# A review of models for effective thermal conductivity of composite materials

### Abstract

The solutions of Maxwell and Rayleigh were the first of many attempts to determine the eective thermal conductivityof heterogeneous material. Early models assumed that no thermal resistance exists between the phases in heterogeneousmaterial. Later studies on solid-liquid and solid-solid boundaries revealed that a temperature drop occurs when heatflows through a boundary between two phases and, as a consequence, the interfacial thermal resistance should be includedin the heat transfer model. This paper is a review of the most popular expressions for predicting the eectivethermal conductivity of composite materials using the properties and volume fractions of constituent phases. Subject toreview were empirical, analytical and numerical models, among others.### References

[1] J. Strutt (Lord Rayleigh).: On the influence of obstacles arranged in rectangular order upon the properties of a medium, Phil. Mag., vol. 34, pp. 481, 1892.

[2] J. C. Maxwell, A treatise on electricity and magnetism, vol. I, 3rd Ed, Oxford University Press, 1904.

[3] A. G. Every, Y. Tzou, D. P. H. Hasselman, R. Raj.: The effect of particie size on the thermal conductivity of ZnS/diamond composites, Acta Metall. Mater., vol. 40, no. 1, pp. 123, 1992.

[4] A. Devpura, P. E. Phelan, R. S. Prasher.: Size effects on the thermal conductivity of polymers laden with highly conductive filler particles, Microscale Thermophysical Engineering, vol. 5, pp. 177, 2001.

[5] D. P. H. Hasselman, L. F. Johnson.: Effective thermal conductivity of composites with interfacial thermal barrier resistance, J. Compos. Mater., vol. 21, no. 6, pp. 508, 1987.

[6] E. T. Swarz, R. O. Pohl.: Thermal boundary resistance, Rev. Mod. Phys., vol. 61, no. 3, pp. 605, 1989.

[7] P. L. Kapitza.: The study of heat transfer in helium II [in Russian], J. Phys. USSR, vol. 4, no. 3, pp.181, 1941.

[8] P. Furmański, T. S. Wiśniewski, J. Banaszek.: Thermal contact resistance and other thermal phenomena at solid-solid interface, Institute of Heat Engineering, Warsaw 2008.

[9] R. S. Prasher, P. E. Phelan.: A scattering-mediated acoustic mismatch model for the prediction of thermal boundary resistance, Journal of Heat Transfer, vol. 123, No. 1, pp. 105, 2001.

[10] G. Granqvist, O. Hunderi.: Conductivity of inhomogeneous materials: Effective-medium theory with dipole-dipole interaction, Phys. Rev. B, vol. 18, no. 4, pp.1554, 1978.

[11] G. Granqvist, O. Hunderi.: Optical properties of Ag-Si02 Cermet films: A comparison of effective-medium theories, Phys. Rev. B, vol. 18, no. 6, pp. 2897, 1978.

[12] A. Devpura, P. E. Phelan, R. S. Prasher.: Percolation theory applied to the analysis of thermal interface materials in flip-chip technology, Thermomechanical Phenomena in Electronic Systems - Proceedings of the Intersociety Conference, Las Vegas, Nevada, Maj 2000, vol. 1, pp. 21.

[13] Q. Xue.: A percolation model of metal–insulator composites, Physica B, vol. 325 s. 195, 2003.

[14] G. Zhang, Y. Xia, H. Wang, Y. Tao, G. Tao, S. Tu, H. Wu.: A percolation model of thermal conductivity for filled polymer composites, Journal of Composite Materiale, vol. 44, no. 8, pp. 963, 2010.

[15] R. B. Bird, W. E. Stewart, E. N. Lightfoot.: Transport phenomena, John Wiley & Sons, 2007.

[16] A. Eucken.: Die Wärmeleitfähigkeit Keramischer, Fester Stoffe – Ihre Berechnung aus der Wärmeleitfähigkeit der Bestandteile, VDI Forschungsheft 353, Beilage zu, Forschung auf dem gebiet des Ingenieurwesens, Ausgabe B, Band 3, 1932.

[17] H. C. Burger.: Das Lertvermogen verdumter Mischkristallfreier Lonsungen, Phys. Zs. Vol. 20, pp. 73, 1915.

[18] R. L. Hamilton, O. K. Crosser, Thermal conductivity of heterogeneous two component systems, Ind. Eng. Chem. Fundamen., vol. 1, no. 3, pp. 187, 1962.

