p:
-
Polymer tunneling resistivity
- w:
-
Weight fraction of carbon black in the nanocomposite
- ρf :
-
Filler density
- ρp :
-
Polymer density
- γp :
-
Polymer surface tension
- γf :
-
Filler surface tension
- γpf :
-
Polymer–filler interfacial tension
- ϕp :
-
Percolation threshold
References
Zhuang, J. et al. Spinning solution viscosity reducing and wet spinning of carbon black-based elastic conductive fibers for sports monitoring and healthcare electrical heating. J. Mater. Res. Technol. 36, 1–12 (2025).
El-Khiyami, S. S., Ali, H., Ismail, A. & Hafez, R. Tunable physical properties and dye removal application of novel Chitosan polyethylene glycol and polypyrrole/carbon black films. Sci. Rep. 15 (1), 20124 (2025).
Fernandez, M. G. C., Hakim, M. L., Alfarros, Z., Santos, G. N. C. & Muflikhun, M. A. Nanoengineered polyaniline/carbon black VXC 72 hybridized with woven Abaca for superior electromagnetic interference shielding. Sci. Rep. 15 (1), 14548 (2025).
Laithong, T., Nampitch, T., Ourapeepon, P. & Phetyim, N. Quality improvement of recycled carbon black from waste tire pyrolysis for replacing carbon black N330. Sci. Rep. 15 (1), 23726 (2025).
Lotfy, V. F., Basta, A. H. & Shafik, E. S. Assessment the performance of chemical constituents of agro wastes in production safety alternative carbon black filler in rubber composite purpose. Sci. Rep. 15 (1), 11035 (2025).
Lee, S-M., Lee, S-H. & Roh, J-S. Analysis of activation process of carbon black based on structural parameters obtained by XRD analysis. Crystals 11 (2), 153 (2021).
Lin, J., Zhao, Q., Chen, H., Li, M. & Yuan, L. A numerical framework for the ITZ percolation, effective fraction and diffusivity of concrete systems considering the nonuniform ITZ. J. Building Eng. 77, 107429 (2023).
Abdollahi, F. et al. A predictive model for electrical conductivity of polymer carbon black nanocomposites. Polym. Compos. 46, 7491–7502 (2025).
Zare, Y., Munir, M. T. & Rhee, K. Y. Assessment of electrical conductivity of polymer nanocomposites containing a deficient interphase around graphene nanosheet. Sci. Rep. 14 (1), 8737 (2024).
Vatani, M., Zare, Y., Munir, M. T. & Rhee, K. Y. Renovating nanocomposite design: A novel conductivity model for polymer graphene systems incorporating interphase and tunneling zones. Polym. Compos. 46, 6933–6943 (2025).
Zare, Y., Munir, M. T., Rhee, K. Y. & Park, S-J. Decrypting of effective resistance for composites of polymer-carbon nanofiber: an applicable approach to regulate the electrical conductivity. J. Mater. Res. Technol. 38, 2105–2112 (2025).
Zare, Y., Naqvi, M., Rhee, K. Y. & Park, S-J. Simulation of electrical conductivity for polymer carbon nanofiber composites assuming an extended nanofiber by interphase depth and tunneling distance. Sci. Rep. 15 (1), 31623 (2025).
Zare, Y., Naqvi, M., Rhee, K. Y. & Park, S-J. Advancing conductivity modeling: A unified framework for polymer carbon black nanocomposites. J. Mater. Res. Technol. 36, 26–33 (2025).
Lin, J. et al. Insight into the diffusivity of particulate composites considering percolation of soft interphases around hard fillers: from spherical to polyhedral particles. Powder Technol. 392, 459–472 (2021).
Lin, J., Zhao, Q., Chen, H., Xue, C. & Li, M. The fraction and percolation of soft interfaces in granular composites containing polyhedral and ovoidal fillers: A theoretical and numerical study. Adv. Powder Technol. 34 (7), 104057 (2023).