[19] B. R. Powell (Jr), G. E. Youngblood, D. P. H. Hasselman, L. D. Bentsen.: Effect of thermal expansion mismatch on the thermal diffusivity of glass-Ni composites, J. Am. Ceram. Soc., vol. 63, pp. 581, 1980.

[20] Y. Benveniste, T. Miloh.: The effective conductivity of composites with imperfect thermal contact at constituent interfaces, Int. J. Eng. Sci., vol. 24, no. 9, pp. 1537, 1986.

[21] Y. Benveniste.: Effective thermal conductivity of composites with a thermal contact resistance between the constituents: Nondilute case, J. Appl. Phys., vol. 61, no. 8 pp. 2840, 1987.

[22] D. A. G. Bruggeman.: Berechnung verschiedener physikalischer konstanten von heterogenen substanzen, Ann. Phys., vol. 24, pp. 636, 1935.

[23] R. Landauer.: The electrical resistance of binary metallic mixtures, J. Appl. Phys., vol. 23, no. 7, pp. 779, 1952.

[24] L. E. Nielsen.: The thermal and electrical conductivity of two-phase systems, Ind. Eng. Chem. Fundam., vol. 13, no. 1, pp. 17, 1974.

[25] A. Bejan, A. D. Kraus.: Heat transfer handbook, John Wiley & Sons, 2003.

[26] Y. Agari, A. Ueda, M. Tanaka, S. Nagai.: Thermal conductivity of a polymer filled with particles in the wide range from low to super-high volume content, Journal of Applied Polymer Science, vol. 40, pp. 929, 1990.

[27] L. C. Davis, B. E. Artz.: Thermal Conductivity of Metal-Matrix Composites, Journal of Applied Physics, vol. 77, no. 10, pp. 4954, 1995.

[28] C. F. Matt, M. E. Cruz.: Effective thermal conductivity of composite materiale with 3-D microstructures and interfacial thermal resistance, Numerical Heat Transfer, Part A, vol. 53, pp. 577, 2008.

[29] Y. Xu, M. Yamazaki, H. Wang, K. Yagi.: Development of an internet system for composite design and thermophysical property prediction, Materials Transactions, vol. 47, no. 8, pp. 1882, 2006.

[30] C. Yue, Y. Zhang, Z. Hu, J. Liu, Z. Cheng.: Modeling of the effective thermal conductivity of composite materials with FEM based on resistor networks approach, Microsyst. Technol. vol. 16, pp. 633, 2010.

[31] J. Yvonnet, Q.-C. He, C. Toulemonde.: Numerical modelling of the effective conductivities of composites with arbitrarily shaped inclusions and highly conducting interface, Composites Science and Technology, vol. 68, pp. 2818, 2008.

[32] H. Jopek, T. Strek.: Optimization of the effective thermal conductivity of a composite [in:] Convection and conduction heat transfer, ed. A. Ahsan, InTech, 2011.

[33] R. Nayak, T. Dora P., A. Satapathy.: A computational and experimental investigation on thermal conductivity of particle reinforced epoxy composites, Computational Materials Science, vol. 48, pp. 576, 2010.

[34] J. Flaquer, A. Rios, A. Martin-Meizoso, S. Nogales, H. Böhm.: Effect of diamond shapes and associated thermal boundary resistance on thermal conductivity of diamond-based composites, Computational Materials Science, vol. 41 pp. 156, 2007.

[35] Ganapathy, K. Singh, P. E. Phelan, R. Prasher.: An effective unit cell approach to compute the thermal conductivity of composites with cylindrical particles, J. Heat Transfer, Vol. 127 pp. 553, 2005.

[36] Y. Benveniste, T. Miloh.: An exact solution for the effective thermal conductivity of cracked bodies with oriented elliptical cracks, J. Appl. Phys., vol. 66, pp 176, 1989.

[37] Y. Benveniste, T. Chen, G. J. Dvorak.: The effective thermal conductivity of composites reinforced by coated cylindrically orthotropic fibers, J. Appl. Phys., vol. 67, pp. 2878, 1990.

[38] Y. Benveniste, T. Miloh.: On the effective thermal conductivity of coated shortfiber composites, J. Appl. Phys., vol. 69, pp. 1337, 1991

[39] J. D. Felske.: Effective thermal conductivity of composite spheres in a continuous medium with contact resistance, International Journal of Heat and Mass Transfer, vol. 47, pp. 3453, 2004.

[40] M. L. Dunn, M. Taya.: The effective thermal conductivity of composites with coated reinforcement and the application to imperfect interfaces, J. Appl. Phys., vol. 73, pp. 1711, 1993.

[41] S. Lu, J. Song.: Effective conductivity of composites with spherical inclusions: Effect of coating and detachment, J. Appl. Phys., vol. 79, pp. 609, 1996.

[42] D. P. H. Hasselman, K. Y. Donaldson, J. R. Thomas Jr.: Effective thermal conductivity of uniaxial composite with cylindrically orthotropic carbon fibers and interfacial thermal barrier, J. Comp. Mater., vol. 27, no. 6, pp. 637, 1993.

[43] C-W. Nan, R. Birringer, D. R. Clarke, H. Gleiter.: Effective thermal conductivity of particulate composites with interfacial thermal resistance, J. Appl. Phys., vol. 81, pp. 6692, 1997.

[44] J. Ordonez-Miranda, Ronggui Yang, J. J. Alvarado-Gil.: A model for the effective thermal conductivity of metal-nonmetal particulate composites, J. Appl. Phys. 111, 044319 (2012).

[45] R. Tavangar, J.M. Molina, L. Weber.: Assessing predictive schemes for thermal conductivity against diamond-reinforced silver matrix composites at intermediate phase contrast, Scripta Materialia, vol. 56, pp.357, 2007.

[46] K. Sebeck, C. Shao, J. Kieffer.: Structure, thermal, and mechanical properties of interfaces in PMC: A molecular simulation study, Proc. 18th International Conf. on Composite Materials, Jeju Island, Korea 2011.

[47] T. C. Clancy, S. J. V. Frankland, J. A. Hinkley.: Prediction of Material Properties of Nanostructured Polymer Composites Using Atomistic Simulations, American Institute of Aeronautics and Astronautics, 2009.

[48] X. Huang, X. Huai, S. Liang, X. Wang.: Thermal transport in Si/Ge nanocomposites, J. Phys. D: Appl. Phys. 42 (2009) 095416 (9pp).

[49] Y. Zhou, B. Anglin, A. Strachan.: Phonon thermal conductivity in nanolaminated composite metals via molecular dynamics, J. Chem. Phys. 127, 184702 (2007).

[2] J. C. Maxwell, A treatise on electricity and magnetism, vol. I, 3rd Ed, Oxford University Press, 1904.

[3] A. G. Every, Y. Tzou, D. P. H. Hasselman, R. Raj.: The effect of particie size on the thermal conductivity of ZnS/diamond composites, Acta Metall. Mater., vol. 40, no. 1, pp. 123, 1992.

[4] A. Devpura, P. E. Phelan, R. S. Prasher.: Size effects on the thermal conductivity of polymers laden with highly conductive filler particles, Microscale Thermophysical Engineering, vol. 5, pp. 177, 2001.

[5] D. P. H. Hasselman, L. F. Johnson.: Effective thermal conductivity of composites with interfacial thermal barrier resistance, J. Compos. Mater., vol. 21, no. 6, pp. 508, 1987.

[6] E. T. Swarz, R. O. Pohl.: Thermal boundary resistance, Rev. Mod. Phys., vol. 61, no. 3, pp. 605, 1989.

[7] P. L. Kapitza.: The study of heat transfer in helium II [in Russian], J. Phys. USSR, vol. 4, no. 3, pp.181, 1941.

[8] P. Furmański, T. S. Wiśniewski, J. Banaszek.: Thermal contact resistance and other thermal phenomena at solid-solid interface, Institute of Heat Engineering, Warsaw 2008.

[9] R. S. Prasher, P. E. Phelan.: A scattering-mediated acoustic mismatch model for the prediction of thermal boundary resistance, Journal of Heat Transfer, vol. 123, No. 1, pp. 105, 2001.

[10] G. Granqvist, O. Hunderi.: Conductivity of inhomogeneous materials: Effective-medium theory with dipole-dipole interaction, Phys. Rev. B, vol. 18, no. 4, pp.1554, 1978.

[11] G. Granqvist, O. Hunderi.: Optical properties of Ag-Si02 Cermet films: A comparison of effective-medium theories, Phys. Rev. B, vol. 18, no. 6, pp. 2897, 1978.