Brosseau, C. Modeling the interface between phases in dense Polymer-Carbon black nanoparticle composites by dielectric spectroscopy: where are we now and what are the opportunities? Macromol. Ther. Simul. 33 (3), 2400009 (2024).
Moronkeji, O. E., Das, D., Lee, S., Chang, K. M. & Chasiotis, I. Local electrical conductivity of carbon black/PDMS nanocomposites subjected to large deformations. J. Compos. Mater. 57 (4), 507–519 (2023).
Wang, Y. et al. Microstructural modeling and simulation of a carbon black-based conductive polymer a template for the virtual design of a composite material. ACS Omega. 7 (33), 28820–28830 (2022).
Albright, T. & Hobeck, J. Investigating the electromechanical properties of carbon black-based conductive polymer composites via stochastic modeling. Nanomaterials 13 (10), 1641 (2023).
Paredes-Madrid, L., Palacio, C. A., Matute, A. & Parra Vargas, C. A. Underlying physics of conductive polymer composites and force sensing resistors (FSRs) under static loading conditions. Sensors 17 (9), 2108 (2017).
Alekseev, A., Wu, T., Van der Ven, L., Van Benthem, R. & de With, G. Global and local conductivity in percolating crosslinked carbon black/epoxy–amine composites. J. Mater. Sci. 55 (21), 8930–8939 (2020).
Casanova, A. et al. Carbon black as conductive additive and structural director of porous carbon gels. Materials 13 (1), 217 (2020).
Deng, F. & Zheng, Q-S. An analytical model of effective electrical conductivity of carbon nanotube composites. Appl. Phys. Lett. 92, 071902 (2008).
Chanda, A., Sinha, S. K. & Datla, N. V. Electrical conductivity of random and aligned nanocomposites: theoretical models and experimental validation. Compos. Part A: Appl. Sci. Manufac. 149, 106543 (2021).
Hipp, J. B., Richards, J. J. & Wagner, N. J. Direct measurements of the microstructural origin of shear-thinning in carbon black suspensions. J. Rheol. 65 (2), 145 (2021).
Chen, L. et al. The recovery of nano-sized carbon black filler structure and its contribution to stress recovery in rubber nanocomposites. Nanoscale 12 (48), 24527–24542 (2020).
Taherian, R. Experimental and analytical model for the electrical conductivity of polymer-based nanocomposites. Compos. Sci. Technol. 123, 17–31 (2016).
Boomhendi, M., Vatani, M. & Zare, Y. Predicting of tunneling conductivity for polymer-carbon black nanocomposites by interphase percolation. Sci. Rep. 15 (1), 42322 (2025).
Jia, L. J. et al. Microcellular conductive carbon black or graphene/PVDF composite foam with 3D conductive channel: a promising lightweight, heat-insulating, and EMI‐shielding material. Macromol. Mater. Eng. 306 (4), 2000759 (2021).
Dang, Z-M. et al. Complementary percolation characteristics of carbon fillers based electrically percolative thermoplastic elastomer composites. Compos. Sci. Technol. 72 (1), 28–35 (2011).
Cheng, H. et al. Enhancement of electromagnetic interference shielding performance and wear resistance of the UHMWPE/PP blend by constructing a segregated hybrid conductive carbon black–polymer network. ACS Omega. 6 (23), 15078–15088 (2021).
Tian, H., Zhou, H., Fu, H., Li, X. & Gong, W. Enhanced electrical and dielectric properties of plasticized soy protein bioplastics through incorporation of nanosized carbon black. Polym. Compos. 41 (12), 5246–5256 (2020).
Brunella, V., Rossatto, B. G., Mastropasqua, C., Cesano, F. & Scarano, D. Thermal/electrical properties and texture of carbon black PC polymer composites near the electrical percolation threshold. J. Compos. Sci. 5 (8), 212 (2021).