[12] A. Devpura, P. E. Phelan, R. S. Prasher.: Percolation theory applied to the analysis of thermal interface materials in flip-chip technology, Thermomechanical Phenomena in Electronic Systems - Proceedings of the Intersociety Conference, Las Vegas, Nevada, Maj 2000, vol. 1, pp. 21.

[13] Q. Xue.: A percolation model of metal–insulator composites, Physica B, vol. 325 s. 195, 2003.

[14] G. Zhang, Y. Xia, H. Wang, Y. Tao, G. Tao, S. Tu, H. Wu.: A percolation model of thermal conductivity for filled polymer composites, Journal of Composite Materiale, vol. 44, no. 8, pp. 963, 2010.

[15] R. B. Bird, W. E. Stewart, E. N. Lightfoot.: Transport phenomena, John Wiley & Sons, 2007.

[16] A. Eucken.: Die Wärmeleitfähigkeit Keramischer, Fester Stoffe – Ihre Berechnung aus der Wärmeleitfähigkeit der Bestandteile, VDI Forschungsheft 353, Beilage zu, Forschung auf dem gebiet des Ingenieurwesens, Ausgabe B, Band 3, 1932.

[17] H. C. Burger.: Das Lertvermogen verdumter Mischkristallfreier Lonsungen, Phys. Zs. Vol. 20, pp. 73, 1915.

[18] R. L. Hamilton, O. K. Crosser, Thermal conductivity of heterogeneous two component systems, Ind. Eng. Chem. Fundamen., vol. 1, no. 3, pp. 187, 1962.

[19] B. R. Powell (Jr), G. E. Youngblood, D. P. H. Hasselman, L. D. Bentsen.: Effect of thermal expansion mismatch on the thermal diffusivity of glass-Ni composites, J. Am. Ceram. Soc., vol. 63, pp. 581, 1980.

[20] Y. Benveniste, T. Miloh.: The effective conductivity of composites with imperfect thermal contact at constituent interfaces, Int. J. Eng. Sci., vol. 24, no. 9, pp. 1537, 1986.

[21] Y. Benveniste.: Effective thermal conductivity of composites with a thermal contact resistance between the constituents: Nondilute case, J. Appl. Phys., vol. 61, no. 8 pp. 2840, 1987.

[22] D. A. G. Bruggeman.: Berechnung verschiedener physikalischer konstanten von heterogenen substanzen, Ann. Phys., vol. 24, pp. 636, 1935.

[23] R. Landauer.: The electrical resistance of binary metallic mixtures, J. Appl. Phys., vol. 23, no. 7, pp. 779, 1952.

[24] L. E. Nielsen.: The thermal and electrical conductivity of two-phase systems, Ind. Eng. Chem. Fundam., vol. 13, no. 1, pp. 17, 1974.

[25] A. Bejan, A. D. Kraus.: Heat transfer handbook, John Wiley & Sons, 2003.

[26] Y. Agari, A. Ueda, M. Tanaka, S. Nagai.: Thermal conductivity of a polymer filled with particles in the wide range from low to super-high volume content, Journal of Applied Polymer Science, vol. 40, pp. 929, 1990.

[27] L. C. Davis, B. E. Artz.: Thermal Conductivity of Metal-Matrix Composites, Journal of Applied Physics, vol. 77, no. 10, pp. 4954, 1995.

[28] C. F. Matt, M. E. Cruz.: Effective thermal conductivity of composite materiale with 3-D microstructures and interfacial thermal resistance, Numerical Heat Transfer, Part A, vol. 53, pp. 577, 2008.

[29] Y. Xu, M. Yamazaki, H. Wang, K. Yagi.: Development of an internet system for composite design and thermophysical property prediction, Materials Transactions, vol. 47, no. 8, pp. 1882, 2006.

[30] C. Yue, Y. Zhang, Z. Hu, J. Liu, Z. Cheng.: Modeling of the effective thermal conductivity of composite materials with FEM based on resistor networks approach, Microsyst. Technol. vol. 16, pp. 633, 2010.

[31] J. Yvonnet, Q.-C. He, C. Toulemonde.: Numerical modelling of the effective conductivities of composites with arbitrarily shaped inclusions and highly conducting interface, Composites Science and Technology, vol. 68, pp. 2818, 2008.

[32] H. Jopek, T. Strek.: Optimization of the effective thermal conductivity of a composite [in:] Convection and conduction heat transfer, ed. A. Ahsan, InTech, 2011.

[33] R. Nayak, T. Dora P., A. Satapathy.: A computational and experimental investigation on thermal conductivity of particle reinforced epoxy composites, Computational Materials Science, vol. 48, pp. 576, 2010.

[34] J. Flaquer, A. Rios, A. Martin-Meizoso, S. Nogales, H. Böhm.: Effect of diamond shapes and associated thermal boundary resistance on thermal conductivity of diamond-based composites, Computational Materials Science, vol. 41 pp. 156, 2007.

[35] Ganapathy, K. Singh, P. E. Phelan, R. Prasher.: An effective unit cell approach to compute the thermal conductivity of composites with cylindrical particles, J. Heat Transfer, Vol. 127 pp. 553, 2005.

[36] Y. Benveniste, T. Miloh.: An exact solution for the effective thermal conductivity of cracked bodies with oriented elliptical cracks, J. Appl. Phys., vol. 66, pp 176, 1989.

[37] Y. Benveniste, T. Chen, G. J. Dvorak.: The effective thermal conductivity of composites reinforced by coated cylindrically orthotropic fibers, J. Appl. Phys., vol. 67, pp. 2878, 1990.

[38] Y. Benveniste, T. Miloh.: On the effective thermal conductivity of coated shortfiber composites, J. Appl. Phys., vol. 69, pp. 1337, 1991

[39] J. D. Felske.: Effective thermal conductivity of composite spheres in a continuous medium with contact resistance, International Journal of Heat and Mass Transfer, vol. 47, pp. 3453, 2004.

[40] M. L. Dunn, M. Taya.: The effective thermal conductivity of composites with coated reinforcement and the application to imperfect interfaces, J. Appl. Phys., vol. 73, pp. 1711, 1993.

[41] S. Lu, J. Song.: Effective conductivity of composites with spherical inclusions: Effect of coating and detachment, J. Appl. Phys., vol. 79, pp. 609, 1996.

[42] D. P. H. Hasselman, K. Y. Donaldson, J. R. Thomas Jr.: Effective thermal conductivity of uniaxial composite with cylindrically orthotropic carbon fibers and interfacial thermal barrier, J. Comp. Mater., vol. 27, no. 6, pp. 637, 1993.

[43] C-W. Nan, R. Birringer, D. R. Clarke, H. Gleiter.: Effective thermal conductivity of particulate composites with interfacial thermal resistance, J. Appl. Phys., vol. 81, pp. 6692, 1997.

[44] J. Ordonez-Miranda, Ronggui Yang, J. J. Alvarado-Gil.: A model for the effective thermal conductivity of metal-nonmetal particulate composites, J. Appl. Phys. 111, 044319 (2012).

[45] R. Tavangar, J.M. Molina, L. Weber.: Assessing predictive schemes for thermal conductivity against diamond-reinforced silver matrix composites at intermediate phase contrast, Scripta Materialia, vol. 56, pp.357, 2007.

[46] K. Sebeck, C. Shao, J. Kieffer.: Structure, thermal, and mechanical properties of interfaces in PMC: A molecular simulation study, Proc. 18th International Conf. on Composite Materials, Jeju Island, Korea 2011.

[47] T. C. Clancy, S. J. V. Frankland, J. A. Hinkley.: Prediction of Material Properties of Nanostructured Polymer Composites Using Atomistic Simulations, American Institute of Aeronautics and Astronautics, 2009.

[48] X. Huang, X. Huai, S. Liang, X. Wang.: Thermal transport in Si/Ge nanocomposites, J. Phys. D: Appl. Phys. 42 (2009) 095416 (9pp).

[49] Y. Zhou, B. Anglin, A. Strachan.: Phonon thermal conductivity in nanolaminated composite metals via molecular dynamics, J. Chem. Phys. 127, 184702 (2007).

Published

2014-12-23

How to Cite

PIETRAK, Karol; WIŚNIEWSKI, Tomasz S..
A review of models for effective thermal conductivity of composite materials.

**Journal of Power Technologies**, [S.l.], v. 95, n. 1, p. 14--24, dec. 2014. ISSN 2083-4195. Available at: <https://papers.itc.pw.edu.pl/index.php/JPT/article/view/463>. Date accessed: 23 july 2021.
Issue

Section

Materials Science

### Keywords

thermal conductivity; interfacial thermal resistance; composite materials

Authors who publish with this journal agree to the following terms:

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).