Kang, M-J., Heo, Y-J., Jin, F-L. & Park, S-J. A review: role of interfacial adhesion between carbon Blacks and elastomeric materials. Carbon Lett. 18 (1), 1–10 (2016).
Uygun, A. & Velasco, J. Electrical Conductivity Modeling of Polypropylene Composites Filled with Carbon Black and Acetylene Black (International Scholarly Research Network ISRN Polymer Science, 2012).
Chen, Y., Wang, S., Pan, F. & Zhang, J. A numerical study on electrical percolation of Polymer-Matrix composites with hybrid fillers of carbon nanotubes and carbon black. J. Nanomaterials. 2014 (1), 614797 (2014).
Konishi, Y. & Cakmak, M. Nanoparticle induced network self-assembly in polymer–carbon black composites. Polymer 47 (15), 5371–5391 (2006).
Hadi, Z. et al. A model for effective conductivity of polymer nanocomposites containing MXene nanosheets. Polym. Compos. 46, 8906–8918 (2025).
Zare, Y., Munir, M. T. & Rhee, K. Y. A new pattern for conductivity of carbon nanofiber polymer composites with interphase and tunneling parameters. Compos. Part A: Appl. Sci. Manufac. 190, 108721 (2025).
Zare, Y., Munir, M. T., Rhee, K. Y. & Park, S-J. Bridging the nano-scale interphase with macro-scale properties: Two-step simulating of electrical conductivity for polymer composites incorporating carbon nanofibers. J. Mater. Res. Technol. 37, 3578–3585 (2025).
Lu, Z. et al. Microcrack-engineered flexible strain sensors enabled by carbon black/graphene/conductive carbon paste ternary nanocomposites for enhanced sensitivity. J. Mater. Sci.: Mater. Electron. 36 (21), 1–12 (2025).
Paleari, L., Bragaglia, M., Fabbrocino, F. & Nanni, F. Structural monitoring of glass fiber/epoxy laminates by means of carbon nanotubes and carbon black self-monitoring plies. Nanomaterials 11 (6), 1543 (2021).
Ghosh, S. K. et al. Combination effect of functionalized high aspect ratio carbonaceous nanofillers and carbon black on electrical, thermal conductivity, dielectric and EMI shielding behavior of co-continuous thermoplastic elastomeric blend composite films. Chem. Eng. J. Adv. 15, 100505 (2023).
Cheng, H. et al. Endowing acceptable mechanical properties of segregated conductive polymer composites with enhanced filler-matrix interfacial interactions by incorporating high specific surface area nanosized carbon black. Nanomaterials 11 (8), 2074 (2021).
Zare, Y. & Rhee, K. Y. Analysis of the connecting effectiveness of the interphase zone on the tensile properties of carbon nanotubes (CNT) reinforced nanocomposite. Polymers 12 (4), 896 (2020).
Zare, Y., Munir, M. T., Rhee, K. Y. & Park, S-J. New insights to effective carbon nanofiber features due to defective interphase for prediction of tunneling conductivity in composites. Sci. Rep. 15 (1), 34786 (2025).
Zare, Y., Naqvi, M., Rhee, K. Y. & Park, S-J. Unraveling the roles of network and tunnels in the conductivity of carbon nanofiber composites. Sci. Rep. 15 (1), 35459 (2025).
Zare, Y., Naqvi, M., Rhee, K. Y. & Park, S-J. Estimating of contact area among carbon nanofibers in nanocomposites by the features of network, tunnel and interphase. Sci. Rep. 15 (1), 37571 (2025).
Funding
No funding was received.
Author information
Authors and Affiliations
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.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
About this article
Cite this article
Boomhendi, M., Vatani, M., Zare, Y. et al. Predictive modeling of conductivity for carbon black nanocomposites: influence of filler features, interfacial effects and network portion. Sci Rep (2026). https://doi.org/10.1038/s41598-026-38296-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41598-026-38296-y
